# Integral Seismic Risk Assessment through Fuzzy Models
## 1 Introduction  
The primary objective of the use of indicators and composite indices is to obtain a scale of measurement to make comparisons among geographical areas. At the same time, a suitable index is expected to reflect a plausible encapsulation of all the different dimensions described by the set of indicators that integrates it. In the case of natural risk, these dimensions could be the social or structural components that are affected by a stressor agent from which a particular level of risk is generated. Different types of indicator-based approaches are used in the assessment of natural hazard risks. A clear division exists between those approaches whose aim is to estimate each of the known risk components separately and those who look for an integrated risk assessment as the convolution of all of the potential risk sources as a single result. The UN World Risk Index [1], or the Disaster Risk Index [2], for example, fall into this last category. Multivariate methodologies to estimate social vulnerability, on the other hand, can be related to the first category [3; 4]. A common point among all these methods can be found in the way they use linear weighted methods to aggregate their indicators to obtain a final index. Many researchers have proposed a variety of indicators to assess earthquake risk to represent the different and multiple components involved, such as the physical, economic, social, and environmental dimensions within an urban environment [5; 6; 7]. However, there is no clear consensuson which are the most suitable proxies to be used in a composite index and, most importantly, to be comparable across borders [8; 9; 10]. In the field of seismic risk assessment, [PERSON]'s equation [11] proposes that seismic risk is the result of physical risk aggravated by social conditions and lack of resilience capacities. This method has been implemented to produce an integrated seismic risk index by different agencies around the world, such as the probabilistic risk model CAPRA [12] or the Global Earthquake Model of the Eucentre in Italy [13]. [PERSON]'s equation is developed as a composite index: after a selection of indicators, importance weights are related to each through a hierarchical process, and then a linear weighted aggregation method is implemented to obtain a final value. The present work proposes a fuzzy modeling approach to represent an estimation of the complex interactions between the physical and social dimensions of seismic risk. This methodology is able to obtain non-linear models that allow a more robust and realistic measure of the influences that each risk component may produce to generate a particular risk level. Therefore, it concedes a clearer understanding of how the disproportional effects of a seismic hazard are distributed over a geographical area. A key area of urban planning is to reduce disaster damage and increase safety [8], a task that can be difficult in scenarios where uncontrolled urbanization trends take place. Combined with inadequate urban planning and structural inner deficiencies, these elements have emerged as obstacles to development and preparedness for complex phenomena such as earthquakes ([14; 15; 16; 17; 18]). In order to emphasize the differences between the two methods of aggregation, we will refer to the fuzzy model as the \"fuzzy method\" and to the [PERSON]'s equation implementation as the \"index method\". The paper is distributed as follows: Section 2 discusses the common methodologies used to build composite indices, and some of their disadvantages are pointed out. The definition of [PERSON]'s linear equation and its inner structure, the basis of the proposed research, is also presented. Section 3 gives a short introduction to fuzzy logic and the properties that make it a suitable method to deal with some of the disadvantages noted in Section 2. Section 4 describes the fuzzy risk model and selected configurations.