Many natural phenomena ranging from climate through to biology are described by complex dynamical systems. Getting information about these phenomena involves filtering noisy data and prediction based on incomplete information (complicated by the sheer number of parameters involved), and often we need to do this in real time, for example for weather forecasting or pollution control. All this is further complicated by the sheer number of parameters involved leading to further problems associated with the 'curse of dimensionality' and the 'curse of small ensemble size'. The authors develop, for the first time in book form, a systematic perspective on all these issues from the standpoint of applied mathematics. The book contains enough background material from filtering, turbulence theory and numerical analysis to make the presentation self-contained and suitable for graduate courses as well as for researchers in a range of disciplines where applied mathematics is required to enlighten observations and models.
Author(s): Andrew J. Majda, John Harlim
Publisher: Cambridge University Press
Year: 2012
Language: English
Pages: x+358
Filtering Complex Turbulent Systems......Page 4
Contents......Page 6
Preface......Page 10
1 Introduction and overview: Mathematical strategies for filtering
turbulent systems......Page 12
1.1 Turbulent dynamical systems and basic filtering......Page 14
1.1.1 Basic filtering......Page 17
1.2 Mathematical guidelines for filtering turbulent dynamical systems......Page 20
1.3 Filtering turbulent dynamical systems......Page 22
Part I: Fundamentals......Page 24
2.1 Kalman filter: One-dimensional complex variable......Page 26
2.1.1 Numerical simulation on a scalar complex Ornstein–Uhlenbeck process......Page 29
2.2 Filtering stability......Page 31
2.3.1 Mean model error......Page 34
2.3.2 Model error covariance......Page 35
2.3.3 Example: Model error through finite difference approximation......Page 36
2.3.4 Information criteria for filtering with model error......Page 37
3.1 The classical N-dimensional Kalman filter......Page 41
3.2 Filter stability......Page 43
3.3.1 Observability and controllability criteria......Page 44
3.3.2 Numerical simulations......Page 45
3.4 Reduced filters for large systems......Page 50
3.5 A priori covariance stability for the unstable mode filter given strong observability......Page 54
4.1 Continuous and discrete Fourier series......Page 58
4.2 Aliasing......Page 60
4.3 Differential and difference operators......Page 63
4.4 Solving initial value problems......Page 64
4.5 Convergence of the difference operator......Page 66
Part II: Mathematical guidelines for filtering turbulent signals......Page 70
5.1 The stochastic test model for turbulent signals......Page 72
5.1.1 The stochastically forced dissipative advection equation......Page 73
5.1.2 Calibrating the noise level for a turbulent signal......Page 75
5.2 Turbulent signals for the damped forced advection–diffusion equation......Page 76
5.3 Statistics of turbulent solutions in physical space......Page 77
5.4 Turbulent Rossby waves......Page 79
Appendix A: Temporal correlation function for each Fourier mode......Page 81
Appendix B: Spatio-temporal correlation function......Page 82
6 Filtering turbulent signals: Plentiful observations......Page 83
6.1 A mathematical theory for Fourier filter reduction......Page 84
6.1.1 The number of observation points equals the number of discrete
mesh points: Mathematical theory......Page 87
6.2 Theoretical guidelines for filter performance under mesh refinement
for turbulent signals......Page 88
6.3 Discrete filtering for the stochastically forced dissipative advection
equation......Page 92
6.3.1 Off-line test criteria......Page 94
6.3.2 Numerical simulations of filter performance......Page 96
7.1 Theory for filtering sparse regularly spaced observations......Page 105
7.1.1 Mathematical theory for sparse irregularly spaced observations......Page 109
7.2 Fourier domain filtering for sparse regular observations......Page 110
7.3 Approximate filters in the Fourier domain......Page 113
7.3.1 The strongly damped approximate filters (SDAF, VSDAF)......Page 114
7.3.3 Comparison of approximate filter algorithms......Page 117
7.4 New phenomena and filter performance for sparse regular
observations......Page 118
7.4.1 Filtering the stochastically forced advection–diffusion equation......Page 119
7.4.2 Filtering the stochastically forced weakly damped advection
equation: Observability and model errors......Page 122
8 Filtering linear stochastic PDE models with instability and model error......Page 127
8.1 Two-state continuous-time Markov process......Page 128
8.2 Idealized spatially extended turbulent systems with instability......Page 130
