dooccttoorraatt dee ll uunniivveerrssiittÉ ddee

Modélisation spatio-temporel des champs de précipitation et des couverts nuageux associés.

Applications à la propagation terre-espace

Ce manuscrit réunit les résultats d’études consacrées à la modélisation spatio-temporelle des affaiblissements de propagation dans la troposphère. La modélisation de ces affaiblissements est utile pour l’optimisation et le dimensionnement des dispositifs adaptatifs de gestion de la ressource des satellites de télécommunication opérant en bande Ka ou Q/V. Ce manuscrit est organisé autour de cinq articles de revues. Les deux premières contributions sont dédiées à la modélisation des distributions statistiques de contenus troposphériques en vapeur d’eau et en eau liquide à partir respectivement de

mesures radiométriques et de bases de données climatiques. La troisième étude touche à l’évaluation des performances d’un système utilisant la diversité de site à partir de données de radar météorologique ; une étude de l’advection des champs précipitations et de son paramétrage par des sorties de modèles de prévision météorologique est également entreprise. Le quatrième article vise à établir une expression analytique de la distribution de la fraction d’une zone, d’une taille comparable à celle d’un faisceau

satellite, affectée par la pluie ; ce modèle est établi en assimilant les champs de précipitations à des champs aléatoires. Le formalisme des champs aléatoires est repris dans le cinquième article qui propose

une modélisation spatio-temporelle des champs de précipitations et des atténuations sous-jacentes à hautes résolutions et à l’échelle d’une couverture satellite continentale. Mots clefs : Propagation troposphérique, Champs aléatoires, FMT, Modèle de canal, Modélisation des précipitations.

Spatio-temporal modelling of rain field and of associated liquid water field.

Application to earth space propagation studies This manuscript gathers together the results studies devoted to the spatio-temporal modeling of propagation impairments through the troposphere. The modelling of the propagation impairments is of

prime importance for the dimensioning and the optimisation of adaptive radio resource management

systems of telecommunication satellite operating at Ka or Q/V band. This manuscript is a compilation of five articles. The two firsts are dedicated to the study of liquid water and water vapor distribution from radiometric and climatologic data. The third article deals with the study of site diversity systems using weather radar data. A study of rain advection and of its parameterization with a meteorological reanalysis database is also presented. The fourth article strives at establishing an analytical distribution of the fraction of an area affected by rain over a given threshold using computations on random fields.

The last publication rely also on this stochastic formalism, in order to propose a space-time modelling of rain field and of the underlying attenuation at high resolution and up to the scale of a continental satellite coverage. Keywords : Tropospheric propagation, Random fields, FMT, Channel Models, Rainfall modelling.

Nic

ola

s J

EA

NN

IN

– M

odéli

sati

on

sp

ati

o-t

em

po

rell

e d

es c

ham

ps d

e p

récip

itati

on

et

des c

ou

verts

nu

ag

eu

x a

sso

cié

s.

Ap

pli

cati

on

s à

la p

ro

pag

ati

on

terre e

sp

ace

THÈSE

En vue de l'obtention du

DDOOCCTTOORRAATT DDEE LL’’UUNNIIVVEERRSSIITTÉÉ DDEE TTOOUULLOOUUSSEE

Délivré par l’Institut Supérieur de l’Aéronautique et de l’Espace

Spécialité : Réseaux et télécommunications

Présentée et soutenue par Nicolas JEANNIN

le 14 novembre 2008

Modélisation spatio-temporelle des champs de précipitation et des couverts nuageux associés.


JURY

M. Michel Bousquet, président

M. Laurent Castanet M. Laurent Féral, co-directeur de thèse

M. Antonio Martellucci M. Aldo Paraboni, rapporteur M. Henri Sauvageot, directeur de thèse

Mme Danielle Vanhoenecker-Janvier, rapporteur M. Philippe Waldteufel, rapporteur

École doctorale : Mathématiques, informatique et télécommunications de Toulouse

Unité de recherche : Équipe d'accueil ISAE-ONERA SCANR

Directeur de thèse : M. Henri Sauvageot

Co-directeur de thèse : M. Laurent Féral

Modélisation spatio-temporel des champs de précipitation et des couverts nuageux associés.


Ce manuscrit réunit les résultats d’études consacrées à la modélisation spatio-temporelle des affaiblissements de propagation dans la troposphère. La modélisation de ces affaiblissements est utile pour l’optimisation et le dimensionnement des dispositifs adaptatifs de gestion de la ressource des satellites de télécommunication opérant en bande Ka ou Q/V. Ce manuscrit est organisé autour de cinq articles de revues. Les deux premières contributions sont dédiées à la modélisation des distributions statistiques de contenus troposphériques en vapeur d’eau et en eau liquide à partir respectivement de

mesures radiométriques et de bases de données climatiques. La troisième étude touche à l’évaluation des performances d’un système utilisant la diversité de site à partir de données de radar météorologique ; une étude de l’advection des champs précipitations et de son paramétrage par des sorties de modèles de prévision météorologique est également entreprise. Le quatrième article vise à établir une expression analytique de la distribution de la fraction d’une zone, d’une taille comparable à celle d’un faisceau

satellite, affectée par la pluie ; ce modèle est établi en assimilant les champs de précipitations à des champs aléatoires. Le formalisme des champs aléatoires est repris dans le cinquième article qui propose

une modélisation spatio-temporelle des champs de précipitations et des atténuations sous-jacentes à hautes résolutions et à l’échelle d’une couverture satellite continentale. Mots clefs : Propagation troposphérique, Champs aléatoires, FMT, Modèle de canal, Modélisation des précipitations.

Spatio-temporal modelling of rain field and of associated liquid water field.

Application to earth space propagation studies This manuscript gathers together the results studies devoted to the spatio-temporal modeling of propagation impairments through the troposphere. The modelling of the propagation impairments is of

prime importance for the dimensioning and the optimisation of adaptive radio resource management

systems of telecommunication satellite operating at Ka or Q/V band. This manuscript is a compilation of five articles. The two firsts are dedicated to the study of liquid water and water vapor distribution from radiometric and climatologic data. The third article deals with the study of site diversity systems using weather radar data. A study of rain advection and of its parameterization with a meteorological reanalysis database is also presented. The fourth article strives at establishing an analytical distribution of the fraction of an area affected by rain over a given threshold using computations on random fields.

The last publication rely also on this stochastic formalism, in order to propose a space-time modelling of rain field and of the underlying attenuation at high resolution and up to the scale of a continental satellite coverage. Keywords : Tropospheric propagation, Random fields, FMT, Channel Models, Rainfall modelling.

N

icola

s J

EA

NN

IN

– M

odéli

sati

on

sp

ati

o-t

em

po

rell

e d

es c

ham

ps d

e p

récip

itati

on

et

des c

ou

verts

nu

ag

eu

x a

sso

cié

s.

Ap

pli

cati

on

s à

la p

ro

pag

ati

on

terre e

sp

ace

THÈSE

En vue de l'obtention du

DDOOCCTTOORRAATT DDEE LL’’UUNNIIVVEERRSSIITTÉÉ DDEE TTOOUULLOOUUSSEE

Délivré par l’Institut Supérieur de l’Aéronautique et de l’Espace

Spécialité : Réseaux et télécommunications

Présentée et soutenue par Nicolas JEANNIN

le 14 novembre 2008

Modélisation spatio-temporelle des champs de précipitation et des couverts nuageux associés.


JURY

M. Michel Bousquet, président

M. Laurent Castanet M. Laurent Féral, co-directeur de thèse

M. Antonio Martellucci M. Aldo Paraboni, rapporteur M. Henri Sauvageot, directeur de thèse

Mme Danielle Vanhoenecker-Janvier, rapporteur M. Philippe Waldteufel, rapporteur

École doctorale : Mathématiques, informatique et télécommunications de Toulouse

Unité de recherche : Équipe d'accueil ISAE-ONERA SCANR

Directeur de thèse : M. Henri Sauvageot

Co-directeur de thèse : M. Laurent Féral

Résumé :

Ce manuscrit réunit les résultats d’une étude consacrée à la modélisation

spatio-temporelle des affaiblissements de propagation dans la troposphère. La

modélisation de ces affaiblissements est utile pour l’optimisation et le

dimensionnement des dispositifs adaptatifs de gestion de la ressource des satellites

de télécommunication opérant en bande Ka ou Q/V. Ce manuscrit est organisé autour

de cinq contributions. Les deux premières contributions sont dédiées à la

modélisation des distributions statistiques de contenus troposphériques en vapeur

d’eau et en eau liquide à partir respectivement de mesures radiométriques et de bases

de données climatiques. La troisième étude touche à l’évaluation des performances

d’un système utilisant la diversité de site à partir de données de radar

météorologique ; une étude de l’advection des champs précipitations et de son

paramétrage par des sorties de modèles de prévision météorologique est également

entreprise. Le quatrième article vise à établir une expression analytique de la

distribution de fraction d’une zone d’une taille comparable à celle d’un faisceau

satellite affectée par la pluie ; ce modèle est établi en assimilant les champs de

précipitations à des champs aléatoires. Le formalisme des champs aléatoires est

repris dans le cinquième article qui propose une modélisation spatio-temporelle des

champs de précipitations et des atténuations sous-jacentes à hautes résolutions et à

l’échelle d’une couverture satellite continentale.

.

Mots clefs : Physique de l’atmosphère – Propagation radioélectrique – Télédétection

atmosphérique – Radar météorologique – Télécommunications spatiales – Champs

aléatoires

Organisation du manuscrit :

Le présent mémoire est constitué de la juxtaposition de cinq articles en cours

de publication dans des revues spécialisées. Ces cinq contributions constituent les

chapitres un à cinq de la deuxième partie de ce manuscrit.

La première partie présente une synthèse en français des divers articles et de

la problématique générale.

Liste des publications de l’auteur

Articles de revues

“Distribution of tropospheric water vapor in clear and cloudy conditions from

microwave radiometric profiling”,

Iassamen A. , H. Sauvageot, N. Jeannin, S. Ameur, Journal of applied

meteorology and climatology, 2009, vol. 48, no3, pp. 600-615.

“Statistical distribution of the fraction of an area affected by rain above a

given threshold”,

Jeannin N, L. Féral, H. Sauvageot, L. Castanet , J. Lemorton, Journal of

Geophysical Research, 2008, vol. 113, D21120, doi:10.1029/2008JD009780.

“Statistical distribution of integrated liquid water and water vapor content

from meteorological reanalysis”,

Jeannin N, L. Féral, H. Sauvageot, L. Castanet , IEEE Transactions on

Antennas and Propagation, 2008, vol. 56, no10, pp. 3350-3355

“Use of weather radar data for site diversity predictions and impact of rain

field advection”,

L. Luini, N. Jeannin, C. Riva, C, C. Capsoni, L. Castanet, J. Lemorton

accepté à International Journal of Satellite Communication.

“A large scale space-time stochastic model of rain attenuation for the design

and optimization of adaptive satellite communication systems operating between 10

and 50GHz”,

Jeannin N, L. Féral, H. Sauvageot, L. Castanet , J. Lemorton, F. Lacoste

soumis à IEEE Transactions on Antennas and Remote Sensing.

Conférences internationales

Jeannin N, L. Féral, H. Sauvageot, L. Castanet, J. Lemorton, K. Leconte and

F. Lacoste, “Modelling of rain fields and attenuation fields spatially correlated over

Europe and US”, in Proceedings 3rd CNES Workshop on Earth - Space Propagation,

(CNES, ed.), Toulouse, France, Sep. 2006.

Jeannin N., L. Féral, H. Sauvageot, L. Castanet, J. Lemorton, F. Lacoste and

K. Leconte, “Modeling of rain fields at large scale”, in International Workshop on

Satellite and Space Communications (IWSSC) 2006, Madrid, Spain, Sep. 2006.

Jeannin N, L. Féral, H. Sauvageot, L. Castanet, J. Lemorton, F. Lacoste

“Stochastic Spatio-Temporal Modelling of Rain Attenuation for Propagation

Studies”, in European Conference on Antennas and Propagation 2007, Edinburgh,

UK, Nov 2007.

Jeannin N., L. Luini, L. Castanet, J. Lemorton, C. Capsoni, C. Riva, A.

Paraboni: “Comparison of the advection of rain fields derived from radar data and

wind outputs from the ERA-40 database”, in Second SatNEx JA-2310 workshop,

Oberpfaffenhofen, Germany, April 2008.

Luini L., N. Jeannin, L. Castanet, J. Lemorton, C. Capsoni, C. Riva, A.

Paraboni: “Simulation of a site diversity system using radar data”, in Second SatNEx

JA-2310 workshop, Oberpfaffenhofen, Germany, April 2008.

Jeannin N., Féral L., Sauvageot H., Castanet L., Lacoste F. “A large scale,

high resolution channel model for propagation impairment techniques design and

optimization”, in International Workshop on Satellite and Space Communications

(IWSSC) 2008, Toulouse, France, Oct. 2008.

Table des matières

0H0HA. Synthèse des travaux 115H1

1H1HProblématique 116H2

2H2HI. Distribution statistique des affaiblissements de propagation 117H12 3H3H1 Introduction 118H12 4H4H2 Données utilisées 119H13 5H5H3 Distribution des intensités de précipitations 120H14

6H6H3.1 Modélisation UIT 121H15 7H7H3.2 Perspectives d’amélioration 122H18

8H8H4 Distribution des contenus intégrés en eau liquide 123H18 9H9H5 Distribution des contenus intégrés en vapeur d’eau 124H21 10H10H6 Conclusion 125H24

11H11HII. Étude de la variabilité spatiale et temporelle des affaiblissements de

propagation 126H26 12H12H1 Introduction 127H26 13H13H2 Etude d'un système de diversité de sites à partir de données radars 128H26 14H14H3 Modélisation spatiale des champs de précipitations 129H30

15H15H3.1 Besoins 130H30 16H16H3.2 Modélisation proposée 131H31 17H17H3.3 Évaluation de la pertinence des résultats obtenus 132H43 18H18H3.4 Limitations 133H48

19H19H4 Modélisation spatio-temporelle des affaiblissements de propagation 134H48 20H20H4.1 Problématique 135H48 21H21H4.2 Méthodologie 136H49 22H22H4.3 Comparaison des caractéristiques des simulations à des données

indépendantes 137H55 23H23H5 Conclusion 138H56

24H24HConclusion et perspectives 139H58

25H25HBibliographie 140H60

26H26HB. Articles 141H67

27H27HStatistical distribution of integrated liquid water and water vapor content

from meteorological reanalysis 142H69 28H28HAbstract 143H69 29H29H1 Introduction 144H69 30H30H2 The data 145H71 31H31H3 Modelling of the ILWC distribution 146H72

32H32H3.1 Analytic formulation and parameter derivation 147H72 33H33H3.2 Model accuracy 148H75

34H34H4 Modelling of the IWVC distribution 149H77 35H35H4.1 Analytic formulation and parameter derivation 150H77 36H36H4.2 Model accuracy 151H80

37H37HConclusion 152H81 38H38HAcknowledgements: 153H82 39H39HReferences 154H82

40H40HDistribution of tropospheric water vapor in clear and cloudy conditions from

microwave radiometric profiling 155H86 41H41HAbstract 156H86 42H42H1 Introduction 157H87 43H43H2 Data 158H89

44H44H2.1 Water vapor parameters 159H89 45H45H2.2 Microwave radiometric measurements 160H90 46H46H2.3 Accuracy and resolution 161H92 47H47H2.4 Site and dataset 162H94

48H48H3 Water vapor content distribution 163H99 49H49H4 Distribution of IWV 164H110 50H50H5 Comparison with operational radiosonde soundings 165H115 51H51H6 Summary and conclusions 166H119 52H52HAcknowledgements 167H120 53H53HReferences 168H120

54H54HWeather radar data for site diversity predictions and evaluation of the

impact of rain field advection 169H125 55H55HAbstract 170H125 56H56H1 Introduction 171H126 57H57H2 The weather radar datasets 172H127 58H58H3 Simulation of site diversity systems using radar images 173H128

59H59H3.1 Gain dependence on the link orientation 174H131 60H60H3.2 Gain dependence on the baseline orientation 175H132 61H61H3.3 Gain dependence on the rain event type 176H135 62H62H3.4 Comparison between the two sites 177H137

63H63H3.5 Comparison with the site diversity model currently recommended by ITU-R

178H138 64H64H4 Comparison between the rain field advection and the ERA-40 wind outputs 179H140

65H65H4.1 Determination of rain advection from radar data 180H141 66H66H4.2 Comparison between radar derived wind speed and direction and ERA-40

wind outputs 181H143 67H67H5 Impact of the wind direction on the rain field spatial anisotropy 182H147 68H68H6 Conclusion 183H149 69H69HAcknowledgements 184H150 70H70HReferences 185H150

71H71HStatistical distribution of the fractional area affected by rain 186H153 72H72HAbstract 187H153 73H73H1 Introduction 188H154 74H74H2 Theoretical considerations on stationary Gaussian random fields 189H157

75H75H2.1 Spatial average M of a Gaussian random field 190H158 76H76H2.2 Spatial variance V of a Gaussian random field 191H159

77H77H3 Model for the CDF of the fractional area over a given threshold 192H161 78H78H3.1 Analytical derivation 193H161 79H79H3.2 Model accuracy and parameter sensitivity 194H162

80H80H4 Application to rain fields 195H165 81H81H4.1 Model extension to rain field 196H165 82H82H4.2 Radar data 197H166 83H83H4.3 Results 198H168 84H84H4.4 Parameterization of the distribution 199H176

85H85H5 Concluding remarks and applications 200H177 86H86HAcknowledgment: 201H178 87H87HBibliography 202H178

88H88HA large scale space-time stochastic model of rain attenuation for the design

and optimization of adaptive satellite communication systems operating between 10

and 50GHz 203H184 89H89HAbstract: 204H185 90H90HIntroduction 205H186 91H91H1 Spatio-temporal modelling of rain fields 206H188

92H92H1.1 Generation of correlated Gaussian random field 207H189 93H93H1.2 Conversion into rain rate fields 208H191 94H94H1.3 Parameterization and refinement of the modeling 209H192 95H95H1.4 Limitations of the presented methodology 210H199

96H96H2 Extension of the model using inputs from meteorological reanalysis database 211H200 97H97H2.1 Description of the ECMWF ERA-40 dataset 212H201 98H98H2.2 Parameterization of the rain amount 213H201 99H99H2.3 Advection 214H208

100H100H3 Conversion into attenuation 215H209

101H101H3.1 Conversion of rain rate fields into attenuation fields 216H209 102H102H3.2 Over sampling of the attenuation time series 217H210

103H103H4 Preliminary validations, limitations 218H211 104H104H4.1 First order statistics 219H211 105H105H4.2 Second order statistics 220H213

106H106H5 Conclusion 221H215 107H107HAcknowledgment: 222H216 108H108HBibliography 223H217

A. Synthèse des travaux

Synthèse des travaux

Problématique

2

Problématique

La demande de plus en plus importante pour des services multimédia nécessitant

un débit et une bande passante importants conduit les systèmes de télécommunications

par satellite à utiliser des bandes de fréquences plus élevées. En effet, les bandes de

fréquences précédemment allouées pour le service fixe, C (4-6 GHz) ou Ku (11-14

GHz), sont engorgées. Les largeurs de bande correspondantes ne suffisent plus pour le

développement de nouveaux systèmes de télécommunication par satellite, qui doivent

présenter des performances comparables à celles offertes par les réseaux terrestres afin

de s’inscrire dans l’infrastructure globale des sytèmes d’information et de

télécommunication.

Par conséquent, de nouveaux systèmes opérant à des bandes de fréquence

supérieures, Ka (20-30 GHz), voire Q/V (40-50 GHz), pour les télécommunications

civiles, ou EHF (20-45 GHz) pour les télécommunications militaires, sont

progressivement mis en service.

A ces fréquences, la propagation des signaux à travers la troposphère joue un rôle

clef dans le dimensionnement des bilans de liaison. Les atténuations rencontrées à ces

fréquences constituent un frein au déploiement de ces systèmes. En effet, lors de la

traversée de la troposphère (couche inférieure de l'atmosphère correspondant en

moyenne aux altitudes 0-12 km), les ondes électromagnétiques subissent une atténuation

causée par les gaz atmosphériques (vapeurs d'eau, oxygène), les nuages et les

hydrométéores (pluie, grêle, neige…) et sont affectées par un effet de scintillation dû à

la turbulence atmosphérique. Les atténuations les plus fortes dans le domaine pour les

fréquences de 10 à 50 GHz sont rencontrées en présence de fortes pluies ou de grêle, et

elles augmentent avec la fréquence de la liaison.

Pour les liaisons terre-espace en bande Ku, ces atténuations restent relativement

faibles et ne dépassent généralement pas 10 dB pendant plus d'une heure par an (0.01%

d’une année) du moins en région tempérée. Afin de tenir compte de cette atténuation

potentielle dans le bilan de liaison des systèmes grand public, une marge de puissance

de quelques dB est suffisante.

Cette solution n'est plus satisfaisante en bande Ka et au dessus car des

atténuations de plusieurs dizaines de dB peuvent être dépassées plusieurs heures par an.


Problématique

3

L'utilisation d'une marge de puissance fixe pour compenser cette atténuation se

traduirait par un coût prohibitif du système et notamment des segments sols.

Toutefois, afin d'assurer une qualité de service et une disponibilité comparables à

celles des systèmes de télécommunications opérant à des gammes de fréquences

inférieures, des techniques de compensation des affaiblissements qui adaptent en temps

réel les caractéristiques du système en fonction de la variabilité spatiale et temporelle

des atténuations troposphériques sont mises en place. Différentes techniques adaptatives

(ou FMT pour Fade Mitigation Techniques) éventuellement cumulables sont

envisageables selon les configurations (liaison montante, liaison descendante, diffusion,

station sols…) du système envisagé [Castanet, 2001]. Le dimensionnement et

l’optimisation de ces systèmes de compensation adaptatifs des affaiblissements de

propagation nécessitent une connaissance fine des variabilités spatiales et temporelles de

ces affaiblissements de propagation qui ne peut se réduire à une simple étude de la

distribution statistique de ces affaiblissements [Cost 255, 2002].

Le dimensionnement des marges statiques de puissance utilisées pour compenser

les atténuations troposphériques, en bande Ku ou à des fréquences inférieures, a donné

lieu à de nombreuses études statistiques des distributions ponctuelles d’atténuation et de

chacun des paramètres radio-climatiques sous-jacents (intensité de précipitation,

contenu intégré en eau liquide, contenu intégré en vapeur d’eau…).

Ces études ont montré une grande disparité de ces distributions d’atténuation

selon les zones climatiques considérées, principalement imputable à la variabilité des

régimes de précipitations. En effet, les pluies fortement convectives des régions

tropicales ou équatoriales génèrent des atténuations supérieures à celles induites par les

précipitations des régions tempérées.

Afin de quantifier aux mieux ces distributions et de prendre en compte cette

dépendance climatique, ces études ont abouti à la création de cartes de paramètres radio-

climatiques et de méthodes permettant d’estimer ces distributions d’atténuation en tout

point du globe. Ces paramètres sont tabulés par des recommandations de l’Union

Internationale des Télécommunications (UIT-R).

Considérant une liaison avec des caractéristiques géographiques et

radioélectriques données, ces recommandations permettent d’estimer les probabilités

d’atténuation sur cette liaison dépassée pendant un certain pourcentage du temps d’une


Problématique

4

année moyenne. A cet effet, les recommandations UIT-R P.618, P.676 et P.840

permettent d’obtenir cette distribution cumulative d’atténuation totale en fonction des

distributions cumulatives d’intensité de précipitations (recommandation UIT-R P.837),

de contenus intégrés en eau liquide (recommandation UIT-R P.840), de contenus

intégrés en vapeur d’eau (recommandation UIT-R P.836) ainsi que de la température et

de l’hygrométrie du sol (recommandation UIT-R P.1511) et de l’altitude de l’isotherme

–2oC (recommandation UIT-R P.839).

Par conséquent, connaissant la spécification de l’indisponibilité nominale que

peut subir une liaison terre-espace compte tenu du service à assurer (généralement

moins de 1 % du temps de fonctionnement pour des terminaux utilisateurs à faible

disponibilité et parfois jusqu’à 0.001 % pour des hubs), la marge statique de puissance

minimale pour compenser les effets atmosphériques dans le bilan de liaison peut être

calculée. Elle correspond à la valeur d’atténuation dépassée pour un temps équivalant à

l’indisponibilité tolérée. La recommandation UIT-R P.618, qui permet d’estimer la

distribution cumulative de l’atténuation totale en fonction du temps quelles que soient

les caractéristiques géographiques, géométriques et radioélectriques de la liaison, ainsi

que la climatologie locale, permet ainsi d’estimer cette marge.

Cependant, l’utilisation de techniques adaptatives de compensation des

affaiblissements de propagation, rendue nécessaire par l’utilisation de fréquences plus

élevées, requiert une connaissance plus fine du canal de propagation. En effet, la seule

connaissance des distributions en un point des affaiblissements de propagation ne

permet pas d’optimiser convenablement ces dispositifs. Par exemple, dans le cas d’une

liaison entre un émetteur et un satellite en présence d’une forte atténuation, la liaison

peut être maintenue en agissant sur la puissance d’émission du terminal, sur la forme

d’onde utilisée, ou encore sur le débit de l’information. L’utilisation de ces différents

mécanismes requiert une connaissance de la dynamique temporelle du canal de

propagation [Castanet et al., 2001].

Dans le cas du contrôle de la puissance émise, il faut ajuster cette puissance aux

variations des affaiblissements de propagation tout en évitant d’interférer sur les liaisons

des spots adjacents réutilisant les mêmes fréquences. Un ajustement de puissance de 3 à

5dB pour les terminaux commerciaux et de 10dB pour les hubs est envisagé pour les

systèmes en cours de déploiement.


Problématique

5

Dans le cas de l’adaptation de la forme d’onde pour laquelle les derniers

standards proposent des modulations résistantes à des atténuations allant jusqu’à 20dB,

il faut également une estimation précise des variations temporelles des atténuations

troposphériques, et en particulier de celles dues à la pluie. En effet, la afin de déterminer

d’une part les différents modes de codages à implémenter et d’autre part les différentes

modulations envisageables [Bolea-Alamañac, 2004].

Par conséquent, en plus d’avoir une connaissance de la statistique des

affaiblissements de propagation affectant la liaison, il est impératif d’avoir une

connaissance des caractéristiques liées à la corrélation temporelle de ces

affaiblissements de propagation pour le dimensionnement et l’optimisation de ces

techniques de compensations adaptatives.

D’autres mécanismes de compensation adaptatifs nécessitent, quant à eux, une

connaissance statistique de la variabilité spatiale des affaiblissements. Par exemple, pour

augmenter la disponibilité d’une liaison entre une station sol et le satellite, une

deuxième station sol, généralement localisée à quelques dizaines de kilomètres et reliée

par un lien terrestre à la première, peut être implantée. Dans le cas où la liaison entre la

première station et le satellite est affectée par une atténuation ne permettant pas la

transmission du signal (en présence de fortes précipitations), l’autre station est utilisée

et le signal est relayé par le réseau terrestre ou une liaison sol dédiée. Cette technique

pour augmenter la disponibilité de la liaison, connue sous le nom de diversité de site,

repose sur l'hétérogénéité spatiale des champs de précipitations [Goldhirch et Robison

1975] .

En effet, considérant deux stations distantes de quelques dizaines de kilomètres,

la probabilité que les liaisons entre le satellite et chacune de ces stations soient

simultanément affectées par un affaiblissement ne permettant pas la transmission du

signal est plus faible que la probabilité qu’une seule liaison subisse un affaiblissement

trop élevé du fait de l’inhomogénéité spatiale des champs de précipitations. La

problématique principale pour le dimensionnement de ces systèmes est celle de la

distance minimale devant séparer les deux sites afin d’obtenir une disponibilité donnée.

La connaissance de la variabilité spatiale des affaiblissements de propagation est

également essentielle du point de vue de la gestion de la ressource embarquée [Pech,


Problématique

6

2003]. En effet, l’allocation des plages de fréquences se fait par groupes d’utilisateurs

situés à l’intérieur d’un même faisceau du satellite, et nécessite d’avoir une estimation

de l’atténuation encourue par chacune des liaisons à l’intérieur de ce faisceau.

Enfin, tenant compte des derniers développements technologiques sur les

antennes et s’appuyant sur le caractère multi-faisceaux des nouveaux systèmes

satellitaires, une reconfiguration du diagramme d’antenne et des gains afférents peut

permettre de contrer les affaiblissements de propagation. En focalisant plus fortement

les antennes sur les zones susceptibles de subir de fortes atténuations, et donc des

précipitations, des marges supplémentaires peuvent être dégagées dans le bilan de

liaison. A cet effet, le mécanisme sous-jacent de reconfiguration des antennes requiert

une connaissance de la répartition spatiale de l’affaiblissement sur la couverture globale

du système (généralement un continent). Cette connaissance doit être statistique pour le

dimensionnement du dispositif, voire prédictive à une échéance de quelques heures pour

la reconfiguration des antennes [Buti et al, 2008].

Certains mécanismes de rétroaction nécessitent une connaissance à la fois

temporelle et spatiale du canal de propagation [Bousquet et al., 2003]. En effet,

l’implémentation de mécanismes de codage et de modulation adaptatifs requiert une

connaissance de l’évolution du canal pour chaque terminal. L’allocation dynamique des

porteuses (dans le cas d’un accès multiple en fréquence) ou des paquets (dans le cas

d’un accès multiple dans le temps) ne peut se faire que globalement à l’échelle d’un

spot, la ressource embarquée étant limitée, et ne peut par conséquent être optimisée

qu’en prenant en considération de manière simultanée l’état du canal de propagation

pour chacun des terminaux.

Pour répondre aux problématiques citées ci-dessus, des modèles statistiques

ont été proposés par l’UIT-R pour donner des indications statistiques sur les variabilités

temporelle et spatiale des affaiblissements de propagation. Ainsi, la recommandation

UIT-R P.1623 propose une méthode d’estimation des statistiques sur la durée des

affaiblissements au-dessus d’un certain niveau d’atténuation, sur le gradient temporel

des affaiblissements ainsi que sur la durée entre deux événements d’atténuation.

D’autres parts, la recommandation UIT - R P.618 permet de calculer le gain de diversité

ou la probabilité jointe d’atténuation simultanément dépassée en fonction de la distance


Problématique

7

entre deux stations. Même si ces méthodes statistiques permettent l’évaluation de

différents paramètres dimensionnant des boucles de contrôle de certaines FMT, elles

s’avèrent relativement imprécises lorsque comparées à des mesures du canal de

propagation. De plus elles ne permettent qu’une description partielle des caractéristiques

du canal de propagation, insuffisante pour résoudre les problèmes liés au

dimensionnement des FMT. Un recours à des données ou méthodes de représentation

différentes doit donc être envisagé.

Diverses expériences de mesure des affaiblissements de propagation sur les

liaisons terre-espace ont été réalisées au cours des dernières décennies (SIRIO [Carassa,

1978], OLYMPUS [OPEX, 1994], ACTS [Crane, 1997], ITALSAT [Riva, 2004] et

bientôt ALPHASAT [Paraboni et al., 2007] et GSAT-4 [Katti et al., 2007]). Elles ont

permis d’obtenir des séries temporelles d’atténuation subie par une liaison terre-espace

en différents points du globe, sur des durées relativement longues, et échantillonnées à

des cadences de l’ordre de la seconde.

Si elles ont contribué à mieux caractériser la dynamique fine du canal de

propagation, ces campagnes de mesure d’atténuation troposphérique ne sont toutefois

pas suffisantes pour résoudre les problèmes posés par le dimensionnement des boucles

de contrôle FMT pour plusieurs raisons. Le nombre de stations de mesure impliquées

dans ces expériences était limité, sans synchronisation entre les stations, ce qui par

conséquent ne donne qu’une idée sommaire de la répartition spatiale des

affaiblissements de propagation. Ensuite, ces expériences ont été menées à des

fréquences, des élévations et dans des régions climatiques bien précises et il n’est pas

aisé de déduire les caractéristiques des affaiblissements pour des liaisons à d’autres

fréquences, élévations et régions climatiques. La prise en compte des affaiblissements

de propagation pour le dimensionnement des boucles FMT doit donc être réalisée par le

biais d’autres moyens.

Une des approches permettant de s’affranchir de ces limitations est l’étude de la

variabilité des affaiblissements de propagation à partir de paramètres météorologiques

causant ces affaiblissements. Comme mentionné précédemment, les affaiblissements de

propagation sont liés à la présence de pluie, de nuages et des gaz atmosphériques

(oxygène et vapeur d’eau). Connaissant les intensités de précipitations, les contenus


Problématique

8

intégrés en eau liquide des nuages et les concentrations en gaz le long d’une liaison

terre-espace, des modèles statistiques reposant sur une base physique ont été développés

pour estimer l’atténuation subie par l’onde électromagnétique le long de cette liaison.

De ce fait, à partir de données ou de prévisions météorologiques portant sur ces

paramètres, il est possible d’estimer l’atténuation subie par la liaison et ce quelles que

soient la configuration géométrique ou la fréquence de la liaison.

Comme les systèmes de télécommunication par satellite qui seront déployés dans

les prochaines années visent l’utilisation de la bande Ka, pour laquelle la contribution de

la pluie à l’atténuation totale est prépondérante, la plupart des études se concentrent sur

les précipitations.

Les données météorologiques permettant l’étude des précipitations sont multiples

[SatNex, 2008]. Les plus appropriées pour calculer l’atténuation sur une liaison satellite

sont celles obtenues à partir des radars météorologiques. En effet, ces derniers

fournissent, à intervalles réguliers, une cartographie des réflectivités induites par les

précipitations. Ces réflectivités radars peuvent être liées aux intensités de précipitations

par diverses relations telles que celles de Marshall-Palmer. Les intensités de

précipitations étant liées à l’atténuation spécifique, l’atténuation totale peut être

calculée à partir de ces données par intégration de l’atténuation spécifique le long de la

liaison. La couverture des réseaux radars météorologiques est cependant insuffisante

pour que ces données soient directement exploitables pour réaliser des simulations

systèmes.

Il serait également envisageable de convertir des sorties de modèle

météorologique méso-échelle contraint par des données de réanalyse en atténuation

troposphérique [Hodges et al., 2006]. Cependant, les temps de calcul et la fiabilité des

prévisions [Lascaux et al., 2003] constituent pour le moment un obstacle à la mise en

œuvre systématique d’un tel dispositif. De plus, la représentativité statistique des

atténuations générées, qui reste un des problèmes majeurs dans le contexte des

télécommunications par satellite, n’est pas assurée par par la description des processus

de microphysiques implémentée dans les modéles météorologiques. Cependant comme

mentionné dans [Hodges et al., 2006], l’utilisation de modèles de prévision méso-

échelle, rest la piste la plus prometteuse pour les prédictions des champs d’atténuations

à une échéances de quelques heures.


Problématique

9

Les autres sources de données météorologiques, (données de télédétection

spatiale, modèles de réanalyse), n'ont pas une résolution suffisante pour rendre compte

précisément des phénomènes donnant lieu à de l'atténuation. L'utilisation de mesures

ponctuelles issues de radio-sondages, de radiomètres ou de réseaux de disdromètres

pourrait également être pertinente, mais, comme pour les mesures de canal de

propagation, le maillage spatial des réseaux de mesures est trop lâche pour fournir une

information spatiale suffisante.

Pour pallier ce manque de données directement et systématiquement utilisables

pour le dimensionnement des FMT, des modèles permettant de générer des conditions de

propagation réalistes ont été développés. Leur élaboration a été permise par l'analyse

statistique des données de propagation issues des campagnes de mesures [Audoire, 2001,

Van de Kamp, 2003, Lacoste, 2005], mais aussi des divers types de données

météorologiques évoquées précédemment. Les efforts se sont dans un premier temps

portés sur des modélisations temporelles du canal de propagation et il existe désormais

des modèles reproduisant de manière assez réaliste la dynamique observée sur des

données expérimentales [Maseng et Bakken, 1981, Gremont et Filip, 2004, Lacoste et

al., 2006, Lemorton et al., 2007].

Des modèles portant sur l'aspect variabilité spatiale des propagations ont ensuite

été étudiés [Casponi et al., 1987a, 1987b], [Féral et al, 2003a, 2003b, 2006],

[Callaghan, 2006]. Ils se sont plus particulièrement attachés à donner des

représentations réalistes des champs de précipitation observés par les radars

météorologiques. Pour le développement de ces modèles, proches de problématiques

relatives à l'hydrologie, différentes pistes ont été explorées. Des modélisations reposant

sur l'individualisation des cellules de précipitation [Féral et al, 2003a], ou sur le

caractère fractal des champs de précipitation ont ainsi été proposées. Des modélisations

de champs de précipitations par des champs aléatoires corrélés dans l'espace ont

également vu le jour.

Le potentiel de ces modèles dépend toutefois de la possibilité de simuler des

champs à des résolutions suffisamment fines (kilomètriques) et sur des étendues

suffisantes (couverture continentale) pour répondre aux besoins exprimés pour les

systèmes de télécommunication par satellite. C'est pourquoi des méthodologies de

généralisation de ces modèles combinant plusieurs approches ou prenant également en


Problématique

10

compte des observations ont été récemment développées pour répondre plus

spécifiquement à ces besoins.

A un degré plus marqué encore, les modèles de propagation troposphérique

prenant en compte les aspects à la fois temporels et spatiaux ne permettent pas pour

l'heure de répondre de manière globale aux problématiques posées par le

dimensionnement des boucles FMT des techniques de gestion de la ressource. Pour

pallier ce manque, l'objectif de cette thèse a été le développement d'un générateur de

champs d'atténuation totale corrélé dans l'espace et dans le temps.

Le présent mémoire réunit différents travaux publiés dans des revues

internationales, et dont l’objectif général est d’arriver à une meilleure connaissance du

canal de propagation dans le temps et dans l'espace en vue du dimensionnement optimal

des systèmes de télécommunications par satellite opérant aux fréquences supérieures à

10 GHz. Ils s’appuient sur une analyse statistique des paramètres météorologiques

influant sur les conditions de propagation.

Le premier article vise à renforcer la connaissance statistique des distributions de

contenus intégrés en eau liquide et en eau vapeur en utilisant des données de réanalyse

météorologique. Cette étude a principalement été motivée par le peu de données et leur

faible qualité concernant la distribution de ces paramètres météorologiques.

Le second article présente une modélisation des distributions de contenus

intégrés en vapeur d'eau à partir de profils radiométriques. Différentes modélisations de

ces distributions en fonction de la présence de nuages ou non et de la température sont

proposées. L'étude de ces distributions peut permettre de mieux connaître les couplages

existant entre les atténuations dues aux nuages et à la vapeur d’eau.

Le troisième article est une étude d’un système de diversité de sites à partir de

données de radars météorologiques. Cette étude montre l’avantage de l’utilisation de

données radars par rapport aux statistiques de l’UIT-R pour le dimensionnement optimal

de ce dispositif. Cette étude évalue également l’influence de différents paramètres sur

les performances du système. En particulier, il est montré que la direction prédominante

du vent a un impact sur l’orientation optimale des stations l’une par rapport à l’autre.

La contribution suivante propose un modèle de distribution statistique de la

fraction d’une zone affectée par des intensités de précipitation dépassant un certain


Problématique

11

seuil, et dont la taille est comparable à celle d’un faisceau satellite. Cette étude permet

d’une part, à l’intérieur d’un faisceau, d’estimer le nombre de récepteurs touchés par une

atténuation au-delà d’un certain seuil, et d’autre part de poser les fondements théoriques

d’un modèle de génération de champs de précipitations reposant sur l’utilisation de

champs aléatoires développés dans le dernier article.

Dans ce dernier article, un modèle de génération de champs d’atténuation due à la

pluie corrélés dans l’espace et dans le temps est proposé. Il repose sur une modélisation

existante des champs de précipitation par le biais de champs aléatoires corrélés dans

l’espace et dans le temps. Une méthode de paramétrage de ce modèle est décrite, et le

modèle lui-même est amélioré afin d’étendre son domaine de validité dans l'espace et

dans le temps. Cette extension du domaine de validité est obtenue en couplant le modèle

avec des données de réanalyse météorologique. Il génère des sorties directement

utilisables pour simuler la couche propagation dans les simulations de systèmes de

télécommunication par satellite.

La première partie de ce mémoire est une synthèse des différentes

problématiques afférentes à la prise en compte des affaiblissements de propagation dans

la troposphère, en vue de dimensionner les systèmes de télécommunication par satellite.

Un premier chapitre est consacré à la modélisation des distributions statistiques

d'atténuation et des paramètres météorologiques sous-jacents. Le deuxième chapitre

traite de la modélisation des variabilités spatiales et spatio-temporelles des

affaiblissements de propagation qui a constitué la plus grande partie de mes travaux de

recherche

La seconde partie du manuscrit reprend le texte des cinq articles, publiés ou en

cours de publication, écrits au cours de la thèse.


