## Transition operator approach
to seismic full-waveform inversion in anisotropic elastic media

**Morten Jakobsen,**
**Ivan Psencik,**
**Einar Iversen** **&**
**Bjorn Ursin**
### Summary

We generalize the existing distorted Born iterative T-matrix
(DBIT) method
to seismic full-waveform inversion (FWI) based on the scalar wave
equation, so that it
can be used for seismic FWI in arbitrary anisotropic elastic media with
variable mass
densities and elastic stiffness tensors. The elastodynamic wave equation
for an arbitrary
anisotropic heterogeneous medium is represented by an integral equation
of
the 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
domain
decomposition and novel variational estimates. The tensorial nonlinear
inverse scattering
problem is solved iteratively by using an expression for the Frechet
derivatives
of the scatteredwavefield 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, it
incorporates approximate Hessian information that is essential for the
reduction of
multi-parameter cross-talk effects. The DBIT method is implemented
efficiently using
a variant of the Levenberg-Marquard method, with adaptive selection of
the regularization
parameter after each iteration. In a series of numerical experiments
based
on synthetic waveform data for transversely isotropic media with
vertical symmetry
axes, we obtained a very good match between the true and inverted models
when
using the traditional Voigt parameterization. This suggests that the
effects of crosstalk
can be sufficiently reduced by the incorporation of Hessian information
and the
use of suitable regularization methods. Since the generalized DBIT
method for FWI
in anisotropic elastic media is naturally target-oriented, it may be
particularly suitable
for applications to seismic reservoir characterization and monitoring.
However, the
theory and method presented here is general.

### Keywords

Waveform inversion, time-lapse seismic, seismic inverse
scattering, target-oriented inversion, Green's function.

### Whole paper

The reprint can be obtained from
Ivan Psencik.

*Commun. computational Phys.*, **28** (2019), 297-327.