Inversion of Fredholm integral equations of the Mellin convolution type arising in atmospheric remote sensing.

Abstract

In this paper, the mathematics associated with simultaneous measurements of two parameters arising in the remote sensing of the atmosphere are cast in a form that permits the explicit solution for the eigenvalues and eigenfunctions. These eigenvalues and eigenfunctions are then used to deal with the ill-posed nature of the problem and ultimately to solve for the unknown atmospheric particle size density based on the information in both sets of measurements. The physical setting for this problem involves the measurement of optical transmission and the angular scattering of intensity. Using the Fraunhofer approximation and the van de Hulst anomalous diffraction approximation, the mathematics reduces to the inversion of Fredholm integral equations of the first kind in the special case known as a Mellin convolution. These equations are shown to be bounded and self-adjoint. Unfortunately, their inverse is seen to be unbounded and consequently the problem is ill-posed. The formulation of these equations to yield consistent, bounded, self-adjoint operators whose eigenfunctions and eigenvalues can be determined and addressing the ill-posed nature of the problem by exploiting the multisource data form the heart of this research.