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Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational Results Observation density and ensemble spread Conclusions Stability of Ensemble Kalman Filters Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley , Heikki Haario and Tuomo Kauranne Lappeenranta University of Technology University of Montana High-Performance Computing in Meteorology, ECMWF October 27, 2014 Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley Stability of Ensemble Kalman Filters

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Page 1: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Stability of Ensemble Kalman Filters

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, AnttiSolonen, John Bardsley†, Heikki Haario and Tuomo

Kauranne

Lappeenranta University of Technology† University of Montana

High-Performance Computing in Meteorology, ECMWFOctober 27, 2014

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 2: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

1 Introduction2 Ensemble Kalman Filtering Methods

The Extended Kalman Filter (EKF)Ensemble Kalman Filters (EnKF)

3 The Variational Ensemble Kalman Filter (VEnKF)4 Stability and Trajectory Shadowing

Regularization implicit in Kalman filteringA CFL like condition on filter stability

5 Computational ResultsThe Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

6 Observation density and ensemble spread7 Conclusions

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 3: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Overview1 Introduction2 Ensemble Kalman Filtering Methods

The Extended Kalman Filter (EKF)Ensemble Kalman Filters (EnKF)

3 The Variational Ensemble Kalman Filter (VEnKF)4 Stability and Trajectory Shadowing

Regularization implicit in Kalman filteringA CFL like condition on filter stability

5 Computational ResultsThe Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

6 Observation density and ensemble spread7 Conclusions

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 4: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Introduction

Data assimilation calls for algorithms that oftenapproximate the Extended Kalman Filter, or EKF, that itselfis too heavy to runIt is essential, but quite non-trivial, that the approximateKalman filters used remain stable over the assimilationperiod.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 5: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Introduction

Stability of a filter is related to the numerical stability of thecorresponding algorithm, but numerical stability alone doesnot guarantee filter stability.It is also mandatory that the filter does not diverge from thetrue state of the system.Yet all filters applied to nonlinear models will diverge ifthere are no observations.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 6: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Introduction

We study the general conditions for filter stability applicableto variational methods, and approximate Kalman filtersmany kinds ;Present several ways to stabilize filters;

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 7: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Introduction

Provide empirical results with a shallow-water model thatillustrate a relation between ensemble spread andtemporal and spatial density of observations thatGeneralizes the well-known Courant-Friedrichs-Lewynumerical stability condition to filter stability in a Hilbertspace setting; andExplain the impact of model bias on filter stability in thiscontext.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 8: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Extended Kalman Filter (EKF)Ensemble Kalman Filters (EnKF)

Overview1 Introduction2 Ensemble Kalman Filtering Methods

The Extended Kalman Filter (EKF)Ensemble Kalman Filters (EnKF)

3 The Variational Ensemble Kalman Filter (VEnKF)4 Stability and Trajectory Shadowing

Regularization implicit in Kalman filteringA CFL like condition on filter stability

5 Computational ResultsThe Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

6 Observation density and ensemble spread7 Conclusions

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 9: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Extended Kalman Filter (EKF)Ensemble Kalman Filters (EnKF)

The Extended Kalman Filter (EKF)

AlgorithmIterate in time

xf (ti) = M(ti , ti−1)(xa(ti−1))

Pfi = MiPa(ti−1)MT

i + Q

Ki = Pf (ti)HTi (HiPf (ti)HT

i + R)−1

xa(ti) = xf (ti) + Ki(yoi − H(xf (ti)))

Pa(ti) = Pf (ti)− KiHiPf (ti)

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 10: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Extended Kalman Filter (EKF)Ensemble Kalman Filters (EnKF)

The Extended Kalman Filter (EKF)

Where

xf (ti) is the prediction at time tixa(ti) is the analysis at time tiPf (ti) is the prediction error covariance matrix at time tiPa(ti) is the analysis error covariance matrix at time tiQ is the model error covariance matrixKi is the Kalman gain matrix at time tiR is the observation error covariance matrixH is the nonlinear observation operatorHi is the linearized observation operator at time tiMi is the linearized weather model at time tiIdrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 11: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Extended Kalman Filter (EKF)Ensemble Kalman Filters (EnKF)

