Doicu / Trautmann / Schreier Numerical Regularization for Atmospheric Inverse Problems

1. Auflage 2010
ISBN: 978-3-642-05439-6
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 426 Seiten

The retrieval problems arising in atmospheric remote sensing belong to the class of the - called discrete ill-posed problems. These problems are unstable under data perturbations, and can be solved by numerical regularization methods, in which the solution is stabilized by taking additional information into account. The goal of this research monograph is to present and analyze numerical algorithms for atmospheric retrieval. The book is aimed at physicists and engineers with some ba- ground in numerical linear algebra and matrix computations. Although there are many practical details in this book, for a robust and ef?cient implementation of all numerical algorithms, the reader should consult the literature cited. The data model adopted in our analysis is semi-stochastic. From a practical point of view, there are no signi?cant differences between a semi-stochastic and a determin- tic framework; the differences are relevant from a theoretical point of view, e.g., in the convergence and convergence rates analysis. After an introductory chapter providing the state of the art in passive atmospheric remote sensing, Chapter 2 introduces the concept of ill-posedness for linear discrete eq- tions. To illustrate the dif?culties associated with the solution of discrete ill-posed pr- lems, we consider the temperature retrieval by nadir sounding and analyze the solvability of the discrete equation by using the singular value decomposition of the forward model matrix.

Autoren/Hrsg.

