# Physical Vulnerability and Landslide Risk Assessment in Tegucigalpa City, Honduras
## 1 Introduction  
### Conceptual Framework  
Disaster risk is defined as \"the potential loss of life, injury, or destroyed or damaged assets which could occur to a system, society or a community in a specific period, determined probabilistically as a function of hazard, exposure and vulnerability\" [1]. This definition can be expressed quantitatively by the following expression [2, 3]:  
\[DR=\sum\bigl{(}H\sum(VE)\bigr{)} \tag{1}\]  
where \(DR\) is the disaster risk; \(H\) is the hazard, expressed as the probability of occurrence of a hazard intensity value [2, 3, 4]; \(V\) is the physical vulnerability of a particular element for a given hazard which, in quantitative risk analyses, takes values between \"0\" (zero effects) and \"1\" (total effects) [5]; and \(E\) is the value of an element exposed to a specific hazard (e.g., number of people and cost of buildings). In the case of infrastructure, the replacement value is typically utilized, which is the value of a similar good at the current market price [6]. Due to the probabilistic nature of disaster risk, uncertainty plays a crucial role in disaster risk assessment models. Uncertainty is typically categorized into two fundamental types: random and epistemological [7]. Random uncertainty stems from the inherent variability of the system under study, while epistemological uncertainty stems from a lack ofknowledge about the system. Probabilistic disaster risk assessment models have developed various techniques to manage random and epistemological uncertainties in each step of the analysis. The losses caused by disasters can be categorized into social, economic, and environmental losses [8]. Social impacts include the loss of human lives, injuries, psychological effects, and the loss of cultural heritage. Economic impacts are divided into direct and indirect losses. Direct losses, referred to as damages, encompass the loss of goods, while indirect losses involve the disruption of the flow of goods and services. Moreover, there are also macroeconomic impacts on variables such as the Gross Domestic Product. Many models and risk assessments focus primarily on direct economic losses or damages, which can be quantified monetarily [8]. This is due to the challenges associated with incorporating intangible losses into the analysis, the controversy over the economic valuation of human life, and the complexity of estimating indirect economic losses or environmental impacts [8]. Loss exceedance curves show the relationship between risk values (expressed as probable losses) and the likelihood that a given loss level will be surpassed in a given year [9; 10; 11]. The integral of the loss exceedance curve is the average annual loss (AAL) [9; 10; 11]. The AAL is crucial for the design of cost-effective disaster risk management policies that involve multiple measures, such as investments in risk reduction, emergency preparedness, emergency funds, and insurance [10; 11]. Slides are a type of slope movement characterized by a displacement along one or multiple failure surfaces with limited deformation of the soil involved. If the surface is parallel to the ground, it is defined as a translational slide, while, when it is cylindrical, it is a rotational slide. Quantitative disaster risk studies for slow-moving translational and rotational landslides are particularly scarce. This is due to the difficulty in estimating their movement intensity (e.g., speed) and characterizing vulnerability due to a lack of data [2; 3; 12; 13]. Landslide hazard can be characterized as follows (adapted from [PERSON] et al. [2]):  
\[\mathbf{H}=\sum\mathbf{P(Mi)}\mathbf{P(Xj|Mi)} \tag{2}\]  
where _P(Mi)_ is the probability that a landslide with magnitude \(i\) occurs and _P(Xj|Mi)_ is the probability that the landslide of magnitude \(i\) reaches a point at a distance \(X\) with intensity j. _P(Mi)_, the probability of the magnitude, depends on a combination of spatial and temporal probabilities. Spatially, it is determined by physical conditions such as the slope, type of soil, and so on.