| Preface |
| Introduction / Chapter 1： |
| The Beginnings / 1.1： |
| Atmospheric Remote Sounding Methods / 1.2： |
| Thermal emission nadir and limb sounders / 1.2.1： |
| Scattered solar radiation / 1.2.2： |
| Absorption of solar radiation / 1.2.3： |
| Active techniques / 1.2.4： |
| Simple Solutions to the Inverse Problem / 1.3： |
| Information Aspects / Chapter 2： |
| Formal Statement of the Problem / 2.1： |
| State and measurement vectors / 2.1.1： |
| The forward model / 2.1.2： |
| Weighting function matrix / 2.1.3： |
| Vector spaces / 2.1.4： |
| Linear Problems without Measurement Error / 2.2： |
| Subspaces of state space / 2.2.1： |
| Identifying the null space and the row space / 2.2.2： |
| Linear Problems with Measurement Error / 2.3： |
| Describing experimental error / 2.3.1： |
| The Bayesian approach to inverse problems / 2.3.2： |
| Bayes' theorem / 2.3.2.1： |
| Example: The Linear problem with Gaussian statistics / 2.3.2.2： |
| Degrees of Freedom / 2.4： |
| How many independent quantities can be measured? / 2.4.1： |
| Degrees of freedom for signal / 2.4.2： |
| Information Content of a Measurement / 2.5： |
| The Fisher information matrix / 2.5.1： |
| Shannon information content / 2.5.2： |
| Entropy of a probability density function / 2.5.2.1： |
| Entropy of a Gaussian distribution / 2.5.2.2： |
| Information content in the linear Gaussian case / 2.5.2.3： |
| The Standard Example: Information Content and Degrees of Freedom / 2.6： |
| Probability Density Functions and the Maximum Entropy Principle / 2.7： |
| Error Analysis and Characterisation / Chapter 3： |
| Characterisation / 3.1： |
| The retrieval method / 3.1.1： |
| The transfer function / 3.1.3： |
| Linearisation of the transfer function / 3.1.4： |
| Interpretation / 3.1.5： |
| Retrieval method parameters / 3.1.6： |
| Error Analysis / 3.2： |
| Smoothing error / 3.2.1： |
| Forward model parameter error / 3.2.2： |
| Forward model error / 3.2.3： |
| Retrieval noise / 3.2.4： |
| Random and systematic error / 3.2.5： |
| Representing covariances / 3.2.6： |
| Resolution / 3.3： |
| The Standard Example: Linear Gaussian Case / 3.4： |
| Averaging kernels / 3.4.1： |
| Error components / 3.4.2： |
| Modelling error / 3.4.3： |
| Optimal Linear Inverse Methods / 3.4.4： |
| The Maximum a Posteriori Solution / 4.1： |
| Several independent measurements / 4.1.1： |
| Independent components of the state vector / 4.1.2： |
| Minimum Variance Solutions / 4.2： |
| Best Estimate of a Function of the State Vector / 4.3： |
| Separately Minimising Error Components / 4.4： |
| Optimising Resolution / 4.5： |
| Optimal Methods for Non-linear Inverse Problems / Chapter 5： |
| Determination of the Degree of Nonlinearity / 5.1： |
| Formulation of the Inverse Problem / 5.2： |
| Newton and Gauss-Newton Methods / 5.3： |
| An Alternative Linearisation / 5.4： |
| Convergence / 5.5： |
| Expected convergence rate / 5.6.1： |
| A popular mistake / 5.6.2： |
| Testing for convergence / 5.6.3： |
| Testing for correct convergence / 5.6.4： |
| Recognising and dealing with slow convergence / 5.6.5： |
| Levenberg-Marquardt Method / 5.7： |
| Numerical Efficiency / 5.8： |
| Which formulation for the linear algebra? / 5.8.1： |
| The n-form / 5.8.1.1： |
| The m-form / 5.8.1.2： |
| Sequential updating / 5.8.1.3： |
| Computation of derivatives / 5.8.2： |
| Optimising representations / 5.8.3： |
| Approximations, Short Cuts and Ad-hoc Methods / Chapter 6： |
| The Constrained Exact Solution / 6.1： |
| Least Squares Solutions / 6.2： |
| The overconstrained case / 6.2.1： |
| The underconstrained case / 6.2.2： |
| Truncated Singular Vector Decomposition / 6.3： |
| Twomey-Tikhonov / 6.4： |
| Approximations for Optimal Methods / 6.5： |
| Approximate a priori and its covariance / 6.5.1： |
| Approximate measurement error covariance / 6.5.2： |
| Approximate weighting functions / 6.5.3： |
| Direct Multiple Regression / 6.6： |
| Linear Relaxation / 6.7： |
| Nonlinear Relaxation / 6.8： |
| Maximum Entropy / 6.9： |
| Onion Peeling / 6.10： |
| The Kalman Filter / Chapter 7： |
| The Basic Linear Filter / 7.1： |
