# Physical Vulnerability and Landslide Risk Assessment in Tegucigalpa City, Honduras
## 2 Materials and Methods
### Temporal Probability  
The temporal probability was calculated by assigning return periods to past landslide events. A landslide event was defined as one or more landslides that occurred on the same date and during the same rainy season [34]. The number of landslides that were part of these historical landslide events was obtained from the DesInventar database, which was supplemented by the event-type inventory compiled by the United States Geological Service (USGS) for Hurricane Mitch [35]. The DesInventar database for the MDC had 237 registers of landslides with information of the day of the event and its location. Only landslide events reported in neighborhoods with colluvium and residual soil cover were considered, as the temporal probability was only estimated for this cover.  
For the analysis of antecedent rainfall, the daily precipitation database of the rainfall station of the National Autonomous University of Honduras (UNAH) was used, which provides continuous data for the period from 1980 to 2019. In the quality control of the database, no missing data were identified. According to [PERSON] et al. [34], episodes where the previous 5 days' precipitation was less than 20 mm were eliminated from the analysis, as the relationship with a precipitation event is considered doubtful in these cases.  
The return period was obtained from the antecedent precipitation data using the methodology proposed by [PERSON] et al. [36]. Background rainfall indicators for 1, 2, 3, 5, 7, 10, 15, 30, and 60 days were selected [30; 36; 37; 38; 39], and the return period for each previous rainfall indicator was calculated by using the Gumbel probability distribution [34; 36; 40; 41] and assigning the longest return period obtained from the previous rainfall indicators to each event [36].  
The area affected by landslides in each episode was estimated using the average area of landslides from the validated inventory [42]. As it is necessary to establish a single return period for any given number of landslides, when several episodes with the same number of landslides were identified, the longest return period was considered, discarding any episodes whose return periods were not consistent with the general trend obtained.  
The Gumbel probability distribution function is as follows.  
\[F(x)=e^{-e^{-\alpha(x-p)}} \tag{5}\]  
The value \(F(x)\) represents the probability that the function will take a value smaller than \(x\), while \(\alpha\) and \(\mu\) are parameters of the function, which can be calculated using different methods (i.e., moments, Chow, least squares, maximum likelihood, or Kimball) [40; 43].  
In accordance with [PERSON] et al. [40], the method of moments was selected. This uses the following equations to estimate the parameters of the probability distribution function.  
\[\alpha=\frac{\pi}{\sigma\sqrt{6}} \tag{6}\]  
\[\mu=\overline{x}-\frac{Y}{\alpha} \tag{7}\]  
Here, \(\Upsilon\) is [PERSON]'s constant (0.5772), \(\overline{x}\) is the average, and \(\sigma\) is the standard deviation.  
The return period (\(T\)) for each antecedent rainfall indicator was obtained with the following formula.  
\[T=\frac{1}{1-e^{-e^{i(x-y)}}} \tag{8}\]