Broad band dispersion curve

The dispersion curve is sampled with 50 points regularly distributed on a log frequency scale and on a wide frequency range from 0.2 to 20 Hz. The parameters are of the same type as for the preceding case with one supplementary layer. Table 4.3 gives the list of parameters and their prior intervals. The $V_p$ profile is imposed to be monotonously increasing by setting positive velocity variations as parameters rather than the absolute value. $V_s$ is kept monotonous by the penalization technique (introdution of section 4.2) on the low velocity zones.

Table 4.3: Parameterized model for three-layer inversions. The "+" sign stands for incremental velocity: the parameter is the velocity gap between the first and the second layer.

Layer	Thickness	$V_p$	$V_s$ /$V_p$	Density
Sediments 1	1 to 50 m	200 to 2,000 m/s	0.01 to 0.707	2 t/m3
Sediments 2	1 to 200 m	+10 to 2,000 m/s	0.01 to 0.707	2 t/m3
Half-space	-	+10 to 3,000 m/s	0.01 to 0.707	2 t/m3

Five independent runs are started with $n_s$ (number of samples per iteration) and $n_r$ (number of cells to resample) being 100. The number of iterations is set arbitrarily to 150. The evolution of the minimum misfit with the number of generated models (not shown) finally proves that values for the tuning parameters are necessary and sufficient. The total number of generated model is hence 75500, with a minimum misfit around 0.02. The $V_p$ and $V_s$ profiles of models (8900) for which the misfit is less than 0.1 are plotted in figures 4.6(a) and 4.6(b), respectively. The corresponding dispersion curves are shown in figure 4.6(c).

Figure 4.6: Inversion with a three-layer model over a broad frequency range. (a) Resulting $V_p$ profiles. (b) Resulting $V_s$ 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 curve used as the target curve during inversion.

On the first ten metres, the inverted profiles are very similar to those obtained with the two-layer parameterization. The velocities of the basement are also relatively well retrieved (below 100 m). The posterior uncertainties of the intermediate layer are higher than the one of the first layer, mainly because of the low sensitivity of the dispersion curve to the intermediate layers (section 3.1.8 on page ). Though Poisson's ratio is left totally free, the uncertainties on $V_p$ and $V_s$ of the intermediate layer are of the same order. The uncertainty on the depth determinations are always high even for the first interface at ten metres (errors up to nearly 40%^4.1). A precise inversion of the depths is possible but requires a very high precision on the dispersion curve.

This case is theoretical. During real experiments, the dispersion is not defined down to 0.2 Hz if the resonance frequency (given by the main peak of the ellipticity or of the measured H/V) is around 5.5 Hz. The effect of such limitation is tested in the next section.