## Data Assimilation: The Ensemble Kalman FilterData 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 |

266 | |

276 | |