An eikonal based formulation for traveltime perturbation with respect to the source location

Introduction

The traveltime field is typically used to describe the phase behavior of the Green's function, a key tool for Kirchhoff modeling and migration. It also is used at the heart of many velocity estimation applications, such as reflection tomography. The traveltime field for a fixed source in a heterogeneous medium is governed by the eikonal equation, derived about 150 years ago by Sir William Rowan Hamilton. Since early 1990s, a direct numerical solution of the eikonal equation has been a popular method of computing traveltimes on regular grids, commonly used in seismic imaging (Podvin and Lecomte, 1991; Vidale, 1990; van Trier and Symes, 1991; Vidale, 1988). Modern methods of traveltime computation include the fast marching method, developed by Sethian (1996) in the general context of level set methods for propagating interfaces. Sethian and Popovici (1999) and Popovici and Sethian (2002) report a successful application of this method in three-dimensional seismic computations. Alkhalifah and Fomel (2001) improved its accuracy using spherical coordinates. Alternative methods include group fast marching (Kim, 2002), fast sweeping (Zhao, 2005), and paraxial marching (Qian and Symes, 2002). Several alternative schemes are reviewed by Kim (2002).

The nonlinear nature of the eikonal partial differential equation was addressed by Aldridge (1994), who linearized the eikonal equation with respect to velocity perturbation, while retaining its first-order nature. Alkhalifah (2002) developed a similar linearization formula for perturbations in anisotropic parameters and solved it numerically using the fast marching method. The linear feature increased the efficiency and stability of the numerical solution, especially in the anisotropic case.

A major drawback of using conventional methods to solve the eikonal equation numerically is that we only evaluate the fastest arrival solution, not necessarily the most energetic one. This results in less than acceptable traveltime computation for imaging in complex media (Geoltrain and Brac, 1993). Eikonal solvers can be extended to image multiple arrivals through semi-recursive Kirchhoff migration (Bevc, 1997), phase-space equations (Fomel and Sethian, 2002), or slowness matching (Symes and Qian, 2003) techniques. The linearization also helps to avoid the first-arrival only limitation, especially when the background traveltime field includes energetic arrivals.

The dependence of the traveltime field on the source location can be empirically evaluated by comparing the shape of the traveltime fields for two different sources when the sources are superimposed on each other. For a medium with no lateral velocity variation, the traveltime field should be source-location independent. Relating the two traveltime fields directly through an equation can provide insights into the dependence of traveltime fields on lateral velocity variations. Such information can serve in developing better traveltime interpolation and velocity estimation.

In this paper, we develop a new eikonal-based partial differential equation that relates traveltime shape changes to changes in the source location. The changes can be described first- or second-order accurate terms and thus used in a Taylor's type expansion to find the traveltime for a nearby source. We test the accuracy of the approximation analytically and numerically through the use of complex synthetic models. In the discussion section, we suggest possible applications for the new equation.

An eikonal based formulation for traveltime perturbation with respect to the source location