Two-layer model

Figures 3.8 and 3.9 show the influence of $V_{s0}$ in the case of a fixed

profile and of a constant Poisson's ratio, respectively. Love and Rayleigh dispersion curves (

) are plotted with plain and dotted lines, respectively. The models and their corresponding dispersion curves are represented by distinct grey levels. In figure 3.8, only $V_{s0}$ is changing from 100 to 1900 m/s. Poisson's ratio varies as well because $V_{p0}$ is held constant (written on the right). In figure 3.9, only the variations of $V_{s0}$ are represented but $V_{p0}$ is also changing for all models to keep a constant Poisson's ratio being 0, 0.25 and 0.45 in figure (a) to (c), respectively. Love and Rayleigh curves are monotonously decreasing with at least one inflexion point. The first derivative of Love curves has always one minimum. For Rayleigh curves, two minima and a maximum may exist in the first derivative, especially for moderate to high Poisson's ratios.

of the first layer changes the limit of the curves at high frequency. The limit at low frequency is not influenced by the properties of the superficial layer. The lower is Poisson's ratio and the higher is $V_{s0}$ , bigger is the difference between Love and Rayleigh dispersion curves at high frequency, in accordance with equation (3.36) for a half-space.

**Figure 3.8:** Influence of $V_{s0}$ with a constant profile. Rayleigh and Love fundamental modes are represented by plain and dotted lines, respectively. The values on the right are the Poisson's ratios corresponding to Rayleigh curves. $V_{s0}$ varies from 100 to 1900 m/s. $V_{p0}$ is 2687 m/s. $V_{s1}$ is 2000 m/s. Poisson's ratio is 0.25 below 50 m. The density is 2 t/m at all depths.
$\includegraphics{fig_chapdispcurve/2layers_vs0-nu.eps}$

**Figure 3.9:** Influence of $V_{s0}$ with a constant Poisson's ratio: (a) $\nu$ =0, (b) $\nu$ =0.25, and (c) $\nu$ =0.45. Rayleigh and Love fundamental modes are represented by plain and dotted lines, respectively. $V_{s0}$ varies from 100 to 1900 m/s. $V_{s1}$ is 2000 m/s. Poisson's ratio is 0.25 below 50 m. The density is 2 t/m at all depths.
$\includegraphics{fig_chapdispcurve/2layers_vs0-nucst.eps}$

**Figure 3.10:** Influence of $V_{p0}$ on the Rayleigh dispersion curve for two cases: (a) $V_{s0}$ =200 m/s, and (b) $V_{s0}$ =1000 m/s. Poisson's ratio varies from 0 (dark) to 0.45 (light). Hence, $V_{p0}$ varies from 283 to 663 m/s (case (a)), and from 1414 to 3316 m/s (case (b)). $V_{s1}$ is 2000 m/s. Poisson's ratio is 0.25 below 50 m. The density is 2 t/m at all depths.
$\includegraphics{fig_chapdispcurve/2layers_vp0.eps}$

**Figure 3.11:** Influence of . Rayleigh and Love fundamental modes are represented by plain and dotted lines, respectively. $V_{s0}$ is 200 m/s. $V_{s1}$ is 2000 m/s. Poisson's ratio is 0.25 at all depths. The density is 2 t/m at all depths.
$\includegraphics{fig_chapdispcurve/2layers_h0.eps}$

In the Rayleigh case, the influence of $V_{p0}$ is checked in figure 3.10 for two distinct $V_{s0}$ values (200 and 1000 m/s). In both cases, Poisson's ratio varies from 0 (dark grey) to 0.45 (light grey).

profiles are shown in the small figure on the left. For case (a), it varies between 280 and 660 m/s, and between 1400 and 3300 m/s for case (b).

has apparently an impact on the dispersion curve when Poisson's ratio is less than a threshold (around 0.27 for case (a) and 0.37 for case (b)) which depends upon $V_{s0}$ . Above this threshold,

looses its influence. For case (a), only three curves are well individualized, those corresponding to

less than 400 m/s. This conditional dependency explains that, in most cases, only a minimum of

can be retrieved from the inversion of dispersion curves (section 4.2).

**Figure 3.12:** Influence of for profile. Rayleigh and Love fundamental modes are represented by plain and dotted lines, respectively. $V_{s0}$ is 200 m/s. $V_{s1}$ is 2000 m/s. Poisson's ratio is (a) 0.00 and (b) 0.25 above 50 m and below 75 m. The density is 2 t/m at all depths. The results for a Poisson's ratio of 0.45 are quite similar to those of 0.25 (not shown).
$\includegraphics{fig_chapdispcurve/2layers_hs0-00-25.eps}$

In figure 3.11, the influence of the thickness of the first layer is tested.

and

profiles are both modified by this parameter. Love and Rayleigh dispersion curves are translated in the same way when the depth is reduced. As

or Poisson's ratio only changes the shape of Rayleigh curves, it is likely that the effects of the thickness are mainly due to the modification of

profile rather than

profile. This is tested hereafter with figures 3.12 and 3.13.

In figure 3.12,

profile is held constant where

profile varies from 50 to 75 m. The same translation as in the general case is observed. For low Poisson's ratios, the velocity at 1.25 Hz is not affected by the changing depth. The third of the wavelength, a common rule of the thumb in surface wave analysis to map frequency scales to depth scales (Tokimatsu (1997)), is about 95 m at 1.25 Hz.

