Spatial auto-correlation method

The signals simulated for the three arrays A, B and C are analysed using the spatial auto-correlation method described in section 5.2. The azimuths and the distances between all couples of stations are shown in figure 6.15. The pairs of grey circles are the selected rings for the spatial auto-correlation computation. Distances are summarized in table 6.4.

Table 6.4: Distance limits for the selected rings for arrays A, B and C. The last column is the number of station couples included in each ring. Distances are expressed in metres.

Array name	Min. radius	Max. radius	Number of pairs
A	7.8	9.4	9
A	12.1	13.2	9
A	15.3	17.0	9
A	21.2	22.5	9
A	24.4	25.3	9
B	12.5	18.0	6
B	22.0	26.3	9
B	34.7	43.3	12
B	49.1	63.8	12
B	73.8	87.3	6
C	33.5	35.0	9
C	48.4	54	9
C	63.9	65.1	9
C	85.6	87.3	9
C	97.5	99.4	9

As in the f-k method (section 6.1.3), the choice of the window length for calculating the auto-correlations is crucial. An example of its influence is presented hereafter. The average auto-correlation ratios are calculated with equation 1.11 for pairs of stations separated by distances between 30 and 40 m. In figure 6.14, the auto-correlation curves are plotted for various window lengths, counted in number of cycles of the central considered frequency ($\omega_0$ ): 10, 25 and 50 (from light to dark grey, respectively). For the three curves, the average values are close to the true auto-correlation curve (black thick line) in the range 3.5 to 5.5 Hz. Below 3.5 Hz, the 10 cycle auto-correlation curve deviates from the correct function, while the two other curves (25 and 50 cycles) are close to it for frequency as low as 2.5 Hz. This discrepancy for short windows is probably due to a lack of source azimuth coverage (Asten et al. (2004)), as the number of acting random sources is inversely proportional to the considered duration. Another explanation might be that the spectral estimates are more influenced by unavoidable side effects generated by cutting signals into time windows. Also, long time window curves are smoother than short ones and exhibit smaller standard deviations (figure 6.14). During this thesis, the 25 cycle time windows are kept for the computation of auto-correlation curves.

Figure 6.14: Influence of time window lengths on auto-correlation curves (average and standard deviations): 10 cycles (light grey), 25 cycles (medium grey) and 50 cycles (dark grey). The thick black line represent the theoretical auto-correlation curve.

Table 6.5: table

Array name	$k_{min}$	$k_{max}$	$f_{min}$	$f_{max}$
A	0.015	-	1.9	$\geq$ 15.0
B	0.012	0.4	1.7	13.4
C	0.011	-	1.5	$\geq$ 15.0

[Apparent limits for auto-correlation methods]For each array the minimum and maximum wavenumbers deduced from the solution density grid (figure 6.17). Also the minimum and maximum frequencies corresponding to the selected samples (Hz).

A total of 15 auto-correlation ratio curves (five by array) are calculated for time windows of 25 cycles. Only one curve per array is shown in figure 6.16 with grey dots and grey errors bars. The consistency of all 15 auto-correlation curves is checked on dispersion curves in figure 6.17(a) to 6.17(c), for arrays A to C, respectively.

The fifteen auto-correlation curves with the selected samples are inverted with five independent runs keeping the same parameterization as for the two preceding methods. The results are shown in figure 6.18. Only three auto-correlation curves among the fifteen are shown in figure 6.18(d) to 6.18(f). A good agreement is found between the calculated curves and the observed auto-correlations (black dots and their error bars) even below 2 Hz. The theoretical dispersion curve is drawn for comparison in figure 6.18(c). The auto-correlation method correctly retrieves the dispersion curve for all frequencies above 2.5 Hz. For lower frequency, a systematic bias is observed in figure 6.18(c). Comparing figures 6.11(b) and 6.13(b), the inversion of auto-correlation offers a little more constraint on Vs at the base of the sediment layer. $V_p$ over the whole column and $V_s$ below the major impedance contrast is not resolved as for the other methods.

Figure 6.15: Azimuth-inter-distance plot: each dot represent one couple of stations. The pairs of grey circles show the limits of the chosen rings. (a) Array A, (b) array B, and (c) array C.

Figure 6.16: Examples of auto-correlation curves obtained for (a) array A, (b) array B, and (c) array C. The black dots and error bars are the samples selected according to criteria of figure 6.17.

Figure 6.17: (a) to (c) grids in frequency-slowness domain representing the density of dispersion curve solutions for arrays A to C, respectively. The plain and the dotted lines are the wavenumber limits deduced from the solution density grids (if any). The dashed and the dot-dashed curves are the wavenumber limits of the apparent dispersion curve or the limits of the area with a high density of solutions.

Figure 6.18: Inversion of the selected 15 auto-correlation curves. Only three of them are presented here. (a) $V_p$ , (b) $V_s$ profiles of generated models. The black lines of figures are the velocity profiles of the true model. (c) The dispersion curves corresponding to model of figures (a) and (b). The thin lines are the theoretical dispersion curves, not used during inversion. (d) to (e) One auto-correlation curve per array, A, B and C respectively. The black dots and their errors bars are the auto-correlation data points to be fit during inversion.