The Extended Kalman Filter (EKF)

PropertiesThe model is not assumed to be perfectModel integrations are carried out forward in time with thenonlinear model for the state estimate andForward and backward in time with the tangent linearmodel and the adjoint model, respectively, for updating theprediction error covariance matrixThere is no minimization, just matrix products andinversionsComputational cost of EKF is prohibitive, because Pf (ti)and Pa(ti) are huge full matrices

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 12: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Extended Kalman Filter (EKF)Ensemble Kalman Filters (EnKF)

Ensemble Kalman Filters (EnKF)

PropertiesEnsemble Kalman Filters are generally simpler to programthan variational assimilation methods or EKF, becauseEnKF codes are based on just the non-linear model and donot require tangent linear or adjoint codes, but theyTend to suffer from slow convergence and thereforeinaccurate analyses because ensemble size is smallcompared to model dimensionOften underestimate analysis error covariance

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 13: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Extended Kalman Filter (EKF)Ensemble Kalman Filters (EnKF)

Ensemble Kalman Filters (EnKF)

PropertiesEnsemble Kalman filters often base analysis errorcovariance on bred vectors, i.e. the difference betweenensemble members and the background, or the ensemblemeanOne family of EnKF methods is based on perturbedobservations, whileAnother family uses explicit linear transforms to build upthe ensemble

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 14: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Extended Kalman Filter (EKF)Ensemble Kalman Filters (EnKF)

EnKF Cost functions

AlgorithmMinimize

(Pf (ti))−1 = (βB0 + (1− β)1N

Xf (ti)Xf (ti)T)−1

AlgorithmMinimize

`(xa(ti)|yoi )

= (yoi −H(xa(ti)))TR−1(yo

i −H(xa(ti)))

+1N

N∑j=1

(xfj (ti)−xa(ti))T(Pf (ti))−1(xf

j (ti)−xa(ti))Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 15: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Overview1 Introduction2 Ensemble Kalman Filtering Methods

The Extended Kalman Filter (EKF)Ensemble Kalman Filters (EnKF)

3 The Variational Ensemble Kalman Filter (VEnKF)4 Stability and Trajectory Shadowing

Regularization implicit in Kalman filteringA CFL like condition on filter stability

5 Computational ResultsThe Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

6 Observation density and ensemble spread7 Conclusions

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 16: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Variational Ensemble Kalman Filter (VEnKF)

AlgorithmIterate in time

Step 0: Select a state xa(t0) and a covariance Pa(t0) andset i = 1

Step 1: Evolve the state and the prior covariance estimate:(i) Compute xf (ti) = M(ti , ti−1)(xa(ti−1));(ii) Compute the ensemble forecastXf (ti) = M(ti , ti−1)(Xa(ti−1));(iii) Minimize from a random initial guess(Pf (ti))−1 = (βB0 + (1− β) 1

N Xf (ti)Xf (ti)T + Qi)−1

by the LBFGS method;Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 17: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Variational Ensemble Kalman Filter (VEnKF)

Algorithm

Step 2: Compute the Variational Ensemble Kalman Filterposterior state and covariance estimates:(i) Minimize`(xa(ti)|yo

i )= (yo

i −H(xa(ti)))TR−1(yoi −H(xa(ti)))

+(xf (ti)−xa(ti))T(Pf (ti))−1(xf (ti)−xa(ti))by the LBFGS method;

(ii) Store the result of the minimization as xa(ti);(iii) Store the limited memory approximation to Pa(ti);(iv) Generate a new ensemble Xa(ti) ∼ N(xa(ti),Pa(ti));

Step 3: Update i := i + 1 and return to Step 1.Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 18: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Variational Ensemble Kalman Filter (VEnKF)

PropertiesFollows the algorithmic structure of VKF, separating thetime evolution from observation processing.A new ensemble is generated every observation stepBred vectors are centered on the mode, not the mean, ofthe ensemble, as in Bayesian estimationLike in VKF, a new ensemble and a new error covariancematrix is generated at every observation timeNo covariance leakageNo tangent linear or adjoint codeAsymptotically equivalent to VKF and therefore EKF whenensemble size increasesIdrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 19: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

