This is the exponential sum formula. It can be regarded as an M term fit to the transmission function, much as the Malkmus model is a two-parameter fit. The Lebesgue integration technique amounts to a simple reshuffling of the terms in the integrand: we collect together all wavenumbers with approximately the same «, compute the transmission for that value, and then weight the result according to "how many" such wavenumbers there are.
Because the absorption coefficient varies over such an enormous range, it is more convenient to work with H(ln«) rather than H(«). A typical result for CO2 is shown in Fig. 4.15, computed for two bands at a pressure of 100mb. The function is quite smooth, and can be reasonably well characterized by ten points or less. In contrast, given that the typical line width at 100mb is only .01cm-1, evaluation of the transmission integral in the Riemann form, Eq. 4.73, would require at least 25000 points in a band of width 25cm-1. Thus, the exponential sum approach is vastly more economical of computer time than a direct line-by-line integration would be.
The decay of the transmission with path length described by Eq. 4.75 is exactly analogous to the decay in time of the concentration of a mixture of radioactive substances with different half-lives. The short-lived things go first, leading to rapid initial decay of concentration; as time goes on, the mixture is increasingly dominated by the long-lived substances, and the decay rate is correspondingly slower. The way the transmission function converges as additional terms are included in the exponential sum formula is illustrated in Figure 4.16. Specifically, we divide the range of absorption coeffients into 20 bins equally spaced in log kg, and then truncate H so as
Figure 4.15: Cumulative probability function of the natural log of the absorption coefficient for CO2. Results are given for the 600-625 cm-1 and 575-600 cm-1 bands, and were computed at a pressure of 100mb and a temperature of 296K.
to keep only the N largest absorption coefficients, with N ranging from 1 (retaining only the strongest absorption) to 20 (retaining all absorption coefficients including the weakest). When only the strongest absorptions are included, the steep decay of transmission for small paths is correctly represented, but the transmission function decays too strongly at large paths. As more of the weaker absorption terms are included, the weaker decay of the transmission is well represented out to larger and larger paths. The ability to represent the decay rate of transmission over a very large range of paths is one of the two advantages of exponential sums over the Malkmus approach, the other advantage being the ability to incorporate scattering effects. An analytical example exploring related features of the exponential integral in Eq. 4.75 is explore in Problem ??.
If it weren't for the dependence of absorption coefficient on pressure and temperature, the exponential sum representation would be exact in the limit of sufficiently many terms. The computational economy of exponential sums comes at a cost, however, which is scrambling the information about which absorption coefficient corresponds to which frequency. This is not a problem if one is dealing with a layer with essentially uniform pressure and temperature, but it becomes a cause for concern in the typical atmospheric case where one is computing the transmission over a layer spanning considerable variations in temperature and pressure. The problem is that changing pressure or temperature changes the shape of the distribution H(k), and there is no rigorously correct way to deal with this within the exponential sum framework. In the discussion of line shapes, for example, we learned that increasing p reduces the peak absorption, but increases absorption between peaks. In terms of the probability distribution of k, this means that the largest and smallest values of k become less prevalent at the same time that the intermediate values become more prevalent. At very large values of pressure where the lines become extremely broad, k becomes a smooth function of frequency within each band and the probability distribution becomes concentrated on a single mean value of k. The effect of temperature on the shape of H(k) can be even more complex, since the temperature-dependence coefficients of line strength can differ greatly even for neighboring lines.
All these problems notwithstanding, experience has shown that one can obtain a reasonably accurate approximation to the band-averaged transmission function by assuming that all the
2000 4000 6000 28000 Path (kg/m2)
2000 4000 6000 28000 Path (kg/m2)
Figure 4.16: Convergence of exponential sum representation of the band averaged transmission function as the number of terms is increased. The calculation is for CO2 in a wavenumber band near the peak absorption.
absorption coefficients within a band have separable scaling of the form
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