J BTdtZAX dz BX T t0 A AA

where the Planck function, B and the transmittance from level z, t depend strongly on A but more weakly on AA, provided AA is narrow. If AA is in a strongly absorbing region, t(0, A, AA) tends to zero, the second term that represents the contribution from the surface to space vanishes and the radiance leaving the top of the atmosphere is just the integral over height of the Planck function weighted by the derivative with respect to height of the atmospheric transmittance along the viewed path. By choosing a series of adjacent spectral intervals along the edge of an atmospheric absorption band, where the absorption coefficient changes rapidly with wavelength, it is possible in principle to select a set of weighting functions that span the atmosphere from top to bottom.

Weighting function flg. 10.11. Weighting functions corresponding to a few of the 2378 spectral channels in the AIRS instrument. Those shown here are in the 1100 cm-1 to 600 cm-1 (about 9 to 16 ¡m) spectral region. The log pressure scale is approximately proportional to height: 100, 10 and 1 mb correspond to about 15, 30 and 50 km above the surface, respectively.

Weighting function flg. 10.11. Weighting functions corresponding to a few of the 2378 spectral channels in the AIRS instrument. Those shown here are in the 1100 cm-1 to 600 cm-1 (about 9 to 16 ¡m) spectral region. The log pressure scale is approximately proportional to height: 100, 10 and 1 mb correspond to about 15, 30 and 50 km above the surface, respectively.

Figure 10.11 shows what is actually achieved by AIRS, which has no less than 2378 infra-red channels sampled every 2.7 s.

The calculation of the weighting functions requires knowledge of the composition of the atmosphere, at least the abundances of the principal absorbers in the spectral intervals chosen for the sounding instrument. This is one of the reasons why wavelengths dominated by CO2, or in the microwave O2, are chosen in general, since these gases are nearly uniformly mixed. However, if a simultaneous set of observations is made at wavelengths dominated by an absorber of unknown concentration, for example water vapour, then a second retrieval can be performed in which the profile of the absorber amount is the unknown to be retrieved. Since most of the radiation originates near the peak of the relevant weighting function, a set of radiance measurements contains information about the variation with temperature with height. A radiometric instrument on a satellite can make such a set of measurements, which can then be used to reconstruct the temperature profile with a vertical resolution that is related to the number and width of the weighting functions. An optimized set of nadir (downward-viewing) radiances can yield profiles with a temperature accuracy of one or two K with a vertical resolution of around 10 km. Limb-viewing instruments have much sharper weighting functions that can resolve layers only 1 to 2 km thick, although of course the horizontal resolution is poorer, in the direction along the line-of-sight (but not perpendicular to that), generally around 200 km. Often, this is an acceptable sacrifice since the most interesting features are those that vary over relatively short distances in the vertical.

The conversion of radiometric measurements into geophysical quantities uses retrieval methods based on radiative-transfer theory to find the best solution for the values of the variables of interest that are consistent with the observed radiation field leaving the top of the atmosphere, and quantify the errors and uncertainties. Several different approaches are possible (see Rodgers 2000) but all involve fitting calculations of the spectrum of the outgoing radiation to the atmospheric parameters that produce a match to the measurements. Most of the retrieval methods used in practice are based on least-squares fitting of measured to calculated radiances, and often include a number of refinements. For example, instead of finding the profile from the measurements alone, statistical 'a priori' information about the expected value of the profile, based on a history of previously measured values ('climatology'), can be folded in. One way to do this is to use the climatological value of the temperature profile as the starting point for a retrieval by iteration, or to use independent data from another instrument (as AIRS does with AMSU and HSB, as discussed above) to tie down the effects of clouds, aerosols and humidity on the observed radiances. These quantities can also be products of the retrieval; the realistic representation of the time-dependent behaviour of clouds, their formation processes and their radiative effects, is a key problem in climate modelling and prediction, and a major source of uncertainty that only more and better data will resolve.

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