Angle-domain common-image gathers in generalized coordinates

Introduction

Angle-domain common-image gathers (ADCIGs) are used increasingly in seismic imaging to examine migration velocity model accuracy (, ). The key idea is that migrating with the correct velocity model leads to flat angle gathers that shift neither vertically nor horizontally as a function of reflection opening angle. Migrating with an incorrect velocity, though, leads to inconsistent angle-domain reflectivity and generates residual curvature in the ADCIG volume. ADCIGs are thus an effective velocity analysis tool and have been incorporated in wave-equation-based inversion schemes to update velocity profiles (, ,).

Wave-equation imaging techniques generate ADCIGs in straightforward manners for both shot-profile (, ,,) and shot-geophone (, ,) migration approaches. In shot-profile migration, one first generates a subsurface-offset axis at each depth step by correlating the source and receiver wavefields at a number of subsurface shifts. The second step involves computing an offset-to-angle domain transformation using, for example, post-imaging Fourier-based operators (, ).

Conventional ADCIG theory usually assumes horizontal wavefield shifts, largely because wavefield extrapolation and imaging are most commonly performed in Cartesian coordinates. However, a number of studies have noted that these ADCIG results degrade for steeply dipping structures, such as salt flanks (, ,). Although this is partially due to problems associated with inaccurate large-angle extrapolation, ADCIGs calculated using horizontal wavefield shifts become decreasingly sensitive for increasingly steep structural dips. () demonstrate that this problem can be addressed by generating ADCIGs with vertical subsurface-offset-domain common-image gathers (VODCIGs); however, this approach is less desirable computationally because it requires storing the larger wavefield volumes required to calculate the VODCIGs in memory.

The introduction of shot-profile migration in more general coordinate systems [e.g. tilted Cartesian (, ) and elliptic meshes (, )] presents an opportunity to circumvent problems associated with generating ADCIGs for steeply dipping structure. In particular, migration domains can be oriented such that geologic structures with steep dips in Cartesian meshes have relatively gentle dip in generalized coordinate systems, thus improving the robustness of the ADCIG calculation. Developing an ADCIG theory capable of handling more arbitrary coordinate meshes, though, requires proper treatment of the effects of non-Cartesian geometries. For example, wavefield extrapolation in non-Cartesian coordinate systems induces local wavenumber stretches, rotations and/or shearing (, ). Similarly, non-uniform wavefield sampling can lead to anisotropic angle-domain stretching. These effects can be corrected using Jacobian change-of-variable transformations.

The goal of this paper is to extend ADCIG theory to non-Cartesian geometries. I demonstrate that ADCIG theory, as developed in a differential sense (, ), remains valid for arbitrary geometries provided that the corresponding derivative operators are properly specified. Non-Cartesian coordinates do, however, introduce space-domain geometric factors that can render Fourier-based offset-to-angle methods unsuitable. However, I show that ADCIGs can be calculated directly in the Fourier domain for all coordinate systems satisfying the Cauchy-Riemann differentiability criteria (, ). Moreover, ADCIGs can be calculated in all situations using the slant-stack approaches discussed in ().

I begin by discussing how to generate subsurface offsets and ADCIGs in Cartesian coordinates. I then provide an extension to generalized coordinate systems based on Jacobian change-of-variable arguments. I examine two canonical coordinate systems, tilted Cartesian and elliptic meshes, where the reflection angle can be explicitly calculated using Fourier-based methods, and a third, polar coordinates, where it cannot. I test the generalized ADCIG theory analytically and numerically using a set of elliptic reflectors, and demonstrate how computing angle gathers in elliptic coordinates can lead to improvements relative to Cartesian coordinates, especially for steeply dipping structure.

Angle-domain common-image gathers in generalized coordinates