Multiparameter Inversion: Cramer's Rule for Pseudodifferential Operators

Rami Nammour,William W Symes

doi:10.1155/2011/780291

Abstract

Linearized multiparameter inversion is a model-driven variant of amplitude-versus-offset analysis, which seeks to separately account for the influences of several model parameters on the seismic response. Previous approaches to this class of problems have included geometric optics-based (Kirchhoff, GRT) inversion and iterative methods suitable for large linear systems. In this paper, we suggest an approach based on the mathematical nature of the normal operator of linearized inversion—it is a scaling operator in phase space—and on a very old idea from linear algebra, namely, Cramer's rule for computing the inverse of a matrix. The approximate solution of the linearized multiparameter problem so produced involves no ray theory computations. It may be sufficiently accurate for some purposes; for others, it can serve as a preconditioner to enhance the convergence of standard iterative methods.

Highlights

The linearized inverse problem in reflection seismology aims at recovering model perturbations from data perturbations, assuming known reference or background model
This method leverages the properties of the normal operator: under some conditions, it is a special type of matrix-valued spatially varying filter [1,2,3,4,5,6,7,8,9]
We summarize the theory as follows: the Hessian is well approximated by a pseudodifferential operator provided (i) the material parameters in the background model vary smoothly on the scale of a wavelength; (ii) diving wave energy is not present in the data or has been muted or dip-filtered out; (iii) the data has been polarized into propagating phase components

Summary

Introduction

The linearized inverse problem in reflection seismology aims at recovering model perturbations from data perturbations, assuming known reference or background model. We present an efficient method to approximate the solution of the normal equations that requires a few applications of the normal operator (one for one-parameter inversion, two for two-parameter inversion, and six for three-parameter inversion) This method leverages the properties of the normal operator: under some conditions (background model parameter fields slowly varying on the wavelength scale, diving wave energy eliminated from data, data polarized by propagating phases), it is a special type of matrix-valued spatially varying filter [1,2,3,4,5,6,7,8,9]. We review our approach to single-parameter inversion (based on constant density acoustic modeling, for instance) in Appendix B It belongs to the genre of scaling methods [13,14,15,16], which approximate the Hessian or its inverse by estimating a filter of some sort from a single application of the normal operator to a migrated image or some other test image. We chose to avoid these issues and focus only on the mathematical/computational issue of multiparameter inversion, given model-consistent data

Theory and Methods

Numerical Examples

Discussion and Conclusion

The Pseudodifferential Operator Algorithm

Findings

The One-Parameter Case