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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 k(l)(omega) for the lth Rayleigh mode to be a single-valued function of frequency.. 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.
The resonance frequency of the transmission response in layered half-space model is important in the study of site effect because it is the frequency where the shake-ability of the ground is enhanced significantly. In practice, it is often determined by the H/V ratio technique in which the peak frequency of recorded H/V spectral ratio is interpreted as the resonance frequency. Despite of its importance, there has not been any formula of the resonance frequency of the layered half-space structure. In this paper, a simple approximate formula of the fundamental resonance frequency is presented after an exact formula in explicit form of the response function of vertically SH incident wave is obtained. The formula is in similar form with the one used in H/V ratio technique but it reflects several major effects of the model to the resonance frequency such as the arrangement of layers, the impedance contrast between layers and the half-space. Therefore, it could be considered as an improved formula used in H/V ratio technique. The formula also reflects the consistency between two approaches of the H/V ratio technique based on SH body waves or Rayleigh surface waves on the peak frequency under high impedance contrast condition. This formula is in explicit form and, therefore, may be used in the direct and inverse problem efficiently. A numerical illustration of the improved formula for an actual layered half-space model already investigated by H/V ratio technique is presented to demonstrate its new features and its improvement to the currently used formula.