# Physical Vulnerability and Landslide Risk Assessment in Tegucigalpa City, Honduras
## 2 Materials and Methods
### Vulnerability to Translational and Rotational Slides  
The disaster risk was only evaluated for single-floor housing, as it is the most common type of housing in the MCD [21]. To assess the susceptibility of homes to translational and rotational landslides, vulnerability functions were developed. Those functions determine the probability of occurrence of different levels of damage, measured as the mean damage ratio (MDR), in the event of a landslide in the area. The MDR ranges from 0 to 1 and represents the proportion of the cost to replace a damaged part of the house to the total replacement cost [11]. Data from surveys conducted by the NGO GOAL between November 2010 and March 2011 were used to construct these functions. This period was characterized by an active winter season with a significant number of landslides. Out of 1206 surveys available, 323 were selected due to being on active landslide sites. These landslides had been characterized as active in the validated inventory prepared in a previous study. This landslide inventory was prepared by the authors using geomorphological mapping, a review of previous inventories (including the interpretation of aerial photos at 1:10,000 scale), analysis of topographic maps, and extensive field validation, with the compilation of 116 points through GPS, and includes landslides which evidenced some level of activity in the period 2010-2016. The landslide inventory comprises 24 translational and 4 rotational landslides, with a total area of 4,119,267 m\({}^{2}\) and an average area of 147,116 m\({}^{2}\) per landslide. The selected surveys were localized on three active landslides in the inventory: (i) a landslide in the La Ulloa and Jose Arturo Duarte neighborhoods, (ii) a landslide in the El Eden neighborhood, and (iii) a landslide in the Reparto Arriba neighborhood (Figure 4). The sampling error was obtained by applying the equation for the calculation of sample sizes that correspond to simple random sampling [44]:  
\[n=\frac{Z^{2}NP(1-P)}{(N-1)e^{2}+Z^{2}P(1-P)} \tag{10}\]  
where \(n\) is the sample size; \(N\) is the population size, \(Z\) is the number of standard deviation units in the normal distribution that will produce the desired confidence level (for a 95% confidence level, this would be 1.96), e is the maximum error (which would be the variable to be estimated), and \(P\) is the proportion of the population that has the study characteristic (in the absence of a pilot study, the conservative hypothesis of 0.5 was assumed beforehand). The population size was assumed to be the total population in the three landslides because, at the time of the survey, these three landslides were the only ones showing activity. The main challenge concerning information about houses affected by landslides is the intermittent nature of landslide activity. To further illustrate this, it is important to mention that, during the study period (2016-2018), there was no landslide activity in the area. These surveys included a field that allowed a binary answer (yes/no) to the question of whether the walls, floor, or ceiling of the houses presented fissures. Considering these three variables, there are 8 possible combinations, which correspond to \"states of damage\": (i) no damage, (ii) fissures in walls, (iii) fissures in the ceiling, (iv) fissures in the floor, (v) fissures in the walls + ceiling (vi) fissures in walls + floor, (vii) fissures in floor + ceiling, and (viii) fissures in walls + floor + ceiling. For each of the types of housing considered, these states of damage were correlated with the value of the MDR, establishing with expert criteria which would be the elements of the house that would have to be replaced if a certain state of damage was reached and defining the cost of replacing these elements for each type of housing.