Short Note Green’s Functions for Surface Waves in a Generic Velocity Structure

Methodologies for calculating surface-wave velocities and the associated displacement/stress eigenfunctions and Green’s functions have been well established for many decades. However, to our knowledge, no one has ever documented a quantitative evaluation of these properties for commonly used empirical scalings. For example, it is currently not possible to take a given power-law dependence of shear-wave velocity on depth and look up the corresponding dependence of phase velocity on frequency, or Green’s function surface displacement. We address this gap in the literature and here provide explicit quantitatively accurate expressions for phase velocities and Green’s function amplitudes for a few commonly used empirical formulas for near-surface velocity structure. These exact expressions are found to be immediately useful in applications that use shallow phase velocities and also in applications that interpret seismic amplitudes or amplitude ratios from near-surface processes such as fluvial transport, icequakes, landslides, and volcanic tremor.