# Integral Seismic Risk Assessment through Fuzzy Models
## 4 An Integral Seismic Risk Fuzzy Model
### Physical Risk Fuzzy Model  
The Physical Risk Fuzzy Model is based on previous work [28] and aims to describe the potential damages caused by an earthquake to different critical urban infrastructure and basic supply lines: Property, LifeLines and Network Supply lines. Three sub-models, each of which is a FIS, were used to represent each type of damage using indicators, as shown in Figure 1: Property Damage (PD), Life Line Sources Damage (LLSD) and Network Damage (ND). In this way, we can handle several indicators at the same time yet narrowing the complexity of the model as the number of fuzzy rules is limited to 2*n (with n = 3)). Table 1 shows a brief description of the indicators used in the Physical Risk model and the resources utilized for their estimation. Each of these indicators was estimated using three FIS sub-models with their antecedent and consequent parts divided into three linguistic classes: Low, Medium, and High. For example, Table 2 \begin{table}
\begin{tabular}{l} \hline \hline
**Property Damage (PD)** \\ \hline Damage Area (The percentage of constructed area that is destroyed, estimated by means of ATC-13 \\ and HAZUS). \\ Dead People (Number of deaths per 1000 inhabitants as estimated by ATC-13 and HAZUS). \\ Injured People (Number of injured per 1000 inhabitants as estimated by ATC-13 and HAZUS). \\ \hline
**LifeLine Sources Damage (LLSD)** \\ \hline Telephonic Substation Affected (number of telephonic stations with high seismic vulnerability \\ according to ATC-21). \\ Electrical Substation Affected (number of electrical substations with high seismic vulnerability \\ according to ATC-21). \\ Damage in Water Mains (number of breaks along the supply water network per kilometer as \\ estimated by ATC-13 and HAZUS). \\ \hline
**Network Damage (ND)** \\ \hline Damage in Gas Network (number of breaks along the gas network per kilometer as estimated by \\ ATC-13 and HAZUS). \\ Fallen Length of Electrical Lines (fallen length of electrical lines per network kilometer as estimated \\ by means of ATC-13 and HAZUS). \\ Damage in Mains Roads (Damage Index). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Indicators used for the Physical Risk Fuzzy Model (References for ATC-13/ATC-21 and HAZUS area [29; 30], respectively). shows the set of fuzzy rules that were used to estimate LifeLine Sources Damage levels (LLSD) using three indicators. To estimate a physical risk level, each sub-model acts as an independent inference system using as inputs their respective original set of indicators. In the subsequent steps, the Physical Risk model uses the estimated inferences obtained from the three FIS sub-models and performs the final estimation through the FIS called Physical Risk. In the case of Physical Risk, its FIS has its antecedent part, divided into a universe of discourse of three linguistic classes: Low, Medium, and High. Each class is associated with a membership function type: Z-shaped, Pi-shaped, and S-shaped, respectively [31]. The FIS consequent part was characterized by five linguistic classes: Low, Medium-Low, Medium-High, High, and Very-High, each related to a membership function type: Z-shaped, Pi-shaped (the three middle ones), and S-shaped, respectively. Figure 2 shows a conceptualization of the membership functions used in the experiments. All membership function shape parameters were chosen to suit both the numerical range and variability of available data. We used the center of gravity (COG) method for the defuzzification of each inference. Following a fuzzy modeling approach allowed us to implement simple fuzzy logic rules to relate each of the indicators (pertaining to a particular FIS) with a specific fuzzy outcome according to their empirical relationship. In this way, each FIS is composed of a set of fuzzy rules (if-then type), which relates the indicator's fuzzified values with an empirical fuzzy consequence.