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

Traveltime behavior due to source perturbations can be
estimated more accurately using higher-order
formulations. Considering that , , and represent
source perturbations in the , , and directions,
respectively, a full representation of the second derivative behavior
is given by the following symmetric matrix

Similarly, higher order approximations in and are given by

respectively, which represent the diagonal terms of the matrix in equation E-1.

To obtain the non-diagonal components of the matrix we differentiate
equation 3 with respect to , instead of ,
yielding:

Substituting the second derivative of traveltime with respect to
source location
into equation E-4 provides us with a first order
linear equation in given by:

or

Similar equations for the rest of the matrix components are given by

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

2013-04-02