An analysis is carried out of the Curtis-Godson approximation to the pressure integral which occurs in the expression for the optical depth for an inhomogeneous atmosphere. This approximation is shown to result from a certain optimization of the integrated Taylor series for the line shape function. In the case of a line strength which is independent of pressure it is also shown to be equivalent to first-order Gaussian quadrature. The Taylor series optimization gives rise to other useful approximations as well as to correction terms for the Curtis-Godson approximation. Calculations are carried out for the various approximations for the case of constant line strength in order to illustrate the magnitude of the errors involved. A discussion is also presented of the application of second-order Gaussian quadrature, which turns out to be the most accurate two-term approximation considered.
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Baxter H. Armstrong (1968) studied this question.