Linearized inversion of multioffset seismic reflection data in the omega -K domain; depth-dependent reference medium

Abstract

Abstract The computation of synthetic seismograms can be linearized with respect to a reference medium that is close to the actual medium. Using a least-squares formulation, the inverse problem can then be set up as a problem of quadratic optimization. The inverse problem is greatly simplified if the reference medium is symmetric. For a homogeneous reference medium, a rigorous and economic solution can be obtained by Fourier transforming all spatial variables. In particular, the solution can be obtained through an explicit formula that does not require the resolution of any linear system (as is the case when not working in the Fourier domain). However, the assumption of a homogeneous reference medium is generally not realistic. In some situations, the reference medium can be depth-dependent. It can then be shown that by Fourier transforming time and all spatial variables except depth, the inverse problem also has an elegant and economic solution. If delta m(x, z) is the (unknown) difference between the reference medium and the true (2-D) medium, the Fourier-transformed solution delta m(k x , z) can be obtained by solving, for each value of the horizontal wavenumber k x , a linear system whose dimension equals the number of depth samples.