Ray-illumination compensation for adjoint-state first-arrival traveltime tomography

Abstract

First-arrival traveltime tomography is an essential method for obtaining near-surface velocity models. The adjoint-state first-arrival traveltime tomography is appealing due to its straightforward implementation, low computational cost, and low memory consumption. Because solving the point-source isotropic eikonal equation by either ray tracers or eikonal solvers intrinsically corresponds to emanating discrete rays from the source point, the resulting traveltime gradient is singular at the source point, and we denote such a singular pattern the imprint of ray-illumination. Because the adjoint-state equation propagates traveltime residuals back to the source point according to the negative traveltime gradient, the resulting adjoint state will inherit such an imprint of ray-illumination, leading to singular gradient-descent directions when updating the velocity model in the adjoint-state traveltime tomography. To mitigate this imprint, we solve the adjoint-state equation twice but with different boundary conditions: one being taken to be regular data residuals and the other taken to be ones uniformly, so that we are able to use the latter adjoint state to normalize the regular adjoint state and we further use the normalized quantity to serve as the gradient direction to update the velocity model; we call this process ray-illumination compensation. To overcome the issue of limited aperture, we have developed a spatially varying regularization method to stabilize the new gradient direction. A synthetic example demonstrates that our method is able to mitigate the imprint of ray-illumination, remove the footprint effect near source points, and provide uniform velocity updates along raypaths. A complex example extracted from the Marmousi2 model and a migration example illustrate that the new method accurately recovers the velocity model and that an offset-dependent inversion strategy can further improve the quality of recovered velocity models.