The inverse problem for the acoustic equation in a~weakly horizontally inhomogeneous medium

A. S. Blagoveshchenskii ; D. A. Fedorenko

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 37, Tome 354 (2008), pp. 81-99

Résumé

The inverse problem of reconstruction of coefficients $A$ and $B$ for equation $$ AU_{tt} =\operatorname{div}(B\operatorname{grad}U) $$ in the half-plane $z>0$ is considered. It is assumed that instantaneous point source at $z=0$ generate wave field $U(t,z,x)$ that is known on the boundary. It is also known that coefficients $A$ and $B$ can be represented in the form \begin{gather*} A=A(z,\varepsilon x)=A_0(z)+\varepsilon xA_1(z)+O(\varepsilon^2),\\ B=B(z,\varepsilon x)=B_0(z)+\varepsilon xB_1(z)+O(\varepsilon^2). \end{gather*} Here $\varepsilon$ is a small parameter. The algorithm for the determination of coefficients $A_0,B_0,A_1,B_1$ with accuracy $O(\varepsilon ^2)$ is constructed. Bibl. – 5 titles.