Kirchhoff migration using eikonal-based computation of traveltime source-derivatives |

Over the years, there have been significant efforts and progress in traveltime computations. The
quality of traveltimes has a direct influence on Kirchhoff-type migrations since it determines the
kinematic behaviors of the imaged wavefields. One can use either ray-tracing approaches or
finite-difference solutions of the eikonal equation. The first option naturally handles multi-arrivals
and can be extended to other wavefield approximations, such as Gaussian beams (Hill, 1990; Albertin et al., 2004; Gray, 2005; Hill, 2001),
but is at the same time usually subject to the necessities of ray-coordinate and migration-grid mapping
and irregular interpolation between rays in the presence of shadow zones in complex velocity media
(Sava and Fomel, 1998). Two popular methods from the second option are the *fast-marching method* (FMM)
(Sethian, 1996; Sethian and Popovici, 1999) and the *fast-sweeping method* (FSM) (Zhao, 2005). They both rely
on an ordered update to recover the causality behind expanding wave-fronts in a general medium, and
are thus limited to first-arrival computations. Several works attempt to overcome the single-arrival
drawback of the finite-difference eikonal solvers, for example multi-phase computation (Engquist and Runborg, 1996),
phase-space escape equations (Fomel and Sethian, 2002), and slowness marching (Symes and Qian, 2003).

In practice, traveltime tables can be pre-computed on coarse grids and saved on disk, then serve as a dictionary when read by Kirchhoff migration algorithms. It is common to carry out a certain interpolation in this process in order to satisfy the needs of depth migration for fine-gridded traveltime tables at a large number of source locations (Alkhalifah, 2011; Vanelle and Gajewski, 2002; Mendes, 2000). Kirchhoff migrations with traveltime tables computed on the fly face the same issue. During the traveltime computation stage, accuracy requirements from eikonal solvers may lead to a fine model sampling. Combined with a large survey, traveltime computation for each shot can be costly. Because all traveltime computations handle one shot at a time, the overall cost increases linearly with the number of sources. Moreover, we need to store a large amount of traveltimes out of a dense source sampling. Therefore a sparse source sampling is preferred. In this paper, we try to address the problem of traveltime table interpolation between sparse source samples. The traveltime table estimated with simple nearest-neighbor or linear interpolation could not provide satisfying accuracy unless the velocity model has small variations. One possible improvement is to include derivatives in interpolation. During ray tracing, traveltime source-derivatives are directly connected to the slowness vector at the source and stay constant along individual rays, thus could be outputted as a by-product of traveltimes. For finite-difference eikonal solvers, such a convenience is not easily available. In these cases, we would like to avoid an extra differentiation on traveltime tables along the source dimension to compute such derivatives (Vanelle and Gajewski, 2002), because its accuracy in turn relies on a dense source sampling and induces additional computations. Alkhalifah and Fomel (2010) derived an equation for the traveltime perturbation with respect to the source location changes. The governing equation is a first-order partial differential equation (PDE) that describes traveltime source-derivatives in a relative coordinate moving along with the source. In this paper, we show that the traveltime source-derivative desired by interpolation is related to this relative-coordinate quantity by a simple subtraction with the slowness vector. Unlike a finite-difference approach, traveltime source-derivatives computed by the PDE method are source-sampling independent. The extra costs are rather inexpensive. In this paper, we apply this method to Kirchhoff migration with first-arrival traveltimes computed by the FMM eikonal solver.

The paper is organized as follows. In the first section, we review the theory and implementation of the eikonal-based traveltime source-derivatives. Next, we use both simple and complex synthetic models to demonstrate the accuracy of a cubic Hermite traveltime table interpolation using the source-derivatives, and show effects of incorporating such an interpolation into Kirchhoff migration. We focus mainly on the kinematics in these experiments by neglecting possible true-amplitude weights in Kirchhoff migration. Finally, we discuss limitations and possible extensions of the proposed method.

Kirchhoff migration using eikonal-based computation of traveltime source-derivatives |

2013-07-26