Inversion problems have existed in various branches of physics and engineering for a long time, but in the past ten or fifteen years they have received far more attention than ever before. The reason, of course, was the arrival on the scene of large computers (which enabled hitherto abstract algebraic concepts such as the solution of linear systems of equations in many unknowns to be achieved arithmetically with real numbers) and the launching of earth-orbiting satellites which viewed the atmosphere from outside and which could only measure atmospheric parameters remotely and indirectly by measuring electromagnetic radiation of some sort and relating this to the atmospheric parameter being sought, very often implying a mathematical inversion problem of some sort.

It was soon found that the dogmas of algebraic theory did not necessarily carry over into the realm of numerical inversion and some of the first numerical inversion s gave results so bad — for example, in one early instance negative absolute temperatures — that the prospects for the remote sensing of atmospheric temperature profiles, water vapor concentration profiles, ozone profiles, and so on, for a time looked very bleak, attractive as they might have appeared when first conceived.

The crux of the difficulty was that numerical inversions were producing results which were physically unacceptable but were mathematically acceptable (in the sense that had they existed they would have given measured values identical or almost identical with what was measured). There were in fact ambiguities — the computer was being "told" to find an f(x) from a set of values for g(y) at prescribed values of y, it was "told" what the mathematical process was which related f(x) and g(y), but it was not "told" that there were many sorts of f(x) — highly oscillatory, negative in some places, or whatever — which, either because of the physical nature of f(x) or because of the way in which direct measurements showed that f(x) usually behaved , would be rejected as impossible or ridiculous by the recipient of the computer's "answer". And yet the computer was often blamed, even though it had done all that had been asked of it and produced an f(x) which via the specified mathematical process led to values of g(y) which were exceedingly close to the initial data for g(y) supplied to the computer. Were it possible for computers to have ulcers or neuroses there is little doubt that most of those on which early numerical inversion attempts were made would have acquired both afflictions.

For a time it was thought that prec1s10n and accuracy in the computer were the core of the problem and more accurate numerical procedures were sought and applied without success. Soon it was realized that the problem was not inaccuracy in calculations, but a fundamental ambiguity in the presence of inevitable experimental inaccuracy — sets of measured quantities differing only minutely from each other could correspond to unknown functions differing very greatly from each other. Thus in the presence of measurement errors (or even in more extreme cases computer roundoff error) there were many, indeed an infinity of possible "solutions". Ambiguity could only be removed if some grounds could be provided, from outside the inversion problem, for selecting one of the possible solutions and rejecting the others. These grounds might be provided by the physical nature of the unknown function, by the likelihood that the unknown function be smooth or that it lie close to some known climatological average. It is important, however, to realize that most physical inversion problems are ambiguous — they do not possess a unique solution and the selection of a preferred unique solution from the infinity of possible solutions is an imposed additional condition. The reasonableness of the imposed condition will dictate whether or not the unique solution is also reasonable. There have been many advances in mathematical procedures for solving inversion problems but these do not remove the fundamental ambiguity (although in some instances they very effectively hide it). Our measurements reduce the number of possible solutions from infinity to a lesser but still infinite selection of possibilities. We rely on some other knowledge (or hope) to select from those a most likely or most acceptable candidate.

S. TWOMEY