Delayed-shot migration in TEC coordinates

Introduction

Wave-equation migration (WEM) methods routinely generate accurate seismic images in areas of complex geology. One common class of WEM approaches is shot-profile migration using one-way wavefield extrapolation. The first shot-profile migration step is to specify source and receiver wavefields that consist of modeled point sources and an individual shot profile, respectively. The migration algorithm propagates these two wavefields through the velocity model and correlates them at each extrapolation step to form an image. Although this procedure generates high-quality migration results, two drawbacks make shot-profile migration a less-than-ideal strategy. The first issue is that each individual shot migration requires a large aperture to propagate energy to wide offsets. The second drawback is that one migrates each shot record individually, which can be computationally expensive for large surveys with a high shot density.

One way to make the shot-profile style of WEM more efficient is to migrate a reduced number of composite source and receiver profiles each covering a broader aperture. For example, one can image a number of shot profiles simultaneously on the same migration domain. The key idea is that one makes a computationally advantageous trade-off of a broader migration aperture for a reduced number of shots. Shot-profile migration with composite wavefields, though, leads to the mixing of information from different shots and generates image crosstalk. A number of authors address this problem using a variety of phase-encoding migration approaches (Sun et al., 2002; Jing et al., 2000; Morton and Ober, 1998; Romero et al., 2000), that minimize the deleterious crosstalk effects.

Plane-wave migration (PWM) is one technique for reducing total migration cost using composite wavefields (Liu et al., 2006,2004; Mosher and Foster, 1998; Duquet et al., 2001; Whitmore, 1995; Zhang et al., 2003). As originally demonstrated by Whitmore (1995), the key idea is to synthesize from the full wavefield volume the set of composite receiver wavefields that would have been recorded were a planar source function used. Generally, the number of synthesized wavefields is fewer than the corresponding number of shot profiles. One generates PWM images by propagating the modeled planar source and composite receiver wavefields through the velocity and computing a (weighted) correlation. Liu et al. (2006) and Duquet and Lailly (2006) demonstrate that PWM is equivalent to shot-profile migration in the limit where one uses many plane waves with well-sampled plane-wave dip spectra. Liu et al. (2006) also prove that 3D PWM is equivalent to conical-wave migration of individual sail lines synthesized as inline composite wavefields. The approach is termed conical wave because the source wavefronts form conic sections (in constant media) for non-zero inline plane-wave ray parameters.

The migration of plane- and conical-wave data, though more efficient than shot-profile migration, is similarly restricted in accuracy by one-way wavefield extrapolation assumptions. The most common limitation is a difficulty in propagating waves at large angles and turning waves by design, both of which are important for accurately imaging salt flanks in complex geologic areas. Shan and Biondi (2004) circumvent this problem by implementing 3D PWM in tilted Cartesian meshes. This coordinate system effectively orients the wavefield extrapolation axis toward the plane-wave take-off vector, enabling more accurate bulk propagation of plane-wave energy. One logistical complication of performing fully 3D PWM is that it requires propagating image-space-sized data volumes on a number of meshes tilting in both the inline and cross-line directions. This leads to a number of computational issues associated with the significant memory footprint.

This paper presents an alternative to the phase-encoding approach of Shan and Biondi (2004), which similarly uses alternative coordinate systems. The key differences between these two approaches are two-fold. The first difference is that I phase encode only according to the inline source coordinate, leading to the inline delayed-shot migration algorithm. This leads to a straightforward coarse-grain parallelization of the migration tasks across individual sail lines, where each migration has a significantly smaller aperture than the corresponding image-space-sized PWM volumes. A second efficiency gain over PWM is a reduction in the total number of migrations, because the number of sail lines is quite often fewer than the required number of cross-line plane waves. Thus, the inline-delayed shot approach has attractive computational advantages over the 3D PWM technique.

The second difference is that I migrate data in tilted elliptical-cylindrical (TEC) coordinates, rather than tilted Cartesian meshes. The key idea is that, because the geometry of the TEC coordinate system closely resembles the shape of a line-source impulse response, TEC meshes afford accurate propagation of most steep-dip and turning waves in all directions. TEC coordinate systems, formed by concatenating a set of the 2D elliptical coordinates (Shragge and Shan, 2008) along the invariant third axis, are thus well-suited for migrating individual sail lines. I extrapolate the inline delay-shot synthesized wavefield volumes outward on a series of elliptical-cylindrical shells. This allows source and receiver wavefields with zero inline dip to overturn in the cross-line direction, if necessary. I introduce an extra degree of freedom that permits the coordinate system to tilt along the invariant inline axis, thus enabling the propagation of turning waves inline. Consequently, inline delayed-shot migration in TEC coordinates allows wavefields with most non-zero dips to propagate and overturn to all azimuths as appropriate.

The paper begins by examining 3D full-plane-wave and inline delayed-shot migration theory. I then introduce the TEC coordinate geometry and develop the corresponding wavenumber that forms the basis of the TEC wavefield extrapolation operator. I discuss the finite-difference extrapolation implementation and present the 3D impulse response. I apply the technique to a 3D wide-azimuth synthetic data set derived from real Gulf of Mexico velocity model to demonstrate the imaging advantages of 3D RWE migration. I then discuss the numerical costs associated with performing inline delayed-shot migration in TEC coordinates relative to Cartesian meshes. The paper concludes with narrow-azimuth migration results demonstrate the applicability of the approach to typical Gulf of Mexico field data.

Delayed-shot migration in TEC coordinates