where To and po are reference temperatures and pressures; and SFB is a factor to be fitted or calculated (see below). The mean absorption coefficient k, may be expanded as k,(T)=k,(To) ('T)^exp

where qr is 1.0 for linear molecules and 1.5 for nonlinear molecules. There are thus five independent parameters in the Goody-Voigt model: k„(T0), y0, 6/a(h, Ei, and SFB, which may either be fitted to laboratory transmission spectra, or alternatively derived from tabulated line listings such as HITRAN and GEISA. These parameters, while usually fitted directly to measured spectra, are related to the properties of the real absorption lines in the spectral band: kp (T0) is the integrated line strength of all the lines in the wavenumber range considered at the standard temperature; y0 is the mean ratio of pressure-broadened to Doppler-broadened linewidths at STP (standard temperature and pressure); S/a% is mean line spacing divided by the Doppler-broadened width at STP; Ei is the mean energy of lower states; and SFB is the mean self-broadening to foreign-broadening ratio of absorption lines.

Once band data have been tabulated, the transmission of homogeneous paths may be rapidly and accurately calculated. However, real atmospheres are inhomo-geneous in that pressure, temperature, and composition vary rapidly with altitude. How then may band models be applied? It may be shown that the mean transmission of a path though an inhomogeneous atmosphere may be well approximated by the mean transmission of an equivalent homogeneous path, whose path amount, mean pressure, and mean temperature are given by:

This is known as the Curtis-Godson approximation. Hence, to use band models to calculate thermal emission spectra, the inhomogeneous atmosphere is represented by a series of equivalent Curtis-Godson paths from space to progressively deeper levels in the atmosphere and the difference between the band-calculated mean transmissions used to find the mean transmission weighting function. In addition, since the absorption lines of different molecules are rarely correlated with each other, the total transmission of Curtis-Godson paths for all the gases concerned is simply found by multiplying the individual gas transmissions together, or equivalently summing the optical thicknesses. The band model approach is very fast, but is found to be useless for multiple scattering calculations and hence is mostly used in the mid-infrared to far-infrared where scattering is generally less important (as we shall see in Section 6.5.2).

Correlated-k approximation

An alternative approach to calculating finite resolution spectra is to use ^-distributions. For a path of absorber amount m in an atmosphere of uniform pressure p and T, the mean transmission is given by where the absorption coefficient at a particular wavelength is the summation of all the individual line contributions. Since the absorption coefficient k(,) is a rapidly varying function of a wavenumber, in order to numerically calculate the mean transmission accurately, a very fine wavenumber step must be chosen. However, when calculating the mean transmission in a spectral interval it does not matter which parts of the interval are actually highly or poorly absorbing. All we need to know is what fraction of the interval has low absorption, what fraction has high absorption, and so on. In other words, if we calculate a high-resolution absorption coefficient spectrum using a regularly spaced high-resolution grid, and then sort the absorption coefficients into order starting with the low absorption coefficients first and then working monotonic-ally through to the high absorption coefficients, the resulting integral of the sorted spectrum is identical to that of the original. The advantage of this approach is that the sorted spectrum, known as the k-distribution, k(g), is a smoothly varying function that is usually expressed in terms of the fraction of the interval g, which varies between 0 and 1. Since it is a smoothly varying function, the integral may be accurately integrated with far fewer quadrature points and thus calculation of mean transmission is very much faster. In practice ten to twenty quadrature points are usually found to be satisfactory and the mean transmission may be approximated by where kt is the k-distribution calculated at each of the N quadrature points; and Agi are the quadrature weights. The k-distributions may be pre-calculated for each gas for a range of temperatures and pressures found in real atmospheres and then stored in look-up tables for rapid interpolation and calculation of mean transmission. Since the absorption lines of different gases may be assumed to a good approximation to be uncorrelated, it is also reasonably straightforward to combine k-distributions together (Lacis and Oinas, 1991). The k-distribution look-up tables may be calculated

either directly from line data or, if the available line data are of poor quality, indirectly from band data using the technique of exponential sums (e.g., Irwin et al., 1996). While we can see that ^-distributions can speed up transmission calculations for homogeneous paths, how can they help us for inhomogeneous paths? For monochromatic calculations, the transmission of an inhomogeneous path is found by splitting the path into small subpaths, calculating the transmission, and then multiplying all the transmissions together. However, for band-averaged transmissions, such as those used by band models, this multiplication is not possible and thus the Curtis-Godson approximation must be used. The Curtis-Godson approximation may also be used with ^-distributions, but there is then no advantage over the band model approach. Instead, it is found that regions of high and low absorption within the spectral band are spectrally correlated between various subpaths within the inhomogeneous path. This correlation exists between the ^-distributions also (Lacis and Oinas, 1991; Goody et al., 1989). Hence, the ^-distributions may effectively be multiplied together almost as though they were monochromatic to determine the mean transmission of the inhomogeneous path where the inhomogeneous path has been split into M subpaths. This is the correlated-k approximation and is found to have a similar accuracy to that of the Curtis-Godson approximation. The great advantage, however, lies in the fact that thermal emission and in particular scattering calculations (discussed in Section 6.6) may also be summed in exactly the same way. Hence, the technique of correlated-k allows for rapid calculation of spectra in multiply scattering atmospheres and is thus used extensively to simulate the near-IR reflectance spectra of the giant planets.

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