Prior information on depth

If the depth of any particular velocity contrast is known from other investigations like a reference borehole or a penetration test, it can be introduced in the parameterization. Such a test is performed on the same dispersion curve as in figure 4.6 with the parameters defined in table 4.4. The depth is supposed to be known with an error of 5 m.

Table 4.4: Parameterized model for three-layer inversions with prior depth. The "+" sign stands for incremental velocity: the parameter is the velocity gap between the first and the second layer.

Layer	Depth	$V_p$	$V_s$ /$V_p$	Density
Sediments 1	1 to 90 m	200 to 2,000 m/s	0.01 to 0.707	2 t/m3
Sediments 2	95 to 105 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 runs are launched generating the models displayed in figure 4.9. The misfit values can be compared directly to the ones of figure 4.7 because the dispersion samples used to calculate them are exactly the same. Reducing the depth prior interval has obviously a positive influence in the inversion process. The main effect is to reduce the uncertainty of the velocities of the intermediate layer.

Figure 4.9: Inversion with a three-layer model with prior depth. (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 4.10: Inversion with a three-layer model at high frequency with prior depth. (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.

In figure 4.9, the improvement of the posterior uncertainty may be due to the strong constraint on the large band dispersion curve. The same parameterization is also tested on the dispersion curve with a narrow frequency band as in figure 4.7. The results are shown in figure 4.10. Forcing the depth of the basement indisputably allows a better retrieval of the velocity in the second layer below 10 m. However, the parameterized model made of three uniform layers imply that the velocity has a constant profile between 10 and 100 m. Stating that the velocity profiles are correctly measured down to 100 m is certainly false. The results at 100 m are influenced by the constraints on $V_s$ between 10 and 25 m. Inversions with one or more supplementary degrees of freedom must be carried out to define the total penetration depth of the method.

In conclusion, any prior information about the depths of the known velocity contrasts help the inversion of the dispersion curves even for incomplete ones. Like any other information source, its reliability must be ensured and the length of the fixed depth interval set according to the data source confidence.