Anisotropic tomography with rock physics constraints

Introduction

Anisotropic model building tries to resolve more than one parameter at each grid point of the subsurface. This number could be 3 for a vertical transverse isotropic (VTI) media, and increase to 5 for a tilted transverse isotropic (TTI) media. Traditional surface seismic tomography may be able to produce accurate isotropic earth models efficiently for a large area when the acquisition is dense and the earth is well-illuminated by rays at a wide range of angles. However surface seismic data inversion becomes ill-posed and highly underdetermined due to the rapidly increasing dimensionalities of the model space with the increasing complexity of the subsurface.

One big disadvantage of the surface seismic tomography is the lack of the depth information. During tomography, not only is the low wavenumber earth model estimated, but the depth of the reflectors is unknown as well. To add the depth dimension into the inversion, several localized tomography experiments around the wells are analyzed (Bakulin et al., 2010d,c). In these studies, joint inversion of surface seismic data and borehole data (check-shots, walkaway VSPs) shows great potential to yield better defined earth models. However, due to the ambiguity among the parameters, even the borehole aided localized tomography has difficulty in resolving a reliable, unique anisotropic model in 3D (Bakulin et al., 2009).

To constrain the inversion further, we need to consider some prior knowledge of the subsurface. This prior knowledge can be characterized by the covariance of the model space and is independent of the data. There are many ways to obtain the covariance information based on different assumptions. For example, we often smooth our earth model horizontally and vertically, which implies a certain user-defined spatial correlation lag. More realistically, we can use the geological information as a prior and shape our estimate accordingly. This model shaping can be posed as a decomposition of the earth model into different layers and horizons before tomography (Bakulin et al., 2010a), or as a regularization/preconditioning operator during tomography (Bakulin et al., 2010b). We can obtain the geological information either by interpreting and picking the horizons or by building a set of steering filters (Clapp, 2000) according to the current subsurface image.

In addition to the spatial covariance, for a multi-parameter estimation, a point-by-point cross-parameter covariance is also needed to fully describe the subsurface. One source of the cross-parameter covariance comes from rock physics study (Sayers, 2004,2010; Hornby et al., 1995; Bachrach, 2010b). In particular, Bachrach (2010a) develops both deterministic and stochastic modeling schemes based on the rock physics effective media models for compacting shale and sandy shale. Along with appropriate laboratory core measurements, the parameters needed by the rock physics model are limited in a certain range, which greatly reduces the correlation lag in the earth model parameters. These rock physics modeling results can be used to construct the initial earth model and the covariance relationships among the earth model parameters. When all of these four ingredients - surface seismic, borehole traveltime measurements, geological information and rock physics priors - are available, we will have a better chance to resolve anisotropic models that both flatten the gathers and follow the geological and rock physics principles at the same time.

In this paper, we assume the spatial covariance and the local cross-parameter covariance can be fully separated and focus on utilizing the rock physics modeling results to constrain the anisotropic tomography. A VSP survey with a common industry geometry is simulated on two different models, one with only shale (sandy shale), and the other with one layer of pure sand (isotropic). We compare the inversion results using the unconstrained tomography, current constrained tomography and the rock physics constrained tomography. Finally, we perform a-posteriori uncertainty analysis (Osypov et al., 2008) for the shale example to evaluate the contribution of rock physics prior knowledge to the reduction of the null-space.

Anisotropic tomography with rock physics constraints