Exact leastsquares and Backus Gilbert solutions

Suppose that we represent a set of measurements (channel radiances, or a whole spectrum measured at a set of discrete wavelengths) by the vector ym of m elements, known as the measurement vector. We may also represent atmospheric conditions with the state vector xn of n elements, which may contain the temperature, composition, cloud abundances, at appropriate levels in the atmosphere, etc. We may represent the radiative transfer model by a forward model F(xn), from which we may calculate the synthetic spectrum yn y„ = F(x„). (7.16)

Retrievals of properties such as temperature may be linearized by expanding the state vector x about an initial first-guess, or a priori, solution x0 and hence y„ = ye + Ay = F(xq)+K(x„ - x>) (7.17)

where K is a matrix containing the rate of range of each element of yc with respect to each element of x; and y0 is the spectrum calculated with the a priori state vector.

If we choose the number of elements n of the state vector to be equal to the number of measured spectral radiances m, then the K-matrix is square and may be inverted. Hence, substituting the measured spectrum ym for the calculated spectrum yn in Equation (7.17), and solving for xn we find x„ = xo + K-1 (ym - yo). (7.18)

This so-called "exact" solution provides a perfect fit to the measured spectrum, but does so at a very heavy price. Such solutions are extremely ill-conditioned and small errors in the measured spectrum may lead to huge errors in xn and hence it is difficult to assess the reliability of the derived solution.

A particularly widespread approach when interpreting giant planet spectra is thus to parameterize the atmospheric temperature and composition profiles with far fewer parameters than the number of spectral points. For example, the abundance of hydrocarbons in the stratosphere is so little known that often all that can be meaningfully retrieved from the data is the approximate mean mole fraction. Assuming a temperature profile derived either from previous radio occultation measurements, or from radiances measured in the nearby 1,300 cm-1 methane band (methane is well-mixed in the atmospheres of the giant planets and its mole fraction is known), synthetic spectra may be generated for a range of mole fractions likely to be consistent with photochemical models and compared with the measured spectrum to find the best fit. Another example is ammonia retrievals for Jupiter and Saturn, where the ammonia profile may be represented by a mean value up to the condensation level, whereupon it is assumed to follow either a relative humidity curve, or alternatively decreases at a certain fractional scale height. Representing the atmospheric profile with fewer points than the number of measurements and solving for xn results in a "least-squares" solution, which minimizes the difference between the measured and calculated spectra by minimizing the "cost function" $

where ^ is simply the sum of the squares of the differences between the measured and calculated spectra.

In cases where more vertical resolution is required, or where the spectra are particularly noisy, it is found that we need to apply some a priori assumptions to prevent meaningless solutions. There are a number of approaches to the problem, which are discussed at length by Hanel et al. (2003), Houghton et al. (1984), and Rodgers (2000). However, all methods basically arrive at the same conclusion: namely, that the precision of a retrieved profile depends on the vertical averaging applied or assumed. Hence, remotely sensed spectra may be inverted to yield very accurate smoothed perturbations to the assumed profile, but increasingly less accurate retrievals as less and less vertical smoothing is applied. There is thus essentially a trade-off between error and vertical resolution, which is formalized by the Backus-Gilbert approach (1970) and further discussed by Hanel et al. (2003) and Rodgers (2000). There are a number of formalisms to solving the inversion problem, of which the two most commonly used for planetary retrievals are the constrained linear inversion technique (Conrath et al., 1998; Hanel et al., 2003) and a modified form of optimal estimation (Rodgers, 2000), which is described here.

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