Data Assimilation: The Ensemble Kalman Filter

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Springer Science & Business Media, Dec 22, 2006 - Science - 280 pages
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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
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About the author (2006)

Geir Evensen obtained his Ph.D. in applied mathematics at the University in Bergen in 1992. Thereafter he has worked as a Research Director at the Nansen Environmental and Remote Sensing Center/Mohn-Sverdrup Center, as Prof. II at the Department of Mathematics at the University in Bergen, and as a Principal Engineer at the Hydro Research Center in Bergen. He is author or coauthor of more that 40 refereed publications related to modelling and data assimilation, and he has been the coordinator of international research projects on the development of data assimilation methodologies and systems.

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