It is well known that waves propagate upward and downward simultaneously. Reverse-time migration Baysal et al. (1983); Biondi and Shan (2002); Whitmore (1983), which solves the full wave equation directly and mimics wave propagation naturally, is still too expensive for today's computing facilities. As a result, downward continuation methods Claerbout (1985), which are based on one-way wave equation wavefield extrapolation and much cheaper than reverse-time migration, are widely used in the industry.
Conventional downward continuation method extrapolates wavefields using the one-way wave equation in Cartesian coordinates. For a medium without lateral variation, the phase-shift method Gazdag (1978) can be applied, and one-way wave equation can model waves propagating in a direction up to 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 the one-way wave equation. A lot of effort has been made to improve the accuracy of the wavefield-extrapolation operator in laterally varying media, including Fourier finite-difference Biondi (2002); Ristow and Ruhl (1994), general screen propagator Huang and Wu (1996); de Hoop (1996), and optimized finite difference Lee and Suh (1985) with phase correction Li (1991). Even if we could model waves accurately up to 90 degree using the one-way wave equation in laterally varying media, overturned waves, which travel downward first and then curve upward, are filtered away during the extrapolation. This is because only down-going waves are allowed in the source wavefield and only up-going waves are allowed in the receiver wavefield in downward continuation. But overturned waves and waves propagating in a high angle direction play a key role in imaging the steeply dipping reflectors. As a consequence, imaging steeply dipping reflectors, such as salt flank and faults, remain a major problem in downward continuation.
Some work has been done to image the steeply dipping reflectors with one-way wave equation 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 Sava and Fomel (2005); Shragge (2006).
Plane-wave source migration Duquet et al. (2001); Liu et al. (2002); Rietveld (1995); Whitmore (1995); Zhang et al. (2005) has been demonstrated as a useful tool in seismic imaging. Shan and Biondi (2004) perform 2D plane-wave migration in tilted coordinates to image steeply dipping reflectors and overturned waves using one-way wave equation. In this paper, we develop 3D full plane-wave migration in tilted coordinates.
This paper is organized as follows: we begin with a brief review of plane-wave migration, then describe the theory of plane-wave migration in tilted coordinates, and finally demonstrate our technique with a real dataset example.