Validation of auto-correlations

The measured auto-correlation curves do not always fit the shape of Bessel's function and the system of auto-correlation equations (of type (3.49) or (3.50)) may have no common solution for all array apertures. Feeding the inversion process with contradictory auto-correlation curves is likely to give an uncontrolled average solution. If the contradiction comes from a defect in the array response (e.g. too wide aperture for the considered wavelength) or in the noise content (e.g. uncorrelated noise due to long distance between sensors for the considered frequency, or insufficient energy level at low frequency), the probability of obtaining an unrealistic solution is high. A selection of the relevant parts of the auto-correlation curves is thus necessary. The problem is complex and there are no objective and commonly applicable rules. Without a prior knowledge of the soil structure, the only reliable features are the array geometry and the auto-correlation curves themselves. From the array geometry, some rough limits can be deduced for a correct response in terms of wavenumber (Asten and Henstridge (1984); Woods and Lintz (1973)), theoretically for the frequency-wavenumber processing only (section 1.1.1 on page ). On the other hand, from the auto-correlation curves for the different rings, we can test the consistency of the system of equations, and discard the samples that are obviously out of the general trend.

Practically, from a very large a priori in terms of apparent velocity (e.g. from 100 to 3000 m/s), all possible solutions $c(\omega)$ of equation (3.49) or (3.50) are calculated independently for each ring. For doing so, we define the function:

$$g(c,\omega)=\overline{\rho_{calc}(r,\omega,c)}-\overline{\rho_{obs}(r,\omega)}$$ (5.1)

where, $\omega$ is the considered frequency band, $\rho_{calc}$ is calculated by equation (3.49) or (3.50), and $\rho_{obs}$ is the auto-correlation ratio calculated on the recorded signals. The roots of function $g(c,\omega)$ are successively bracketed by a coarse grid search starting from the lowest velocity, and then refined by an iterative scheme based on the Lagrange polynomial constructed by the Neville's method (Press et al. (1992)). The same algorithm as for the internal computation of dispersion curves is used (section 3.1.5). In a second stage, we construct a grid for each ring in the frequency-slowness domain. The grid cells are filled with 1 if at least one solution exists within the cell, with 0 in the contrary case. All the grids are stacked and the values in each cell give the

Figure 5.12: Grids in frequency-slowness domain representing the density of dispersion curve solutions. (a) Solutions of equation (3.50) for the perfect auto-correlation curves of figure 5.11. The theoretical dispersion curve is represented by a plain line.

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number of consistent rings for a particular couple frequency-slowness. If the auto-correlation curves are consistent, the cells where the density of solutions is maximum should delineate the corresponding dispersion curve. From this plot, we determine the minimum and the maximum slowness for each frequency, as well as the minimum and the maximum wavenumber for which we observe a focused dispersion curve. To reduce the subjectivity of the selection, zones where no clear consistency between auto-correlation curves is observed are systematically rejected. Once the dispersion curve limits are set, it is straightforward to reject the contradictory data on the auto-correlation curves. This procedure is tested on the pure synthetic case (figure 5.11) where no contradictory samples are present in the auto-correlation curves. Figure 5.12 shows the resulting frequency-slowness grid obtained after seeking for all possible solutions. The dispersion curve can be entirely retrieved from the auto-correlation curves between 1 and 10 Hz. When the auto-correlation value is less than 0.025 (arbitrary threshold to avoid an infinite number of solutions), no solution is calculated. This is why, for high frequency, the large apertures provide no points and hence the density vanishes to one or two occurrences only.