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Author Doicu, Adrian.

Title Numerical regularization for atmospheric inverse problems / by Adrian Doicu, Thomas Trautmann, Franz Schreier
Published Berlin ; London : Springer, 2010
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Description 1 online resource
Series Springer Praxis Books
Springer-Praxis books in environmental sciences.
Contents 1. Remote sensing of the atmosphere -- 1.1 The atmosphere facts and problems -- 1.1.1 Greenhouse gases -- 1.1.2 Air pollution -- 1.1.3 Tropospheric ozone -- 1.1.4 Stratospheric ozone -- 1.2 Atmospheric remote sensing -- 1.3 Radiative transfer -- 1.3.1 Definitions -- 1.3.2 Equation of radiative transfer -- 1.3.3 Radiative transfer in the UV -- 1.3.4 Radiative transfer in the IR and microwave -- 1.3.5 Instrument aspects -- 1.3.6 Derivatives -- 1.4 Inverse problems -- 2 Ill-posedness of linear problems -- 2.1 An illustrative example -- 2.2 Concept of ill-posedness -- 2.3 Analysis of linear discrete equations -- 2.3.1 Singular value decomposition -- 2.3.2 Solvability and ill-posedness -- 2.3.3 Numerical example -- 3 Tikhonov regularization for linear problems -- 3.1 Formulation -- 3.2 Regularization matrices -- 3.3 Generalized singular value decomposition and regularized solution -- 3.4 Iterated Tikhonov regularization -- 3.5 Analysis tools -- 3.5.1 Filter factors -- 3.5.2 Error characterization -- 3.5.3 Mean square error matrix -- 3.5.4 Resolution matrix and averaging kernels -- 3.5.5 Discrete Picard condition -- 3.5.6 Graphical tools -- 3.6 Regularization parameter choice methods -- 3.6.1 A priori parameter choice methods -- 3.6.2 A posteriori parameter choice methods -- 3.6.3 Error-free parameter choice methods -- 3.7 Numerical analysis of regularization parameter choice methods -- 3.8 Multi-parameter regularization methods -- 3.8.1 Complete multi-parameter regularization methods -- 3.8.2 Incomplete multi-parameter regularization methods -- 3.9 Mathematical results and further reading -- 4 Statistical inversion theory -- 4.1 Bayes theorem and estimators -- 4.2 Gaussian densities -- 4.2.1 Estimators -- 4.2.2 Error characterization -- 4.2.3 Degrees of freedom -- 4.2.4 Information content -- 4.3 Regularization parameter choice methods -- 4.3.1 Expected error estimation method -- 4.3.2 Discrepancy principle -- 4.3.2. Discrepancy principle -- 4.3.3. Hierarchical models -- 4.3.4. Maximum likelihood estimation -- 4.3.5. Expectation minimization -- 4.3.6. general regularization parameter choice method -- 4.3.7. Noise variance estimators -- 4.4. Marginalizing method -- 5.1. Landweber iteration -- 5.2. Semi-iterative regularization methods -- 5.3. Conjugate gradient method -- 5.4. Stopping rules and preconditioning -- 5.4.1. Stopping rules -- 5.4.2. Preconditioning -- 5.5. Numerical analysis -- 5.6. Mathematical results and further reading -- 6.1. Four retrieval test problems -- 6.1.1. Forward models and degree of nonlinearity -- 6.1.2. Sensitivity analysis -- 6.1.3. Prewhitening -- 6.2. Optimization methods for the Tikhonov function -- 6.2.1. Step-length methods -- 6.2.2. Trust-region methods -- 6.2.3. Termination criteria -- 6.2.4. Software packages -- 6.3. Practical methods for computing the new iterate -- 6.4. Error characterization -- 6.4.1. Gauss[-]Newton method -- 6.4.2. Newton method -- 6.5. Regularization parameter choice methods -- 6.5.1. priori parameter choice methods -- 6.5.2. Selection criteria with variable regularization parameters -- 6.5.3. Selection criteria with constant regularization parameters -- 6.6. Iterated Tikhonov regularization -- 6.7. Constrained Tikhonov regularization -- 6.8. Mathematical results and further reading -- 7.1. Nonlinear Landweber iteration -- 7.2. Newton-type methods -- 7.2.1. Iteratively regularized Gauss[-]Newton method -- 7.2.2. Regularizing Levenberg[-]Marquardt method -- 7.2.3. Newton[-]CG method -- 7.3. Asymptotic regularization -- 7.4. Mathematical results and further reading -- 8.1. Formulation -- 8.2. Truncated total least squares -- 8.3. Regularized total least squares for linear problems
Note continued: 8.4. Regularized total least squares for nonlinear problems -- 9.1. Backus[-]Gilbert method -- 9.2. Maximum entropy regularization -- A.1. Elements of functional analysis -- A.2. Least squares solution and generalized inverse -- A.3. Singular value expansion of a compact operator -- A.4. Solvability and ill-posedness of the linear equation -- B.1. Explicit transformations -- B.2. Implicit transformations -- C.1. Basic assumptions -- C.2. Source condition -- C.3. Error estimates -- C.4. priori parameter choice method -- C.5. Discrepancy principle -- C.6. Generalized discrepancy principle -- C.7. Error-free parameter choice methods -- E.1. Linear regularization methods -- E.2. Conjugate gradient method -- E.2.1. CG-polynomials -- E.2.2. Discrepancy principle -- G.1. Error estimates -- G.2. priori parameter choice method -- G.3. Discrepancy principle -- H.1. Newton-type methods with a priori information -- H.1.1. Error estimates -- H.1.2. priori stopping rule -- H.1.3. Discrepancy principle -- H.2. Newton-type methods without a priori information -- J.1. Equality constraints -- J.2. Inequality constraints
Summary The subject of this book is a hot topic with currently no monographic support. It is more advanced, specialized and mathematical than its competitors, and a comprehensive book on regularization techniques for atmospheric science is much needed for further development in this field. Written by brilliant mathematicians, this research monograph presents and analyzes numerical algorithms for atmospheric retrieval, pulling together all the relevant material in a consistent, very powerful manner. The first chapter presents the typical retrieval problems encountered in atmospheric remote sensing. Chapter 2 introduces the concept of ill-posedness for linear discrete equations, illustrating the difficulties associated with the solution of the problems by considering a temperature retrieval test problem and analyzing the solvability of the discrete equation by using the singular value decomposition of the corresponding matrix. A detailed description of the Tikhonov regularization for linear problems is the subject of Chapter 3, in which the authors introduce a set of mathematical and graphical tools to characterize the regularized solution. The goal of Chapter 4 is to reveal the similitude between Tikhonov regularization and statistical inversion regarding the regularized solution representation, the error analysis, and the design of parameter choice methods. The following chapter briefly surveys some classical iterative regularization methods such as the Landweber iteration and semi-iterative methods, and then treats the regularization effect of the conjugate gradient method applied to the normal equations. Having set the stage in the first part of the book, the remaining chapters dealing with nonlinear ill-posed problems. The authors introduce four test problems that are used throughout the rest of the book to illustrate the behaviour of the numerical algorithms and tools. These deal with the retrieval of ozone and BrO in the visible spectral region, and of CO and temperature in the infared spectral domain. Chapter 6 looks at the practical aspects of Tikhonov regularization for nonlinear problems, while Chapter 7 presents the relevant iterative regularization methods for nonlinear problems. The following chapter reviews the truncated and the regularized total least squares method for solving linear ill--posed problems, and include the similarity with the Tikhonov regularization. Chapter 9 brings the list of nonlinear methods to a close. It describes the Backus-Gilbert approach as a representative member of mollifier methods and finally, addresses the maximum entropy regularization. For the sake of completeness and in order to emphasize the mathematical techniques which are used in the classical regularization theory, five appendices at the end of the book present direct and iterative methods for solving linear and nonlinear ill-posed problems
Bibliography Includes bibliographical references (pages 407-422) and index
Notes Print version record
Subject Atmosphere -- Remote sensing.
Atmosphere -- Mathematical models.
Inverse problems (Differential equations)
Form Electronic book
Author Trautmann, Thomas.
Schreier, Franz.
LC no. 2010920974
ISBN 3642054390
9783642054396
(print)
(print)