I. Distribution statistique des affaiblissements de propagation

12


1 Introduction

Les études de bilan de liaison en vue du déploiement des systèmes de

télécommunication par satellite ont conduit à de nombreux travaux sur la distribution

statistique des affaiblissements de propagation à travers l’atmosphère. Ces études ont

débouché sur des modélisations communément acceptées au travers de recommandations

de l’UIT-R qui permettent d’obtenir une estimation de la probabilité d’occurrence d’une

atténuation pour une liaison donnée comme illsutré figure 1.

Figure 1: Affaiblissement dépassé durant P% d’une année moyenne à Paris calculé à

partir de la Recommandation UIT-R P.618-9. Fréquence = 40 GHz, élévation = 30°

Ces distributions d’atténuation sont obtenues en établissant un lien physique ou

statistique avec les distributions des paramètres météorologiques sous-jacents.



13

Néanmoins, une amélioration de ces modélisations est encore nécessaire. En effet, il

serait intéressant de pouvoir estimer les variabilités de ces distributions (saisonnières,

diurnes, interannuelles) lesquelles ne sont pas quantifiées à l’heure actuelle. D’autre

part, une attention toute particulière doit être portée à l’atténuation induite par les gaz et

les nuages, qui, aux bandes Q/V ne dépasse certe pas quelaues dB et pour lesquels les

modélisations actuelles sont imprécises. Les articles “Statistical distribution of

integrated liquid water and water vapor content from meteorological reanalysis” et

“Distribution of tropospheric water vapor in clear and cloudy conditions from

microwave radiometric profiling” sont consacrés à cette problématique.

La connaissance de ces distributions est un point de départ incontournable non

seulement pour le dimensionnement du bilan de liaison des satellites de

télécommunication opérant aux fréquences supérieures à 5 GHz, mais aussi pour les

modèles permettant de générerle comportement fin du canal de propagation dans

l’éspace et dans le temps. L’amélioration de la description statistique des

affaiblissements a, par conséquent, un effet bénéfique sur le réalisme des modèles s’y

référant.

2 Données utilisées

La modélisation des distributions d'atténuation telle que proposée par l’UIT-R

repose sur l’utilisation de deux sortes de données :

- des données expérimentales de référence, ponctuelles, collectées en différents

points du globe mais dont le nombre est relativement restreint ;

- des bases de données mondiales de paramètres radio-climatiques qui servent

d’entrée au modèle proposé.

La démarche générale a consisté à élaborer des modèles statistiques, prenant en

entrée les paramètres radio-climatiques des bases de données mondiales, et à optimiser

les paramètres de ces modèles afin qu’ils reproduisent le plus fidèlement possible les

statistiques obtenues à partir des données expérimentales de référence.

Différents types de données de référence ont été considérés pour la dérivation des

distributions de paramètres météorologiques ou d’atténuation. Les données des



14

campagnes de mesures de propagation sont évidemment considérées comme les données

expérimentales de référence pour les effets de propagation à savoir l’atténuation, la

scintillation ou la dépolarisation. Les données expérimentales utilisées pour la

modélisation des précipitations sont issues de mesures disdrométriques ou

pluviométriques, celles pour la modélisation des contenus en eau liquide ou en vapeur

d’eau sont issues de mesures effectuées à partir de radiomètres micro-ondes ou de radio-

sondages. Les mesures servant de référence ont été recueillies sur plusieurs décennies à

partir de différentes instrumentations dont les résultats ne sont pas nécessairement

comparables entre-elles.

Les bases de données mondiales sont quant à elles généralement dérivées de

données de réanalyse météorologique. Ces données, représentant l’état passé de

l’atmosphère à l’échelle mondiale, sont issues de modèles de prévision numérique,

contraints par des observations (radio-sondages, données satellitaires, barométriques,

anémométriques…).

Dans le cadre des études de propagation menées jusqu’ici, les bases de données

de réanalyse utilisées émanent majoritairement de l’ECMWF (Centre Européen de

Prévision Météorologique à Moyen Terme). Le processus de réanalyse porte uniquement

sur certains paramètres atmosphériques, à savoir : la pression, la température, l’humidité

relative et le vent horizontal. Les paramètres importants pour l’étude de la propagation,

comme les précipitations ou les contenus intégrés en eau liquide des nuages, sont

déduits de ces paramètres internes et ne sont pas recalibrés sur des observations.

3 Distribution des intensités de précipitations

De nombreuses hypothèses et pistes ont été explorées pour modéliser les

statistiques des intensités de précipitation. Ainsi des lois, hyperbolique, Weibull, log-

normale, Gamma, ont été proposées, pour modéliser cette distribution, principalement

dans le domaine de l’hydrologie. Aucun consensus absolu ne se dégage, bien que

certaines modélisations semblent plus adaptées que d’autres pour décrire la majorité des

distributions d’intensités de précipitation observées et du fait des hypothèses physiques

sous-jacentes. La modélisation ayant cours dans le domaine des télécommunications par



15

satellite ne repose sur aucun de ces modèles. Toutefois, elle présente l’intérêt de fournir

une modélisation de l’intensité des précipitations en tout point du globe.

3.1 Modélisation UIT

La recommandation UIT-R P.837-5 ([Poaires-Baptista and Salonen, 1997] mis à

jour dans [Castanet et al., 2007]) propose le modèle suivant pour la CCDF (fonction de

répartition complémentaire) d’intensité de précipitations en tout point du globe :

*

**

11

0* )( Rc

RbRa

ePRRP +

+−

=> (1)

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−=

−−

6

)1(0079.0

60 1 r

T

PM

r ePPβ

(2)

où P0 représente la probabilité de pluie en un point, avec :

bcP

Mb

a

T

02.2621797

09.1

0

=

=

=

(3)

Trois paramètres calculés à partir de la réanalyse météorologique ERA-40 de

l’ECMWF [Uppala et al., 2005], servent d’entrée au modèle (1). La base de données

ERA-40 couvre une quarantaine d’années (1957-2001). Comme ces données comportent

des erreurs d’estimation, elles ont été recalibrées en utilisant les bases de données

GPCC [Rudolf et al., 2003] et GPCP [Huffman et al., 1997] élaborées à partir de

données pluviométriques mensuelles.

Les paramètres de cette distribution sont :

- MT, le cumul annuel moyen de précipitation obtenu en un point de la base de

donnée ERA-40 ;

- β, le rapport entre le cumul annuel moyen de précipitations convectives et le

cumul annuel moyen de précipitation totale déduit de la base de données ERA-40 ;

- Pr6, la probabilité (en pourcentage) en un point d’avoir un cumul de

précipitation supérieur à 0.1 mm par tranches de 6 heures.



16

Les cartes mondiales de ces paramètres sont présentées respectivement aux

figures 1, 2 et 3. Les coefficients a, b et c sont obtenus en minimisant l’écart

quadratique entre la base de données DBSG-3 de l’UIT-R de CCDF d’intensité de

précipitations issue de données pluviométriques et le modèle appliqué aux endroits où

les mesures ont été effectuées.

Figure 1: Carte mondiale du paramètre Mt

Figure 2: Carte mondiale du paramètre β

La prise en compte des cartes de paramètres géo-climatiques permet de modéliser

de manière réaliste la disparité des intensités de précipitations et de ce fait d’identifier

les zones climatiques les plus sujettes aux affaiblissements dus à la pluie, comme illustré

figure 4.



17

Figure 3: Carte mondiale du paramètre Pr6

Figure 4: Carte mondiale d’intensité de précipitations dépassée durant, 0.01 % d’une

année moyenne



18

3.2 Perspectives d’amélioration

Si la recommandation actuelle proposée par l’UIT-R est le modèle de référence

car simple d’utilisation, il présente quelques limitations dont certaines pourraient être

levées en exploitant mieux le potentiel de la base de données ERA-40.

Par exemple, les statistiques de précipitations de l’UIT-R sont données en termes

de « pourcentage du temps d’une année moyenne ». Ce concept d’année moyenne est

assez vague compte tenu des variabilités inter-annuelles annuelles des cumuls de

précipitations qui sont de l’ordre de 30%. Certains cycles observés pour les régimes de

précipitations présentent des durées de l’ordre d’une dizaine d’années [Poiares Batista

et al., 1987], ce qui est à rapprocher de la durée de vie des satellites géostationnaires. Il

n’est dès lors pas évident que les statistiques de précipitations calculées sur la durée de

vie du satellite soient représentatives de cette année moyenne avec les implications que

cela peut avoir sur la disponibilité résultante des liaisons, et cela sans tenir compte

d’éventuelles tendances long terme. Une analyse plus fine des variabilités inter-

annuelles des précipitations rendrait possible l'élaboration d'une marge de confiance par

rapport aux distributions calculées pour une « année moyenne ». Les quarante-trois

années de données de la base de réanalyse ERA-40 devraient permettre de quantifier

plus précisément cette variabilité, et de donner un intervalle de confiance associé à la

distribution moyenne des intensités de précipitations. Il devrait également être possible

de donner une indication sur les variations mensuelles voire diurnes de ces distributions.

4 Distribution des contenus intégrés en eau liquide

Jusqu’à présent, il n’existe pas de modèle similaire à celui proposé par l’UIT-R

pour les distributions d’intensité de précipitations, pour quantifier les distributions de

contenus intégrés en eau liquide. L’UIT-R propose des cartes statistiques de contenus

intégrés en eau liquide dépassés pour différents pourcentages du temps, issues de la base

de données de réanalyse NA-4 de l’ECMWF portant sur 2 années. Cette

recommandation UIT-R présente deux carences importantes :

- La représentativité statistique des cartes calculées sur deux années de données

est insuffisante,



19

-L’absence de modèle analytique des distributions de contenus intégré en eau

liquide constitue une entrave au développement de modèles tentant de représenter des

champs de contenus intégrés en eau liquide corrélés dans l’espace et dans le temps, qui

ont besoin d’une description analytique de ce paramètre.

Pour remédier à ces défauts, la première partie de l’article « Statistical

distribution of integrated liquid water and water vapor content from meteorological

reanalysis » propose un modèle analytique de cette distribution du contenu intégré en

eau liquide reposant sur des cartes de paramètres géoclimatiques dérivés des quinze

années de données de la base de réanalyse ERA-15 de l’ECMWF. Parmi plusieurs

modèles de distributions testés, la distribution mixte Dirac log-normale est la mieux

appropriée. L’utilisation de cette distribution suppose que le contenu en eau liquide est

distribué de manière log-normale avec une probabilité Pcloud et nul avec une

la probabilité 1-Pcloud . L’expression analytique de cette distribution est la suivante :

( )

1)0(

0L if ²2

²lnexp21)( **

*

=≥

>⎥⎦⎤

⎢⎣⎡ −−=≥ ∫

+∞

LP

dlll

PLLPL

cloud σμ

σπ , (4)

où L représente le contenu intégré en eau liquide, μ, σ sont les paramètres de la

loi log-normale qui représente la distribution des contenus en eau liquide strictement

positifs. Pcloud représente la probabilité en un point d’avoir un contenu intégré en eau

liquide strictement positif. Les cartes mondiales des paramètres obtenus en régressant le

modèle (4) sur les cartes statistiques dérivées de la base de données ERA-15 sont

représentées aux figures 5, 6, 7.



20

Figure 5 : Carte mondiale de Pcloud régressée à partir des données ERA-15.

Figure 6 : Carte mondiale de μ régressée à partir des données ERA-15.



21

Figure 7: Carte mondiale de σ regressée à partir des données ERA-15.

Le modèle de contenu intégré en eau liquide (4) combiné avec les cartes des

figures 5, 6 et 7 permet de caractériser en tout point du globe les distributions de

contenus intégrés en eau liquide et d’en déduire les atténuations sous-jacentes en

utilisant la Recommandation UIT-R P. 840-3.

5 Distribution des contenus intégrés en vapeur d’eau

La problématique rencontrée pour les distributions du contenu intégré en vapeur

d’eau est exactement la même que celle pour les distributions du contenu intégré en eau

liquide. En effet, la recommandation UIT-R P.836-3 actuelle ne propose pas de modèle

analytique et les cartes statistiques proposées ont une représentativité limitée du fait de

l’utilisation de la base de données NA-4 de l’ECMWF. La deuxième partie de

l’article “Statistical distribution of integrated liquid water and water vapor content from

meteorological reanalysis” propose un modèle de distribution pour ces contenus intégrés

en vapeur d'eau reposant sur l’utilisation de statistiques dérivées de la base de réanalyse

ERA-15. Cet article montre que la distribution la plus appropriée est une distribution de

Weibull dont la CCDF est donnée par :

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=≥

kVVVPλ

** exp)( (5)



22

où V représente le contenu intégré en vapeur d’eau, et k et λ sont les paramètres

de la loi.

Les cartes mondiales de ces paramètres ont été obtenues en minimisant l’erreur

quadratique du modèle (5) avec les distributions statistiques dérivées d'ERA-15.

Figure 8 : Cartes mondiales du paramètre λ de la loi de Weibull représentant les

contenus intégrés en vapeur d'eau.

Figure 9: Cartes mondiales du paramètre k de la loi de Weibull représentant les

contenus intégrés en vapeur d'eau



23

Combinés avec les paramètres représentés sur les figures 8 et 9, le modèle (5)

permet d'obtenir en tout point du globe une estimation de la distribution des contenus

intégrés en vapeur d'eau.

La connaissance des distributions de chacun des paramètres dont découle

l'atténuation n'est pas suffisante pour décrire la combinaison des effets. A l'heure

actuelle, des méthodes statistiques approchées permettent de combiner les différents

effets afin d'obtenir la distribution de l'atténuation totale [Castanet et al., 2001].

Cependant, une estimation plus précise de cette distribution nécessiterait une meilleure

connaissance des corrélations entre les différents paramètres. Ainsi, une partie de

l'article "Distribution of tropospheric water vapor in clear and cloudy conditions from

microwave radiometric profiling", se propose d'étudier la distribution des contenus

intégrés en vapeur d'eau en présence de nuages et en air clair. Ces distributions sont

obtenues à partir de profils radiométriques pour différents sites situés dans le Sud-ouest

de la France. Elles soulignent la forte dépendance entre distribution des contenus

intégrés en vapeur d'eau et présence ou non de nuages, conformément à la figure 10.



24

Figure 10 : Distribution de contenus intégrés en vapeur d'eau dérivés des données

radiométriques conditionnée à la présence de nuages, et en air clair.

L'étude de la distribution des contenus intégrés en vapeur d'eau à partir des

données du radiomètre micro-ondes de l’ONERA montre également que le modèle (5) et

le paramétrage associé est valable dans le sud-ouest de la France.

6 Conclusion

La modélisation statistique des distributions d’atténuation troposphérique et des

quantités météorologiques sous-jacentes est une thématique qui a été largement abordée

au cours des études de propagation pour les liaisons Terre-satellite opérant aux

fréquences supérieures à 10 GHz. Des modèles performants ont été élaborés pour les



25

précipitations. Ils permettent de caractériser de manière relativement satisfaisante les

distributions d’intensité de précipitation et d’atténuation due à la pluie, même si la

quantification de l'incertitude sur ces distributions serait une amélioration considérable.

En effet, cela permettrait d’en déduire un intervalle de confiance pour les valeurs

d'indisponibilité des liaisons.

Les distributions statistiques des contenus intégrés en vapeur d’eau et en eau

liquide, rendues indispensables pour les études de propagation en bande Ka, mais

surtout en EHF ou dans la bande Q/V, doivent être améliorées. Un modèle, plus

pertinent que la recommandation UIT-R actuelle, a par conséquent été développé et testé

dans le cadre de ce travail de thèse. Cependant, la question de la corrélation des

différents paramètres et de leur combinaison reste ouverte.

L'amélioration des modèles décrivant les distributions des paramètres

météorologiques reste intéressante pour le dimensionnement des liaisons terrestres ou

terre-espace aux fréquences supérieures à 20 GHz. La majorité des modèles de canal

développés pour optimiser la gestion de la ressource prennent en entrée ces statistiques.

Aussi une plus grande précision sur ces distributions améliorera la précision des

modèles de canal.


II. Étude de la variabilité spatiale et temporelle des affaiblissements de propagation

26


1 Introduction

Comme expliqué dans la problématique générale, le dimensionnement et

l’optimisation des systèmes de gestion adaptatifs de la ressource, rendus nécessaires par

l’utilisation de fréquences supérieures à 20 GHz, requièrent une connaissance des

variabilités spatiales et temporelles des affaiblissements de propagation. Considérant la

diversité des techniques qui peuvent être employées pour contrer ces affaiblissements,

les besoins en termes de modélisation du canal de propagation peuvent prendre

différentes formes. Ce chapitre regroupe une description des réponses envisagées pour

répondre aux diverses problématiques posées par le dimensionnement des techniques de

compensation. Ces réponses peuvent englober l'utilisation directe de données, de

modèles statistiques ou de modélisations. Ces différents aspects sont détaillés dans les

sections suivantes. L’utilisation de données radars pour le dimensionnement d’un

système de diversité est dans un premier temps présentée. Une modélisation spatiale des

champs d’atténuation totale et une modélisation spatio-temporelle des affaiblissements

de propagation sont ensuite détaillées.

2 Etude d'un système de diversité de sites à partir de données radars

Certains problèmes liés au dimensionnement des systèmes adaptatifs de

compensation des affaiblissements peuvent être résolus par une exploitation directe des

données météorologiques existantes. Par exemple, pour le dimensionnement des

systèmes utilisant la diversité de sites, les données radars peuvent constituer une

solution relativement simple si une archive de données radars couvre la zone d'intérêt

sur une période relativement longue (au moins une année). La diversité de sites est

utilisée dans le cas où il est nécessaire d'avoir des liaisons à très haute disponibilité

(stations d’ancrage, stations de contrôle, gateways).Une ou plusieurs autres stations sont



27

implantées, généralement à quelques dizaines de kilomètres les unes des autres, comme

illustré figure 11, si le bilan de liaison d'une station ne permet pas d'assurer la

disponibilité requise. Le but est de tirer partie de l'inhomogénéité spatiale des champs de

précipitation.

Figure 11: Principe des systèmes de diversité de sites. Dans le cas où une des stations

subit une atténuation trop forte pour assurer la liaison, l'autre station assure le relais

de l'information.

En effet, plus les stations sont éloignées les unes des autres, plus les atténuations

affectant les différentes liaisons sont décorrélées. En cas de décorrélation complète de

deux liaisons, la probabilité d'indisponibilité (probabilité que les deux liaisons subissent

une atténuation supérieure à un seuil A* du système de diversité résultant) s'exprimerait

en fonction des indisponibilités affectant les deux liaisons par :

P(A1>A*,A2>A*) = P(A1>A*)P(A2>A*) (6)

Ainsi, supposant que les deux liaisons aient une probabilité de 0.1 %, la

probabilité que les deux stations soient simultanément affectées serait de 0.0001 %. En



28

pratique, les champs d’atténuation due aux précipitations présentent une corrélation

spatiale décroissant avec la distance. La problématique de dimensionnement se résume

donc à trouver la distance minimale entre les deux sites permettant d'assurer la

disponibilité requise. L'augmentation de disponibilité induite par l'utilisation de deux

sites (par rapport à l'utilisation d'un seul), est quantifiée par l’avantage de diversité

comme illustré figure 12. Un autre paramètre, le gain de diversité permet aussi d’estimer

les performances de la diversité de site. Il se definit comme le différentiel d’atténuation,

pour une disponibilité donnée, entre l’atténuation affectant un site et l’atténuation

affectant simultanément deux sites:

),()(),( PDAPAPDG js −= (7)

où As est l’atténuation dépassée pendant le pourcentage P du temps et Aj est

l’atténuation jointe dépassée par les deux station en diversité séparées par une distance

D pendant le pourcentage P du temps, comme illustré figure 12. Il est également

possible d’exprimer le gain de diversité G(D,As) en fonction de la distance et de

l’atténuation As dépassée pendant un pourcentage du temp P pour un des sites comme

illustré à la figure 13 du fait de la bijectivité de la relation liant les deux quantitées.

0 5 10 15 20 25 30 35 40

10-1

100

Attenuation [dB]

Per

cent

age

[%]

Spino d'Adda - f = 30 GHz - distance = 8 km

Single-site attenuation CDFJoint two-site attenuation CDF (D = 8 km)

PG

AsAj

Figure 12 : Lien entre gain de diversité, loi jointe d’atténuation et loi d’atténuation sur

un site

Avantage



29

La connaissance du gain de diversité en fonction de la distance permet

d'optimiser la distance nécessaire entre deux sites afin d'assurer la disponibilité requise.

Les données radar converties en atténuation grâce à la recommandation UIT-R

P.618 permettent d'estimer la loi jointe d'atténuation simultanément dépassée pour deux

sites séparés d'une distance D, ainsi que la distribution d'atténuation pour un seul site.

Par conséquent, il est possible de calculer le gain de diversité en fonction de la distance

comme illustré sur la figure 13 où le gain de diversité a été calculé à partir d'une année

de données du radar météorologique du réseau ARAMIS de Météo-France situé à

Bordeaux.

0 10 20 30 40 50 600

5

10

15

20

25

30

Distance [km]

Gai

n [d

B]

Bordeaux

4 dB 8 dB12 dB16 dB20 dB24 dB28 dB32 dB

Figure 13: Gain de diversité fonction de la distance, calculé à partir des données radars

(lignes pleines) et à partir de la recommandation UIT-R P.618 (lignes pointillées) pour

différentes marges d’atténuation sur une liaison.

La figure 13 montre qu'il existe des différences importantes entre le gain de

diversité dérivé des données radars et le modèle statistique de l'UIT-R. Les possibilités

offertes par les données de radars météorologiques sont détaillées dans l'article “Use of

weather radar data for site diversity predictions and impact of rain field advection”,

présent dans la deuxième partie de ce manuscrit.



30

L'utilisation d'une base de données de champs de précipitation observés par des

radars météorologiques constitue certainement la meilleure solution pour optimiser les

mécanismes de gestion adaptatifs de la ressource, même si les problèmes inhérents à la

mesure radar peuvent donner lieu à des pré-traitements relativement complexes.

Cependant, la couverture des réseaux radars est relativement restreinte à l'échelle

mondiale, et de qualité inhomogène (présence d’échos de sols, méthodes de calibration

différentes, résolutions différentes…). Cela implique un recours à la modélisation pour

pallier ce manque de données disponibles. Le réseau américain NEXRAD [Klazura et

Imy, 1993] et, dans une moindre mesure, les réseaux européens homogénéisés dans le

cadre du projet en cours OPERA2 [Huuskonen A. et al., 2007], constituent néanmoins

des entrées de première importance pour la modélisation.

3 Modélisation spatiale des champs de précipitations

3.1 Besoins

Les données radars existantes étant trop peu nombreuses, il est nécessaire de

modéliser les champs d'atténuation afin de servir d'entrée aux simulateurs systèmes. Ces

champs doivent être générés sur la couverture continentale du satellite de

télécommunication afin de simuler de manière aussi réaliste que possible l’allocation de

la ressource embarquée à l’intérieur d’un groupe d’utilisateurs ou d’un spot, mais aussi

entre les différents groupes d’utilisateurs ou entre spots. Par conséquent, il est important

que les champs d’atténuation et de précipitation simulés soient représentatifs des

caractéristiques statistiques déduites des données ayant servi de base à la modélisation.

Ainsi, il est nécessaire que les distributions d'atténuation en un point, calculées à partir

d'un grand nombre de simulations, soient aussi proches que possible de celles évaluées à

partir de jeux de données indépendants. La répartition spatiale des affaiblissements doit

elle aussi être représentative de celle observée sur les données.

D’autre part, considérant l'évolution des systèmes de télécommunication vers

l’utilisation de bandes de fréquences de plus en plus élevées, il n'est plus possible de

s'intéresser uniquement aux atténuations dues aux précipitations car l'importance des



31

atténuations dues aux nuages et à la vapeur d'eau est croissante. Même si comme illustré

figure 1, l’atténuation due aux nuages et à la vapeur d’eau en l’absence de précipitation

est de l’ordre de 3-4 dB à 40GHz, la compensation adaptatives de ces effets, eux aussi

extrêmement variables dans l’espace, peut permettre d’augmenter de manière

significative les performances des systèmes de télécommunications à ces bandes de

fréquence. En effet l’utilisation de modulations efficaces en air clair peut augmenter

sensiblement les capacités de transport globales du système.

Aussi, une méthode de génération de champs d'atténuation totale à l'échelle

continentale est présentée dans cette section.

3.2 Modélisation proposée

La modélisation proposée [Jeannin et al., 2006a, 2006b] pour répondre aux

besoins s'appuie sur la méthodologie HYCELL de génération de champs de

précipitations spatialement corrélés à l’échelle de la France [Féral et al. 2003a, 2003b,

2006]. Le modèle développé permet de générer des champs de paramètres

météorologiques (précipitation, eau liquide, eau vapeur) à l’échelle continentale sur

l’Europe, les États-Unis, ou l’Afrique respectant la climatologie locale et de les

transformer, considérant une liaison radioélectrique donnée, en champs d’atténuation

totale. Cette section reprend les diverses étapes ayant permis de s’acheminer vers

l’élaboration de ce modèle.

3.2.1 HYCELL

Les principales hypothèses du modèle HYCELL développé dans [Féral et al.

2003a, 2003b] sont rappelées dans cette section. Cette méthodologie, établie à partir de

l'étude de données radars en région tempérée, permet de générer des champs de

précipitations spatialement corrélés à moyenne échelle (~150×150 km²), connaissant la

proportion de la zone de moyenne échelle affectée par des précipitations.

Cette méthode repose sur la modélisation des champs de précipitations comme

une conglomération de cellules convectives. Les cellules de précipitation sont

modélisées par un cœur convectif à décroissance Gaussienne et un étalement stratiforme



32

à décroissance exponentielle comme illustré figure 14. Les cellules de précipitation sont

supposées de section horizontale circulaire. En effet, si d’un point de vue

observationnel, la majorité des cellules de pluie présente plutôt une forme elliptique,

l’équiprobabilité de leur distribution conduit finalement à considérer des cellules de

forme circulaire.

Figure 14 : Modélisation d'une cellule de précipitation par HYCELL.

Comme illustré sur la figure 14, la décroissance de la partie Gaussienne de la

cellule est caractérisée par le paramètre aG. La décroissance de la partie exponentielle

est quant à elle caractérisée par le paramètre aE. L’intensité de précipitations maximale

de la cellule est atteinte pour l’intensité pic de la partie gaussienne RG. L’intensité de

précipitation notée R1 sur la figure 14 est l’intensité de précipitation limite entre zones

de définition Gaussienne et exponentielle. Les intensités de précipitation inférieures à

Rm=1 mm h-1 (valeur conforme à la littérature permettant d’isoler les cellules de pluie de

leur environnement stratiforme) ne sont pas représentées par HYCELL. Le paramètre RE

correspond à l’intensité de précipitation maximale de la cellule si la cellule était

uniquement modélisée par une exponentielle.

Une étude des cellules de précipitation observées par radar en zone tempérée a

permis d’établir un lien statistique entre leur surface à Rm=1 mm h-1 et leur intensité pic.



33

Par ailleurs, une dépendance statistique entre les paramètres RG et R1 de la modélisation

HYCELL a également été mise en évidence. Compte tenu des relations existant entre les

différents paramètres (équation de continuité en R=R1), un paramétrage générique des

cellules a ainsi pu être défini à partir de la connaissance de la surface de la cellule à Rm

et de l’intensité de précipitation maximale RG.

En utilisant cette modélisation générique de la cellule de précipitation, une

méthode de génération de champs de précipitation sur une zone de simulation de

moyenne échelle (150×150 km²) a été développée [Féral et al., 2003b]. Connaissant la

distribution cumulative (CDF) locale de l’intensité de précipitation ainsi que la fraction

de la zone de moyenne échelle affectée par les précipitations, les cellules de pluie sont

générées sur cette zone de simulation de manière itérative, par intensité de précipitation

décroissante. La surface occupée à Rm=1 mm.h-1 par chacune des cellules est tirée

suivant une distribution exponentielle conformément aux études déjà menées [Denis et

Fernald, 1963], [Goldhirsh et Musiani, 1986], [Mesnard et Sauvageot, 2003]. La surface

totale occupée à la fin du processus par l’ensemble des cellules générées correspond à la

surface de la zone de moyenne échelle affectée par des précipitations supérieures à 1

mm.h-1. La modélisation HYCELL de chaque cellule est conduite de manière à ce que la

distribution spatiale des intensités de précipitation reproduise la distribution d’intensité

de précipitation temporelle locale conformément à l’hypothèse d’ergodicité des champs

de pluie formulée dans [Nzeukou et Sauvageot, 2002].

Afin de localiser les cellules ainsi générées à l’intérieur de la zone de moyenne

échelle, les cellules sont positionnées par une marche aléatoire comme illustré figure 15.

A partir d’observations radar, il a été proposé de modéliser la localisation spatiale des

cellules de pluie à moyenne échelle par agrégats. Leurs caractéristiques statistiques en

termes de nombre de cellules par agrégat, de distances intercellulaires ou de distance

inter-agrégats ont été mises en évidence à partir des données du radar de Bordeaux

[Féral et al., 2003b]. Dés lors, la marche aléatoire permet de localiser les agrégats et les

différentes cellules qui les composent dans une zone de simulation de moyenne échelle.

Considérant un grand nombre de champs de précipitation générés avec la marche

aléatoire, la distribution des distances intercellulaires obtenue reproduit celle dérivée des

observations radar. Cette procédure de localisation des cellules introduit donc une

corrélation spatiale sur les champs de précipitations simulés.



34

Figure 15 : A gauche, champ de précipitation modélisé par Hycell avec une localisation

complètement aléatoire des cellules. A droite, champ de précipitation dont les cellules

ont été localisées par la marche aléatoire.

3.2.2 Extension de la zone de simulation à l’échelle de la France

Afin d’étendre cette modélisation à l’échelle de la France métropolitaine, [Féral

et al., 2006] proposent de coupler cette modélisation cellulaire du champ de

précipitation à moyenne échelle avec une modélisation spatiale des zones précipitantes

(R>0 mm.h-1) à l’échelle de la France. Considérant les observations du réseau de radar

ARAMIS de Météo France, il a été montré que les zones affectées par les précipitations

(R>0 mm.h-1) à l’échelle de la France pouvaient être correctement représentées par un

champ aléatoire stationnaire Gaussien G seuillé, de corrélation :

)))sin()cos(())sin()cos((exp(),( 22

XYG L

xyL

yxyxc θθθθ −+

+−= , (8)

avec LX =200 km, LY=100 km et θ est l'orientation de la structure frontale associée.

Cette fonction de corrélation a été déterminée en définissant une relation entre la

corrélation des champs binaires de précipitation (pluie/non-pluie) calculée à partir des

données radar et la corrélation du champ Gaussien sous-jacent. A partir des observations

ARAMIS sur la France, la distribution de probabilité de la fraction f du territoire

national affectée par la pluie a été dérivée. Celle-ci a été modélisée par une loi

exponentielle de moyenne 8.8%. Afin de localiser les zones affectées par des



35

précipitations sur les simulations, le champ Gaussien G est transformé par seuillage en

un champ binaire de précipitation B de manière à ce que la fraction de la zone affectée

par des précipitations soit f conformément à la figure 16.

Figure 16 : Champ Gaussien généré sur la France et Champ binaire de précipitation

seuillé afin d’obtenir une occupation des précipitations de f=15%.

Le couplage avec la modélisatin HYCELL est immédiat dès lors que le champ de

précipitation binaire à large échelle est découpé en sous-zones de moyenne échelle

conformément à la figure 17. Finalement, cette approche permet de générer des champs

de précipitation corrélés dans l’espace à l’échelle de la France ( 224HFigure 17).

Figure 17 : Division de la couverture en sous zones de moyenne échelle et couplage

avec la modélisation cellulaire pour chaque zone de moyenne échelle

θ

x

O

x’

y

y’



36

Toutefois, la modélisation des zones affectées par les précipitations n’est pas

entièrement satisfaisante car elle ne permet pas de prendre en compte la variabilité de la

probabilité d’apparition de la pluie à l’échelle de la France laquelle présente pourtant

des disparités importantes. De ce fait, il n’est pas possible d’étendre cette modélisation à

l’échelle d’une couverture satellitaire (i.e. échelle continentale).

3.2.3 Extension de la zone de simulation à l’échelle d’une couverture satellitaire

L’inhomogénéité spatiale des champs de précipitation (illustrée figure 18) ne

permet pas d’appliquer la modélisation proposée ci-dessus à l’ensemble d’une

couverture satellitaire. Une modification de cette modélisation a par conséquent été

proposée [Jeannin et al., 2006a] afin de tenir compte de cette variabilité.

Figure 18: Probabilité d’apparition de la pluie sur l’Europe dérivée de la Recommandation

UIT-R P.837

La conversion du champ aléatoire Gaussien G en un champ binaire de précipitation B est

maintenant réalisée en utilisant un seuil dépendant en tout point de la probabilité locale

d’apparition de la pluie. Ainsi, si de nombreux champs sont générés, la probabilité

d’obtenir des précipitations en un point des simulations tendra vers la probabilité

d’apparition de la pluie de la Rec UIT-R P.837. Si ce seuil tenant compte de la

probabilité d’apparition de la pluie permet de s’affranchir du problème posé par



37

l’inhomogénéité des probabilités d’apparition de la pluie, la fraction de la zone affectée

par les précipitations n’est elle plus contrôlée. En utilisant la fonction de corrélation

donnée par (8), il apparaît que la fraction de la zone de couverture affectée par la pluie

sur l’ensemble de la couverture est très peu variable, proche de la moyenne des

probabilités d’apparition de la pluie sur la zone, alors qu’une étude des champs de

précipitations à partir de données TRMM (Tropical Rainfall Measurement Mission)

[Kummerow et al., 1998] montre qu’il existe une variabilité importante de ce paramètre

[Jeannin et al., 2006b]. Afin d’une part de reproduire la probabilité d’apparition de la

pluie en tout point et d’autre part d’approcher la distribution de la fraction de la

couverture occupée par les précipitations, une modification de la fonction de corrélation

(8) a été proposée :

2

2222

1

)800

exp())130

)sin()cos(()200

)sin()cos((exp(),(

Z

Z

GL

yxLxyyx

yxc+

+−+

−+

+−

=

θθθθ

, (9)

.

avec LZ valant 0.25 sur l’Europe et 0.20 sur l’Amérique du Nord. L’introduction d’une

composante de corrélation à décroissance lente permet d’augmenter la variabilité de la

distribution de la fraction de la zone de couverture affectée par les précipitations comme

expliqué plus en détail dans l’article « Statistical Distribution of the fractional area

affected by rain » inclus dans la deuxième partie de ce manuscrit. Le paramètre LZ a été

ajusté de manière à reproduire aussi fidèlement que possible la distribution de la fraction

de la zone affectée par la pluie déduite des données TRMM. Ainsi l’utilisation de la

forme de corrélation (9) permet à la fois de reproduire en tout point la probabilité

d’apparition de la pluie dérivée de la recommandation ITU-R P.837 ainsi que la

distribution de la fraction de la zone de couverture affectée par les précipitations déduite

des observations TRMM.

Une fois le champ binaire de précipitation généré, la méthodologie Hycell est appliquée

sur chaque sous zone de moyenne échelle de (150×150 km2) conformément à la 225HFigure

17, ce qui permet de générer des champs de précipitation sur l’ensemble de la couverture

satellite comme illustré figure 19.



38

Figure 19: Champ de précipitation généré sur l’Europe avec la méthodologie HYCELL.

Cette modélisation a été utilisée pour générer des champs de précipitation

spatialement corrélés sur l’Europe, l’Amérique du Nord et l’Afrique. Afin de tenir

compte d’une possible dépendance climatique de la modélisation cellulaire, les relations

statistiques utilisées pour définir la forme générique des cellules ainsi que pour le

paramétrage de la marche aléatoire ont été étudiées sur différents jeux de données radar.

Des études ont ainsi été menées sur des couvertures radar en Europe (France,

Scandinavie, Europe centrale), aux États-Unis, (Louisiane, Floride, Californie, New

Jersey, Dakota), et en zone tropicale (Martinique, Sénégal). Une modélisation un peu

différente pour prendre en compte la saisonnalité extrêmement marqué des régimes de

précipitation sur l’Afrique a été proposée. Ces différents points sont présentés à la

section 226H 3.3.

3.2.4 Génération des champs de contenus intégrés en vapeur d’eau et en eau liquide

Afin de prendre en compte également les affaiblissements dus à la vapeur d'eau et

aux nuages, importants aux bandes de fréquences supérieures à 20 GHz, des champs de

contenus intégrés en eau liquide et en vapeur d’eau corrélés aux champs de précipitation

doivent être générés. La solution retenue, développée dans le cadre de ce travail de



39

thèse, repose sur une modélisation des champs météorologiques d’intérêt par des champs

aléatoires stationnaires. La méthodologie de génération des champs est similaire à celle

utilisée pour générer les champs binaires de précipitation. Un champ gaussien

stationnaire est généré dans l'espace et, en tout point, la variable gaussienne tirée dans

une loi normale centrée réduite est convertie, en utilisant une équivalence des fonctions

de répartition, en une variable suivant la distribution des contenus intégrés en eau

liquide des nuages ou en vapeur d'eau, présentées au chapitre précédent. Dès lors, reste à

déterminer la corrélation spatiale de ces champs. Des données de contenus intégrés en

vapeur d'eau issues des réanalyses ERA-40 ont été utilisées pour déterminer la fonction

de corrélation du champ aléatoire gaussien générateur des champs de contenus intégrés

en vapeur d'eau. Quant à la fonction de corrélation des contenus en eau liquide, elle a été

obtenue à partir de données infrarouges de Météosat seuillées à une température de

brillance de -38°C. Une forme analytique approchée de ces fonctions de corrélation a été

déterminée. Pour les contenus en vapeur d'eau la fonction de corrélation du champ

Gaussien utilisée est:

))385

)sin()cos(()565

)sin()cos((exp(),( 22 θθθθ xyyxyxcG−

++

−= , (10)

où l'angle θ représente l'orientation de la structure frontale associée. Pour les champs de

contenu intégré en eau liquide des nuages, la fonction de corrélation déterminée a été

trouvée identique à celle déterminée pour la génération des champs gaussiens

générateurs des champs binaires de précipitation (9).

Pour coupler les champs de contenus intégrés en vapeur d'eau et les champs de

contenus en eau liquide aux champs de précipitations, il est supposé qu'il existe une

corrélation maximale entre contenus en eau liquide des nuages, contenus en vapeur d'eau

et intensité de précipitation. Il n’existe pas de données pour étayer cette hypothèse.

Cependant les mesures radiométriques simultanées, de contenu en eau nuageuse, de

contenu en vapeur d’eau et de présence de précipitation présentées sur la figure 1 de

l’article «Distribution of tropospheric water vapor in clear and cloudy conditions from

microwave radiometric profiling » montre que cette corrélation est importante, puisque

l’apparition de précipitation est associée à une augmentation de contenu en eau liquide

et de contenu en vapeur d’eau. Faute d’autres données, cette modélisation simpliste a été

adoptée.



40

Figure 20 : Génération des champs aléatoires générateurs des divers champs

météorologiques d’intérêt.

Ce couplage est obtenu en utilisant la même graine aléatoire pour la génération

des différents champs gaussiens comme illustré à la figure 20. Les champs de contenu

intégré en eau liquide englobent par construction les zones de précipitations (figure 21)

car la probabilité d’avoir un contenu en eau liquide positif est supérieure à celle d’avoir

des précipitations.

Graine aléatoire

Corrélation (10)

Corrélation (9)

Champ gaussien générateur d’un champ de contenu intégré en

vapeur d’eau

Champ gaussien générateur d’un champ binaire de précipitation et d’un

champ de contenu intégré en eau liquide



41

Figure 21: Champ de précipitation (rouge) et couvert nuageux avec eau liquide(bleu)

Le couplage des différents champs permet d'obtenir des champs d'atténuation

totale comme illustré à la figure 22.



42

c

Figure 22 : Exemple de champs météorologiques générés sur une zone de moyenne

échelle et champs d’atténuation résultant. De gauche à droite et de bas en haut: champs

de précipitation, de contenu intégré en eau liquide, de contenu intégré en vapeur d'eau,

d’atténuation due aux précipitations, d'atténuation due aux nuages et à la vapeur d’eau,

champ d’atténuation due à l’oxygène et enfin, champ d’atténuation totale.

La méthodologie présentée permet de générer des champs d’atténuation totale

statistiquement réalistes à l’échelle de la couverture continentale d’un satellite de

télécommunication. Le réalisme de ces simulations est évalué dans la section suivante

permettant ainsi d’identifier les limites d’utilisation de ce modèle.

Champs d’atténuation



43

3.3 Évaluation de la pertinence des résultats obtenus

Pour évaluer le réalisme de la méthodologie de simulation employée, les

caractéristiques statistiques des divers champs générés ont été comparées à celles de

données réelles issues de radars météorologiques ou de données de télédétection. Pour

cela une base de données de 100 000 simulations a été générée sur diverses zones

climatiques [Jeannin et al., 2006b].