Overview1 Introduction2 Ensemble Kalman Filtering Methods

The Extended Kalman Filter (EKF)Ensemble Kalman Filters (EnKF)

3 The Variational Ensemble Kalman Filter (VEnKF)4 Stability and Trajectory Shadowing

Regularization implicit in Kalman filteringA CFL like condition on filter stability

5 Computational ResultsThe Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

6 Observation density and ensemble spread7 Conclusions

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 20: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

Kalman filtering with a biased model

Having a biased model means that the model produces aforecast error with non-zero mean. In this case, our modelequations:

x(ti) = M(ti , ti−1)(x(ti−1)) + η(ti)

entail that the expectation of model error is non-zero, butgenerally unknown, and can locally at ti be approximatedby a linear error.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 21: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

Kalman filtering with a biased model

This reads as

E(η(ti)) = b(ti − ti−1) 6= 0

If the bias b is known, there are various ways tocompensate for it, such as the ones presented by Dee(2005) and Trémolet (2005).

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 22: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

Kalman filters on combined state and observationspace

Let us denote the time interval between observations by∆t .To second order accuracy in ∆t , we can derive a two termform for model evolution, when looking at it over shortobservation intervals ∆t .This form separates the smoothly evolving model bias fromstochastic Gaussian model noise.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 23: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

Kalman filters on combined state and observationspace

x0(ti + ∆t) = M(ti , ti−1)(x(ti)) + b∆t + η̂(ti−1) +O(∆t2)

where x0 is the true future state and η̂(ti) is Gaussianmodel noise with zero mean.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 24: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

Kalman filters on combined state and observationspace

In this error decomposition, the smooth model bias termb∆t represents drift, see e.g. Orrell (2005), Orrell et al.(2001).It indicates a tendency of unknown direction.But the maximum speed ||b|| of the expected state of animperfect model to drift away from the true evolution of thestate of the system can be estimated from statistics offorecast systematic errors.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 25: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

Kalman filters on combined state and observationspace

Let us recall the innovation form of the Kalman filter thatwe have used.

Pf (ti) = Mi−1Pa(ti−1)MTi−1 + Qi

Ki = Pf (ti)HTi (HtPf (ti)HT

i + Ri)−1.

xa(ti) = xf (ti) + Ki(yoi − Hi(xf (ti))).

(1)

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 26: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

From the last equation, we see that the state incrementδx(ti) satisfies

δx(ti) = xa(ti)− xf (ti) (2)

and is therefore computed from the innovation vector di

di = yoi − Hi(xf (ti)) (3)

by solving a linear equation with the inverse of the Kalmangain matrix Ki :

K−1i δx(ti) = di (4)

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 27: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

Kalman filters on combined state and observationspace

Replacing the Kalman gain here with its definition in thesecond equation in the Kalman group above, we see thatthe two last equations are equivalent to the followingsystem of two equations:

(HiPf (ti)HTi + Ri)δzi = di

δx(ti) = Pf (ti)HTi δzi

(5)

where δzi is the information vector.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 28: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

Kalman filters on combined state and observationspace

Inserting the first and third equations from our Kalmangroup of equations into the first equation above, we get alinear operator equation, local in time, whose operator weshall denote by Ai .This operator can aptly be called the symmetric Kalmanfilter operator and it defines the information form of theKalman filter.

Aiδzi =

(HiMi−1Pa(ti−1)MTi−1HT

i + HiQiHTi + Ri)δzi = di

(6)

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 29: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

Kalman filters on combined state and observationspace

The operator Ai above that defines the Kalman filter isapplied to measurements sampled from a stochasticprocess, but it is itself a deterministic linear operator.Ai defines the metric in the quadratic form, in which theKalman filter produces a least squares estimate that canalso be interpreted as the maximum a posteriori estimateaccording to the Bayes theorem.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 30: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

Kalman filters on combined state and observationspace

The equation (6) is an equation defined on the space ofobservations.Its various component operators are defined on differentspaces as well:Pa(ti−1) is defined on the state space at time ti−1,Qi is defined on the state space at time ti andRi is defined on the observation space at time ti .

