Linearization of the P‐wave eikonal equation for weak vertical transverse isotropy

Abstract

I develop approximate P‐wave and SV‐wave eikonal equations for weak vertical transversely isotropic (VTI) media and then use perturbation theory to solve the P‐wave eikonal equation. I proceed by expressing the exact VTI eikonal equations for P‐waves and SV‐waves in terms of the intuitive “Thomsen” anisotropy parameters and then linearizing the results with respect to these parameters. Next, I apply perturbation theory (with a heterogeneous, isotropic “reference” medium) to the P‐wave weak VTI eikonal equation. This results in a linear partial differential equation for the traveltime perturbation, which I solve analytically along raypaths in the isotropic reference medium. Traveltimes in the weak VTI medium may then be obtained by simply integrating the reference slowness function plus the (weighted) anisotropy perturbations along raypaths in the isotropic reference medium. This simple solution may be incorporated into standard isotropic ray tracing or traveltime generation methods to produce traveltimes for weak VTI media. I analytically incorporated the method into an isotropic eikonal equation solver and tested it on a vertically heterogeneous weak VTI model, where the strength of the anisotropy was 5–15%. Comparison against results from an exact VTI ray tracing code showed that, for lateral distances up to twice the depth, the weak VTI traveltimes and phase angles were accurate to within 1.3% and 2.2%, respectively. This linear theory yields a physical understanding for how the anisotropy parameters cause traveltimes in VTI media to differ from those in isotropic media. Applications of the theory also include traveltime computation and insights into linear tomographic velocity estimation of reflection seismic data in weak VTI media.