Wave-equation tomography for anisotropic parameters

Introduction

Since first reported in exploration seismology in the 1930s (McCollum and Snell, 1932), the importance of anisotropy has been continuously increased in seismic imaging and exploration. This is partially due to acquisition with increasingly longer offsets and exploration in areas with strong geological deformation. Until now, the transverse isotropic (TI) model has been the most commonly used anisotropic model in seismic imaging. Postma (1955) and Helbig (1956) showed that a sequence of isotropic layers on a scale much smaller than the wavelength leads to an anisotropic medium. For the case of horizontal layers, the medium can be described by an equivalent vertical transverse isotropic (VTI) medium. When dip is present, the medium develops a tilted transverse anisotropy (TTI). Many authors (Shan, 2009; Fei and Liner, 2008; Zhang and Zhang, 2009; Fletcher et al., 2009) have developed various migration and processing schemes for VTI and TTI medium; however, the estimation of the anisotropy model is still challenging.

The existing anisotropic model-building schemes are mostly based on measuring the non-hyperbolic moveout along the traveltime curve to flatten the common image gathers (CIG) (Zhou et al., 2003; Cai et al., 2009; Zhou et al., 2004; Yuan et al., 2006). However, traveltime-based methods are prone to errors and unrealistic results when multi-pathing exists in areas of complex overburden.

Wave-equation tomography (WETom) has been widely investigated in isotropic velocity building, and can be implemented either in the data space (Woodward, 1992; Tarantola, 1984) or in the image space (Shen, 2004; Sava and Biondi, 2004b; Shen and Symes, 2008; Sava and Biondi, 2004a; Guerra et al., 2009). Several advantages drive us to use the image-space WETom instead of the data-space version: First, the migration image is often much cleaner than the recorded wavefields. Second, we can use cheaper one-way extrapolators in the image space, compared with expensive two-way extrapolators in the data space. Third, the objective function is directly related to the final image. Therefore, we choose to extend image-space WETom from isotropic velocity building to anisotropic model building.

In this paper, we first explain the parametrization of the inversion problem and then extend image-space WETom from the isotropic medium to the anisotropic medium. We show that theoretically the gradient of the tomographic objective functional for the anisotropic medium is similar to its isotropic version, with an extra term for the additional parameter. Then, we test the anisotropic WETom operator using a model with a localized anomaly. Finally, we invert for a 2-D VTI model using the proposed anisotropic WETom operator.

Wave-equation tomography for anisotropic parameters