Traditionally, the problem of dealing with the joint effects of scattering and gaseous absorption has been considered to be one of the scariest problems in radiative transfer. The problem appears scary only if one intends to mount an attack on it by some kind of modification of the band-averaged transmission function approach. The problem here stems from the fact that band-averaged transmission functions do not satisfy the multiplicative property, so that the path that one feeds to the transmission function involves the entire past history of the radiation between being the time it is emitted (or injected by solar radiation) and the time it leaves the atmosphere. When there is no scattering, the path is simple, but multiple reflection leads to an ensemble of very complex paths. For example, consider an isothermal layer of a nongrey gas sandwiched between a perfectly reflecting ground and a cloud that reflects half of all energy incident on it from below. Suppose the layer has a mass path I. Upward radiation emitted from near the center of the layer will be attenuated according to a path length 1Í by the time it reaches the cloud. Half of this will escape to space, but the other half will be reflected downward. To get out, it reaches the bottom boundary, where it is reflected upward. By the time it hits the top again, the path length is |Í, and the beam has been attenuated accordingly. Half of this escapes while the other half is reflected downward, and the process continues until there is essentially no beam left. If T is the transmission function written as a function of the path, the escaping flux is
where Io is the initial upward radiation emitted from the center of the layer. If the transmission function had the multiplicative property, then we would have t((2 + n + i)e) = t((2 + n)e)T(e) (5.57)
and the problem could be done as an iteration in which the fate of each reflection of flux hitting the cloud is independent of how many bounces it took before it got there. For band averaged transmission functions, which do not retain the multiplicative property, this is not the case and one needs to perform the sum over all past histories of paths taken by radiation. For the two-reflector case this is not too bad, but when one considers a more realistic problem in which absorption and scattering occur between a continuous array of pairs of layers of the atmosphere, one does indeed begin to quake in ones boots, if just a bit.
Most of the fear can be dispelled, however, through use of the exponential sums approach and it variants.
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