The authors describe a procedure they believe to be optimum for determining the path of a ray through an atmosphere described by a radially varying refractive index profile on the Earth. The result is exact for patched quadratic profiles inverse problem and is suitable for calculating ray paths in complex profiles. The method also facilitates greatly the inverse problem.
Brett D. Nener,
"Analytical models of optical refraction paths in the lower atmosphere (Invited Paper)", Proc. SPIE 5793, Atmospheric Propagation II, (25 May 2005); doi: 10.1117/12.604295; https://doi.org/10.1117/12.604295