Elastic Wave Propagation in Anisotropic Media: Source Theory, Traveltime Computations and Migration

The availability of the Green’s tensor in analytical form enables one to obtain numerical solutions for sources in anisotropic media. First, we show that the radiation field of an explosion has the following new features: 1) quasi-transverse waves are created with four- and eight-lobe patterns; 2) quasi-longitudinal waves are generated for the collatitudinal displacement with four-lobe patterns; 3) the energy ratio SV/P may reach the value of 20 for more than 50% of the azimuths in crustal structures such as tuff and shales; and 4) radiation patterns for vertical shear waves are created that are indistinguishable from corresponding waves produced by earthquake faults. For the special case of azimuthally isotropic media, we present an alternative representation of the Green’s tensor and the displacement fields in the form of an exact Hankel transform over the horizontal wave number variable. The total field is specified in terms of two potentials: an SH potential and a mixed quasi-transverse/quasi- longitudinal potential, both which assume the role of two scalar Green’s functions. A Haskell-type matrix algorithm for a multilayered azimuthally isotropic half-space can then be established, enabling us to calculate body waves and surface waves in real-earth crustal models.

Next, we derive analytical expressions for the traveltime-offset curves for multilayered, weakly azimuthally isotropic and anisotropic media in terms of the elastic properties of each layer. This method is based upon an approximate skewed hyperbolic moveout formula involving three measured bulk velocities for each reflector. The primary benefits of this technique are: 1) it allows for fast traveltime computation; 2) it makes possible an extremely rapid estimation of the interval elastic parameters; and 3) it provides physical insight into wave propagation in anisotropic media.

Based on forward modeling discussed above, we develop a traveltime inversion algorithm that estimates the five elastic constants together with the orientation of the axis of symmetry for each layer. In the isotropic limit, this algorithm reduces to the conventional one used for determining the interval velocities from stacking velocity measurements. The inversion technique is applied to surface seismic measurements and VSP field surveys. In both cases, the method provides very good estimates of the six parameters. The results also show that the orientation of the horizontal axis of symmetry, for a given azimuthally anisotropic layer at depth, can be obtained using only quasi-P wave information. This technique can be easily incorporated into conventional velocity analysis algorithms.

Finally, we combine the theoretical ray amplitudes together with the traveltime equations to generate a suitable Green’s tensor to perform Kirchhoff migration in anisotropic media. This imaging scheme is applied to the case of azimuthal isotropy. An anisotropic velocity analysis scheme is also established in order to generate an appropriate velocity (elastic constant) model for migration in azimuthal isotropic media for non converted and converted qP-qSV waves. Synthetic examples showing migration of qP-qP and converted qP-qSV sections are presented. In both cases, the method provides accurate images of the subsurface. In quasi-compressional qP=qP case, we show that, even with a weak to moderate percentage of anisotropy, an isotropic migration algorithm cannot handle the anisotropy properly. In a real data example from South Texas using converted qP-qSV waves, our anisotropic migration scheme improves the delineation of a fault plane and the lateral continuity of the flat reflectors. Furthermore, the anisotropic Kirchhoff migration algorithm enables us to manipulate multicomponent data with an arbitrary geometry of sources and receivers. In addition, the proper handling of anisotropy, together with the combination of converted and nonconverted waves, provides more geological information about the subsurface and better delineation of potential hydrocarbon reservoirs.