Les hypothèses relatives au modèle cellulaire ont été évaluées sur des données

radar européennes, américaines, africaines et de Martinique. Elles se révèlent

appropriées quelle que soit la zone climatique considérée. Ceci est illustré à la figure 23

pour la distribution des distances intercellulaires et à la figure 24 pour la distribution des

diamètres des cellules de précipitation.

Figure 23 : Distribution des distances intercellulaires pour la Martinique, Dakar et Bordeaux



44

Figure 24 : Distribution du diamètre des cellules observées sur différentes couvertures

radar des Etats-Unis

La distribution d'atténuation totale en chaque point approche précisément la

distribution d'atténuation totale dérivée de la recommandation ITU-R P.618-9. Un

exemple est donné à la figure 26.

Figure 25: CDF d’atténuation totale 30 GHz stimulant une liaison avec OLYMPUS.



45

La description de la variabilité spatiale des champs d’atténuation par le modèle

semble cohérente comme l’atteste la figure 26.

Figure 26: Gain de diversité en fonction de la distance calculé sur New York en

simulant une liaison avec ACTS à 27.5 GHz à partir de données radars, de simulations,

et de différents modèles pour une 0.03% du temps

Des validations de la fonction de corrélation utilisée pour la génération des

champs binaires de précipitation ont aussi été entreprises. Les distributions

d'occupations spatiales des précipitations ont également été comparées à celles dérivées

des données radars. Toutes ces validations sont expliquées plus en détail dans [Jeannin

et al., 2006b].

Afin de s'adapter aux caractéristiques des pluies tropicales, des probabilités

mensuelles de précipitations ont été calculées sur l'Afrique à partir des données TRMM

(voir figure 27). La prise en compte de ces probabilités mensuelles de précipitation

permet de modéliser le phénomène de mousson, caractéristique des régimes de

précipitation des zones équatoriales et tropicales.



46



47

Figure 27: Cartes mensuelles de probabilité de précipitation déduites des données TRMM et de la Recommandation ITU-R P.837



48

3.4 Limitations

Diverses limitations sont apparues lors de l’utilisation et de la validation de ce

modèle.

-La première est inhérente à la modélisation cellulaire employée: les intensités de

précipitation générées sont uniquement celles supérieures à 1 mm.h-1, faussant donc le

réalisme des simulations pour les faibles valeurs d’atténuation.

-L'hypothèse de stationnarité pour des champs aléatoires sur de larges étendues

est peu réaliste

-La complexité du modèle cellulaire induit des temps de calcul très importants.

-Le nombre de paramètres du modèle cellulaire HYCELL rend difficile une

inclusion de la dimension temporelle dans la modélisation.

Du fait de ces diverses limitations, l'approche suivie pour la modélisation spatio-

temporelle des affaiblissements de propagation dus aux précipitations a reposé sur

d'autres fondements théoriques.

4 Modélisation spatio-temporelle des affaiblissements de propagation

4.1 Problématique

Le dimensionnement des dispositifs de gestion adaptative de la ressource

embarquée utilisant des codages et modulations adaptatifs, nécessite d'avoir une

représentation finedu canal de propagation d’une part dans le temps, afin de pouvoir

commuter d'un mode à l'autre et d’autre part dans l'espace, afin de répartir les porteuses

ou les paquets de données aux utilisateurs, au mieux des capacités du satellite et du

réseau. Peu de modèles existent dans la littérature pour générer des conditions de

propagation corrélées dans l'espace et dans le temps [Gremont et Filip, 2004],

[Bertorelli et Paraboni, 2005]. De plus ces modèles ne permettent pas de simuler des

conditions de propagation réalistes sur de larges étendues et des périodes suffisemment

longues pour obtenir des statistiques stables. Le modèle de génération de champs



49

d'atténuation due à la pluie corrélés dans l'espace et dans le temps présenté en détail

dans "A large scale space-time stochastic model of rain attenuation for the design and

optimization of adaptive satellite communication systems operating between 10 and

50GHz " a pour objet de simuler des conditions de propagation corrélées dans le temps

et dans l'espace à l'échelle d’une couverture satellitaire continentale. La modélisation

s'appuie sur une modélisation existante [Bell, 1987] des champs de précipitation par des

champs aléatoires corrélés dans l'espace et dans le temps. L'évolution majeure apportée

à cette modélisation est l'extension du domaine de validité dans l'espace (à l’échelle

continentale putôt que sur des zones de 100x100 km2) et dans le temps (jusqu’à

plusieurs années plutôt que sur des durées de quelques heures) en contraignant le modèle

à reproduire les sorties du modèle de réanalyse météorologique ERA-40. Une

méthodologie de paramétrage de la corrélation du modèle à partir de jeux de données

radar est également présentée.

4.2 Méthodologie

4.2.1 Modélisation des champs de précipitations par des champs aléatoires

La modélisation proposée dans [Bell, 1987], repose sur la modélisation de

champs de précipitation corrélés dans l'espace et dans le temps R(x,t) par une

transformation non linéaire d'un champ gaussien stationnaire G(x,t) lui-même corrélé

dans l'espace et dans le temps tel que :

)),((),( txGtxR ψ= . (11)

La transformation ψ est paramétrée de telle manière que la distribution statistique

des intensités de précipitations strictement positives générées soit log-normale.

L'évolution temporelle du champ est régie par une évolution des fréquences du

spectre spatial des précipitations suivant un processus de Markov. Le champ de

précipitation est également soumis à une advection prise en compte par une rotation de

phase dans le domaine de Fourier du champ aléatoire Gaussien. Le paramétrage du

modèle a été effectué à partir des données du radar de Bordeaux.

Cette modélisation répond aux besoins exprimés de génération de champs de

précipitations corrélés dans l’espace et dans le temps. Cependant, l'hypothèse de

stationnarité des champs limite le réalisme de la modélisation à des zones de quelques



50

centaines de kilomètres de côté. L'évolution temporelle des champs de précipitation

manque elle aussi de réalisme au-delà de quelques heures. En effet, la fraction de la

zone affectée par la pluie reste aproximativement constante au cours des simulations ce

qui empèche de modéliser l'alternance d'évènements pluvieux et de périodes de ciel

clair. De plus la direction et la vitesse d'advection restent constantes contrairement aux

observations. Ces limitations sur l’extension spatiale et temporelle du domaine de

simulation, restreignent en l'état l'intérêt de ce modèle pour les études de propagation.

4.2.2 Couplage de la modélisation avec des sorties de modèles de réanalyse

Afin de lever les limitations sur le domaine de validité des simulations, dans

l'espace et dans le temps, une méthode pour contraindre l'évolution du modèle par des

sorties de données de réanalyses météorologiques a été développée. Elle s'articule autour

de deux axes:

-Une relation statistique connue sous le nom d'aires fractionnaires [Donneaud et

al., 1984, Chiu 1988, Eltahir et Bras 1993, Sauvageot 1994] est utilisée pour déduire la

fraction d'une cellule de résolution du modèle de réanalyse affectée par la pluie

-Une relation théorique est établie en utilisant la modélisation de [Bell, 1987]

entre la fraction d'une zone de simulation affectée par la pluie et la moyenne du champ

générateur Gaussien. La justification de cette relation est dévelopée dans l'article

"Statistical distribution of the fractional area affected by rain".

Ainsi, considérant une série temporelle de cumul de précipitation, il est possible

de déduire une série temporelle de la valeur moyenne du champ aléatoire comme illustré

à la figure 28. En utilisant la méthodologie de [Bell, 1987] et en contraignant la

moyenne du champ aléatoire, il est possible de générer des champs d'atténuation corrélés

dans l'espace à l’échelle continentale et dans le temps sur de très longues durées

(plusieurs années).



51

Figure 28: Déduction de la série temporelle de la moyenne du champ Gaussien à partir

des cumuls de précipitation de la base de donnée ERA-40

Les simulations ont alors la taille de la cellule de résolution des données ERA-40

utilisées à savoir 2.5°x2.5°. Pour obtenir une simulation sur de larges étendues la

méthodologie est appliquée à plusieurs cellules de résolution de la base de données

ERA-40. Une méthode pour assurer la continuité des champs générés sans en modifier

les caractéristiques statistiques a été dévelopée et permet de générer des champs sur

l'ensemble d'une couverture satellite comme montré à la figure 29. L'advection des

champs de précipitation est également paramétrée à partir des données ERA-40.

Effectivement, il est montré dans l'article "Use of weather radar data for site diversity

predictions and impact of rain field advection" que le vent horizontal extrait au niveau

700 hPa de la base de donnée ERA-40 est fortement corrélé en terme de vitesse et de

direction à l'advection des précipitations.



52

Figure 29: Exemple de champ de précipitation généré sur la couverture Européenne et

sorties concurrentes du modèle ERA-40.

4.2.3 Génération de séries temporelles d'atténuation corrélées dans l'espace et dans le temps

La procédure de conversion des champs de précipitations en champ d'atténuation

est détaillée dans [Féral et al., 2006]. Cette conversion en atténuation se fait par

intégration de l'atténuation spécifique le long du trajet de la liaison sous la pluie. Celui-

ci dépend de la hauteur de pluie qui est assimilée à la hauteur de l'isotherme -2oC

(Conformément à la Recommandation UIT-R P.839) qui peut être déduite des profils de

température de la base de donnée ERA-40 concurrents au champ de précipitation généré.

A ce stade les champs d'atténuation dus aux précipitations sont générés avec une

résolution spatiale de 1 km et une résolution temporelle de 0.1 h, comparable à la



53

résolution des jeux de données radar utilisés pour développer le modèle. Toutefois, cette

résolution temporelle est insuffisante pour certaines applications puisque les fluctuations

rapides du canal de propagation peuvent avoir une influence significative sur le

dimensionnement des dispositifs de codage ou de modulation adaptatifs ou sur les règles

de commutation d’un système utilisant la diversité de sites. Pour cette raison le modèle

d'interpolation stochastique "on demand" [Lacoste et al., 2006, Satnex 2008] est utilisé

pour améliorer la résolution temporelle des simulations. Comme illustré à la figure 30,

ce modèle est appliqué aux séries temporelles d'atténuation à basse résolution (0.1 h).

Sur la série temporelle basse résolution, les évènements de précipitation (échantillons

consécutifs d'atténuation strictement positive) sont isolés. Pour chacun de ces

événements, la méthodologie d'interpolation stochastique du modèle on-demand est

appliquée comme illustré à la figure 30. Cette méthode d’interpolation stochastique

repose sur une hypothèse de Markovianité du canal de propagation.

Figure 30 : Interpolation haute résolution des séries temporelles d’atténuation faible

résolution (points rouges)

En localisant plusieurs points dans la zone de simulation, correspondant chacun à

la position d'un terminal, et en appliquant la procédure décrite ci dessous, des séries

temporelles corrélées dans l'espace et dans le temps sont générées, comme illustré à la

figure 31.



54

Figure 31: Exemple de simulation de champs sur plusieurs heures et couplage des

séries temporelles obtenues sur différents sites à faible résolution avec le modèle

stochastique de canal



55

4.3 Comparaison des caractéristiques des simulations à des données indépendantes

Les caractéristiques statistiques des simulations ont été confrontées à des

statistiques déduites soit de campagnes de mesures d’affaiblissement, soit de données

radars.

Par exemple, considérant la campagne de mesure de diversité sur deux sites

effectuée par le CNET (Ex France Télécom R&D) près de Paris [OPEX, 1994], des

séries temporelles de 9 mois d’affaiblissements dus à la pluie ont été générées avec les

mêmes caractéristiques (durée, localisation, caractéristiques de la liaison). Les CCDF

d’atténuation obtenues à partir des simulations et des données radar sont présentées à la

figure 32.

Figure 32: CCDF d’atténuation obtenues lors des mesures de la campagne OLYMPUS

et à partir des simulations (Fréquence 20GHz)

La figure 32 montre que les simulations reproduisent de manière convenable les

distributions d’atténuation expérimentales. Afin d’évaluer la capacité de la modélisation

à reproduire la variabilité spatiale des affaiblissements, la loi jointe d’atténuation

simultanément dépassée sur les deux sites déduite des séries temporelles simulées est



56

comparée à celle dérivée de la campagne de mesure. La figure 33 présente les résultats

obtenus et montre l’adéquation entre simulation et expérimentation.

Figure 33: CCDF d’atténuation simultanément dépassée obtenues à partir des données

de la campagne OLYMPUS et à partir des simulations

Des comparaisons simulations / expérimentations sont présentées de manière plus

exhaustive dans la dernière section de l'article "A large scale space-time stochastic

model of rain attenuation for the design and optimization of adaptive satellite

communication systems operating between 10 and 50 GHz".

5 Conclusion

Le modèle de génération de champs d'atténuation corrélés dans l'espace et dans le

temps fournit les entrées requises pour les simulations des dispositifs de gestion

adaptative de la ressource. En effet, il permet de générer des séries temporelles

d'atténuation sur de longues durées (plusieurs années), en reflétant les fluctuations

rapides du canal (résolution temporelle de 1 s), et sur l'étendue d'une couverture satellite

(échelle continentale). La modélisation proposée repose sur une modélisation existante



57

des champs de précipitation à moyenne échelle contrainte par des sorties de modèles de

réanalyses météorologiques. Cette utilisation des données de réanalyse peut permettre

une étude des champs d’atténuation en fonction des conditions météorologiques. Il est

ainsi envisageable de simuler les cas les plus défavorables, d'étudier les variations

diurnes ou saisonnières des précipitations.

Diverses caractéristiques statistiques de simulation ont été comparées à celles de

jeux de données indépendants, accréditant ainsi le réalisme des simulations effectuées.

Différents points restent à approfondir pour fournir une modélisation complètement

satisfaisante. En particulier la sensibilité du paramétrage des corrélations vis-à-vis de la

climatologie locale n'est à l'heure actuelle pas évaluée de manière satisfaisante. Il de

plus est vraisemblable que la méthodologie proposée ne permettra pas de modéliser

convenablement les systèmes précipitants aux faibles latitudes. En effet les cyclones

tropicaux ou les lignes de grains caractéristiques des régimes de précipitation à ces

latitudes présentent une organisation spatiale qui ne peut pas être représentée de manière

réaliste par un processus aléatoire stationnaire. L’impact de ce manque de réalisme des

simulations aux basses latitudes doit être évalué.

D'autre part, des efforts restent à fournir afin d'inclure la scintillation, les

couverts nuageux et la vapeur d'eau dans la modélisation afin d'avoir une représentation

de l'affaiblissement total. Le point délicat reste la corrélation entre les différents effets

qui pourra vraisemblablement être étudiée à partir de l'observation simultanée de radar

et de radiomètres spatio-portés comme ceux embarqués sur le satellite MODIS.


Conclusion et perspectives

58


Cette thèse a été centrée sur la modélisation spatio-temporelle des

affaiblissements de propagation à partir de diverses données météorologiques. Cette

modélisation a requis dans un premier temps l'extraction des différentes caractéristiques

statistiques des affaiblissements de propagation. Après une étude de la variabilité

spatiale des affaiblissements, un couplage entre la dimension spatiale et la dimension

temporelle a été entrepris. Comme le modèle de représentation des variabilités spatiales

des affaiblissements développés dans un premier temps a été jugé inadapté pour

l'adjonction de la dimension temporelle, une modélisation couplant une représentation

stochastique des champs d'affaiblissement avec des données de réanalyse

météorologique a par la suite été développée. Les sorties de ce modèle peuvent être

directement utilisées pour des simulations de dispositifs de gestion adaptative de la

ressource même si l'atténuation due à l'eau liquide et à la vapeur d'eau font encore

défaut. Le modèle développé devrait ainsi permettre de répondre à certains problèmes

posés par le dimensionnement des systèmes de gestion adaptative de la ressource

embarquée.

L'utilisation de données de réanalyse météorologique ouvre des perspectives

intéressantes pour la simulation des conditions de propagation sur des situations ciblées

qui peuvent être intéressantes d'un point de vue de la gestion de la ressource ainsi que

pour la modélisation des systèmes précipitants.

Dans la continuité des travaux présentés, différents axes d'étude pourraient

enrichir les modélisations déjà développées et constituer des compléments intéressants.

Toujours orienté vers la même problématique des télécommunications par

satellite, l'exploitation de nouvelles données issues de campagnes de mesures de

propagation rendra possible un affinement des modèles.

- L'expérience GSAT-4 [Katti et al., 2007] (charge utile de propagation en bande

Ka prévue pour 2009), mettant en jeu cinq stations de réception centrées sur l'Inde et des

moyens météorologiques abondants (radars, pluviomètres, radiomètres), devrait

permettre de mieux caractériser le canal de propagation dans les régions tropicales



59

- Le satellite Alphasat [Paraboni et al., 2007] (charge utile de propagation en

bande Ka et Q/V prévue pour 2011), devant impliquer plusieurs dizaines de terminaux

sur l'Europe, a pour but d’effectuer directement des mesures synchronisées des

affaiblissements de propagation et devrait donc permettre d'améliorer la connaissance de

la dynamique spatio-temporelle des affaiblissements de propagation, tout en

expérimentant des méthodes de compensation adaptatives.

Néanmoins, avant de chercher des améliorations aux modélisations de canal

proposées, il serait utile de quantifier l'impact des imprécisions dans la modélisation du

canal de propagation, sur les performances sur des systèmes de télécommunication, afin

d’apporter une réponse pertinente aux problèmes posés par le dimensionnement des

mécanismes de gestion adaptative de la ressource.

Si les modélisations présentées ont été développées pour des applications dans le

domaine des télécommunications par satellite, elles peuvent aussi répondre à des besoins

exprimés dans d'autres champs d'applications. En effet, le modèle de génération de

champs d'atténuation corrélés dans l'espace peut être utilisé pour calculer les

déformations des échos des radars altimétriques spatio-portés en bande Ka, ou simuler

l'atténuation subie sur une liaison micro-ondes entre un drone et une station sol.

Ce type de modélisation pourrait également trouver des applications dans les

domaines de l'hydrologie ou de la modélisation climatique, pour lesquels la modélisation

des systèmes précipitants est encore très approximative.


Bibliographie

60

Bibliographie

Audoire B. , "Modélisation fine du canal de propagation Terre-satellite dans les

bandes Ka et V", Ph’D of University Paul Sabatier, Toulouse, France, 20th

December 2001.

Bell, T.L. “A space-time stochastic-dynamic model of rainfall for satellite remote

sensing studies.” J. Geophys. Res., 92(D8), 9631–9643, 1987.

Bertorelli S. and Paraboni A. "Simulation of Joint Statistics of Rain Attenuation

in Multiple Sites Across Wide Areas Using ITALSAT Data.", IEEE Antenna and Prop.,

53(8):2611-2622, 2005.

Bolea-Alamañac A. "Fade Mitigation techniques to optimise the radio resource

in a fixed satellite communications link", Ph’D of SUPAERO, Toulouse, France, 15th

December 2004.

Bousquet, M., L. Castanet , L. Féral, P. Pech, and J. Lemorton, “Application of a

Model of Spatial Correlated Time-Series, into a Simulation Platform of Adaptive

Resource Management for Ka-band OBP Satellite Systems”, 3rd COST 280 workshop,

May 2003, Noordwijk, The Netherlands

Buti, M., A. Paraboni and, P. Gabellini, “Analysis of Performances of a Ka Band

Reconfigurable Satellite Antenna Front-End in a Broadcasting Application over the

European Territory”, IWSSC 2008,,Toulouse, France, 2008.

Callaghan S.A., “Fractal Modelling of Rain Fields: From Event-On-Demand to

Annual Statistics”, Proceedings of The European Conference on Antennas and

Propagation: EuCAP 2006 (ESA SP-626). 6-10 November 2006, Nice, France.


Bibliographie

61

Capsoni C., F. Fedi, and A. Paraboni, "A comprehensive meteorologically

oriented methodology for the prediction of wave propagation parameters in

telecommunication applications beyond 10 GHz." Radio Sci., 22(3), 387-393, 1987a.

Capsoni C., F. Fedi, C. Magistroni, A. Paraboni, and A. Pawlina, Data and theory

for a new model of the horizontal structure of rain cells for propagation applications.

Radio Sci., 22(3), 395-404, 1987b.

Castanet L., M. Bousquet, and D. Mertens, “Simulation of the performance of a

Ka-band VSAT videoconferencing system with uplink power control and data rate

reduction to mitigate atmospheric propagation effects.” International . Journal of

Satellite. Communication, 20(4), 231-249,2001a.

Castanet L. : "Fade Mitigation Techniques for new SatCom systems operating at

Ka and V bands", Ph’D of SUPAERO, Toulouse, France, 18th December 2001b.

Castanet L., Lemorton J., Konefal T., Shukla A., Watson P., and Wrench C.,

"Comparison of combined propagation models for predicting loss in low-availability

systems that operate in the 20 GHz to 50 GHz frequency range", Int. J. Sat. Com., 19,

317-334, 2001c.

Castanet L., C. Capsoni , G. Blarzino, D. Ferraro, and A. Martellucci,

“Development of a new global rainfall rate model based on ERA40, TRMM and GPCC

products”, International Symposium on Antennas and Propagation, ISAP 2007, Niigata,

Japan, 20-24 August 2007.

Carassa, F. “The SIRIO programme and its propagation and communication

experiment”, Aha Freq., 47,65-71, Specialissue on the SIRIO programme, 1978.

Chiu, L. S., “Estimating areal rainfall from rain area.” Tropical Rainfall

Measurements, J.S Theon and N. Fugono, Eds., A. Deepak, 361-367, 1988.


Bibliographie

62

Crane R., X. H Wang, D.B Westenhaver, and W.J. Vogel, “ACTS propagation

experiment: Experiment design, calibration and data preparation and archival”, Proc

IEEE, 85(6), 863-878, 1997

Cost 255: “Radiowave propagation modelling for new SatCom services at Ku-

band and above”, final report, ESA publication division, Noordwijk, The Netherlands,

2002.

Dennis A.S., F.G Fernald, “Frequency distribution of shower size”, J. Apl.

Meteor., 2, 767-769, 1963

Donneaud, A. A., S. I. Niscov, D. L. Priegnitz, and P. L. Smith , “The area-time

integral as an indicator for convective rain volume.” J. Appl. Meteor., 23(4),

1984,10.1175/1520-0450(1984)023<0555:TATIAA>2.0.CO;2.

Eltahir, E. A. B. and R. L. Bras , “Estimation of the Fractional Coverage of

Rainfall in Climate Models”, Journal of Climate, 6(4), 639-644, 1993, doi: DOI:

10.1175/1520-0442(1993)006<0639:EOTFCO>2.0.CO;2.

Féral L., "Etude des variabilités spatiales des précipitations pour optimiser les

systèmes de télécommunication par satellite", Ph’D of University Paul Sabatier,

Toulouse, France, 11th December 2002.

Féral, L., H. Sauvageot, L. Castanet, and J. Lemorton, “HYCELL—A new hybrid

model of the rain horizontal distribution for propagation studies: 1. Modeling of the rain

cell”, Radio Sci., 38(3), 1056,2003 doi:10.1029/2002RS002802.

Féral, L., H. Sauvageot, L. Castanet, and J. Lemorton (2003), HYCELL—A new

hybrid model of the rain horizontal distribution for propagation studies: 2. Statistical

modeling of the rain rate field, Radio Sci., 38(3), 1057, doi:10.1029/2002RS002803.


Bibliographie

63

Féral L., H. Sauvageot, L. Castanet, J. Lemorton, F. Cornet, and K. Leconte

(2006), Large-scale modeling of rain fields from a rain cell deterministic model, Radio

Sci., 41, RS2010,2006, doi:10.1029/2005RS003312.

Goldhirsh, J., and F. L. Robison, “Attenuation and space diversity statistics

calculated from radar reflectivity data of rain”, IEEE Transactions on Antenna and

Propagation, 23(2), 221-227,1975.

Goldhirsh J., and B. Musiani, “Rain cell size statistics derived from radar

observations at Wallops Island, Virginia” IEEE trans. Geosci. And Remote Sensing,

24(6),947-954, 1986.

Huffman G. J., et al., “The Global Precipitation Climatology Project (GPCP)

Combined Precipitation Dataset”. Bull. Amer. Meteor. Soc., 78, 5–20, 1997.

Hodges D. D., R J Watson and G Wyman: "An attenuation time series model for

propagation forecasting", IEEE transactions on Antennas and Propagation, 54(6), 2006.

Huuskonen A., et al. "EUMETNET OPERA PROGRAMME", Operational

Programme for the Exchange of Weather Radar Information Final report, March 2007,

109H109Hhttp://www.knmi.nl/opera/opera2/OPERA_final_report.pdf.

ITU-R Software and database website

110H110Hhttp://www.itu.int/ITU-R/study-groups/software/rsg3-p836-water-vapour-

columnar-data.zip

http://www.itu.int/ITU-R/study-groups/software/rsg3-p840-cloud-liquid.zip

ITU-R P.837-5, “Characteristics of precipitation for propagation modeling“, ITU-

R P Series Recommendations-Radiowave Propagation, 2007.

ITU-R P.836-3, “Water vapour: surface density and total columnar content “,

ITU-R P Series Recommendations - Radiowave Propagation, 2001.


Bibliographie

64

ITU-R P.676-7, “Attenuation by atmospheric gases”, ITU-R P Series

Recommendations-Radiowave Propagation, 2007.

ITU-R P.840-3, “Attenuation due to clouds and fog”, ITU-R P Series


Jeannin N., Castanet L., Lemorton J., Feral L., and Sauvageot H. "Modeling of

rain fields at large scale", First IEEE International Workshop on Satellite and Space

Communications, IWSSC 2006, Madrid, Spain, September 2006a.

Jeannin N., Féral L., Castanet L., Lemorton J., Sauvageot H., Cornet F., and

Leconte K. "Modelling of rain fields and attenuation fields spatially correlated over

Europe and USA", 3rd CNES workshop on Earth-space propagation, Toulouse, France,

September 2006b.

Katti V. R., K. Thyagarajan, K. N. Shankara and A. S. Kiran Kumar, " Spacecraft

technology", Current Science, 93(12), 2007.

Klazura, G.E., and D.A. Imy, “A Description of the Initial Set of Analysis

Products Available from the NEXRAD WSR-88D System.” Bull. Amer. Meteor. Soc.,

74, 1293–1311, 1993.

Kummerow, C., W. Barnes, T. Kozu, J. Shiue, and J. Simpson, “The Tropical

Rainfall Measuring Mission (TRMM) Sensor Package.” J. Atmos. Oceanic Technol., 15,

809–817, 1998.

Lacoste F., "Modelling of the dynamics of the Earth-space propagation channel at

Ka and EHF bands", Ph’D of SUPAERO, Toulouse, France, September 2005.

Lacoste F., M. Bousquet, F. Cornet, L. Castanet and J. Lemorton, “Classical and

On-Demand Rain Attenuation Time Series Synthesis: Principle and Applications”,

AIAA-2006-5325, 24th AIAA International Communications Satellite. Systems

Conference, San Diego, California, June 11-14, 2006.


Bibliographie

65

Lascaux F., E. Richard, C. Keil and O. Bock, 2003: Numerical simulations of the

precipitation event observed during the MAP IOP2a: Sensitivity to initial

conditions.International Conference on Alpine Meteorology, Brig, Switzerland, 19-23

May, 2003, 12-15.

Lemorton J., L. Castanet , F. Lacoste, C. Riva , E. Matricciani, U-C. Fiebig,

M.M.J.L. Van de Kamp, and A. Martellucci : "Development and validation of time

series synthesizers of rain attenuation for Ka-band and Q/V-band satellite

communication systems", International Journal of Satellite Communications and

Networking, 25, 575-601, 2007.

Maseng T., and P.M Bakken, “A stochastic dynamic model of rain attenuation”,

IEEE Trans. on Commun., 29, 660-669, 1981.

Mesnard, F., and H. Sauvageot, “Structural characteristics of rain fields”, J.

Geophys. Res., 108(D13), 4385, 2003, doi:10.1029/2002JD002808.

Nzeukou, A., and H. Sauvageot “Distribution of rainfall parameters near the

coast of France and senegal.” J. Appl. Meteor., Vol. 41, n°1, 69-82, 2002.

OPEX: “Reference Book on Attenuation Measurement and Prediction”, 2nd

Workshop of the OLYMPUS Propagation Experimenters (OPEX), Doc ESA-ESTEC-

WPP-083 volume 1, Noordwijk, The Netherlands, 1994.

Paraboni, A., A. Vernucci, L. Zuliani, E. Colzi, and A. Martellucci, "A New

Satellite Experiment in the Q/V Band for the Verification of Fade Countermeasures

based on the Spatial Non-Uniformity of Attenuation", in European Conference on

Antennas and Propagation 2007, Edinburgh, UK, Nov 2007.

Pech, P. “Incidence de la prise en compte des mécanismes de lutte contre les

affaiblissements (FMT) en bande Ka sur la gestion de la ressource dans un système

d’accès multimédia par satellite géostationnaire. ”, Ph’D of SUPAERO, Toulouse,

France, Décembre 2003.


Bibliographie

66

Poiares Baptista J.P.V, Z. W. Zhang, and N. J. Mcewan, “Stability of rain rate

cumulative distribution”, Electronics Letters, 22(7), 350-352, 1987.

Poiares-Baptista J.P.V, and Salonen E.T. "Review of rainfall rate modelling and

mapping", Proc. of URSI Climpara'98 Comm. F Open Symposium on "Climatic

Parameters innRadiowave Prediction", Ottawa, Canada, 35-44, April 1998.

Rudolf, B., T. Fuchs, U. Schneider, and A. Meyer-Christoffer, “Introduction of

the Global Precipitation Climatology Centre (GPCC)”. Deutscher Wetterdienst Rep.,

Offenbach am Main, Germany, 16 pp, 2003.

SatNEx JA-2310 book, Castanet L. (editor) et al. : “Influence of the variability of

the propagation channel on mobile, fixed multimedia and optical satellite

communications”, ISBN 978-3-8322-6904-3, Shaker, 2008.

Sauvageot, H., “The probability density function of rain rate and the estimation

of rainfall by area integrals.”, J. Appl. Meteor., 33(11), 1255-1262, 1994,doi:

10.1175/1520-0450(1994)033<1255:TPDFOR>2.0.CO;2.

Riva, C., “Seasonal and diurnal variations of total attenuation measured with the

ITALSAT satellite at Spino d’Adda at 18.7, 39.6 and 49.5 GHz,” International Journal

of Satellite Communications and Networking, 22, 449–476, 2004.

Uppala S. et al., “The ERA-40 re-analysis”. Quart. J. R. Meteorol. Soc., 131,

2961-3012, 2005.

Van de Kamp M.M.J.L., "Rain attenuation as a Markov process, using two

samples", Post-doctorate thesis, Document ONERA RT 1/06733.04 DEMR, Toulouse,

France, March 2003.

67

B. Articles

Articles

Statistical distribution of integrated liquid water and water vapor content from meteorological

reanalysis

68

Articles


reanalysis

69

Statistical distribution of integrated liquid water and water vapor content from meteorological reanalysis

Nicolas Jeannin (1), Laurent Feral (2), Henri Sauvageot (3), Laurent Castanet (1) (1)ONERA, Département ElectroMagnétisme et Radar, Toulouse, France.

(2) Université Paul Sabatier, Laboratoire Antennes Dispositifs et Matériaux

Micro-ondes (AD2M), Toulouse, France. (3) Université Paul Sabatier, Observatoire Midi-Pyrénées, Laboratoire

d’Aérologie, Toulouse, France.

Abstract

In this paper a worldwide modelling of integrated liquid water and water vapor

content distributions is proposed and evaluated. The knowledge of those distributions is

valuable to predict attenuation for Earth-space communication systems operating at

frequencies higher than 10GHz.

1 Introduction

The prediction of tropospheric attenuation is of major interest for satellite

communication purposes. Up to now, most of the studies have focused on the dominant

effect: the attenuation due to rain. Nevertheless, with the use of higher frequencies such

as Ka band (20-30 GHz) and Q/V band (40-50 GHz), tropospheric attenuation can not be

limited to rain any longer.. For instance, as shown by Liebe (1985), Liebe et al. (1989)

or Pujol et al. (2006), attenuation due to clouds (i.e. liquid water) or water vapour can

exceed 5 dB and 6 dB, respectively, at 50 GHz, for 0.01% of the time. Therefore, system

engineers have now to consider these tropospheric components in addition to rain to

optimally design the next generation of communication systems.

Articles


reanalysis

70

the field of radio-communications, the Cumulative Distribution Function (CDF)

of tropospheric attenuation is usually computed from statistical distributions of

meteorological parameters. The latter are derived from worldwide meteorological

databases obtained from reanalysis models. Obviously, it is not convenient to handle

with these huge datasets that are moreover not always available to the community.

Therefore models have been proposed to derive worldwide distributions of

meteorological parameters from few inputs. For instance, the local rain rate CDF is

modelled by Rec. ITU-R P.837-3 from three inputs given by ECMWF (European Centre

for Medium-Range Weather Forecast) reanalysis meteorological database ERA-15

(Salonen et al., 1994): the probability of rain during 6 hour periods, the total yearly

amount of convective rain and the total yearly amount of stratiform rain. A double

exponential prediction method is then used to derive rainfall rate CDF characteristic of

the geographical area considered (Baptista and Salonen, 1998). Such a model is of prime

importance as it provides inputs to rain field and rain attenuation field simulators

anywhere in the world, while accounting for the local characteristics of rain (Capsoni et

al., 1987a, 1987b, Féral et al., 2003a, 2003b, 2006, Grémont and Filip, 2004).

Unfortunately, such a model and its associated worldwide parameterization are not

currently available for the CDFs of the Integrated Liquid Water Content (ILWC, i.e. the

mass of liquid water contained in a columnar section of 1 square meter) and the

Integrated Water Vapor Content (IWVC, i.e. the weight of water vapor contained in a

columnar section of 1 square meter).

This paper aims at providing the community with worldwide maps of parameters

that allow the modelling of ILWC and IWVC CDFs anywhere in the world. To achieve

this, 15 years of reanalysis meteorological data (ERA-15) are considered. They are

presented in Section 2. Section 3 is devoted to the analytical modelling of the ILWC

CDF, to the assessment of the parameters and to the model accuracy. The same work is

performed in Section 4 to model the IWVC CDF.

Articles


reanalysis

71

2 The data

So as to derive the analytic formulations of the ILWC and IWVC CDFs, Rec.

ITU-R P.840-3 (1999) and P.836-3 (2001) should have been considered. Indeed, the

latter give the time percentages during which a given value of ILWC and IWVC is

exceeded respectively.

Nevertheless, recalling that ILWC and IWVC values are derived from profiles of

pressure P, temperature T and humidity H (Salonen and Uppala, 1991), the P, T, H

profile database has to be selected with care. Particularly, Rec. ITU-R P.840-3 and

P.836-3 were derived from NA-4, an ECMWF database that provides 2920 P, T, H

profiles collected worldwide from 1992 to 1994. In the framework of the present study,

the ERA-15 database, also produced by the ECMWF, has been selected for statistical

stability reasons. Indeed, ERA-15 consists in 15 years (1979-1994) of reanalysed

meteorological data corresponding to 21900 samples (one sample every 6h) located all

around the world. Therefore, for each ERA-15 grid point (resolution 1.5°×1.5°), the

ILWC and IWVC values exceeded for 18 successive time percentages 18,..,1iiP = =99, 95,

90, 80, 70, 60, 50, 30, 20, 10, 5, 3, 2, 1, 0.5, 0.3, 0.2, 0.1 have been derived worldwide

from ERA-15 P, T, H profiles (Riva et al., 2006).

Now, when dealing with the ERA-15 database, some shortcomings have to be

kept in mind. First, the data are the outputs of a reanalysis process and do not

correspond to physical measurements. They are the result of a mathematical model

constrained to minimize the error with respect to local measurements derived from

remote sensing, radio-soundings, or ground stations. Second, it should be noticed that

the data from radio-soundings or radiometers are not available when it is raining, i.e.

precisely when the highest values of water vapor and liquid water content are reached

(Snider, 2000). Therefore, the confidence in the tail of ILWC and IWVC distributions

has to be limited. Nevertheless, as shown by Riva et al. (2006), distributions of ILWC

and IWVC derived from local radio-sounding compare satisfactorily with these derived

from ERA-15. Indeed, for 24 radio-sounding sites located in various climatic areas, the

average value of the RMS relative error between the CDFs of ILWC derived from ERA-

15 and the ones derived from radio-sounding is of 28%. For IWVC CDFs, this value is

around 10%. It has to be noticed that the confidence in the ILWC values has to be lower

Articles


reanalysis

72

than the one in the IWVC values as they result from a semi-empirical model applied on

P, T, H profiles (Salonen and Uppala, 1991) and not from a direct integration of the

profiles.

3 Modelling of the ILWC distribution

3.1 Analytic formulation and parameter derivation

From aircraft and radiometric measurements, Gultepe and Isaac (1997) and

Fairall et al. (1990) have showed that liquid water content LWC and ILWC conditional

distributions (i.e. knowing that LWC>0 and ILWC>0, respectively) are well represented

by lognormal distributions. Consequently, the experimental CDFs of ILWC derived

from ERA-15 are first approximated by a lognormal law whenever the ILWC value is

strictly positive:

( )

1)0(

0L if ²2

²lnexp21)( **

*

=≥

>⎥⎦⎤

⎢⎣⎡ −−=≥ ∫

+∞

LP

dlll

PLLPL

cloud σμ

σπ (1)

In (1), L is the ILWC, Pcloud the probability to have a non zero ILWC, μ and σ the

lognormal parameters of the conditional distribution of ILWC )0/( * >≥ LLLP . For

each point of the ERA-15 grid, Pcloud, μ and σ are determined worldwide by minimizing

the error with respect to the local ILWC CDF derived from ERA-15 (1979-1994), for the

standard time percentages 18,..,jiiP = and associated ILWC 18,..,jiiL = defined in Section 2:

( )∑=

=⎥⎦

⎤⎢⎣

⎡ −18 2

,,

,,,min

i

ji i

icloudi

P LLPPF

cloud

σμσμ

. (2)

In (2), ( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−×σ+μ=σμ −

cloud

i1cloudi P

P21erf2expP,,,PF is the inverse of the

cumulative distribution function and erf(x) is the error function. Worldwide maps of

Pcloud, μ and σ are given in Figures 1, 2 and 3, respectively. Importantly, in (2),

Articles


reanalysis

73

subscript j (j≥1) indicates that only ILWC values Li greater than 0.1kg/m2 are considered

in the minimisation process. In some cases, it prevents Pcloud , μ and σ from reaching

unrealistic values.

Figure 1: Worldwide map of Pcloud (probability to have a non zero ILWC) regressed

from the ERA-15 database.

Figure 2: Worldwide map of parameter μ regressed from the ERA-15 database.

Articles


reanalysis

74

Figure 3: Worldwide map of parameter σ regressed from the ERA-15 database.

According to Figs. 1, 2 and 3, Pcloud, μ and σ show a strong climatic dependency.

As expected, the probability to have non zero ILWC is higher over sea than over land. It

is also higher for equatorial areas than for arid or temperate areas. Moreover, in the

presence of cloud, μ is higher at low latitudes. These trends are confirmed by Table 1

which summarizes some ILWC values exceeded for different time percentages and

various locations. As shown in Table 1, ILWC values are much higher in equatorial and

tropical areas, than in mid-latitudes areas.

Articles


reanalysis

75

ILWC (kg/m2) exceeded for x% of time Location 10% 1% 0.1%

ERA-15 0.25 0.75 1.20 Moscow

Model 0.25 0.76 1.18

ERA-15 0.81 2.31 3.27 Rio

Model 0.94 2.24 3.60

ERA-15 0.44 1.27 1.71 Paris

Model 0.47 1.25 1.87

ERA-15 2.10 3.50 3.89 Singapore

Model 2.12 3.41 4.09

ERA-15 0 0.31 0.60 Denver

Model 0.05 0.31 0.59

Table 1: Values of ILWC exceeded for different time percentages and various locations

from ERA-15 and from the model.

3.2 Model accuracy

In order to assess the model accuracy, the RMS error ε between the modelled

CDFs and those derived from ERA-15 has been computed according to:

( ) ( )( ) 100PL

PLPL)1j(18

1 18

ji

2

i15ERA

i15ERA

iMODEL

×⎥⎦

⎤⎢⎣

⎡ −−−

=ε ∑=

−

−

. (3)

So as to assess whether the lognormal approach is the best one to model the

conditional CDF of ILWC, Weibull and Gamma conditional distributions have been also

considered. Their optimal parameterisation has been obtained following the same

minimisation process used to derive the lognormal parameters μ and σ.

Besides, Weibull and Gamma distributions have been chosen due to their limited

number of parameters and because they may have a large tail, in compliance with ILWC

data derived from ERA-15.

