Wave-equation time migration

Introduction

Seismic imaging has the ultimate goal of creating an image of subsurface structures in depth (Biondi, 2006; Gray et al., 2001; Etgen et al., 2009; Bednar, 2005). However, it often falls short of this goal in practice because of the requirement of having an accurate velocity model (Glogovsky et al., 2009,2008). That is possibly the main reason why time-domain imaging methods continue to play an important role in practical seismic data analysis.

Time migration takes a different route from depth migration by reducing the problem of velocity estimation to a parameter-picking problem. Each image point in a time-migrated image is associated with its own velocity parameter, which can be determined either by scanning different velocities (Yilmaz et al., 2001) or by wave extrapolation in the image-velocity space (Fomel, 2003). The computational advantage of time-domain imaging comes at the expense of two main flaws:

1. Time migration uses approximate Green's functions (Zhang and Zhang, 1998) that typically rely on 1-D velocity models and hyperbolic or slightly non-hyperbolic traveltime approximations.
2. In the case of lateral velocity variations, the transformation between the image-ray coordinates and Cartesian coordinates gets distorted in places where the image rays cross. In such areas, there is no longer a one-to-one mapping between image-ray coordinates and Cartesian coordinates, and the coordinate transformation will also have a zero determinant (at the caustics of the image-ray field) (Hubral, 1977).

Image rays are seismic rays orthogonal to the surface of observation. These rays remain straight in the absence of lateral velocity variations but bend when they meet lateral heterogeneities. Cameron et al. (2008b,2007,2009) extend the image-ray theory to establish an exact theoretical connection between depth- and time-migration velocities and an inversion algorithm for converting the latter to the former. In the absence of lateral velocities variations, the time-to-depth conversion is accomplished by Dix inversion Dix (1955). As shown by Cameron et al. (2007) and Iversen and Tygel (2008), an additional correction is required when velocities vary laterally that is the correction related to the geometrical spreading of image rays. Li and Fomel (2013,2015) develop a robust algorithm for time-to-depth conversion including a geometrical-spreading correction in the presence of lateral variations. Sripanich and Fomel (2018) develop a fast version of the time-to-depth conversion algorithm in the case of weak lateral variations.

In this paper, we propose to apply wave-equation imaging to create accurate seismic images in image-ray coordinates but without relying on Green's function approximations and thus avoiding any inaccuracy issues associated with time migration (Fomel, 2013). We use the method of Sava and Fomel (2005) to define wave propagation in an alternative coordinate system and to connect it with the theory that relates time-migration velocities to velocity models in depth (Cameron et al., 2007). We show that, when the wave equation is transformed into the image-ray coordinate system, its coefficients are simply related to ideal time-migration velocities. Therefore, accurate one-way or two-way wave-equation imaging can be accomplished by using information that is readily obtainable from conventional time-domain processing. This observation leads to a new imaging workflow, which provides a seamless transition from wave-equation time migration in time domain to wave-equation time migration in depth domain along with velocity model building in depth domain for depth imaging. We test the proposed approach using simple synthetic and field data examples.

Wave-equation time migration