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 2D ) as
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
In the next section we show three numerical examples of Hessians estimated with the proposed target-oriented approach.