Articles


reanalysis

76

The results are given in Table 2, in terms of mean <ε>, standard deviation εstd

and maximum value of the error εmax computed for all the pixels of the ERA-15

database.

Weibull law Lognormal law Gamma law <ε> 24% 9% 19% εstd 20% 11% 23% εmax 85% 51% 70%

Table 2: Worldwide mean <ε>, standard deviation εstd and maximum value εmax of the

RMS error ε between the ILWC conditional CDFs derived from ERA-15 and the 3

models considered (lognormal, Weibull and Gamma).

In compliance with Table 2, the lognormal model minimises the error with

respect to ILWC conditional CDFs derived from ERA-15. The highest errors can be

observed to be mainly located over tropical oceans or highly mountainous areas, i.e. for

places where the ERA-15 database is known to show some inaccuracies (Allan et al.,

2001). Over temperate areas and over land, the error is low. To conclude, the lognormal

distribution is appropriate to model the ILWC conditional CDF worldwide.

As an example, Fig. 4 shows the ILWC CDF derived from ERA-15 for Dublin

(Ireland) and the associated lognormal modelling. The RMS error (3) is equal to 6%.

Articles


reanalysis

77

Figure 4: ILWC CDF derived from ERA-15 for Dublin (Ireland) and associated

lognormal modelling.

4 Modelling of the IWVC distribution

4.1 Analytic formulation and parameter derivation

From data collected by aircraft, radio-sounding and remote sensing, Zhang et al.

(2003) have shown the possible existence of a bimodal behaviour of the IWVC

Probability Distribution Function (PDF) in tropical areas due to alternate dry and moist

regimes, disappearing when integrating over a wide area and a long time. Foster et al.

(2006) from GPS delay and radio-sounding measurements propose a log-normal model

and a reverse log-normal model in tropical oceans for precipitable water. However

IWVC CDFs derived from ERA-15 do not offer such a bimodal or log-normal

behaviour. They rather show that the IWVC CDFs can be appropriately represented, as

shown in the next section, by Weibull distributions:

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=≥

kVVVPλ

** exp)( , (4)

Articles


reanalysis

78

where V is the IWVC, k and λ are the shape and scale parameters of the Weibull

distribution, respectively. Considering the 18 values 18,..,1iiV = of IWVC exceeded a

fraction 18,..,1iiP = of the time, the Weibull approximations of the IWVC CDFs derived

from ERA-15 are determined by solving:

( )∑=

=⎥⎦

⎤⎢⎣

⎡ −18

1

2

,

,,min

i

i i

ii

k VVkPH λ

λ, (5)

where ( ) kii PkPH /1)(ln,, λλ −= is the inverse of the cumulative distribution

function. The regression is performed on each grid pixel of the ERA-15 database, using

the same minimization algorithm as the one used to derive ILWC CDF parameters.

Worldwide maps of scale parameter λ and shape parameter k are presented in Figures 5

and 6, respectively.

Figure 5: Regressed values of the scale parameter λ of the Weibull distribution of IWVC

from the ERA-15 database (1979-1994).

Articles


reanalysis

79

Figure 6: Regressed values of the shape parameter k of the Weibull distribution of

IWVC from the ERA-15 database (1979-1994).

The parameters show a significant latitude dependency. Indeed, as can be seen in

Fig.5, the scale parameter λ, that drives the mean of the distribution, is higher at low

latitudes because the saturation water vapor content and the evaporation becomes higher

with higher temperatures. In other respects, the shape parameter k that drives the

skewness is higher in places where the troposphere saturation in water vapor is frequent.

As can be observed in Fig.1, saturation occurs more often in areas where there is a high

probability to have liquid clouds. Since values of IWVC can not exceed the saturation,

the distribution becomes negatively skewed (large λ).

Moreover, λ and k are also affected by orography (the atmosphere is thinner) and

land-sea transitions. Table 3 gives IWVC values exceeded for different time

percentages, various locations and computed from the model or derived from ERA-15.

According to Table 3 and as expected, the values of IWVC are higher in equatorial or

tropical areas than in temperate areas.

Articles


reanalysis

80

IWVC (kg/m2) exceeded for x% of time

Location 10% 1% 0.1%

ERA-15 25.3 33.3 38.2 Moscow

Model 23.9 32.6 40.9

ERA-15 48.2 58.6 63.4 Rio

Model 50.1 58.8 64.5

ERA-15 28.4 35.4 41.2 Paris

Model 27.8 35.4 40.7

ERA-15 53.4 63.4 69.6 Singapore

Model 55.1 63.1 67.5

ERA-15 16.1 25.2 28.4 Denver

Model 15.9 24.3 30.1

Table 3: Values of IWVC exceeded for different time percentages, various locations and

derived from ERA-15 or computed from the model.

4.2 Model accuracy

The RMS error ε committed by approximating the IWVC CDF derived from

ERA-15 by a Weibull distribution has been computed according to (3). As in section 3,

so as to quantitatively assess whether the Weibull distribution is the best one to model

the IWVC CDF, the RMS error ε has also been computed considering 3 other

distributions potentially able to fit the CDFs derived from ERA-15, namely normal,

lognormal and gamma distributions. The results are given in Table 4, in terms of mean

<ε>, standard deviation εstd and maximum value of error εmax computed for all the pixels

of the ERA-15 database. As shown in Table 4, the Weibull model minimises the error

with respect to IWVC CDFs derived from ERA-15, with a mean error <ε> lower than

5% worldwide. Maximum errors are reached in specific places such as the Andes

(~30%), around the Bering sea (~20%) and in southern Indian Ocean (~15%), i.e. in

places where problems have been reported with reanalysis data (Allan et al., 2001).

Articles


reanalysis

81

Weibull law Normal law Lognormal law Gamma law <ε> 4.3% 23% 27% 24% εstd 5.1% 19% 12% 14% εmax 37% 198% 152% 77%

Table 4: Worldwide mean <ε>, standard deviation εstd and maximum value εmax of the

RMS error ε between the IWVC CDFs derived from ERA-15 and the 4 models

considered (Weibull, normal, lognormal and Gamma).

As an example, Fig. 7 shows the IWVC CDF derived from ERA-15 for Dublin

(Ireland) and the associated lognormal modeling. The RMS error (3) is equal to 4%.

Figure 7: IWVC CDF derived from ERA-15 for Dublin (Ireland) and the associated

Weibull modelling.

Conclusion

Worldwide maps of parameters that allow the modelling of ILWC and IWVC

CDFs anywhere in the world have been proposed. They allow the computation of cloud

and water vapour attenuation CDFs worldwide from a few numbers of parameters.

Articles


reanalysis

82

To do so, empirical CDFs of ILWC and IWVC have been derived from the

ECMWF ERA-15 database. The ILWC conditional CDFs derived from ERA-15 are

showed to be best modelled by a lognormal law anywhere in the world, with an average

RMS error lower than 10%. Highest errors are found over tropical oceans and highly

mountainous areas, i.e. in places where ERA-15 database is known to have some

inaccuracies. Concerning the IWVC CDFs, the Weibull distribution is shown to be the

one that best describes the worldwide IWVC CDFs derived from ERA-15, with an

average RMS error lower than 5%.

To model the IWVC and ILWC CDFs anywhere in the world, the worldwide

maps of parameters obtained in the framework of the present paper can be downloaded

at the URL: 111H111Hhttp://www-mip.onera.fr/ILWC_IWVC_distributions/IWVC_ILWC_map.

zip. Further studies will be possible with the forthcoming ERA-40 database whose

duration (40 years) ensures a sufficient amount of data to study the monthly dependency

of many meteorological parameters of interest.

Acknowledgements:

This research was partly supported by CNES (contract noA88550) and has been

partly carried out in the framework of the European Network of Excellence SatNex. The

authors are very grateful to Antonio Martellucci from ESA (The Netherlands) and Carlo

Riva from Politecnico di Milano (Italy) for kindly providing the statistical distributions

of ILWC and IWVC derived from ERA-15 and computed in the framework of

ESA/ESTEC Contract N°17760/03/NL/JA

References

Allan, R., A. Slingo and V. Ramaswamy, “Analysis of moisture variability in the

European Centre for Medium-Range Weather Forecasts 15-year reanalysis over the

tropical oceans”, J. Geophys. Res., 107(D15), 2002.

Articles


reanalysis

83

Capsoni, C., F. Fedi, and A. Paraboni, “A comprehensive meteorologically

oriented methodology for the prediction of wave propagation parameters in

telecommunication applications beyond 10 GHz”, Radio Sci., 22(3) , 387-393, 1987a.

Capsoni, C., F. Fedi , C. Magistroni, A. Paraboni and A. Pawlina, “Data and

theory for a new model of the horizontal structure of rain cells for propagation

applications”, Radio Sci., 22(3), 395-404, 1987b.

ERA: ECMWF Re-analaysis project report series, “1. ERA-15 Description”,

Version 2, European Centre for Medium-Range Weather Forecasts, 1999.

Féral L., H. Sauvageot, L. Castanet and J. Lemorton, “HYCELL: A new hybrid

model of the rain horizontal distribution for propagation studies. Part 1: Modeling of the

rain cell”, Radio Sci., 38(3), 2003a.

Féral L., H. Sauvageot, L. Castanet and J. Lemorton, “HYCELL: A new hybrid

model of the rain horizontal distribution for propagation studies. Part 2: Statistical

modeling of the rain cell”, Radio Sci., 38(3), 2003b.

Féral L., H. Sauvageot, L. Castanet, J. Lemorton, F. Cornet and K. Leconte,

“Large-scale modeling of rain fields from a rain cell deterministic model”, Radio Sci.,

41(2), 2006.

Fairall, C.W., J.E. Hare and J.B. Snider, “An eight month sample of marine

stratocumulus cloud fraction, albedo and integrated liquid water”, J. Climate, 3, 847-

864, 1990.

Foster, J. M. Bevis and W. Raymond, “Precipitable water and the log-normal

distribution”, J. Geophys. Res., 111, D15102.

Gremont, B.C. and M. Filip, “Spatio-Temporal rain attenuation Model for

application to FMT”, IEEE Trans Ant. Prop. , 52, 5, 2004.

Articles


reanalysis

84

Gultepe, I., and G.A. Isaac, “Liquid water content and temperature relationship

from aircraft observations and its applicability to GCMs”, J. Climate, 10, 446-452,

1997.

ITU-R Software and database website

112H112Hhttp://www.itu.int/ITU-R/study-groups/software/rsg3-p836-water-vapour-

columnar-data.zip

http://www.itu.int/ITU-R/study-groups/software/rsg3-p840-cloud-liquid.zip

ITU-R P.837-4, “Characteristics of precipitation for propagation modeling“, ITU-

R P Series Recommendations-Radiowave Propagation, 2003.

ITU-R P.836-3, “Water vapour: surface density and total columnar content “,

ITU-R P Series Recommendations - Radiowave Propagation, 2001.

ITU-R P.676-5, “Attenuation by atmospheric gases”, ITU-R P Series


ITU-R P.840-3, “Attenuation due to clouds and fog”, ITU-R P Series


Liebe, H.J., “An updated model for millimeter-wave propagation in moist air”,

Radio Sci. , 5, 1069-1089, 1985.

Liebe, H.J., T. Manabe, and G.A. Hufford, “Millimetre wave attenuation and

delay rates due to fog and cloud conditions”, IEEE Trans Ant. Prop, 37, 1617-1623,

1989.

Poiares Baptista J. P. V., and E. T. Salonen, “Review of rainfall rate modeling

and mapping”, Proceedings of URSI-F Open Symposium, CLIMPARA'98, Ottawa, 27-

29 June 1998, 35-44.

Articles


reanalysis

85

Pujol, O., J-F. Georgis, L. Féral, and H. Sauvageot, “Degradation of radar

reflectivity by cloud attenuation at microwave frequency”, J. Atmos. Oceanic. Technol.,

24, 640-657, 2007.

Riva, C., C. Capsoni, M. D’Amico, L. Luini, E. Mattricciani, A. Paraboni, L.

Castanet, T. Deloues,V. Fabbro, L. Féral, F. Lacoste, J. Lemorton, E. Kubista, T.

Prechtl, and M. Schönhuber,. “Characterisation and Modeling of propagation effects in

20-50GHz band”, Final report ESA/ESTEC/ Contract N°17760/03/NL/JA, 2006.

Salonen, E., S. Karhu, S. Uppala, and R. Hyvönen, “Study of improved

propagation predictions”, Final Report for ESA/ESTEC Contract 9455/91/NL/LC(SC),

Helsinki University of Technology and Finnish Meteorological Institute, 1994.

Salonen, E., and S. Uppala, “New prediction method of cloud attenuation”,

Electronic Letters, 27(12), 1108-1109, 1991.

Snider, J.B., “Long-term observations of cloud liquid, water vapour, and cloud

base temperature in the North-Atlantic Ocean”, J. Atmos. Oceanic Technol., 17, 928-

939, 2000.

Zhang, C., B. Mapes, and B.J. Soden, ”Bimodality in tropical water vapour”, Q.

J. R. Meteor. Soc., 129, 2847-2866, 2003.

Articles

Distribution of tropospheric water vapor in clear and cloudy conditions from microwave

radiometric profiling

86

Distribution of tropospheric water vapor in clear and cloudy conditions from microwave radiometric

profiling

Alia IASSAMEN(1)(3), Henri SAUVAGEOT(1),

Nicolas JEANNIN(2), and Soltane AMEUR(3)

(1) Laboratoire d'Aérologie, Observatoire Midi-Pyrénées, Université Paul

Sabatier, Toulouse, France.

(2) Département Electromagnétisme et Radar, Office National d'Etudes et de

Recherches Aérospatiales, Toulouse, France.

(3) LAMPA, Département d'Electronique, Université Mouloud Mammeri, Tizi

Ouzou, Algeria.

Abstract

A dataset gathered over 369 days in various midlatitude sites with a 12 frequency

microwave radiometric profiler is used to analyze the statistical distribution of

tropospheric water vapor content (WVC) in clear and cloudy conditions. The WVC

distribution inside intervals of temperature is analyzed. WVC is found to be well fitted

by a Weibull distribution. The two Weibull parameters, the scale (λ) and shape (k), are

temperature (T) dependent. k is almost constant, around 2.6, for clear conditions. For

cloudy conditions, at T < -10°C, k is close to 2.6. For T > -10°C, k displays a maximum

in such a way that skewness, which is positive in most conditions, reverses at negative

in a temperature region approximately centred around 0°C, i.e. at a level where the

occurrence of cumulus clouds is high. Analytical λ(T) and k(T) relations are proposed.

The WVC spatial distribution can thus be described as a function of T. The mean WVC

vertical profiles for clear and cloudy conditions are well described by a function of

temperature of the same form as the Clausius-Clapeyron equation. The

WVCcloudy/WVCclear ratio is shown to be a linear function of temperature. The

Articles



87

vertically integrated WV (IWV) is found to follow a Weibull distribution. The IWV

Weibull distribution parameters retrieved from the microwave radiometric profiler agree

very well with the ones calculated from the European reanalysis meteorological database

ERA 15. The radiometric retrievals compare fairly well with the corresponding values

calculated from an operational radiosonde sounding dataset.

1 Introduction

Water vapor is a component of paramount importance of the atmosphere in

relation with weather and climate sensitivity research. It is the main greenhouse gas and

plays a predominant role in cloud dynamics through the release of wide amounts of

latent heat with condensation. Water vapor content distribution (WVCD) is also very

important in scientific fields related with the electromagnetic propagation in the

microwave domain. Knowing the WVCD is thus essential for numerous environmental

sciences (e.g. Peixoto et al. 1992, Chahine 1992, Webster 1994, IPCC 2001, Soden and

Held 2006, Sherwood et al. 2006, among others). Points of particular interest concern

the parametrization of WVCD and the quantitative differences of WVCD between clear

and cloudy atmosphere. Climatological column water vapor content varies only slightly

with the cloud cover in tropical regions but is significantly lower in clear sky than in

cloudy sky at midlatitudes and the variation is not simply due to differences in

atmospheric temperature distribution, as shown by Gaffen and Elliott (1993). The shape

of WVCD has not been discussed and that of integrated water vapor (IWV) is, again, a

point in discussion (e.g. Zhang et al. 2003, Foster et al. 2006).

Atmospheric water vapor is very difficult to quantify and describe due to the

various physical and dynamical processes affecting its distribution, notably on short

time and space scales. The in situ data used to analyze the tropospheric WVCD were

mainly collected by radiosonde observations (Liu et al. 1991, Gaffen and Elliott 1993,

Foster et al. 2006, Miloshevich et al. 2006). However, radiosonde observations do not

permit to associate tightly the presence of clouds and the WVC profiles. Some

fragmentary, in space and time, in situ observations were also gathered from aircraft

(Spichtinger et al. 2004, Korolev and Isaac 2006). In clear air and in the daytime, the

Articles



88

total column water vapor from the surface was measured with simple near-infrared sun

photometer (Brooks et al. 2007). Water vapor in upper atmospheric layers was observed

by remote sensing from geostationary operational meteorological satellite using the 6.7

μm channel (e.g. Soden and Bretherton 1993) and from infrared sounder such as AIRS

(Atmospheric InfraRed Sounder). Observations were also made using microwave

sensors such as UARS (Upper Atmospheric Research Satellite) microwave limb sounder

(Stone et al. 2000) or from AMSU (Advanced Microwave Sounding Unit) instruments

and SSM/I (Special Sensor Microwave Imager) instruments or by combining the two

kinds of data (Kidder and Jones 2007). Vertically integrated (column) water vapor

(IWV), or precipitable water, data were obtained from Global Positioning System (GPS)

signals (e.g. Rocken et al. 1997, Liou et al. 2001, Van Baelen et al. 2005, Foster et al.

2006). Standard dual-channel ground-based microwave radiometers, with a frequency

close to the 22.235 GHz water vapor line and the other in a window at higher frequency,

were used to measure IWV and LWP (Liquid Water Path) simultaneously (e.g. Hogg et

al. 1983, Snider 2000, Westwater et al. 2004, Meijgaard and Crewell 2005).

Tropospheric vertical profile of water vapor mixing ratio was also observed by

active remote sensing, notably by Raman lidar by Sakai et al. (2007) in the night-time

from a ground-based device, and by Whiteman et al. (2007) in the daytime with an

airborne spectroscopic lidar. Upper-troposphere water vapor was measured by

differential absorption lidar technique (Ferrare et al. 2004). However, lidar

measurements work only in the absence of optically thick clouds. Information on

humidity profiles have been retrieved with wind profiling radar from refractive index

gradient inferred from the echo power with variational assimilation (Furumoto et al.

2007) and by combining wind profiler and ground-based radiometer (Bianco et al. 2005)

or with GPS (Furumoto et al. 2003). The above quoted techniques are not able to

provide the data required to analyse the WVC distribution on short time and space

scales, in clear and cloudy conditions.

More recently, a ground-based multi-channel microwave radiometric technique

was proposed for the retrieval of temperature, water vapor, and liquid water profiles

(Solheim et al. 1998, Solheim and Godwin 1998, Güldner and Spänkuch 2001, Ware et

al. 2003). This new technique has not yet gained the confidence of long time used

methods and its accuracy is not that of in situ measurements, but it offers some

Articles



89

opportunities to capture the differences between clear and cloudy vertical profiles which

are still poorly known and documented, due to a lack of relevant data.

The object of the present paper is to discuss the form of the WVCDs and to

propose their parametrization as a function of the temperature in order to quantify the

differences between clear and cloudy sky conditions. The paper is based on a set of data

collected with a multichannel microwave radiometric profiler.

2 Data

2.1 Water vapor parameters

The quantity retrieved by the radiometer (cf. Section 2b) is the water vapor

density of the moist air, or absolute humidity, ρ. Water vapor approximately behaves as

an ideal gas, with each mole of gas obeying an equation of state that can be written:

TRe

w

ww =ρ , (1)

where ew is the partial pressure of the water vapor in hPa, Rw is the gas constant

for water vapor (that is Rw = R/mw, where R is the universal gas constant and mw is the

mass of one mole of vapor), and T is the absolute temperature in K. Usually ρw is

multiplied by 106 and expressed in g m-3.

At saturation, (1) is:

sw

w,sw,s TR

e=ρ , (2)

where the subscript s refers to saturation. The temperature variation of the

pressure at which the phase transition takes place is given by the Clausius-Clapeyron

equation which can be written for equilibrium over water and over ice. A large number

of equations were proposed to calculate the saturation vapor pressure over a surface of

liquid water or ice. Murphy and Koop (2005) review the vapor pressure of ice and

supercooled water for atmospheric applications. They conclude that all the commonly

used expressions for the vapor pressure of water and ice are very close to each other for

tropospheric temperature. In the present work, a formulation derived from the Magnus

Articles



90

equation, convenient for routine computation, is used (cf. Pruppacher and Klett 1997, p.

117, Eq. 4-83 and 4-84 and Appendix A-4-8). No enhancement factor (defined as the

ratio of the saturation vapor pressure of moist air to that of pure water vapor (eg. Buck

1981) is used. This factor is very small and depends on air pressure. It gives:

For water

( )⎥⎦

⎤⎢⎣

⎡−

−=

w

0w0w,sw,s BT

TTAexp)T(e)T(e , (3)

with es,w in hPa, T in K, es,w(T0) = 6.1070 hPa, T0 = 237.15 K,

Aw = 17.15 and Bw = 38.25 K.

For ice

( )⎥⎦

⎤⎢⎣

⎡−

−=

BiTTTA

exp)T(e)T(e 0i0i,si,s , (4)

with es,i in hPa, T in K, es,i(T0) = 6.1064 hPa, T0 = 273.15 K, Ai = 21.88 and Bi =

7.65 K.At the triple point, for Ts = T0, we have Pa154.611ee i,sw,s == ,with

3

w

s 1032.1Re

×= and 11w gKJ463.0R −−= .

Using (3) and (4) in (2) gives the vapor density at saturation over water and ice

respectively, namely, with the numeric coefficients:

( )⎥⎦

⎤⎢⎣

⎡−

−×=ρ

25.38T15.273T15.17

expT

1032.1

s

s

s

3w,s (5)

and

( )⎥⎦

⎤⎢⎣

⎡−

−×=ρ

65.7T15.273T88.21

expT

1032.1

s

s

s

3i,s (6)

with ρ in g m-3 and T in K.

2.2 Microwave radiometric measurements

The data were collected with a microwave radiometric profiler TP/WVP-3000

manufactured by Radiometrics. This radiometer is described in several papers (e.g.

Solheim and Godwin 1998, Ware et al. 2003, Liljegren 2004) and on the web site

Articles



91

113H113Hwww.radiometrics.com. In short, this instrument measures 12 radiometric brightness

temperatures through a sequential scan of 12 frequencies inside the microwave

spectrum, 5 in the K band, between 22.035 and 30.000 GHz, on the flank of the 22 GHz

water vapor absorption line, and 7 in the V band, between 51.250 and 58.800 GHz, on

the flank of the 60 GHz molecular oxygen complex. The bandwidth for each channel is

300 MHz. The beamwidth is 5 to 6 degrees at 22-30 GHz and 2 to 3 degrees at 51-59

GHz.

The radiometer also includes some in situ sensors for the ground level

measurement of temperature, pressure, and humidity, as well as a zenith looking infrared

radiometer with a 5° beamwidth and a liquid water (precipitation) detector. The

combination of the cloud base temperature provided by the IR sensor with the retrieved

temperature profile gives an estimate of the cloud base height. The cloud base height

can be associated with warm liquid or ice (e.g. cirrus), or with mixed clouds. The

combination of temperature and WVC enables the calculation of the relative humidity at

each level.

The 22-30 GHz channels are calibrated to 0.3 K rms by automated tipping

procedures. The 51-59 GHz channels are calibrated to 0.5 K RMS (Root Mean Square)

with a liquid nitrogen target (Han and Westwater 2000, Westwater et al. 2001). Liquid

nitrogen calibration is performed regularly (about each month) and when the radiometer

is moved.

The time cycle of the whole 12-channel scanning process is about 23 s. However

for the present study, the time interval between profiles is only 92 s, which means that

only one profile out of four was retained to reduce statistical redundancy.

Inputs include the 12 microwave brightness temperature measurements, one

infrared temperature, as well as temperature, humidity, and pressure at surface level,

representing 16 inputs. By inversion of the radiances measured at the different channels

through a neural network application, the radiometer retrieves, up to a height of 10 km,

the vertical profiles of temperature and water vapor, cloud base temperature and height,

vertically integrated water vapor (IWV) and liquid water (ILW). The neural network

profile retrieval is based on a training from historical series of local radiosondes

(Schroeder and Westwater 1991, Solheim et al. 1998, Liljegren 2004). Brightness

Articles



92

temperatures in the five K band channels are calculated with the revised absorption

model proposed by Liljegren et al. (2004).

2.3 Accuracy and resolution

The profiler retrieval accuracy was discussed and estimated from a comparison

with simultaneous radiosonde data (Güldner and Spänkuch 2001, Liljegren et al. 2001,

Ware et al. 2003, Liljegren 2004). Such comparisons are approximate due to spatial and

volume sampling differences between radiosonde and radiometer measurements. The

neural network method is found slightly less accurate than a statistical regression

method (see Güldner and Spänkuch, 2001, for the resulting vertical profiles of

accuracy). The information content of the measurements degrades with altitude. The

vertical resolution in retrieved temperature linearly changes from about 0.1 km near the

surface to ~ 6 km at a height of 8 km. For vapor density, the resolution falls from 0.3 km

near the surface to 1.5 km at a height of 6 km and 3.5 km at 10 km (Güldner and

Spänkuch 2001, Liljegren 2004).

Accuracy is degraded when there is liquid water on the radome of the radiometer

and in the presence of heavy precipitation above the radiometer, like with other

microwave radiometers. Retrievals are not applicable in this case. Light precipitation

aloft with a dry radome is acceptable. A field campaign was performed in March and

April 2007 at Aire-sur-Adour (Table 1 and Fig. 1) in order to compare the TP/WVP-

3000 radiometric profiler with radiosounding. The analysis of the results is out of the

scope of the present paper and will be discussed elsewhere. However, it can be briefly

stated that, for accuracy and resolution, the results for the TP/WVP 3000 radiometer are

in agreement with the conclusions of the previous studies quoted above.

Articles



93

Figure 1: Radiometer-observed time series of vertically integrated liquid water (ILW)

and water vapor (IWV), rain detection (binary), cloud base height, and surface air

temperature for 9 days, from 8 to 16 April 2004.

Figure 1 shows, as an example, the temporal series of ILW, IWV, binary

detection of rain presence at the surface, cloud base height as estimated by the IR

temperature combined to temperature profile, and air temperature measured at surface

Articles



94

level for 9 days, from 8 to 16 April 2004 at Toulouse (cf. Section 2.d for location). The

air temperature displays the usual diurnal variations with a trend towards warming from

day 4 to day 8. For day 9, the diurnal oscillation is very small because the air

temperature is dominated by a wide rainy system sweeping the site during the last 14

hours of the series.

In the presence of rain reaching the surface in Fig. 1, the microwave radiometer

is found not to work correctly. ILW, notably, displays peaks which always exceed 2 mm

(this threshold is never reached in the absence of rain), and the cloud base height is seen

near or at the surface. The IWV is less sensitive to the presence of rain. All that justifies

the removal of the profiles with ILW > 2 mm (cf. Section 2d). In clear sky, the IR

thermometer displays C8.49 °− , an arbitrary rounded down value meaning that there is

no cloud above. The height of the corresponding level can be seen to increase from day

4 to day 9. ILW and IWV are strongly correlated but neither of them is correlated with

surface air temperature. In other words, there is no significant diurnal variation of ILW

and IWV, which are dominated by the general circulation of the synoptic scale

structures.

To assess how the results presented in the present paper are biased by the

retrieval uncertainty, we need to know the RMS error vector affecting the retrieved

profiles. Because we do not know this vector, we have used the RMS error vectors

computed from a similar 12 channel microwave radiometer operated by the Atmospheric

Radiation Measurement (ARM) Program in Germany between March and December

2007, namely the error vector for "Black Forest Germany" available on the site

114H114Hhttp://www.archive.arm.gov. We believe that this error vector is representative for our

west European climatic area and convenient for the use done in the present paper. To

compute the RMS error on the retrievals, we have added to each retrieved profile a

Gaussian noise with zero mean and a standard deviation equal to the RMS error for each

layer.

2.4 Site and dataset

The data were collected at three different sites located in the South-West of

France during six periods distributed over the four seasons as specified in Fig. 2 and

Table 1. This dataset is considered to be representative of coastal oceanic mid-latitude

Articles



95

climate (e.g. Martyn 1992, see also arguments in Section 4). During the data acquisition

at the various sites, it rained for 7.5% of the time (the long term mean in this area is

8%), clouds were present 72% of the time and clear sky 28% of the time.

Figure 2: Map showing the location of the data collection sites and radiosounding

station (Bordeaux).

Articles

Distribution of tropospheric water vapor in clear and cloudy conditions from microwave radiometric profiling

96

Table 1. Dataset.

Subset location and designation Observation periods Duration

(day)

Profiles rejected

(%) Number of clear sky profiles Number of cloudy sky profiles

2007 March 13-15

2007 March 29-31 Aire-sur-l'Adour

2007 April 1-15

21 4.81 6166

32.85%

12603

67.15

2006 December 1-18 Toulouse 1

2007 January 1-20 38 9.31

6650

20.55%

25710

79.45%

2008 January 21-31

2008 February 1-29

2008 March 1-31 Toulouse 2

2008 April 01-06

77 9.72 23754

36.39%

41521

63.61%

2007 January 26-31 Lannemezan 1

2007 February 1-27 32 7.35

13970

50.18%

13869

49.82%

2007 April 24-30

2007 May 1-29 Lannemezan 2

2007 June 5-28

60 11.78 8131

16.36%

41572

83.64%

2007 July 2-31

2007 August 1-31

2007 September 1-30

2007 October 1-31

Lannemezan 3

2007 November 1-19

141 12.40 31384

27.06%

84597

72.94%

Total 369 10.55 90055

29.39%

219872

70.61%

Articles



97

Profile retrievals are obtained at 250 m intervals from the surface to 10 km,

which represents 41 levels (or range bins) including the surface level where temperature,

humidity, and pressure are measured by in situ sensors.

The training of the neural network was performed with 10 years of radiosonde

data (2 per day) from Bordeaux-Mérignac, a meteorological station located on the

Atlantic Coast (Fig. 2). Upper air data at Bordeaux-Mérignac are assumed to be

representative of the atmosphere over the coastal, oceanic, midlatitude Western Europe.

One profile, corresponding to one 23 s measurement cycle, was retained each 92

s, which gives 309927 profiles for the whole 369 days of the observing period. One

profile includes in fact the three 41 level basic profiles of temperature, WVC, and RH,

the two integrated values IWV and ILW, and the surface parameters. Profiles associated

with the presence of rain as detected by the rain detector of the radiometer were

removed, as well as the profiles associated with an ILW higher than 2 mm. ILW > 2 mm

is due to the presence of precipitation aloft that can bias the retrievals (Snider 2000).

The remaining profiles (89.45%) were classified in two subsets considering the cloud

base temperature (tbase in °Celsius) data from the IR radiometer, namely:

- clear sky profiles when tbase ≤ -49.8°C. 90055 profiles were included in this

subset.

- cloudy sky profiles, when tbase > -49.8°C, made up of 219872 profiles.

When a profile was removed, for example for not satisfying the ILW condition,

all the other data of the profile were rejected.

Differences between beamwidth of the IR radiometer and microwave channels

may cause a classification in clear category for profiles with microwave beam partly

filled with cloud. This case in encountered in the presence of a non uniform cloud cover,

notably with cumulus clouds, when clouds go into or out of the beam while they are

advected over the radiometer. The profiles with tbase ≤ -49.8°C and ILW higher than the

minimum detectable value of the radiometer, have thus been looked for. Less than 1% of

profiles were found to be concerned. They were classified as cloudy.

Mean vertical profiles of WVC and relative humidity as a function of

temperature, for clear and cloudy conditions, over the subset Toulouse 2 (winter), are

presented in Fig. 3 with the standard deviation bars. For vertical coordinate, temperature

Articles



98

was preferred to altitude because water vapor content depends only on temperature.

Using altitude as vertical coordinate blurs the representation of vertical profiles of WV

parameters. The upper limit of the profiles has been taken at -45°C because at colder

temperature WVC can fall below the minimum detectable which is about 0.02 g m-3. The

mean vertical profile of temperature versus altitude averaged over the dataset (almost

one year) is not presented because devoid of interest. In fact it shows a quasi linear

decrease of T from 10°C at the surface to -55°C at 10 km of altitude, that is the standard

atmospheric lapse rate of -6.5°C km-1. At a given altitude, temperature for cloudy

conditions is slightly warmer (about 0.5°C) than the one for clear conditions.

Figure 3 shows that the clear and cloudy profiles for WVC present a monotonic

decreases from 6-7 g m-3 near the surface to < 0.1 g m-3 at temperature < -40°C, with

some irregularities for temperature > 20°C. These irregularities (inversion of the vertical

gradient) are thought to be associated with the convective boundary layer structure.

Profiles of RH increase from the surface up to about 0°C then decreases above. This

variation with T can be seen as resulting from the combination of the effect of cloud

liquid water vertical distribution, which displays a maximum at mid tropospheric levels

(e.g. Mazin 1995, Gultepe and Isaac 1997), and of the monotonic decrease of WVC with

temperature.

Articles



99

Figure 3: Mean vertical profiles of water vapor content (WVC) and relative humidity

retrieved from the microwave radiometric profiler for clear and cloudy conditions for

the subset Toulouse 2 (2008 January 21 to 2008 March 31).

3 Water vapor content distribution

To analyse the WVC distribution, three datasets are considered: clear sky, cloudy

sky, and all – i.e. the sum of clear and cloudy sky subsets – in order to emphasize the

differences between clear and cloudy. Probability density functions (pdf) were

calculated by temperature class of width ΔT = 10°C, WVC bin of 0.2 g m-3, between 20

Articles



100

and -45°C, as was the fitting for two analytical forms: lognormal and Weibull. These

forms were selected among others from empirical tests and from physical and

bibliographical considerations. They are frequently used for atmospheric and cloud

characteristic representation because they are related to physical processes relevant for

atmospheric phenomena. The lognormal distribution is associated with the statistical

process of proportionate effects (see Aitchison and Brown 1966, p. 22, and Crow and

Shimizu 1988): the change in the variate at any step of the process is a random

proportion of the previous value of the variate. The lognormal form is found to be

convenient for many cloud characteristics such as rain cell size distributions (Mesnard

and Sauvageot 2003), rain rate distributions (Atlas et al. 1990, Sauvageot 1994),

precipitable water (Foster et al. 2006), and relative humidity (Soden and Bretherton

1993, Yang and Pierrehumbert 1994). The Weibull form is found convenient for

variables whose distribution is limited by extreme values, for example life variables or

wind when the velocity is limited by turbulence. For the WVC distribution the upper

limit is the vapor density at saturation ρs in the presence of condensation (cloudy case).

In clear air there is no upper limit.

The lognormal probability density function (pdf) is written:

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

σ

μ−−

σπ=σμ

2

y

y

yyy

y21exp

x21,;xf , (7)

where xny l= . The two parameters of the distribution are the mean μy and

variance 2yσ of y, defined as:

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛μσ

+μ=μ

2/12

x

xxy 1nl (8)

and

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛μσ

+=σ2

x

x2y 1nl . (9)

The Weibull pdf is:

Articles



101

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

λ−⎟

⎠⎞

⎜⎝⎛

λλ=λ

− k1k xexpxk,k;xf . (10)

The two parameters of the distribution are the scale λ and the shape k. The mean

μx and variance 2xσ are defined as:

( )k11x +Γλ=μ , (11)

and

( ) 222x k

21 μ−+Γλ=σ . (12)

Γ is the gamma function.

Distribution Variable Parameter x mean μx variance 2

xσ lognormal (logn) y = ln x mean μy variance 2

yσ Weibull (wbl) x scale λ shape k

Table 2. Definition of the parameters used in the present paper.

Figure 4 shows the probability density of WVC, data points, best fitting curves,

and error bars for 4 classes of temperature out of 12 (for the sake of clarity). The

numeric values of the fitting parameters and correlation coefficients are given in Table

3. For the clear case, lognormal and Weibull distributions are almost equivalent.

Lognormal is slightly better for T < -15°C and Weibull slightly better for T > -15°C. For

the cloudy case, Weibull is better than lognormal at temperatures between –20 and 10°C

and the two distributions are almost equivalent at T < -20°C and >10°C. It is the same

for all, which is dominated by the cloudy case. The Weibull fitting gives better results

because the WVC fluctuations have limits related with the condensation process when

the WVC reaches saturation.

Articles

Distribution of tropospheric water vapor in clear and cloudy conditions from microwave radiometric profiling

102

Clear Cloudy All Lognormal Weibull Lognormal Weibull Lognormal Weibull

Temperature class (°C) μy σy r λ K r μy σy r λ K r μy σy r λ k R [-45; -35] -2.50 0.51 1.00 0.10 2.19 1.00 -2.38 0.57 1.00 0.12 2.07 1.00 -2.42 0.55 1.00 0.12 2.08 1.00 [-40; -30] -2.05 0.49 0.99 0.16 2.18 0.81 -1.83 0.51 0.93 0.20 2.25 0.96 -1.91 0.51 0.87 0.19 2.17 0.93 [-35; -25] -1.62 0.49 0.98 0.25 2.11 0.99 -1.34 0.48 0.99 0.30 2.34 0.98 -1.43 0.50 0.99 0.30 2.20 0.99 [-30; -20] -1.18 0.52 0.99 0.40 2.04 0.97 -0.88 0.48 0.99 0.52 2.39 0.98 -0.98 0.51 0.99 0.48 2.20 0.98 [-25; -15] -0.73 0.54 0.99 0.63 2.01 0.98 -0.40 0.49 0.97 0.80 2.41 0.98 -0.55 0.52 0.97 0.74 2.21 0.98 [-20; -10] -0.27 0.56 0.98 1.00 2.02 0.97 -0.03 0.49 0.96 1.22 2.45 0.99 -0.12 0.53 0.96 1.14 2.25 0.98 [-15; -5] 0.20 0.56 0.95 1.59 2.15 0.98 0.42 0.49 0.94 1.91 2.55 0.99 0.34 0.52 0.94 1.79 2.35 0.99 [-10; 0] 0.62 0.52 0.88 2.37 2.48 0.98 0.90 0.45 0.82 3.02 2.99 0.97 0.81 0.50 0.85 2.80 2.68 0.98 [-5; 5] 0.99 0.46 0.89 3.29 2.92 0.99 1.28 0.36 0.85 4.19 3.85 0.99 1.19 0.42 0.85 3.93 3.34 0.99 [0; 10] 1.29 0.39 0.94 4.35 3.23 0.99 1.57 0.30 0.96 5.51 4.09 0.98 1.49 0.35 0.94 5.19 3.61 0.99 [5; 15] 1.51 0.33 0.98 5.27 3.59 0.98 1.81 0.28 0.99 6.99 3.86 0.96 1.73 0.32 0.98 6.58 3.51 0.97 [10; 20] 1.67 0.32 0.97 6.14 3.71 0.97 1.98 0.29 0.99 8.34 3.89 0.98 1.93 0.32 0.98 7.98 3.59 0.98

Table 3. Fitting parameters for the water vapor content pdf (Fig. 4). μy and σy are the mean and standard deviation of the lognormal

distribution, λ and k are the scale and shape of the Weibull distribution, respectively. r is the correlation coefficient. WVC unit is g m-3

Articles



103

The various coefficients of Table 3 are not very sensitive to the width of the

temperature classes ΔT. Similar results are obtained for ΔT up to 15 degrees but beyond

it degrades. On the other hand, the coefficients display large differences for clear and

cloudy cases.

Articles



104

Figure 4: Probability distribution function of WVC for 4 temperature classes between 10

and -30°C and Weibull best fitting for the three datasets: clear sky, cloudy sky, and all.

The corresponding numerical values are given in Table 3. The error bars were

computed with the ARM error vectors. Note that coordinate scales are different for left

and right parts.

Articles



105

Figure 5 upper part shows the variation as a function of temperature of the WVC

pdf Weibull parameters calculated for 24 temperature classes of width ΔT = 5°C,

between -45°C and 20°C. Figure 5 lower part displays the corresponding Fisher's

coefficients of skewness (γ1) and kurtosis (γ2). For γ1 > 0, skewness is on the left side

(the mode is lower than the average) and there is a tail on the right side. For γ1 < 0, it is

the reverse, that is to say skewness is on the right and tail on the left. γ1 and k convey

partly the same kind of information. For k < 2.6, the Weibull pdfs are positively skewed

(with a right side tail). For 2.6 < k < 3.6, skewness coefficient approaches zero (no tail),

that is to say pdf is quasi normal. For k > 3.6, Weibull pdfs are negatively skewed, and

the tail is on the left. It is what Fig. 5 shows. At cold temperature, WVC pdfs are

positively skewed, more for clear than for cloudy. While temperature increases,

skewness diminishes and WVC pdf evolves toward a normal shape. For temperature

between about -2°C and 12°C, k for cloudy pdfs is higher than 3.6, that is to say slightly

negatively skewed. Clear case is positively skewed everywhere. For T > 10°C, clear and

cloudy cases evolve toward a same value since the corresponding atmospheric level is

clear for the two cases (i.e. below cloud base). Kurtosis variations are small around the

normal value. Clear case is leptokurtic at cold temperature. Platykurtic character

increases with temperature up to about -5°C, then, at warmer temperature, a peak of

leptokurtic character correlated with skewness is observed.

