On the determination of the velocity and elastic parameters of the focal zone medium from the earthquake hodographs

We consider an inverse kinematic problem of seismic (IKPS) with internal sources. It consists in determining the velocities of longitudinal and transverse waves by the travel times from earthquake sources, occurring in the focal zone, to a group of seismic stations. We propose an algorithm for the numerical solution to the problem, based on the eikonal equation and the technology of smoothing multidimensional splines, which give an approximation of the velocity structure of the focal zone. The paper presents theoretical results that substantiate the algorithm for solving the problem by approximation methods by smoothing with multidimensional splines from data on irregular grids. We describe the results of the numerical solution to the problem, the calculations with real data on earthquakes in the focal zone, and give the estimates of the velocity and elastic parameters of the medium.