Fundamental and first higher modes

To check these conclusions, the fundamental mode below 5 Hz and the first higher mode above 5 Hz are jointly inverted in figure 5.4. The black dots in figure 5.4(c) are the sample points of the inverted dispersion curve. The thin dotted lines are the theoretical dispersion curves of the unconstrained parts of the dispersion curves. These later ones do not show any special spreading of the calculated dispersion curves, proving that they contain redundant information.

Table 5.1: Parameterized model with a basement between 30 and 120 m. 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 20 m	200 to 2,000 m/s	0.01 to 0.707	2 t/m3
Sediments 2	30 to 120 m	+10 to 2,000 m/s	0.01 to 0.707	2 t/m3
Half-space	-	4000 to 5,000 m/s	0.65 to 0.68	2 t/m3

Table 5.2: Parameterized model with a basement between 15 and 30 m. 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 15 m	200 to 2,000 m/s	0.01 to 0.707	2 t/m3
Sediments 2	15 to 30 m	+10 to 2,000 m/s	0.01 to 0.707	2 t/m3
Half-space	-	4000 to 5,000 m/s	0.65 to 0.68	2 t/m3

In a real case, the fundamental dispersion curve is rarely available down to 0.2 Hz when the first peak of the ellipticity is at 5 Hz. Usually, one can expect to get a reliable dispersion curve only below 5 Hz, which is redundant with the high frequency part of the first higher mode. As a last example, the inversion of the narrow band dispersion curve shown in figure 4.7 is re-started adding the first higher mode as a supplementary constraint. Like the fundamental mode, the higher mode is rarely well defined at low frequencies. In this case, the first higher mode is supposed to be observed down to 9 Hz. Five runs are launched with the same parameterization as the inversion of figure 4.7. This parameterization contains very little prior information as reported by table 4.3. The majority of the models generated by the neighbourhood algorithm inside this parameter space have a $V_s$ below 1500 m/s down to 120 m which can give the illusion that the inversion with the first high mode really offers a better constraint. But three other inversions (two with the parameters of table 5.1 and one with table 5.2) are also run to force the generation of models with a high $V_s$ at shallow depths. The results displayed in figure 5.5 gather all the models of the eight runs. Comparing with figure 4.7, it clearly shows that the first higher mode does not provide any special information about deeper layers, because it is possible to find models with a very good misfit having almost any $V_s$ values below 10 or 15 m.

The same composite dispersion curve is also inverted with a prior depth information like in the inversion shown in figure 4.10 (not shown here). There is no significant improvement of the solution induced by the use of the first higher mode.

Figure 5.5: Inversion of the fundamental and the first higher mode: narrow band. (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 grey curves are the calculated fundamental mode (lowest curves) and the first higher mode (highest curves). The black dots are the theoretical dispersion curves used as the target curve during inversion