Uniqueness theorem of solution the integral geometry problem for the family curves

Abstract

In this article the following class of integral geometry problems is considered: about the function reconstruction, shaped by the integrals on some set of curves. These problems are correlated with several applications. In order to study the internal earth structure, the multiple explosions are held on Earth surface. Then, the fluctuation regimes of earth surface are measured on equipment for each explosion. The goal of research is to determine the distribution of physical parameters inside the Earth according to equipment measurements, correlated with laws on dissemination of seismic waves. The most clear functional of such equipment is the arrival time of seismic wave, which exactly serves as a base for interpretation practice. It is known that lineriazed problem of seismic-exploration data interpretation is actually the integral geometry problem. An integral geometry also includes the problems related to the radiography, particularly the interpretation problem of X-ray examination. For instance, a X-ray film darkening functionally correlated with the absorption integral along the X-ray from the source to point on the film. Thus, determination problem of spatial distribution for the absorption coefficient is also actually an integral geometry problem. In this case, it is required to determine the function if the integrals of this function on set of rays were set. The integral geometry problem is studied in this work. The solution uniqueness theorem is proved for the considered integral geometry problem.