Physical Vulnerability and Landslide Risk Assessment in Tegucigalpa City, Honduras

Abstract

Quantitative disaster risk studies for slow-moving rotational and translational landslides in small regions (e.g., cities and watersheds) are very scarce. The limitations of risk modeling associated with these hazards include (i) the lack of data for physical modeling, (ii) methodological restrictions on estimating landslide intensity with statistical models and determining the temporal probability of landslides, and (iii) the absence of characterizations of the physical vulnerability of exposed assets. The present study combines and updates different methodologies to overcome these limitations for quantitative landslide disaster risk estimation, creating a novel methodological approach that was applied in a pilot study in Tegucigalpa city, Honduras. Tegucigalpa, the capital city of Honduras, has the highest number of recorded landslides in the country. In a previous study, landslides were found to be mainly concentrated in areas with colluvium and residual soils. As an input for the disaster risk assessment, this study generated landslide risk vulnerability functions based on empirical data. The application of the proposed methodology allowed us to estimate the average annual loss (AAL) caused by landslides in the study area—a key disaster risk metric that is lacking in other landslide disaster risk studies—enabling comparisons with disaster risk estimates associated with other hazards. In particular, the AAL value obtained for the study region was USD 7.26 million.

1. Introduction

1.1. 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]:

$$DR={\displaystyle \sum }\left(H{\displaystyle \sum }(VE)\right)$$

(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 of knowledge 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 Corominas et al. [2]):

$$\mathit{H}={\displaystyle \sum }\mathit{P}\left(\mathit{M}\mathit{i}\right)\mathit{P}\left(\mathit{X}\mathit{j}|\mathit{M}\mathit{i}\right)$$

(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. Temporally, it is based on the likelihood of a triggering event happening.

1.2. Research Background

Most landslide hazard studies have focused on using statistical techniques to estimate landslide susceptibility—that is, the spatial probability of the occurrence of a landslide—with a recent trend toward the use of Machine Learning (ML) [14,15,16], as promising classification results have been obtained using ML algorithms such as the Maximum Entropy Model [17,18]. Some authors argue that deterministic methods, based on physical laws controlling landslides, may be more suitable for estimating the spatial probability for limited regions such as watersheds and cities, as they can predict new landslides independently of historic inventories [3,19]. However, the lack of data on soil mechanical properties and their spatial variability limits the application of these methods [3,16]. Furthermore, there is a significant scarcity of studies estimating the probability of landslide magnitude P(Mi), in particular for slow-moving landslides [2,3,15], and landslide vulnerability [2,5,14,20]. This absence of methodologies for estimating intensity and vulnerability contributes to the overall scarcity of quantitative slow-moving landslide risk assessment studies. It can be concluded that there is currently no widely disseminated methodology or platform for quantitative estimation of the probable losses due to slow-moving landslides [14,15,16].

1.3. Study Area

Tegucigalpa, the Municipality of the Central District (MCD), is the municipality in Honduras with the highest number of landslides. The city has undergone rapid expansion in recent years, resulting in the location of neighborhoods of the lowest social classes on the outskirts of the urban area. These areas are often on steeply sloping terrain, which makes them more susceptible to landslides [21]. In the MCD, single-story housing predominates and housing construction on several levels is still very scarce [21].

In the period between 1966 and 2016, the MCD experienced 36% (237 events) of the total landslides nationwide, affecting 13,644 people, resulting in 55 casualties and the destruction of 875 homes [22]. Especially devastating were the landslides associated with Hurricane Mitch, which occurred in the last few days of October 1998 and caused events such as the El Berrinche landslide. These landslide events were both rotational and translational, and directly or indirectly affected the entire population of the city [23].

After Hurricane Mitch, several studies were carried out to characterize landslide-prone areas. A critical review of landslide studies was conducted in a previous publication [24], resulting in the development of a new inventory of slow-moving translational and rotational landslides. The focus on slow-moving translational and rotational events was emphasized, due to the historical concentration of losses in the study area associated with these types of landslides. The study found that slow-moving translational and rotational landslides were primarily concentrated in colluvium deposits and residual soils. Consequently, a susceptibility map applying a bivariate statistical approach was developed, based on the colluvium and residual soil coverage. Using the success curve method, the study found that this susceptibility map was more effective in explaining the spatial distribution of translational and rotational landslides than previous susceptibility maps, with an area under the curve (AUC) of 88.64.

Although different landslide studies have been conducted in the MCD, the city lacks access to a map identifying landslide risk [21]. This information is crucial for evaluating the significance of the issues related to these hazards and to analyze the costs and benefits of implementing measures to mitigate or manage them.

The present study aims to assess the risk of landslides in the MCD and estimate the potential direct economic losses associated with them, taking into account physical vulnerability.

The study area, which is shown in Figure 1, spans over 260 km² and includes the entire MCD urban sprawl. The MCD is situated on a plateau surrounded by steep slopes, with 62% of the study area having slopes greater than 20% [21]. The region experiences rainfall mainly between May and October, with an average annual rainfall of 870 mm [25].

