Angle-domain illumination gathers by wave-equation-based methods

introduction

Seismic image quality obtained using prestack depth migration is highly dependent on subsurface illumination, which can be affected by many factors, e.g., the limited acquisition geometry, the complex velocity model, and the frequency content of seismic waves. Uneven illumination causes distorted seismic images due to the fact that migration operator is nonunitary (Tang, 2009; Clapp, 2005; Nemeth et al., 1999; Valenciano, 2008). Seismic illumination analysis quantifies such image distortion and provides ways to optimally restore the reflectivity. More accurate amplitude-versus-angle (AVA) analysis can be obtained if the angle-domain common-image gather is corrected with angle-dependent illumination. Angle-domain velocity analysis may also benefit from angle-dependent illumination weighting. It can weight up well-illuminated angular images, while weight down poorly-illuminated ones for robust residual parameter estimation, reducing the uncertainty caused by poor illumination.

Angle-domain illumination analysis is often carried out using ray-based method due to its efficiency (Bear et al., 2000; Schneider and Winbow, 1999). However, the infinite-frequency approximation and the caustics inherent in ray theory prevent ray-based method from accurately modeling complex wave phenomena. Wave-equation-based method, on the other hand, more accurately describes bandlimited wave phenomena, and therefore more suitable for complex geologies, e.g., subsalt regions with complex overburdens. Wave-equation-based angle-domain illumination analysis is proposed by Xie et al. (2006), where local plane-wave decomposition has been employed to extract the directional information for both source and receiver Green's functions before spatial and temporal correlations.

In this paper, we present an alternative way of computing the angle-dependent illumination. Instead of performing local plane-wave decomposition for each Green's function, our method transforms a subsurface-offset-indexed sensitivity kernel into the angle domain and then correlating the corresponding angle-domain sensitivity kernel to produce the angle-dependent illumination. The procedure closely resembles the method of Sava and Fomel (2003), which computes angle-dependent reflectivity image after imaging using a Fourier-domain mapping. We demonstrate in two dimensions that our method generates the scattering-angle illumination suitable for point scatterers if only horizontal subsurface offset has been computed. For planar reflectors, however, dip-dependent scattering-angle illumination is necessary, instead of scattering-angle illumination that averages over all dips. We show that the dip-dependent scattering-angle-domain illumination gather can be obtained by mapping either midpoint wavenumbers or subsurface-offset wavenumbers that contain both horizontal and vertical subsurface offsets.

In fact, the subsurface illumination that we often refer to is only a subset, or more precisely, the diagonal part of the imaging Hessian matrix. The off-diagonal components of the Hessian have been proven useful when applied to improve images of extremely poor illumination, e.g., subsalt regions with shadow zones (Tang, 2009; Valenciano, 2008). Our method discussed here also allows us to compute the off-diagonal components of the angle-domain imaging Hessian in a target-oriented fashion, hence it can be used to invert for the angle-domain reflectivity in a model-domain least-squares migration/inversion. A naive implementation of the proposed method, however, can be prohibitively expensive with the computational cost proportional to the number of sources, receivers and frequencies. We show how the phase-encoding technique (Tang, 2009,2008a,b) is able to make the method more cost effective.

This paper is organized as follows: we first review the theory of computing the illumination, or more generally, the imaging Hessian, in the subsurface-offset domain. Then we demonstrate how to transform the sensitivity kernel into angle-domain and how to compute the scattering-angle-domain Hessian, as well as the dip-dependent scattering-angle-domain Hessian. Finally we apply our method to the Sigsbee2A model.

Angle-domain illumination gathers by wave-equation-based methods