Ellipticity

The H/V Method is a common tool used for site-effect investigations (Nogoshi and Igarashi (1970); Nakamura (1989); Bard (1998)). The horizontal (H) and vertical (V) components are simultaneously recorded at one single point. The ratio of H over V generally exhibits a peak, that corresponds more or less to the fundamental frequency of the site ( $f_0=\frac{V_s}{4h}$ , Bonnefoy (2004)). However, the ambient wavefield is composed of unknown parts of body and surface waves. In the first case, the ratio is mainly influenced by $S_H$ resonance in the superficial layers. On the other hand, if Rayleigh surface waves predominate, the theoretical ellipticity dictates the observed curves (Scherbaum et al. (2003); Nogoshi and Igarashi (1970); Fäh et al. (2003); Fäh (2001)). Real data peaks usually fit the frequency of the theoretical curves but the amplitude is rarely stable and reliable. Malischewsky and Scherbaum (2004) developed an analytical formulation for two-layer models. They plotted the differences of the peak frequency between the two aforementioned assumptions versus the magnitude of the velocity contrast. At intermediate and low contrasts (below a factor of 4 between $V_{s0}$ and $V_{s1}$ ), a drastic gap may exist between the two interpretations. In this case, Bonnefoy (2004) showed that the observed H/V peak better fits with the extrema of the $S_H$ transfer function.

H/V spectrum contains valuable information about the underlying structure, especially a particular relationship between $V_s$ of the sediments and their thickness (Scherbaum et al. (2003); Boore and Toksöz (1969)). Because the absolute amplitude of the curve cannot be directly compared to the amplitude of the $S_H$ transfer function or the ellipticity, only the frequency of the peak is considered here. Some preliminary tests showed that using ellipticity amplitude offers a very good constraint even on $V_p$ profile. However, wrong assumptions on the amplitude also lead to completely biased results. Nevertheless, Fäh et al. (2003) invert the amplitude between the peak and the trough by means of assumptions about the energy partition between Love, Rayleigh and body waves. This alternative has been discarded during our work. The next sections focused on how to calculate the ellipticity of Rayleigh waves and how to calculate the exact frequencies of the peaks.