An inverse two-dimensional problem for determining two unknowns in equation of memory type for a weakly horizontally inhomogeneous medium

A two-dimensional inverse coefficient problem of determining two unknowns - the coefficient and the kernel of the integral convolution operator in the elasticity equation with memory in a three-dimensional half-space, is presented. The coefficient, which depends on two spatial variables, represents the velocity of wave propagation in a weakly horizontally inhomogeneous medium. The kernel of the integral convolution operator depends on a time and spatial variable. The direct initial boundary value problem is the problem of determining the displacement function for zero initial data and the Neumann boundary condition of a special kind. The source of perturbation of elastic waves is a point instantaneous source, which is a product of Dirac delta functions. As additional information, the Fourier image of the displacement function of the points of the medium at the boundary of the half-space is given. It is assumed that the unknowns of the inverse problem and the displacement function decompose into asymptotic series by degrees of a small parameter. In this paper, a method is constructed for finding the coefficient and the kernel, depending on two variables, with an accuracy of correction having the order of O(ε2). It is shown that the inverse problem is equivalent to a closed system of Volterra integral equations of the second kind. The theorems of global unique solvability and stability of the solution of the inverse problem are proved.