# Hierarchical Fuzzy MCDA Multi-Risk Model for Detecting Critical Urban Areas in Climate Scenarios
## 2 Materials and Methods
### The Heatwave and Pluvial Flooding Vulnerability Model  
In [5], to evaluate the vulnerability of an urban area of study, the urban settlement is composed of three sub-systems: the physical and constructive sub-systems of buildings and open spaces, and the social sub-system. The vulnerability of each subsystem to heatwave and pluvial flooding phenomena is assessed as a weighted average of a set of intermediate indicators whose values are all normalized on a scale from 1 to 5. Each intermediate indicator is evaluated on a set of measures based on the characteristics of the elements of the urban system. The weight of an intermediate indicator was determined based on a calibration process applied using simulation tools applied to sample elements. In Table 2, the intermediate indicators used to assess the vulnerability of the sub-systems of buildings and open spaces to the heatwave and pluvial flooding phenomena are shown. The last column shows the weights assigned to the indicators. Regarding the vulnerability of buildings to heatwaves, the thermal lag and the thermal decrement indicators measure the performance of the building envelope during the overheating of the external environment. The building volume indicator is evaluated by \begin{table}
\begin{tabular}{c c c} \hline \hline
**Phenomenon** & **Subsystem** & **Intermediate Indicator** & **Weight** \\ \hline \multirow{6}{*}{Heatwave} & \multirow{4}{*}{Residential Buildings} & Thermal lag & 5 \\  & & Thermal decrement factor & 4 \\  & & Building volume & 2 \\  & & Solar exposure of building envelope & 5 \\ \cline{2-3}  & & Sky View Factor & 5 \\  & & Albedo & 2 \\  & & Solar exposure of open spaces & 4 \\  & & Normalized Difference Vegetation Index (NDVI) & 3 \\ \hline \multirow{6}{*}{Pluvial flooding} & \multirow{4}{*}{Residential Buildings} & Building coverage ratio & 1 \\  & & Percentage of building on the sidewalk & 4 \\  & & Ground floor activities & 5 \\  & & Roof typology & 2 \\ \cline{2-3}  & & Open space area & 2 \\  & & Soil permeability & 4 \\  & & Flow capacity of storm drainage system & 5 \\  & & Maintenance of storm drainage system & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Intermediate indicators used in [5] to evaluate the vulnerabilities of buildings and open spaces to heatwave and pluvial flooding phenomena. measuring the volume of the entire building envelope. The solar exposure of a building envelope is evaluated based on the exposure of the facades and roofing to solar radiation on summer days. The vulnerability of open spaces to heatwaves is assessed, taking into consideration the effect of solar radiation on the surface and on the material that makes up the open space. The sky view factor indicator measures the portion of the sky visible from open space. Albedo measures the reflective power of the surface covering the open space. The solar exposure of an open space is evaluated based on the exposure of the open space to solar radiation on summer days. NDVI measures the relevance of live green vegetation in an open space; it is obtained from the red and near infrared remotely sensed multispectral images. As regards the vulnerability of buildings to pluvial flooding, the building coverage ratio indicator measures the presence of uncovered internal areas that can cause flooding of ground floors. The percentage of buildings on the sidewalk evaluates the presence of sidewalks around built-up areas; the more a building is surrounded by sidewalks, the less likely it is that rainwater will spill from the street into the internal areas of the building. The ground floor activities indicator evaluates the presence of activities and services on the ground floor of buildings. Roof typology evaluates the ability of the roofing slab to facilitate the outflow of rainwater, avoiding the development of infiltrations on the walls and floors below. The vulnerability of open spaces to pluvial flooding is assessed by analyzing the physical characteristics of the open space that affect its ability to collect and absorb extreme meteorological flows.