Migration/inversion for transversely isotropic elastic media

Abstract

SUMMARY Seismic migration and inversion are closely related processes for obtaining images of the subsurface. Both techniques attempt to infer petrophysical and structural parameters from seismic data, and both are driven by an underlying mathematical model for wave propagation in the earth. In this sense, migration can be regarded as the first step in’ a general linearized-inversion scheme; we adopt this viewpoint here in formulating a joint migration/inversion method for anisotropic elastic media. Our derivation is based on the distorted-wave Born approximation, where approximate Green’s functions for the background medium are determined using asymptotic ray theory. We also make use of a stationary-phase correction to account for out-of-plane scattering in a 2-D earth model. The inversion is cast as a discrete, generalized l2 optimization problem, which we regularize using a priori model variances. An approximate solution is obtained in one or more iterations using a quasi-Newton technique. Our implementation is tailored for weakly anisotropic elastic media possessing transversely isotropic (TI) symmetry, and so is well suited for investigations of earth models characterized by a singe set of parallel fractures or periodic thin layering. Several TI model parametrization schemes are evaluated for a vertical-incidence narrow-aperture recording configuration. By applying singular-value decomposition to the approximate Hessian operator used in the inversion, we find that when the symmetry axis of the medium is vertically oriented the condition number is minimized by choosing vertical impedance parameters, density and Thomsen’s (1986) anisotropy parameters. Elastic stiffnesses and density may be a better parametrization choice if the symmetry axis is either horizontal or unknown. The small effective rank (2-3) of the Hessian in our narrow-aperture tests implies that migration inversion (without a priori constraints) of surface-reflection data may be inadequate to fully characterize a TI medium, which would require six parameters. However, this result does not in itself justify the use of simpler isotropic inversion schemes, since the principal eigenvectors determined here represent a blend of ‘isotropic’ and ‘anisotropic’ information about the earth. A suite of point-diffractor tests shows that individual perturbation amplitudes recovered in the inversion are smaller, but cover a larger region of the image, than those used in the original model, consistent with the band-limited nature of the input signal. Parameter cross-coupling problems affect the inversion results for all parameters, but are more prominent in the inversion images for anisotropy parameters. In our tests, the linearized-inversion results converge to a local minimum after three iterations, and the scatterer distributions in the inversion images are essentially fixed after a single iteration.