Sensitivity kernels for finite-frequency surface waves

Sensitivity kernels for fundamental mode surface waves at finite frequency for 2-D phase speed and 3-D shear wave speed are constructed based on the Born and Rytov approximations working with a potential representation for surface waves. The use of asymptotic Green's functions for scalar wave equations provides an efficient way to calculate the Born or Rytov kernels. The 2-D sensitivity kernels enable us to incorporate the finite-frequency effects of surface waves, as well as off-great-circle propagation, in tomographic inversions for phase-speed structures. We derive examples of the 2-D sensitivity kernels both for a homogeneous background model (or a spherically symmetric model), and for a laterally heterogeneous model. The resulting distortions of the shape of the sensitivity kernels for a heterogeneous background model indicate the importance of the use of proper kernels to account of the heterogeneity in the real Earth. By combining a set of 2-D sensitivity kernels with 1-D vertical sensitivity kernels for a particular frequency range and taking the inverse Fourier transform, we can derive 3-D sensitivity kernels for surface waves in the time domain. Such 3-D kernels are useful for efficient forward modelling of surface waveforms incorporating finite-frequency effects, and will also enable us to perform direct inversion of surface waveforms into 3-D structure taking account of finite-frequency effects.