On the problem of determining the parameters of a layered elastic medium and an impulse source

Abstract

Considering the linear system of elasticity equations describing the wave propagation in the half-space ℝ+3 = {x ∈ ℝ3 | x3 > 0} we address the problem of determining the density and elastic parameters which are piecewise constant functions of x3. The shape is unknown of a point-like impulse source that excites elastic oscillations in the half-space. We show that under certain assumptions on the source shape and the parameters of the elastic medium the displacements of the boundary points of the half-space for some finite time interval (0, T) uniquely determine the normalized density (with respect to the first layer) and the elastic Lame parameters for x3 ∈ [0, H], where H = H(T). We give an algorithmic procedure for constructing the required parameters.