The effect of velocity anisotropy on wave propagation in homogeneous and heterogeneous media has been the subject of numerous publications. Careful forward modeling has helped interpreters to understand how velocity anisotropy manifests itself in field data. Few attempts have been made, however, to estimate the parameters that describe the complexity of velocity anisotropy, namely, the elastic constants. Estimation of the elastic constants is important because it can aid in lithologic discrimination and fracture orientation, reveal anisotropic properties of the medium not obvious in the data, and provide further imaging or full waveform inversion algorithms with background models that can be refined iteratively.
Elastic constants have been estimated primarily in the lab where the samples can be assumed to be homogeneous. When the assumption of homogeneity is not valid, the problem is more difficult because the effects of anisotropy and heterogeneity are coupled in the data. This coupling problem has been addressed by making prior assumptions about the nature of both the anisotropy and heterogeneity. For example, by assuming that the rock samples are homogeneous, Arts et al. (1991) estimate, from lab measurements, the 21 elastic constants that characterize a general triclinic system. By assuming isotropic velocities, McMechan (1983) estimates tomographically arbitrary spatial variations using 2-D and 3-D models. These two papers are examples of different trade-offs between the complexity of velocity and heterogeneity. On the one hand, the first paper uses the simplest model for heterogeneities (homogeneous) without making any assumption about the type of velocity anisotropy. On the other hand, the second paper uses the simplest model for the velocity (isotropic) without making any assumption about the nature of the heterogeneity.
A model commonly used to describe velocity anisotropy is transversely isotropic (TI). This is because transverse isotropy is the most common form of anisotropy in the subsurface. Several authors have addressed the issue of estimating elastic constants that characterize this type of anisotropy. White et al. (1983) estimate the five TI elastic constants of a homogeneous formation using a VSP geometry. Hake et al. (1984) approximate the traveltimes curves of layered models with a three-term Taylor series expansion in which the coefficient are a function of the elastic constants. Winterstein and Paulsson (1990) estimate the elastic constants from VSP and cross-well measurements in a medium with velocity linearly increasing with depth. Byun and Corrigan, (1990) propose an iterative model-based optimization scheme to invert traveltimes for the five elastic constants of a TI-layered media. More recently, Sena (1991) proposed a variant of this method in which all the calculations are done analytically, without having to go through the semblance analysis needed in Byun and Corrigan's method.
Near the axes of symmetry, a TI medium looks like an elliptically anisotropic medium. This property of transverse isotropy is used by Muir (1990a) and Dellinger et al. (1992) to approximate P- and SV-wave slowness surfaces and impulse responses with ellipses fitted near both horizontal and vertical axes. In a later paper, Muir (1990b) suggests the transformation of the elliptical parameters into elastic constants by using well known expressions that relate them (Levin, 1979; Levin, 1980). However, Muir implicitly assumes in this paper that traveltimes near both axes are available, which doesn't happen often.
Unfortunately, all the preceding methods fail when the data are not wide aperture or when measurements along both axes are not available. This is often the case with VSP and cross-well experiments.
I show in this paper how to obtain the elastic constants that control P- and SV-wave propagation in TI media from limited aperture traveltimes, either from VSP or from cross-well geometries. I start by fitting the traveltimes for P- and SV-waves with elliptical time-distance relations near a single axis (either vertical or horizontal). The result is four velocities: two based on the time-of-arrival and distance along a symmetry axis (the direct velocities) and two based on the differential traveltime and differential distance as the direction is perturbed (the normal moveout velocities). These four elliptical parameters are used to solve analytically a system of four equations and four unknown elastic constants. Since the procedure is based on fitting the data with elliptical velocity models, it is exact when estimating elastic constants from SH-wave traveltimes.
The data aperture is constrained in two different ways. First, it should not be too small to ensure that there is enough curvature to estimate the normal moveout velocities. Second, it should not be too wide to ensure that the elliptical fit remains accurate for the given wave type. I show in this paper that there is an intermediate range of ray angles that satisfy these two requirements when estimating elastic constants from either VSP or cross-well geometries.
The calculations presented here are valid for homogeneous media. When the model is heterogeneous, it can be described as a superposition of homogeneous regions, and the elliptical parameters needed at each region can be estimated tomographically, as explained by Michelena et al. (1992) and Michelena (1992a). Once the elliptical parameters at each cell are estimated, the procedure developed here for homogeneous media can be applied at each cell to obtain 2-D maps of elastic constants.
The equations I use in this paper to transform elastic constants into elliptical parameters (forward mapping) are not new. They are the same as the ones summarized by Muir (1990b), which can also be found in Levin (1979) and Levin (1980). What is new is the simultaneous solution of these equations near each axis to obtain elastic constants as a function of elliptical parameters (inverse mapping).
I start by rederiving the basic equations of the forward mapping from the expression of P- and SV-wave phase velocities in TI media. The calculations are done near both the horizontal and the vertical axes. Then, using these expressions, I solve the inverse mapping analytically. The final section illustrates the use of the technique when estimating the elastic constants of a homogeneous medium from impulse responses sampled near either the vertical axis or the horizontal axis, to simulate VSP and cross-well configurations, respectively.