Angle-domain common-image gathers in generalized coordinates

Introduction

Angle-domain common image gathers, or ADCIGs, are used increasingly in seismic imaging to examine migration velocity model accuracy (, ). The key idea is that migrating with correct velocity models leads to ADCIGs that do not shift vertically as a function of reflection opening angle (i.e., flat gathers). Migrating with incorrect velocity models, though, leads to inconsistent reflector depths and generates reflector smiles or frowns. ADCIGs are thus an effective analysis tool and have been incorporated in wave-equation-based inversion schemes to update velocity profiles (Sava and Biondi, 2004b,a).

ADCIGs can be generated during wave-equation imaging in a straightforward manner for both shot-profile (, ; Sava and Fomel, 2003) and shot-geophone (Prucha et al., 1999; Mosher and Foster, 2000) migration approaches. In shot-profile migration, one can generate 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 through a post-imaging, Fourier-based stretch (Sava and Fomel, 2003).

ADCIG theory is usually developed assuming horizontal and uniformly spaced wavefield shifts, largely because wavefield extrapolation and imaging are more commonly performed in Cartesian coordinates. The introduction of shot-profile migration in non-Cartesian coordinate systems (, ), though, warrants the development of a general ADCIG theory able to handle more arbitrary geometries and wavefield shifts. Generalizing ADCIG theory requires properly handling the effects of non-Cartesian geometry. For example, wavefield propagation in non-Cartesian coordinate systems induces local stretches, rotations and/or shearing of the wavenumbers (, ). Similarly, nonlinear and non-horizontal shifts can lead to angle-domain stretches.

The goal of this paper is to extend ADCIG theory to non-Cartesian geometries and nonlinear subsurface offset sampling. I demonstrate that ADCIG theory - as developed in a differential sense (Sava and Fomel, 2003) - remains valid for arbitrary geometry. Non-Cartesian coordinates do, though, introduce space-domain geometric factors that can render Fourier-based offset-to-angle methods unsuitable. I begin with a review of Cartesian ADCIG theory and provide an extension to generalized coordinate systems. I examine three canonical coordinate systems where the reflection angle can be explicitly calculated. I show how nonlinear shifting can modify the subsurface offset axis such that Fourier-based ADCIG calculation methods remain applicable.

Angle-domain common-image gathers in generalized coordinates