Delayed‐shot 3‐D prestack depth migration

Abstract

Summary We present a formulation for delayed-shot migration of marine data in 2-D (plane-wave sources) and in 3-D (linear sources and planar sources). We present speedup factors for these delayed- shot migrations over common-shot migration, and we discuss some sampling theory issues associated with the formation of delayed-shot records. On both synthetic and real data examples, delayed-shot migration has produced images comparable to those from common-shot migration. Introduction The increasing demands of imaging complex geologic structures, for example beneath salt bodies, has led the industry to explore wave equation based prestack depth migration methods that do not suffer from the high frequency and multipathing limitations of Kirchhoff migration. However, common- shot migrations based on wavefield extrapolation are typically more computationally intensive, especially when 3-D migrated common image gathers (CIG's) are output. This relative inefficiency has spurred researchers to seek various ways to speed up their wave-equation migration programs. The computational cost of common-shot migration is roughly the cost of migrating a single shot record multiplied by the number of migrated shots. Reducing the number of shots, and consequently the number of migrations, is an obvious way to improve the total migration efficiency, although it is not obvious that simply decimating shots will allow one to maintain the fidelity of the migrated image. A different approach to reducing the number of shots without decimation is based on the linearity of the wave equation: a linear stacking of wavefields initiated at different shotpoints satisfies the same wave equation as each of the individual wavefields. Therefore, migration can be applied to the superposition of different shot records, allowing the total number of migrations to be reduced. This idea has led to the migration of phase-encoded shot records (Romero et al., 2000), in which a subset of all the shot records are linearly combined together by applying some phase functions chosen to reduce the cross-term artifacts. A specialization of this idea, called delayed-shot migration, has also appeared (Whitmore, 1995; Rietveld, 1995; Duquet et al., 2001; Liu, 2002). In this method, a linear time delay, based on the distance from the shotpoints from some reference location, is used to combine different shot records. The surface data are transformed from the response to point sources to the response to linear or planar sources. In this paper, we review our formulation of delayed-shot migration for 3-D prestack imaging of marine data, discuss its realistic cost impact and illustrate its applicability with a synthetic data example. Delayed-shot migration versus common-shot migration Common-shot migration is performed on individual common-shot records, and the individually migrated records are typically stacked to form the final image of the Earth's subsurface. Each migration is performed by using the wave equation to downward continue both the wavefield recorded at the receiver locations and the wavefield initiated at the shotpoint, and combining these downward-continued wavefields with an imaging condition. Since the shotpoint is localized in space, it acts (in 3-D) as a point source, emitting waves that are spherical, at least near the shotpoint. Although a localized source distribution near the Earth's surface can be considered as a point source, numerical simulation of 2-D wave behavior using finite difference methods typically contains the tacit assumption that the source is a line source in 3-D, with no spreading in the out-of-plane direction. Then it is possible to perform a decomposition of the recorded data into plane-wave components by using some variant of slant stack processing. This involves applying a linear time delay to each shot record (Figure 1). Specifically, each receiver gather is transformed into plane waves as: