Wave-equation migration velocity analysis by residual moveout fitting |

The derivatives of 4 with respect to the vector of moveout parameters is easily evaluated using the following expression:

The linear operator can be represented as a matrix. The elements of this matrix are given by:

The first term (I) is given by the depth-derivative of the image after moveout. This term can be numerically evaluated by applying to the moved-out image a finite-difference approximation of the first-derivative operator. The second term (II) is different from zero only when the spatial coordinate of the image element is the same as the coordinate corresponding to the moveout parameter . When they do, and for the choice of moveout parameters expressed in equation 7, we have .

The preceding expression simplifies when the gradient is evaluated for . This simplifying condition is actually always fulfilled unless the optimization algorithm includes inner iterations for fitting the moveout parameters using a linearized approach. Under this condition, equation 8 becomes

and equation 9 becomes

Wave-equation migration velocity analysis by residual moveout fitting |

2010-11-26