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

Considering the linear system of elasticity equations describing the wave propagation in the half-space $\mathbb R^3_+=\{x\in\mathbb R^3\mid x_3>0\}$ we address the problem of determining the density and elastic parameters which are piecewise constant functions of $x_3$. 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 Lamé parameters for $x_3\in[0,H]$, where $H=H(T)$. We give an algorithmic procedure for constructing the required parameters.