A unified approach for relationships among Green's function, normal modes and dispersion spectrum in layered elastic half-space, with corrected misconceptions on surface wave dispersion and testing

Abstract

SUMMARY In this paper, the global stiffness matrix [K] and the Fourier–Bessel series methods are proposed to derive the accurate Green's function and dynamic response in a form that is directly related to the dispersion curve and experimental dispersion spectrum. Detailed analyses are carried out for the two-layered half-space with different velocity profiles, including the homogeneous half-space as a special case. Our studies indicate that, in Rayleigh wave analysis, the original Rayleigh equation, instead of the rationalized Rayleigh equation as previously derived and used, should be used since the latter would contain extra non-physical roots. We further reveal and characterize three distinct types of leaky waves: the intrinsic surface leaky wave, the apparent Rayleigh mode with a frequency gap associated with a low-velocity half-space and the fast-guided P–SV wave in the layered medium with a high VS contrast between the upper layer and the lower half-space. All leaky modes can be captured by local minima of |det[K]| instead of tracing complex roots in other existing approaches. In the experimental estimation of dispersion curves for practical applications, we have observed that the truncation effect is the major source of uncertainty regardless of the wavefield transformation method utilized. Furthermore, the truncation effect is both location- and model-dependent, without a unique optimal near offset. As such, in order to reduce the uncertainty from the truncation effect, the receiver layout should be considered in the inversion of dynamic response, instead of relying on ensuring a minimum near offset. This becomes possible with the present fast and accurate complete dynamic Green's function by which all wave phenomena (including different types of leaky waves) and receiver locations can be considered in the wavefield transformation.