Articles



106

Figure 5: Variation as a function of the temperature of the two parameters of the

Weibull distribution, shape k and scale λ, and of skewness and kurtosis for the pdf of

WVC calculated for 24 classes of temperature of width ΔT = 5°C between -45 and 20°C.

The error bars were computed with the ARM error vectors. For the sake of clarity, only

the error bars for the curves All are drawn. In the upper panel, horizontal lines at k =

2.6 and 3.6 show the domain where pdfs are quasi normal.

Articles



107

This evolution of the WVC Weibull pdf parameters with temperature can be

explained, at least in part, by considering the WVC limit of saturation with respect to

water and ice. In the absence of limit, WVC is lognormally distributed as most of

dynamically controlled atmospheric parameters (cf. Crow and Shimizu 1988). Clear

profiles are in this case. Limits linked to saturation create a truncation on the right side

of the pdfs and thus induces a reverse skewness. The bump observed on the k curve, for

-10°C < T < 15°C, corresponds tightly to the average liquid cloud water vertical

distribution associated with mid latitude cumulus clouds (e.g. Mazin 1993, Gultepe and

Isaac 1997).

Curves k(T) of Fig. 5 can be fitted with a general model of Gauss2 form, namely:

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −−+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

2

2

22

2

1

11 c

bTexpac

bTexpa)T(k (13)

with coefficients given in Table 4.

T > 0°C a1 b1 c1 a2 b2 c2 r clear 3.70 285 17.3 0.543 273 2.74 0.99 cloudy 3.11 274 8.21 3.68 288 11.7 0.99 all 1.91 273 7.58 3.51 288 15.1 0.97 T ≤ 0°C a1 b1 c1 a2 b2 c2 r clear 2.02 282 15.1 2.22 227 74.1 0.99 cloudy 4.71 287 13.6 2.42 252 50.2 0.99 all 1.59 279 9.6 2.23 256 100 0.99

Table 4. Fitting parameters for the shape curve of Fig. 5 (Eq. 13). r is the correlation

coefficient. Upper part T > 0°C, lower part T ≤ 0°C.

Curves λ(T) of Fig. 5 can be fitted very well with a general model similar to (5),

that is:

( )⎥⎦

⎤⎢⎣

⎡−

−=λ

3

33cT

15.273Tbexp

Ta

)T( (14)

with coefficients given in Table 5 and T in K.

Articles



108

T > 0°C a3 b3 c3 r clear 884 1.81 252 0.99 cloudy 1103 2.14 249 0.99 all 1039 2.43 245 0.99

T ≤ 0°C a3 b3 c3 r clear 913 10.8 124 0.99 cloudy 1147 16.8 54.9 0.99 all 1079 17.2 51.9 0.99

Table 5. Fitting parameters for the scale curve of Fig. 5 (Eq. 14). r is the correlation

coefficient. Upper part T > 0°C, lower part T ≤ 0°C.

Because it is more representative physically, the WVC mean, μWVC, has also been

plotted and fitted as shown Fig. 6. The coefficients of the fitting with a function of the

form (5), namely:

( )⎟⎠⎞

⎜⎝⎛

−−

=μCT

15.273TBexpTA)T(WVC , (15)

are given in Table 6. Of course these coefficients are not far from those of Table

5. In Fig. 6, the saturation curves ρs,w and ρs,i given by (5) and (6) are also drawn.

Articles



109

Figure 6: Variation as a function of the temperature of the mean WVC, of the ratio R =

WVC cloudy/WVC clear and best fitted curves. Curves ρsw and ρsi give the water vapor

density at saturation with respect to liquid water and ice respectively. The error bars

were computed with the ARM error vectors.

T > 0°C A B C r clear 787 1.76 253 0.99 cloudy 998 2.05 250 0.99 all 933 2.35 246 0.99

Articles



110

T ≤ 0°C A B C r clear 814 11.5 117 0.99 cloudy 1035 20.1 18.9 0.99 all 968 19.6 26.3 0.99

Table 6. Fitting parameters of the mean WVC for the curves of Fig. 6 (Eq. 15). r is the

correlation coefficient. Upper part T > 0°C, lower part T ≤ 0°C.

Curves of Figs 5 and 6 can be fitted acceptably with a single function for all the

temperature range considered in the present work (-40°C, +20°C). However, similarly

with the saturation water vapor density equations (5 and 6), a fitting slightly better is

obtained when considering separately the parts <0°C and >0°C. That is why Tables 4, 5,

and 6 present these two parts.

To sum up, the mean of WVC in clear and cloudy conditions depends only on the

temperature and is described accurately by (15) with the coefficients of Table 6.

From (15), using the coefficients of Table 6, the ratio of the mean of WVC for

cloudy and clear conditions can be written:

clearWVC

cloudyWVCRμ

μ= 27.1T0061.0 += for T > 0°C (16)

27.1T0007.0 += for T ≤ 0°C.

with T temperature in °Celsius.

Using (13) and (14) in (10) enables the writing of the WVC pdf as a function of

the sole temperature.

4 Distribution of IWV

As for WVC, pdf of vertically integrated WVC (IWV) was calculated and fitted

with lognormal and Weibull distributions for clear, cloudy and all datasets. Curves are

shown in Fig. 7. Fitting coefficients and correlation coefficients are given in Table 7.

Articles



111

For clear conditions, the lognormal distribution gives a correlation coefficient slightly

better than the Weibull one, but for cloudy and for all, the Weibull distribution is the

best. Using a wide radiosounding and GPS (Global Positioning System) dataset, Foster

et al. (2006) found that precipitable water (that is IWV) distribution can be well fitted

with a lognormal distribution. They emphasize that, for tropical oceanic environments

IWV tends to exhibit negatively skewed histograms for which fitting with a reverse

lognormal distribution is proposed.

Articles



112

Figure 7: Pdf of vertically integrated water vapor for the three datasets and lognormal

and Weibull best fitting.

Articles



113

The numerical values are given in Table 7.Using 15 years of ECMWF (European

Centre for Medium-range Weather Forecast) reanalysis meteorological database ERA

15, the Weibull parameters k and λ for the IWV distribution were calculated. Isolines of

k and λ over the western European area, where the radiometric data were collected (as

shown in Fig. 2), are presented in Fig. 8. The fields of k and λ values are rather

homogeneous showing the validity of our assumption of Section 2d that this area can be

considered as a single climatological entity for WVC distribution. Averaged values over

the area represented in Fig. 8 are k = 2.5 and λ = 1.9 which compare very well with the

values for all of Table 7, that is k = 2.8 and λ = 1.9.

lognormal Weibull

μ σ r λ k r

clear 0.10 0.34 0.93 1.30 3.45 0.92

cloudy 0.59 0.34 0.97 2.12 3.44 0.98

all 0.45 0.41 0.96 1.90 2.86 0.98 Table 7. Fitting parameters for the IWV curve of Fig. 7. r is the correlation coefficient.

This agreement can be seen as supporting the validity of the IWV radiometric

measurements. In a companion paper analysing the world wide IWV distribution from

the ERA 15 database (Jeannin et al. 2008), we found that k for tropical environment is

negatively skewed, what agrees with Foster et al. (2006) results. For latitude lower than

about 10°, k is higher than 6 with peak at 10 in some very humid equatorial areas (e.g.

Amazonia and Indonesia).

Articles



114

Figure 8: Isolines of the Weibull distribution parameters for the fitting table pdf of IWV

calculated from ERA 15.

Articles



115

5 Comparison with operational radiosonde soundings

Operational radiosonde soundings (RS) provide data on the vertical WVC

distribution along slanted trajectories two times a day, at 00 and 12 UTC. It is not

possible to distinguish clear and cloudy vertical RS profiles in the same way that we do

with the radiometric profiles. Besides, the RS sampling frequency does not permit to

capture some important aspect of the WVC distribution related with the WVC diurnal

cycle. However a rough comparison can be made between the radiometer retrieved

WVC pdfs and profiles for the All case and those provided by a radiosonde dataset. To

perform this comparison, we have used the two daily RSs gathered at Bordeaux during

the period of 2006, 2007, and 2008 when the radiometer was active (Table 1). The RS

data considered for the statistic are taken over the altitude domain cover by the

radiometer. For example, for the period when the radiometer was operated from

Lannemezan (altitude 600 m), the RS data from Bordeaux (altitude 50 m) are taken

above 600 m.

All Lognormal Weibull

Temperature class (°C) μy σy r λ k r

[-45; -35] -2.43 0.52 1.00 0.11 2.33 1.00 [-40; -30] -2.10 0.61 1.00 0.16 1.91 1.00 [-35; -25] -1.59 0.61 0.90 0.27 1.96 0.98 [-30; -20] -1.13 0.63 0.94 0.43 1.93 0.99 [-25; -15] -0.67 0.62 0.91 0.68 2.01 0.98 [-20; -10] -0.25 0.62 0.91 1.03 2.00 0.97 [-15; -5] 0.17 0.63 0.89 1.60 1.98 0.95 [-10; 0] 0.60 0.63 0.80 2.45 2.06 0.91 [-5; 5] 1.01 0.60 0.66 3.60 2.30 0.88 [0; 10] 1.38 0.56 0.66 5.08 2.52 0.91 [5; 15] 1.60 0.50 0.68 6.66 2.83 0.94

[10; 20] 1.90 0.44 0.77 8.13 3.24 0.97 Table 8. Same as Table 3 calculated from the radiosonde dataset for the All case.

Table 8 gives the coefficients of the WVC pdfs calculated from the RS data and

fitted with lognormal and Weibull functions. The Weibull fitting is clearly very good

Articles



116

and better than the lognormal one. Coefficients λ and k of the RS pdfs versus

temperature are represented in Fig. 9.

Figure 9: The dots show the variation as a function of the temperature of the two

parameters of the Weibull distribution, shape k and scale λ, and of the skewness and

kurtosis for the pdf of WVC calculated from the RS dataset for 24 classes of temperature

of width ΔT = 5°C between -45 and 20°C. The curves and error bars are the radiometer

retrievals for the All case redrawn from Fig. 5 for comparison with the RS values.

Articles



117

In Fig. 9, we have redrawn the curves of Fig. 5 for the All case with their error

bars and added the values calculated form the RS data. Scale λ from RS is very close to

the radiometer retrieval one. Shape k from RS is, as the radiometric one, almost constant

and close to the radiometric values for T < 5°C. For T > 5°C it is lower than the

radiometric values but enclosed between 2.6 and 3.6, that is quasi normal. Skewness and

kurtosis also are very similar for RS and radiometer.

Fig. 10 shows the WVC mean versus temperature. In Fig 10 the curve of Fig. 6

for All with the error bars have been redrawn and the values calculated from RS have

been superimposed. A detailed discussion of the differences between the radiometric

retrievals and the RS values is out of the scope of the present paper. That is why by way

of conclusion we limit to note that the agreement for the pdf shape and the mean WVC

variation with temperature is rather good.

Articles



118

Figure 10: The dots show the variation as a function of the temperature of the mean

WVC calculated from the RS dataset for 24 classes of temperature of width ΔT = 5°C

between -45 and 20°C. The curves and error bars are the radiometer retrievals for the

All case redrawn from Fig. 6.

Articles



119

6 Summary and conclusions

The distribution of the vapor phase of tropospheric water was studied from 369

days of observations collected in various sites of Western Europe with a 12 frequency

microwave radiometric profiler. Data were regrouped in two subsets, clear sky and

cloudy sky. The profiles associated with the presence of rain were rejected.

Several forms of distribution were considered to fit the pdf of the atmospheric

water vapor profiles observed with the profiler. For the sake of clarity, only lognormal

and Weibull distributions were discussed in the present paper because they proved to be

the most efficient for the pdf fitting. The lognormal function is associated with the

statistical process of proportionate effect, frequently identified in the atmosphere. The

Weilbull function is convenient for random variables whose distributions are limited by

extreme values such as water vapor with condensation.

The WVC pdf inside 10°C temperature classes between 20 and -45°C are found

to be accurately fitted by Weibull distributions. The two parameters of the Weibull

distribution, the shape k and scale λ, are shown to be well described by analytical

functions of temperature t. Using k(T) and λ(T) in the pdf enables the writing of the

WVC spatial distribution as a function of T. The mean WVC vertical profile can be

represented by a function of the temperature of the same form than the Clausius-

Clapeyron equation. The ratio of mean WVC for cloudy to mean WVC for clear

conditions is linearly dependent on T, with a mean value of 1.27.

The pdf of the vertically integrated WVC, or precipitable water, is found to be

Weibull distributed rather than lognormal. The values retrieved from the microwave

radiometric profiler compare very well with the ones calculated from the ERA 15

reanalysis meteorological database.

A rough comparison between the radiometric retrievals and the values provided

by an operational radiosonde sounding dataset shows that the agreement for the pdf

shape and the mean WVC profile as a function of temperature is fairly good.

Articles



120

Acknowledgements

The authors would like to thank all the contributors to the radiometer dataset,

notably Mrs Edith Gimonet and Mr Giulio Blarzino from ONERA (Office National

d'Etudes et de Recherche Aérospatiales). Thanks are also due to one anonymous

reviewer for helpful comments and suggestions and to the ARM Program for providing

the radiometer error vector used in the present paper.

References

Aitchison, J., and J.A. Brown, 1966: The lognormal distribution. Cambridge

University Press, 176 pp.

Atlas, D., D. Rosenfeld, and D.A. Short, 1990: The estimation of convective

rainfall by area integrals: 1. The theoretical and empirical basis. J. Geophys. Res., 95

(D3), 2153-2160.

Bianco, L., D. Cimini, F.S. Marzano, and R. Ware, 2005: Combining microwave

radiometer and wind profiler radar measurements for high-resolution atmospheric

humidity profiling. J. Atmos. Oceanic Technol., 22, 949-965.

Brooks, D.R., F.M. Mims III, and R. Roettger, 2007: Inexpensive near-IR sun

photometer for measuring total column water vapor. J. Atmos. Oceanic Technol., 24,

1268-1276.

Buck, A.L., 1981: New equations for computing vapor pressure and enhancement

factor. J. Appl. Meteor., 20, 1527-1532.

Chahine, M.T., 1992: The hydrological cycle and its influence on climate.

Nature, 359, 373-380.

Crow, E.L., and K. Shimizu, 1988: Lognormal distributions. CRC Press, Boca

Raton, Fla., 387 pp.

Ferrare, R.A., and coauthors, 2004: Characterization of upper-troposphere water

vapor measurements during AFWEX using LASE. J. Atmos. Oceanic Technol., 21,

1790-1808.

Articles



121

Foster, J., M. Bevis, and W. Raymond, 2006: Precipitable water and the

lognormal distribution. J. Geophys. Res., 111, D15102, doi: 10.1029/2005JD006731.

Furumoto, J.I., K. Kurimoto, and T. Tsuda, 2003: Continuous observations of

humidity profiles with the MU radar-RASS combined with GPS or radiosonde

measurements. J. Atmos. Oceanic Technol., 20, 23-41.

Furumoto, J.I., S. Imura, T. Tsuda, H. Seko, T. Tsuyuki, and K. Saito, 2007: The

variational assimilation method for the retrieval of humidity profiles with the wind-

profiling radar. J. Atmos. Oceanic Technol., 24, 1525-1545.

Gaffen, D.J., and W.P. Elliott, 1993: Column water vapor content in clear and

cloudy skies. J. Climate, 6, 2278-2287.

Güldner, J., and D. Spänkuch, 2001: Remote sensing of the thermodynamic state

of the atmospheric boundary layer by ground-based microwave radiometry. J. Atmos.

Ocean. Tech., 18, 925-933.

Gultepe, I., and G.A. Isaac, 1997: Liquid water content and temperature

relationship from aircraft observations and its applicability to GCMs. J. Climate, 10,

446-452.

Han, Y., and E. Westwater, 2000: Analysis and improvement of tipping

calibration for ground-based microwave radiometers. IEEE Trans. Geosci. Remote Sens.,

38, 1260-1276.

Hogg, D.C., F.O. Guiraud, J.B. Snider, M.T. Decker, and E.R. Westwater, 1983:

A steerable dual-channel microwave radiometer for measurement of water vapor and

liquid in the troposphere. J. Climate Appl. Meteor., 22, 789-806.

IPCC (Intergouvernmental Panel on Climatic Change), 2001: Climate change:

synthesis report. Cambridge University Press, London, UK, 700 pp.

Jeannin, N., L. Féral, H. Sauvageot, and L. Castanet, 2008: Statistical distribution

of integrated liquid water and water vapor content from meteorological reanalysis. IEEE

Transactions on Antenna and Propagation. In revision.

Kidder, S.Q., and A.S. Jones, 2007: A blended satellite total precipitable water

product for operational forecasting. J. Atmos. Oceanic Technol., 24, 74-81.

Korolev, A., and G.A. Isaac, 2006: Relative humidity in liquid, mixed-phase, and

ice clouds. J. Atmos. Sci., 63, 2865-2880.

Articles



122

Liljegren, J.C., B. Lesht, E.E. Clotiaux, and S. Kato, 2001: Initial evaluation of

profiles of temperature, water vapor and cloud liquid water from a new microwave

profiling radiometer. Proc. 5th Symp. Integrated Observing Systems, Albuquerque, NM,

Amer. Meteor. Soc. CD-ROM, 4.7.

Liljegren, J.C., S.A. Boukabara, K. Cady-Pereira, and S.A. Clough, 2004: The

effect of the half-width of the 22 GHz water vapor line on retrievals of temperature and

water vapor profiles with a twelve-channel microwave radiometer. IEEE Trans. on

Geos. and Remote Sensing. 10, 1-7.

Liou, Y.A., Y.T. Teng, T. Van Hove, and J.C. Liljegren, 2001: Comparison of

precipitable water observations in the near tropics by GPS, microwave radiometer, and

radiosondes. J. Appl. Meteor., 40, 5-15.

Liu, W.T., W. Tang and P.P. Nüler, 1991: Humidity profiles over the ocean. J.

Climate, 4, 1023-1034.

Martyn, D., 1992: Climates of the world. Elsevier. Developments in Atmos. Sci.,

18, 435 pp.

Mazin, I.P., 1995: Cloud water content in continental clouds of middle latitudes.

Atmos. Res., 35, 283-297.

Meijgaard, E. van, and S. Crewell, 2005: Comparison of model predicted liquid

water path with ground-based measurements during CLIWA-NET. Atmos. Res., 75, 201-

226.

Mesnard, F., and H. Sauvageot, 2003: Structural characteristics of rain fields. J.

Geophys. Res., 108 (D13), 4385, doi: 10.1029/2002JD002808.

Miloshevich, L.M., H. Vömel, D.N. Whiteman, B.M. Lesht, F.J. Schmidlin, and

F. Russo, 2006: Absolute accuracy of water vapor measurements from six operational

radiosonde types launched during AWEX-G, and implications for AIRS validation. J.

Geophys. Res., 111, D09S10, doi: 10.1029/2005JD006083.

Murphy, D.M., and T. Koop, 2005: Review of the vapour pressures of ice and

supercooled water for atmospheric applications. Q.J.R. Meteor. Soc., 131, 1539-1565.

Peixoto, J.P., A.H. Oort, and E.N. Lorenz, 1992: Physics of climate. American

Institute of Physics, New-York, 520 pp.

Pruppacher, H.R., and J.D. Klett, 1997: Microphysics of clouds and precipitation.

Kluwer Academic Publishers, 954 pp.

Articles



123

Rocken, C., T. Van Hore, and R. Ware, 1997: Near real-time GPS sensing of

atmospheric water vapor. Geophys. Res. Lett., 24, 3221-3224.

Sakai, T., T. Nagai, M. Nakazato, T. Matsumura, N. Orikasa, and Y. Shoji, 2007:

Comparison of Raman lidar measurements of tropospheric water vapor profiles with

radiosondes, hygrometers on the meteorological observation tower, and GPS at Tsukuba,

Japan. J. Atmos. Oceanic Technol., 24, 1407-1423.

Sauvageot, H., 1994: The probability density function of rain rate and the

estimation of rainfall by area integrals. J. Appl. Meteor., 33, 1255-1262.

Schroeder, J., and E. Westwater, 1991: Users's guide to WPL microwave

radiative transfer software. NOAA Technical Memorandum ERL WPL-213.

Sherwood, S.C., E.R. Kursinski, and W.G. Read, 2006: A distribution law for

free-tropospheric relative humidity. J. Climate, 19, 6267-6277.

Snider, J.B., 2000: Long-term observations of cloud liquid, water vapor, and

cloud-base temperature in the North Atlantic Ocean. J. Atmos. Oceanic Technol., 17,

928-939.

Soden, B.J., and F.P. Bretherton, 1993: Upper tropospheric relative humidity

from the GOES 6.7 μm channel: method and climatology for July 1987. J. Geophys.

Res., 98, D9, 16,669-16,688.

Soden, B.J., and I.M. Held, 2006: An assessment of climate feedbacks in coupled

ocean-atmosphère models. J. Climate, 19, 3354-3360.

Solheim, F., J. Godwin, E. Westwater, Y. Han, S. Keihm, K. Marsh, and R.

Ware, 1998: Radiometric profiling of temperature, water vapor, and liquid water using

various inversion methods. Radio Sci., 33, 393-404.

Solheim, F., and J. Godwin, 1998: Passive ground-based remote sensing of

atmospheric temperature, water vapor, and cloud liquid profiles by a frequency

synthesized microwave radiometer. Meteor. Z., 7, 370-376.

Spichtinger, P., K. Gierens, H.G.J. Smit, J. Ovarlez, and J.F. Gayet, 2004: On the

distribution of relative humidity in cirrus clouds. Atmos. Chem. Phys., 4, 639-647.

Stone, E.M., L. Pan, B.J. Sandor, W.G. Read, and J.W. Waters, 2000: Spatial

distributions of upper tropospheric water vapor measurements from UARS microwave

limb sounder. J. Geophys. Res., 105, 12149-12161.

Articles



124

Van Baelen, J., J.P. Aubagnac, and A. Dabas, 2005: Comparison of near-real

time estimates of integrated water vapor derived with GPS, radiosondes, and microwave

radiometer. J. Atmos. Oceanic Technol., 22, 201-210.

Ware, R., R. Carpenter, J. Güldner, J. Liljegren, T. Nehrkorn, F. Solheim, and F.

Vandenberghe, 2003: A multi-channel radiometric profiler of temperature, humidity and

cloud liquid. Radio Sci., 38, 8079, doi: 10.1029/2002 RS002856.

Webster, P.J., 1994: The role of hydrological processes in ocean-atmosphere

interactions. Rev. Geophys., 32(4), 427-476.

Westwater, E.R., Y. Han, M. Shupe, and S. Matrosov, 2001: Analysis of

integrated cloud liquid and precipitable water vapor retrievals from microwave

radiometers during the Surface Heat Budget of the Artic Ocean Project. J. Geophys.

Res., 106, 32,019-32,030.

Westwater, E.R., S. Crewell, and C. Mätzler, 2004: A review of surface-based

microwave and millimetre-wave radiometric remote sensing of the troposphere. Radio

Sc. Bull., 310, 59-80.

Whiteman, D.N., I. Veselovskii, M. Cadirola, K. Rush, J. Comer, J.R. Potter, and

R. Tola, 2007: Demonstration measurements of water vapor, cirrus clouds, and carbon

dioxide using a high-performance Raman lidar. J. Atmos. Oceanic Technol., 24, 1377-

1388.

Yang, H., and R.T. Pierrehumbert, 1994: Production of dry air by isentropic

mixing. J. Atmos. Sci., 51, 3437-3454.

Zhang, C., B. Mapes, and B.J. Soden, 2003: Bimodality in tropical water vapour.

Q. J. R. Meteor. Soc., 129, 2847-2866.

Articles

Weather radar data for site diversity predictions and evaluation of the impact of rain field

advection

125

Weather radar data for site diversity predictions and evaluation of the impact of rain field advection

Lorenzo Luini(1), Nicolas Jeannin(2), Carlo Capsoni(1), Aldo Paraboni(1), Carlo

Riva(1), Laurent Castanet(2), Joël Lemorton(2) (1)DEI – Dipartimento di Elettronica e Informazione

Politecnico di Milano

Milano, Italy (2)DEMR – Département de Électromagnétisme et radar

ONERA

Toulouse, France

Manuscript submited to International Journal of Satellite Communication

Abstract

This paper presents the analysis on two weather radar datasets, collected at Spino

d’Adda (Italy) and Bordeaux (France), for the simulation and the performance

evaluation of a site diversity system. Results from the two locations are compared and

the impact of different factors such as the baseline and link orientation is assessed and

related to the local climatologic characteristics. The results obtained are then compared

with the model currently recommended by the ITU-R for the estimation of the site

diversity performance. A linkage is then established between the preferable baseline

orientation and the predominant direction of the rain field advection. The rain field

displacement is finally shown to be well approximated by the wind speed and direction

relative to the 700 hPa isobar extracted from the ECMWF ERA-40 meteorological

database.

Articles


advection

126

1 Introduction

The evolution of telecommunication systems is always pushing towards the use

of higher transmission frequencies owing to the congestion of lower bands, to the

always increasing demand of bandwidth from the users and, sometimes, to the limits

imposed on the system dimension (especially on the antenna). As widely known,

telecommunication systems operating at Ka, Q and V bands (roughly from 20 to 50

GHz) are strongly affected by attenuation phenomena due to atmospheric impairments

[Riva, 2004]. Still, they are expected to provide real-time multimedia services, and

consequently, to be reliable and guarantee the desired system availability. In this

environment, strong signal fades can no longer be overcome making use of static power

margins, but require the application of Fade Mitigation Techniques (FMTs) as a viable

solution [Fukuchi and Saito, 2004]. To this end, telecommunication systems based on

site diversity can be envisaged [Goldhirsh et al., 1997]: in fact, the use of multiple

receiving stations permits to take advantage of the spatial variability of rain, so that a

distance of the order of tens of kilometers between the stations significantly reduces the

probability that both stations are undergoing an outage and, therefore, that the system is

unavailable [Capsoni et al., 2007]. The design of a site diversity system (of FMTs in

general), however, requires the evaluation of the advantages (for example in terms of

outage probability) deriving from the implementation of such a countermeasure to signal

fades. For that purpose, weather radar data, where available, are the most suitable source

of information, as they inherently reflect the influence of the local climatology and

topography on the rain field spatial distribution, which, on turn, determines the

effectiveness of a site diversity solution [Goldhirsh and Robison, 1975].

This paper presents the analysis performed on two weather radar datasets,

collected at Spino d’Adda (Italy) and Bordeaux (France), for the simulation and the

performance evaluation of a site diversity system with tunable characteristics. After a

brief description of the two radar datasets, the site diversity gain is assessed for the two

locations and, afterwards, the impact of different factors such as link and baseline

orientation is investigated. Moreover, as expected, the study clearly shows the

dependence of the system performance on the type of rain event (stratiform or

convective) affecting the stations. The comparison with the model currently

Articles


advection

127

recommended by the ITU-R for the estimation of the site diversity performance is shown

for both locations. Afterwards the local climatologic features, namely, the predominant

direction of the rain fields advection, are linked to the trends observed for different

baselines orientation. To this end, a comparative analysis is devised between the rain

advection speed and direction and the wind outputs extracted from the ERA-40 ECMWF

(European Center for Medium-range Weather Forecast) database [Uppala et al., 2005].

Specifically, the analysis strives at determining whether ERA-40 wind data are suitable

to determine the most advantageous baseline direction for a site diversity system, as

well as whether they are adequate to parameterize global space-time rain models such as

those proposed in [Gremont and Filip 2004] and [Jeannin et al., 2007]. Lastly, a

preliminary analysis is presented to assess the linkage between the rain field advection

direction and its spatial anisotropy.

2 The weather radar datasets

The two radar datasets utilized in this study have been collected in two temperate

sites, namely Spino d’Adda, Italy (45.4° N, 9.5° E) and Bordeaux (44.5° N, -0.34° E).

They have the following characteristics:

• Spino d’Adda: approximately 15000 CAPPI (Constant Altitude Plane

Position Indicator) radar images, extracted from rain events (in the period

from 1988 to 1992) which have proven to be fully representative of the

local yearly rainfall statistics [Capsoni et al., 2008]. The maximum

operational range of the radar considered in this study is 40 km in order to

avoid the inclusion of clutter pixels due to the surrounding mountains. The

spatial resolution of the radar scans is 0.5 km × 0.5 km and the interval

between consecutive images is 77 seconds.

• Bordeaux: approximately 30000 CAPPI radar images, extracted from one

year (1996) of continuous operation (also non-rainy images have been

recorded). The maximum operational range of the radar is 100 km, the

spatial resolution of the radar scans is 1 km × 1 km and the interval

between consecutive images is 5 minutes.

Articles


advection

128

Despite some differences between the two radar datasets (period of data

acquisition, number of images at disposal, spatial and temporal resolution), the work

presented in this paper suggests that they permit to derive fully comparable results,

obviously as long as the same image dimension is considered. For that purpose, a

circular area with a 40-km radius has been selected in the South-Western portion of each

original Bordeaux radar scan, as it is a flat zone, over which reliable data, seldom

affected by clutter and by anomalous propagation effects, are obtained.

3 Simulation of site diversity systems using radar images

The radar derived rainfall images have been used for the simulation of an Earth-

satellite site diversity system with tunable characteristics. Both in Spino d’Adda and in

Bordeaux, the rainfall snapshots have been converted into maps of attenuation

experienced by a radio link pointing to a geostationary satellite, under a fixed yearly

mean rain height Hr, derived from the ITU-R Rec. P.839-3 [ITU-R, 2001]. The path

attenuation A has been calculated through the numerical integration of:

dllRkAL∫= α)( [dB] (1)

where L = Hr/sin(θ) is the path length affected by rain, θ is the link elevation, k

and α, provided by the ITU-R Rec. P.838-3 [ITU-R, 2003], are rain-to-attenuation

conversion coefficients that depend on the link elevation and on the radio wave

frequency and polarization (always vertical in this work). Obviously, R(l), indicating the

rain intensity value at position l impairing the transmission link, is strongly dependent

on the local precipitation characteristics and therefore is expected to be tightly bound to

the system performance. In this work, all attenuation maps have been calculated at the

reference frequency of 30 GHz, although results could be easily extended to different

frequencies and link geometries with a little effort.

The gain G offered by a two-site diversity system, with separation D between the

stations, has been calculated as:

),()(),( PDAPAADG jss −= [dB] (1)

Articles


advection

129

where As and Aj are the attenuation values of the cumulative distribution functions

(CDFs) (both for the same probability level P), respectively relative to a single station

and to a two-site diversity system (depending on D), for which the minimum attenuation

value is always selected (see Figure 1).

0 5 10 15 20 25 30 35 40

10-1

100

Attenuation [dB]

Per

cent

age

[%]

Spino d'Adda - f = 30 GHz - distance = 8 km

Single-site attenuation CDFJoint two-site attenuation CDF (D = 8 km)

PG

AsAj

Figure 1 Example of site diversity gain (Spino d’Adda, 30 GHz, D = 8 km).

The calculation of G has been performed according to the following options:

• reference attenuation level As: from 4 to 32 dB with a 4-dB step, so that the minimum

percentage level of the single-site CDF is approximately 10-1 % at 30 GHz;

• distance D between the receiving stations: from 4 to 56 km in 4-km step;

• link orientation φ , defined as in Figure 2 : -20° (towards the East), 0° (coinciding with

the South) and 20° (towards the West). Each value of φ corresponds to a rain-to-

attenuation conversion procedure, as different rain rates R(l) affect the link. For a

geostationary satellite, the link elevation θ is univocally defined by φ ;

• baseline orientations, defined as in Figure 3: horizontal (H), vertical (V) and 45° (HV)

with respect to the horizontal direction;

• type of rain event: stratiform or convective.

Articles


advection

130

W E

S

N

θ

φ

L

W E

S

N

θ

φ

L

Figure 2: Geometry of the link (indicated by the red solid line); θ is the elevation angle,

is the orientation angle, L is the path length affected by rain

45°

D

DDW E

N

S

45°

D

DDW E

N

S

Figure 3: Geometry of the baseline between the receiving stations (dashed lines): horizontal (H),

vertical (V) and 45° (HV) from the West-East direction. The blue solid line indicates the reference

link, with respect to which site diversity statistics are calculated

Considering all maps, the single-site CDF has been calculated starting from the

attenuation value relative to the reference pixel (blue link in Figure 3), whereas the two-

site joint attenuation CDF (for given D, θ, φ ) has been obtained from the minimum

attenuation value between the one experienced by the reference pixel and the one

relative to the associated site diversity link. In order to improve the statistic robustness

of the analysis and to prevent results from being dependent only on the climatologic

and/or topographic peculiarities of specific areas, attenuation CDFs consist of all the

values obtained by moving the stations pattern depicted in Figure 3 across the whole

map.

Articles


advection

131

3.1 Gain dependence on the link orientation

In this section, the dependence of the system performance on the link orientation

is analyzed. 227H

lists the link characteristics as a function of φ , for both sites. Figure 4 shows the

site diversity gain comparison in Spino d’Adda, for the three selected values of φ : -20°,

0°, 20°. In this case, the baseline orientation is horizontal, but very similar results

(omitted here for brevity) have been obtained also for baselines V and HV (the same

applies to Bordeaux): in all cases, the dependence of the system gain on the link

orientation is clearly negligible. Such conclusion is a positive one, if a real system is

concerned: in fact, the link orientation usually is not a design parameter, as it obviously

depends on the relative position between the satellite and the receiving stations (e.g.,

broadcasting systems). Thus, a system designer could disregard the impact of φ in

estimating the system performance through prediction methods.

θ [°] Hr [km] L [km] φ [°]

Bordeaux Spino Bordeaux Spino Bordeaux Spino -20 34.6 33.5 2.86 3.3 5 5.9 0 38.7 37.7 2.86 3.3 4.6 5.3 20 34.6 33.5 2.86 3.3 5 5.9

Table 1: Elevation angles and rain heights relative to the considered links in Spino d’Adda and

Bordeaux.

Articles


advection

132

0 10 20 30 40 50 600

5

10

15

20

25

30

Distance [km]

Gai

n [d

B]

Spino d'Adda


Figure 4 : Spino d’Adda, site diversity gain comparison for different link orientations:

-20° (solid lines), 0° (dashed lines with circles), 20° (dashed lines with crosses). The

baseline orientation is horizontal.

3.2 Gain dependence on the baseline orientation

If the dependence of G on φ is negligible (therefore, hereinafter, φ will always

be set to 0°), the same conclusion can not be drawn for the baseline orientation. Figure 5

depicts the comparison of the system gain, respectively for Spino d’Adda and Bordeaux,

both for H (solid lines) and V (dashed lines with circles) baseline orientations. The gain

results relative to the HV baseline orientation have been omitted as they are less

interesting: in both cases, they are comprised between the ones relative to the horizontal

and the vertical baselines. Figure 5 clearly points out how the choice of the system

baseline may affect the achievable site diversity gain. Specifically results show that for

Bordeaux, an horizontal baseline between the stations would improve G (w.r.t an

horizontal baseline), whereas the opposite is observed for Spino d’Adda, where a

vertical baseline would assure a better performance. It is worth noting that for D = 4 km,

in both cases, a horizontal baseline provides a higher gain. In fact, making reference to

the geometry in Figure 3, when the links point towards the South, the correlation

between the associated attenuation values is higher (and therefore, the gain is obviously

lower) in the case of a vertical baseline because links tend to overlap. For longer values

Articles


advection

133

of D, this explanation no longer holds as attenuation values get uncorrelated after some

tens of kilometers. The local preferable baseline orientation can be roughly linked to the

prevalent elongation direction of the rain field. In fact, a visual inspection of radar data

suggests that, most of the time, rain structures tend to stretch orthogonally to the rain

field advection direction. More details about this effect will be given in section 4, which

also suggests how to derive the preferential direction of the rain field advection from the

ERA-40 database.

0 10 20 30 40 50 600

5

10

15

20

25

30

Distance [km]

Gai

n [d

B]

Spino d'Adda


0 10 20 30 40 50 600

5

10

15

20

25

30

Distance [km]

Gai

n [d

B]

Bordeaux


Figure 5 : Site diversity gain comparison for different baselines: H (solid lines) and V (dashed lines

with circles). φ = 0°.

In order to quantify such difference, let us define the following error figure:

s

sSNsEWsbl A

ADGADGAD ),(),(100),( −− −=ε (3)

Articles


advection

134

where GW-E and GN-S are the system gains, respectively for the W-E and the N-S

baseline, relative to the stations distance D and to the reference single-site attenuation

As. 228HFigure depicts the trend of blε with D, calculated as the average value of blε relative

to all the As values at the same distance D.

0 10 20 30 40 50 60-6

-4

-2

0

2

4

6

8

Distance [km]

E[ ε

bl] [

%]

Spino d'AddaBordeaux

Figure 6:Trend of the average value of blε relative to all the As values at the same

distance D (both for Spino d’Adda and Bordeaux). φ is set to 0°

Both Figure 5 and 229HFigure clearly point out how the choice of the system baseline

may affect the achievable gain. Specifically results show that for Bordeaux, a W-E

baseline between the stations would improve G, whereas the opposite is true for Spino

d’Adda, where a N-S baseline would assure a better performance. It is worth noting that

for D = 4 km, in both cases, a W-E baseline provides a higher gain. In fact, making

reference to the geometry in Figure 3, when the links point towards the South, the

correlation between the associated attenuation values is higher (and therefore, the gain is

obviously lower) in the case of a N-S baseline because links tend to overlap. For higher

values of D, this effect is no longer visible as

Articles


advection

135

3.3 Gain dependence on the rain event type

With the aim of assessing the impact of the rain event type on the system

performance, in both sites, rain events have been classified as stratiform or convective if

belonging respectively to “colder” (from November to April) or to “warmer” (from May

to October) months. Although this is quite a rough selection criterion, nevertheless, in

temperate zones, it can provide at least a first approximated discrimination between

weak and widespread rain phenomena and intense and spatially limited convective

precipitations. If single-site stratiform rain attenuation CDFs are considered, strong

fades tend to be less probable. For this reason, in order to deal with reliable statistics, in

this section, the reference attenuation As ranges from 4 to 14 dB using a 2-dB step.

Results are reported in 230HFigure 7, where solid and dashed lines depict G relative to

stratiform and convective events, respectively. As expected, the system performance

increases when convective rain events affect the stations: in fact, convective phenomena

are characterized by intense rain rate values but cover limited areas. Obviously, both

these characteristics concur to reduce the spatial correlation of rain. As a consequence,

the site diversity gain obtained during convective events shows a steep increase,

especially within the first 15 km, while the one relative to stratiform events definitely

increases more gradually with distance.

Articles


advection

136

0 10 20 30 40 50 600

2

4

6

8

10

12

14

Distance [km]

Gai

n [d

B]

Spino d'Adda

4 dB 6 dB 8 dB10 dB12 dB14 dB

0 10 20 30 40 50 600

2

4

6

8

10

12

14

Distance [km]

Gai

n [d

B]

Bordeaux

4 dB 6 dB 8 dB10 dB12 dB14 dB

Figure 6. Site diversity gain comparison between stratiform (solid lines) and convective

(dashed lines with circles) rain events. Ф= 0°, H baseline.

0 20 40 60 80 100 120 140 160 18010-4

10-3

10-2

10-1

100

101

Rain rate [mm/h]

Per

cent

age

[%]


Figure 8: Comparison between the radar-derived rainfall rate CDFs at Spino d’Adda

and Bordeaux.

Articles


advection

137

The differential gain is more marked in Bordeaux than in Spino d’Adda, which

may be due both to the fact that the monthly classification of rain events into stratiform

and convective is more effective in Bordeaux and/or to the higher convectivity in

Bordeaux, probably linked to the proximity of the Atlantic ocean and to its influence on

the rain structures. The higher convectivity seems to be confirmed also by Figure 8,

where the radar-derived rainfall rate CDF relative to Bordeaux denotes a higher

convective contribution (i.e. more probable intense rain rate values) if compared to the

Spino d’Adda CDF.

3.4 Comparison between the two sites

In this section, the system performance calculated for Spino d’Adda and

Bordeaux is compared. However, since G depends on the chosen baseline, as pointed out

in section 3.2, results relative to the H, V and HV orientations have been averaged and

are compared in Figure 9. The higher gain values in Bordeaux seem to confirm the

strongest convectivity of that site, as already mentioned in the previous section.

