In the case of wave equation migration or inversion, the operator is expensive to apply. Thus, iteratively applying this operator and its transpose is sometimes prohibitive. The computational cost is proportional to the number of depth steps the wavefields need to be propagated Audebert (1994), and the number of iterations, among other factors.
Since reflection amplitudes are 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. Instead of recursively computing the Green functions at each depth step and at each inversion iteration, we can compute them from the surface to the target and from the target to the surface during the first iteration, store them, and reuse them in subsequent iterations. By storing the Green functions to the target and from the target, we add a new problem, since we then require approximately twice the disk space for data storage; however, we save computing time.
In the next sections, we write a target-oriented, one-way approximation to the scalar wave equation operator.