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 31: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

Stabilizing Kalman filters

Because the symmetric Kalman filter operator (6) isdefined on the observation space at time ti , it imposes itsimplicit optimality condition as a final time observationspace control.For small enough analysis increments δx(ti), nonlinearKalman filtering for smoothly evolving dynamical systemswill be locally stable, if one of the following conditions isfulfilled:

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 32: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

Stabilizing Kalman filters

1 An explicit or implicit static prior or background term with apositive weight is used, or

2 The state space is completely observable and theobservation operator is a projection operator. Theseconditions imply that the spectrum of the error covarianceoperator Pa(ti) is both bounded and positively boundedfrom below, or

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 33: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

Stabilizing Kalman filters

1 The model is not perfect on any observable subspace ofthe model state space, which implies that the spectrum ofthe error covariance operator Qi is positively bounded frombelow, or

2 All observations are noisy and there are no rigidconstraints between errors in different observations, whichimplies that the spectrum of the error covariance operatorRi is positively bounded from below

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 34: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

Stabilizing Kalman filters

The list above is formulated in terms of a small enoughanalysis increment δx(ti), rather than a small time betweenobservations ∆t .It can be seen that the latter is a special case of the former.The condition of smallness of analysis increments coverssmall perturbations in any direction in the state space, andnot just the ones parameterized by the time variable.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 35: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

Stabilizing Kalman filters

In both cases, the validity of the above statementsdepends on the smoothness assumption of the modelevolution on the state space, so that the model evolutionoperator converges to identity as the magnitude of aperturbation decreases to zero.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 36: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

Stabilizing Kalman filters

TheoremStability property of VKF under Gaussian model noise. TheVariational Kalman Filter algorithm for smoothly evolvingdynamical systems is stable under Gaussian model noise, ifany of the conditions in the list above is fulfilled, and theiterations employed in the VKF algorithm are carried out untilconvergence to a limit set by the lower bound ε on the spectrumof the symmetric Kalman Filter operator Ai . This will take afinite number of steps independent of model resolution.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 37: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

A CFL like condition for stability

Let us now look at the impact of model bias on the stabilityanalysis above. We have denoted a local model biasvector in unit time by b.It will thus represent the direction and speed of model driftb∆t .We shall assume that the dynamics of both the trueoperatorM and the biased discrete model M are smooth.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 38: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

A CFL like condition for stability

This assumption implies that the accumulation of bias inthe model state will be proportional to the duration ofmodel evolution to second order accuracy, in the form

x(t + ∆t)− x0(t + ∆t) =

M(t + ∆t , t)(x(t))−M(t + ∆t , t)(x(t)) =

b∆t + η̂(t) +O(∆t2)

(7)

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 39: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

A CFL like condition for stability

where x0(t) denotes the model state evolved with theunbiased true modelM andwhere we have continued to assume that model noise η̂(t)is Gaussian and Markov.Because of the smoothness of model evolution, for smallenough ∆t , the drift b∆t will stay beneath ε, no matterwhat is the direction of the bias b.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 40: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

A CFL like condition for stability

But as we have seen, any innovation direction that ismodified by the local Kalman operator with a factor lessthan ε away from the identity will be suppressed by thestatic prior plus the noise term in the Variational KalmanFilter.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 41: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

A CFL like condition for stability

We can state this result as a conditional stability propertyof the Variational Kalman Filter against model bias

TheoremConditional stability under bias of VKF. The VariationalKalman Filter is stable for smoothly evolving dynamicalsystems, if there is a sufficient temporal and spatial density ofobservations available, with an observation bias uncorrelatedwith the model bias, and such that the span of the temporallylocal model bias is observed.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 42: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

A CFL like condition for stability

The conditional stability property above can be seen as akind of Courant-Friedrichs-Lewy (CFL) stability condition,only in state space and not in the computational domain,for Kalman filtering algorithms.It means that if the model bias drives model evolution awayfrom the true trajectory, the bias will not accumulatebeyond a given threshold, if the corresponding drift can becountered fast enough with an observation with an errorthat is uncorrelated with model bias, before the drift hasincreased beyond the assumed model noise level.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 43: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

