Introduction

Migration methods based on wave-equation extrapolation are gaining popularity in the industry. They can potentially better handle multi-pathing caused by complex geological structure. In nature, wave propagation has no directional preference: up-going and down-going waves propagate simultaneously. However, two-way wave-equation-based methods, such as reverse time migration Baysal et al. (1983); Biondi and Shan (2002); Whitmore (1983), are still too expensive for today's computing resources. As a result, downward continuation Claerbout (1985), based on the one-way wave equation, is usually used. In downward-continuation methods, only down-going waves are allowed in the source wavefield, and only up-going waves are allowed in the receiver wavefield. Overturned waves, which travel downward first and then upward, are totally eliminated by the one-way wave equation. Even for waves only propagating upward or downward, it is difficult for downward-continuation methods to handle waves propagating far from the vertical direction in laterally varying media. As a result, downward-continuation methods have difficulty imaging steeply dipping reflectors.

A lot of effort has been made to improve the accuracy of one-way wave equation extrapolators in laterally varying media Huang and Wu (1996); Lee and Suh (1985); Ristow and Ruhl (1994); de Hoop (1996). At the same time, people also design coordinates to make the extrapolation direction close to the propagation direction of waves, thereby reducing the accuracy requirement of the extrapolator. These include tilted coordinates Etgen (2002); Higginbotham et al. (1985), beam migration Brandsberg-Dahl and Etgen (2003), and Riemannian coordinates Sava and Fomel (2005).

 

impulse
Figure 1 Impulse response of two-way wave equation (a), downward continuation (b), and the plane-wave migration in tilted coordinates (c).

In this paper, we review plane-wave migration in tilted coordinates Shan and Biondi (2004). We present plane-wave migration in tilted coordinates for both isotropic and anisotropic media. Isotropic media are still isotropic after rotating the coordinates, while VTI media change to tilted TI media after rotating the coordinates. We use the one-way wave-equation extrapolation operator for tilted TI media proposed by Shan and Biondi (2005). We apply plane-wave migration in tilted coordinates to the BP synthetic velocity benchmark dataset Billette and Brandsberg-Dahl (2005), a synthetic dataset for VTI media, and an anisotropic real dataset.