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Information content and optimization of high-spectral-resolution measurements

A remote measurement of the atmosphere with high spectral resolution form the ground or from space, such as might be made form one of the new generation of instruments such as the Fourier transform or grating spectrometers AIRS, MIPAS, TES, or IASI, would appear to contain a large amount of information about the atmosphere, but it is not immediately obvious how to quantify it or to use it efficiently or effectively. The usefulness of a particular remote measurement can be described by a wide range of parameters including the spatial resolution, precision and accuracy of the retrieved product. For optimization of a measurement method, however, a single scalar figure of merit is required, whereas all of the above are vectors because they are functions of altitude and/or quantity retrieved. Spatial averages of resolution, precision and accuracy could be used, but there are two quantities of general applicability which can be defined without reference to the specific retrieval method, and are therefore more straightforward in use. These are the information content, and the number of degrees of freedom for signal, the latter being a measure of the number of statistically independent quantities in any one measurement. It can be shown that both of these quantities are functions of the eigenvalues of a certain matrix, and are closely related. The information content and the degrees of freedom provide single parameters which can be used in the automated optimization of for example, instrument design parameters, the selection of microwindows or subsets of channels for retrieval, the optimization of retrieval strategy, and the understanding of the information content of a spectrum. Some such applications are illustrated by means of simulated spectra for the AIRS instrument.