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

The accuracy of the above formulations are first order in source
perturbation, which is valid for small perturbation distances. To
obtain a higher-order accuracy, we differentiate
equation 3 again with respect to yielding:

Substituting the second derivative of traveltime with respect to
source location
into
equation 18 provides us with a first order linear
partial differential equation in given by:

This equation is similar in form to the first order equations, but with a different source function. Of course, must be evaluated first using equation 4 to solve equation 20.

Based on Taylor's series expansion, the traveltime for a source at is approximated by

Using an infinite series representation by defining poles to eliminate
the most pronounced transient behavior using Shanks transforms (Bender and Orszag, 1978), we can
represent the second order Taylor's expansion in
equation 21 as follows

Similar equations for expansions in 3-D are obtained with the help of the following matrix

The higher-order equations provide better approximations of the traveltime perturbation. However, they require both the velocity and its derivative to be continuous in the direction of the source perturbation.

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

2013-04-02