First-break traveltime tomography is based on the eikonal equation. Since the
eikonal equation is solved at fixed shot positions and only receiver positions can move along the ray-path,
the adjoint-state tomography relies on inversion to resolve possible contradicting information between
independent shots. The double-square-root 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 shot-wise approach in the traditional
eikonal-based framework. The double-square-root eikonal equation is singular for the horizontal waves, which
require special handling. Although it is possible to recover all branches of the solution through
post-processing, our current forward modeling and tomography focus on the diving wave branch only. We consider
two upwind discretizations of the double-square-root eikonal equation and show that the explicit scheme is only
conditionally convergent and relies on non-physical 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 marching method thus obtaining first-break traveltimes both efficiently and
accurately. To compare the new double-square-root 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.
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, Alexander Vladimirsky
and Sergey Fomel
Bureau of Economic Geology
Department of Mathematics