8.3 The mean stochastic model for filtering......Page 134
8.4 Numerical performance of the filters with and without model error......Page 138
Part III: Filtering turbulent nonlinear dynamical systems......Page 142
9 Strategies for filtering nonlinear systems......Page 144
9.1 The extended Kalman filter......Page 145
9.2 The ensemble Kalman filter......Page 147
9.3.1 The ensemble transform Kalman filter......Page 150
9.3.2 The ensemble adjustment Kalman filter......Page 152
9.4 Ensemble filters on the Lorenz-63 model......Page 154
9.5 Ensemble square-root filters on stochastically forced linear systems......Page 160
9.6 Advantages and disadvantages with finite ensemble strategies......Page 162
10.1 The nonlinear test model for filtering slow–fast systems with strong
fast forcing: An overview......Page 164
10.1.1 NEKF and linear filtering algorithms with model
error for the slow–fast test model......Page 167
10.2 Exact solutions and exactly solvable statistics in the nonlinear test
model......Page 170
10.2.1 Path-wise solution of the model......Page 171
10.2.2 Invariant measure and choice of parameters......Page 172
10.2.3 Exact statistical solutions: Mean and covariance......Page 175
10.3 Nonlinear extended Kalman filter (NEKF)......Page 182
10.4.1 Linear filter with model error......Page 185
10.5 Filter performance......Page 188
10.6 Summary......Page 201
11.1 The L-96 model......Page 203
11.2 Ensemble square-root filters on the L-96 model......Page 206
11.3 Catastrophic filter divergence......Page 211
11.4 The two-layer quasi-geostrophic model......Page 215
11.5 Local least-square EAKF on the QG model......Page 221
12 Filtering turbulent nonlinear dynamical systems by linear
stochastic models......Page 225
12.1 Linear stochastic models for the L-96 model......Page 226
12.1.1 Mean stochastic model 1 (MSM1)......Page 227
12.1.2 Mean stochastic model 2 (MSM2)......Page 229
12.1.3 Observation time model error......Page 230
12.2 Filter performance with plentiful observation......Page 231
12.3 Filter performance with regularly spaced sparse observations......Page 234
12.3.1 Weakly chaotic regime......Page 238
12.3.2 Strongly chaotic regime......Page 240
12.3.3 Fully turbulent regime......Page 243
12.3.4 Super-long observation times......Page 245
13 Stochastic parametrized extended Kalman filter for filtering turbulent
signals with model error......Page 247
13.1 Nonlinear filtering with additive and multiplicative biases:
One-mode prototype test model......Page 249
13.1.1 Exact statistics for the nonlinear combined model......Page 250
13.1.2 The stochastic parametrized extended Kalman filter (SPEKF)......Page 254
13.1.3 Filtering one mode of a turbulent signal with instability with SPEKF......Page 255
13.2 Filtering spatially extended turbulent systems with SPEKF......Page 262
13.2.1 Correctly specified forcing......Page 266
13.2.2 Unspecified forcing......Page 268
13.2.3 Robustness and sensitivity to stochastic parameters and observation
error variances......Page 270
13.3 Application of SPEKF to the two-layer QG model......Page 274
A.1. Var(u(t))......Page 280
A.2. Cov(u(t), u∗(t))......Page 282
A.3. Cov(u(t), γ(t))......Page 284
A.5. Cov(u(t), b∗(t))......Page 285
14 Filtering turbulent tracers from partial observations: An exactly
solvable test model......Page 287
14.1 Model description......Page 289
14.2.1 Tracer statistics for a general linear Gaussian velocity field......Page 290
14.2.2 A particular choice of the Gaussian velocity field and its statistics......Page 291
14.2.3 Closed equation for the eddy diffusivity......Page 296
14.2.4 Properties of the model......Page 297
14.3.1 Classical Kalman filter......Page 303
14.3.2 Nonlinear extended Kalman filter......Page 306
14.3.3 Observations......Page 307
14.4 Filter performance......Page 308
14.4.1 Filtering individual trajectories in physical space......Page 309
14.4.2 Recovery of turbulent spectra......Page 312
14.4.3 Recovery of the fat tail tracer probability distributions......Page 317
14.4.4 Improvement of the filtering skill by adding just one observation......Page 319
15 The search for efficient skillful particle filters for high-dimensional
turbulent dynamical systems......Page 327
15.1 The basic idea of a particle filter......Page 328
15.2.1 Rank histogram particle filter (RHF)......Page 330
15.2.2 Maximum entropy particle filter (MEPF)......Page 332
15.2.4 Dynamic range A......Page 335
15.3 Filter performance on the L-63 model......Page 337
15.3.1 Regime where EAKF is superior......Page 341
15.3.2 Regime where non-Gaussian filters (RHF and MEPF) are superior......Page 343
15.3.3 Nonlinear observations......Page 345
15.4 Filter performance on the L-96 model......Page 350
15.5 Discussion......Page 357
References......Page 360
Index......Page 367
