Analytical solutions to the refraction integrals appropriate for ray trajectories along slant paths through the atmosphere are derived in this paper. This type of geometry is commonly encountered in remote-sensing applications utilizing an occultation technique. The solutions are obtained by evaluating higher-order terms from expansion of the refraction integral and are dependent on the vertical temperature distribution. Refraction parameters such as total refraction angles, air masses, and path lengths can be accurately computed. It is also shown that the method can be used for computing refraction parameters in astronomical refraction geometry for large zenith angles.
William P. Chu, "Calculations of atmospheric refraction for spacecraft remote-sensing applications," Appl. Opt. 22, 721-725 (1983)