• adjoint methods;
  • Fréchet derivatives;
  • scattering;
  • seismic anisotropy;
  • sensitivity;
  • surface waves


We calculate finite-frequency anisotropic traveltime sensitivity kernels for Rayleigh and Love waves using the recently developed combination of the adjoint method with spectral-element modelling of seismic wave propagation. We describe anisotropy following the ‘natural’ 13 elastic parameters for surface waves (A, C, F, L, N, Bc,s, Hc,s, Gc,s and Ec,s) complemented by eight ‘body-wave parameters’ (Jc,s, Kc,s, Mc,s and Dc,s). Along the ray path, the adjoint spectral-element computations agree well with asymptotic theory, but also expose the limitations of the asymptotic description. The adjoint spectral-element method is an efficient and flexible numerical tool, but it does not allow one to identify the various wave propagation phenomena contributing to the observed sensitivity. To decipher the numerical results, we apply Born scattering theory together with a surface-wave mode-coupling formulation. We identify a strong effect due to mode coupling. The sensitivity of Rayleigh waves for some of the anisotropic parameters is affected by Love–Rayleigh coupling, while Love-wave sensitivity is affected by cross-branch coupling. In addition, and very specific to anisotropy, the directional dependence of the sensitivity to azimuthal anisotropy may strongly distort the kernels, rendering them highly path-dependent. Because of these combined effects, the anisotropic sensitivity kernels can deviate substantially from the simple elliptical kernels used in recent isotropic finite-frequency tomography.