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

Abstract

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 eikonalbased 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.

A robust approach to time‐to‐depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations

“…For this purpose, we choose the upwind finite-difference scheme (Franklin and Harris 2001;Li, Fomel and Vladimirsky 2011) based on t 0 for both L t 0 and L x 0 . As shown by Li, Vladimirsky and Fomel (2013), applying J and its transpose involves only sparse triangularized matrix-vector multiplications and is therefore inexpensive. For example, at each grid point, L −1 t 0 relies only on its upwind neighbours.…”

Obtaining low frequencies for Full Waveform Inversion by using augmented physics

“…Since the travel time is an integral of the model over the ray path, its Jacobian with respect to the model contains mainly low frequencies. Thus, a way to overcome the lack of low frequencies in the data, d, is to extract it from the travel time, T. Travel time tomography has been considered by Benaichouche et al, (2015) and by Li et al, (2013). Here we use the Fast Marching method to solve the forward problem and compute the sensitivities directly.…”

Full Waveform Inversion Guided by Travel Time Tomography

“…Since the FM algorithm uses only known variables for determining each new variable, by reordering the unknowns according to the FM order we obtain a lower triangular system which can be solved efficiently in one forward substitution sweep in O(n) operations. For more information on the solution of travel time tomography using FM see [23,48]. Applying the operator (2.18) is required for each source in (2.7), but storing storing the sparse matrix in (2.19) for each source may be highly memory consuming at large scales.…”

“…In this paper we present this approach using travel time tomography. This problem is computationally attractive when based on the eikonal equation and its factored version [23,5,48] for forward modeling. Since the solution of the eikonal equation is based on integration of the slowness along rays, the tomographic travel time data contains low frequency information [58,55].…”