Sensitivity kernels for finite-frequency surface waves

With the assumption of scalar wave propagation in laterally heterogeneous structure, two-dimensional sensitivity kernels for surface-wave phase speeds for a single frequency can be constructed using Born and Rytov approximations with asymptotic Green's functions. A full treatment requires extensive two-point ray tracing and requires too much computation to be practical for regular use in tomographic studies. Fortunately, the paraxial ray approximation provides a means to estimate such asymptotic kernels efficiently, and works quite well for most cases where the velocity distributions are smoothly varying. The paraxial theory gives a symmetric kernel around the ray path of the surface wave. In regions with strong lateral heterogeneity and large velocity gradients, the exact kernels differ noticeably from the paraxial results with considerable asymmetry with respect to the central ray. However, within the influence zone [Yoshizawa and Kennett, 2002], over which the surface wave field is expected to be coherent, the paraxial kernels are a very good approximation even in the presence of strong heterogeneity. This influence zone is approximately one-third of the width of the first Fresnel zone. The extension of the results to finite bandwidth in terms of the three-dimensional sensitivity to the shear wave speed structures can be derived from a combination of the two-dimensional sensitivity kernels to phase speed structures and the one-dimensional sensitivity kernels to a reference Earth model. This set of sensitivity kernels enable us to incorporate the finite-frequency effects of surface waves, as well as off-great-circle propagation in tomographic inversion for phase speed and shear wave speed structures.