Data Assimilation (eBook)

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.

Preface 6
Contents 7
List of symbols 13
1 Introduction 20
2 Statistical definitions 24
2.1 Probability density function 24
2.2 Statistical moments 27
2.3 Working with samples from a distribution 28
2.4 Statistics of random fields 29
2.5 Bias 30
2.6 Central limit theorem 31
3 Analysis scheme 32
3.1 Scalar case 32
3.2 Extension to spatial dimensions 35
3.3 Discrete form 43
4 Sequential data assimilation 45
4.1 Linear Dynamics 45
4.2 Nonlinear dynamics 50
4.3 Ensemble Kalman filter 56
5 Variational inverse problems 64
5.1 Simple illustration 64
5.2 Linear inverse problem 67
5.3 Representer method with an Ekman model 74
5.4 Comments on the representer method 84
6 Nonlinear variational inverse problems 87
6.1 Extension to nonlinear dynamics 87
6.2 Example with the Lorenz equations 98
7 Probabilistic formulation 110
7.1 Joint parameter and state estimation 110
7.2 Model equations and measurements 111
7.3 Bayesian formulation 112
.0 .1 .2 .3 .4 .5 .6 .7 113
.k 113
d1 d2 d3 113
dJ 113
dj 113
.i 113
7.4 Summary 116
8 Generalized Inverse 117
8.1 Generalized inverse formulation 117
8.2 Solution methods for the generalized inverse problem 122
8.3 Parameter estimation in the Ekman flow model 127
8.4 Summary 131
9 Ensemble methods 132
9.1 Introductory remarks 132
9.2 Linear ensemble analysis update 134
9.3 Ensemble representation of error statistics 135
9.4 Ensemble representation for measurements 137
9.5 Ensemble Smoother (ES) 137
9.6 Ensemble Kalman Smoother (EnKS) 139
9.7 Ensemble Kalman Filter (EnKF) 142
9.8 Example with the Lorenz equations 144
9.9 Discussion 150
10 Statistical optimization 151
10.1 Definition of the minimization problem 151
10.2 Bayesian formalism 153
10.3 Solution by ensemble methods 154
10.4 Examples 157
10.5 Discussion 166
11 Sampling strategies for the EnKF 168
11.1 Introduction 168
11.2 Simulation of realizations 169
11.3 Simulating correlated fields 173
11.4 Improved sampling scheme 174
11.5 Experiments 178
12 Model errors 186
12.1 Simulation of model errors 186
12.2 Scalar model 191
12.3 Variational inverse problem 192
12.4 Formulation as a stochastic model 196
12.5 Examples 196
13 Square Root Analysis schemes 206
13.1 Square root algorithm for the EnKF analysis 206
13.2 Experiments 212
14 Rank issues 217
14.1 Pseudo inverse of C 217
14.2 Efficient subspace pseudo inversion 222
14.3 Subspace inversion using a low-rank C 228
14.4 Implementation of the analysis schemes 230
14.5 Rank issues related to the use of a low-rank C 231
14.6 Experiments with m N 234
14.7 Summary 239
15 An ocean prediction system 240
15.1 Introduction 240
15.2 System configuration and EnKF implementation 241
15.3 Nested regional models 244
15.4 Summary 245
16 Estimation in an oil reservoir simulator 247
16.1 Introduction 247
16.2 Experiment 249
16.3 Results 253
16.4 Summary 256
A Other EnKF issues 257
A.1 Local analysis 257
A.2 Nonlinear measurements in the EnKF 259
A.3 Assimilation of non-synoptic measurements 261
A.4 Time difference data 262
A.5 Ensemble Optimal Interpolation (EnOI) 263
A.6 Chronology of ensemble assimilation developments 263
References 274
Index 283