Time-shift imaging condition for converted waves

Introduction

A key challenge for imaging in complex areas is accurate determination of a velocity model in the area under investigation. Migration velocity analysis is based on the principle that image accuracy indicators are optimized when data are correctly imaged. A common procedure for velocity analysis is to examine the alignment of images created with multi-offset data. An optimal choice of image analysis can be done in the angle domain which is free of complicated artifacts present in surface offset gathers in complex areas (Stolk and Symes, 2004).

Migration velocity analysis after migration by wavefield extrapolation requires image decomposition in scattering angles relative to reflector normals. Several methods have been proposed for such decompositions (Soubaras, 2003; Rickett and Sava, 2002; Xie and Wu, 2002; Biondi et al., 2003; Fomel, 2004; Prucha et al., 1999; Mosher and Foster, 2000; de Bruin et al., 1990; Sava and Fomel, 2003). These procedures require decomposition of extrapolated wavefields in variables that are related to the reflection angle.

A key component of such image decompositions is the imaging condition. A careful implementation of the imaging condition preserves all information necessary to decompose images in their angle-dependent components. The challenge is efficient and reliable construction of these angle-dependent images for velocity or amplitude analysis.

In migration with wavefield extrapolation, a prestack imaging condition based on spatial shifts of the source and receiver wavefields allows for angle-decomposition (Rickett and Sava, 2002; Sava and Fomel, 2005a). Such formed angle-gathers describe reflectivity as a function of reflection angles and are powerful tools for migration velocity analysis (MVA) or amplitude versus angle analysis (AVA). However, due to the large expense of space-time cross-correlations, especially in three dimensions, this imaging methodology is not yet used routinely in data processing.

A different form of imaging condition involves time-shifts instead of space-shifts between wavefields computed from sources and receivers (Sava and Fomel, 2006). Similarly to the space-shift imaging condition, an image is built by space-time cross-correlations of subsurface wavefields, and multiple lags of the time cross-correlation are preserved in the image. Time-shifts have physical meaning that can be related directly to reflection geometry, similarly to the procedure used for space-shifts. Furthermore, time-shift imaging is cheaper to apply than space-shift imaging, and thus it might alleviate some of the difficulties posed by costly cross-correlations in 3D space-shift imaging condition.

The time-shift imaging concept is applicable to Kirchhoff migration, migration by wavefield extrapolation, or reverse-time migration. This concept is also applicable to migration of single-mode (PP) or converted-mode (PS) waves. In this paper, we develop the theory and show examples of angle decomposition after time-shift imaging of converted waves. All formulas developed for this purpose reduce to the previously-derived formulas for decomposition of single-mode images (Sava and Fomel, 2006,2005a).

Time-shift imaging condition for converted waves