Nonlinear 2D Elastic Inversion of Multi-Offset Seismic Data
, by Peter Mora
The treatment of multi-offset seismic data as an acoustic wavefield
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 also
treated as undesirable noise when they should rather be used to provide
extra information about the sub-surface 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 has been provided by Tarantola (1984) who described
how to do an elastic inversion of seismic data in very
general situations for the Lame's parameters and density. In this paper,
new equations have been 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 the observed data and the modeled data corresponding to these
earth parameters. The iterations are based on the preconditioned conjugate gradient algorithm. The fundamental requirement of such a least
squares algorithm is the gradient direction which tells how to
update the 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 the 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.
Three main weaknesses of this technique are that it requires fairly accurate knowledge of the low frequency velocity model beforehand, it assumes Gaussian model statistics and that it is very computer intensive. All these problems seem
surmountable. The low frequency information can be obtained either by
a prior tomographic step, or by the conventional NMO method or by
adding an additional inversion step for low frequencies to each iteration;
the statistics can be altered by preconditioning the gradient direction perhaps to make the solution blocky in appearance like well logs; and with some improvements to the algorithm and more parallel computers, it is hoped the technique will soon become routinely feasible.
STANFORD