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PrefaceThis book is aimed at graduate students and at post-doctoral and other researchers who need to solve inverse problems for atmospheric measurements. The intention is both to provide a well-founded background to the inverse problem and its solution, with which the student will develop a good intuition about the nature of the prob lem, and to give practical recipes for solving real problems. It has developed from various courses that I have given over many years, including informal seminars to research groups at the National Center for Atmospheric Research in Boulder, Col orado, the Subdepartment of Atmospheric, Oceanic and Atmospheric Physics at Oxford, the National Institute for Water and Air at Lauder, New Zealand, and formal courses given at the 1995 NATO Advanced Study Institute at Val Morin, Canada (Brasseur, 1997) and the 1999 Oxford/RAL Spring Schools on Quantitative Earth Observation, which is to be repeated in 2000 and future years. The usual approach to solving an atmospheric retrieval problem will consist of several stages: design a forward model to describe the instrument and the physics of the measurement; determine the criterion by which a solution is acceptable as valid; construct a numerical method to find a solution which satisfies the criterion; carry out an error analysis; validate the process by reference to internal diagnostics and independent measurements; and finally attempt to understand how the result obtained is related to reality and examine how much information has been obtained. In this book I go through these stages more or less in the reverse order to try to develop the reader's intuition about the nature of the inverse problem. I first discuss the information content of remote measurements and the corresponding retrievals, introducing the Bayesian approach as a conceptual background for the problem, then carry out a formal characterisation and error analysis of a generic retrieval to show how retrievals are related to reality, and how their errors can be described. The characterisation and error analysis provides a basis from which we can understand what is meant by 'optimality' in designing an inverse method. Next I discuss numerical methods for solving both linear and nonlinear problems, optimally and otherwise. After a detour into related issues of Kalman filtering and assimilation of meteorological observations, both of which are concerned with the VII Inverse Methods for Atmospheric Sounding Downloaded from www.worldscientific.com by 54.245.55.244 on 05/12/18. For personal use only. VlllPreface time variation of measured quantities, I discuss forward modelling, prior information and systematic approaches to the design and validation of remote sounding data analysis systems. A set of exercises for the student are given within the iext. Many of these are part of the algebraic or conceptual development of the topic being described, so they should be attempted. Full answers are given in Appendix B.There is a range of mathematical tools and concepts which are of particular va...

[1] When intercomparing measurements made by remote sounders, it is necessary to make due allowance for the differing characteristics of the observing systems, particularly their averaging kernels and error covariances. We develop the methods required to do this, applicable to any kind of retrieval method, not only to optimal estimators. We show how profiles and derived quantities such as the total column of a constituent may be properly compared, yielding different averaging kernels. We find that the effect of different averaging kernels can be reduced if the retrieval or the derived quantity of one instrument is simulated using the retrieval of the other. We also show how combinations of measured signals can be found, which can be compared directly. To illustrate these methods, we apply them to two real instruments, calculating the expected amplitudes and variabilities of the diagnostics for a comparison of CO measurements made by a ground-based Fourier Transform spectrometer (FTIR) and the ''measurement of pollution in the troposphere'' instrument (MOPITT), which is mounted on the EOS Terra platform. The main conclusions for this case are the following: (1) Direct comparison of retrieved profiles is not satisfactory, because the expected standard deviation of the difference is around half of the expected natural variability of the true atmospheric profiles. (2) Comparison of the MOPITT profile retrieval with a simulation using FTIR is much more useful, though still not ideal, with expected standard deviation of differences of around 20% of the expected natural variability. (3) Direct comparison of total columns gives an expected standard deviation of about 9%, while comparison of MOPITT with a simulation derived from FTIR improved this to 8%. (4) There is only one combination of measured signals that can be usefully compared. The difference is expected to have a standard deviation of about 5.5% of the expected natural variability, which is mostly due to noise.

This paper reviews the methods which may be used to estimate the state of the atmosphere, i.e., the distribution of temperature and composition, from measurements of emitted thermal radiation such as are made by remote sounding instruments on satellites. The principles of estimation theory are applied to a linear version of the problem, and it is shown that many of the apparently different methods to be found in the literature are particular cases of the same general method. As an aid to understanding, the optimum linear solution is described in terms of the geometry of n dimensions, with n = 3 for illustration. In generalizing the approach to the nonlinear problem there are two stages: (1) finding any member of the infinite family of possible solutions, which may be done by any convenient iterative method, and (2) finding the optimum solution by satisfying appropriate constraints.

The characterization and error analysis of profiles retrieved from remote measurements present conceptual problems, particularly concerning interlevel correlations between errors, the smoothing effect of remote sounding and the contribution of a priori information to profile. A formal analysis for profile retrieval is developed which is independent of the nature of the retrieval method, provided that the measurement process can be characterized adequately. The relationship between the retrieved and true profiles is expressed in terms of a smoothing function which can be straightforwardly calculated. The retrieval error separates naturally into three components, (1) random error due to measurement noise, (2) systematic error due to uncertain model parameters and inverse model bias, and (3) null‐space error due to the inherent finite vertical resolution of the observing system. A recipe is given for evaluating each of the components in any particular case. Most of the error terms appear as covariance matrices, rather than simple error variances. These matrices can be interpreted in terms of “error patterns”, which are statistically independent contributions to the total error. They are the multidimensional equivalent of “error bars”. An approach is described which clarifies the relation of a priori data to the retrieved profile, and identifies a priori in cases where it is not an explicit part of the retrieval.