The study area is characterized by three main geological formations [26]: (i) the Valle de Ángeles group, which is from the High Cretaceous period; consists of shale, sandstone, and reddish conglomerate; and is further divided into two formations—namely, the Villanueva Formation and the Rio Chiquito Formation; (ii) the Padre Miguel group, which is from the Middle Miocene period and consists of tuff and ignimbrite; and (iii) andesite and basalt, most likely from the Holocene period. Deposits of quaternary alluviums and colluviums are also found atop these formations, as shown in Figure 2.

2. Materials and Methods

2.1. Landslide Hazard Estimation

Rotational and translational landslide hazards can be evaluated by analyzing landslide inventories by means of a statistical spatial and temporal probability model [27,28,29,30]. These models estimate the term P(Mi) of the landslide hazard equation. When the landslide is spatially well defined and its velocity is slow, the term P(Mi) can be considered as being equivalent to the landslide hazard [2]. These models combine the spatial and temporal probability of landslides by considering them as two separate conditional probabilities [27,28,29,30]. The hazard is estimated by combining susceptibility (calculated using a statistical model), the probability of the magnitude (calculated using a power law), and the temporal probability (determined using a landslide inventory) [27,28,29,30]. However, this approach has some limitations. The power law is not based on empirical data and the methodology produces different maps for each landslide magnitude (area) per return period, which is impractical for risk communication [29]. Additionally, the spatial and temporal probabilities are estimated using the inventory as an input, potentially affecting the independence between the two probabilities—a precondition of this method [29]. The present study proposes some modifications to this conceptual model to overcome these limitations, including simplifying the formula by eliminating the landslide magnitude probability term. This is feasible as the combination of landslide intensity and vulnerability is characterized by a damage function. Furthermore, the use of a physical model to estimate the spatial probability contributes to the independence between spatial and temporal probabilities. The formula proposed to integrate these probabilities is as follows:

$$H=P_{t } · P_s$$

(3)

where H is the landslide hazard, P_t is the temporal probability, and P_s is the spatial probability.

Considering that a previous study showed that translational and rotational landslides were concentrated in the colluvium and residual soil layer, the landslide risk analysis was limited to this layer (Figure 3), which covers 39 km² (or 15%) of the study area.

2.2. Spatial Probability

A physical model was applied to characterize the spatial probability (Pt). For this purpose, the PISA-m program was utilized [31]. This program integrates a physical model based on the infinite slope and treats the uncertainty of input parameters using the first-order second-moment method (FOSM). This method estimates the probability that the safety factor (SF) is less than 1 in each pixel. The equation used for the infinite slope is as follows [31]:

$$SF={\displaystyle \frac{c_{r }+c_{s }+\left[q_{t }+{{\rm Y}}_{m}D+\left({{\rm Y}}_{sat}-{{\rm Y}}_w-{{\rm Y}}_m\right)H_{w}D \right]co{s}^2\mathsf{\beta } \tan Ø}{\left[q_{t }+{{\rm Y}}_{m}D+\left({{\rm Y}}_{sat}-{{\rm Y}}_{m }\right)H_{w}D\right]\sin \mathsf{\beta } \cos \mathsf{\beta }}}$$

(4)

where SF is the safety factor; c_r is the cohesive strength of tree roots (strength/area); c_s is the cohesive strength of the soil (strength/area); q_t is the uniform load due to the weight of vegetation (strength/area); Υ_m is the unit weight of wet soil (weight/volume); Υ_sat is the unit weight of saturated soil below the water table (weight/volume); Υ_w is the unit weight of water; D is the thickness of the soil above the breaking surface (length); H_w is the height of the water table above the breaking surface normalized by the thickness of the soil (dimensionless); β is the slope (degrees); and Ø is the internal friction angle (degrees).

In order to use the PISA-m program, three raster coverages were created, all having the same extension and pixel size. These included (i) Digital Elevation Model (DEM) coverage; (ii) “soil” coverage, containing the colluvium and residual soil layer; and (iii) “trees” coverage, showing the different types of vegetation cover in the study area. The tree resistance properties were considered to be zero, as there is no significant tree coverage in the study area. The DEM was created with contour intervals of 10 m and a pixel size of 10 × 10 m. The contour data were provided by the municipality of Tegucigalpa and covers the entire study area. The layer of colluvium and residual soil was produced through geomorphological mapping, involving the classification of landforms via photo interpretation of aerial photos at a 1:10,000 scale, topographic analysis, and fieldwork [17]. In regard to those coverages, there were no missing data.

In addition, a parameter file was generated for each type of colluvium and residual soil. These geotechnical parameters were determined using average values of local laboratory tests collected in the study area for each type of soil. It was assumed that these parameters followed a normal distribution, which is one of the requirements for applying the FOSM method [32]. The value of H_w was assumed to be 1, indicating the most unfavorable condition, where the ground is completely saturated.

