Numerical Fr\'echet derivatives of the displacement tensor for 2.5-D frequency-domain seismic full-waveform inversion in viscoelastic TTI media

TL;DR

This paper derives explicit formulas for the derivatives of the displacement tensor in 2.5-D seismic modeling, enabling improved full-waveform inversion in viscoelastic TTI media with realistic wavefield simulations.

Contribution

It provides the first explicit analytic expressions for displacement tensor derivatives in 2.5-D frequency-domain seismic inversion in viscoelastic anisotropic media.

Findings

Derived explicit formulas for derivatives in viscoelastic isotropic media.

Numerical calculations for TTI solids demonstrate sensitivity patterns.

Comparison of derivatives across different models shows varied sensitivity patterns.

Abstract

Derivatives of the displacement tensor with respect to the independent model parameters of the subsurface, also called Fr\'echet derivatives (or sensitivity kernels), are a key ingredient for seismic full-waveform inversion with a local-search optimization algorithm. They provide a quantitative measure of the expected changes in the seismograms due to perturbations of the subsurface model parameters for a given survey geometry. Since 2.5-D wavefield modeling involves a real point source in a 2-D geological model with 3D (spherical) wave properties, it yields synthetic data much closer to the actual practical field data than the commonly used 2-D wave simulation does, which uses an unrealistic line source in which the waves spread cylindrically. Based on our recently developed general 2.5-D wavefield modeling scheme, we apply the perturbation method to obtain explicit analytic expressions for the derivatives of the displacement tensor for 2.5-D/2-D frequency-domain seismic full-waveform inversion in general viscoelastic anisotropic media. We then demonstrate the numerical calculations of all these derivatives in two common cases: (i) viscoelastic isotropic and (ii) viscoelastic tilted transversely isotropic (TTI) solids. Examples of the differing sensitivity patterns for the various derivatives are investigated and compared for four different homogeneous models involving 2-D and 2.5-D modeling.