Single source wavefield

The f-k method is first applied to a wavefield produced by a single source of the aforementioned dataset, situated at about 650 m (310$^\circ$ counted clockwise from the North or Y

Figure 6.4: Single source wavefield measured by the vertical sensors of array B.

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axis) from the centre of arrays A, B and C. The source is punctual with a force vector oriented along direction 293$^\circ$ and inclined at 50$^\circ$ from the vertical axis. From its orientation and its position far from the arrays, Rayleigh waves are supposed to be mainly recorded. The signals computed at the ten receivers of array B are shown in figure 6.4. Their energy is spread over a 6 second period for a total calculated duration of 360 seconds.

The array response is calculated for single windows of varying durations: 20 s, 6 s, and 3 s, all centred around the most energetic part of the signal. The velocity at the semblance maximum is plot for all frequency bands in figures 6.5(a) to 6.5(c), for the duration 20 s, 6 s and 3 s, respectively. The theoretical dispersion curves (the fundamental and the first three higher modes) are plotted on the same graphs for comparison. The validity curves are drawn at constant $k_{min}$ (plain lines), $k_{max}/2$ (dotted lines) and $k_{max}$ (dashed lines). For the long time window, the velocity determination is nearly perfect for the whole frequency range except below 2 Hz, which corresponds to 40 cycles (20 s times 2 Hz). When decreasing the time length of the processed signal, the dispersion curve quality is degrading at low frequency. We define a minimum threshold frequency in each case indicating where the calculated dispersion curve leaves the theoretical curve. For the six-second window this threshold frequency is around 2.5 Hz (15 cycles) and around 5 Hz (15 cycles) for the three-second case. Comparing to the response of the arrays A and C, figures 6.6(a) and 6.6(b), respectively (six-second time window), it can be observed that the limit of 2.5 Hz is independent of the array aperture or array geometry. Array A provides a correct velocity estimation, though being far outside the valid wavenumber range. This frequency limit is linked to the energy content of the vertical spectrum shown in figure 6.2(a) as reported by Scherbaum et al. (2003).

Figure 6.5: Frequency-wavenumber analysis for array B with various time windows: (a) 20 s., (b) 6 s., and (c) 3 s. For each plot, the thin lines are the theoretical dispersion curves for the original ground model (first modal curves of figure 6.1). The three exponential curves represent constant wavenumber curves values of which are deduced from theoretical frequency-wavenumber response (figure 6.3): minimum (continuous line), half the maximum (dots) and maximum wavenumber (dashed).

The calculated curves in figures 6.5(b), 6.5(c), 6.6(a) and 6.6(b) show at least two major defects: one located at 4 Hz where the velocity increase is not retrieved and the other between 6 and 9 Hz, especially obvious for array A. The first one is not present on the 20 s. results (figure 6.5(a)), proving that the choice of a long enough window is crucial to correctly process the signals. The second defect may be investigated by examining the responses of arrays A and B in the plane $(k_x,k_y)$ (figures 6.7(a) and 6.7(b)). Below 6 Hz (not shown here) the shape of the array response is quite similar to the theoretical response, supporting the assumption of a single dominant surface wave. Above 6 Hz, the general shape is changing with the apparition of a secondary main peak at higher velocity, as shown by figure 6.7 (calculated at 6.5 Hz). Because of its relative low resolution limit ( $[k_{min}]_B < [k_{min}]_A$ ), array A cannot distinguish the two peaks and the exact position of the fundamental peak is shifted erroneously towards a higher velocity. This explains the velocity bump on the dispersion curve of figure 6.6(a).

Figure 6.6: Frequency-wavenumber analysis for arrays A and C in figures (a) and (b), respectively. The window length is 6 seconds.

Figure 6.7: Array responses for arrays A and B in figures (a) and (b), respectively, calculated at 6.5 Hz.

The signal processing shows that the simulated vibrations are mostly composed of surface wave which dispersion curve is perfectly retrieved in figure 6.5(a) above 2.5 Hz. Waves travelling at a higher velocity are detected between 6 and 9 Hz (figures 6.6(a) and 6.7(b)), probably corresponding to the first higher mode. Comparison of arrays with different resolving power allows the rejection of non trusted samples. The parameters of the signal processing, particularly the choice of a too short time window, may introduce undesirable effect on the dispersion curve construction. The results obtained with a single source suggest the use of windows of at least 15 to 40 periods. In the following, a complex wavefield is analysed by the means of three processing techniques (frequency-wavenumber, high resolution and auto-correlation methods).