# Data Assimilation: The Ensemble Kalman Filter

Springer Science & Business Media, Dec 22, 2006 - Science - 280 pages

Data Assimilation comprehensively covers data assimilation and inverse methods, including both traditional state estimation and parameter estimation. This text and reference focuses on various popular data assimilation methods, such as weak and strong constraint variational methods and ensemble filters and smoothers. It is demonstrated how the different methods can be derived from a common theoretical basis, as well as how they differ and/or are related to each other, and which properties characterize them, using several examples.

Rather than emphasize a particular discipline such as oceanography or meteorology, it presents the mathematical framework and derivations in a way which is common for any discipline where dynamics is merged with measurements. The mathematics level is modest, although it requires knowledge of basic spatial statistics, Bayesian statistics, and calculus of variations. Readers will also appreciate the introduction to the mathematical methods used and detailed derivations, which should be easy to follow, are given throughout the book. The codes used in several of the data assimilation experiments are available on a web page. In particular, this webpage contains a complete ensemble Kalman filter assimilation system, which forms an ideal starting point for a user who wants to implement the ensemble Kalman filter with his/her own dynamical model.

The focus on ensemble methods, such as the ensemble Kalman filter and smoother, also makes it a solid reference to the derivation, implementation and application of such techniques. Much new material, in particular related to the formulation and solution of combined parameter and state estimation problems and the general properties of the ensemble algorithms, is available here for the first time.

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### Contents

 Introduction 1 Statistical deﬁnitions 5 22 Statistical moments 8 223 Covariance 9 232 Sample variance 10 243 Sample covariance 11 26 Central limit theorem 12 Analysis scheme 13
 97 Ensemble Kalman Filter EnKF 129 972 EnKS using EnKF as a prior 130 98 Example with the Lorenz equations 131 982 Assimilation Experiment 132 99 Discussion 137 Statistical optimization 139 1011 Parameters 140 1014 Cost function 141

 312 Bayesian formulation 15 32 Extension to spatial dimensions 16 322 EulerLagrange equation 17 323 Representer solution 19 324 Representer matrix 20 326 Uniqueness of the solution 22 327 Minimization of the penalty function 23 328 Prior and posterior value of the penalty function 24 Sequential data assimilation 27 411 Kalman filter for a scalar case 28 412 Kalman ﬁlter for a vector state 29 42 Nonlinear dynamics 32 422 Extended Kalman filter in matrix form 33 423 Example using the extended Kalman ﬁlter 35 424 Extended Kalman ﬁlter for the mean 36 425 Discussion 37 43 Ensemble Kalman ﬁlter 38 432 Prediction of error statistics 39 433 Analysis scheme 41 434 Discussion 43 435 Example with a QG model 44 Variational inverse problems 46 52 Linear inverse problem 50 521 Model and observations 51 524 Statistical hypothesis 52 526 Extremum of the penalty function 53 527 EulerLagrange equations 54 528 Strong constraint approximation 55 529 Solution by representer expansions 56 53 Representer method with an Ekman model 57 531 Inverse problem 58 533 EulerLagrange equations 59 534 Representer solution 60 535 Example experiment 61 536 Assimilation of real measurements 64 54 Comments on the representer method 67 Nonlinear variational inverse problems 71 611 Generalized inverse for the Lorenz equations 72 612 Strong constraint assumption 73 613 Solution of the weak constraint problem 76 614 Minimization by the gradient descent method 77 615 Minimization by genetic algorithms 78 62 Example with the Lorenz equations 82 622 Time correlation of the model error covariance 83 623 Inversion experiments 84 624 Discussion 92 Probabilistic formulation 94 72 Model equations and measurements 96 73 Bayesian formulation 97 731 Discrete formulation 98 732 Sequential processing of measurements 99 74 Summary 101 Generalized Inverse 103 812 Prior density for the initial conditions 104 814 Prior density for the measurements 105 816 Conditional joint density 107 82 Solution methods for the generalized inverse problem 108 822 EulerLagrange equations 109 823 Iteration in α 111 83 Parameter estimation in the Ekman ﬂow model 113 84 Summary 117 Ensemble methods 118 92 Linear ensemble analysis update 121 93 Ensemble representation of error statistics 122 94 Ensemble representation for measurements 124 96 Ensemble Kalman Smoother EnKS 126
 103 Solution by ensemble methods 142 1031 Variance minimizing solution 144 104 Examples 145 105 Discussion 154 Sampling strategies for the EnKF 156 112 Simulation of realizations 158 1121 Inverse Fourier transform 159 1123 Specification of covariance and variance 160 113 Simulating correlated fields 162 114 Improved sampling scheme 163 115 Experiments 167 1152 Impact from ensemble size 170 1153 Impact of improved sampling for the initial ensemble 171 1155 Evolution of ensemble singular spectra 173 1156 Summary 174 Model errors 175 1212 Physical model 176 1214 Updating model noise using measurements 180 123 Variational inverse problem 181 1232 Penalty function 182 1235 Solution by representer expansions 183 1236 Variance growth due to model errors 184 124 Formulation as a stochastic model 185 1251 Case A0 186 1253 Case B 189 1254 Case C 192 1255 Discussion 193 Square Root Analysis schemes 195 1311 Updating the ensemble mean 196 1313 Randomization of the analysis update 197 1314 Final update equation in the square root algorithms 200 132 Experiments 201 1322 Impact of the square root analysis algorithm 203 Rank issues 207 1411 Pseudo inverse 208 1412 Interpretation 209 142 Efficient subspace pseudo inversion 212 1422 Analysis schemes based on the subspace pseudo inverse 216 1423 An interpretation of the subspace pseudo inversion 217 143 Subspace inversion using a lowrank Cϵϵ 218 1432 Analysis schemes using a lowrank Cee 219 144 Implementation of the analysis schemes 220 145 Rank issues related to the use of a lowrank C 221 146 Experiments with m N 224 147 Summary 229 An ocean prediction system 230 152 System conﬁguration and EnKF implementation 232 153 Nested regional models 235 154 Summary 236 Estimation in an oil reservoir simulator 239 162 Experiment 241 1621 Parameterization 242 1622 State vector 243 163 Results 245 164 Summary 248 Other EnKF issues 249 A2 Nonlinear measurements in the EnKF 251 A3 Assimilation of nonsynoptic measurements 253 A4 Time difference data 254 A5 Ensemble Optimal Interpolation EnOI 255 A62 Other ensemble based ﬁlters 264 A65 Nonlinear ﬁlters and smoothers 265 References 266 Index 276 Copyright