Under the stationary phase approximation, the equation is in the
standard linear form: . This has a well-known damped least-squares
Gauss-Newton solution:
, where
and are the gradient and Hessian of *E* respectively, and
is the adjoint operator. In this case, the gradient can be derived as

(4) |

(5) |

(6) |

The approximate diagonal Hessian solution (6) costs little more to implement than a standard prestack migration, but requires twice the memory in order to store the gradient and Hessian images separately. In particular, (6) involves two separate but simultaneous images to be evaluated, the gradient (numerator) being a weighted Kirchhoff prestack depth migration, and the approximate Hessian (denominator) being an accumulation of the squared migration weights.

11/17/1997