Mathematical statement of the inverse kinematic problem of seismics for 3D inhomogeneous medium. Part I

The inverse kinematic problem (IKP) for 2D inhomogeneous half-plane with a 2D hodograph of refracted waves given at its boundary is reduced to the initial problem for the non-classical evolutional partial differential equation of the first order. In the present paper, the method considered earlier is extended to the 3D case of the IKP. It is necessary to determine the velocity distribution of elastic waves in a 3D spatial subdomain adjacent to the aperture for the 3D inhomogeneous half-space using the 4D hodograph of refracted waves given at the 2D aperture of its boundary. In the derivation of the equation, the differentiation rule of hodograph is rigorously substantiated for the case, when the coordinates of the source and the receiver coincide.