Rami Nammour

Multiparameter Inversion: Cramer's Rule for Pseudodifferential Operators

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, Cramers 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.

Approximate Multi-Parameter Inverse Scattering: Cramer's Rule And Phase Space Scaling

We describe a computationally efficient method of approximate muli-parameter linearized inversion. The method assumes that an accurate background model is known, and that the recorded traces approximate perturbational (Born) data. Approximate material parameters are extracted from migrated data in two steps: (i) one or more (depending on the number of parameters) additional applications of Born modeling followed by wave equation migration are combined to form modified images, one for each parameter; and (ii) a previously developed scaling method corrects amplitudes and spacially deconvolves the modified images to obtain estimates of material parameter perturbations. Neither iterative inversion nor explicit geometric optics computations are required. We illustrate the method by recovering velocity and density perturbations in an acoustic model, from synthetic prestack data.