# Integral Seismic Risk Assessment through Fuzzy Models
## 5 Results and Discussion
### Results for Barcelona  
The city of Barcelona is divided into 10 administrative districts and 73 neighborhoods. For statistical purposes, the city is also split into an arbitrary division of 273 small-sized areas called ZEP (small statistical areas, in Spanish; this division was recently replaced with a new group of 233 \"basic statistical areas\" but their statistical objectives remain the same). The left-hand side plots of Figure 4 show the spatial distribution of aggravation levels estimated for the 10 administrative districts of Barcelona according to the proposed fuzzy model (upper plot) and [PERSON]'s equation (lower plot). Both estimations show that aggravation levels in Barcelona are divided into two main areas: the highest aggravation is spread mostly over the northeast part of the city, whilst lower aggravation levels are contained in the southwest. It is interesting to note, however, how both models differ in representing the shift between the two extreme labels of these categories (from low to very high). The index method estimates the higher levels of aggravation for Districts (1), (8), (9), and (10), while the proposed method estimates the same districts, plus District (7), but with much larger aggravation magnitudes. These results, which effectively create a spatial frontier that indicates where significant aggravation levels start to grow, are important when the final risk level is estimated. Figure 5 shows the numerical aggravation levels per district obtained by the two models. Figure 4: Risk components obtained for Barcelona city by the fuzzy model (upper row) and the index model (lower part). From left to right: social aggravation, physical risk, and total seismic risk. Significant values of each component have been highlighted to show a measure of how the assumption of compensability can lead towards different risk spatial patterns. Regardless of the resemblances between the spatial configurations estimated by the two models for social aggravation and physical risk levels, it is clear how a linear aggregation produces a different total risk pattern. Figure 5: Aggravation values obtained for the 10 districts of Barcelona with the index method and the proposed fuzzy model. The center plots of Figure 4 show the estimated spatial patterns of physical risk levels for the 248 administrative zones (ZEP) utilizing the fuzzy model and [PERSON]'s equation, respectively. Both approaches show that the highest levels of physical risk are mostly contained in the central part of the city (districts: (1) Ciutat Vella and (2) Example). The most obvious difference is again in the magnitude of physical risk, as shown in Figure 6.  
[PERSON]'s equation estimates the highest magnitudes of physical risk at ZEPs contained in Districts (1) and (2). A magnitude of medium-low was estimated for a large extension of Districts (2) and (3), which seems to create a corridor of physical risk between these two districts. In contrast, the fuzzy model estimates that this same magnitude would be contained mainly towards the south and that the same corridor may exist but involves a reduced number of ZEPs. The right-hand-side plots of Figure 4 show the total risk spatial distribution of the levels calculated from both approaches. [PERSON]'s equation estimates spatial distribution of total risk with its highest values contained in the central area of the city, namely, over Districts (1) and (2). The fuzzy model also estimates higher levels in these areas but, additionally, another large area is estimated at the northeast of the city, having values labeled as medium-low. These spatial differences are worth examining more carefully since they are not just a matter of value difference but are related to how both methodologies, [PERSON]'s and fuzzy, perform the aggregation of the two dimensions of risk. We will further discuss these implications in Sections 4 and 5. Figure 7 shows total risk numerical values for the different ZEPs, where we can note the distribution of similar values between the fuzzy and Moncho methods.