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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. 

Seismic waves in earth media usually undergo attenuation, causing energy losses and phase distortions. In the regime of high-frequency asymptotics, a complex-valued eikonal is an essential ingredient for describing wave propagation in attenuating media, where the real and imaginary parts of the eikonal function capture dispersion effects and amplitude attenuation of seismic waves, respectively. Conventionally, such a complex-valued eikonal is mainly computed either by tracing rays exactly in complex space or by tracing rays approximately in real space so that the resulting eikonal is distributed irregularly in real space. However, seismic data processing methods, such as prestack depth migration and tomography, usually require uniformly distributed complex-valued eikonals. Therefore, we have developed a unified framework to Eulerianize several popular approximate real-space ray-tracing methods for complex-valued eikonals so that the real and imaginary parts of the eikonal function satisfy the classic real-space eikonal equation and a novel real-space advection equation, respectively, and we dub the resulting method the Eulerian partial-differential-equation method. We further develop highly efficient high-order methods to solve these two equations by using the factorization idea and the Lax-Friedrichs weighted essentially nonoscillatory schemes. Numerical examples demonstrate that our method yields highly accurate complex-valued eikonals, analogous to those from ray-tracing methods. Our methods can be useful for migration and tomography in attenuating media.