Spatial auto-correlation method

Figure 6.35: Examples of auto-correlation curves obtained for array A. The black dots and error bars are the samples selected according to criteria of figure 6.37.

Figure 6.36: Azimuth-inter-distance plot for array A: each dot represent one couple of stations. The pairs of grey circles show the limits of the chosen rings. Figure 6.37: Grid in frequency-slowness domain representing the density of dispersion curve solutions for array A. The plain and the dotted lines are the wavenumber limits deduced from the solution density grid. The dashed and the dot-dashed curves are the wavenumber limits of the apparent dispersion curve.

Figure 6.38: Inversion of the selected samples of 10 auto-correlation curves for array A. Only three of them are presented here. (a) $V_p$ , (b) $V_s$ of generated models and (c) corresponding dispersion curves. The dispersion curve observed for frequency wavenumber method is plotted for comparison in grey. (d) to (e) Auto-correlation curves for three rings with the observed ones (black dots and error bars).

This method is applied to the four arrays but only array A is shown here. Arrays B, C, and D are not shown here because the obtained auto-correlation curves are not consistent according to the test described in section 5.2.3. The array geometries are probably not well adapted for the auto-correlation method. For non circular arrays, the spatial auto-correlation method requires the definition of rings (Bettig et al. (2001)) and the ten rings chosen for array A are shown in figure 6.36. The auto-correlation curves are calculated using the method described in section 6.1. Three of the ten auto-correlation curves are presented in figure 6.35 (black and grey dots). The consistency of all 10 auto-correlation curves is checked in figure 6.37 with the grid method described in section 5.2.3. From 5 to 12 Hz, all rings are consistent with each other and a common dispersion curve is delineated by the dashed lines and the wavenumber limits (plain and dotted lines). The data outside of those limits are considered as incoherent and are discarded. They are marked with grey dots on the auto-correlation curves of figure 6.35. The data selected from the ten auto-correlation curves are inverted together with the neighbourhood algorithm as in section 6.1. Five runs are used with the same parameterization as for the frequency wavenumber method (table 6.8). The $V_s$ profiles are shown in figure 6.38. The three of the ten calculated auto-correlation curves are shown in figures 6.38(d) to 6.38(f) with the experimental black dots and the error bars shown in figure 6.35. The minimum misfit found is 0.65. This relatively high value is due to the residual inconsistencies between the ten auto-correlation curves. Above 8 Hz, both frequency wavenumber and auto-correlation methods give the same dispersion curve. If we assume that a misfit of one is a good threshold to select all models within the experimental uncertainties, this example shows that the final $V_s$ uncertainty range for all depth above 8 m is larger for the auto-correlation method than for the wavenumber method. For frequencies below 8 Hz, a huge gap is observed between the two methods, with much lower velocity estimates for the auto-correlation results. Moreover, the experimental auto-correlation curves are not correctly fit below 6 Hz. In figure 6.38(f), the experimental points below 8 Hz are located on the left side of the calculated curves, which correspond to an increase of the velocity and to a better agreement with f-k methods. On the contrary, in figure 6.38(e), fitting the experimental points below 6 Hz would imply an even lower velocity. The same type of deviation is observed for the high resolution method (figure 6.34(a)). This difference has a strong influence over the inversion as shown by figure 6.38(b).