Seismic wavefield redatuming with regularized multi-dimensional deconvolution

In seismic imaging the aim is to obtain an image of the subsurface using reflection data. The reflection data are generated using sound waves and the sources and receivers are placed at the surface. The target zone, for example an oil or gas reservoir, lies relatively deep in the subsurface below several layers. The area above the target zone is called the overburden. This overburden will have an imprint on the image. Wavefield redatuming is an approach that removes the imprint of the overburden on the image by creating so-called virtual sources and receivers above the target zone. The virtual sources are obtained by determining the impulse response, or Green's function, in the subsurface. The impulse response is obtained by deconvolving all up- and downgoing wavefields at the desired location. In this paper, we pose this deconvolution problem as a constrained least-squares problem. We describe the constraints that are involved in the deconvolution and show that they are associated with orthogonal projection operators. We show different optimization strategies to solve the constrained least-squares problem and provide an explicit relation between them, showing that they are in a sense equivalent. We show that the constrained least-squares problem remains ill-posed and that additional regularization has to be provided. We show that Tikhonov regularization leads to improved resolution and a stable optimization procedure, but that we cannot estimate the correct regularization parameter using standard parameter selection methods. We also show that the constrained least-squares can be posed in such a way that additional nonlinear regularization is possible.