Analytic solutions to the refraction and extinction integrals are presented for the case of a horizontally or spherically stratified medium. These solutions are not only useful for the calculation of the images that would be seen through a lens with a continuously varying index of refraction, such as the atmosphere, they also provide a solution to the inverse problem of determining the refractive structure from measurements of the image. A remarkably simple inversion scheme is presented for determining the refractive (temperature) structure of the earth’s atmosphere by observations of the setting sun. The same scheme works for determination of vertical profiles of the extinction coefficient.
Alistair B. Fraser, "Solutions of the refraction and extinction integrals for use in inversions and image formation," Appl. Opt. 16, 160-165 (1977)