High resolution method

For the three arrays A, B, and C, the dispersion curves have been calculated by searching the maximum of the high-resolution frequency wavenumber estimator defined by Capon (1969) and Ohrnberger et al. (2004). The estimator depends upon the cross spectral matrix averaged over the 6 minutes of available signals. The results are shown in figures 6.12(a) to 6.12(c), for arrays A, B, and C, respectively. The limits $k_{min}$ and $k_{max}/2$ validated for the f-k method are shown in grey. Theoretically, the resolving power of the high-resolution method should be better than the f-k method, and estimates of velocity may be reliable even outside those restrictive limits. From the observation of the stability of the high-resolution results and the comparison with the theoretical dispersion curve, we define apparent limits of the high-resolution valid for this particular case (table 6.3). This task is not possible for a real experiment. From a careful examination of figures 6.9 and 6.12, the high resolution method provides correct answers below $k_{min}$ , extending the frequency range by approximately 0.5 Hz. The poor resolution of array A between 6 and 9 Hz is not significantly improved by the high resolution approach. At high frequency, array B gives nearly perfect results up to its $k_{max}$ , in contrast with array C which shows a lot of instabilities above $k_{max}/2$ . After selecting the points between the validity curves, an average dispersion curve is calculated to feed the inversion algorithm.

Table 6.3: For each array, the minimum and maximum wavenumbers deduced from the comparison of the high resolution results to the theoretical dispersion curve (rad/m). Also are given the minimum and maximum frequencies corresponding to those wavenumbers (Hz).

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Array name	$k_{min}$	$k_{max}$	$f_{min}$	$f_{max}$
A	0.069	-	3.9	$\geq$ 15.0
B	0.023	0.46	2.4	14.2
C	0.023	0.22	2.4	6.9

Figure 6.12: Results of the high resolution frequency-wavenumber method applied to arrays A (a), B (b) and C (c). The grey exponential curves are the minimum and half maximum wavenumber limits deduced from theoretical array response. The black lines with dots obtained from computations are compared to the theoretical dispersion curves (thin plain lines).

Figure 6.13: Results from inversion of the dispersion curve obtained with the high-resolution frequency wavenumber method. (a) $V_p$ , (b) $V_s$ of generated models and (c) corresponding dispersion curves. The dots represent the experimental dispersion curves to which the calculated dispersion curves are compared. The black lines of figures (a) and (b) are the velocity profiles of the true model.

We performed exactly the same inversion processes as for the f-k results (figure 6.13). As we do not have error estimation on the dispersion curve, the model selection is based on the misfit threshold (0.075) for which the dispersion curve uncertainty includes the data scattering. As for f-k method, the $V_s$ profile up to the major impedance contrast can be determined. $V_p$ over the whole column and $V_s$ below 25 m are not defined by analysing the vertical component of the ambient vibrations. The slightly extended frequency range compared to f-k method does not induce a significant difference in the inverted $V_s$ profiles.