In figure 3.13,

profile is held constant. Rayleigh dispersion curve is nearly not influenced except for low Poisson's ratio. For some other cases with higher Poisson's ratios (not shown here), the only affected part of the dispersion is the curvature close to the maximum Rayleigh velocity. Uncoupling depth limits of

and

is one of the perspectives offered by the conditional neighbourhood algorithm (section 2.4).

**Figure 3.13:** Influence of for profile. Rayleigh and Love fundamental modes are represented by plain and dotted lines, respectively. $V_{s0}$ is 200 m/s. $V_{s1}$ is 2000 m/s. Poisson's ratio is (a) 0.00 and (b) 0.25 above 25 m and below 50 m. The density is 2 t/m at all depths. The results for a Poisson's ratio of 0.45 are quite similar to those of 0.25 (not shown).
$\includegraphics{fig_chapdispcurve/2layers_hp0-00-25.eps}$

The density of the first layer has a low influence on the dispersion curves (either Love or Rayleigh) as shown by figure 3.14. The density is changed from 1 to 3 t/m

with $V_{s0}$ being 200 and 1000 m/s, case (a) and (b), respectively. The effects clearly depend upon $V_{s0}$ . $V_{s1}$ is the same for both cases, hence the velocity contrast is also modified between (a) and (b). The effect of the density is only visible for case (b). In case (a), the density has almost no influence except around 1 Hz. Only the shape is modified, not the low and high frequency limits. The considered interval (from 1 to 3 t/m

) is probably larger than usual prior uncertainty on density. Hence, this parameter is generally fixed to a constant value during inversions of dispersion curve (section 4.2).

**Figure 3.14:** Influence of $\rho _0$ for two cases: (a) $V_{s0}$ =200 m/s, and (b) $V_{s0}$ =1000 m/s. Rayleigh and Love fundamental modes are represented by plain and dotted lines, respectively. $\rho _0$ varies from 1 to 3 t/m . $V_{s1}$ is 2000 m/s. Poisson's ratio is 0.25 at all depths.
$\includegraphics{fig_chapdispcurve/2layers_rho0.eps}$

**Figure 3.15:** Influence of $V_{s1}$ with a constant profile. Rayleigh and Love fundamental modes are represented by plain and dotted lines, respectively. The values on the left are the Poisson's ratios corresponding to Rayleigh curves. $V_{s1}$ varies from 300 to 2000 m/s. $V_{s0}$ is 200 m/s. Poisson's ratio is 0.25 above 50 m. The density is 2 t/m at all depths.
$\includegraphics{fig_chapdispcurve/2layers_vs1-nu.eps}$

**Figure 3.16:** Influence of $V_{p1}$ on the Rayleigh dispersion curve for two cases: (a) $V_{s1}$ =500 m/s, and (b) $V_{s1}$ =2000 m/s. Poisson's ratio varies from 0 (dark) to 0.45 (light). $V_{s0}$ is 200 m/s. Poisson's ratio is 0.25 above 50 m. The density is 2 t/m at all depths.
$\includegraphics{fig_chapdispcurve/2layers_vp1.eps}$

**Figure 3.17:** Influence of $\rho _1$ for two cases: (a) $V_{s1}$ =500 m/s, and (b) $V_{s1}$ =2000 m/s. Rayleigh and Love fundamental modes are represented by plain and dotted lines, respectively. $\rho _1$ varies from 1 to 3 t/m . $V_{s0}$ is 200 m/s. Poisson's ratio is 0.25 at all depths.
$\includegraphics{fig_chapdispcurve/2layers_rho1.eps}$

The same sensitivity analysis is carried out for the parameters of the bottom half-space. The influence of $V_{s1}$ is estimated in figure 3.15.

profile is held constant. $V_{s1}$ and Poisson's ratio in the bottom half space vary from 300 to 2000 m/s, and from 0.5 to 0, respectively. It acts exactly like $V_{s0}$ replacing high by low frequencies and vice-versa. The difference between Love and Rayleigh curves at low frequency increases like $V_{s1}$ , and it is maximum for Poisson's ratio equal to 0. Above 2 Hz, no effect can be observed. Poisson's ratio has a little effect on the shape of the dispersion between the low and high frequency limits. The magnitude of the effect is much smaller than the effect of superficial Poisson's ratio.

To corroborate this observation, the effect of $V_{p1}$ alone is measured in figure 3.16. $V_{s1}$ is fixed to a constant value equal to 500 m/s and 2000 m/s, for cases (a) and (b), respectively. In a similar way as for $V_{p0}$ , all curves appear to be merged together for all $V_{p1}$ greater than a particular threshold (around 4000 m/s for case (b)).

Finally, the influence of the density is shown in figure 3.17. Two cases are chosen with $V_{s1}$ fixed to 500 and 2000 m/s, noted by (a) and (b), respectively. The density varies from 1 to 3 t/m

. Comparing figures 3.14 and 3.17, the densities of the first layer and of the half-space do not affect the dispersion in the same way. The last one reduces the apparent velocity when the density decreases. Like the density of the superficial layer, the interval of variation is probably larger than the prior uncertainties. Hence, the effects of $\rho$ are generally negligible.