Refractive index inversion using analytic models of the atmosphere

An extremely accurate but simple asymptotic description (with known error, and with a rational improvement scheme) is obtained for the path of a ray propagating over a curved Earth with radial variations in refractive index. The result is sufficiently simple that analytic solutions for the path can be obtained for linear, quadratic and exponential refractive index profiles over a curved Earth. As well as rendering the inverse problem trivial for these profiles this formulation displays the results that images remain undistorted when viewed through a quadratic refractive index profile with explicit analytic results for the dependence of the displacement and magnification of the image on the refractive index profile parameters. Distortions occur for higher order refractive index profiles.