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 $O(\varepsilon^2)$. 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.