A CFL like condition for stability

If the drift has grown too large, VKF (and EKF) may chooseto believe the biased forecast, rather than the contradictingobservation.In the terminology or Orrell et al. (2001), the stability underbias property above gives a sufficient condition for theKalman Filter to guarantee that the sequence of analysesproduced by VKF will continue to shadow the truth with abound that corresponds to the level of model errorcovariance ||Pa(ti)||, or ||B0 + –Pf (ti) + Qi || in VKF notation.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 44: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

A CFL like condition for stability

Shadowing cannot similarly be guaranteed for the strongconstraint 4D-Var without a background term, because theabsence of a model error term and the strictness of themodel constraint prevent the strong constraint 4D-Varoperator from being a Fredholm operator.There will always be directions in the state space that arenot observable.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 45: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

A CFL like condition for stability

The background term does stabilize the filter, but if thebackground has been produced by the same biased model,the ensuing analysis will be bias-blind, in the terminologyof Dee (2005), and continue to suffer from the same bias.This was also empirically observed by Orrell (2005).Strong-constraint 4D-Var is therefore not stable againstmodel bias.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 46: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

A CFL like condition for stability

The CFL condition for a numerical model of the advectionequation reads

Theorem

∆t ≤ ∆x/||v|| (8)

where ∆x is the shortest spatial grid length and ||v|| is thefastest advection velocity in the system.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 47: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

A CFL like condition for stability

The corresponding expression for the shadowing conditionabove reads

Theorem

∆t ≤ ε/||b|| (9)

where ε is the amplitude of Gaussian noise used in theKalman filter and ||b|| the speed of growth of forecast biasVerbally, the above formula says that we must havecorrecting observations in the direction of the bias b beforethe corresponding drift has become larger than the noiselevel of model error.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 48: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

A CFL like condition for stability

In practice, the assumptions of the stability under modelbias property may not be fulfilled and models will exhibitbias, especially in poorly observed areas of their statespace, such as the stratosphere.This can be countered with covariance inflation.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 49: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Regularization implicit in Kalman filteringA CFL like condition on filter stability

A CFL like condition for stability

The shadowing condition with covariance inflationbecomes

Theorem

∆t ≤ ||B0 + Qi ||/||b|| (10)

or, more generally,

Theorem

||B0 + Qi || ≥ ||b||||δx(ti)|| (11)

for any state increment δx(ti).

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 50: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

Overview1 Introduction2 Ensemble Kalman Filtering Methods

The Extended Kalman Filter (EKF)Ensemble Kalman Filters (EnKF)

3 The Variational Ensemble Kalman Filter (VEnKF)4 Stability and Trajectory Shadowing

Regularization implicit in Kalman filteringA CFL like condition on filter stability

5 Computational ResultsThe Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

6 Observation density and ensemble spread7 Conclusions

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 51: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

The Shallow Water Model

MOD_FreeSurf2D by Martin and GorelickFinite-volume, semi-implicit, semi-Lagrangian MATLABcodeUsed to simulate a physical laboratory model of a DamBreak experiment along a 400 m river reach in IdahoThe model consists of a system of coupled partialdifferential equations

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 52: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

The Shallow Water Model - 1

Shallow Water Equations

∂U∂t

+ U∂U∂x

+ V∂U∂y

= −g∂η

∂x+ ε

(∂2U∂x2 +

∂2U∂y2

)+γT (Ua − U)

H

−g√

U2 + V 2

Cz2 U + fV ,

∂V∂t

+ U∂V∂x

+ V∂V∂y

= −g∂η

∂y+ ε

(∂2V∂x2 +

∂2V∂y2

)+γT (Va − V )

H

−g√

U2 + V 2

Cz2 V − fU,

∂η

∂t+∂HU∂x

+∂HV∂y

= 0Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 53: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

The Shallow Water Model - 2

WhereU is the depth-averaged x-direction velocityV is the depth-averaged y-direction velocityη is the free surface elevationg is the gravitational constantε is the horizontal eddy viscosity coefficientγT is the wind stress coefficientUa and Va are the reference wind components for topboundary frictionH is the total water depthCz is the Chezy coefficient for bottom frictionf is the Coriolis parameter

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 54: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