However, it is worth noting that, despite the reduced temporal (5 min) and spatial ( 11×

km2) resolution of the Bordeaux dataset, such radar scans are clearly suitable for site

diversity simulations, as they provide results that are fully comparable with those

obtained from a dataset with a finer temporal (77 sec) and spatial ( 5.05.0 × km2)

resolution.

Articles


advection

138

0 10 20 30 40 50 600

5

10

15

20

25

30

Distance [km]

Gai

n [d

B]

Spino d'Adda and Bordeaux


Figure 9: Site diversity gain comparison between Spino d’Adda (solid lines) and

Bordeaux (dashed lines with circles). Results refer to average values of H, V and HV

baseline orientations.

3.5 Comparison with the site diversity model currently recommended by ITU-R

The ITU-R has recently included into the ITU-R Rec. P.618-9 [ITU-R, 2007] a

new model aiming to predict the site diversity gain on a global basis. The methodology,

developed by Paraboni and Barbaliscia and whose rationale is provided in [Paraboni

and Barbaliscia, 2002], is based on the assumption that the logarithm of the rain

intensity R and the one of the associated attenuation A follow a multivariate Gaussian

distribution. The spatial cross-correlations of the logarithm of R and A are expressed by

the following double decreasing exponential forms, respectively:

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−+⎥⎦

⎤⎢⎣⎡−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−+⎥⎦

⎤⎢⎣⎡−=

2

2

500exp06.0

30exp94.0)(

700exp3.0

60exp7.0)(

ddd

ddd

A

R

ρ

ρ

(4)

where d is the distance (in km) between the two stations.

In this section, the radar-derived diversity gains are compared with those

estimated by the above mentioned model: concerning radar data, results of the W-E, N-S

Articles


advection

139

and NW-SE baseline orientations have been averaged. The comparison is shown Figure

10, respectively for Spino d’Adda and Bordeaux.

0 10 20 30 40 50 600

5

10

15

20

25

30

Distance [km]

Gai

n [d

B]

Spino d'Adda

0 10 20 30 40 50 600

5

10

15

20

25

30

Distance [km]

Gai

n [d

B]

Bordeaux


Figure 10: site diversity gain comparison between radar-derived values (solid lines) and

the ones estimated by the ITU-R Rec. P.618-9. φ = 0°, average values of horizontal,

vertical and 45° baseline orientations.

The prediction model tends to underestimate the diversity gain, especially for

high attenuation levels As. This may be due to the assumption at the basis of the model

(multivariate Gaussian distribution), which is valid as long as not too high values of rain

intensity (<15-20 mm/h) are concerned. The estimation accuracy of the model has been

quantified by defining the following error figure:

),(),(),(100),(

sr

srsmsm ADG

ADGADGAD −=ε [%] (2)

where Gm and Gr are the system gains, respectively estimated by the ITU-R Rec.

P.618-9 and derived from the radar images, relative to the stations distance D and to the

reference single-site attenuation As. Figure 11 depicts the trend of mε , calculated as the

average value of mε conditioned to D.

Articles


advection

140

0 10 20 30 40 50 60-35

-30

-25

-20

-15

-10

-5

Distance [km]

E[ ε

m] [

%]


Figure 11: mean value of the percentage error εm conditioned to D, for Spino d’Adda

and Bordeaux.

Results indicate a particularly high gain underestimation of the prediction model

for small distances (up to 20 km), for which the average error mε ranges approximately

from -30% to -20%. For greater distances, the error gradually decreases and tends to a

stable value (between -15% and -10%). It is worth pointing out that the ITU-R Rec.

P.618-9 is independent of the baseline orientation, as it assumes an isotropic modeling

of the rainfall spatial correlation [Paraboni and Barbaliscia, 2002]. It is therefore clear,

that, even if the model provided an accurate estimation of the diversity gain, it would

not be able to determine the most appropriate baseline orientation aimed at maximizing

the system performance. From this standpoint, radar data are certainly very useful and

offer unique chances of investigating some specific properties of the site diversity gain.

4 Comparison between the rain field advection and the ERA-40 wind outputs

As underlined in section 231H 3.2, the diversity gain at a given location exhibits

significant changes with the baseline orientation. The results obtained in this section

suggest that, in both sites, a linkage can be established between the rain front orientation

and the predominant rain advection direction. Indeed, a simple visual inspection of radar

Articles


advection

141

derived rain fields shows that most of the rain structures seem to elongate orthogonally

to their motion direction. Consequently, the knowledge of the rainfall prevailing

advection direction could help in the choice of the most advantageous baseline

orientation. To this aim, in this section, the rain advection speed and direction derived

from radar data are compared with the ERA-40 wind outputs. In addition, this study can

furthermore help to determine whether ERA-40 data are suitable for the

parameterization of the rain advection in space-time channel model such as those

proposed in [Gremont and Filip, 2004] and [Jeannin et al., 2008].

4.1 Determination of rain advection from radar data

The rain field advection direction and speed has been determined from successive

radar scans at time steps t and t+dt. It was implicitly assumed that the field advection

has a single direction and magnitude on the whole radar coverage.

Let R(x,y,t) be the rain fields at a given time t and position (x,y). The advection is

considered to be the vector [xa/dt ; ya/dt] such that R(x-xa,y-ya,t+dt) and R(x,y,t) exhibit

the maximum cross-correlation. This methodology, commonly used for this kind of

analysis [Kitzmiller et al., 2002], allows the retrieval of the rain field advection from

radar images in a simple, yet very effective way. The procedure assumes that the

temporal evolution of the rain field can be decomposed into two terms: one

corresponding to the actual modification of the rain structures within a Lagrangian

referential frame and the other one corresponding to displacement of such a frame

[Gremont and Filip, 2004]. This latter term will in the following be considered as the

advection of the field. The cross-correlation ρ between two successive maps at time t

and t+dt can be expressed as:

)]([Var)](Var[)]([)]([],,(),,([),(

dttRtRdttREtREdttyyxxRtyxREyx aa

aa+

+−+−−=ρ (6)

Where [...]E and Var[…] denote respectively the average value and the variance

of the field. The cross-correlation ρ is computed for all the xa and ya values that

correspond to reasonable displacements of the rain field, i.e. such that the motion of the

field will in general be slower than 150 km/h.

Articles


advection

142

An example of the cross-correlation between two consecutive rain maps of

Bordeaux (Figure 12) is shown in 232HFigure . The peak of ρ plotted in Figure 13 was found

for ya = 5 km (North to South) and for xa = 4 km (West to East): such displacement is

thus considered to be the advection of the field in the whole 5-minute interval. The

algorithm outlined above performs accurately if the correlation peak is higher than 0.7.

This condition implies that the time interval between the two radar observations is short

enough so as to prevent the rain field modifications from degrading too much the peak

value of ρ. Empirically, with a spatial resolution of 1 km, a time interval of 10 minutes

between consecutive images was found to be the upper bound guaranteeing a reliable

estimation of the advection. Thus, the time step between two successive images used to

compute the advection is set to 5 minutes for Bordeaux and to 3.5 minutes for Spino

d’Adda (one every three maps). Moreover, in order to obtain a sufficiently robust

estimation of ρ, the fraction of the observation area affected by rain has to be

significant. For this reason, radar observations for which the rainy area is smaller than

2% of the whole radar coverage (area with a 100-km radius) have been discarded

leading thus to consider 17% of the dataset.

Figure 12: Evaluation of the displacement on two consecutive radar maps taken at time

steps t and t+dt (Bordeaux, dt = 5 minutes)

Articles


advection

143

Figure 13: Cross-correlation for different displacement vectors between two

consecutive rain maps of Bordeaux. The maximum correlation corresponds to the most

plausible advection between the two images

4.2 Comparison between radar derived wind speed and direction and ERA-40 wind outputs

The ERA-40 data considered in this study are the wind values, relative to

different isobars, that are freely available on the web with a latitude-longitude regular

grid of 2.5°×2.5° . The period of radar data acquisition (1988-1992 for Spino d’Adda

and 1996 for Bordeaux) falls within the one of the ERA-40 reanalysis database (1957-

2002), so that a comparison of the rain advection estimated from both sources is

possible by taking into account concurrent years. Data of winds (flowing both from North to South (N-S) and from West to East

(W-E)) relative to different pressure levels have been considered in this study, namely

850, 775, 700, 600, 500 hPa. Such levels roughly correspond to reasonable mid-height

values of cloudy structures typical of temperate climates.

In order to compare the rain advection speed and direction retrieved from radar

data with the ERA-40 winds outputs, the following methodology has been applied. The

ERA-40 wind data for both locations have been derived by applying a bilinear

Articles


advection

144

interpolation of the values relative to the four closest ERA-40 pixels, as indicated Figure

14.

Figure 14: Position of the observation areas and of the surrounding ERA-40 cells

As an example, Figure 15 illustrates the wind direction statistics relative to the 775

hPa isobar and calculated from one year of ERA-40 data, both for Spino d’Adda and

Bordeaux: in the latter location, that lies in the proximity of the ocean and whose area is

not characterized by a significant orography, the wind direction is mainly eastward; on

the contrary, in Spino d’Adda, air fluxes from the North are mainly blocked by the Alps

and consequently, most of the time, air streams flow from West or South.

Figure 15: Histogram of the wind direction relative to the 775 hPa isobar computed

from one year of ERA-40 data for Spino d’Adda and Bordeaux

Articles


advection

145

In order to allow a direct comparison between the two datasets, the radar derived

rain advection speeds and directions have been averaged for each 6-hour period,

according to the time resolution of ERA-40 data. Considering the averaging operation

and the constraints illustrated in the previous section for the calculation of the rain field

advection from radar data, 160 and 64 samples have been respectively obtained for

Bordeaux and Spino d’Adda. The correlation coefficient between the rain advection

derived from the radar datasets and from the ERA-40 wind outputs at several pressure

levels is listed in Table 2, both for Bordeaux and Spino.

Direction Magnitude Pressure level (hPa)

Bordeaux Spino d’Adda Bordeaux Spino d’Adda

850 0.66 0.71 0.85 0.58

775 0.82 0.83 0.89 0.77

700 0.90 0.83 0.92 0.84

600 0.84 0.77 0.85 0.85

500 0.82 0.76 0.77 0.82 Table 2. Correlation between the rain advection derived from radar data and from ERA-

40 wind outputs for Bordeaux and Spino d’Adda

Among all the levels reported in Table 2, the 700 hPa isobar presents the highest

overall correlation. This pressure level was already found to be the most representative

of the rain field advection for different general circulation models as those in [Kitzmiler

et al., 2002] and [Sokol, 2006]. Lower levels (850 hPa, 775 hPa) underestimate the rain

advection speed and the proximity to the ground perturbs the direction estimation,

whereas, on the contrary, the advection velocity is overestimated at higher layers (600

hPa, 500 hPa).

The sample by sample comparison between the rain advection speed and the

ERA-40 wind velocity at 700 hPa is presented in Figure 16, both for Bordeaux

(asterisks) and Spino d’Adda (dots). The correlation between the two types of data is

slightly poorer for Spino d’Adda, as also observed in the comparison concerning the

wind direction reported in Figure 17. This trend may be explained by the large impact of

Articles


advection

146

the mountainous area surrounding Spino (see Figure 14) that probably leads to a less

reliable estimation of wind flows provided by the ERA-40. In fact, the mean height

associated to the four ERA-40 cells surrounding Spino d’Adda is over 1000 m a.m.s.l.

whereas the site lies only at 84 m a.m.s.l.

0 20 40 60 80 100 120 1400

20

40

60

80

100

120

140

Rain advection speed from radar data averaged over 6h [km/h]

Win

d sp

eed

from

ER

A 4

0 [k

m/h

]

Wind speed from ERA-40 and advection speed from radar

BordeauxSpino d'Adda

Figure 16: ERA-40 wind speed relative to the 700hPa isobar versus rain advection

speed for Bordeaux (asterisks) and Spino d’Adda (dots)

0 50 100 150 200 250 300 3500

50

100

150

200

250

300

350

Rain advection direction deduced from radar averaged over 6h [°]

Win

d di

rect

ion

from

ER

A-4

0 [°]

Wind direction from ERA-40 and advection direction from radar

BordeauxSpino d'Adda

Figure 17: ERA-40 wind speed at 700h Pa versus rain advection speed for Bordeaux and Spino

d’Adda

Articles


advection

147

The results obtained so far show that the rain field advection (both direction and

speed) can be approximated with a relatively high confidence by the ERA-40 wind

outputs relative to the 700 hPa isobar. As a consequence, such data can serve for the

correct parameterization of global space-time rain models such as those proposed in

[Gremont and Filip, 2004] and [Jeannin et al., 2007].

5 Impact of the wind direction on the rain field spatial anisotropy

As suggested in [Goldirsh et Robison, 1975], the local predominant wind

direction seems to be linked to the preferable baseline orientation of site diversity

systems. A confirmation of this trend is also given by the results obtained in this paper:

indeed, the diversity gain computed from the radar data of Bordeaux and Spino d’Adda

for the W-E and the N-S baselines shows sizeable differences (refer to Figure 5). In the

light of the predominant wind directions depicted in Figure 5 for the two sites, this result

suggests that, on the average, rain fields are elongated towards the N-S direction in

Bordeaux, whereas in Spino they are rather stretched towards the W-E direction: as a

result, in both locations, the most advantageous baseline orientation is parallel to the

predominant wind direction.

The effect of the wind flows on rain structures is confirmed also by Figure 18,

where the overall spatial correlation has been calculated from all the rain maps of

Bordeaux using the spatial spectrum technique [Jeannin et al., 2008]: indeed, results

indicate a slight anisotropy of the field, which actually tends to be more correlated along

the N-S direction than along the W-E one.

Articles


advection

148

Figure 18: Overall spatial correlation of the rain field computed using one year of Bordeaux radar data

It can furthermore be wondered if the influence of wind flows on the rain field

spatial correlation holds also on instantaneous basis. The analysis of several rain maps

has shown that this conclusion is not always valid: consider, for instance, Figure 19,

where it can be clearly noticed that, if in most of the areas the wind is orthogonal to the

rain bands, in some other zones, especially on the South-Western edge of the field, the

rainy front and the wind are almost parallel. A more detailed analysis of pressure fields

and of the underlying air streams may explain more physically these considerations.

Figure 19: Rain field over France acquired by the French radar Network Aramis in

October 2000 and concurrent ERA-40 wind outputs relative to the 700 hPa isobar

Articles


advection

149

6 Conclusion

In this work, weather radar data collected at two sites have been used to evaluate

the performance of Earth-satellite site diversity systems with different link and baseline

orientations. Results have shown a negligible dependence of the system performance on

the link orientation, whereas, on the contrary, the system gain has proven to be tightly

linked to the choice of the baseline direction. This result seems to be related to the

overall anisotropy of rain structures which generally tend to elongate perpendicularly to

the local prevalent direction of the wind: a baseline orientation parallel to the prevailing

wind direction (i.e. from North to South for Spino d’Adda and from West to East for

Bordeaux) has shown to maximize the system gain. As a further result of this study, a

rough classification of rain events into stratiform and convective according to “colder”

and “warmer” months has allowed to confirm the strong dependence of the system

performance on the type of precipitation, due to the different spatial correlation

characteristics associated to stratiform and convective phenomena. The site diversity

results obtained from radar data have been afterwards compared with those provided by

the model currently recommended by the ITU-R for the estimation of the site diversity

performance: the ITU-R recommendation has shown to underestimate the diversity gain,

especially when low reference attenuation levels are considered.

More in general, the analysis has shown that, despite the reduced temporal (5

minutes) and spatial (1 km × 1 km) resolution of the Bordeaux dataset, those radar scans

have proven to be adequate for site diversity simulations, as they provide results that are

fully comparable with the ones obtained from a dataset with a finer temporal (77

seconds) and spatial (0.5 km × 0.5 km) resolution (Spino d’Adda data). Specifically, this

study underlines the usefulness of radar data for the simulation and the performance

evaluation of a site diversity system, as they reflect the local climatologic and

topographic characteristics, which, on turn, may reveal some properties on the diversity

hardly caught by analytical estimation models.

With the aim of establishing a linkage between the most advantageous site

diversity baseline orientation and the local predominant wind direction, the rain field

advection has been estimated from successive radar maps (both in Spino d’Adda and

Bordeaux). Results have confirmed that, in both sites, rain structures generally tend to

Articles


advection

150

elongate perpendicularly to the local prevalent direction of the wind. In addition, ERA-

40 wind outputs relative to different isobars have been extracted and compared to the

radar derived rain field advection. Data relative to the 700 hPa isobar have proven to be

the most correlated both with the rain advection direction and speed: not only ERA-40

data can provide a useful indication for the choice of the optimum baseline orientation in

the design of site diversity systems, but, in addition, they can be used for the

parameterization of space-time rainfall models such as those presented in [Gremont and

Filip, 2004] and [Jeannin et al., 2007], in which the knowledge of the correct rain field

advection is of key importance..

Acknowledgements

This work has been partially developed in the framework of the European

Network of Excellence SatNEx.

References

Capsoni, C., M. D’Amico, “Performance of small-scale multiple-site diversity

systems investigated through radar simulations”, Radio Science, 42, 2007

Capsoni, C., M. D'Amico, P. Locatelli, “Statistical properties of rain cells in the

Padana Valley,” Journal of Atmospheric and Oceanic Technology (JTECH). In Press.

Castanet L., Bousquet M., Filip M., Gallois P., Gremont B., De Haro L.,

Lemorton J., Paraboni A., Schnell M. : "Impairment mitigation and performance

restoration", COST 255 Final Report, Chapter 5.3, ESA Publications Division, SP-1252,

March 2002.

Articles


advection

151

Fukuchi H., T. Saito, “Novel mitigation technologies for rain attenuation in

broadband satellite communication system using from Ka- to W-band,” 6th International

Conference on Information, Communications & Signal Processing, 10-13 Dec., 2007.

Goldhirsh, J. F. L. Robison, “Attenuation and space diversity statistics calculated from radar

reflectivity data of rain”, IEEE Transactions on Antenna and Propagation, 23(2), 221-227,1975.

Goldhirsh, J., B.H. Musiani, A.W. Dissanayake, L. Kuan-Ting, “Three-site

space-diversity experiment at 20 GHz using ACTS in the Eastern United States,” Proc.

IEEE, 85, 970-980, June, 1997.

Gremont, B., and M. Filip, “Spatio-temporal rain attenuation model for

application to fade mitigation techniques”, IEEE Trans. Ant. Prop., 52, 1245-1256,

2004.

ITU-R: “Rain height model for prediction methods,” Propagation in Non-Ionized

Media, Recommendation P.839-3, Geneva, 2001.

ITU-R: “Specific attenuation model for rain for use in prediction methods,”

Propagation in Non-Ionized Media, Recommendation P.838-3, Geneva, 2003.

ITU-R: “Propagation data and prediction methods required for the design of

Earth-space telecommunication systems,” Propagation in Non-Ionized Media,

Recommendation P.618-9, Geneva, 2007.

Jeannin N., L. Féral, H. Sauvageot, L. Castanet, J. Lemorton, F. Lacoste,

“Stochastic Spatio-Temporal Modelling of Rain Attenuation for Propagation Studies”, EUCAP

2007, Edinburgh UK, Nov 2007.

Kitzmiller, H. D., G. F. Samplatsky, C. Mello, “Probabilistic Forecasts Of Severe

Local Storms In The 0-3 Hour Timeframe From An Advective-Statistical Technique”,

19th Conf. on weather Analysis and Forecasting/15th Conf. on Numerical Weather

Prediction, San-Antonio, August 2002.

Articles


advection

152

Paraboni, A., F. Barbaliscia, “Multiple site attenuation prediction models based

on the rainfall structures (meso or synoptic scales) for advanced TLC or broadcasting

systems,” Proceedings of the URSI GA, Maastricht, August 2002.

Riva, C., “Seasonal and diurnal variations of total attenuation measured with the

ITALSAT satellite at Spino d’Adda at 18.7, 39.6 and 49.5 GHz,” International Journal

of Satellite Communications and Networking, 22, 449–476, 2004.

Sokol, Z., “Nowcasting of 1-h precipitation using radar and NWP data”. Journal

of Hydrology, 328, 200-211, 2006.

Uppala, S. et al., “The ERA-40 re-analysis”. Quart. J. R. Meteorol. Soc., 131, 2961-3012,

2005.

Articles

Statistical distribution of the fractional area affected by rain

153


Nicolas Jeannin(1), Laurent Féral (2), Henri Sauvageot (2), Laurent Castanet (1), and

Joël Lemorton (1)

(1)ONERA, Département Electromagnétisme et Radar, Toulouse, France

(2)Université Paul Sabatier, Laboratoire LAME, Toulouse, France. (3)Université Paul Sabatier, Observatoire Midi-Pyrénées, Laboratoire

d’Aérologie, Toulouse, France

Abstract

The knowledge of the fraction of an area that is affected by rain (or fractional

area) is of prime interest for hydrologic studies or for rainfall field modeling. Up to now

the statistical distribution of this parameter has been poorly studied. In the present

paper, a model of the statistical distribution of the fraction of an area affected by rain

over a given rainfall rate is proposed. It takes into account at the same time the size of

the area and the local climatology. The analytic formulation of the distribution is

established considering that rainfall fields can be obtained from a non-linear filtering of

a Gaussian random field. As the analytic derivation of the distribution lies on some

assumptions, the model accuracy is first evaluated from numerical simulations. It is then

shown that the model reproduces accurately the distribution of fractional areas derived

from radar observations of rain fields, for various rain thresholds, sizes of area, and

climatologies. A generic parameterization is then proposed for areas ranging from

100×100km2 to 300×300km2.

Articles


154

1 Introduction

In a typical General Circulation Model (GCM), the spatial resolution of the

outputs is generally not sufficient to describe rain at a scale of interest for hydrologic

processes such as interception and runoff [Pitman, 1991; Thomas and Henderson-

Sellers, 1991]. Therefore, a representation of rainfall spatial variability at a finer scale is

necessary. Thus, many models were developed to assess the spatial variability of rain at

coarser scales, using either fractal approaches [Lovejoy and Schertzer, 1985; Over and

Gupta, 1996], stochastic approaches [Mejia and Rodriguez-Iturbe, 1974; Bell, 1987;

Lebel et al.,1998] or cellular approaches [Le Cam, 1961; Capsoni et al., 1987a, 1987b;

Goldhirsh 2000, Feral et al., 2003a, 2003b]. For most of these approaches, a key

parameter that has to be specified is the fraction f of the simulation area affected by rain,

often called lacunarity or intermittency. Moreover, it was shown that surface hydrology

exhibits a strong sensitivity to this parameter [Pitman et al., 1990]. Indeed, if the rain

fractional coverage over an area of interest and for a given rain amount decreases, the

climatology turns from evaporation dominated to runoff dominated, thus showing the

importance of this quantity on climate simulations.

In other respects, the knowledge of this parameter is also valuable to evaluate the

performances of earth-space telecommunication systems. Indeed, the attenuation

undergone by an earth space satellite link operating at a frequency above 20 GHz is

mainly driven by the rainfall rate along the path of the link (Castanet et al. [2001]).

High attenuations due to rain will result in an unavailability of the satellite link or will

require the use of adaptive fade mitigation techniques (Castanet et al. [2002], Neely et

al. [2003]). Considering a satellite spot beam whose diameter is typically of 300 km, the

knowledge of the fraction of the area affected by a rain rate over a given threshold gives

an estimate of the fraction of the user-terminals in the satellite spot beam that will

undergo a given level of attenuation. Consequently, this parameter gives an estimate,

considering the link budget, either of the number of the terminals that will be in outage

or of the additional margin that has to be provided to ensure a sufficient quality of

service.

A convenient way to prescribe the fractional area f was addressed from the

estimation of rainfall from space. Indeed, Donneaud et al. [1984], Chiu [1988], Braud et

al. [1993], and Oki et al. [1997] observed that the rain fractional coverage over a preset

Articles


155

threshold is tightly correlated with the spatially averaged rain rate over the area of

interest. Particularly, whenever the local Probability Density Function (PDF) of rain is

known, Atlas et al. [1990], Kedem et al. [1990] and Sauvageot [1994] have shown that

the proportionality coefficient between the fractional area and the spatially averaged rain

rate can be determined from the rain conditional PDF (i.e. knowing that it is raining).

Invoking an ergodicity assumption of the rainfall process, Eltahir and Bras [1993] gave

a procedure to deduce the fractional area affected by rain from GCM outputs accounting

for the GCM spatial and temporal resolutions. Nevertheless, for rain field simulation

purposes, either as an input of the model [Feral et al., 2006] or as a tool to evaluate the

model accuracy [Guillot, 1999; Guillot and Lebel, 1999], it could be interesting to have

a simple way to get the statistical distribution of the rain fractional coverage for various

thresholds, while accounting not only for the local climatology but also for the size of

the simulation area.

This was attempted by Onof and Wheater [1996] who tried, from weather radar

observations over Wales, to approximate the PDF of the fractional area affected by rain

with a parabolic distribution. In other respects, from radar observations over the French

territory, Feral et al. [2006] approximated the distribution of the fractional area affected

by rain with an exponential law with mean 8.8%. Nevertheless, in both studies, the

dependence on the local climatology and the size of the area are not evaluated.

Moreover, only the rain/no-rain threshold is considered, so that the results highly

depend on the radar sensitivity.

From an analysis of radar data on various climatic zones and for various sizes of

observation area, this paper proposes an analytical expression of the Cumulative

Distribution Function (CDF) of the fractional area affected by rain over a preset

threshold. The formulation is derived from properties of rain fields generated using a

non-linear transformation on stationary Gaussian random fields, a generation scheme

that has already been reported in the literature (Bell 1987, Guillot 1999). This approach

has been theoretically justified in Ferraris and al., [2002]. The latter is based on two

steps. First, a correlated, stationary, homogeneous Gaussian random field is generated

on a lattice. Second, on each grid point, the Gaussian random variables are turned into

rain rate values R. More specifically, in Bell [1987] or Guillot [1999], the random values

of the Gaussian field lower than a value α driven by the local probability of rain are put

to 0 while the random values greater than α are transformed so that they follow the rain

Articles


156

rate local conditional PDF )0R|R(p > . Nevertheless, as underlined by Guillot [1999],

the main defect of such kind of modeling is that the fraction of area affected by rain on

one realization is not prescribed. Furthermore, Guillot [1999] pointed out that if a long

range dependence is not introduced in the correlation function of the Gaussian random

field, the fraction of the area affected by rain converges towards the local temporal

expectancy to have rain, as a second order stationary random field is ergodic. This

means that the variance of the fractional coverage of rain computed from a large number

of simulated fields tends to 0. However, and as shown by Eltahir and Bras [1993], the

fractional coverage of rain is a highly variable parameter. Therefore, rain fields

simulated with the methodology described above must have a long range dependence to

be realistic. This long range dependence can be quantified by what Lantuejoul [1991]

called the “integral range”, related to the average value of the correlation function over

the domain of interest.

Section 2 of the present paper precisely addresses the effect of the long range

dependence on the statistical properties of the spatial average and spatial variance for

homogeneous stationary Gaussian fields generated on a finite grid. Particularly, it is

shown that, under some assumptions, the spatial distribution of the samples defining one

realization of a Gaussian field generated on a finite grid is approximately normal with

mean M and variance 1-σ², where σ² is the average value of the correlation matrix that

drives the dependencies between all the points of the grid. Additionally, the spatial

mean M is shown to be a random variable that follows a centered normal distribution

with variance σ². From those considerations, in section 3, an analytical formulation is

proposed to model the CDF of the fractional area of a stationary Gaussian random field

over a preset threshold α. The model accuracy is first evaluated numerically, from

simulated Gaussian random fields. Then, in section 4, the model is compared with

fractional area CDFs derived from true rain fields observed by weather radar over

various places. A set of parameters is then proposed in order to reproduce the CDF of

the fractional coverage of rain for various climatic zones and various sizes of area,

ranging from 100×100 km² to 300×300 km².

Articles


157

2 Theoretical considerations on stationary Gaussian random fields

As mentioned in the introductive part, the simulation of rain field using a non

linear transformation of a Gaussian field has been already widely studied (Bell [1987] or

Guillot [1999]) and was seen to perform satisfactorily in numerous situations. In those

works, a methodology to derive the correlation function of the underlying Gaussian field

from observed rain fields on a range of a few hundreds of kilometers was developed.

The correlation function of the Gaussian field computed from rain fall observations

(radar, rain gauges) was found to have a steep decay in the first tens of kilometers and

was then only very slowly decreasing without reaching 0 at the end of the observation

range of some hundreds of kilometers. Considering rain gauges spread across Italy,

Barbaliscia et al [1992] and Bertorelli and Paraboni [2005] identified three scales of

evolution for the correlation of rain fields:

Below 100 km the correlation was shown to have a steep decay,

From 100 to 700 km the correlation was shown to be almost steady or very

slowly decreasing with distance,

After 700 km the correlation decreases slowly until total decorrelation.

This long range dependence can be explained by the non uniform repartition in

space over wide areas of the rainy structures and of the underlying pressure and

humidity field. This shape of correlation function is awkward because it implies that the

rain fields are stationary over wide areas. As rain fields display some preferential path

due for instance to the orography or to land sea transitions, this assumption would

probably collapse for some hundreds of kilometers. The range of validity of this

stationarity hypothesis is hardly assessed as the requirements to apply a formal

stationarity test for spatial stochastic processes as described in Fuentes [2005] are not

matched by rain fields. Consequently, the areas of side greater than 300×300 km² for

which the stationary hypothesis of the rain fields is likely to be unrealistic will not be

considered in the present paper.

The aim of the rest of this section is to derive from those models an expression of

the distribution of the fraction of an area f (or fractional area) over a given rain rate

threshold using the properties of the underlying Gaussian field. The methodology

developed in the following strives at finding an approximation of the spatial distribution

of one realization of a Gaussian field on a finite grid considering that it has a correlation

Articles


158

function of the shape described above. In section 3, from this spatial distribution, an

expression of the fraction of the area of one realization of a Gaussian field is first

derived. Then, the statistical distribution of the fractional area considering a large

number of independent realizations of the Gaussian field is proposed, validated and

transposed to rain fields in section 4.

2.1 Spatial average M of a Gaussian random field

A stationary, homogeneous, standard, and centered Gaussian random field g

generated on a bidimensional grid L with size N×N can be interpreted as a set of

standard centered Gaussian random variables ( ))s(g),...,s(g),s(gg 2N21= correlated with

each other, and indexed by their position si on the grid. Let Σ be the N2×N2 correlation

matrix of the Gaussian random field g with general term )]s(g)s(g[E jiij =Σ , where E[]

is the expectancy operator. The spatial average M of the random field g over the grid L

is defined by:

)s(gN1M

2N

1ii2 ∑

=

= . (1)

As all the random variables g(si) follow a standard centered normal distribution,

the random variable M is also normal with an expectation of 0. The variance of M is

defined by:

2N

1i

N

1jij4

²N

1i

N²

1jji4

2N²

1ii4

2 2

N1

)s(g)s(gEN1

)s(gEN1)M(Var

σΣ ==

⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛=

∑∑

∑∑

∑

= =

= =

=

. (2)

Equation (2) shows that the variance Var(M) of the spatial average M is the mean

σ² of the terms of the correlation matrix Σ. At this stage we demonstrate that the spatial

average M of the field is normally distributed with mean 0 and variance σ².

Articles


159

2.2 Spatial variance V of a Gaussian random field

Similarly, the spatial variance V of the Gaussian random field g over the grid is

given by:

( )∑=

−=2N

1i

2i2 M)s(g

N1V . (3)

The spatial variance V is a random variable and its expectation can be defined

as :

[ ] [ ] [ ]

[ ] [ ] [ ]

2

N

1i

N

1jij4

N

1iji

N

1j2i2

2N

1ii

2i2

2i

N

1i2

1)M(Var1

N12)M(Var1

)s(g)s(g(EN2MVar)s(gVar

N1

MEM)s(gE2)s(gEN1

])M)s(g([EN1]V[E

2 2

2 2

2

2

σ

Σ

−=−=

−+=

⎭⎬⎫

⎩⎨⎧

−+=

+−=

−=

∑∑

∑ ∑

∑

∑

= =

= =

=

=

. (4)

Consequently, the samples 2N..1ii ),s(g =ω of one realization g(ω) of the Gaussian

random field g are characterized by their spatial mean M(ω) and by their spatial variance

V. In compliance with section 2.1, the spatial mean M(ω) is drawn from a normal

distribution whose mean is 0 and whose variance σ² is the mean of the correlation

function Σ. In such conditions, the expectancy E[V] of the spatial variance V is 1-σ², as

shown by (4). However, the spatial distribution of the N² random values 2N..1ii ),s(g =ω

is still unknown. Nevertheless, if the correlation function decrease sufficiently fast with

regards to the size of the lattice (ie 02 →σ see (Lantuejoul [1991]), the distribution of

the 2N..1ii ),s(g =ω tends to be normal if the size of the lattice is large enough. As

mentioned in the preliminary part of this section, the correlation function of the

underlying Gaussian field used to simulate rain fields exhibits a very slow decay with

distance. In that case, considering a lattice on which the decorrelation of the sample is

Articles


160

not reached from one side to the other, the normality of 2N..1ii ),s(g =ω is not ensured

because σ² will not tend to 0. Nevertheless, if between all the points of the grid there

exists a residual positive correlation denoted by a², the correlation gc of the stationary

Gaussian field g can be rewritten:

)ss(cba)ss(c 21g~22

21g −+=− , with 1ba 22 =+ . (5)

The random field g can thus be considered as the weighted sum of a standard

centered Gaussian random field g~ whose correlation is gc ~ and weight b with a standard

centered Gaussian random variable y and weight a. Consequently, for each point s of the

grid:

ay)s(g~b)s(g += . (6)

If g~c rapidly decreases with regards to the size of the lattice, the spatial

distribution of 2N..1ii ),s(g~ =ω tends to be normal. Moreover, as y does not vary in space,

the spatial distribution of 2N..1ii ),s(g =ω also tend to be normal. Considering the

correlation functions proposed for rain field simulation by Barbaliscia et al. 1992 or

Guillot and Lebel [1999], and grid sizes ranging from 100×100 to 300×300 km2, the

steep decay g~c with regards to the size of the grid is ensured and the normality of

2N..1ii ),s(g =ω is a reasonable assumption. As no means were found to quantify the

derivation from normality of 2N..1ii ),s(g =ω due to the finiteness of the lattice, the effect

of this approximation on the accuracy of the model proposed in section 3.1 is assessed

from simulated Gaussian fields in section 3.2.

Articles


161

3 Model for the CDF of the fractional area over a given threshold

3.1 Analytical derivation

As discussed in section 2.2, for one realization g(ω) of a stationary standard

centered Gaussian random field g with suitable shape of correlation function, the spatial

distribution of 2N..1ii ),s(g =ω tends to be normal if the size of the grid L is sufficiently

large. For the purpose of this study, the spatial distribution 2N..1ii ),s(g =ω is considered

as a normal distribution with mean M and variance V=1-σ². Besides, as shown in section

2.1, M is random and follows a centered normal distribution with variance σ². Therefore,

considering one realization ω of the homogeneous random field g, the fractional area

fα(ω) of g over which the threshold value α is exceeded is given by:

⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−=

)1(2)(Merfc

21)(f

2σωαωα , (7)

where erfc denotes the complementary error function.

The CDF of the fractional area f exceeding the threshold α can then be derived

considering the CDF of M:

( ))2(erfc)1(2P

)1(2erfc

21P)(P

*12

*

2

*

fM

fMff

−−−>=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛>

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−=>

σα

σα

α. (8)

Recalling that M follows a centred normal law with variance σ², we have:

⎟⎠

⎞⎜⎝

⎛=>σ2

erfc21)(P xxM , (9)

so that:

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−=>

−

σσα

α 2)2(erfc)1(2

erfc21)(P

*12* f

ff . (10)

Derivation of (10) gives the PDF pα(f) of the fractional area f above α:

[ ] ⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−−

−−=

−−

σσα

σσ

α 2)1(2)2(erfc

exp1)2(erfcexp)(212

21 fffp (11)

Articles


162

The analytical formulation of the fractional area average value <f>α is difficult to

obtain from (11). Nevertheless, it can be accurately approximated by:

)2

(erfc21)( ααα =>=>< GPf . (12)

(12) is obtained by considering that, over several realizations of the Gaussian

random field g, the fraction f of field g above the threshold value α is simply the overall

number of points above α divided by the overall number of points. As g is

homogeneous, the expectancy to exceed α is the same everywhere in the lattice so that

<f>α amounts to (12), i.e. to the probability for a Gaussian centred standard variable G

to exceed α. The numerical computation of <f>α from the PDF (11) with different

parameterizations complies satisfactorily with (12), i.e. with a maximum error lower

than 1%.

In compliance with (10) and (11), the statistical distribution of the fractional area

f depends on 2 parameters. The first one, α, is related to the average value of f by (12).

The second one, σ², is the average value of the N²×N² Gaussian field correlation matrix

Σ. Obviously, σ² strongly depends on the size N×N of the simulation grid. So as to

quantitatively assess the model accuracy and the parameter sensitivity, (10) is compared

to the fractional area CDFs derived from Gaussian field simulation.

3.2 Model accuracy and parameter sensitivity

For each configuration detailed hereafter, 6000 stationary Gaussian fields are

generated using the spectral method described by Bell [1987]. Further details on this

approach can be found in Shinozuka and Jan [1972], Meija and Rodriguez-Iturbe

[1974], and Borgman et al. [1984]. Lattice size varies from N×N=50×50 km² to

N×N=200×200 km², in compliance with the typical coverage of the operational radars

considered in section 4. Moreover, to evaluate the model sensitivity, two analytical

formulations of the correlation function are considered for the Gaussian field g: 30/d

1,g e)d(c −= and 800/d30/d2,g e5.0e5.0)d(c −− += , where d is the distance in km. Such

exponential formulations of the correlation function are commonly accepted to simulate

rain fields (Guillot and Lebel [1999], Barbaliscia et al. [1992], Feral et al. [2006]). In

Articles


163

such conditions, considering )d(c 1,g and equation (2), σ varies from 0.71 to 0.34 when

N×N varies from 50×50 km² to N×N=200×200 km², respectively. Similarly, considering

)d(c 2,g , σ varies from 0.85 to 0.72 when N×N varies from 50×50 km² to N×N=200×200

km², respectively. Six threshold values α are considered successively to compute the

fractional area CDFs from the simulated Gaussian fields, namely α=0.5, 1, 1.5, 2, 2.5, 3.

The distributions thus obtained are compared with (10). The results are shown in Fig. 1

and 2. It then appears that, whatever the configuration, model (10) accurately reproduces

the fractional area CDFs derived from the simulated fields. As all the realization of the

simulated field are independent the values f obtained for each realization of the fields

are independent and common statistical test can therefore be applied to test the validity

of (10). Particularly, the null hypothesis “the CDF )(P *ff >α derived from the

simulated Gaussian fields is (10)” is never rejected by a unilateral Smirnov-Kolmogorov

test [Chakravarti et al., 1967] with a confidence level of 0.05. To conclude, model (10)

accurately reproduces the Gaussian field fractional area CDFs whatever the threshold α,

the lattice size between 50×50 km² and 200×200 km², and the standard formulations

)d(c 1,g or )d(c 2,g of the correlation function.

Articles


164

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CDF of the fraction of the field over α

f*

P(f

<f* )

α=0.5α=1α=1.5α=2α=2.5α=3

Figure 1: CDFs of the fractional area over the threshold α computed from model (11)

and derived from the simulation of Gaussian fields: size N×N=200×200 km², correlation

function )800/exp(5.0)30/exp(5.0)( dddcG −+−= .

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CDF of the fraction of the field over α

f*

P(f

<f* )

α=0.5α=1α=1.5α=2α=2.5α=3

Figure 2: CDFs of the fractional area over the threshold α computed from model (11)

and derived from the simulation of Gaussian fields: size N×N=50×50 km², correlation

function )800/exp(5.0)30/exp(5.0)( dddcG −+−= .

Articles


165

4 Application to rain fields

4.1 Model extension to rain field

In the previous section, a mathematical framework to describe the CDF of the

fractional area of a homogeneous Gaussian field that exceeds a preset threshold α has

been developed. The same methodology applies to rain fields whenever the latter are

derived from a non-linear transformation of a stationary homogeneous Gaussian field

[Bell, 1987; Guillot, 1999]. Indeed, considering that the rain rate R conditional PDF is

lognormal (mean Rμ , standard deviation Rσ ) as it is commonly accepted [Bell, 1987;

Sauvageot, 1994; Feral et al., 2006], and assuming that it rains a fraction P0 of the time,

the transformation ψ that converts the Gaussian field g into a rain field r is:

[ ]

[ ]0

0

P0

1RR

P

)s(gfiP

)2/)s(g(erfcerfc2exp)s(gψ)s(r

)s(gif0)s(gψ)s(r

ασμ

α

≥⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+==

<==

−,(13)

where

)P2(erfc2 01

P0

−=α . (14)

By construction, the expectation for a Gaussian value to be converted into rain is

exactly P0. Moreover, the computation of the CDF of the fraction f of a rain field

affected by a rain rate R exceeding the threshold R* amounts to the computation of the

fraction of a standard Gaussian field exceeding the threshold )(ψ *1* RR

−=α . As

[ ] [ ]*R* )s(gPR)s(rP α>=> , we have:

[ ] *1R

R)s(rP2erfc2* >= −α . (15)

If we assume that the rain field r(s) is homogeneous, the probability of rain P0 is

the same all over the grid and [ ]*R)s(rP > does not depend on the location s so that (15)

reduces to:

[ ])(P2erfc2 *1* RR

R>= −α , (16)

where P(R>R*) is the rain absolute CDF intrinsic to the simulation area. (16)

gives a convenient way to determine *Rα from radar data.

