Some results of the group approach in the kinematic seismic problem (Q2760701)

The article is devoted to some theoretical and applied questions related to studying the two-dimensional (direct and inverse) kinematic seismic problem (geometrical optics). In this study, the eikonal equation plays an important role. Using the group bundle \(G\) of the eikonal equation, the author presents a new description of the direct and inverse kinematic seismic problems. The resultant quasilinear wave equation is the resolving system of the group bundle of the eikonal equation and admits the Lax representation as an \(L\)--\(A\)-pair.NEWLINENEWLINENEWLINEThe author exposes a collection of results on the behavior of the geometric divergence of rays \(D(t,x,y)\) and its properties. In particular, the author demonstrates a method for calculating \(D(t,x,y)\) in the direct problem, the connection between the invariant \(K(x,y)\) of the group \(G\) and \(D(t,x,y)\). Moreover, the author obtains upper and lower estimates for \(D\), \(D^*\), and their derivatives, and also compares the estimates for \(D\), \(D^*\) in different points of the same ray. In addition, a series of integral formulas in the local inverse problem is presented. These formulas provide an expression for some functionals in terms of the data to the inverse kinematic problem. Explicit formulas in the quadratures for the variables \(S = (D^*_{\tau})^2\) and \(\tau_{tt}\) are given as functions of the new independent variable \(z = z(\tau)\) along the ray and as solutions to the equation \(S_z + K = 0\). Some results on the connection of the equations with the differential and Riemann geometries are also presented.