The Dam Break laboratory experiment

WhereThe 400 m long river stretch has been scaled down to 21.2mWater depth is 0.20 m above the damThe dam is placed at the most narrow point of the riverThe riverbed downstream from the dam is initially dryIn the experiment the dam is broken instantly and a floodwave sweeps downstreamThe total duration of the laboratory experiment is 130seconds

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 55: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

The observations

WhereThe flow is measured with eight wave meters for waterdepth, placed irregularly at the approximate flume mid-lineup and downstream from the damWave meters report the depth of water at 1 Hz, so with 1 stime intervalsComputational time step is 0.103 s

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 56: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

Flume geometry and wave meters

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 57: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

Vertical profile of flume

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 58: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

VEnKF applied to shallow-water equations

WhereEnsemble size 100Observations are interpolated in space and timeA new ensemble is therefore generated every time step

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 59: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

Interpolating kernel

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 60: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

Observation interpolation in space

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 61: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

Observation interpolation in time

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 62: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

Model vs. hydrographs - 1

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 63: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

Model vs. hydrographs - 2

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 64: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

The Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

VEnKF vs. hydrographs

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 65: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Overview1 Introduction2 Ensemble Kalman Filtering Methods

The Extended Kalman Filter (EKF)Ensemble Kalman Filters (EnKF)

3 The Variational Ensemble Kalman Filter (VEnKF)4 Stability and Trajectory Shadowing

Regularization implicit in Kalman filteringA CFL like condition on filter stability

5 Computational ResultsThe Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

6 Observation density and ensemble spread7 Conclusions

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 66: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Evidence of the CFL condition in Hilbert space

When observations are interpolated to appear on everytime step, or less frequentlyThe VEnKF algorithm always stays numerically stable, butWith long time intervals between observations,Fails to capture waves present in the solution.Moreover, the empirical relationship between observationinterval and filter divergence is linear

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 67: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Ensemble spread vs. observation frequency

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 68: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Overview1 Introduction2 Ensemble Kalman Filtering Methods

The Extended Kalman Filter (EKF)Ensemble Kalman Filters (EnKF)

3 The Variational Ensemble Kalman Filter (VEnKF)4 Stability and Trajectory Shadowing

Regularization implicit in Kalman filteringA CFL like condition on filter stability

5 Computational ResultsThe Shallow Water Equations - Dam Break ExperimentLaboratory and numerical geometry

6 Observation density and ensemble spread7 Conclusions

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 69: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Conclusions - 1

Kalman filters stabilize their estimates by a variety ofmeansThe most reliable way is to have abundant and frequentobservations combined withVery short assimilation windows - even just one numericaltime step

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 70: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Conclusions - 2

Generating a new ensemble every time step is optimal,becauseThe more frequent the inter-linked updates of theensemble and the error covariance estimate, the moreaccurate the analysisThe local linearity assumption implicit in all Kalman filtersremains more valid

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 71: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Bibliography

Martin, M. and Gorelick, S. M.: MOD_FreeSurf2D: AMATLAB surface fluid flow model for rivers and streams.Computers & Geosciences 31 (2005), pp. 929-946.Auvinen, H., Bardsley J., Haario, H. and Kauranne, T.:The Variational Kalman Filter and an efficientimplementation using limited memory BFGS. Int. J. Numer.Meth. Fluids 64(3)2010, pp. 314-335(22).

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 72: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Bibliography

Solonen, A., Haario, H., Hakkarainen, J., Auvinen, H.,Amour, I. and Kauranne, T.: Variational ensembleKalman filtering using limited memory BFGS, ElectronicTransactions on Numerical Analysis, 39(2012), pp-271-285(15).Orrell, D.: The Future of Everything, Thunder’s MouthPress, 2007.

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters

Page 73: Stability of Ensemble Kalman Filters - ECMWF · Introduction Ensemble Kalman Filtering Methods The Variational Ensemble Kalman Filter (VEnKF) Stability and Trajectory Shadowing Computational

IntroductionEnsemble Kalman Filtering Methods

The Variational Ensemble Kalman Filter (VEnKF)Stability and Trajectory Shadowing

Computational ResultsObservation density and ensemble spread

Conclusions

Thank You

Thank You!

Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley†, Heikki Haario and Tuomo KauranneStability of Ensemble Kalman Filters