Articles


166

Finally, the CDF of the fractional area f of a rain field affected by a rain rate

exceeding the threshold value R* derives from (10):

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−=>

−

σσα2

)2(erfc)1(2erfc

21)(P

*12*

R

*

*

fff R , (17)

under the hypothesis made to establish (10) and (17), namely the homogeneity of

the field (ie. the pdf of rainfall rate dos not vary inside the considered area) and a

correlation function with a fast and a slow decay component with respect to the lattice

size as advocated by Barbaliscia et al., [1992] or Guillot and Lebel [1999].

4.2 Radar data

Two yearly radar datasets are considered. The first one was collected in 1996 and

comes from the weather radar of Bordeaux that is part of the French operational radar

network managed by Météo-France. The second one refers to 2004 and consists of

composite radar images from the US NEXRAD network. For both datasets, the temporal

sampling is one observation every 5 min. The data were projected on a Cartesian grid

within a uniform pixel size of 1×1 km². Images including ground clutter or melting layer

echoes were removed from the dataset so that the used radar data only refer to rainfall

fields. Lastly, reflectivity fields were converted into rain fields using the standard Z-R

relation:

baRZ = , (18)

where a=300, b=1.35, and Z is the radar reflectivity factor in mm6 m-3 and R the

rain rate in mm h-1.

As shown in Fig. 3 and 4, areas of interest offering clearly distinct climatologies

have been selected inside the radar coverage areas. Moreover, for each selected location,

five area sizes are considered successively, namely 100×100, 150×150, 200×200,

250×250 and 300×300 km².

Articles


167

Figure 3: Geographical position of the 3 areas of interest selected over the US. The

background image is a rain field observation collected by the NEXRAD radar network

on 08/27/04 at 17:00 UTC.

Articles


168

Figure 4: Geographical position of the area of interest over Bordeaux (France). The

background image is an example of rain field observation collected by the ARAMIS

radar network on 09/129/01 at 16:00 UTC.

4.3 Results

The CDFs of the fractional area affected by rain were computed for thresholds

R*=0.5, 1, 2.4, 5.7, 13 mm h-1 for the areas of interest over the US and R*=1, 1.4, 2.2,

3.5 mm h-1 for the ones over Bordeaux (the thresholds considered are not exactly the

same for both datasets due to some differences in the quantization steps between the

radar of Bordeaux and the NEXRAD network).

First, the rain rate absolute climatological probabilities P(R>R*) are computed by

counting the number of pixels over R* on the whole dataset and by dividing it by the

overall number of pixels. These distributions may encounter more or less large

Articles


169

fluctuations on the same geographical zone for the different area sizes. This may be due

to some defects in radar measurements (particularly attenuation with increasing range)

or to some climatologic inhomogeneities of rain processes in squares of some hundreds

of kilometres of side [Onof and Wheater, 1996]. Then, for the successive values of R*,

the corresponding *Rα is deduced from (16). The results are given in table 1.

South Dakota

R*=0.5 mm h-1 R*=1 mm h-1 R*=2.4 mm h-1 R*=5.7 mm h-1 R*=13 mm h-1 Area size km2

P(R>R*) *Rα P(R>R*) *R

α P(R>R*) *Rα P(R>R*) *R

α P(R>R*) *Rα

100×100 0.49% 2.6 0.25% 2.8 0.10% 3.1 0.04% 3.3 0.01% 3.6 200×200 0.56% 2.5 0.28% 2.8 0.12% 3 0.05% 3.3 0.02% 3.5 300×300 0.65% 2.5 0.33% 2.7 0.15% 3 0.07% 3.2 0.03% 3.4

Mississipi

R*=0.5 mm h-1 R*=1 mm h-1 R*=2.4 mm h-1 R*=5.7 mm h-1 R*=13mm h-1

Area

size (km²) P(R>R*) *R

α

P(R>R*) *Rα

P(R>R*) *R

α

P(R>R*) *Rα

P(R>R*) *R

α

100×100 1.8% 2.1 1.1% 2.3 0.63% 2.5 0.29% 2.8 0.12% 3

200×200 1.8% 2.1 1.1% 2.3 0.60% 2.5 0.29% 2.8 0.13% 3

300×300 1.8% 2.1 1.1% 2.3 0.61% 2.5 0.29% 2.8 0.13% 3

Ohio

R*=0.5 mm h-1 R*=1 mm h-1 R*=2.4 mm h-1 R*=5.7 mm h-1 R*=13 mm h-1

Area

size (km²) P(R>R*) *R

α

P(R>R*) *Rα

P(R>R*) *R

α

P(R>R*) *Rα

P(R>R*) *R

α

100×100 1.8% 2.1 1.1% 2.3 0.60% 2.5 0.26% 2.8 0.10% 3.1

200×200 2.0% 2 1.3% 2.2 0.72% 2.5 0.30% 2.7 0.10% 3.1

300×300 2.1% 2 1.4% 2.2 0.73% 2.4 0.28% 2.8 0.09% 3.1

Bordeaux

R*=1 mm h-1 R*=1.4 mm h-1 R*=2.2 mm h-1 R*=3.4 mm h-1

Area size (km²) P(R>R*) *Rα

P(R>R*) *Rα

P(R>R*) *Rα

P(R>R*) *Rα

100×100 0.8% 2.4 0.5% 2.6 0.2% 3 0.1% 3.2

200×200 1.2% 2.3 0.8% 2.4 0.4% 2.7 0.2% 3

Table 1: Rain rate absolute CDFs P(R>R*) derived from radar observations for the

successive areas of interest and associated value of *Rα computed from (17).

Articles


170

For R*=1 mm h-1 only, model (17) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−=

−

σσα

σα2

)2(erfc)1(2erfc

21),,F(f

121

1iif

is regressed with respect to the fractional area CDF P1(f>f*) derived from radar

observations to obtain parameters σ reported in table 3 (first line, area size

N×N=100×100 km² located over the US and over Bordeaux). The values of σ were

obtained solving (19) using a non linear least square algorithm:

∑ ⎥⎦

⎤⎢⎣

⎡>

>−

i i

ii

ffPffPfF

2

1

11

)()(),,(minarg σα

σ, (19)

where α1 is determined from (16) and the fi are regularly sampled from 0.01 to

fmax by step of 0.01. fmax is defined as the 30th largest value of rain occupation found on

each radar data set for each considered threshold R*. This choice of fmax is made in

order to ensure a sufficient statistical reliability to the empirical CDFs especially for the

greatest values of occupation. This regression procedure allows defining a relative error

criterion between the model (17) and the CDF regressed from radar data:

∑⎥⎥⎦

⎤

⎢⎢⎣

⎡

>

>−×=

i iR

iRRi

ffPffPfF

N

2

)()(),,(1100

*

** σαε (20)

(20) quantifies the error made when approximating the empirical occupation CDF

by model (19) in terms of mean relative RMS error.

The values of σ, obtained using this regression procedure for each dataset are

used for all the other values of R* as σ should not depend on the threshold whenever rain

fields are stationary and homogeneous. Besides, R*=1 mm h-1 is the value considered in

the regression process because, first, it is the lowest value common to both datasets, and

second, it complies with Krajewski et al., [1992] that advocates the use of a relatively

low threshold to better estimate the fractional area from real data.

For the successive threshold values R*, Fig. 5 and 6 show the fractional area

CDFs derived from model (17) and those derived from the radar observations over

Articles


171

Bordeaux (France), for area sizes N×N=100×100 km² (σ regressed for R*=1 mm h-1 is

0.88) and N×N=200×200 km² (σ regressed for R*=1 mm h-1 is 0.77), respectively.

Similarly, the results obtained for Ohio (US) are shown in Fig. 7, for N×N=100×100

km² (σ regressed for R*=1 mm h-1 is 0.9). Lastly, Fig. 8 and 9 underline the fractional

area CDF dependence on the geographical location and the size of the area, respectively.

The impact of the location on the distribution is mainly driven by the local rain rate

CDF while increasing the area size lowers the average value σ² of the correlation

function.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.96

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1CDF of the fraction of the area with a rain rate exceeding R

f*

P(f

<f* )

1mm/h1.4 mm/h2.2 mm/h3.4mm/h

Figure 5: CDFs of the fractional area f affected by rain over a given threshold R*

computed from (18) (markers) and derived from the radar of Bordeaux (solid lines)

N×N=100×100 km².

Articles


172

0 0.1 0.2 0.3 0.4 0.50.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1CDF of the fraction of the area with a rain rate exceeding R

f*

P(f

<f* )

1mm/h1.4 mm/h2.2 mm/h3.4mm/h


computed from (18) (markers) and derived from the radar of Bordeaux (solid lines)

N×N=200×200 km².

Nevertheless, in compliance with Fig. 5 to 9, model (17) satisfactorily compares

with the fractional area CDFs derived from radar observations whatever the threshold

R*, the location, or the area size N×N ranging 100×100 to 300×300 km². However, some

trends can be noticed considering the errors computed according to (20) for different

areas, rain rate thresholds and locations as shown in table 2. Firstly, the larger are the

considered areas, the higher the errors between the two distributions. Secondly, for

threshold values ≥*R 5 mm h-1 for US radar and ≥*R 3.4 mm h-1 for Bordeaux, the

model tends to overestimate the probability of having a large fraction of the area

affected by rain.

Articles


173

Rain rate threshold 100x100 km2 200x200 km2 300x300 km2 Bordeaux R=1 mm h-1 0.54 0.82 1.9 R=3.4 mm h-1 2.4 2.3 8.4 Ohio R=1 mm h-1 1.5 3.2 8.5 R=5 mm h-1 7.1 14 21 Dakota R=1 mm h-1 0.65 1.2 3.9 R=5 mm h-1 6.7 14 13 Mississippi R=1 mm h-1 2.5 4.0 4.6 R=5 mm h-1 8.1 13 17

Table 2: Mean relative RMS error as defined by (21) between the model (18) and the

empirical CDF deduced from radar observations (%) for different rain rate thresholds

and different areas.

0 0.1 0.2 0.3 0.4 0.5 0.60.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

f*

P(f

<f* )

CDF of the fraction of the areaaffected by rain rates over R

0.5 mm/h1 mm/h2.4 mm/h5.7 mm/h13 mm/h


computed from (18) (markers) and derived from radar observations over Ohio (solid

lines) N×N=100×100 km².

Articles


174

0 0.1 0.2 0.3 0.4 0.50.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

f*

P(f

<f* )

CDF of the fraction of the areaaffected by a rain rate over 1mm/h

DakotaMississipiOhioBordeaux

Figure 8: CDFs of the fractional area with a rain rate over 1 mm.h-1 for areas of

100×100 km² located in the 4 successive geographical places of interest.

0 0.2 0.4 0.60.9

0.92

0.94

0.96

0.98

1

f*

P(f

<f* )

CDF of the fraction of the areaaffected by rain rates over 1mm/h

100km radar100km model300km radar300km model

Figure 9: CDFs of the fractional area with a rain rate over 1 mm.h-1 for areas with size

100×100 km² and 300×300 km² located over Mississippi.

Articles


175

These two trends question the rain field homogeneity hypothesis used to derive

(10) and (17). Indeed, over areas of hundreds of square kilometres, the rain rate local

CDF may differ from one point to the other. In such conditions, the model no longer

holds because the stationarity and homogeneity of the field become an unrealistic

assumption. In other respects, the differences observed for increasing values of R* may

be linked to the use of a constant parameter to describe the average value σ² of the

correlation function. Indeed, high rain rates result from a convective process. Now, the

radar observation of convective rain fields shows clustered structures, whose spatial

extent is clearly lower than that observed for stratiform rain events. This point is

partially confirmed when regressing model (17) for R*=5 mm h-1 (US radar) and R*=3.4

mm h-1 (Bordeaux) to determine σ. Indeed, the results reported in table 3 show that,

whatever the location, σ is slightly lower than the value obtained for R*=1 mm h-1,

suggesting (as expected) that high rain rates are less correlated in space than lower rain

rates because the average of the correlation of the underlying Gaussian fields for rain

fields thresholded with high values is lower than the one found for lower thresholds.

This may be due to a faster decay of the correlation considering rain fields with a more

significant proportion of convective rain.

Dakota Mississipi Ohio Bordeaux R*=1 mm h-1 0.86 0.90 0.92 0.88 R*=5 mm h-1 0.83 0.87 0.90 0.85 (R*=3.4 mm h-1)

Table 3: Regressed value of σ for 2 rain rate thresholds R* and for a 100×100 km² area

of interest.

Articles


176

4.4 Parameterization of the distribution

The results obtained for the parameter σ of the distribution display an interesting

stability as shown in Fig. 10.

100 150 200 250 3000.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

Length of the side

σ

MississipiOhioSouth DakotaBordeauxBest regression

Figure 10: Dependence of the parameter σ on the size of the considered area for

various locations.

Indeed, for different climatic regions, this parameter seems to have

approximately the same behaviour with regards to the evolution of the size of the area.

A generic parameterization of σ independent from the location can hence be regressed

and given as an indication for areas ranging from 100×100 to 300×300 km2:

L0007.094.0 −=σ , (21)

where L is the length of the side in km. The relatively high values found for σ

argue in favour of the existence of a long range correlation confirming the hypothesis

made in the second section of this study. (21) allows to get for an arbitrary location an

approximation of the distribution of the fraction of an area L×L affected by a rain rate

over a given threshold, whenever the local distribution of rain rates is known. Indeed,

from Rμ , Rσ and 0P where Rμ , Rσ are the parameters of the lognormal pdf that

Articles


177

characterizes the conditional distribution of rain rates and 0P is the probability of rain,

the parameter α needed to compute the distribution of the fraction of the area affected

by rain can be deduced from (16).

When combined with (17), parameters α and σ obtained that way allow to

approximate the CDF of the fraction of an area affected by rain for areas ranging from

100×100 to 300×300km2. The lower bound of this domain is driven by the requirements

on the size of the area with regards to the alleged underlying correlation function. The

upper bound is set considering the hypothesis of stationarity of the rain fields and of

spatial invariance of the point rainfall rate distribution across the areas that were

considered to establish the model. It should be noticed that the data used in this study

have been projected on a grid of 1×1 km2. Considering data with a lower spatial

resolution, the parameterization proposed will no longer hold. First, the standard

deviation of the distribution of rainfall rate that is used to determine the thresholds will

change. Second, the parameter 2σ that corresponds to the spatial average of the

correlation is likely to be slightly higher as the reduction of the resolution will smooth

the data and consequently generates higher correlation.

5 Concluding remarks and applications

From considerations on stationary Gaussian random fields, a model that

reproduces the CDF of the fractional area affected by rain over a preset threshold has

been proposed for areas ranging from 100×100 to 300×300 km2. It has been developed

assuming that rain fields can be seen as a non-linear transformation of a stationary

Gaussian random field.

First, the model was shown to match very accurately the fractional area CDF

derived from simulated Gaussian random fields. Second, when confronted to

distributions obtained from radar observations of rain field, the model satisfactorily

reproduces the CDF of the fractional area above a rain threshold R*. This model

requires two inputs. The first one α is related to the local distribution of rain rates, the

second one σ to the mean of the correlation function of the underlying Gaussian field.

A parameterization accounting for the local climatology and for the size of the area was

proposed for this parameter. The optimal value of σ was found to be slightly dependent

Articles


178

on the threshold considered on radar data, questioning the hypothesis of the stationarity

of the field.

The high values found in any case for the parameter σ put forward the long

range dependence that has to be introduced in the correlation function of the Gaussian

field when generating rainfall fields. Important conclusions can be drawn from those

results for simulation purposes. For models such as those described by Bell [1987] or

Guillot and Lebel [1999], the use of a correlation function that has a mean value of 2σ

allows, when generating a large collection of simulated fields, to reproduce the local

distribution of rain rates and the distribution of the fraction of the area affected by rain

over a given threshold. Additionally, the distribution of the fraction of the area affected

by rain is required by rainfall field generators such as the one described in Feral et al.,

[2006]. The methodology presented here provides a convenient mean to approximate

this distribution without computing it from radar data.

Acknowledgment:

The authors are very grateful to Météo-France and the US NEXRAD network for

providing freely radar images. This research was partly supported by CNES (contract

n°05/CNES/2038/00) and Thales Alenia Space (contract n°A88550) and has been partly

carried out in the framework of the European Network of Excellence SatNEx.

Bibliography

Atlas, D., D. Rosenfeld and D.A. Short (1990), The estimation of convective

rainfall by area integrals, 1, the theoretical and empirical basis. J. Geophys. Res.,

95(D3), 2153-2160.

Barbaliscia, F., G. Ravaioli and A. Paraboni (1992), Characteristics of the spatial

statistical dependence of rainfall rate over large areas. IEEE Trans. Antenn. and Prop.,

40(1), 8 – 12, doi:10.1109/8.123347.

Articles


179

Bell, T.L. (1987), A space-time stochastic-dynamic model of rainfall for satellite

remote sensing studies. J. Geophys. Res., 92(D8), 9631–9643.

Bertorelli, S., and A. Paraboni (2005), Simulation of Joint Statistics of Rain

Attenuation in Multiple Sites Across Wide Areas Using ITALSAT data. IEEE Trans.

Antenn. and Prop., 53(8), 2611-2622.

Borgman, L., M. Taheri, and R. Hagan (1984), Three-dimensional frequency-

domain simulations of geological variables, Geostatistics for Natural Resource

Characterization, G.Verly, M.David, A.G.Journel, and A.Marechal, 517-541.

Braud, I., J. Creutin, and C. Barancourt (1993), The relation between the mean

areal rainfall and the fractional area where it rain above a given threshold. J. Appl.

Meteor., 32(2), 193-202, doi: 10.1175/1520-0450(1993)032<0193:TRBTMA>2.0.CO;2

Capsoni, C., F. Fedi, and A. Paraboni (1987a), A comprehensive

meteorologically oriented methodology for the prediction of wave propagation

parameters in telecommunication applications beyond 10 GHz. Radio Sci., 22(3), 387-

393.

Capsoni, C., F. Fedi, C. Magistroni, A. Paraboni, and A. Pawlina (1987b), Data

and theory for a new model of the horizontal structure of rain cells for propagation

applications. Radio Sci., 22(3), 395-404.

Castanet, L., J. Lemorton, T. Konefal, A. Shukla, P. Watson, and C. Wrench

(2001), Comparison of combined propagation models for predicting loss in low-

availability systems that operate in the 20GHz to 50GHz frequency range. Int. J. Satell.

Commun, 19(3), 317-334. Int. J. Satell. Commun, 20(4), 231-249.

Castanet, L., D. Mertens and M. Bousquet (2002), Simulation of the performance

of a Ka-band VSAT videocomferencing system with uplink power control and data rate

reduction to mitigate atmospheric propagation effects.

Articles


180

Chakravarti I. M., R. C. Laha, and J. Roy (1967), Handbook of Methods of

Applied Statistics, Vol I, John Wiley and Sons, 392-394, Hoboken, NJ.

Chiu, L. S. (1988), Estimating areal rainfall from rain area. Tropical Rainfall

Measurements, J.S Theon and N. Fugono, Eds., A. Deepak, 361-367.

Donneaud, A. A., S. I. Niscov, D. L. Priegnitz, and P. L. Smith (1984), The area-

time integral as an indicator for convective rain volume. J. Appl. Meteor., 23(4),

10.1175/1520-0450(1984)023<0555:TATIAA>2.0.CO;2.

Eltahir, E. A. B. and R. L. Bras (1993), Estimation of the Fractional Coverage of

Rainfall in Climate Models, Journal of Climate, 6(4), 639-644, doi: DOI: 10.1175/1520-

0442(1993)006<0639:EOTFCO>2.0.CO;2.

Féral, L., H. Sauvageot, L. Castanet, and J. Lemorton (2003a), HYCELL—A new

hybrid model of the rain horizontal distribution for propagation studies: 1. Modeling of

the rain cell, Radio Sci., 38(3), 1056, doi:10.1029/2002RS002802.

Féral, L., H. Sauvageot, L. Castanet, and J. Lemorton (2003b), HYCELL—A new

hybrid model of the rain horizontal distribution for propagation studies: 2. Statistical

modeling of the rain rate field, Radio Sci., 38(3), 1057, doi: 10.1029/2002RS002803.

Féral L., H. Sauvageot, L. Castanet, J. Lemorton, F. Cornet, and K. Leconte

(2006), Large-scale modeling of rain fields from a rain cell deterministic model, Radio

Sci., 41, RS2010, doi:10.1029/2005RS003312.

Ferraris, L., S. Gabellani, U. Parodi, N. Rebora, J. Von Hardenberg, and A.

Provenzale (2002), Revisiting Multifractality in Rainfall Fields. J. of Hydromet., 4(3),

544-551, doi: 10.1175/1525-7541(2003)004.

Fuentes, M. (2005), A formal test for nonstationarity of spatial stochastic

processes. J. Multivar. Anal., 96(1), 30-54.

Articles


181

Goldhirsh, J. (2000), Two-dimension visualization of rain cell structures, Radio

Sci., 35(3), 713–729.

Guillot, G. (1999), Approximation of Sahelian rainfall fields with meta-Gaussian

random functions. Part 1: Model definition and methodology. Stoch. Environ. Res. Risk

Ass., 13(1), doi:100-112, doi:10.1007/s004770050034.

Guillot, G., T. Lebel (1999), Approximation of Sahelian rainfall fields with meta-

Gaussian random functions. Part 2: Parameter estimation and comparison to data. Stoch.

Environ. Res. Risk Ass., 13(1), 113-130, doi :10.1007/s004770050035.

Kedem, B., L. S. Chiu, and Z. Karni (1990), An analysis of the threshold method

for measuring area-average rainfall. J. Appl. Meteor, 29(1), 3-21, doi :10.1175/1520-

0450.

Krajewski, W. F., M. L. Morrissey, J. A. Smith, and D. T. Rexroth (1992), The

accuracy of the area-threshold method: A model-based simulation study. J. Appl.

Meteor., 31(12), 1396–1406, doi :10.1175/1520-0450.

Lantuejoul, C. (1991), Ergodicity and integral range, Journal of Microscopy,

161(3), 387-404.

Lebel, T., I. Braud and J-D. Creutin (1998), A space-time disaggregation model

adapted to Sahelian squall lines. Water Resour. Res., 34(7), 1711-1726, 1998,

doi:10.1029/98WR00434.

Le Cam, L. (1961), A stochastic description of precipitation, in Proceedings of

the fourth Berkeley Symposium on Mathematical Statistics and Probability, vol III,

edited by J. Neyman, pp. 165-186, University of California Press, Berkeley, California.

Lovejoy, S., and D. Schertzer (1985), Generalized scale invariance in the

atmosphere and fractal models of rain, Water Resour. Res., 21, 1233-1250.

Articles


182

Mejia, J. and I. Rodriguez-Iturbe (1974), On the synthesis of random field

sampling from the spectrum : an application to the generation of hydrologic spatial

process. Water Resour. Res., 10, 705-711.

Neely, M. J., E. Modiano, and C. E. Rohrs (2003), Power Allocation and Routing

in Multi-Beam Satellites with Time Varying Channels, IEEE Trans. on Networking, 11(

1), 138-152.

Oki, R., A. Sumi, and D.A. Short (1997), TRMM Sampling of Radar–AMeDAS

Rainfall Using the Threshold Method. J. Appl. Meteor., 36(11), doi:10.1175/1520-

0450(1997)036<1480:TSORAR>2.0.CO;2.

Onof, C., and H. S. Wheater, (1996), Analysis of the spatial coverage of British

rainfall fields, Journal of Hydrology , 176(1), 97 – 113, doi:10.1016/0022-

1694(95)02770-X.

Over, T. , and V. K. Gupta (1996), A space-time theory of mesoscale rainfall

using random cascades. J. Geophys. Res., 101(D21), 26319-26331.

Pitman, A. J. (1991), Sensitivity of the land surface sub-grid scale process

implications for climate simulations. Plant Ecology, 91(1), 121-134,

doi:10.1007/BF00036052.

Pitman, A. J., and A. Henderson-Sellers (1990), Z-L. Yang: Sensitivity of

regional climates to localized precipitation in global models. Nature, 346, 734-737,

doi:10.1038/346734a0.

Thomas, G, and A. Henderson-Sellers (1991), An evaluation of proposed

representations of subgrid hydrologic processes in climate models. J. Clim, 4(9), 898-

910, doi: 10.1175/1520-0442(1991)004<0898:AEOPRO>2.0.CO;2.

Articles


183

Sauvageot, H. (1994), The probability density function of rain rate and the

estimation of rainfall by area integrals. J. Appl. Meteor., 33(11), 1255-1262, doi:

10.1175/1520-0450(1994)033<1255:TPDFOR>2.0.CO;2.

Shinozuka, M., and C. M. Jan (1972), Digital simulation of random processes and

its applications. Journal of Sound and Vibration, 25(1), 111-128, doi :10.1016/0022-

460X(72)90600-1.

Articles

A large scale space-time stochastic model of rain attenuation for the design and optimization of

adaptive satellite communication systems operating between 10 and 50GHz

184



Nicolas Jeannin (1), Laurent Féral (2), Henri Sauvageot (3),

Laurent Castanet (1), Frédéric Lacoste (4).

(1)ONERA, Département Electromagnétisme et Radar, Toulouse, France

(2)Université Paul Sabatier, LAboratoire Micro-ondes et Electromagnétisme, Toulouse,

France (3)Université Paul Sabatier, Laboratoire d’Aérologie, Toulouse, France

(4)Centre National d’Études Spatiales (CNES), Toulouse, France

Manuscript submitted to IEEE Antennas and Propagation

Articles



185

Abstract:

The design and optimization of propagation impairment techniques for space

telecommunication systems operating at frequencies above 20 GHz require a precise

knowledge of the propagation channel both in space and time. For that purpose, space

time channel models have to be developed. In this paper the description of a model for

the simulation of long term rain attenuation time series correlated both in space and time

is described. It relies on the use of a stochastic rain field simulator constrained by the

rain amount outputs of the ERA-40 reanalysis meteorological database. With this

methodology, realistic propagation conditions in terms of spatio-temporal correlation

can be generated at the scale of satellite coverage (i.e. over Europe or USA). To increase

the temporal resolution, a stochastic interpolation algorithm is used to generate spatially

correlated time series sampled at 1 Hz, providing that way valuable inputs for the study

of the performances of propagation impairment techniques required for adaptive SatCom

systems operating at Ka band and above.

Articles



186

Introduction

With the congestion of conventional frequency bands such as C (4-6 GHz) or Ku

(11-14 GHz) band and the need to convey higher data rates for multimedia services, new

Sat Com systems are progressively pushed towards the use of higher frequency bands

such as Ka (20-30 GHz) or Q/V band (40-50 GHz) where larger bandwidth are available.

However, at those frequencies, strong tropospheric impairments occur on Earth-space

links with a significant impact on the system quality of service. The latter are due to

rain, clouds, water vapor and oxygen. The resulting attenuation can not be mitigated

simply acting on the margin taken on the overall link budget as it will result in a major

loss of efficiency. In order to perform a good trade off between quality of service,

system capacity and link availability, adaptive resource allocation mechanisms using

Fade Mitigation Techniques (FMT) have to be implemented [Cost 255, 2002].

FMT implementation leads also to consider the system resource allocation and

therefore upper layer issues that can be studied through protocol simulations. Network

simulations for SatCom systems operating at Ka and Q/V-bands have to take into

account the influence of the propagation channel, not only in terms of dynamics but also

in terms of the spatial variations [Bousquet et al., 2003]. In order to assess FMT

efficiency or the resulting system availability, the propagation channel has to be

simulated.

To emulate channel dynamic, attenuation time series synthesizers [Masseng and

Bakken, 1981], [Gremont and Filip, 2004], [Carrier et al., 2008] have been developed.

They are crucial to design and optimize Up Link Power Control (ULPC) systems, or

data rate reduction mechanisms [Castanet et al., 2001]. Nevertheless, the knowledge of

the temporal dynamic of the propagation channel is not sufficient to implement all the

FMTs. For instance, the optimization of site diversity systems requires also the

knowledge of the spatial variability of the propagation impairments.

Several models and methods exist to depict the spatial variability of tropospheric

impairments, especially the attenuation due to rain. Indeed, attenuation fields and the

underlying rain fields can be modeled by rain cells [Capsoni et al., 1988a, 1988b],

Articles



187

[Féral et al., 2003a, 2003b, 2006] [Montopoli and Marzano, 2008], by fractals

[Callaghan, 2006] or random fields [Gremont and Filip, 2004]. Even if the spatio

temporal description of the propagation channel is of great interest for the design and

optimization of the above mentioned FMTs, it is not adapted to solve the issues raised

by on-board radio resource management for which a description both in space and in

time of the propagation channel over the whole satellite coverage is required.

Up to now, few models exist to combine the temporal and the spatial description

of the propagation channel. [Gremont and Filip, 2004] have adapted the model of rain

fields correlated in space and in time developed by [Bell, 1987], while [Bertorelli and

Paraboni, 2005] have proposed to correlate scaled time series from measurements made

during the ITALSAT campaign. The main limitation of those models lies in their

temporal or spatial range of validity. Indeed, the model proposed by [Gremont and Filip,

2004] is realistic for short durations (some hours) and small areas (a few hundreds of

square kilometers) due to the stationarity assumption necessary to construct the random

field in the Fourier plane. The model of correlated time series presented in [Bertorelli

and Paraboni, 2005] is limited to duration of one hour. As network simulations, to be

statistically reliable, require attenuation data during several years and over the satellite

coverage as a whole, there is a need for propagation inputs correlated in space and in

time for long durations and large areas.

The aim of this paper is to extend the range of validity of the stochastic rainfall

field simulator described in [Bell, 1987] and adapted for rain attenuation in [Gremont

and Fillip, 2004], using rain amount time series provided by general circulation models.

Indeed, those stochastic models are shown to reproduce appropriately the spatial and

temporal characteristics of observed rain fields, but for limited areas and durations. To

overcome these limitations, a methodology to constrain the model with inputs from the

low-resolution reanalysis database ERA-40 provided by the ECMWF (European Center

for Medium-range Weather Forecast) is presented. The use of ERA-40 data ensures a

realistic long term evolution of the rain fields as they reproduce large-scale conditions

from past meteorological situations.

The basics and limitations of the stochastic modelling of rain rate field presented

in [Bell, 1987] are recalled in the first part of the paper. A set of parameters more

suitable for temperate areas is also determined from a radar dataset acquired by the

Articles



188

weather radar from Bordeaux (France). In a second part of the paper, a methodology to

parameterize the model with outputs from the ECMWF ERA-40 reanalysis database is

developed. Some statistical properties of the simulations are lastly compared to statistics

derived from weather radar datasets, ITU-R Recommendations and to statistics obtained

from beacon measurements.

1 Spatio-temporal modelling of rain fields

In this section, the basics of the rain field space-time stochastic modelling firstly

presented in [Bell, 1987] are detailed. This model was originally developed to study the

performances of satellite rainfall sensors (i.e the precipitation radar of TRMM) from

simulated rain fields which statistical characteristics must be as close as possible to the

ones of real rain fields. It was constructed to mimic the correlation structure and the

local climatologic Cumulative Distribution Function (CDF) of rain rates given as input

parameter.

The rain fields R(x,t) are generated on a domain with size N×N (grid step Δx)

during a time interval T (time step Δt) where x is the spatial index and t the temporal

one. It is assumed that for each spatial location and each time index, the rain rate is the

realization of a random variable. Using this methodology, the simulation is conducted in

two main steps. First, a stationary Gaussian random field G(x) with correlation structure

cG is generated in space. Second, to account for the temporal evolution, each spatial

frequency k of G(x) is constrained to follow a first order Markov process which

innovation depends on the spatial frequency. Moreover, the field advection is modelled

by a phase shift in the frequency domain.

Then, the Gaussian field is turned into a rainfall field converting the samples

from the Gaussian field G(x,t) into rainfall rate samples R(x,t). For that purpose, the

rainfall rate distribution is assumed to be zero with probability 1-P0 and lognormal with

probability P0, where P0 denotes the local probability of rain.

The different steps of this modelling are detailed in this section. A methodology

to derive the correlation parameters from weather radar observations is also presented. It

will be applied to the radar observations of Bordeaux.

Articles



189

1.1 Generation of correlated Gaussian random field

A convenient framework to simulate a stationary random field R that has an

arbitrary distribution is to define a monotonic transformation ψ that converts a Gaussian

field G into the field R [Guillot, 1999]. It allows taking advantage of efficient methods

for the simulation of Gaussian fields and obtaining the desired field R from ψ(G). If the

correlation cR is estimated from realizations of the field R, different methodologies

described in section 1.3.2 exist to define a correlation cG such that R=ψ(G) will have the

correlation cR.

1.1.1 Generation of the Gaussian field in space

A Gaussian random field can be interpreted as a collection of Gaussian random

variables correlated the ones with each other. A Gaussian random field is said to be

stationary if its correlation function cG between two points x and y depends only on the

distance between x and y:

)yx(c)y,x(c GG −= (1)

This property of stationarity of the field is essential from a computational point

of view. Indeed, it allows reducing the size of the correlation matrix from N2×N2 to N×N

to use efficient algorithms such as turning band methods [Mantoglou and Wilson, 1982]

or circular embedded correlation matrices [Kozintsev and Kedem, 2000].

Another convenient methodology to generate a stationary Gaussian field G is to

start from its spectrum s that is directly obtained from the Fourier transform of the

correlation function cG [Bell, 1987]. The N×N stationary Gaussian random field G can

be expressed for each point ),( 21 xxx = of the simulation grid by its Fourier expansion:

∑ ∑−

=

−

=

=1

0

1

01 2

)/2exp()(N

k

N

k

Tk NxkiaxG π , (2)

where ),( 21 kkk = represents the spatial frequency. Coefficients ak are shown to

be equal to:

kkk esa = , (3)

where sk is the spectral density corresponding to the spatial frequency k and the ek

are independent standard centred complex Gaussian random variables. (2) and (3) show

Articles



190

that the knowledge of the power spectrum of the field or reciprocally of its correlation

function cG is sufficient to generate a spatially correlated Gaussian field.

1.1.2 Addition of the temporal dimension

The rain field temporal evolution can be decomposed into two components. The

first one is related to the advection of the rain fields due to air streams. The second one

corresponds to the rain cell evolution along their path. In [Bell, 1987], both terms are

taken into account separately in the simulation algorithm. The temporal evolution of the

rain field is introduced through the spatial spectrum of the Gaussian field. Considering

that each spatial frequency evolves according to a first order Markov process, the

relation between the spatial spectral components ak defined by (3) of the field at time

step t and at time step t+Δt is given by:

( ) ttkkkkT

tkttk estVikaa Δ+Δ+ −+Δ−= ,2

,, 1)exp( ββ , (4)

where V represents the advection vector, sk the spectral density of the spatial

process corresponding to the spatial frequency k and ek,t+Δt are uncorrelated complex

standard centered Gaussian random variables. βk represents the autocorrelation lag of the

Markov process corresponding to each spatial frequency. In (4), the term

)exp(, tVika Ttk Δ− represents the field spectral component at spatial frequency k and time

step t but with a phase shift that corresponds to the rain advection between the time step

t and the time step t+Δt. The term ttkk es Δ+, corresponds to the innovation of the

process. The variance of ak,t is assumed to be constant in time, thus preserving the

spatial correlation of the Gaussian field when performing the Fourier expansion (2). The

parameterization proposed in [Bell, 1987] (i.e. )/exp( kk t τβ Δ−= with kτ representing

the time autocorrelation-lag for each spatial frequency) complies with the definition of a

Markov process. This parameterization of the temporal evolution induces a non-

separable spatio-temporal correlation function as it can not be expressed as the product

of a spatial correlation function by a temporal one [Fuentes, 2005]. The

parameterization and the choice of this temporal evolution will be discussed in Section

2.

Articles



191

1.2 Conversion into rain rate fields

In order to convert the Gaussian random fields obtained from (2) following the

temporal evolution driven by (4) into rain fields, the local statistical distribution of rain

rate has to be known. This distribution is highly dependent on the geographical area. For

that purpose, Rec. ITU-R P.837-5 aims at giving an estimation of the rainfall rate

Cumulative Distribution Function (CDF) all over the world [Poiares Baptista and

Salonen, 1998]. The methodology followed to derive this CDF can be found in and was

updated in [Castanet et al., 2007]. It relies on input parameters computed from the ERA-

40 ECMWF database. However the analytical shape of the model is rather sophisticated.

For our purpose, the use of a lognormal approximation [Sauvageot, 1994] for the

conditional rainfall rate distribution was found to be more relevant as the moments of

the distribution can be easily computed. For temperate and terrestrial areas, the

difference between the CCDF (Complementary CDF) of rainfall rate derived from Rec.

ITU-R P.837-5 and the log normal model is very weak as illustrated on 233HFigure 1.

10−2

10−1

100

101

0

5

10

15

20

25

30

35

40

45

P(R>R*)

R*

mm

h−1

CDF of rainfall rate from ITU−R Recommandation

BordeauxHelsinkiMadridLondon

Figure 1: CCDF of rain fall rate from Rec. ITU-R P.837-5 (crosses) and best lognormal

regression (solid line) for various places.

Consequently, in the following, the rainfall rate CCDF P(R≥R*) at any location

will be approximated by:

Articles



192

( )

1)0R(P

0R if dr2

²rlnexpr2

1P)RR(P *

R2R

R2R

0*

*

=≥

>⎥⎦

⎤⎢⎣

⎡ −−=≥ ∫

+∞

σμ

σπ , (5)

where Po=P(R>0) is obtained directly from Rec. ITU-R P.837-5, and where μR

and σR are regressed by least square fitting from the ITU-R conditional CCDF

P(R≥R*|R>0). In order to convert the generated Gaussian field into rainfall rate fields,

[Bell, 1987] advocates the use of the following transformation:

[ ]

[ ] ασμϕ

αϕ

≥⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+==

<==

− )x(G if P/2

)x(Gerfcerfc 2exp)x(G)x(R

)x(G if 0)x(G)x(R

01

RR

, (6)

where )2(2 01 Perfc−=α with erfc the complementary error function and erfc-1

its inverse, and where μR, σR and Po are the parameter of the lognormal law in (5). The

transformation (6) is established considering that )GG(P)RR(P *1*1 ≥≥= −− oϕ where

)RR(P *1 ≥− is the inverse of the CCDF of rainfall rate given by (5) and )GG(P *≥

represents the CCDF of a standard centered normal random variable. Therefore, (6)

allows the conversion of a Gaussian random variable into a random variable whose

distribution is given by (5). However, transformation (6) is shown not to perform

completely satisfactorily and a refinement of this transformation will be proposed in the

next section.

1.3 Parameterization and refinement of the modeling

The spatial and temporal correlations in the model proposed by [Bell, 1987] are

the key parameters that need to be accurately assessed in order to obtain a realistic

description of the statistical dependence of rain rates in space and in time. In the initial

work of [Bell, 1987] those correlations were determined from radar data sets collected

over tropical oceans during the experimental campaign GATE. As the spatial and

temporal characteristics of the rain fields over oceanic tropical areas are likely to be

different from the ones over continental temperate areas, a retrieval of those

characteristics from mid-latitude datasets is undertaken. For that purpose, the spatial

correlation function and the parameters that drive the temporal evolution are determined

Articles



193

from a weather radar dataset collected over Bordeaux (France) whose main

characteristics are described hereafter in. A slight modification of transformation (6)

that turns the Gaussian fields into rain rate fields while preserving features observed on

the radar data is also presented.

1.3.1 Description of the dataset

The data come from the weather radar located at Bordeaux (44.5oN, -0.5oE),

South western France, in the vicinity of the Atlantic Ocean. The radar is a S-band radar

which is part of the French operational radar network managed by Météo France. The

polar scans acquired each 5 mn are projected into a Cartesian grid with a pixel size of

1×1 km2. The dataset contains 35286 scans acquired from January to December 1996.

Images including ground clutter or melting layer echoes were removed from the data set

so that the used radar data only refer to rainfields. The conversion of the reflectivity into

rain rate was made using the standard Z-R relation for mid-latitudes:

Z=300×R1.5,

where Z is the radar reflectivity factor in mm6.m-3 and R the rainfall rate in

mm.h-1.

