Semi-analytical solution for the elastic wave propagation due to a dislocation source in a transversely isotropic half-space

SUMMARYIn this paper, the 3-D theoretical seismogram at the free surface of an elastic half-space with ‘vertical transverse isotropy (VTI)’ and an ‘elliptic anisotropy‘ is synthesized via semi-analytical formulation. To this end, the time-domain solution is obtained via the Hankel and Laplace integral transforms and the Cagniard contour integration scheme. The formulations include complete Green's functions expressed in compact and elegant formulations in terms of some elementary line integrals over the finite interval (0, π/2), due to the general point force and point double-couple source of arbitrary orientation, varying with time as Heaviside step function. This solution is further implemented to model seismic waves generated from moment tensor source with a ramp-type slip function. The solution clearly delineates compressional P waves, shear SV waves and SH waves, diffracted SPwaves and surface Rayleigh waves, with two critical distances in which the mode conversion happens. The first marks the conversion of the total reflection of SV wave into the diffracted SPwave travelling on the free surface with the velocity of compressional wave. The second, however, marks a location where the order of arrival times of SP and SH waves is reversed. The interesting phenomenon of shear wave splitting by the apparent time lag between the arrivals of SV and SHwaves is further demonstrated. Studied also is the effect of non-double-couple (non-DC) components of the moment tensor of the shear fault. Particularly, it is shown that non-DC components may lead to large amplitude of the P wave in the presence of anisotropy, resulting in the changed polarity of P wave at the onset of the motion. It is found that for the elliptic anisotropy, and for oblique-thrust faults, the isotropic moment (ISO) is always negative if the elastic constants satisfy C11 > C33 and C44 > C66, whereas it is always positive if C11 < C33 and C44 < C66 (for oblique-reverse faults, the situation is reversed). It is shown that for some sedimentary rocks, the decomposition of moment tensor in the general VTI and in its elliptic approximation are close to each other, and that the moment tensor decomposition in VTI media is very sensitive to the elastic coefficients, such that a small change in the elastic constants may lead to a remarkable change in the moment tensor decomposition. Particularly, the effect of elastic constant C13 is shown to be significant. While the time-domain solution obtained in this paper can be also degenerated to the solution in the corresponding isotropic half-space, the Mathematica code of the solution provided can be served as benchmark to other numerical solutions, applicable for the computation of the theoretical seismogram in the involved media.