Nonlinear algorithm to solve an inverse kinematic seismic problem in the plane (Q5942501)

An inverse kinematic seismic problem is considered. Let a simply connected plane domain \(D\) with the boundary \(\partial D\) be given and a distribution of velocities \(\mathbf v (\mathbf r)\) (\(\mathbf r\) is the radius-vector) in the domain \(D\) be known. Additionally, let for any pair of points \(P\) and \(Q\) (\(P \ni \partial D, Q \ni \partial D\)) the time of transmission of the seismic signal \(T(P,Q)\) along a trajectory \(\gamma(P,Q)\) connecting the points \(P\) and \(Q\) be given by the expression \[ T(P,Q) = \int_{\gamma (P,Q)} \frac{ds}{|\mathbf v (\mathbf r)|} . \] It is necessary to reconstruct the velocity distribution \(\mathbf v(\mathbf r)\) if the function \(T(P,Q)\) is known. A nonlinear algorithm for solving the above mentioned inverse kinematic seismic problem on the plane is presented in the article. For the realization of the algorithm theoretical investigations of the Riemann space with conformal metric are used.