Inverse problems of ray tomography

The ray tomographic problems, where it is necessary to reconstruct the structure of the medium from measurements of the linear integrals for a system of rays intersecting a certain fixed region, are encountered in a large of various fields of science and technology, for example, in seismic tomography, optical tomography of atmosphere, acoustic tomography of the ocean, radiotomography (RT) of the ionosphere, etc. The approaches in the closest fields of tomography of geophysical structures are described and systematized. The choice of a ray tomography method to be used determines the form linear integrals. Usually the problems of ray tomography are small-angle (small foreshortening) problems which can be conveniently solved in algebraic form without using integral transformations of the Radon type. As a rule most of these methods lead to problems of linear tomography, where the rays are straight and independent on the function to be reconstructed. The general problems of linear ray tomography and some limitations are discussed. Having been subjected to appropriate mathematical treatment (discretization), such problems lead to systems of linear equations (SLE). The methods of the operator construction are primary, but SLE solution algorithms are secondary. The authors consider different variants of the inverse problem of ray tomography