Plane-wave migration in tilted coordinates

Introduction

Kirchhoff migration has been widely applied in seismic processing due to its relatively low cost and flexibility. However, it cannot provide reliable images where multi-pathing occurs. Wave-equation migration, which is performed by recursive wavefield extrapolation, has been demonstrated to overcome these limitations and produce better images in areas of complex geology.

It is well known that in a single-shot experiment waves propagate upward and downward simultaneously. Reverse-time migration (Biondi and Shan, 2002; Whitmore, 1983; Baysal et al., 1983), which solves the full wave equation directly and mimics wave propagation naturally, is expensive for routine use in today's computing facilities. As a consequence, downward continuation migration Claerbout (1985), which are based on one-way wave-equation wavefield extrapolation and are much cheaper than reverse-time migration, are widely used in the industry.

Conventional downward-continuation methods extrapolate wavefields using the one-way wave equation in vertical Cartesian coordinates. For a medium without lateral velocity variation, the phase-shift method (Gazdag, 1978) can be applied, and the one-way wave-equation can model waves propagating in a direction up to $90^\circ$ away from the extrapolation direction. But in a laterally varying medium, it is very difficult to model waves propagating in a direction far from the extrapolation direction using a one-way wavefield extrapolator. Many methods have been developed to improve the accuracy of the one-way wavefield extrapolator in laterally varying media, such as Fourier finite-difference (Biondi, 2002; Ristow and Ruhl, 1994), the general screen propagator (de Hoop, 1996; Huang and Wu, 1996) and optimized finite difference (Lee and Suh, 1985) with a phase correction (Li, 1991). Even if we could model waves accurately up to $90^\circ$ using the one-way wavefield extrapolator in laterally varying media, overturned waves, which travel downward first and then curve upward, are filtered away during the wavefield extrapolation because of the assumption that the waves propagate vertically only in one direction: downward for source wavefields and upward for receiver wavefields. However, overturned waves and waves propagating at high angles play a key role in imaging steeply dipping reflectors, such as salt flank and faults. As a consequence, imaging these steeply dipping reflectors remains a major problem in downward continuation migration.

Work has been done to image the steeply dipping reflectors with one-way wavefield extrapolators by coordinate transformation. This includes tilted coordinates (Etgen, 2002; Higginbotham et al., 1985), the combination of downward continuation and horizontal continuation (Zhang and McMechan, 1997), or wavefield extrapolation in general coordinates, such as ray coordinates (Nichols, 1994) and Riemannian coordinates (Shragge, 2006; Sava and Fomel, 2005).

In tilted coordinates, waves traveling along the extrapolation direction are most accurately modeled, and the maximum angle of their propagation direction from the extrapolation direction that can be handled is determined by the accuracy of the wavefield extrapolator. For a point source, where waves travel in all directions from a point, it is impossible for one tilted coordinate system to cover all these directions. But for a plane-wave source, waves travel in a similar direction from all spacial points at the surface, and thus most of them can be modeled accurately in a tilted coordinate system with a well-chosen tilting direction. In this paper, we apply plane-wave migration (Rietveld, 1995; Duquet et al., 2001; Liu et al., 2002; Whitmore, 1995; Zhang et al., 2005) in tilted coordinates. Plane-wave migration has been demonstrated to be a useful tool in seismic imaging. By slant-stacking, the recorded surface data are synthesized into areal plane-wave-source gathers, which are what would be recorded if plane-wave sources were excited at the surface. A plane-wave source is characterized by a ray parameter, and its take-off angle can be calculated from the ray parameter, given the velocity at the surface. Each areal plane-source gather is migrated independently, similar to shot-profile migration, and the image is formed by stacking the images of all possible plane-wave sources. Given a plane-wave source, we tilt the coordinate system according to its take-off angle. For most waves, the resulting extrapolation direction is closer to the propagation direction, and thus we can image steeply dipping reflectors correctly using one-way wavefield extrapolators. Plane-wave migration is potentially more efficient than shot-profile migration (Etgen, 2005; Zhang et al., 2005). To image steeply dipping reflectors or overturned waves, a large migration aperture is required to cover the whole propagation path of source and receiver waves. In shot-profile migration, this requires large padding in space. In contrast, plane-wave migration uses the whole seismic survey as the migration aperture. It is well known that one-way wave-equation shot-profile migration is much cheaper than reverse-time migration. Compared to conventional plane-wave migration, the cost of plane-wave migration in tilted coordinates is a little higher because of the data and velocity model interpolation, but it is still much lower than reverse-time migration.

This paper is organized as follows: we begin with a brief review of one-way wave-equation migration and plane-wave migration. Then we introduce how to extrapolate the wavefield in a tilted coordinate system and describe plane-wave migration in tilted coordinates. Finally, we demonstrate our technique with synthetic data examples.

Plane-wave migration in tilted coordinates