# NUMERICAL REGULARIZATION FOR ATMOSPHERIC INVERSE PROBLEMS

### Adrian Doicu, Thomas Trautmann and Frank Schreier

The retrieval problems arising in atmospheric remote sensing belong to the class of the so-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 accounting on additional information.

*Numerical Regularization for Atmospheric Inverse Problems* is the first book to present and analyze in depth numerical algorithms for atmospheric retrieval. Its emphasis is on the practical aspects of regularization theory and on the design of numerical algorithms. The book

- focuses on computational aspects but also provides some theoretical results;
- surveys the state-of-the-art numerical methods for solving discrete ill-posed problems;
- illustrates the various methods in action with examples from atmospheric remote sensing;
- presents the relevant inverse methods from classical regularization theory, together with approaches from statistical inversion.

This comprehensive and specialised coverage of the subject will lay the foundations for future work in this important area of environmental science, physics and mathematics.

**Table of Contents**
Preface

1. Remote sensing of the atmosphere

2. Ill-posedness of linear problems

3. Tikhonov regularization for linear problems

4. Statistical inversion theory

5. Iterative regularization methods for linear problems

6. Tikhonov regularization for nonlinear problems

7. Iterative regularization methods for nonlinear problems

8. Total least squares

9. Two direct regularization methods

A. Analysis of continuous ill-posed problems

B. Standard-form transformation for rectangular regularization matrices

C. A general direct regularization method for linear problems

D. Chi-square distribution

E. A general iterative regularization method for linear problems

F. Residual polynomials of the LSQR method

G. A general direct regularization method for nonlinear problems

H. A general iterative regularization method for nonlinear problems

I. Filter factors of the truncated total least squares method

J. Quadratic programming

References

Index

**Extent: **440 pages

**Binding:** Hardback

**Published:** 2010

**ISBN: **978-3-642-05438-9

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