Doicu, Adrian

Trautmann, Thomas

Schreier, Franz

Inhaltsverzeichnis

1;Table of Contents;8
2;Preface;12
3;1 Remote sensing of the atmosphere;16
3.1;1.1 The atmosphere – facts and problems;16
3.1.1;1.1.1 Greenhouse gases;18
3.1.2;1.1.2 Air pollution;19
3.1.3;1.1.3 Tropospheric ozone;19
3.1.4;1.1.4 Stratospheric ozone;19
3.2;1.2 Atmospheric remote sensing;19
3.3;1.3 Radiative transfer;23
3.3.1;1.3.1 Definitions;24
3.3.2;1.3.2 Equation of radiative transfer;24
3.3.3;1.3.3 Radiative transfer in the UV;25
3.3.4;1.3.4 Radiative transfer in the IR and microwave;29
3.3.5;1.3.5 Instrument aspects;32
3.3.6;1.3.6 Derivatives;32
3.4;1.4 Inverse problems;33
4;2 Ill-posedness of linear problems;37
4.1;2.1 An illustrative example;37
4.2;2.2 Concept of ill-posedness;41
4.3;2.3 Analysis of linear discrete equations;42
4.3.1;2.3.1 Singular value decomposition;42
4.3.2;2.3.2 Solvability and ill-posedness;43
4.3.3;2.3.3 Numerical example;46
5;3 Tikhonov regularization for linear problems;53
5.1;3.1 Formulation;53
5.2;3.2 Regularization matrices;55
5.3;3.3 Generalized singular value decomposition and regularized solution;59
5.4;3.4 Iterated Tikhonov regularization;63
5.5;3.5 Analysis tools;64
5.5.1;3.5.1 Filter factors;64
5.5.2;3.5.2 Error characterization;65
5.5.3;3.5.3 Mean square error matrix;70
5.5.4;3.5.4 Resolution matrix and averaging kernels;71
5.5.5;3.5.5 Discrete Picard condition;72
5.5.6;3.5.6 Graphical tools;75
5.6;3.6 Regularization parameter choice methods;80
5.6.1;3.6.1 A priori parameter choice methods;81
5.6.2;3.6.2 A posteriori parameter choice methods;82
5.6.3;3.6.3 Error-free parameter choice methods;88
5.7;3.7 Numerical analysis of regularization parameter choice methods;97
5.8;3.8 Multi-parameter regularization methods;107
5.8.1;3.8.1 Complete multi-parameter regularization methods;108
5.8.2;3.8.2 Incomplete multi-parameter regularization methods;112
5.9;3.9 Mathematical results and further reading;117
6;4 Statistical inversion theory;121
6.1;4.1 Bayes theorem and estimators;121
6.2;4.2 Gaussian densities;123
6.2.1;4.2.1 Estimators;124
6.2.2;4.2.2 Error characterization;126
6.2.3;4.2.3 Degrees of freedom;127
6.2.4;4.2.4 Information content;132
6.3;4.3 Regularization parameter choice methods;135
6.3.1;4.3.1 Expected error estimation method;135
6.3.2;4.3.2 Discrepancy principle;138
6.3.3;4.3.3 Hierarchical models;139
6.3.4;4.3.4 Maximum likelihood estimation;140
6.3.5;4.3.5 Expectation minimization;142
6.3.6;4.3.6 A general regularization parameter choice method;144
6.3.7;4.3.7 Noise variance estimators;149
6.4;4.4 Marginalizing method;151
7;5 Iterative regularization methodsfor linear problems;155
7.1;5.1 Landweber iteration;155
7.2;5.2 Semi-iterative regularization methods;158
7.3;5.3 Conjugate gradient method;160
7.4;5.4 Stopping rules and preconditioning;168
7.4.1;5.4.1 Stopping rules;169
7.4.2;5.4.2 Preconditioning;170
7.5;5.5 Numerical analysis;174
7.6;5.6 Mathematical results and further reading;176
8;6 Tikhonov regularizationfor nonlinear problems;177
8.1;6.1 Four retrieval test problems;178
8.1.1;6.1.1 Forward models and degree of nonlinearity;178
8.1.2;6.1.2 Sensitivity analysis;183
8.1.3;6.1.3 Prewhitening;185
8.2;6.2 Optimization methods for the Tikhonov function;187
8.2.1;6.2.1 Step-length methods;188
8.2.2;6.2.2 Trust-region methods;192
8.2.3;6.2.3 Termination criteria;193
8.2.4;6.2.4 Software packages;197
8.3;6.3 Practical methods for computing the new iterate;197
8.4;6.4 Error characterization;204
8.4.1;6.4.1 Gauss–Newton method;205
8.4.2;6.4.2 Newton method;210
8.5;6.5 Regularization parameter choice methods;213
8.5.1;6.5.1 A priori parameter choice methods;214
8.5.2;6.5.2 Selection criteria with variable regularization parameters;217
8.5.3;6.5.3 Selection criteria with constant regularization parameters;220
8.6;6.6 Iterated Tikhonov regularization;223
8.7;6.7 Constrained Tikhonov regularization;226
8.8;6.8 Mathematical results and further reading;231
9;7 Iterative regularization methodsfor nonlinear problems;235
9.1;7.1 Nonlinear Landweber iteration;236
9.2;7.2 Newton-type methods;236
9.2.1;7.2.1 Iteratively regularized Gauss–Newton method;237
9.2.2;7.2.2 Regularizing Levenberg–Marquardt method;246
9.2.3;7.2.3 Newton–CG method;251
9.3;7.3 Asymptotic regularization;253
9.4;7.4 Mathematical results and further reading;260
10;8 Total least squares;265
10.1;8.1 Formulation;266
10.2;8.2 Truncated total least squares;268
10.3;8.3 Regularized total least squares for linear problems;272
10.4;8.4 Regularized total least squares for nonlinear problems;281
11;9 Two direct regularization methods;285
11.1;9.1 Backus–Gilbert method;285
11.2;9.2 Maximum entropy regularization;294
12;A Analysis of continuous ill-posed problems;299
12.1;A.1 Elements of functional analysis;299
12.2;A.2 Least squares solution and generalized inverse;302
12.3;A.3 Singular value expansion of a compact operator;304
12.4;A.4 Solvability and ill-posedness of the linear equation;305
13;B Standard-form transformationfor rectangular regularization matrices;308
13.1;B.1 Explicit transformations;308
13.2;B.2 Implicit transformations;312
14;C A general direct regularization methodfor linear problems;315
14.1;C.1 Basic assumptions;315
14.2;C.2 Source condition;317
14.3;C.3 Error estimates;318
14.4;C.4 A priori parameter choice method;318
14.5;C.5 Discrepancy principle;319
14.6;C.6 Generalized discrepancy principle;322
14.7;C.7 Error-free parameter choice methods;325
15;D Chi-square distribution;330
16;E A general iterative regularization methodfor linear problems;334
16.1;E.1 Linear regularization methods;334
16.2;E.2 Conjugate gradient method;338
16.3;E.2.1 CG-polynomials;339
16.4;E.2.2 Discrepancy principle;343
17;F Residual polynomials of the LSQR method;353
18;G A general direct regularization methodfor nonlinear problems;358
18.1;G.1 Error estimates;359
18.2;G.2 A priori parameter choice method;362
18.3;G.3 Discrepancy principle;363
19;H A general iterative regularization methodfor nonlinear problems;373
19.1;H.1 Newton-type methods with a priori information;373
19.1.1;H.1.1 Error estimates;376
19.1.2;H.1.2 A priori stopping rule;376
19.1.3;H.1.3 Discrepancy principle;378
19.2;H.2 Newton-type methods without a priori information;381
20;I Filter factors of the truncated total leastsquares method;392
21;J Quadratic programming;398
21.1;J.1 Equality constraints;398
21.2;J.2 Inequality constraints;401
22;References;413
23;Index;429