RECOVERY OF TWO-DIMENSIONAL PERTURBATIONS OF THE VELOCITY OF A VERTICALLY INHOMOGENEOUS MEDIUM FROM MULTICOVERAGE DATA (linearized formulation)

We consider a linearized problem of recovery of local two-dimensional perturbations of a vertically inhomogeneous medium of a given structure from multicoverage data of an ideal system (with the sources and receivers filling a straight line completely). After a Fourier transform is made with regard to time and source coordinates, it reduces to a decomposable system of Fredholm integral equations of the first kind with a continuous kernel relative to Fourier transform components with regard to the horizontal variable of the function to be sought. This paper reports results of numerical SVD analysis of linear finite-dimensional operators that appear in the process of discretization of this system in a realistic model for a vertically inhomogeneous enclosing medium. It is shown that the concept of “r”-solution to this system – a solution obtained by truncating SVD of the matrices representing these linear finite-dimensional operators in some basis – is meaningful even at small values of parameter “r” (r – is the number of singular vectors corresponding to largest singular values and retained upon truncating the SVD). Sensitivity of “r”-solutions to errors in specification of a vertically inhomogeneous enclosing medium is investigated numerically. They are also compared with the ideal multicoverage data on prestack migration into the enclosing vertically inhomogeneous medium.