Arbitrary profile

The model is made of 11 layers with one parameter per layer ($V_{sp}$ , the ratio of $V_s$ over $V_p$ ). Table 4.6 summarizes the properties of each layer.

Table 4.6: Parameterized model for N-layer inversions.

Layer	Depth	$V_p$	$V_s$ /$V_p$	Density
0	2 m	375 m/s	0.01 to 0.707	2 t/m3
1	5 m	375 m/s	0.01 to 0.707	2 t/m3
2	10 m	375 m/s	0.01 to 0.707	2 t/m3
3	18 m	1750 m/s	0.01 to 0.707	2 t/m3
4	30 m	1750 m/s	0.01 to 0.707	2 t/m3
5	47 m	1750 m/s	0.01 to 0.707	2 t/m3
6	70 m	1750 m/s	0.01 to 0.707	2 t/m3
7	100 m	1750 m/s	0.01 to 0.707	2 t/m3
8	138 m	4500 m/s	0.01 to 0.707	2 t/m3
9	185 m	4500 m/s	0.01 to 0.707	2 t/m3
Half-space	-	4500 m/s	0.01 to 0.707	2 t/m3

Poisson's ratios are totally independent and $V_s$ profiles might be generated with eventually various LVZs. The inversion is started with five distinct random seeds. The number of new models per iteration is 100 ($n_s$ ) and the number of cells resampled is 100 ($n_r$ ). 150 iterations are successively performed to obtain a total of 75,500 models. The results are shown in figure 4.13. The minimum misfit is around 0.005.

Figure 4.13: Inversion with a N-layer model accepting LVZ ($V_s$ only). (a) Resulting $V_s$ profiles. The black lines are the theoretical velocity profiles. (b) Dispersion curves corresponding to models of figure (a). The black dots are the theoretical dispersion curve used as the target curve during inversion.

The $V_s$ profiles in figure 4.13(a) can be directly compared with figure 4.11(b) also obtained with a fixed $V_p$ profile and on the same dispersion curve. The presence of LVZs slightly increases the non-uniqueness of the problem. The effect of a very slow layer may be thwarted when overlying a faster layer.

In figure 4.13, the fixed $V_p$ profile prevents from generating a number of additional models. Another inversion case is then proposed with varying $V_p$ and $V_s$ profiles. The range of $V_p$ values inside each layer is set to $[200, 6000]$  m/s. The inversion is started with 20 distinct random seeds to obtain a total of 202,000 models. The results are shown in figure 4.14. The minimum misfit is around 0.012. 90,000 models have misfit less than 0.1.

Figure 4.14: Inversion with a N-layer model accepting LVZ ($V_p$ and $V_s$ ). (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 figure (a) and (b). The black dots are the theoretical dispersion curve used as the target curve during inversion.

The $V_s$ profiles in figure 4.14(b) can be directly compared with figure 4.6(b) obtained on the same dispersion curve. In this case, the presence of LVZs drastically increases the non-uniqueness of the problem. From figure 4.14, no information can be retrieved between 10 m and 185 m. By contrast, figure 4.6 shows for the same dispersion curve that interesting information can be extracted by assuming that no LVZ are present. However, in figure 4.6, the velocities just below 10 m and just above the contrast around 100 m must be the same, which is probably too restrictive. In the next sections, various approaches are proposed to allow velocity variations inside layers and avoiding LVZs with the standard neighbourhood algorithm. However, a simpler solution can be implemented with the conditional neighbourhood algorithm.