3D shot-profile migration in ellipsoidal coordinates

Introduction

Imaging steeply dipping structure, such as salt flanks, in complex geologic environments remains an issue in 3D seismic migration. Many difficulties arise due to the relate problems of incorrect velocity models and poor downgoing illumination. One strategy is to use turning-wave energy arriving from all azimuths to image the steep dip structure that remains unilluminated by downgoing energy alone. A number of studies have examined this approach with varying degrees of success (, ,,).

The practical imaging improvements afforded by imaging turning-wave energy, though, are unavoidably linked to data acquisition geometry and velocity model structure. For example, narrow-azimuth migrations of data sets containing predominantly inline turning-wave energy are usually constrained to have minimal crossline aperture. This restriction precludes imaging turning-wave energy originating from crossline structure arriving at near-zero offsets. The increasing popularity of wide-azimuth acquisition - well suited for recording turning-wave energy from all azimuths - suggests the need for improved wide-aperture wavefield propagation techniques. Determining which seismic imaging methods optimally realize these requirements, both physically and computationally, remains an open research question.

Wave-equation migration (WEM) techniques generally generate superior images relative to other approaches in complex geologic environments. Reverse-time migration, which solves the full acoustic wave equation, is one class of WEM techniques able to propagate turning waves in all directions (, ,). The computational costs associated with 3D wave propagation and imaging remain significant, though, especially in wide-azimuth contexts. A second class of WEM approaches, based on one-way wavefield extrapolation, rapidly generates solutions to approximate one-way wave equations. The computational advantages of one-way wavefield extrapolation, relative to reverse-time migration, are less obvious when considering the lower accuracy of high-angle propagation and an inherent inability to propagate turning waves by design.

Formulating the seismic imaging problem in more generalized coordinate systems is one way to exploit the computational advantages of one-way wavefield extrapolation while reducing the steep-dip limitations. The general strategy involves extrapolating source and receiver wavefields on meshes oriented toward the wave-propagation direction, generating local images through cross-correlation in the transformed coordinate system, and interpolating the result back to the global image volume. This process is repeated for all source-receiver wavefield pairs. Coordinate transforms proved to be an effective strategy for 2D and 3D plane-wave migration when using tilted Cartesian coordinate systems oriented toward the take-off vector (, ,). () apply this strategy in developing a 2D shot-profile migration in confocal elliptic coordinates.

In this paper, we apply the coordinate transform strategy to 3D ellipsoidal meshes. We extrapolate source and receiver wavefields outward in confocal ellipsoidal shells and perform cross-correlations to form local images. In most circumstances the wave-propagation direction is conformal to the ellipsoidal shells, permitting the imaging of turning waves with one-way operators. A second advantage of ellipsoidal coordinate systems is that the inline/crossline aspect ratio is controlled by a single parameter, and meshes can be rescaled to fit either narrow or wide-azimuth geometries. Ellipsoidal coordinates also can be defined using integral transforms that leave the corresponding dispersion relationship no more complicated than that of elliptically anisotropic media. Wavefield extrapolation is thus achieved using numerical approaches similar to optimized elliptically anisotropic finite-difference extrapolation (, ).

The paper begins with a discussion regarding two definitions of an ellipsoidal coordinate system. We generate the corresponding extrapolation wavenumber appropriate for performing wavefield continuation in ellipsoidal coordinates. We then discuss the implicit 3D finite-difference implementation used to propagate source and receiver wavefields, and give the results of impulse response tests. The paper ends with discussions on future work and the computational overhead associated with performing shot-profile migration in ellipsoidal coordinates.

3D shot-profile migration in ellipsoidal coordinates