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E-book: Inverse Methods For Atmospheric Sounding: Theory And Practice

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(Univ Of Oxford, Uk)
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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.
Preface vii
Introduction
1(12)
The Beginnings
2(1)
Atmospheric Remote Sounding Methods
3(4)
Thermal emission nadir and limb sounders
3(1)
Scattered solar radiation
4(2)
Absorption of solar radiation
6(1)
Active techniques
6(1)
Simple Solutions to the Inverse Problem
7(6)
Information Aspects
13(30)
Formal Statement of the Problem
13(4)
State and measurement vectors
13(1)
The forward model
14(1)
Weighting function matrix
15(1)
Vector spaces
15(2)
Linear Problems without Measurement Error
17(3)
Subspaces of state space
17(1)
Identifying the null space and the row space
18(2)
Linear Problems with Measurement Error
20(7)
Describing experimental error
20(1)
The Bayesian approach to inverse problems
21(1)
Bayes' theorem
22(2)
Example: The Linear problem with Gaussian statistics
24(3)
Degrees of Freedom
27(5)
How many independent quantities can be measured?
27(2)
Degrees of freedom for signal
29(3)
Information Content of a Measurement
32(5)
The Fisher information matrix
32(1)
Shannon information content
33(1)
Entropy of a probability density function
33(1)
Entropy of a Gaussian distribution
34(2)
Information content in the linear Gaussian case
36(1)
The Standard Example: Information Content and Degrees of Freedom
37(3)
Probability Density Functions and the Maximum Entropy Principle
40(3)
Error Analysis and Characterisation
43(22)
Characterisation
43(5)
The forward model
43(1)
The retrieval method
44(1)
The transfer function
45(1)
Linearisation of the transfer function
45(1)
Interpretation
46(1)
Retrieval method parameters
47(1)
Error Analysis
48(4)
Smoothing error
48(1)
Forward model parameter error
49(1)
Forward model error
50(1)
Retrieval noise
50(1)
Random and systematic error
50(1)
Representing covariances
51(1)
Resolution
52(3)
The Standard Example: Linear Gaussian Case
55(10)
Averaging kernels
56(2)
Error components
58(2)
Modelling error
60(1)
Resolution
61(4)
Optimal Linear Inverse Methods
65(16)
The Maximum a Posteriori Solution
66(5)
Several independent measurements
68(1)
Independent components of the state vector
69(2)
Minimum Variance Solutions
71(2)
Best Estimate of a Function of the State Vector
73(1)
Separately Minimising Error Components
73(1)
Optimising Resolution
74(7)
Optimal Methods for Non-linear Inverse Problems
81(20)
Determination of the Degree of Nonlinearity
82(1)
Formulation of the Inverse Problem
83(2)
Newton and Gauss-Newton Methods
85(1)
An Alternative Linearisation
86(1)
Error Analysis and Characterisation
86(1)
Convergence
87(5)
Expected convergence rate
87(1)
A Popular mistake
88(1)
Testing for convergence
89(1)
Testing for correct convergence
90(1)
Recognising and dealing with slow convergence
91(1)
Levenberg-Marquardt Method
92(1)
Numerical Efficiency
93(8)
Which formulation for the linear algebra?
