Coefficient reconstruction problem for the two-dimensional viscoelasticity equation in a weakly horizontally inhomogeneous medium

We discuss the inverse problem of successively finding two unknowns (a one-dimensional integral operator kernel and a two-dimensional wave propagation velocity) for the viscoelasticity equation in a weakly horizontally inhomogeneous medium. The direct initial boundary value problem for the displacement function contains zero initial data and the Neumann boundary condition of special form. Additional information consists in the Fourier transform of the displacement function at $x_3=0$. We assume that the unknown functions are expanded in an asymptotic power series in a small parameter. We prove theorems on the global unique solvability and stability of the inverse problem solution.