Theoretical and numerical investigations of global and regional seismic wave propagation in weakly anisotropic earth models

Smith and Dahlen demonstrated that in a weakly anisotropic earth model the relative surface wave phase-speed perturbation delta c/c may be written in the form delta c/c = Sigma(n=0,2,4)(A(n) cos n zeta + B-n sin n zeta), where A(n) and B-n are frequency-dependent depth integrals and zeta denotes the ray azimuth. In this approximation, surface wave anisotropy is governed by 13 elastic parameters and the azimuthal dependence of the phase speed is represented by an even Fourier series in zeta involving degrees zero (five elastic parameters), two (six elastic parameters), and four (two elastic parameters). Jech and Psencik demonstrated that in such a weakly anisotropic earth model the relative compressional-wave phase-speed perturbation may be expressed as delta c/c = (2c(2))(-1) B-33, whereas the relative shear wave phase-speed perturbations are given by delta c/c = (4c(2))(-1){B-11 + B-22 +/- [(B-11 - B-22)(2) + 4B(12)(2)](1/2)}. We demonstrate that the coefficients B-33, B-11, B-22, and B-12 may be written in the generic form B-lm = Sigma(4)(n=0) [a(n)(i) cos n zeta + b(n)(i) sin n zeta], where. denotes the local azimuth and i the local angle of incidence. For B-11, B-22 and B-33 the coefficients an(i) and bn(i) are an even Fourier series of degrees zero, two and four in i, but for B-12 they are an odd Fourier series of degrees one and three. Like surface waves, the azimuthal (zeta) dependence of body waves involves even degrees zero (five elastic parameters), two (six elastic parameters), and four (two elastic parameters), but, unlike surfacewaves, it also involves the odd degrees one (six elastic parameters) and three (two elastic parameters). Thus, weakly anisotropic body-wave propagation involves all 21 independent elastic parameters. We use spectral-element simulations of global and regional seismic wave propagation to assess the validity of these asymptotic body and surface wave results. The numerical simulations and asymptotic predictions are in good agreement for anisotropy at the 5 per cent level.