Model description

The synthetic ground model is composed of a soil layer with a thickness of 25 m overlying an infinite bedrock. The properties of each layer are specified in table 6.1.

Table 6.1: Properties of the reference model.

Thickness	$V_p$	$V_s$	Density	$Q_p$	$Q_s$
25 m	1350 m/s	200 m/s	1.9 t/m$^3$	50	25
-	2000 m/s	1000 m/s	2.5 t/m$^3$	100	50

The theoretical Rayleigh dispersion curves (the fundamental and the first four higher modes in the case of an elastic media) for this model are shown in figure 6.1(a) between 1 and 15 Hz. For frequencies below 15 Hz, only the first five modes are present and appear to be well separated in terms of velocity. Figure 6.1(b) shows the fundamental ellipticity and the $S_H$ transfer function. The fundamental resonance frequency is 2 Hz while the peak of the fundamental ellipticity is at 1.9 Hz. This frequency difference, recently studied by Malischewsky and Scherbaum (2004), is mainly influenced by the magnitude of the velocity contrast between the sediments and the bedrock.

Figure 6.1: Theoretical model for synthetic ambient vibrations. (a) Dispersion curves for Rayleigh modes calculated with the synthetic model: fundamental model (thick plain line), first (thick dashed line), second (thin plain line), third (thin dashed line), and fourth higher mode (thin dotted line). Other modes do not exist in the plotted range. (b) Theoretical $S_H$ transfer function for the synthetic model (plain line) and fundamental Rayleigh ellipticity curve (dotted line).

Figure 6.2: Spectral curves of the central station of array A to C. The plain line is the average and the dashed lines are located at one standard deviation. (a) Amplitude spectrum of the vertical component. (b) Amplitude spectrum of one horizontal component. (c) Spectral ratio Horizontal to Vertical (H/V). Grey bands indicate the average and standard deviation of the frequency peak values observed for each individual time window.

Synthetic ambient vibrations have been computed during 6 minutes using the method proposed by Hisada (1994 and 1995), and Bonnefoy-Claudet et al. (2004) which is valid for a one-dimensional model with sources and receivers placed at any depth. This dataset includes 333 source points randomly distributed from 140 to 750 m from the central receiver. Sources are punctual forces with delta-like functions of random amplitudes and directions. All types of waves existing in such media are modelled generating a wave field containing body, Love and Rayleigh waves. The frequency spectrum of generated waves is limited to 15 Hz in order to reduce CPU time. The spectra of the vertical (V) and one horizontal (H) component of the central station is shown in figure 6.2(a) and 6.2(b), as well as the H over V ratio (figure 6.2(c)). The frequency of the H/V peak (2 Hz) matches the resonance frequency of the soft layer (figure 6.1(b)). The Fourier spectra show that the energy of the vertical component vanishes in the vicinity of and below the fundamental frequency as reported by Scherbaum et al. (2003), while the energy on the horizontal component decreases below 1.5 Hz.

Figure 6.3: Array geometries and their f-k responses. (a), (d) and (g) Geometries of arrays for arrays A, B, and C, respectively. (b), (e) and (h) Their corresponding theoretical frequency-wavenumber responses. The circles correspond to the chosen wavenumber limits detailed in table 6.2. (c), (f) and (i) Sections across several azimuths for the theoretical frequency-wavenumber grids of arrays A, B, and C, respectively. The black curve is oriented along the line drawn in figures (b), (e) and (h).

Table 6.2: Properties of the array geometries. For each array the minimum and maximum distance between sensors. The minimum and maximum wavenumbers deduced from the theoretical frequency-wavenumber responses in figures 6.3(b), 6.3(e) and 6.3(h). Also the minimum and maximum frequencies corresponding to those wavenumbers (Hz).

Array name	Min. dist.	Max. dist.	$k_{min}$	$k_{max}$	$f_{min}$	$f_{max}$
A	8 m	25 m	0.095	1.50	4.4	$\geq$ 15.0
B	13 m	87 m	0.037	0.495	3.25	7.6
C	34 m	99 m	0.024	0.39	3.0	7.2

On this model we set up three arrays (labelled A, B and C) the geometries of which are plotted in figures 6.3(a), 6.3(c) and 6.3(e), respectively. Array A is composed of nine sensors roughly distributed around a central sensor, with an approximate aperture of 25 metres. Array B is made of three triangles approximately rotated by 40$^{\circ}$ and with increasing aperture up to 90 metres. Finally, array C is made of nine sensors roughly distributed around a central sensor, with an approximate aperture of 100 metres. Theoretical f-k responses (section 1.1.1 on page ) for arrays A, B, and C are shown in figures 6.3(b), 6.3(d) and 6.3(f), respectively. The resolution and aliasing limits deduced from Woods and Lintz (1973) and Asten and Henstridge (1984) criteria are marked by circles and are summarized in table 6.2. Sections are made across each of them along several azimuths (628) and they are plotted by grey curves in figures 6.3(c), 6.3(e) and 6.3(g). The bold black curves correspond to the minimum aliasing azimuths which are marked by black lines in figures 6.3(b), (d) and (f). From equation (1.2), a wave travelling at $k_{max}$ appears in the semblance map with the main peak right on the aliasing limit and the lateral aliasing peaks greater than 0.5 are located on a circle crossing the origin. In the case of a complex wavefield with waves travelling in several directions, there are lot of chances to confuse the true peak with sums of secondary aliasing peaks that do not correspond to the correct velocity. Hence, a safe approach would be to limit the valid range to $k_{max}/2$ , which is illustrated by the results of the next sections.