A single Rayleigh mode may exist with multiple values of phase-velocity at one frequency

Abstract

SUMMARY Other than commonly assumed in seismology, the phase velocity of Rayleigh waves is not necessarily a single-valued function of frequency. In fact, a single Rayleigh mode can exist with three different values of phase velocity at one frequency. We demonstrate this for the first higher mode on a realistic shallow seismic structure of a homogeneous layer of unconsolidated sediments on top of a half-space of solid rock (LOH). In the case of LOH a significant contrast to the half-space is required to produce the phenomenon. In a simpler structure of a homogeneous layer with fixed (rigid) bottom (LFB) the phenomenon exists for values of Poisson’s ratio between 0.19 and 0.5 and is most pronounced for P-wave velocity being three times S-wave velocity (Poisson’s ratio of 0.4375). A pavement-like structure (PAV) of two layers on top of a half-space produces the multivaluedness for the fundamental mode. Programs for the computation of synthetic dispersion curves are prone to trouble in such cases. Many of them use mode-follower algorithms which loose track of the dispersion curve and miss the multivalued section. We show results for well established programs. Their inability to properly handle these cases might be one reason why the phenomenon of multivaluedness went unnoticed in seismological Rayleigh wave research for so long. For the very same reason methods of dispersion analysis must fail if they imply wave number kl(ω) for the lth Rayleigh mode to be a single-valued function of frequency ω. This applies in particular to deconvolution methods like phase-matched filters. We demonstrate that a slant-stack analysis fails in the multivalued section, while a Fourier–Bessel transformation captures the complete Rayleigh-wave signal. Waves of finite bandwidth in the multivalued section propagate with positive group-velocity and negative phase-velocity. Their eigenfunctions appear conventional and contain no conspicuous feature.

May Rayleigh waves propagate with group- and phase-velocities of opposite sign in the valley of Mexico City?

“…Following Forbriger et al (2020) these effects described here for the first higher mode occur for the fundamental mode in certain more complex models as well.…”

Rayleigh Wave Dispersion and Inversion for Shallow Surface with a High-velocity Rigid Pavement

“…The secular function is derived for the model (table 1) with a 40cm-thickness rigid pavement layer and the energy map is plotted in figure 2(a). We can observe the first higher mode appear multiple different values of phase velocity at one frequency, which is called 'reversal of the dispersion curve' by Forbriger 7 . We use the method of searching the absolute minima of the moduli of the secular function to obtain the first higher mode dispersion curves in the frequency range 22-28 Hz (figure 2(b)).…”

“…We observed an overestimation of phase velocity comparing with that without top pavement structure. We then study the multi-valued dispersion relation of the single Rayleigh mode by calculating the secular function energy map based on the generalized reflection and transmission (R/T) coefficient method [7]. For special dispersion characteristic, we propose a new misfit function for inversion and quantify the bias due to ignoring the rigid pavement.…”