Ellipsoidal approximation of the elastic wavefront in monoclinic symmetry media

Abstract

In this work, an ellipsoidal approximation is introduced for the phase velocities of the longitudinal -P, and transversal -S1 and -S2 modes in media with monoclinic symmetry. This new approach is valid for small polar angles near the vertical direction, but without restrictions in the azimuth angle and for arbitrary degrees of anisotropy. From an analytical treatment of Christoffel’s equation in terms of slowness, it is possible to obtain mathematical expressions for the phase velocities in monoclinic media that turn out to be 3D ellipsoids rotated with respect to the symmetry axes. To establish the degree of validity of these expressions, the Christoffel equation is solved numerically and the exact and approximate wavefronts are displayed, and it is found that near
the vertical axis of symmetry they give similar results. Although monoclinic models can be presented in double fractured geological formations, they have been little used in multicomponent seismic methods for fracture detection mainly due to the large amount of elastic parameters present. The ellipsoidal approximations of the wavefront can be used for the calculation of transit times, as well as the modeling and inversion of the elastic constants in double fractured environments and for vertical and/ or multi-acimutal seismic profile type acquisition geometries.