Two layers

The shape of the fundamental Rayleigh dispersion curve in figure 4.1(b) has a complex shape. However, we first test if it is possible to invert it with a simple model made of one layer overlying a half space. The curve is resampled with 50 points regularly distributed on a log frequency scale. The utilized parameters are detailed in table 4.2.

Table 4.2: Parameterized model for two-layer inversions. The "+" sign stands for incremental velocity: the parameter is the velocity gap between the first and the second layer.

Layer	Thickness	$V_p$	$V_s$ /$V_p$	Density
Sediments	1 to 200 m	200 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

Figure 4.2: Inversion of the full dispersion curve with a two-layer mode. (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 neighbourhood algorithm is tuned to be as exploratory as possible, generating 100 models per iteration ($n_s$ ) in the current 100 best cells ($n_r$ ). Runs of 50 iterations are started with five distinct random seeds (chapter 2) to test the robustness of the results. These parameters are usually adjusted by trial and error. The dimension of the parameter space is 5. Each individual process generates an ensemble of 5100 possible solutions ranked by their misfit values. The results of the inversion are shown in figure 4.2 in terms of velocity profiles. Only the models with a misfit less than 0.1 are selected. The shape of the dispersion curve at low frequency (figure 4.1) is obviously too complex to be correctly inverted with a simple model made of two layers. A more complex structure has to be assumed in order to invert the dispersion curve between 0.2 and 20 Hz (section 4.2.2). However, the $V_s$ profile below 8 m is well retrieved. The shapes of the reference and the calculated dispersion curve at high frequency (above 5 Hz) are similar. The low frequency part of the curve prevents the misfit from being improved and it influences the error on the depth and on $V_p$ . In the next paragraph, better results can be achieved by considering only the dispersion curve at high frequency.

The fundamental Rayleigh dispersion curve between 5.5 and 15 Hz, resampled with 30 points regularly distributed on a log frequency scale and described in section 4.1 (figure 4.1) is inverted in the same conditions as above. Figure 4.3 shows the minimum misfit evolution with the number of generated models. The curve is never regular as already noticed by Sambridge (1999a). But in general, the variations are progressively damped if the number of generated models is sufficient.

Figure 4.3: Inversion with a two-layer model: variation of the minimum misfit with the inversion advance. The five curves correspond to the five inversion processes initiated.

l0cm

In figure 4.4, each generated model is represented by a dot with a grey scale depending on the misfit value. Figure 4.4(a) is a projection of the five dimension parameter space on the plane $z_1-V_{s0}$ , while the other plots (figures 4.4(b) to 4.4(f)) show the one-dimensional variation of each parameter versus the misfit value. The minimum achieved misfit is around 0.01. All generated models are plotted in figure 4.4(a). The shapes for lowest misfit values in figures 4.4(b) to 4.4(f) give valuable information about the posterior marginal uncertainties of one parameter. For instance, accepting a level of error on the experimental curve of 0.05, all values of $V_{p0}$ between 300 and 2000 m/s ensure a good fit of the data curve. $V_{p0}$ (theoretical value is 375 m/s) is better resolved only if misfits below 0.03 are considered. In figures 4.4(d) to 4.4(f), it is clear that the inversion algorithm is not exploratory enough to sample the whole parameter space for poorly resolved parameters.

Figure 4.4: Inversion with a two-layer model: parameter space. (a) Projection of model points on the plane $z_1-V_{s0}$ . One-dimensional marginal for (b) $z_1$ , (c) $V_{s0}$ , (d) $V_{s1}$ , (e) $V_{p0}$ , and (f) $V_{p1}$ .

For well constrained parameters ($z_1$ and $V_{s0}$ ), the results are approximately the same for all runs. For other parameters, each additional run may brought some new solutions, improving the global sampling of the parameter space. Even with only five parameters, the complexity of the parameter space is such that an exhaustive sampling would be prohibitive.

The results of the inversion are shown in figure 4.5 in terms of velocity profiles. Only the models with a misfit less than 0.1 are selected ($\approx$ 25000 models).

Figure 4.5: Inversion with a two-layer model: velocity profiles. (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.

Retrieved $V_p$ and $V_s$ profiles are visible in figures 4.5(a) and 4.5(b). The black lines are the theoretical velocity profiles. The dispersion curve calculated for profiles of figures 4.5(a) and 4.5(b) are shown in figure 4.5(c) where the black dots are the simulated experimental curve defined on a restricted range (section 4.1). According to the level of confidence on the experimental curve, darkest models may be discarded. The lightest models (misfit $<$ 0.3) fit nicely with the theoretical model except for $V_p$ within the basement. $V_s$ is well retrieved for the first 8 m whereas a wide range of $V_p$ values may explain the observed dispersion curve. Even for $V_s$ , the uncertainties greatly increase from 8 m, below the depth of the velocity contrast. However, if the dispersion is known with a very good confidence and a good precision, $V_{p0}$ can be correctly estimated because it is not possible to find any model with $V_{p0}>500 m/s$ and a misfit below 0.03.

A common solution to improve the precision for deeper structure is to enlarge the frequency range of the dispersion curve. For a two-layer parameterization, broader frequency ranges lead to badly resolved structures with a minimum achievable misfit above 0.1 (figure 4.2). Hence, it is not possible to find an equivalent two-layer model for the more complex soil structure. In a real situation, when a two-layer parameterization gives worse results than a more complex parameterization, it is a piece of evidence that the structure is probably not simply made of homogeneous sediments overlying a hard-rock basement. In the next section, a three-layer parameterization is used and the influence of the frequency range is checked.