1.3.2 Refinement of the conversion of Gaussian fields into rain fields

If transformation (6) proposed by [Bell, 1987] for the conversion of Gaussian

fields into rain fields allows preserving the local distribution of rain rate, it introduces

an unrealistic feature on the generated rain fields. Indeed, (6) introduces on the

generated field a non-realistic link between the fraction fR of a simulated rain field that

is affected by rain and the spatial average of the strictly positive rain rates values

<(R|R>0)> as illustrated on 234HFigure 2. This is in contradiction with the ergodicity

assumption of rain rate fields [Eltahir and Bras, 1993], [Sauvageot, 1994], [Nzeukou

and Sauvageot, 2002] that implies that <(R|R>0)> does not change with respect to the

fractional area fR affected by rain. For that purpose, a modification of (6) is proposed. It

keeps unchanged the point distribution of rainfall rates whenever a large number of

realizations are considered while suppressing this unrealistic link. Considering a

realization of the random field G, the spatial distribution of random variables

Articles



194

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−

01 P/

2)x(Gerfcerfc2 with G(x) above the threshold α is approximately normal with

mean μoff and with standard deviation σoff. Consequently, the transformation Ψ defined

by:

[ ]

[ ] ασσ

σμ

μψ

αψ

≥⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+−==

<==

− )x(G if P/2

)x(G erfcerfc2exp)x(G)x(R

)x(G if 0)x(G)y,x(R

01

off

R

off

offR

(7)

induces that the strictly positive samples of the rain rate field are lognormally

distributed with mean μR and standard deviation σR. Moreover, considering one point of

the simulation area, the rain rate R conditional distribution P(R≥R*|R>0) is also

lognormal with parameters μR and σR. (7) guarantees the statistical independence

between the fraction of the simulation area fR that is affected by rain and the conditional

mean rain rate <(R|R>0)> as illustrated on 235HFigure 2.

Figure 2: Relation between the fraction of the area that is affected by rain and

the conditional mean rain rate computed from 35 000 radar scans and from 35 000

simulations using the transformations φ and ψ.

Articles



195

1.3.3 Parameterization of the spatial correlation

The spatial correlation of the Gaussian field can be determined from the analysis

of radar data. Actually, each radar image can be assimilated to a realization of a rain rate

field R(x) where x is the spatial index in the horizontal plane. Considering

transformation (6), the correlation cG of the underlying Gaussian field G can be linked to

the covariance of the rainfall rate field R. In the following, r1 and r2 denote random

variables representing rainfall rates that are separated by a distance |x|. Moreover, we

assume that )( 11 gr ψ= and )( 22 gr ψ= , where g1 and g2 are standard centered variable

whose correlation is cG(|x|). In such conditions, the joint Probability Density Function

p(g1, g2) of g1 and g2 is given by:

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−+

−−

=)x(c1

gg)x(c2gg21exp

)x(c121)g,g(p 2

21g22

21

221gg

π (8)

Recalling that )2(2 01 Perfc−=α , the cross expectation of r1 and r2 can be linked

to the correlation between the two Gaussian random variables g1 and g2 by:

∫ ∫

∫ ∫∞ ∞

∞

∞−

∞

∞−

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−+

−−

=

=>=<

α α πψψ

ψψψψ

21221g

22

21

221

2121212121

dgdg)x(c1

gg)x(c2gg21exp

)x(c121)g()g(

dgdg)g,g(p)g()g()g()g(rr

gg

. (9)

(9) allows to get a relation between the rain rate cross expectation <r1r2> and the

correlation of the underlying Gaussian variables cG(|x|). As the numerical integration of

(9) is inaccurate and requires a significant computation time, the link between the

covariance of the rain field and the correlation of the Gaussian field is established using

a method based on Hermit polynoms, detailed in [Rivoirard, 1995] or [Guillot, 1999]:

kg

k

k xck

rrrr )(!1

2

2121 ∑∞

=

=−><ψ

, (10)

where kψ are the coefficients of the Hermit polynomial expansion of ψ . In order

to reduce the inaccuracy due to the finite number of terms in the sum (10) and its slow

convergence rate [Guillot, 1999], it was chosen to evaluate the covariance of the field

G~ . G~ is obtained from field R by the transformation )(~~ 1 RG −=ψ with )()(~ 11 xx −− =ψψ

Articles



196

if x>0 and αψ =− )(~ 1 x otherwise. Thus, G~ is a Gaussian field thresholded to α. It is

linked to the Gaussian field G by the relation: )G(G~ Ω= with Ω(x)=α if x≤α and

Ω(x)=x if x>α. The main advantage of this transformation is that the terms of the

Hermite expansion of φ are bounded and the series (11) converges rapidly thus allowing

a fast numerical computation:

kg

1k

2k

2121 )x(c!k

g~g~g~g~ ∑∞

=

=−><Ω . (11)

In order to estimate the covariance of the underlying Gaussian field,

transformation 1~ −ψ is first applied to the radar data, thus converting rain rate values

)(xR into values of a truncated Gaussian field )(~ xG . Afterwards, the covariance of the

truncated Gaussian field G~ is computed performing the inverse Fourier transform of the

spatial power spectrum of the field for each radar map. Finally, the correlation cG of the

Gaussian field G is computed inverting (11), considering the average covariance of field

G~ over all the radar maps. The resulting correlation function is presented on Figure 3 for

the radar data of Bordeaux. The obtained correlation is slightly anisotropic with a larger

correlation along a north south axis. Nevertheless, in the following, the correlation is

assumed to be isotropic.

Figure 3: Radar derived correlation function cG of the Gaussian field G and its

analytical approximation by (12).

Articles



197

This correlation function can be appropriately approximated by:

)800/exp(41.0)31/exp(59.0)( xxxcG −+−= . (12)

The analytical expression of the correlation function is close to the one derived

by [Paraboni and Barbaliscia, 1992] from rain gauge measurement across Italy. The

slowly decaying part of this correlation function denotes the existence of a long range

dependence of rain rates.

1.3.4 Parameterization of the temporal evolution

As mentioned in section 1.1.2, the temporal evolution of the rain fields proposed

by [Bell, 1987] can be decomposed into one term related to the advection of the rain

fields and the other related to the rain rate field dynamics along their tracks. [Bell, 1987]

has proposed to parameterize the evolution of the underlying Gaussain field G(x,t) given

by (4) according to:

)exp(k

kt

τβ Δ

−= , (13)

where τk=0.24×(π/k)2/3 is the autocorrelation time of the Fourier coefficients ak,t

of G. (13) has been regressed directly by [Bell, 1987] on successive radar maps.

Therefore, it corresponds to the time auto-correlation of rain fields and not to the one of

the underlying Gaussian field. Moreover, the effect of the advection was not isolated,

biasing the analysis. However, a rigorous parameterization of the temporal evolution of

the Gaussian field G(x,t) assuming model (4) can be derived from a radar dataset. First,

the rain fields have to be corrected for advection. To do it, the average advection vector

is computed from two successive radar observations of rain field, by maximizing their

spatial cross-correlation. It hence allows defining rain fields ),(ˆ txR whose average

motion is null. This correction of the advection allows getting rid of the shift phase

)exp( tiVkΔ− in the evolution equation (4). The temporal evolution of the Gaussian fields

),(ˆ txG such that [ ] )t,x(R)t,x(G =ψ should consequently obey to:

( ) ttkkkktkttk esaa Δ+Δ+ −+= ,2

,, 1 ββ , (15)

where ak,t are the coefficients of the spatial Fourier expansion of ),(ˆ txG , that are

also the coefficients of the Fourier expansion of the field G(x,t), and where the

Articles



198

coefficients βk are the one defined in (4). From (15), recalling that ek,t is normal,

standard, centered and uncorrelated, βk can be obtained considering the temporal

autocorrelation of the components of the spatial Fourier expansion of ),(ˆ txG :

ktktktkttk aaaa β,,,, =Δ+ . (16)

The term tkttk aa ,, Δ+ corresponds to the spatial cross spectral density of the

spatial processes )t,x(G and )tt,x(G Δ+ . The remaining step to retrieve the parameters

βk lies in the definition of a correspondence between the characteristics of the field

),(ˆ txR with the ones of the field ),(ˆ txG . For that purpose, an approach quite similar to

the one used to infer the spatial correlation of the Gaussian field is used. The advection

corrected rain rate fields ),(ˆ txR are transformed into truncated Gaussian fields applying

the transformation 1~ −ψ , giving truncated Gaussian fields ),(ˆ txGT corrected for

advection. The space-time correlation function ),(ˆ txcTG , can then be deduced from the

spatio-temporal power spectrum of the successive fields. (11) gives a method to define a

correspondence between the spatio-temporal covariance function of ),(ˆ txGT and the one

of ),(ˆ txG . The Fourier inversion in space of the correlation function ),(ˆ

txcG

gives the

spatial cross-spectral density tktkk aas ,,, ττ += for different time-lags τ. Its evaluation for

a time-lag τ=Δt allows the retrieval of βk for each spatial frequency and thus an estimate

of τk:

)log()log()log( ,,,, tktktkttkkk aaaa

tt−

Δ−=

Δ−=

Δ+βτ . (17)

To evaluate the value of τk from radar data, successive rainy maps corrected for

advection have been used. As the radar spatial coverage is limited, the correction of the

advection on the radar data has been restricted to areas of 100×100 km² and has not been

considered for duration longer than 2 hours. 25 rainy events R(x,t) of 2 hours corrected

for advection have been selected in the database. For each event, the value of τk has been

computed according to the methodology describes above. As shown in Figure 4, the τk

average dependency with respect to the spatial frequency k can be approximated by: 94.0)/1(06.0 kk =τ . (18)

Articles



199

Figure 4 : Evolution of the parameter τk as a function of the spatial frequency k.

(18) was found assuming that τk follows a power-law function of the spatial

frequency k. In compliance with Figure 4, the dependency of τk with respect to k

suggests, as expected, that the evolution of large features (rain field stratiform

component associated to small k) is slower (higher auto-correlation time τk) that the one

of smaller features (rain field convective component with limited spatial extent,

associated to highest k values).

1.4 Limitations of the presented methodology

Several hypotheses limit the validity range of the modeling proposed by [Bell,

1987]. The most significant one is the assumption of stationarity of the random field.

Even if [Ferraris et al, 2003] have showed that the stationarity of rain fields is realistic

for areas of some hundreds of square kilometers, this assumption is not likely to hold for

larger areas as orography or climatology related effects will introduce unstationarity in

the rain field. For that reason, the model presented above should be limited to areas of

some hundreds of square kilometers. Therefore, its use for areas of the size of a typical

satellite regional coverage (such as Europe or USA) is not appropriate. The same kind of

restriction applies for the simulation duration. Indeed, considering (13) and (18), the 0

Spatial frequency k (km-1)

Articles



200

frequency term of the temporal evolution coefficient β0 is equal to 1. Consequently, the

spatial average of the Gaussian field M=a0 remains constant. It hence prevents the

evolution of integrated values of the simulation such as the average rain amount during a

given duration or the fraction of the area that is affected by rain. As a consequence, the

alternation of clear sky and rainy period can not be reproduced by the simulated rain rate

fields as the amount of rain is approximately constant during the simulation. In order to

overcome those limitations a mean to constrain this stochastic model with inputs from

meteorological reanalysis database is described in Section 2. It will ensure to benefit

from the statistical robustness of this stochastic model for small scale features combined

with a realistic repartition of the generated rain amount over long durations and large

areas.

2 Extension of the model using inputs from meteorological reanalysis database

To cope with the problem of the temporal evolution of the fraction of the area

affected by rain and hence being able to generate realistic long-term correlated time

series of rain rate or rain attenuation, the use of GCM (General Circulation Model)

outputs is proposed. For that purpose, the use of ECMWF ERA-40 reanalysis database

[Uppala et al., 2005] is chosen as it offers a worldwide estimation of rainfall amounts

for durations long enough to be statistically representative. Though the spatial resolution

of this database is not fine enough to be directly converted into attenuation for earth-

space links or terrestrial links, some of its outputs are of great interest to be used as

input parameters for space time rain rate and rain attenuation field simulators. In the

following, the use of such a database is shown to overcome the main limitations of the

model presented in [Bell, 1987] or [Gremont and Filip, 2004], namely the unrealistic

temporal evolution of rain fields and the distribution of the rainy areas at large scales.

The purpose of this work is not to forecast propagation conditions from outputs of

numerical weather prediction as proposed by [Hogdes et al., 2008] but rather to generate

statistically plausible propagation scenarios from realistic past meteorological situations.

Articles



201

2.1 Description of the ECMWF ERA-40 dataset

The dataset used in this study is a subset from the ECMWF ERA-40 reanalysis

database freely available on the ECMWF website. This database strives at reproducing

the state of the atmosphere in the past (the considered period ranges from 1957 to 2002),

constraining a numerical weather forecast model with observations. The model internal

variables are pressure, temperature, wind and humidity profiles from which several other

parameters such as rain amounts are deduced. This database is provided with a spatial

resolution of 2.5o×2.5o with one sample each 6 hours. Nevertheless, some limitations

have been identified, especially the rainfall estimation. Indeed, as the vertical wind

information is not directly included in the meteorological model, the estimation of

convective rain is not reliable [Uppala et al., 2005]. In addition, the low resolution of

the model over land and different assimilation procedures between land and sea led to

some bias in costal and mountainous areas.

2.2 Parameterization of the rain amount

A relation between the rain amount of one ERA-40 resolution cell with the

fraction of this resolution cell that is affected by rain is presented in Section 2.2.1. Then,

this information related to the rain fractional area is introduced in the stochastic

approach developed in Section 1. Proceeding that way, a realistic (since driven by ERA-

40 output) spatio-temporal evolution of the simulated rain fields is insured at large scale

(continental scale) and for long durations (several hours).

2.2.1 Link between the rain amount from a GCM and the fraction of the area affected by rain

Many studies on rainfall remote sensing have put forward that if a sufficiently

large number of rain cells at different stages of their life-cycle are observed over an area

(between 1000 and 100000 km2), a tight link exists between the fraction fR of the area

that is affected by rain above a given threshold (the fractional area) and the area

averaged rain rate <R> [Donneaud, 1984], [Chiu, 1988]. As an example, 236HFigure 5 shows

the linear dependency of <R> with respect to the fractional area fR deduced from the

radar data of Bordeaux.

Articles



202

Figure 5 : Area averaged rain rate <R> as a function of the fractional area fR

above 0.5mm.h-1 from the radar data set of Bordeaux. The correlation coefficient with

the linear regression (red line) is 0.85.

This relation is all the tighter because a large number of rain cells are observed

simultaneously in the rain field, giving an exhaustive representation (in a statistical

sense) of the climatological rain rate distribution [Atlas and Bell, 1992].

Besides, [Eltahir and Bras, 1993] have shown that considering rainfall outputs

from low resolution GCM, the same kind of relation holds between the rain amount,

which corresponds to a rainfall rate averaged in space and in time, and the fraction of

the GCM pixel that is under rain:

TrVfR Δ

= , (19)

where V is the rain amount over the GCM pixel, r is the conditional mean rain

rate (i.e. knowing that it is raining) on the GCM considered area and TΔ is the time

interval during which the GCM rain amount is computed. From (5), the conditional

Articles



203

mean rain rate r can be obtained considering the lognormal local distribution of rainfall

rates:

r = )2

exp()0RR(2R

Rσμ +>=>< . (20)

Therefore, from the rain amount time series given by ERA-40, the associated

time series of the fractional area fR can be inferred worldwide from (19) and (20), every

TΔ =6 hours, over cells with size 2.5o×2.5o.

2.2.2 Parameterization of the mean value of the Gaussian field

In order to include the information derived from the reanalysis database in the

high resolution stochastic rain model, a link has to be found between the rain fractional

area fR and the simulation parameters. Recently, [Jeannin et al., 2008] have proposed a

model to link the fraction fG of the samples of one Gaussian field realization that are

above a given threshold with the average value MG of this realization. Under some

assumptions on the shape of the correlation function, the authors have shown that the

spatial distribution of the samples of one Gaussian field realization simulated on a finite

grid with size N×N is normal, with mean MG and standard deviation 1- 2Gσ , where 2

Gσ is

the spatial average of the Gaussian field correlation function over the N×N simulation

grid. In such conditions, [Jeannin et al., 2008] have shown that the fraction fG of a

simulated Gaussian field that is over a preset threshold αG is given by:

))1(2

M(erfc21f

2G

GGG

σ

α

−

−= . (21)

From the Gaussian modeling of rain fields developed in Section 1, αG is linked to

the rain no-rain threshold through the equation )P2(erfc2 01

G−=α where P0 is the

probability of rain defined in Section 1.2. Obviously fG=fR so that the mean value MG of

the Gaussian field can be related to the rain fractional area fR by:

)f2(erfc)1(2)P2(erfc2M R12

G01

G−− ×−−= σ (22)

By definition of the Fourier transform, the 0 frequency term a0 of the Fourier

expansion of the Gaussian field G is equal to its mean value MG. Therefore, whenever a0

is set to MG, the fraction of the simulation area that undergoes rain is fR, as expected.

Articles



204

In the foregoing, the spatio-temporal evolution of the simulated rain fields is

derived from ERA-40 rain amount time series which temporal sampling ΔT is 6 hours. It

is thus mandatory to find an interpolation scheme to get values of the rain fractional area

fR at a finer time resolution. This high resolution rain fractional area time series denoted

by fHR(t) and sampled at Δt (Δt<ΔT) has to respect different constraints such as

continuity. But first and foremost, its average value over ΔT=6 hours must be equal to

the low resolution rain fractional area derived each 6 hours from ERA-40 and denoted

by fLR(t) hereafter in. Therefore, considering that fLR(t) derived from ERA-40 holds in

the temporal interval [-T/2 T/2], fHR(t) is defined as a piecewise linear function such as:

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎦⎤

⎢⎣⎡∈+=

⎥⎦⎤

⎢⎣⎡−∈+=

2;0)(

0;2

)(

2

1

Ttbtatf

Ttbtatf

HR

HR

. (23)

To ensure the continuity at the edge of the interval, we set:

⎪⎩

⎪⎨

⎧

=

−=−

)0()()2

(

)0()()2

(

LRLRHR

LRLRHR

fTfTf

fTfTf, (24)

, combined with the constraint on the average value on [-T/2 T/2]:

)0()(1)(1)(1 2/

02

0

2/1

2/

2/ LR

T

T

T

T HR fdtbtaT

dtbtaT

dttfT

=+++= ∫∫∫−

− (25)

(23), (24), (25) define a set of three equations with three unknown coefficients

a1, a2 and b. The effect of this interpolation procedure is illustrated on 237HFigure 6, where it

is shown to reproduce accurately the rain fractional area derived from ERA-40.

Nevertheless, the effect on the simulations of this piecewise linear interpolation is not

well assessed. It will be shown further that it has not a significant impact on the realism

of the simulations in terms of temporal auto-correlation.

Articles



205

Figure 6 : Interpolation at higher temporal resolution of the rain fractional area

time series derived from ERA-40 (ΔT=6h). The black dashed line results from the rain

fractional area computed from rain field spatio-temporal simulations with high

resolution parameters (i.e. Δt=6mn).

238HFigure 7 sums up the step by step methodology developed to parameterize the

stochastic model from ERA-40 rain amount time series. Particularly, from the ERA-40

low resolution (ΔT=6 hours) rain amount time series, a low resolution (ΔT=6 hours)

time-series of the rain fractional area is deduced from (19). The latter is then

interpolated with a lower time step using the interpolation methodology defined above

(Δt=6mn). At this stage, a time-series of the Gaussian field mean value MG(t)=a0(t) is

derived from (22). Finally, the model generates realistic rain fields with long-term

evolution at the scale of one ERA-40 resolution cell (2.5o×2.5o) with a spatial resolution

of 1×1 km² and with a temporal resolution of 6 mn .

Articles



206

Figure 7: Conversion of the rain amount time series given by ERA-40 into rain

fractional area and into spatial average of the Gaussian field MG=a0.

2.2.3 Extension to several cells of the GCM

The extension of the simulation domain to areas larger than 2.5o×2.5o requires

taking into account more than one ERA-40 cell. It is not possible to juxtapose directly

the 2.5o×2.5o rain fields obtained from the methodology described above for one ERA-

40 resolution cell because the rain field continuity between two adjoining cells is not

ensured. For that purpose, an interpolation procedure that preserves the statistical

features of each 2.5o×2.5o sub-field while ensuring the continuity of the global rain field

has been developed. Particularly, for each ERA-40 cell contained in the simulation area,

a Gaussian field with size 2N×2N is simulated according to the methodology described

in Section 1. Then, for each point (x,y) of the simulation area, the value of the global

Gaussian field G results from the weighted sum of the Gaussian fields G1, G2, G3, G4

generated for each of the ERA-40 adjacent resolution cells:

dyxGxyyxGyxdyxGydxyxGydxd

yxG),(),()(),()(),())((

),( 4321 +−+−+−−= , (26)

Articles



207

where d is the distance between two ERA-40 resolution cell (d=N). Figure 8

gives an overview of the interpolation scheme.

Figure 8 : Spatial extension of the modelling to several resolution cells of the

ERA-40 reanalysis database. Each gray square figures an ERA-40 resolution cell. The

interpolation procedure (26) is applied inside the white square.

Common interpolation methodologies such as bilinear or cubic spline

interpolation are not suitable in our case because they do not preserve the Gaussian field

statistical features in terms of variance and correlation. On the contrary, the

interpolation procedure (26) does not change the variance and the correlation of the

Gaussian sub-fields Gi. Moreover, by construction, the model reproduces both the rain

amount given by ERA-40 for each resolution cell (as a laplacian filtering is applied to

the coefficient a0 of each subfield) the (local) lognormal rain rate distribution given as

input parameters. Figure 9 gives an example of rain rate field simulated at large scale,

over Europe, on 24/02/1999 at 19:30 UTC (i.e. using the ERA-40 rain amount data

provided at this date).

G1 G2

G4 G3

M(x,y)

G

Articles



208

Figure 9 : Example of rain rate field simulated over Europe on 24/02/1999 at 19:30

UTC (i.e. using the ERA-40 rain amount data provided at this date).

2.3 Advection

Whenever the rain field spatio-temporal simulation lasts a few hours, a common

and practical approach is to consider the advection vector as constant both in speed and

direction. However, this frozen storm hypothesis is not satisfying if long-durations or

large-scale simulations are considered. Indeed, the wind field and the resulting rain field

motion may evolve considerably with distance or with time. In such conditions, wind

data from the ERA-40 database can provide realistic inputs for this parameter. Indeed,

comparing radar derived advection with wind outputs from meteorological model,

[Kitzmiler et al., 2002], [Luini et al., 2008] have shown that the rain field motion can be

accurately derived from the ERA-40 wind data at the 700 hPa pressure level.

Consequently, for each ERA-40 resolution cell making part of the large scale simulation

Articles



209

area, the advection vector V defined in (4) is driven by the ERA-40 concurrent wind data

at a pressure level of 700 hPa. Lastly, in order to get a smooth temporal evolution of the

rain advection, both component of the winds are interpolated by cubic splines with a

time step Δt of 6 mn.

3 Conversion into attenuation

3.1 Conversion of rain rate fields into attenuation fields

Rain field simulations are converted into attenuation by integration of the specific

attenuation along the link path. The attenuation (dB) endured by the electromagnetic

wave during its propagation through rain is given by:

∫=sL

0

dx)x(kRA α , (27)

where R(x) is the rain rate intensity (mm.h-1) at x and Ls is the slant path (km)

through rain. k and α are coefficients, function of the elevation angle, the frequency and

the polarization of the electromagnetic wave. Their values are given by Rec. ITU-R

P.838. The definition of the geometric (latitude, longitude, altitude of the satellite) and

radiowave (frequency, polarization) characteristics of the telecommunication link allows

the conversion of the rainfall field into an attenuation field. Here, the altitude of the

N×N points composing the simulation grid is derived from Rec. ITU-R P.1511. The rain

height is the -2 oC isotherm height given by the ERA-40 temperature profiles. The use of

ERA-40 data, sampled each 6 hours, to define the -2oC isotherm height allows a better

description of the rain height with respect to Rec. ITU-R P.839 that provides only the

yearly average of the -2 oC isotherm height as shown in Figure 10.

Articles



210

Figure 10: Height of the -2oC isotherm deduced from ERA-40 temperature profiles for

Bordeaux (France) in 2000.

3.2 Over sampling of the attenuation time series

For adaptive SatCom systems using adaptive coding, adaptive modulation or

other FMTs, it is crucial to get an estimation of the evolution of the propagation channel

at a time resolution of about 1 second. However, the temporal resolution of the model

developed in Section 2 is 6 mn. Therefore, it is mandatory to simulate the fast dynamic

fluctuations of the propagation channel to increase the model temporal resolution. For

that purpose, the on-demand time series synthesizer fully described in [Lacoste et al.,

2006] and [Carrier and Lacoste, 2008] is used. The latter makes a 1 s stochastic

interpolation of the 6 mn attenuation time series extracted from the space time model,

generating that way high frequency (1 Hz) attenuation time series as illustrated in Figure

11.

Figure 11 : 1 Hz attenuation time series simulated on two sites located 7 km

apart, at 20 GHz. The green circles represent the values of the initial time-series

extracted from the spatio-temporal model (Δt=6 mn).

Articles



211

The high resolution interpolation describes in [Lacoste et al., 2006] and [Carrier

et al., 2008] holds whenever attenuation time series are Markov and follow a log normal

distribution which parameters can be derived from Rec. ITU-R P.618. The temporal

autocorrelation time-lag is supposed to be 10-4 s whatever the link configuration

[Carrier et al., 2008]. Coupled with the spatio-temporal model, the stochastic

interpolation enables to simulate 1 Hz spatially correlated attenuation time series for

thousands of links disposed across the satellite coverage, for long durations (several

years).

4 Preliminary validations, limitations

4.1 First order statistics

Preliminary analysis shows that the rain amount from the input reanalysis

database is reproduced by the simulations for each 6h period with an RMS error of about

15% as illustrated in Figure 12.

Figure 12: 6h rain amount derived from ERA-40 database on 12/24/1999 at 08:00 UTC

(left) and corresponding 6h rain amount computed from 6 mn rain field spatio-temporal

simulations (right).

By construction, the model reproduces the rain rate lognormal approximation of

Rec. ITU-R P.837-5 on each point of the simulation grid. In addition, long term

Articles



212

attenuation time series reproduce Rec. ITU-R P.618 CCDF of attenuation. From

OLYMPUS satellite measurements at 20 GHz led in 1992 near Paris (France), the model

ability to reproduce attenuation statistics derived from beacon measurements has been

also investigated. High resolution (1 Hz) attenuation time series have been then

synthesised from the spatio-temporal model starting from ERA-40 data concurrent to

OLYMPUS measurement period and assuming a link with the geostationary satellite

OLYMPUS (19°W) at 20 GHz. Figure 13 shows the results in terms of attenuation

CCDF derived from simulations and from OLYMPUS measurements.

Figure 13: Attenuation CCDF at 20 GHz measured during the OLYMPUS

campaign (1992) in two sites near Paris (La-Folie-Bessin and Gometz-la-ville, color

lines) and attenuation CCDF derived from simulated attenuation time series (black

lines). The two sites are located 7 km apart.

According to Figure 13, statistics derived from spatio temporal simulations of

attenuation fields are comparable to the experimental ones up to small time percentages.

Nevertheless, it is noticeable that the variability of the experimental statistics is higher

than the one derived from the simulations.

Articles



213

4.2 Second order statistics

In order to assess the model ability to reproduce the spatial variability of the

attenuation due to rain, the diversity gain has been computed, first, from the radar data

of Bordeaux collected in 1996 every 5 mn [Luini et al., 2008]. As the latter refer to rain

fields, they have been converted to attenuation fields assuming a 30 GHz link with

OLYMPUS. On the other hand, the same exercise has been conducted from the rain

fields simulated by the spatio temporal model each 6 mn using ERA-40 data of 1996.

The results are presented in Figure 14, for single site attenuation values ranging from 4

dB to 32 dB. Figure 14 shows that the spatio temporal model reproduces satisfactorily

the diversity gain derived from radar data, even if a trend to slightly underestimate the

diversity gain for the highest single site attenuation values and for low distances can be

observed.

Figure 14: Diversity gain as a function of distance for different single site

attenuation values derived from radar data (blue curves) and derived from simulations

(red curves).

A similar test has been carried out with the 1 Hz over-sampled time series.

Indeed, the joint attenuation CCDF has been computed first from OLYMPUS [OPEX,

1994] measurements at La-Folie-Bessin and Gometz-La-Ville that are 7 km apart and,

second, from the 1 Hz time series derived at both locations from the spatio temporal

model. According to Figure 15, the experimental and model based distributions match

Articles



214

satisfactorily, showing thus the model ability to account for the rain attenuation spatial

variability.

Figure 15: Joint attenuation CCDF derived from simulations and derived from

OLYMPUS measurements at 20 GHz, at La-Folie-Bessin and Gometz-La-Ville (7 km

apart).

To evaluate the relevance of the model temporal parameterization, the

autocorrelation function of attenuation time series has been evaluated first from yearly

attenuation time series extracted from the radar observations at Bordeaux in 1996, and,

second, from the 6 mn attenuation time series simulated over Bordeaux in 1996.

Figure 16: Temporal autocorrelation estimated from radar derived attenuation

time series and from simulations.

Articles



215

The results, reported in Figure 16, show that the temporal auto-correlation

function computed from simulations is close to the one derived from radar.

Nevertheless, the temporal validation of the model has to be deepened, notably to assess

the ability of the simulated time series to reproduce fade slopes, fade duration or inter-

fade durations statistics derived from beacon measurements.

5 Conclusion

A model to generate spatially and temporally correlated rain fields or rain

attenuation fields for propagation studies has been presented. It lies on a non-linear

transformation of random Gaussian fields constructed in the Fourier plane.

After a brief description of the mathematical framework, a method to derive the

spatio temporal correlation parameters from radar data has been proposed. It has been

applied to the weather radar data of Bordeaux (France). As the initial approach proposed

by [Bell, 1987] was not intended to describe the rain rate fields at large scale and for

long durations, a theoretical framework to enlarge the validity domain of the model has

been developed. A methodology to constrain the model with rain outputs from the ERA-

40 reanalysis database has then been presented. Particularly, the rain amount generated

each 6h by the spatio temporal model is constrained to be the one given by ERA-40. The

rain advection is also modelled by ERA-40 wind data. Coupled with a large scale

interpolation scheme, the spatio temporal model is thus able to generate realistic rain

fields for long durations (several years) and large areas (i.e. over Europe or USA, the

size of a typical SatCom system coverage) with a spatial resolution of 1×1 km² and a

temporal resolution of 6 mn. Then, defining the geometric and radio electric

characteristics of a link, the rain rate fields are turned into attenuation fields considering

the wave path through rain. Here, the stochastic interpolation scheme described in

[Carrier et al., 2008] allows reducing the temporal resolution up to 1 s.

The model has been shown to reproduce several statistical characteristics

observed on different and independent sources of data. In addition to the first order

statistics of rain and rain attenuation, the model has shown its ability to reproduce the

Articles



216

spatial repartition of attenuation due to rain over the coverage area (diversity gain, joint

attenuation CDF). From the temporal point of view, first validations in terms of

temporal autocorrelation function are promising but further enquiries are needed to

consolidate the results.

Reliable inputs for FMTs and RRM (Radio Resource Management) design and

optimization may be obtained from the presented model, for several configurations,

since the simulation area can reach the size of a typical satellite coverage with a

temporal resolution that can be reduced down to 1 s for several years. Particularly,

considering complex network simulations that involve thousands of links dispersed over

the whole coverage, the spatio temporal model is able to provide realistic propagation

conditions, both correlated in space and in time.

However, the use of the ERA-40 reanalysis database may lead to some bias in the

simulations, especially in coastal or mountainous areas, i.e. in places where the quality

of the reanalysis database is questionable. Nevertheless, the use of reanalysis data turn

out to be promising as it allows simulating test-cases from specific past weather

conditions. For instance, this can provide a good opportunity to study diurnal cycles,

seasonal cycles or inter-annual variability but also worst cases identified on the

reanalysis data.

A logical following to this work is the inclusion, using the same approach, of the

gaseous and liquid water attenuation as they become critical for SatCom system

operating at Q or V band or even at Ka band for other applications such as radar

altimetry.

Acknowledgment:

The authors are very grateful to Météo-France for providing the radar data from

Bordeaux and to ECMWF for providing ERA-40 data.

Articles



217

Bibliography

Barbaliscia, F., G. Ravaioli and A. Paraboni, “Characteristics of the spatial statistical

dependence of rainfall rate over large areas.” IEEE Trans. Antenn. and Prop., 40(1),

8–12, 1992, doi:10.1109/8.123347.

Bell, T.L. “A space-time stochastic-dynamic model of rainfall for satellite remote

sensing studies.” J. Geophys. Res., 92(D8), 9631–9643, 1987.

Bertorelli S. and Paraboni A. "Simulation of Joint Statistics of Rain Attenuation in

Multiple Sites Across Wide Areas Using ITALSAT Data.", IEEE Antenna and

Prop., 53(8):2611-2622, 2005.

Bousquet, M., L. Castantet , L. Féral, P. Pech, J. Lemorton, Application of a Model of

Spatial Correlated Time-Series, into a Simulation Platform of Adaptive Resource

Management for Ka-band OBP Satellite Systems”, 3rd COST 280 workshop, May

2003, Noordwijk, The Netherlands.

Calaghan S.A., “Fractal Modelling of Rain Fields: From Event-On-Demand to Annual

Statistics”, Proceedings of The European Conference on Antennas and Propagation:

EuCAP 2006 (ESA SP-626). 6-10 November 2006, Nice, France.

Capsoni C., F. Fedi, and A. Paraboni, "A comprehensive meteorologically oriented

methodology for the prediction of wave propagation parameters in

telecommunication applications beyond 10 GHz." Radio Sci., 22(3), 387-393,

1987a.

Capsoni C., F. Fedi, C. Magistroni, A. Paraboni, and A. Pawlina, Data and theory for a

new model of the horizontal structure of rain cells for propagation applications.

Radio Sci., 22(3), 395-404, 1987b.

Articles



218

Carrie G., Lacoste F., Castanet L. : ”New “on-demand” channel model to synthesise rain

attenuation time series at Ku-, Ka- and Q/V-bands”, submitted to International

Journal of Satellite Communications and Networking, Special issue on Channel

Modelling and Propagation Impairment Simulation.

Castanet L., M. Bousquet, D. Mertens, “Simulation of the performance of a Ka-band

VSAT videoconferencing system with uplink power control and data rate reduction

to mitigate atmospheric propagation effects.” International . Journal of Satellite.

Communication, 20(4), 231-249,2001.

Castanet L., C. Capsoni , G. Blarzino, D. Ferraro, A. Martellucci, “Development of a

new global rainfall rate model based on ERA40, TRMM and GPCC products”,

International Symposium on Antennas and Propagation, ISAP 2007, Niigata, Japan,

20-24 August 2007.

Chiu, L. S. , “Estimating areal rainfall from rain area.” Tropical Rainfall Measurements,

J.S Theon and N. Fugono, Eds., A. Deepak, 361-367, 1988.

Cost 255, 2002: “Radiowave propagation modelling for new SatCom services at Ku-

band and above”, final report, ESA publication division, Noordwijk, The

Netherlands, 1994.

Donneaud, A. A., S. I. Niscov, D. L. Priegnitz, and P. L. Smith , “The area-time integral

as an indicator for convective rain volume.” J. Appl. Meteor., 23(4),

1984,10.1175/1520-0450(1984)023<0555:TATIAA>2.0.CO;2.

Gremont B.C., M. Filip. “Spatio–Temporal Rain Attenuation Model for Application to

Fade Mitigation Techniques.” IEEE Antenn. and Prop., 52(5),1245-1256, 2004.

Eltahir, E. A. B. and R. L. Bras , “Estimation of the Fractional Coverage of Rainfall in

Climate Models”, Journal of Climate, 6(4), 639-644, 1993, doi: DOI: 10.1175/1520-

0442(1993)006<0639:EOTFCO>2.0.CO;2.

Articles



219

Féral, L., H. Sauvageot, L. Castanet, and J. Lemorton, “HYCELL—A new hybrid model

of the rain horizontal distribution for propagation studies: 1. Modeling of the rain

cell”, Radio Sci., 38(3), 1056, doi:10.1029/2002RS002802, 2003a.

Féral, L., H. Sauvageot, L. Castanet, and J. Lemorton (2003), HYCELL—A new hybrid

model of the rain horizontal distribution for propagation studies: 2. Statistical

modeling of the rain rate field, Radio Sci., 38(3), 1057, doi:10.1029/2002RS002803,

2003b.

Féral L., H. Sauvageot, L. Castanet, J. Lemorton, F. Cornet, and K. Leconte (2006),

Large-scale modeling of rain fields from a rain cell deterministic model, Radio Sci.,

41, RS2010, 2006, doi:10.1029/2005RS003312.

Ferraris, L., S. Gabellani, U. Parodi, N. Rebora, J. Von Hardenberg, and A. Provenzale

,” Revisiting Multifractality in Rainfall Fields.” J. of Hydromet., 4(3), 544-551,

2003, doi: 10.1175/1525-7541(2003)004.

Fuentes, M., “Testing for separability of spatial-temporal covariance functions”. Journal

of Statistical Planning and Inference, 136, 447-466, 2005.

Guillot, G., “Approximation of Sahelian rainfall fields with meta-Gaussian random

functions. Part 1: Model definition and methodology.” Stoch. Environ. Res. Risk

Ass., 13(1), 1999, doi:100-112, doi:10.1007/s004770050034.

Hodges D. D., R J Watson and G Wyman: "An attenuation time series model for

propagation forecasting", IEEE transactions on Antennas and Propagation, 54(6),

2006.

Jeannin N., L. Féral, H. Sauvageot, L. Castanet and J. Lemorton, "Statistical distribution

of the fractional area affected by rain", Journal of Geophysical Reasearch, In Press.

Articles



220

Kitzmiller H.D, G. F. Samplatsky, C. Mello, “Probabilistic Forecasts Of Severe Local

Storms In The 0-3 Hour Timeframe From An Advective-Statistical Technique”, 19th

Conf. on weather Analysis and Forecasting/15th Conf. on Numerical Weather

Prediction, San-Antonio, August 2002.

Kozintsev B. and B. Kedem: “Generation of "Similar" Images from a Given Discrete

Image”, Journal of Computational and Graphical Statistics, 9(2), 286-302, 2000.

Lacoste F., M. Bousquet, F. Cornet, L. Castanet and J. Lemorton, “Classical and On-

Demand Rain Attenuation Time Series Synthesis: Principle and Applications”,

AIAA-2006-5325, 24th AIAA International Communications Satellite. Systems

Conference, San Diego, California, June 11-14, 2006.

Luini L., N. Jeannin, C. Capsoni, C. Riva, L. Castanet, J. Lemorton, "Simulation of a

site diversity system from weather radar data", submitted to International Journal of

Satellite Communication

Mantoglou A. and J. L. Wilson, “The turning bands method for simulation of random

fields using line generation by a spectral method”, Water Resour. Res., 18 , 1379–

1394, 1982.

Maseng T., P.M Bakken, “A stochastic dynamic model of rain attenuation”, IEEE Trans.

on Commun., 29, 660-669, 1981.

Montopoli, M., F. S. Marzano, “Maximum-likelihood Retrieval of Modelled Convective

Rainfall Patterns From Midlatitude C-Band Weather Radar Data”, IEEE trans. on

geoscience and Remote Sensing, 45(7), 2403-2416, 2007.

Nzeukou, A., and H. Sauvageot: Distribution of rainfall parameters near the coast of

France and senegal. J. Appl. Meteor., Vol. 41, n°1, 69-82, 2002.

Articles



221

OPEX: “Reference Book on Attenuation Measurement and Prediction”, 2nd Workshop of

the OLYMPUS Propagation Experimenters (OPEX), Doc ESA-ESTEC-WPP-083

volume 1, Noordwijk, The Netherlands, 1994.

Poiares-Baptista J.P.V, Salonen E.T. : "Review of rainfall rate modelling and mapping",

Proc. of URSI Climpara'98 Comm. F Open Symposium on "Climatic Parameters

innRadiowave Prediction", Ottawa, Canada, April 1998, 35-44.

Rivoirard J., “Introduction to Disjunctive Kriging and non Linear Geostatistics”, Oxford

Clarendon Press, 1994.

Uppala S. et al., “The ERA-40 re-analysis”. Quart. J. R. Meteorol. Soc., 131, 2961-3012,

2005.

Sauvageot, H., “The probability density function of rain rate and the estimation of

rainfall by area integrals.”, J. Appl. Meteor., 33(11), 1255-1262, 1994,doi:

10.1175/1520-0450(1994)033<1255:TPDFOR>2.0.CO;2.

dooccttoorraatt dee ll uunniivveerrssiittÉ ddee

Documents