Inverse Methods For Atmospheric Sounding: Theory And Practice

by Clive D Rodgers

Overview

Remote sounding of the atmosphere has proved to be a fruitful method of obtaining global information about the atmospheres of the earth and other planets. This book treats comprehensively the inverse problem of remote sounding, and discusses a wide range of retrieval methods for extracting atmospheric parameters of interest from the quantities (thermal emission, for example) that can be measured remotely. Inverse theory is treated in depth from an estimation-theory point of view, but practical questions are also emphasized, such as designing observing systems to obtain the maximum quantity of information, efficient numerical implementation of algorithms for processing large quantities of data, error analysis and approaches to the validation of the resulting retrievals. The book is targeted at graduate students as well as scientists.

Product Details

ISBN-13:	9789810227401
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	07/17/2000
Series:	Series On Atmospheric, Oceanic And Planetary Physics , #2
Edition description:	New Edition
Pages:	256
Product dimensions:	6.60(w) x 9.90(h) x 0.80(d)

Table of Contents

	Preface	vii
Chapter 1	Introduction	1
1.1	The Beginnings	2
1.2	Atmospheric Remote Sounding Methods	3
1.2.1	Thermal emission nadir and limb sounders	3
1.2.2	Scattered solar radiation	4
1.2.3	Absorption of solar radiation	6
1.2.4	Active techniques	6
1.3	Simple Solutions to the Inverse Problem	7
Chapter 2	Information Aspects	13
2.1	Formal Statement of the Problem	13
2.1.1	State and measurement vectors	13
2.1.2	The forward model	14
2.1.3	Weighting function matrix	15
2.1.4	Vector spaces	15
2.2	Linear Problems without Measurement Error	17
2.2.1	Subspaces of state space	17
2.2.2	Identifying the null space and the row space	18
2.3	Linear Problems with Measurement Error	20
2.3.1	Describing experimental error	20
2.3.2	The Bayesian approach to inverse problems	21
2.3.2.1	Bayes' theorem	22
2.3.2.2	Example: The Linear problem with Gaussian statistics	24
2.4	Degrees of Freedom	27
2.4.1	How many independent quantities can be measured?	27
2.4.2	Degrees of freedom for signal	29
2.5	Information Content of a Measurement	32
2.5.1	The Fisher information matrix	32
2.5.2	Shannon information content	33
2.5.2.1	Entropy of a probability density function	33
2.5.2.2	Entropy of a Gaussian distribution	34
2.5.2.3	Information content in the linear Gaussian case	36
2.6	The Standard Example: Information Content and Degrees of Freedom	37
2.7	Probability Density Functions and the Maximum Entropy Principle	40
Chapter 3	Error Analysis and Characterisation	43
3.1	Characterisation	43
3.1.1	The forward model	43
3.1.2	The retrieval method	44
3.1.3	The transfer function	45
3.1.4	Linearisation of the transfer function	45
3.1.5	Interpretation	46
3.1.6	Retrieval method parameters	47
3.2	Error Analysis	48
3.2.1	Smoothing error	48
3.2.2	Forward model parameter error	49
3.2.3	Forward model error	50
3.2.4	Retrieval noise	50
3.2.5	Random and systematic error	50
3.2.6	Representing covariances	51
3.3	Resolution	52
3.4	The Standard Example: Linear Gaussian Case	55
3.4.1	Averaging kernels	56
3.4.2	Error components	58
3.4.3	Modelling error	60
3.4.4	Resolution	61
Chapter 4	Optimal Linear Inverse Methods	65
4.1	The Maximum a Posteriori Solution	66
4.1.1	Several independent measurements	68
4.1.2	Independent components of the state vector	69
4.2	Minimum Variance Solutions	71
4.3	Best Estimate of a Function of the State Vector	73
4.4	Separately Minimising Error Components	73
4.5	Optimising Resolution	74
Chapter 5	Optimal Methods for Non-linear Inverse Problems	81
5.1	Determination of the Degree of Nonlinearity	82
5.2	Formulation of the Inverse Problem	83
5.3	Newton and Gauss-Newton Methods	85
5.4	An Alternative Linearisation	86
5.5	Error Analysis and Characterisation	86
5.6	Convergence	87
5.6.1	Expected convergence rate	87
5.6.2	A popular mistake	88
5.6.3	Testing for convergence	89
5.6.4	Testing for correct convergence	90
5.6.5	Recognising and dealing with slow convergence	91
5.7	Levenberg-Marquardt Method	92
5.8	Numerical Efficiency	93
5.8.1	Which formulation for the linear algebra?	93
5.8.1.1	The n-form	94
5.8.1.2	The m-form	97
5.8.1.3	Sequential updating	97
5.8.2	Computation of derivatives	98
5.8.3	Optimising representations	99
Chapter 6	Approximations, Short Cuts and Ad-hoc Methods	101
6.1	The Constrained Exact Solution	101
6.2	Least Squares Solutions	105
6.2.1	The overconstrained case	105
6.2.2	The underconstrained case	106
6.3	Truncated Singular Vector Decomposition	107
6.4	Twomey-Tikhonov	108
