Nonlinear elastic inversion of multioffset seismic data

The treatment of multioffset seismic data as an acoustic wave field is becoming increasingly disturbing to many geophysicists who see a multitude of wave phenomena such as amplitude-offset variations or shear-wave events, which can only be explained by using the more correct physical equation, namely the elastic wave equation. Not only are these phenomena ignored by acoustic theory, but they are treated as undesirable noise when they should rather be used to provide extra information about the subsurface such as S-velocity. The problems of using the conventional acoustic wave equation approach can be eliminated by starting afresh with an elastic approach. One framework was provided by Tarantola (1984) who described how to do an elastic inversion of seismic data in very general situations for the Lame' parameters and density. New equations were derived to perform an inversion for P-velocity, S-velocity and density as well as the P-impedance, S-impedance and density since these are better resolved than the Lame parameters. The inversion is based on nonlinear least-squares and proceeds by iteratively updating the earth parameters until a good fit is achieved between observed and modeled data corresponding to these earth parameters. The iterations are based on a preconditioned conjugate gradient least-squares algorithm. The fundamental requirement of such a least-squares algorithm is the gradient direction which tells how to update model parameters. This can be derived directly from the wave equation and it may be computed by several wave propagations. Although any scheme could in principle be chosen to perform wave propagations, the elastic finite-difference method is used because it directly simulates the elastic wave equation and can handle complex and thus realistic distributions of elastic parameters. This method of inversion is costly since it is similar to an iterative prestack shot-profile migration but has greater power than any migration since it solves for the P-velocity, S-velocity and density and can handle very general situations including transmission problems (see Gauthier and Tarantola, 1985, for an acoustic transmission problem example using the same technique).