Isotropic angle-domain elastic reverse-time migration

Introduction

Seismic processing is usually based on acoustic wave equations, which assume that the Earth represents a liquid that propagates only compressional waves. Although useful in practice, this assumption is not theoretically valid. Earth materials allow for both compressional and shear wave propagation in the subsurface. Shear waves, either generated at the source or converted from compressional waves at various interfaces in the subsurface, are detected by multicomponent receivers. Shear waves are usually stronger at large incidence and reflection angles, often corresponding to large offsets. However, for complex geological structures near the surface, shear waves can be quite significant even at small offsets. Conventional single-component imaging ignores shear wave modes, which often leads to incorrect characterization of wave-propagation, incomplete illumination of the subsurface and poor amplitude characterization.

Even when multicomponent data are used for imaging, they are usually not processed with specifically designed procedures. Instead, those data are processed with ad-hoc procedures borrowed from acoustic wave equation imaging algorithms. For isotropic media, a typical assumption is that the recorded vertical and in plane horizontal components are good approximations for the P- and S-wave modes, respectively, which can be imaged independently. This assumption is not always correct, leading to errors and noise in the images, since P- and S-wave modes are normally mixed on all recorded components. Also, since P and S modes are mixed on all components, true-amplitude imaging is questionable no matter how accurate the wavefield reconstruction and imaging condition are.

Multicomponent imaging has long been an active research area for exploration geophysicists. Techniques proposed in the literature perform imaging by using time extrapolation, e.g. by Kirchhoff migration (Hokstad, 2000; Kuo and Dai, 1984a) and reverse-time migration (Chang and McMechan, 1994; Whitmore, 1995; Chang and McMechan, 1986) adapted for multicomponent data. The reason for working in the time domain, as opposed to the depth domain, is that the coupling of displacements in different directions in elastic wave equations makes it difficult to derive a dispersion relation that can be used to extrapolate wavefields in depth (Clayton and Brown, 1979; Clayton, 1981).

Early attempts at multicomponent imaging used the Kirchhoff framework and involve wave-mode separation on the surface prior to wave-equation imaging (Wapenaar and Haimé, 1990; Wapenaar et al., 1987). Kuo and Dai (1984b) perform shot-profile elastic Kirchhoff migration, and Hokstad (2000) performs survey-sinking elastic Kirchhoff migration. Although these techniques represent different migration procedures, they compute travel-times for both PP and PS reflections, and sum data along these travel time trajectories. This approach is equivalent to distinguishing between PP reflection and PS reflections, and applying acoustic Kirchhoff migration for each mode separately. When geology is complex, the elastic Kirchhoff migration technique suffers from drawbacks similar to those of acoustic Kirchhoff migration because ray theory breaks down (Gray et al., 2001).

There are two main difficulties with independently imaging P and S wave modes separated on the surface. The first is that conventional elastic migration techniques either consider vertical and horizontal components of recorded data as P and S modes, which is not always accurate, or separate these wave modes on the recording surface using approximations, e.g. polarization (Pestana et al., 1989) or elastic potentials (Zhe and Greenhalgh, 1997; Etgen, 1988) or wavefield extrapolation in the vicinity of the acquisition surface (Wapenaar et al., 1990; Admundsen and Reitan, 1995). Other elastic reverse time migration techniques do not separate wave modes on the surface and reconstruct vector fields, but use imaging conditions based on ray tracing (Chang and McMechan, 1994,1986) that are not always robust in complex geology. The second difficulty is that images produced independently from P and S modes are hard to interpret together, since often they do not line-up consistently, thus requiring image post processing, e.g. by manual or automatic registration of the images  (Gaiser, 1996; Nickel and Sonneland, 2004; Fomel and Backus, 2003).

We advocate an alternative procedure for imaging elastic wavefield data. Instead of separating wavefields into scalar wave modes on the acquisition surface followed by scalar imaging of each mode independently, we use the entire vector wavefields for wavefield reconstruction and imaging. The vector wavefields are reconstructed using the multicomponent vector data as boundary conditions for a numerical solution to the elastic wave equation. The key component of such a migration procedure is the imaging condition which evaluates the match between wavefields reconstructed from the source and receiver. For vector wavefields, a simple component-by-component cross-correlation between the two wavefields leads to artifacts caused by crosstalk between the unseparated wave modes, i.e. all P and S modes from the source wavefield correlate with all P and S modes from the receiver wavefield. This problem can be alleviated by using separated elastic wavefields, with the imaging condition implemented as cross-correlation of wave modes instead of cross-correlation of the Cartesian components of the wavefield. This approach leads to images that are cleaner and easier to interpret since they represent reflections of single wave modes at interfaces of physical properties.

As for imaging with acoustic wavefields, the elastic imaging condition can be formulated conventionally (cross-correlation with zero lag in space and time), as well as extended to non-zero space lags. The elastic images produced by extended imaging condition can be used for angle decomposition of PP and PS reflectivity. Angle gathers have many applications, including migration velocity analysis (MVA) and amplitude versus angle (AVA) analysis.

The advantage of imaging with multicomponent seismic data is that the physics of wave propagation is better represented, and resulting seismic images more accurately characterize the subsurface. Multicomponent images have many applications. For example they can be used to provide reflection images where the P-wave reflectivity is small, image through gas clouds where the P-wave signal is attenuated, validate bright spot reflections and provide parameter estimation for this media, Poisson's ratio estimates, and detect fractures through shear-wave splitting for anisotropic media (Simmons and Backus, 2003; Stewart et al., 2003; Gaiser et al., 2001; Li, 1998; Knapp et al., 2001; Zhu et al., 1999). Assuming no attenuation in the subsurface, converted wave images also have higher resolution than pure-mode images in shallow part of sections, because S-waves have shorter wavelengths than P-waves. Modeling and migrating multicomponent data with elastic migration algorithms enables us to make full use of information provided by elastic data and correctly position geologic structures.

This paper presents a method for angle-domain imaging of elastic wavefield data using reverse-time migration (RTM). In order to limit the scope of our paper, we ignore several practical issues related to data acquisition and pre-processing for wave-equation migration. For example, our methodology ignores the presence of surface waves, e.g. Rayleigh and Love waves, the relatively poor spatial sampling when imaging with multicomponent elastic data, e.g. for OBC acquisition, the presence of anisotropy in the subsurface and all amplitude considerations related to the directionality of the seismic source. All these issues are important for elastic imaging and need to be part of a practical data processing application. We restrict in this paper our attention to the problem of wave-mode separation after wavefield extrapolation and angle-decomposition after the imaging condition. These issues are addressed in more detail in a later section of the paper.

We begin by summarizing wavefield imaging methodology, focusing on reverse-time migration for wavefield multicomponent migration. Then, we describe different options for wavefield multicomponent imaging conditions, e.g. based on vector displacements and vector potentials. Finally, we describe the application of extended imaging conditions to multicomponent data and corresponding angle decomposition. We illustrate the wavefield imaging techniques using data simulated from the Marmousi II model (Martin et al., 2002).

Isotropic angle-domain elastic reverse-time migration