Wave-equation tomography by beam focusing |

The computation of the derivatives of 3 with respect to each vector of local-moveout parameters is easily evaluated using the following expression:

The linear operator has the dimensions and is given by

where , with being the time derivative of the recorded-data traces. For the choice of moveout parameters expressed in equation 6 we have .

Similarly, the evaluation of the derivatives of 7 with respect to each shift parameter is easily carried out by the following:

where the linear operator is given by

When the moveout parameters are simple trace-by-trace phase shifts, as defined in equation 10, it results that .

On a practical note, the preceding expressions look more daunting than they are in practice. They greatly simplify in the important case when the gradient is evaluated for and . This simplifying condition is actually always fulfilled unless the optimization algorithm includes inner iterations for fitting the moveout parameters using a linearized approach. Under these conditions, equations 12 and 13 become, respectively,

and

Similarly, equations 14 and 15 become

and

Wave-equation tomography by beam focusing |

2010-05-19