The temperature scaling function S would be computed separately for each band. The use of linear scaling in pressure is justified by the fact that most of the spectrum is far from line centers, so absorption scales like the strong-line approximation as long as the pressure is not so large that line widths become comparable to the width of the band under consideration. The use of a single temperature-scaling function is harder to justify theoretically, but seems to be supported by numerical experiment.
When qG is not small, there is one additional complication to take into account when defining the equivalent path. Namely, the absorption coefficient for self-broadened collisions is generally different from that for foreign-broadened collisions, so one must scale the absorption coefficient according to the proportion of self vs. foreign collisions. Like the temperature scaling factor, the ratio of self to foreign broadening can vary considerably even amongst nearby lines. Typically, though, one simply chooses a representative ratio of self to foreign broadening which is assumed constant within each band over which the distribution H is computed. Let's call this ratio aself. The molar concentration of the greenhouse gas is (M/MG)qG, where M is the mean molecular weight of the mixture; hence the proportion of collisions which are self-collisions is (M/MG)qG while the proportion of foreign collisions is 1 — (M/MG)qG. Then, if H is computed using the foreign-broadened absorption at the standard pressure, the appropriate equivalent path to use in computing the transmission is
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