93(1)
The n-form
94(3)
The m-form
97(1)
Sequential updating
97(1)
Computation of derivatives
98(1)
Optimising representations
99(2)
Approximations, Short Cuts and Ad-hoc Methods
101(20)
The Constrained Exact Solution
101(4)
Least Squares Solutions
105(2)
The overconstrained case
105(1)
The underconstrained case
106(1)
Truncated Singular Vector Decomposition
107(1)
Twomey-Tikhonov
108(2)
Approximations for Optimal Methods
110(3)
Approximate a priori and its covariance
110(1)
Approximate measurement error covariance
111(1)
Approximate weighting Functions
111(2)
Direct Multiple Regression
113(1)
Linear Relaxation
114(2)
Nonlinear Relaxation
116(2)
Maximum Entropy
118(1)
Onion Peeling
119(2)
The Kalman Filter
121(8)
The Basic Linear Filter
122(2)
The Kalman Smoother
124(1)
The Extended Filter
125(1)
Characterisation and Error Analysis
126(1)
Validation
127(2)
Global Data Assimilation
129(12)
Assimilation as a Inverse Problem
129(1)
Methods for Data Assimilation
130(5)
Successive correction methods
130(1)
Optimal interpolation
131(1)
Adjoint methods
132(2)
Kalman filtering
134(1)
Preparation of Indirect Measurements for Assimilation
135(6)
Choice of profile representation
137(1)
Linearised measurements
137(1)
Systematic errors
138(1)
Transformation of a characterised retrieval
139(2)
Numerical Methods for Forward Models and Jacobians
141(18)
The Equation of Radiative Transfer
141(2)
The Radiative Transfer Integration
143(2)
Derivatives of Forward Models: Analytic Jacobians
145(2)
Ray Tracing
147(5)
Choosing a coordinate system
148(1)
Ray tracing in radial coordinates
149(1)
Horizontally homogeneous case
149(2)
The general case
151(1)
Transmittance Modelling
152(1)
Line-by-line modelling
153(1)
Band transmittance
154(1)
Inhomogeneous paths
155(1)
Curtis--Godson approximation
155(1)
Emissivity growth approximation
156(1)
McMillin--Fleming method
156(1)
Multiple absorbers
157(2)
Construction and Use of Prior Constraints
159(16)
Nature of a Priori
159(2)
Effect of Prior Constraints on a Retrieval
161(1)
Choice of Prior Constraints
162(3)
Retrieval grid
162(1)
Transformation between grids
162(1)
Choice of grid for maximum likelihood retrieval
163(1)
Choice of grid for maximum a priori retrieval
164(1)
Ad hoc Soft constraints
165(1)
Smoothness constraints
165(1)
Markov process
165(1)
Estimating a priori from real data
166(2)
Estimating a Priori from independent sources
166(1)
Maximum entropy and the estimation of a priori
166(2)
Validating and improving Priori with indirect measurements
168(3)
The nearly linear case
169(1)
The moderately non-linear case
170(1)
Using Retrievals Which Contain a Priori
171(4)
Taking averages of sets of retrievals
172(1)
Removing a Priori
172(3)
Designing an Observing System
175(10)
Design and Optimisation of Instruments
175(4)
Forward model construction
176(1)
Retrieval method and diagnostics
177(1)
Optimisation
178(1)
Specifying requirements for the accuracy of parameters
179(1)
Operational Retrieval Design
179(6)
Forward model construction
180(1)
State vector choice
180(1)
Choice of vertical grid coordinate
181(1)
Choice of parameters describing constituents
182(1)
A priori information
183(1)
Retrieval method
183(1)
Diagnostics
183(2)
Testing and Validating an Observing System
185(12)
Error Analysis and Characterisation
186(1)
The x2 Test
187(1)
Quantities to be Compared and Tested
188(4)
Internal consistency
188(1)
Does the retrieval agree with the measurement?
189(1)
Consistency with the a priori
190(1)
Measured signal and a priori
190(1)
Retrieval and a priori
191(1)
Comparison of the retrieved signal and the a priori
191(1)
Intercomparison of Different Instruments
192(5)
Basic requirements for intercomparison
192(1)
Direct comparison of indirect measurements
193(1)
Comparison of linear functions of measurements
194(3)
Appendix A Algebra of Matrices and Vectors 197(8)
A.1 Vector Spaces
197(2)
A.2 Eigenvectors and Eigenvalues
199(1)
A.3 Principal Axes of a Quadratic Form
200(1)
A.4 Singular Vector Decomposition
201(2)
A.5 Determinant a d Trace
203(1)
A.6 Calculus with Matrices and Vectors
203(2)
Appendix B Answers to Exercises 205(18)
Appendix C Terminology and Notation 223(6)
C.1 Summary of Terminology
223(2)
C.2 List of Symbols Used
225(4)
Bibliography 229(6)
Index 235


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