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

Abstract

Derivatives of the displacement tensor with respect to the independent model parameters of the subsurface, also called Frechet derivatives (or sensitivity kernels), are a key ingredient for seismic full-waveform inversion (FWI) 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. Because 2.5-D wavefield modelling involves a real point source in a 2-D geological model with 3-D (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 modelling 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 FWI in general viscoelastic anisotropic media. We then demonstrate the numerical calculations of all these derivatives in two common cases: (1) viscoelastic isotropic; and (2) 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 modelling. Moreover, the numerical results are verified against the analytic solutions for homogeneous models. We further validate the numerical derivatives in a 2-D heterogeneous viscoelastic TTI case by conducting a synthetic data experiment of frequency-domain FWI to individually recover the 12 independent model parameters (density, dip angle, 5 elastic moduli and 5 corresponding Q-factors) in a simple model comprising an anomalous square box target embedded in a uniform background. Another 2.5-D multi-target model experiment presenting impacts from four common seismic surveying geometries validates the Frechet derivatives again.