# Green's Function, Earthquakes, and a Fast Ambient Noise Tomography Methodology
## 3 Results and Discussion
### Inversion for a 3D \(V_{s}\) Model and Discussion  
To invert the Rayleigh wave group velocity data to obtain 3D shear velocity models, we applied the algorithm proposed by [PERSON] et al. [40]. This algorithm does not make the inherent assumption of straight ray propagation which is commonly used in most ANT studies, but instead solves the eikonal equation with the fast-marching method [41] at each period, to reproduce a more appropriate ray path between source and receiver. This algorithm avoids the intermediate step of inversion for the group velocity map. The initial solution proposed (along with the depth constraint associated with it) was the 1D V\({}_{\text{s}}\) profile in Figure 5.  
We have assessed the validity of the inversion algorithm by performing a checkerboard test, in which the synthetic travel time data produced by the checkerboard model are inverted (after adding 2% Gaussian noise). The checkerboard model (Figure 6a) is produced by the spatially symmetric perturbation of our initial solution and shows a good correspondence with the model produced by the inversion of the synthetic data (Figure 6b), inside the area covered by the stations. The value \(\lambda=4\) was chosen as the value of the Lagrange multiplier of the inversion algorithm, as it produced the best result during the checkerboard test compared to lower values (i.e., with less smoothing). Figure 5: The 1D V\({}_{\text{s}}\) model obtained in this study by applying the Neighborhood Algorithm. Note that the deviation of the best model (blue line) from the  
The iteratively reweighted least squares method was applied until convergence, at which point the objective function was close to zero, and the final model produced is shown in Figure 7. Four layers are assumed (their number is a priori determined during parameterization) with increasing velocity from 1.7 \(\pm\) 0.3 km/s at the surface up to 2.8 \(\pm\) 0.2 km/s at 4.5 km depth. During the inversion process the wave speed was constrained between 1 and 3 km/s. Our results are consistent with previous studies in northern and eastern neighboring regions [42,43]. Considering the known geology derived from borehole data in neighboring regions, which showed that approximately for the first half-kilometer there are Neogene sediments while the Vigla and Pantokrator Limestones extend below [38], the final 3D model, and known shear velocity-soil type relations (i.e., in brief, that the value of V\({}_{\text{s}}\) below 2 km/s corresponds to sediments while the value above it corresponds to sound limestones [44]), we infer the existence of loose sedimentary rocks until a depth of 500 m, while below we infer the existence of limestones. The curvature of the layers lies outside the area covered by the stations, so it is not considered. The uncertainty of the final model is determined by the application of a Monte Carlo error propagation technique [45]. After producing synthetic travel times from the final model, we inverted them (after adding 2% Gaussian noise) many times and measured the standard deviation of the resultant set of models (Figure 8). Inside the area defined by the outermost stations, the uncertainty of the wave speed model is small (\(\approx\)\(\pm\)0.2 km/s). To further investigate if the model is well constrained in terms of depth, we calculated using disba [46] the sensitivity of the surface wave group velocity to the shear wave speed for different periods, from the final 1D model (Figure 9a). The latter was calculated from the mean and standard deviation of different horizontal slices of the final 3D model, including only data inside the area defined by the seismic stations. Apart from the final 1D model, the corresponding compressional wave speed and density profile were also required. We assumed a constant density of 2.000 kg/m\({}^{3}\) and a V\({}_{\text{p}}\)/V\({}_{\text{s}}\) ratio of 2. As expected, the larger periods are more sensitive to larger depths.