SUMMARY
Green’s functions provide an efficient way to model surface-wave propagation and estimate physical quantities for near-surface processes. Several surface-wave Green’s function approximations (far-field, no mode conversions and no higher mode surface waves) have been employed for numerous applications such as estimating sediment flux in rivers, determining the properties of landslides, identifying the seismic signature of debris flows or to study seismic noise through cross-correlations. Based on those approximations, simple empirical scalings exist to derive phase velocities and amplitudes for pure power-law velocity structures providing an exact relationship between the velocity model and the Green’s functions. However, no quantitative estimates of the accuracy of these simple scalings have been reported for impulsive sources in complex velocity structures. In this paper, we address this gap by comparing the theoretical predictions to high-order numerical solutions for the vertical component of the wavefield. The Green’s functions computation shows that attenuation-induced dispersion of phase and group velocity plays an important role and should be carefully taken into account to correctly describe how surface-wave amplitudes decay with distance. The comparisons confirm the general reliability of the semi-analytic model for power-law and realistic shear velocity structures to describe fundamental-mode Rayleigh waves in terms of characteristic frequencies, amplitudes and envelopes. At short distances from the source, and for large near-surface velocity gradients or high Q values, the low-frequency energy can be dominated by higher mode surface waves that can be captured by introducing additional higher mode Rayleigh-wave power-law scalings. We also find that the energy spectral density for realistic shear-velocity models close to piecewise power-law models can be accurately modelled using the same non-dimensional scalings. The frequency range of validity of each power-law scaling can be derived from the corresponding phase velocities. Finally, highly discontinuous near-surface velocity profiles can also be approximated by a combination of power-law scalings. Analytical Green’s functions derived from the non-dimensionalization provide a good estimate of the amplitude and variations of the energy distribution, although the predictions are quite poor around the frequency bounds of each power-law scaling.
Study on the Nonlinear Site Response Using the Green's Functions of a Near-Surface Layer Estimated for Weak Motions-Borehole Array Recordings at the IWTH25 Site for the 2008 Iwate-Miyagi Inland Earthquake (M7.2)-