Inversion for Anisotropic Velocity Parameter

Abstract

The problem under study concerns the robust computation of a certain parameter of anisotropy from observed travel-times of a seismic shear wave propagating through a geological medium.

We have obtained an exact mathematical description of a geoseismic signal propagating through an anisotropic medium using a constant coefficient wave equation as the basic model. This model captures exactly the elliptical velocity profile required in the formulation of the geophysical model from which we obtained exact formulas describing the travel-time through a two layer geological structure, and an exact inversion formula for computing the anisotropic velocity parameter (gamma). A robust numerical method based on a minimization technique was presented as an accurate method of computing both travel-time and the inverted gamma.

The exact formulas and robust numerical methods are significant improvements over the approximations and root finding methods discussed in the background material, and we note our formulation is no more difficult than these background methods.

We derived asymptotic formulas valid for the near vertical case, which describe accurately the high sensitivity of gamma to the input parameters in this case. Our numerical work also confirms this sensitivity, even using exact formulas and robust numerical methods.

We conclude that the computation of the anisotropic velocity parameter (gamma) for the given physical measurements from a series of surface signals and single borehole receiver is intrinsically unstable. By changing to the alpha,beta velocity parameter space, we obtain an inversion method that is much less sensitive to input errors. For certain geophysical problems, the alpha,beta parameters may suffice for an accurate description of the material.

When the anisotropic velocity parameter (gamma) is needed directly, a different measurement technique is required. This route will require further investigation, and we have proposed a number of promising possibilities involving a differential time measure.