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

Abstract

AbstractDerivatives of the displacement tensor with respect to the independent model parameters of the subsurface, also called Fréchet 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 Fréchet derivatives again.