# Integral Seismic Risk Assessment through Fuzzy Models
## 5 Results and Discussion
### Comparison Surfaces of the Fuzzy Model and the Index Method in a Synthetic Dataset  
An analysis of the differences between the solution surfaces of [PERSON]'s equation and the fuzzy model is performed. A synthetic database [33] is used to show how the linearity of [PERSON]'s equation generates a smooth surface, while a fuzzy model displays a rough surface more suitable to describe a non-linear response of the simultaneous influences of different indicators. These surfaces are shown in Figure 3.  
Figure 3: [PERSON]’s equation Surface (**left**) and Fuzzy Decision Surface (**right**) generated from a synthetic database of 100 numerical values ranging from 0 to 1 and taking finite steps of 0.02 units [23].  
\begin{table}
\begin{tabular}{l} \hline
10. If (Ag is VH) and (PhR is L) then (TR is MH) \\
11. If (Ag is L) and (PhR is MH) then (TR is ML) \\
12. If (Ag is ML) and (PhR is MH) then (TR is MH) \\
13. If (Ag is MH) and (PhR is MH) then (TR is MH) \\
14. If (Ag is H) and (PhR is MH) then (TR is MH) \\
15. If (Ag is VH) and (PhR is MH) then (TR is H) \\
16. If (Ag is L) and (PhR is H) then (TR is MH) \\
17. If (Ag is ML) and (PhR is H) then (TR is VH) \\
18. If (Ag is MH) and (PhR is H) then (TR is VH) \\
19. If (Ag is H) and (PhR is H) then (TR is H) \\
20. If (Ag is VH) and (PhR is H) then (TR is VH) \\
21. If (Ag is L) and (PhR is VH) then (TR is VH) \\
22. If (Ag is L) and (PhR is VH) then (TR is VH) \\
23. If (Ag is MH) and (PhR is VH) then (TR is H) \\
24. If (Ag is H) and (PhR is VH) then (TR is H) \\
25. If (Ag is VH) and (PhR is VH) then (TR is VH) \\ \hline \end{tabular}
\end{table}
Table 5: _Cont_.  
The lineal assumption made in [PERSON]'s equation generates a non-balanced contribution between aggravation and physical risk to the overall composite, underestimating the influence of the aggravation, particularly in its lower part.  
Let us first analyze the case in which the physical risk is zero. In this case, [PERSON]'s equation always obtains a constant total risk of zero regardless of the aggravation value. The same happens not only with a value of zero but also for small values of physical risk, with the total risk remaining practically constant regardless of the aggravation taking large values. It is important to underline that both physical and aggravation are risk factors that must have an impact on total risk. The total risk should not be null just because there is no physical risk. The previous sentence is a fundamental point that we consider to be a strong limitation of the approaches that use [PERSON]'s equation.  
In the case of the fuzzy model, the apparent dependence on physical risk values and the inherent diminishing of the influence of aggravation is no longer observed on its solution surface. Hence, the way higher aggravation values combine with physical risk to generate total risk levels is consequent and more natural, with the main assumption made in an integral risk estimation in which both dimensions should be shaping any total risk level. As we will see later, these characteristics have a vital significance in terms of the final interpretation of the role that each indicator is playing to generate a particular total risk level.  
Another interesting aspect to comment on is the shape of the surfaces. The surface obtained with [PERSON]'s equation has a very pronounced profile due, on the one hand, to the fact that for low physical risk values, the total risk remains low regardless of the aggravation values. On the other hand, for high values of physical risk, the influence on the total risk of the aggravation is very high, e.g., the total risk is doubled when physical risk and aggravation have maximum values. Contrarily, the surface derived from the fuzzy model has a softer shape, with a moderate progression. This allows obtaining more balanced and sensible model responses for close values of physical risk and aggravation.