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 δc/c may be written in the form, δc/c = ∑_n=0,2,4 (A_n cos nζ + B_n sin nζ) where A_n and B_n are frequency-dependent depth integrals and ζ 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 ζ involving degrees zero (five elastic parameters), two (six elastic parameters), and four (two elastic parameters). Jech and Pšenčík demonstrated that in such a weakly anisotropic earth model the relative compressional-wave phase-speed perturbation may be expressed as δc/c = (2c²)^-1B₃₃, whereas the relative shear wave phase-speed perturbations are given by δc/c = (4c²)^-1{B₁₁ + B₂₂ ± [(B₁₁ - B₂₂)² + 4B₁₂²]^1/2. We demonstrate that the coefficients B₃₃ B₁₁ B₂₂, and B₁₂ may be written in the generic form, B_lm = ∑_n=0⁴[a_n(i) cos nζ+b_n(i) sin nζ], where ζ denotes the local azimuth and i the local angle of incidence. For B₁₁, B₂₂ and B₃₃ the coefficients a_n (i) and b_n(i) are an even Fourier series of degrees zero, two and four in i, but for B₁₂ they are an odd Fourier series of degrees one and three. Like surface waves, the azimuthal (ζ) dependence of body waves involves even degrees zero (five elastic parameters), two (six elastic parameters), and four (two elastic parameters), but, unlike surface waves, 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.