## 2 Neural Network Approach and Employed Features  
ANNs are based on the concept of the single artificial neuron, the 'Perceptron', introduced by [20] to solve problems in the area of character recognition [21]. Using supervised learning, with the Error-Correction Learning (ECL) rule for network weights adjustment, those networks can learn to map from one data space to another using examples [22]. One of the most common and reliable learning techniques is the back-propagation (BP) algorithm [23]. BP consists of two phases: in the feedforward pass, an input vector is presented to the network and propagated forward to the output; in the back-propagation phase, the network output is compared to a desired output; network weights are then adjusted in accordance with an ECL rule [23, 24, 25]. Cross validation can be used to detect when over-fitting starts during supervised training of a neural network; training is then stopped before convergence to avoid over-fitting (early stopping). Early stopping using cross validation was done by splitting the training data into a training set, a cross-validation set, and a test set, and then training the networks only using the training set and evaluating the per-example error on the test set on a sample basis after a defined number of epochs. Finally, training was stopped when the error--the difference between neural network output and target--on the cross validation set was higher than the previous error value [26]. The performance of a trained ANN is generally assessed by computing the root mean square error (RMSE) between expected values and activation values at the output nodes or, in the case of classification, a percentage of correctly classified examples of the validation set. At the time our study was carried out, the ground truth was unavailable for Amatrice earthquake, so a visual inspection using high resolution images has been adopted. From studies of past literature and development activities performed in a European research project (APhoRISM, www.aphorism-project.eu), we identified and employed a set of features that demonstrated a good sensitivity to damage at object scale. Regarding optical data, we used the NDI, KLD and MI change indexes. As far as SAR data is concerned, we considered the KLD and ICD parameters. Note that all change indexes have been calculated at object scale, i.e., by considering polygons that refer to city blocks. These latter have been extracted through the free geo-database of the Open Street Map project. The NDI parameter is defined as  
\[NDI_{i}=\frac{POST_{i}-PRE_{i}}{POST_{i}+PRE_{i}} \tag{1}\]  
where \(PRE_{i}\) and \(POST_{i}\) indicate the mean values of intensity, respectively for pre- and post-seismic images, associated to \(i\)-th polygon (see also Figure 2 as example). The intensity values are obtained averaging the Top of Atmosphere (TOA) Red-Green-Blue (RGB) reflectance and corrected by applying the Flat Field procedure for atmospheric correction [27]. The MI index measures the correlation loss between pre- and post-seismic images (see Equation (2)). \[MI_{i}=-\ln\Bigl{(}1-r_{i}^{2}\Bigr{)}\times 0.5 \tag{2}\]  
where \(r_{i}^{2}\) is the correlation between the pre- and post-seismic pixels within each polygon. Correlation is obtained from intensity and backscattering values for optical and SAR data, respectively. \(MI\) is inversely proportional to the damage grade. The KLD parameter is defined as  
\[KLD_{i}=\frac{\left(PRE_{i}-POST_{i}\right)^{2}+Var(PRE_{i})+Var(POST_{i})}{2} \times\left(\frac{1}{Var(PRE_{i})}+\frac{1}{Var(POST_{i})}\right)-2 \tag{3}\]  
where \(PRE_{i}\) and \(POST_{i}\) are the same parameters in the Equation (1), and \(Var(PRE_{i})\) and \(Var(POST_{i})\) are their variances within the \(i\)-th polygon. The \(KLD\) parameter has the same behavior of NDI: KLD increasing values correspond to increasing damage level.