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

Algorithm

All the traveltime source perturbation equations developed above are linear first order partial differential equations that can be solved using any of the many upwind numerical methods. Similar to Franklin and Harris (2001) and Alkhalifah (2002), we will rely on the fast marching method Sethian (1996) to solve such linear equations.

An update procedure for such a method is based on an upwind first or second-order approximation to the new equations. In simple terms, the procedure starts with selecting one or more (up to three) neighboring points around the updated point. The traveltime values at the selected neighboring points need to be smaller than the current value. After the selection, one solves the discrete version of the linear partial differential equation for $D_x$. We add this perturbation value multiplied by the perturbation distance to the background traveltime. As the result of the updating, either a FarAway point is marked as NarrowBand or a NarrowBand point gets assigned a new value. This process is repeated until we run out of points in the narrow band.

In all cases, we will need the traveltime field for a given source obtained using the eikonal equation or ray-based methods. This traveltime field serves as the background field for predicting the traveltime for other sources. For the first-order accuracy expansion, we have to only solve the linear source differential eikonal partial differential equation once. However, for the second order expansion or its shank transform representation, we will need $D_{xx}$, and thus need to solve an equivalent linear differential equation again.

The critical part of solving these equations is the need to evaluate the first and second order derivatives of the velocity field or equivalently the second- and third-order derivatives of the background traveltime field all with respect to the direction in which the source is perturbed. This poses a challenge in media where the velocity changes abruptly in that direction. Therefore, some smoothing may be required for the velocity field in the source perturbation direction.

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