Prior information on $V_p$

$V_p$ profiles may also be measured by other means not related to surface wave properties. Refraction tests, borehole logging, cross-hole, ...may bring valuable information about $V_p$ . Like the depth, the prior information about $V_p$ is introduced in the parameterization itself. In the above sections, the $V_p$ profile is left as totally free in a very large interval. Here, we fix it in a deterministic way, removing $V_p$ from the parameter list. Table 4.5 details the remaining parameters. The dimension of the parameter space reduces from 8 to 5.

Table 4.5: Parameterized model for three-layer inversions with prior $V_p$ .

Layer	Thickness	$V_p$	$V_s$ /$V_p$	Density
Sediments 1	1 to 200 m	375 m/s	0.01 to 0.707	2 t/m3
Sediments 2	1 to 200 m	1750 m/s	0.01 to 0.707	2 t/m3
Half-space	-	4500 m/s	0.01 to 0.707	2 t/m3

Using the standard implementation of the neighbourhood algorithm, it is not possible to disconnect the depths of the $V_s$ and $V_p$ profiles. Hence, a real $V_p$ profile cannot be fixed without forcing the $V_s$ profile to have interfaces at the same depths. For this test, the depths of the $V_p$ profile are left as free parameters and they follow the depths of the $V_s$ profile. The conditional neighbourhood algorithm (section 2.4) would allow totally independent profiles for $V_s$ and $V_p$ . Consequently, the $V_p$ profile could be fixed without affecting directly the inversion of $V_s$ .

Figure 4.11: Inversion with a three-layer model with prior $V_p$ . (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.

Figure 4.12: Inversion with a three-layer model at high frequency with prior $V_p$ . (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.

The results are shown in figures 4.11 and 4.12 for a dispersion curve defined over a broad and a narrow frequency band, respectively (five distinct inversion processes in each case). The minimum misfit is around 0.002 for both cases. In figure 4.11, 31000 models have a misfit lower than 0.1 (23000 in figure 4.12), the threshold used to select model.

Comparing figures 4.6 and 4.11, the uncertainty of $V_s$ on the intermediate layer is greatly reduced, showing a direct effect of the fixing $V_{p0}$ . However, fixing $V_p$ has also an effect on the depth error of the deepest contrast. Other tests with wrong prior $V_p$ values show that the final $V_s$ results are weakly affected by over-estimated $V_p$ profiles. In contrast, any under-estimation of $V_p$ completely ruins the inversion of $V_s$ because the maximum of $V_s$ is automatically set to $\frac{\sqrt{2}}{2}V_p$ . This is why $V_p$ values can be fixed only when reliable data exist. Tests with and without the prior information must be carried out. When there is no pre-existing data about $V_p$ , the best option is to include it in the parameterization like in preceding section, with a very large prior interval.

The parameterization used for generating figure 4.12 is a particular case of the more general parameterization relating to figure 4.7. Hence, if the investigation of the parameter space was perfect for figure 4.7, all models appearing in figure 4.12 would be also generated by the inversion process illustrated in figure 4.7. Clearly, the introduction of reliable prior information about $V_p$ also makes the inversion more efficient leading to a better parameter space investigation. From figure 4.12, if the dispersion curve is known with a sufficient precision (acceptable misfit at 0.2), $V_{s1}$ can be determined with a precision of 200 m/s ($\approx$ 20%) down to 20 or 30 m. Without the $V_p$ information this uncertainty is greater than 200 m/s (case of figure 4.7).