## 2 Materials and Methods
### Geostatistical Correlation Analysis Methodology  
The statistical correlation at the spatial level between the seven indicators described and the importance of the damage produced by the earthquake will be evaluated through geo-process functions from ArcGIS and geostatistical tools from ArcGIS Pro Desktop (GIS database was previously elaborated through GvSIG program). First, the level of statistical significance of the distribution of each of the seven earthquake indicators will be analyzed through a geo-process based on the Global Moran's I statistic [29]. Global Moran's I statistic will analyze the degree of spatial autocorrelation of the different geolocated data obtained from the distribution of the indicators in the city of Lorca, and the sign of this autocorrelation (positive or negative). Its statistical formula for spatial autocorrelation is given as:  
\[I=\frac{n}{S_{0}}\frac{\sum_{i=1}^{n}\sum_{j=1}^{n}w_{i,j}z_{i}z_{j}}{\sum_{ i=1}^{n}z_{i}^{2}} \tag{5}\]  
where \(z_{i}\) is the deviation of an attribute for feature \(i\) from its mean \(\left(x_{i}-\overline{X}\right)\); \(w_{i,j}\) is the spatial weight between feature \(i\) and \(j\); \(n\) is equal to the total number of features; and \(S_{0}\) is the aggregate of all the spatial weights:  
\[S_{0}=\sum\
olimits_{i=1}^{n}\sum\
olimits_{j=1}^{n}w_{i,j} \tag{6}\]  
The \(z_{l}\) -score for the statistic is calculated by the GIS program as:  
\[z_{l}=\frac{I-E[l]}{\sqrt{V[l]}} \tag{7}\]  
where:  
\[E[l]=-1/(n-1) \tag{8}\]  
\[V[l]=EI^{2}-E[l]^{2} \tag{9}\]  
Global Moran's I statistic returns three values: the Moran's I Index, \(z\)-score, and \(p\)-value. Given a set of features and an associated attribute, the ArcGIS geo-process evaluates whether the pattern expressed is clustered, dispersed, or random. When the \(z\)-score or \(p\)-value indicates statistical significance, a positive Moran's I index value indicates a trend toward clustering while a negative Moran's I index value indicates a trend toward dispersion. The \(z\)-score and \(p\)-value are measures of statistical significance which inform us whether or not to reject the null hypothesis. For this statistical geo-process, the null hypothesis states that the values associated with features are spatially randomly distributed. Secondly, the level of spatial correlation between each of the indicators and the severity of the damage detected will be evaluated. We will use two tools to numerically calculate the two-dimensional correlation between each indicator and the observed damage, and on the other hand spatially distribute the intensity of said correlation. [PERSON]'s I [30] will analyze cluster and outlier numerical bivariate analysis. This statistic of spatial association is given as:  
\[I_{i}=\frac{x_{i}-\overline{X}}{S_{i}^{2}}\sum\
olimits_{j=1,j=i}^{n}w_{i,j} \left(x_{j}-\overline{X}\right) \tag{10}\]  
where \(x_{i}\) is an attribute for feature \(i\); \(\overline{X}\) is the mean of the corresponding attribute; \(w_{i,j}\) is the spatial weight between feature \(i\) and \(j\), and:  
\[S_{i}^{2}=\frac{\sum\
olimits_{j=1,j=i}^{n}\left(\[z_{l_{i}}=\frac{I-E[I_{i}]}{\sqrt{V[l_{i}]}} \tag{12}\]  
where (see Appendix B for further mathematical details):  
\[E[I_{i}]=-\frac{\sum_{j=1,j=i}^{n}w_{i,j}}{n-1} \tag{13}\]  
\[V[I_{i}]=EI_{i}^{2}-E[I_{i}]^{2} \tag{14}\]  
Finally, for those indicators that reflect a significant statistical correlation with the observed damages numerically and at an aggregate level, an intensity spatial analysis will be carried out through the segmentation of cold and hot spots Getis-Ord Gi [31].