We have seen in Chapter 6 how the electromagnetic spectra of the planets have many absorption and reflection features that are unique to particular constituents. Using measured and estimated absorption coefficients and assumed atmospheric profiles of temperature, cloud, and composition, radiative transfer models may be constructed which can simulate the spectrum of a planet as seen from the Earth or a passing spacecraft. These synthetic spectra may then be compared with measured spectra and any differences interpreted in terms of how the assumed profiles need to be modified in order to achieve the best possible fit between the two. This is the essence of the retrieval theory, and at first sight appears relatively straightforward. For example, consider a region of the thermal-infrared where atmospheric absorption is due only to a well-mixed gas. The 600 cm-1 to 700 cm-1 spectrum of the giant planets is a good example of such a region since absorption here is due almost entirely to H2-H2 and H2-He collision-induced absorption. Suppose our simulated spectrum was too bright at some wavelength in this range, then we would correctly deduce that our assumed, or a priori, temperature profile was too warm at roughly the altitude where the weighting function (Section 6.4) peaked. Hence, we would need to slightly "cool" our model profile at this level in order to improve the fit between synthetic and measured spectra.
Although this seems straightforward it is in fact very difficult for a number of reasons. First of all, it can be seen from the radiative transfer equation (Section 6.4.1 and Equation 6.40) that radiance at any particular wavelength is actually the weighted average of thermal emission from a continuous range of altitudes governed by the transmission weighting function. The width of the transmission weighting function for nadir-viewing is approximately one scale height and so information on the vertical temperature structure is considerably vertically smoothed. Hence, there is an infinitely wide range of possible temperature profiles whose synthetic spectra would fit the measured spectrum equally well! Fortunately, using spectral data from a range of wavelengths, or a range of emission angles, such that the peaks of the weighting functions cover a certain vertical span, reduces this ambiguity somewhat, but even then it must be remembered that the spectra are measured at a finite number of wavelengths (or emission angles), whereas an atmospheric profile is a continuous function. Hence, the retrieval problem is in effect one of attempting to calculate an infinite set of parameters using a finite set of measurements, which is known as an ill-posed problem. While there are techniques for solving such ill-posed problems, retrievals also suffer from a further complication in that they are ill-conditioned. This means that, without care, experimental noise in the measured spectrum may be greatly amplified by the fitting process leading to wholly unreliable vertical oscillations in the fitted atmospheric profiles.
Since there is literally an infinite number of possible atmospheric profiles whose simulated spectra will fit a measured spectrum to within the measurement error, how then may we hope to extract any meaningful information from the measurement of planets' electromagnetic spectra? Fortunately nature comes to our rescue since, although atmospheric properties such as temperature are continuous functions of height, we know that in practice they are also generally smoothly varying functions. Hence, by imposing sufficient vertical smoothing, meaningful vertical profiles may be determined from the measurements, as we shall see in Sections 7.10.1 to 7.10.4.
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