## 5 Multidimensional Definition of SR  
Following the previous [33], a multidimensional formulation of SR needs to be defined. In particular, the calculation of SR needs the application of the loss model and the recovery model:  
1. The loss model describes the reduction in functionality due to the initial impact of an event on a system. The quantification of the losses that are at the base of this model may be a challenging issue since several typologies of losses need to be considered, such as direct and indirect losses, as described in [20]. 2. The recovery model aims to assess the ability of the system to recover from the impact by considering the process of recovery that is generally represented with an analytical formulation. Such formulation needs to describe several variables that are challenging to be defined and generally depend on the preparedness of a community, the level of technological know-how, and the distribution of economic funds for the recovery process. In this background, herein is extended what is described in Section 3 to assess the resilience to earthquake hazard by considering the presence of DT. SR is formulated on the basis of the recovery function \(Q_{N}(t)\) that describes the recovery process to return an estimated level of functionality:  
\[Q=Q(t,\;\lambda,Q_{1},\ldots,\;Q_{N}) \tag{4}\]  
which is defined in the R\({}^{\text{N}}\) space and is a function of time (t), of the coefficient \(\lambda\) and all the various dimensions \(Q_{i}\) of the problem/system. Consequently, SR may be computed by the following [24]:  
\[\text{SR}=\int_{t0F}^{t0F+RT}\frac{Q(t,Q_{1},\ldots,\;Q_{N})}{RT}dt \tag{5}\]  
It is worth noticing that this formulation has the same shape as that proposed above, and it expresses the dependency of SR not only on time but also on the N dimensions with that the system may be described. Graphically, Equation (4) represents the normalized volume underneath the recovery function QN(t) and the plane t = RT (Figure 2). It is worth noting that in Figure 2, the values of RT could be different for the various dimensions. \begin{table}
\begin{tabular}{c c c} \hline
**Level** & **Accuracy** & **SR** \\ \hline L1 & Slight & 0.502 \\ L2 & Moderate & 0.531 \\ L3 & Extensive & 0.644 \\ L4 & Complete & 0.951 \\ \hline \end{tabular}
\end{table} 
Table 2: Case Study (Hp: linear repair function). It is worth noting that for every time \(t=k\), with \(k<RT\), it is possible to consider the plane:  
\[Q(t=k,Q_{1},\ldots,\ Q_{N}) \tag{6}\]  
which represents a plain surface connecting the points on the recovery function corresponding at \(t=k\) for all the variable Qi. Therefore, Equation (5) describes the section of the cone whose perimeter represents the sum of the functionalities Qi reached by the total system at the selected time \(t=k\). Figure 3 shows the contour lines (lines joining points at the same time \(t=k_{i}\)). Following [1], in the present paper, the simplification of the linear repair function has been considered realistic when there is not sufficient information regarding the preparedness of the community or the consistency of the resources. Therefore, the solid assumes a conical shape with different angles with the \(t\)-axis, depending on the coefficient ci:  
\[ci=\frac{Q_{F,i}}{RT_{i}} \tag{7}\]  
Figure 3: Contour lines of the functionality \(Q_{N}\). Figure 2: Multidimensional formulation of FR. where \(Q_{F,i}\) is the value of the functionality after the complete recovery of the system \(i\). Such value may vary from 1% to 100% with respect to the original functionality but may also be bigger than 100% when the repair function allows improvements to the original functionality. Herein it was assumed that the loss is 100%, meaning that the value of functionality just after the earthquake is 0.