## 2 Data and Methods
### Coda Attenuation 3D Imaging  
In order to obtain 3D coda attenuation images, we used the approach discussed by [PERSON] and [PERSON] [11], based on the scattering kernels described by [PERSON] and [PERSON] [15]. [PERSON] and [PERSON] [11] computed the polynomial approximation of the analytically calculated sensitivity kernels, which noticeably reduces the calculation time costs. In particular, we used the sensitivity kernel as a weighting function, heuristically giving them a probabilistic meaning. The \(Q_{c}\) weighting functions ([PERSON] and [PERSON], 2020) are here denoted as:  
\[w_{ij}\big{(}x_{j},\;y_{j},\;z_{j},x_{si},\;y_{si},\;z_{si},x_{ri},\;y_{ri}\big{)} \tag{4}\]  
and represent the probability that, for the \(i\)-th source-receiver couple positioned, respectively, at \([x_{si},\;y_{si},\;z_{si}]\) and \(\{x_{ri},\;y_{ri}\}\), the measured value of \(Q_{c}\) effectively corresponds to its true value at the space point with the coordinates \(\{x_{j},\;y_{j},\;z_{j}\}\). The subscript \(i\) spans from 1 to \(N_{s-r}\) (\(N_{s-r}\) is the number of source-receiver couples in the data set); \(j\) spans from 1 to \(M_{pix}\) (\(M_{pix}\) is the number of cells in which the space has been subdivided, here called \"pixels\").  
We indicated with \(Q_{ci}\) the measure of \(Q_{c}\) for the \(i\)-th source-receiver pair, and with \(Q_{cj}\) the measure of \(Q_{c}\) for the \(j\)-th pixel. In this heuristic scheme, \(w_{ij}\) represents the un-normalized probability that the \(i\)-th measure, \(Q_{ci}\), is associated with the \(j\)-th pixel. Averaging over the events and weighting for the kernel, at the single position \(j\), the method yields:  
\[Q_{cj}=\frac{\sum_{i}Q_{ci}w_{ij}}{\sum_{i}w_{ij}} \tag{5}\]The measure set thus produces for any pixel, \(j\), a set of point values of \(Q_{ci}\) multiplied by their probability, \(w_{ij}\). We consider their weighted average as the characteristic value of \(Q_{c}\) at the specific point. In other words, we simply back-project the \(Q_{c}\) measure by attributing the measure to a space point of given coordinates, as weighted for the weighting functions.  
By applying the ordinary error propagation equation to Equation (5) and by assuming \(\sigma_{i}\) the standard deviation of \(Q_{ci}\), we obtain the standard deviation of the \(j\)-th pixel:  
\[\sigma_{j}=\frac{\sum_{i}\left(\sigma_{i}w_{ij}\right)^{2}}{\left(\sum_{i}w_{ ij}\right)^{2}} \tag{6}\]  
We used Equation (6) to estimate the error on \(Q_{cj}\) in any \(j\)-th pixel. Since \(\sigma_{j}\) represents the uncertainty of \(Q_{cj}\) in the \(j\)-th pixel, and the diffusive waves are not related to seismic rays, we do not need to evaluate the ray crossing inside each \(j\)-th pixel. In fact, the weighting functions used are peaked at source and receiver. Consequently, the sole distribution of sources and receivers gives a sufficient idea of the areas where the coda radiation has the maximum probability of passing through. On the other hand, the coda tomography concept goes beyond the ray tracing assumption, taking into account the energy space distribution more than the wave front of the direct (ballistic) radiation, depicted by the ray trace space distribution. [20].