# Physical Vulnerability and Landslide Risk Assessment in Tegucigalpa City, Honduras
## 2 Materials and Methods
### Risk Estimation  
The values per m\({}^{2}\) of housing and the population in areas with colluvium and residual soils were projected for the year 2020. This was carried out by projecting the prices per square meter of construction and the population for different MCD social categories of neighborhoods from the year 2014. The 2014 population values [21] were projected for 2020 by allowing for the annual population growth rate proposed in the same study (1.5%), and the construction (square meter) values were adjusted for 2020 by applying the annual depreciation. In total, 30 units of homogeneous neighborhoods were identified, corresponding to the categories of residential neighborhood (4 units), middle class (4), popular (9), and precarious (13) [21], all of which are located on the layer of colluvium and residual soils (Figure 5). For each of the 30 units, the expected value of the economic loss in the event of a landslide was estimated using the following formulas:  
\[\left(P\left|l\right)_{i}=e(\sum N_{xj}V_{xj})\right) \tag{14}\]  
\[E[L\left|l\right|]=\sum_{i}^{N}(L\left|l\right)_{i}F_{i} \tag{15}\]  
where \(\left(P\left|l\right)_{i}\) is the loss conditioned to the occurrence of a landslide d in unit \(i\); e represents the constant exposed value of the unit (in USD); \(N_{xj}\) is the proportion (ranging from 0 to 1) of the area of the unit that corresponds to a particular type of housing \(xj\), where the sum of the proportions for each type of housing is equal to 1 for each unit; \(V_{xj}\) is the vulnerability of different housing types present in the unit, ranging from 0 to 1; \(F_{i}\) is the frequency of landslide occurrence; and \(E\) is the expected value. For each unit, a thousand \((P\,|\,l)_{i}\) calculations were performed using the Monte Carlo method. The vulnerability \((V_{xj})\) was assumed to be equivalent to the MDR. MDR was characterized as a random variable following a Beta-type probability density distribution function. The parameters of the Beta function for each type of housing were obtained from the surveys. The proportions \((N_{xj})\) of different types of housing (popular brick, precarious brick, block, and wooden) were assumed to be constant for each type of neighborhood (residential, middle class, popular, and precarious). This assumption was made based on census data and expert criteria. _E[_1_ _l_] is the expected value of the losses for the unit given the occurrence of landslide \(l\) and is estimated as the sum of the loss values obtained with the Monte Carlo model multiplied by its frequency \((F_{i})\). To estimate the frequency, the values obtained from the Monte Carlo model were divided into 20 equal intervals and the frequency \((F_{i})\) for each interval was obtained [48]. The value of the potential loss for each pixel of the unit was estimated by applying the following equation:  
\[(L\,|\,l)\,\,px_{j}=\frac{E[L\,|\,l]}{N_{px}} \tag{16}\]  
where \((L\,|\,l)\,\,px_{j}\) is the potential loss in pixel \(j\) given that landslide \(l\) occurs and \(N_{px}\) is the number of pixels in the unit. The economic loss at the pixel level given a precipitation event was estimated with the following equation (adapted from [13]):  
\[Lpx_{ji}\,\,=\,Fpx_{j}l_{i}\,\,(L\,|\,l)px_{j} \tag{17}\]  
where _Lpxji_ is the loss in pixel \(j\) given precipitation event \(i\); _Fpxji_ is the conditional probability that pixel \(j\) is affected by landslide \(l\) in precipitation event \(i\), which integrates spatial and temporal probabilities; and \((L\,|\,l)px_{j}\) are the potential losses in pixel \(j\), given the occurrence of landslide \(l\). This equation enables the integration of the conditional probability that a pixel will be impacted by a landslide, given a precipitation event, with the value of the economic loss conditioned on the occurrence of a landslide. To apply this equation, the spatial and temporal probability and probable loss maps were multiplied to obtain the probable losses for each of the return periods considered.