Inversion

A two-layer model is considered with the parameter ranges specified in table 5.4. In the shallow layer, the velocity can increase with a power law relation, and the parameters are four ($V_p$ , $V_s/V_p$ , the thickness and the $V_p$ increase between the top and the bottom). The constant velocity layer corresponding to the true model is a particular realization of the parameterization. The bedrock parameters are two ($V_p$ increase, and $V_s/V_p$ ). The neighbourhood algorithm has been started using 3 independent runs with distinct random seeds, generating a total of 30,000 models. Among them about 13,500 have a misfit less than 1 and are plotted in figure 5.13. The lowest misfit is 0.03.

Table 5.4: Parameters for auto-correlation inversion. The ``+'' sign stands for incremental velocity: the parameter is the velocity gap between the first and the second layer. The power law gradient across the first layer is represented by a stack of 5 sub-layers. The value of the parameter is the total velocity variation across the layer.

Layer	Thickness	$V_p$	$V_s/V_p$	Density	$V_p$ variation
Sediments	10 to 50 m	200 to 2,000 m/s	0.01 to 0.707	2 t/m3	10 to 1,000 m/s
Half-space	-	+10 to 3,000 m/s	0.01 to 0.707	2 t/m3	-

Figure 5.13: Inversion of the auto-correlation curves. (a) $V_p$ profiles: true average model (plain line), true standard deviations (dotted lines) and inverted models ranked by their auto-correlation misfit (common grey scale). (b) $V_s$ Profiles: same legend as for $V_p$ . (c) Dispersion curves of generated models. (d) to (h) Auto-correlation ratios for chosen rings plotted against frequency, average and standard deviation of data points to be fitted (dots).

The $V_s$ and $V_p$ models resulting from the auto-correlation inversion are plotted in figures 5.13(a) and 5.13(b) with their misfit value. On these figures, is drawn the theoretical model of figure 5.11. Most of the solutions with a misfit lower than 0.4 are able to explain in a consistent way the auto-correlation data given their standard deviations (figures 5.13(a) and 5.13(b)). In figure 5.13(c) are plotted the corresponding dispersion curves. The $V_s$ profile (figure 5.13(b)) is very well constrained from 6 to 20 metres deep. The very superficial layers (less than 6 m) are at a depth lower than one third of the minimum wave length (20 m) and $V_s$ values are less constrained, resulting from the limited bandwidth at high frequency. Below 35 metre, $V_s$ values are well retrieved due to the wide low frequency range of the auto-correlation curves. In real data, this well constrained velocity in the bedrock is usually missing due to the site high-pass filter of the Rayleigh waves below the fundamental frequency (Scherbaum et al. (2003), chapter 6). The dispersion curves computed for the best fitting models compare very well with the theoretical one (figures 5.11(c) and 5.13(c)). The resolution is relatively poor between 22 m and 35 m: a velocity jump at 22 metres gives a misfit value equivalent to the one for a contrast at 35 metres. Other inversion tests (not presented here) have shown that this lack of resolution results from the uncertainties considered on the auto-correlation data. However, the lowest misfit model correctly finds an interface around 25 m depth.

Usually, $V_p$ has a low influence on the dispersion curve, and hence on the auto-correlation curves. Boore and Toksöz (1969) proved for a five-layer model that the influence of $V_p$ on the dispersion curve is about one tenth the influence of $V_s$ . However, for low Poisson's ratios, $V_p$ has more influence. In this latter situation, the final $V_s$ profile depends upon the correctness of the $V_p$ profile. In classical iterative inversions (least-square scheme), $V_s/V_p$ or Poisson's ratio is kept constant because the small influence of $V_p$ on the auto-correlation curves generally leads to unrealistic velocities. For the neighbourhood algorithm inversions, the parameterization is easily adjusted to fit the physical limits of $V_p$ and the prior information, for instance, about the superficial values of $V_p$ . When no information is available about $V_p$ , it is still used as a parameter with large prior intervals to prevent from altering the final result with unreliable assumptions. For this inversion test, we assumed that no prior information exists on $V_p$ . As the Poisson's ratio for the theoretical model is 0.49, the compressional-wave velocity ($V_p$ ) profile is badly recovered. Equivalent models are found for the whole prior $V_p$ range (from 200 to 2000 m/s in the upper layer).