Remote determination of the refractive index structure parameter from experimental data by Tatarski's integral equation requires numerical inversion of a compact linear operator. Such problems are known to be ill posed; the consequent ill conditioning of the inversions has led to a large degree of uncertainty in reported reconstructions. In this paper we use the singular value decomposition of a compact operator to define a measure of the maximum amount of recoverable information in such an inversion, terming it the essential dimension of the operator. We propose the use of filtered singular value decomposition as the numerical algorithm that will recover most of the information and minimize uncertainty. A detailed study of the operators appearing in determination of horizontal (wave front is spherically symmetric) and vertical (wave front is plane) profiles of the atmosphere is under-taken to determine their essential dimensions by both analysis and computation. The results indicate that for typical parameter values, both operators have small essential dimensions, with vertical profiles being harder to reconstruct than horizontal profiles.