Ellipticity inversion

The principles and the solutions developed for the inversion of the Rayleigh ellipticity are discussed in section 3.2. The ellipticity shown in figure 4.1(d) is first inverted alone with a simple model made of one layer overlaying an infinite half-space. The shape of the ellipticity curves is not inverted but only the frequency of peak, which is exactly found at 5.63 Hz. The secondary peak at 3 Hz is not considered here. It is not possible to retrieve a complete ground structure only from the frequency of the ellipticity peak. Hence, a model with only two parameters (thickness and $V_{s0}$ ) is used in the inversion, detailed in table 5.5. The

value in the half space is fixed to ensure a constant

of 1000 m/s. Five runs are launched with ten iterations each generating a total of 5500 models. The minimum misfit achieved is 0, because only one single frequency is fit with a precision of $10^{-3}$ Hz. The results are shown in figure 5.14. A clear relationship between the thickness and $V_{s0}$ is found corroborating the conclusions of Scherbaum et al. (2003) about the inversion of the frequency of the ellipticity peak for a two-layer model. The theoretical model has a

of 10 m and $V_{s0}$ is 200 m/s.

Table 5.5: Parameters for ellipticity alone inversion.

Layer	Thickness			Density
Sediments	1 to 50 m	375 m/s	0.01 to 0.707	2 t/m3
Half-space	-	1750 m/s	0.57143	2 t/m3

**Figure 5.14:** Inversion of the ellipticity alone showing the trade-off between the depth of the velocity contrast and .
r0cm $\includegraphics{fig_chapenhanced/ell_alone.eps}$

The last example confirmed that the frequency of the ellipticity peak contains pertinent information about the thickness and the shear velocity of the first layer. The ellipticity target is then added to a usual dispersion curve inversion to test its ability to improve the final solution. The case of a narrow frequency band and a two-layer parameterization inverted in figure 4.5 is utilized again. As detailed in section 3.2, the misfit is calculated by a weighted sum of the dispersion and the ellipticity misfits. 10 and 90 % weights were chosen for the dispersion and the ellipticity misfits, respectively. This ensures that nearly all generated model are complying with an ellipticity peak at 5.63 Hz. Consequently, to achieve a comparable good fit of the dispersion curve as in figure 4.5, the misfit scale is divided by 10. Five runs are launched with the parameters described in table 4.2. To make sure that the parameter space is sufficiently sampled in terms of

(depth of the top of the half space), two more inversion processes are started with the depth restricted to

m and

m, respectively. The ensemble of all models with a misfit less than 0.01 is plotted in figure 5.15.

**Figure 5.15:** Join inversion of the dispersion curve and the ellipticity peak. (a) Resulting profiles. (b) Resulting profiles. The black lines are the theoretical velocity profiles. (c) Dispersion curves corresponding to models of figures (a) and (b). The black dots are the theoretical dispersion curves used as the target curve during inversion. (d) Ellipticity curves calculated for models of figures (a) and (b). The black dots are the theoretical ellipticity curve but only the frequency of the main peak is used as the inversion target.
$\includegraphics{fig_chapenhanced/ell_disp.eps}$

Compared to figure 4.5, the posterior error obtained for the depth of the basement interface is greatly reduced. According to the level of confidence put into the dispersion curve, the depth is known with a one-metre precision whereas the uncertainty in figure 4.5 is much greater. However the velocity in deeper layer is not retrieved as in the first inversion. Tests were also conducted with a three-layer parameterization but no significant improvement has been observed. Other inversions could have been started with the low frequency secondary peak appearing in figure 4.1(d) at 3 Hz, but there are few chances for this peak to be detected with a real experiment.