Simulation study on vibration signals of surface Green's function based on time singular point parameter perturbation algorithm

SUMMARY The Lamb problem stands as a classic issue in theoretical seismology aimed at obtaining solutions for the Green's functions of point sources in elastic half-spaces. It serves as the foundation for studying vibrational signals from many sources such as walking and driving, bearing significant theoretical and practical value. While the analytical solutions exist for the Lamb problem when both excitation and reception occur on the ground, the presence of singularity makes the numerical stability of calculating the Green's functions from these analytical solutions a challenge. In this study, we propose a stable algorithm that circumvents the impact of time singularity in the analytical solutions of the Lamb problem by introducing a tiny time parameter perturbation and judiciously selecting the starting position for time discretization sampling. This means that the zero time point (i.e. the excitation time of the source pulse) and the starting time of discretization sampling may not be coincident. The advantages of this method lie in its stability, simplicity and practical accuracy, with the calculation results aligning consistently with the theoretical geometric decay of surface waves. Additionally, analysis of field data demonstrates that our stable algorithm effectively captures the amplitude characteristics of measured footstep responses and vehicle signals. Building upon the foundation of obtaining stable discrete solutions, we further elaborate on the process of transforming the discrete sampling starting point to approach the actual zero time point infinitely, even though this tiny time parameter perturbation does not affect the simulation results.