I introduce a prestack generalization of exploding-reflectors modeling that has the potential to drastically reduce the computational cost of Migration Velocity Analysis (MVA) based on wavefield-continuation migration and modeling. The method aims at synthesizing, starting form a prestack migrated image, a limited number of independent experiments that contain all the velocity information needed for MVA.
The basic element of the method is the modeling of one isolated Subsurface-Offset Domain Common Image Gather (SODCIG) by using the prestack image as an initial condition. Taking advantage of the limited range of the subsurface offsets in the migrated image, we can combine many SODCIGs in the same modeling experiment without diminishing the velocity information contained in the data. The number of independent experiments required depends on how close we are to the correct migration velocity. Theoretically, this number is the same as the number of subsurface offsets needed to represent the prestack image from which we started the modeling. Several numerical examples illustrate the basic concepts of the proposed method, and demonstrate its potential usefulness for MVA.
We would like to estimate migration-velocity models based on iterative wavefield-continuation migrations, in contrast with the conventional use of iterative Kirchhoff migrations, to take full advantage of the superior imaging capabilities of wavefield-continuation migration. If computational cost were not a consideration, we also should be updating the velocity field using a wavefield tomographic operator Pratt (1999); Sava and Biondi (2004a,b); Symes and Carazzone (1991) instead of the computationally cheaper conventional ray tomography. Unfortunately, the high computational cost of wavefield migration (and modeling) stands on the way to the practical achievements of these goals.
In this paper I present a method that has the potential of substantially reducing the computational cost of Migration Velocity Analysis (MVA) based on wavefield migration. The method could be applied to both MVA based on conventional ray tomography as well as MVA based on wavefield operators. The computational saving is achieved by synthesizing a new data set that has a number of independent experiments lower than the number of shot gathers in the original data set, but that contains all the information needed to estimate migration velocity. This new data set is synthesized from a prestack image obtained using a wavefield-continuation migration. This initial prestack image is assumed to be only partially focused, with the defocusing related to an inaccurate migration velocity. In principle, the better focused the initial image is, the fewer independent experiments are needed for the new data set to contain all the information needed by MVA.
I propose to synthesize the new data set by applying a prestack generalization of exploding-reflectors modeling. Conceptually, the starting point is the modeling of an isolated Subsurface-Offset Domain Common Image Gathers (SODCIGs). The image in the SODCIGs is used to generate both the source and receiver wavefields at zero time (exploding reflectors). The wavefields are then propagated toward the surface and recorded there by aerial arrays covering the whole surface of the model. The receiver wavefield is propagated forward in time, whereas the source wavefield is propagated backward in time. Any numerical scheme can be used for wave propagation. In particular, both the one-way wave equation and the two-way wave equations can be used for this purpose. The numerical examples I show in this paper are obtained by solving the two-way wave equation in the time domain, because the visual inspection of propagating wavefields provides an intuitive understanding of the process.
Generating an independent experiment for every SODCIG in the prestack image would actually create a data set even larger than the original data set. However, the fundamental observation is that we can simultaneously model several SODCIGs without hampering the velocity analysis based on this reduced data set, as long as we are careful to combine SODCIGs that are sufficiently uncorrelated. If the SODCIGs have been focused by migration, albeit imperfectly because of velocity errors, decorrelation can be achieved by taking advantage of the limited range along the subsurface offset axis of the partially focused image.
When the SODCIGs are uncorrelated, the modeling for many SODCIGs can be combined. The data resulting from modeling the combined SODCIGs can be imaged without the contributions of the individual SODCIGs correlating with each other and diminishing the velocity information contained in the migrated image. In this paper I take a simple approach to assure decorrelation; I model SODCIGs defined on several coarse grids which are shifted with respect to each other. In other words, I obtain the initial condition for modeling each independent experiment by multiplying the prestack image by a spatial comb function. I then shift the comb function until the image space is entirely covered. I show that if the distance between adjacent SODCIGs is equal, or larger, than the effective range of offsets in the SODCIGs, the velocity information is preserved. This process reduces to the conventional zero-offset exploding reflectors modeling in the limit that we model all SODCIGs together and assume that they are different from zero only at zero offset. From this viewpoint, the implicit assumption underlying zero-offset exploding-reflectors modeling is that the image is perfectly focused and all the energy is concentrated at zero subsurface offset.
The proposed method can be theoretically analyzed by using concepts that are similar to the ones used to analyze the ``cross-talks'' in phase-encoding Romero et al. (2000) and Montecarlo Cazzola et al. (2004) migration. This theoretical insight suggests that even further saving might be achieved by using temporally uncorrelated and spatially varying source functions. This direction is promising and deserves further work, but is beyond the scope of this paper.