Wave-equation tomography by beam focusing |

In this appendix I present the analytical development needed to derive equations 22-23 from equations 20-21.

Equation 20 can be rewritten as

where,

and in which . Given the moveout parametrization expressed in 6, and the previous expression simplifies into the following:

Consequently, the general expression for the gradient of the local moveout parameters with respect to the slowness model is:

When , the general expression further simplifies into:

Similar derivation can be developed for the derivative of the global moveout parameters with respect to slowness. Equation 21 can be rewritten as:

where,

Given the moveout parametrization in expressed in 10, and the previous expression simplifies into:

The general expression for the gradient of the global moveout parameters with respect to the slowness model is:

When and the general expression further simplifies into:

Wave-equation tomography by beam focusing |

2010-05-19