Since accurate imaging of reflectors is more important in the neighborhood of the reservoir, it makes sense to apply a target-oriented strategy to reduce the number of depth steps. A way to achieve this objective is to write the modeling operator in a target-oriented fashion and explicitly compute the Hessian.

In general, the synthetic data for one frequency, a shot positioned at and a receiver positioned at can be given by a linear operator acting on the full model space with ( in 2*D* ) as

(4) |

In equation (4), two important properties have been used Ehinger et al. (1996): first, the Green functions are computed by means of the one-way wave equation, and second, the extrapolation is performed by using the adequate paraxial wave equations (flux conservation) Bamberger et al. (1988).

The quadratic cost function is

(5) |

(6) | ||

(7) |

(8) |

In the next section we show three numerical examples of Hessians estimated with the proposed target-oriented approach.

10/23/2004