Propagation of waves in a randomly stratified medium: an inverse problem

We pose and solve an inverse problem of finding a coefficient in the wave equation in the inhomogeneous semispace on the scattering data of a plane wave incident from the homogeneous semispace. The unknown coefficient is a sum of a deterministic summand of one variable (the “depth” $z$) and a small random summand $\alpha(x,z)$. We look for the deterministic summand, the expectation $E(\alpha(x,z))=:m(z)$, and the second moment $r(x_1-x_2,z_1,z_2):=E(\alpha(x_1,z_1)\alpha(x_2,z_2))$. Here the symbol $E(\cdot)$ stands for expectation. The stratification property of a medium means that (i) the deterministic summand depends only on $z$, (ii) $m(z)$ depends only on $z$, and (iii) the second moment for fixed $z_1$ and $z_2$ depends only on $x_1-x_2$.