| The Kalman Smoother / 7.2： |
| The Extended Filter / 7.3： |
| Characterisation and Error Analysis / 7.4： |
| Validation / 7.5： |
| Global Data Assimilation / Chapter 8： |
| Assimilation as a Inverse Problem / 8.1： |
| Methods for Data Assimilation / 8.2： |
| Successive correction methods / 8.2.1： |
| Optimal interpolation / 8.2.2： |
| Adjoint methods / 8.2.3： |
| Kalman filtering / 8.2.4： |
| Preparation of Indirect Measurements for Assimilation / 8.3： |
| Choice of profile representation / 8.3.1： |
| Linearised measurements / 8.3.2： |
| Systematic errors / 8.3.3： |
| Transformation of a characterised retrieval / 8.3.4： |
| Numerical Methods for Forward Models and Jacobians / Chapter 9： |
| The Equation of Radiative Transfer / 9.1： |
| The Radiative Transfer Integration / 9.2： |
| Derivatives of Forward Models: Analytic Jacobians / 9.3： |
| Ray Tracing / 9.4： |
| Choosing a coordinate system / 9.4.1： |
| Ray tracing in radial coordinates / 9.4.2： |
| Horizontally homogeneous case / 9.4.3： |
| The general case / 9.4.4： |
| Transmittance Modelling / 9.5： |
| Line-by-line modelling / 9.5.1： |
| Band transmittance / 9.5.2： |
| Inhomogeneous paths / 9.5.3： |
| Curtis--Godson approximation / 9.5.3.1： |
| Emissivity growth approximation / 9.5.3.2： |
| McMillin--Fleming method / 9.5.3.3： |
| Multiple absorbers / 9.5.3.4： |
| Construction and Use of Prior Constraints / Chapter 10： |
| Nature of a Priori / 10.1： |
| Effect of Prior Constraints on a Retrieval / 10.2： |
| Choice of Prior Constraints / 10.3： |
| Retrieval grid / 10.3.1： |
| Transformation between grids / 10.3.1.1： |
| Choice of grid for maximum likelihood retrieval / 10.3.1.2： |
| Choice of grid for maximum a priori retrieval / 10.3.1.3： |
| Ad hoc Soft constraints / 10.3.2： |
| Smoothness constraints / 10.3.2.1： |
| Markov process / 10.3.2.2： |
| Estimating a priori from real data / 10.3.3： |
| Estimating a priori from independent sources / 10.3.3.1： |
| Maximum entropy and the estimation of a priori / 10.3.3.2： |
| Validating and improving a priori with indirect measurements / 10.3.4： |
| The nearly linear case / 10.3.4.1： |
| The moderately non-linear case / 10.3.4.2： |
| Using Retrievals Which Contain a Priori / 10.4： |
| Taking averages of sets of retrievals / 10.4.1： |
| Removing a priori / 10.4.2： |
| Designing an Observing System / Chapter 11： |
| Design and Optimisation of Instruments / 11.1： |
| Forward model construction / 11.1.1： |
| Retrieval method and diagnostics / 11.1.2： |
| Optimisation / 11.1.3： |
| Specifying requirements for the accuracy of parameters / 11.1.4： |
| Operational Retrieval Design / 11.2： |
| State vector choice / 11.2.1： |
| Choice of vertical grid coordinate / 11.2.3： |
| Choice of parameters describing constitutents / 11.2.3.1： |
| A priori information / 11.2.4： |
| Retrieval method / 11.2.5： |
| Diagnostics / 11.2.6： |
| Testing and Validating an Observing System / Chapter 12： |
| The X[superscript 2] Test / 12.1： |
| Quantities to be Compared and Tested / 12.3： |
| Internal consistency / 12.3.1： |
| Does the retrieval agree with the measurement? / 12.3.2： |
| Consistency with the a priori / 12.3.3： |
| Measured signal and a priori / 12.3.3.1： |
| Retrieval and a priori / 12.3.3.2： |
| Comparison of the retrieved signal and the a priori / 12.3.3.3： |
| Intercomparison of Different Instruments / 12.4： |
| Basic requirements for intercomparison / 12.4.1： |
| Direct comparison of indirect measurements / 12.4.2： |
| Comparison of linear functions of measurements / 12.4.3： |
| Algebra of Matrices and Vectors / Appendix A： |
| Vector Spaces / A.1： |
| Eigenvectors and Eigenvalues / A.2： |
| Principal Axes of a Quadratic Form / A.3： |
| Singular Vector Decomposition / A.4： |
| Determinant and Trace / A.5： |
| Calculus with Matrices and Vectors / A.6： |
| Answers to Exercises / Appendix B： |
| Terminology and Notation / Appendix C： |
| Summary of Terminology / C.1： |
| List of Symbols Used / C.2： |
| Bibliography |
| Index |
| Preface |
| Introduction / Chapter 1： |
| The Beginnings / 1.1： |
