Transition Operator Approach to Seismic Full-Waveform Inversion in Arbitrary Anisotropic Elastic Media (Communications in Computational Physics, 2020)
We generalize the existing distorted Born iterative T-matrix (DBIT) methodto seismic full-waveform inversion (FWI) based on the scalar wave equation, so that itcan be used for seismic FWI in arbitrary anisotropic elastic media with variable massdensities and elastic stiffness tensors. The elastodynamic wave equation for an arbitrary anisotropic heterogeneous medium is represented by an integral equation ofthe Lippmann-Schwinger type, with a 9-dimensional wave state (displacement-strain)vector. We solve this higher-dimensional Lippmann-Schwinger equation using a transition operator formalism used in quantum scattering theory. This allows for domaindecomposition and novel variational estimates. The tensorial nonlinear inverse scattering problem is solved iteratively by using an expression for the Fréchet derivativesof the scattered wavefield with respect to elastic stiffness tensor fields in terms of modified Green's functions and wave state vectors that are updated after each iteration.Since the generalized DBIT method is consistent with the Gauss-Newton method, itincorporates approximate Hessian information that is essential for the reduction ofmulti-parameter cross-talk effects. The DBIT method is implemented efficiently usinga variant of the Levenberg-Marquard method, with adaptive selection of the regularization parameter after each iteration. In a series of numerical experiments basedon synthetic waveform data for transversely isotropic media with vertical symmetryaxes, we obtained a very good match between the true and inverted models whenusing the traditional Voigt parameterization. This suggests that the effects of cross-talk can be sufficiently reduced by the incorporation of Hessian information and theuse of suitable regularization methods. Since the generalized DBIT method for FWIin anisotropic elastic media is naturally target-oriented, it may be particularly suitablefor applications to seismic reservoir characterization and monitoring. However, thetheory and method presented here is general.