Sensitivity

For a two-layer model the influence of $V_{s0}$ is shown in figures 3.21 and 3.22, for a constant $V_p$ profile and a fixed Poisson's ratio, respectively. Hence, the ellipticity of a two-layer model has in most cases a root (at 1 Hz for the darkest curve) and a singular point (the maximum at 0.5 Hz for the darkest curve) but it is not always true as demonstrated in figure 3.22. Even for a two-layer model, a secondary maximum may be encountered (at 0.8 and 2.2 Hz in figure 3.22). There is always one frequency band (narrow or large) where the ellipticity is maximum.

When the number of layers increases, several singularities are sometimes observed but it is not a constant feature. Figure 3.23 illustrates the variation of the ellipticity with $V_s$ of the intermediate layer for a three-layer case. At high frequency for the darkest model (with $V_{s1}$ being 100 m/s), usual precision is not sufficient to achieve a correct computation of the ellipticity curve. An experimental algorithm with high precision arithmetics has been developed for this particular case. A striking feature of the ellipticity curve of the two darkest models of figure 3.23, both having a thin hard ground at the surface, is that the ellipticity ratio at high frequency does not tend to the value predicted by equation (3.43). All other models follow equation (3.43) at high frequency. Physically, this could be explained by the trapping of energy within the intermediate layer which alters the classical development of surface waves.

Figure 3.21: Influence of $V_{s0}$ with a constant $V_p$ profile. $V_{s0}$ varies from 100 to 1900 m/s. $V_{p0}$ is 2687 m/s hence, Poisson's ratio varies from 0.499 (dark) to 0 (light) like in figure 3.8. $V_{s1}$ is 2000 m/s. Poisson's ratio is 0.25 below 50 m. The density is 2 t/m$^3$ at all depths.

Figure 3.22: Influence of $V_{s0}$ with a constant Poisson's ratio. $V_{s0}$ varies from 100 to 1900 m/s. $V_{s1}$ is 2000 m/s. Poisson's ratio is 0.25 and the density is 2 t/m$^3$ at all depths.

Scherbaum et al. (2003) showed for a two-layer model that an inversion of the frequency of the main peak can bring valuable information. The generalisation to $n$ layers is not straightforward because the shape of the ellipticity curve appears to be very sensitive to the model parameters. In this context, the determination of the frequency of the main peak is not univocal in all cases even for the simplest models. Also, the inversion of the absolute amplitude of the experimental H/V curves with the Rayleigh fundamental ellipticity in the general case of $n$ layers may not be reliable. If experimental H/V curves may present several peaks, there is no strong evidence of a relationship between those real peaks and the various peaks of the fundamental Rayleigh curve. The ellipticity of the higher modes or body wave resonance may be also suspected. Without a clear agreement on the physical model to explain multiple peaks of the experimental H/V curves, a conservative option, detailed in the next section, is kept to avoid the introduction biased prior information.

Figure 3.23: Influence of $V_{s1}$ with a constant Poisson's ratio. $V_{s1}$ varies from 100 to 2500 m/s. $V_{s0}$ =200 m/s. $V_{s2}$ =2000 m/s. Poisson's ratio is 0.25 and the density is 2 t/m$^3$ at all depths.