An extremely accurate but simple asymptotic description for the path of a light ray propagating over a curved Earth with steady radial variations in refractive index is derived using simple scaling arguments. It is used to determine effectively exact analytic solutions for the path of rays through refractive-index profiles described in terms of patched quadratics. Such patched quadratics can be used to accurately describe almost all refractive-index profiles of practical interest. The results show that images generated by rays passing through a quadratic refractive-index profile are uniformly magnified in the vertical direction, and magnification and displacement observations can be used to determine the refractive-index profile parameters. For patched quadratics, observations of critical rays can be used to determine the thicknesses of the quadratic layers.An effectively exact solution is also obtained for exponential index profiles and this is used to determine the path of rays through a thin boundary layer attached to the Earth; an inferior mirage situation.