# Green's Function, Earthquakes, and a Fast Ambient Noise Tomography Methodology
## 3 Results and Discussion
### Frequency Time Analysis (FTAN) and 1D Inversion  
After the extraction of each EGF, FTAN [26] was applied to obtain the group velocity as a function of the period, using the EGFAnalysisTimeFreeq dispersion software [27]. The group velocity dispersion analysis uses a Gaussian window \(G\) in the frequency domain, which is applied as a narrow band-pass filter at each central frequency. Considering \(f\) as the frequency, \(f_{c}\) as the central frequency, and \(A\) as an interstation-distance-dependent coefficient, where \(A\) is equal to 5 for interstation distances less than 100 km and \(A\) is equal to 8 otherwise, we obtain the following:  
\[G(f,f_{c},A)=e^{\frac{-A(f-f_{c})^{2}}{f_{c}^{2}}} \tag{4}\]  
Figure 2: The normalized EGF for each pair of stations, plotted in increasing interstation distance with respect to time.  
To obtain reliable dispersion measurements, we accept a frequency-dependent velocity of the fundamental mode, provided that its SNR is at least 5 and its ratio of the interstation distance over its wavelength is at least 2 [24]. Here, the SNR is frequency-dependent and for each central frequency, it is equal to the maximum amplitude of the envelope in the signal window divided by the mean amplitude of the envelope of the 150 s long noise window right after the signal window. The signal window is such that the group velocity lies between 1.8 and 5.5 km/s, according to the interstation distance. The period range under investigation is between 1 and 10 s, after examining the peak of the probabilistic power spectral densities of several stations for the given time frame. An example of such a measurement for a pair of stations is shown in Figure 3. All dispersion curves were later averaged for periods between 1 and 8 s and the average was smoothed by applying a moving mean scheme (Figure 4).  
In general, the effect of a water layer can make it difficult to measure unambiguously the fundamental-mode dispersion velocities at frequencies above 0.1 Hz [28]. However, since here this is not the case (Figure 3), we assume the effect of the water layer to be minimal, due to its small depth which is at most 400 m in the study area [29].  
Figure 4: Average group velocity dispersion curve, for periods between 1 and 8 s, smoothed by applying a 15-point moving average.  
Figure 3: A selected group velocity curve of the fundamental mode for the EVGI-LKD2 station pair. Only the red circles to the left of the vertical red line are considered valid measurements, as only those points satisfy the condition that the interstation distance over wavelength ratio is larger than 2. Similarly, the dashed cyan line is drawn when such a ratio becomes equal to 3. Brown colors indicate high amplitudes, while blue colors denote low amplitudes.  
To assess which 1D shear velocity model best fits the average dispersion curve (Figure 4), we applied a Neighborhood Algorithm [30] as an inversion technique, using the software Geopsy v3.4.2 [31]. This technique improves upon the efficiency of a standard Monte Carlo method by introducing a sampling preference over the more promising--based on the results obtained so far--subsets of the parameter space [32]. The forward problem solution [33] is based on the theoretical elastic computation of a dispersion curve for a stack of horizontal and homogeneous layers [34; 35; 36; 37]. The initial model consists of four layers from the surface up to infinite depth with increasing velocity, V\({}_{\text{p}}\) from 0.2 to 5 km/s, V\({}_{\text{s}}\) from 0.5 to 3.5 km/s, and Poisson's ratio from 0.25 to 0.5 to cover all possible solutions. A constant density of 2000 kg/m\({}^{3}\) was chosen, due to the fact that the study area mainly consists of sediments and limestones [38], which indicate a density range typically between 1600 and 2750 kg/m\({}^{3}\)[39].