First-break traveltime tomography with the double-square-root eikonal equation

First-break traveltime tomography is based on the eikonal equation. Because the eikonal equation is solved at fixed-shot positions and only receiver positions can move along the raypath, the adjoint-state tomography relies on inversion to resolve possible contradicting information between independent shots. The double-square-root (DSR) eikonal equation allows not only the receivers but also the shots to change position, and thus describes the prestack survey as a whole. Consequently, its linearized tomographic operator naturally handles all shots together, in contrast with the shotwise approach in the traditional eikonal-based framework. The DSR eikonal equation is singular for the horizontal waves, which require special handling. Although it is possible to recover all branches of the solution through postprocessing, our current forward modeling and tomography focuses on the diving wave branch only. We consider two upwind discretizations of the DSR eikonal equation and show that the explicit scheme is only conditionally convergent and relies on nonphysical stability conditions. We then prove that an implicit upwind discretization is unconditionally convergent and monotonically causal. The latter property makes it possible to introduce a modified fast matching method thus obtaining first-break traveltimes efficiently and accurately. To compare the new DSR eikonal-based tomography and traditional eikonal-based tomography, we perform linearizations and apply the same adjoint-state formulation and upwind finite-differences implementation to both approaches. Synthetic model examples justify that the proposed approach converges faster and is more robust than the traditional one.