6.5	Approximations for Optimal Methods	110
6.5.1	Approximate a priori and its covariance	110
6.5.2	Approximate measurement error covariance	111
6.5.3	Approximate weighting functions	111
6.6	Direct Multiple Regression	113
6.7	Linear Relaxation	114
6.8	Nonlinear Relaxation	116
6.9	Maximum Entropy	118
6.10	Onion Peeling	119
Chapter 7	The Kalman Filter	121
7.1	The Basic Linear Filter	122
7.2	The Kalman Smoother	124
7.3	The Extended Filter	125
7.4	Characterisation and Error Analysis	126
7.5	Validation	127
Chapter 8	Global Data Assimilation	129
8.1	Assimilation as a Inverse Problem	129
8.2	Methods for Data Assimilation	130
8.2.1	Successive correction methods	130
8.2.2	Optimal interpolation	131
8.2.3	Adjoint methods	132
8.2.4	Kalman filtering	134
8.3	Preparation of Indirect Measurements for Assimilation	135
8.3.1	Choice of profile representation	137
8.3.2	Linearised measurements	137
8.3.3	Systematic errors	138
8.3.4	Transformation of a characterised retrieval	139
Chapter 9	Numerical Methods for Forward Models and Jacobians	141
9.1	The Equation of Radiative Transfer	141
9.2	The Radiative Transfer Integration	143
9.3	Derivatives of Forward Models: Analytic Jacobians	145
9.4	Ray Tracing	147
9.4.1	Choosing a coordinate system	148
9.4.2	Ray tracing in radial coordinates	149
9.4.3	Horizontally homogeneous case	149
9.4.4	The general case	151
9.5	Transmittance Modelling	152
9.5.1	Line-by-line modelling	153
9.5.2	Band transmittance	154
9.5.3	Inhomogeneous paths	155
9.5.3.1	Curtis--Godson approximation	155
9.5.3.2	Emissivity growth approximation	156
9.5.3.3	McMillin--Fleming method	156
9.5.3.4	Multiple absorbers	157
Chapter 10	Construction and Use of Prior Constraints	159
10.1	Nature of a Priori	159
10.2	Effect of Prior Constraints on a Retrieval	161
10.3	Choice of Prior Constraints	162
10.3.1	Retrieval grid	162
10.3.1.1	Transformation between grids	162
10.3.1.2	Choice of grid for maximum likelihood retrieval	163
10.3.1.3	Choice of grid for maximum a priori retrieval	164
10.3.2	Ad hoc Soft constraints	165
10.3.2.1	Smoothness constraints	165
10.3.2.2	Markov process	165
10.3.3	Estimating a priori from real data	166
10.3.3.1	Estimating a priori from independent sources	166
10.3.3.2	Maximum entropy and the estimation of a priori	166
10.3.4	Validating and improving a priori with indirect measurements	168
10.3.4.1	The nearly linear case	169
10.3.4.2	The moderately non-linear case	170
10.4	Using Retrievals Which Contain a Priori	171
10.4.1	Taking averages of sets of retrievals	172
10.4.2	Removing a priori	172
Chapter 11	Designing an Observing System	175
11.1	Design and Optimisation of Instruments	175
11.1.1	Forward model construction	176
11.1.2	Retrieval method and diagnostics	177
11.1.3	Optimisation	178
11.1.4	Specifying requirements for the accuracy of parameters	179
11.2	Operational Retrieval Design	179
11.2.1	Forward model construction	180
11.2.2	State vector choice	180
11.2.3	Choice of vertical grid coordinate	181
11.2.3.1	Choice of parameters describing constitutents	182
11.2.4	A priori information	183
11.2.5	Retrieval method	183
11.2.6	Diagnostics	183
Chapter 12	Testing and Validating an Observing System	185
12.1	Error Analysis and Characterisation	186
12.2	The X[superscript 2] Test	187
12.3	Quantities to be Compared and Tested	188
12.3.1	Internal consistency	188
12.3.2	Does the retrieval agree with the measurement?	189
12.3.3	Consistency with the a priori	190
12.3.3.1	Measured signal and a priori	190
12.3.3.2	Retrieval and a priori	191
12.3.3.3	Comparison of the retrieved signal and the a priori	191
12.4	Intercomparison of Different Instruments	192
12.4.1	Basic requirements for intercomparison	192
12.4.2	Direct comparison of indirect measurements	193
12.4.3	Comparison of linear functions of measurements	194
Appendix A	Algebra of Matrices and Vectors	197
A.1	Vector Spaces	197
A.2	Eigenvectors and Eigenvalues	199
A.3	Principal Axes of a Quadratic Form	200
A.4	Singular Vector Decomposition	201
A.5	Determinant and Trace	203
A.6	Calculus with Matrices and Vectors	203
Appendix B	Answers to Exercises	205
Appendix C	Terminology and Notation	223
C.1	Summary of Terminology	223
C.2	List of Symbols Used	225
	Bibliography	229
	Index	235

Editorial Reviews

Rodgers (U. of Oxford) provides graduate students and other researchers a background to the inverse problem and its solution, with applications relating to atmospheric measurements. He introduces the stages in the reverse order than the usual approach in order to develop the learner's intuition about the nature of the inverse problem. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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