In a previous publication [33], the infinite slope model has been used to characterize the SF in the study area. The results indicated that the model tends to underestimate landslide areas in low-slope terrains. Therefore, a review of the modeling results was carried out using expert criteria, in order to manually include potential landslide zones that were classified as stable by the model. The applied methodology was geomorphological mapping to classify certain landforms as landslides by means of expert criteria, combining aerial photo interpretation, topographic maps, and field observations.

2.3. 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 Valenzuela 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 Zêzere 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.

$$F\left(x\right)=e^{-e^{-\alpha \left(x-\mu \right)}}$$

(5)

The value F(x) represents the probability that the function will take a value smaller than x, while α and μ 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 Khan et al. [40], the method of moments was selected. This uses the following equations to estimate the parameters of the probability distribution function.

$$\alpha ={\displaystyle \frac{\pi }{\sigma \sqrt{6}}}$$

(6)

$$\mu =\overline{x}-{\displaystyle \frac{{\rm Y}}{\alpha }}$$

(7)

Here, Υ is Euler’s constant (0.5772), $\overline{x}$ is the average, and σ is the standard deviation.

The return period (T) for each antecedent rainfall indicator was obtained with the following formula.

$$T={\displaystyle \frac{1}{1-e^{-e^{\alpha \left(x-\mu \right)}}}}$$

(8)

2.4. Integration of Spatial and Temporal Probability

To determine the probability of a landslide being triggered, an equation for integrating the probability of SF < 1 and the temporal probability was adapted [36]. The temporal probability is represented as the average likelihood of triggering a landslide in a pixel located in a high-susceptibility area and differs for each precipitation return period. This probability is assumed to be equal for all pixels within the high susceptibility area. On the other hand, the spatial probability is the likelihood of a pixel being affected by a landslide based on its location, and it was assumed to be equal to the probability of SF < 1. The original formulation [36] used a statistical model to estimate the spatial probability. The revised approach involves using a physical model for this estimation. The applied formula is as follows:

$$Fp{x}_j{l}_i={\displaystyle \frac{T{a}_i}{Ts}} F\left(\mathrm{SF}<1\right)$$

(9)

where $Fp{x}_j{l}_{i }$ is the conditional probability that pixel j will be affected by landslide l in precipitation event i; Ta_i is the number of pixels affected by landslides in precipitation event i (based on historical data); T_s is the number of pixels susceptible to landslides, which was considered equivalent to the entire area covered by colluviums and residual soils; and F(SF < 1) is the frequency at which pixel j presents SF < 1, as the result of probabilistic modeling with PISA-m. The term ${\displaystyle \frac{T{a}_i}{Ts}}$ represents the temporary probability that a slide will be triggered (P_t), and the term F(SF < 1) represents the spatial probability (P_e).

2.5. 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 m² and an average area of 147,116 m² per landslide. The selected surveys were localized on three active landslides in the inventory: (i) a landslide in the La Ulloa and José Arturo Duarte neighborhoods, (ii) a landslide in the El Edén 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]:

$$n={\displaystyle \frac{Z^{2}NP\left(1-P\right)}{\left(N-1\right)e^2+Z^{2}P\left(1-P\right)}}$$

(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.

Based on the types of housing materials identified in the surveys and the classification of neighborhoods according to their social status, as defined by BID [21], the MDR was estimated for the 8 states of damage for 4 types of one-story housing: (i) popular brick, (ii) precarious brick, (iii) block, and (iv) wood. Based on the survey data, the mean and standard deviation of the MDR for each type of housing were obtained.

Vulnerability functions should consider various sources of epistemological and random uncertainty [13,45]. The epistemological uncertainties include those related to the simplifications made to characterize the intensity/magnitude of the movements or the models used to describe the response of the structures. Random uncertainties are related to variability associated with (i) the intensity/magnitude of the movement and (ii) the response of the infrastructures located on landslides. The variability of the landslide’s intensity and its impact on overlaying infrastructures is particularly significant for slow-moving rotational and translational landslides. In slow-moving landslides, although the entire landslide might be in motion, the speed and the associated damage can vary significantly on different areas of the landslide body. Furthermore, the degree of damage will depend on whether the landslide shear zone intersects the infrastructure [3,13]. If the landslide displacement occurs in blocks with practically no deformation, the overlying infrastructure may experience limited damage [3,13].

To manage these uncertainties, the damage expressed by the MDR for the condition of a landslide occurring was considered as a random variable, assuming that it follows a Beta probability density function [11,46] and using the values of the mean and standard deviation of the MDR obtained to estimate the parameters of the Beta function (i.e., α and β). Treating damage as a random variable makes it possible to manage not only the uncertainty of how houses will respond to a specific landslide intensity but also the uncertainty of the intensity of the movement within the landslide itself. The formulas considered were as follows [47]:

$$B \left(\alpha , \beta \right)={{\displaystyle \int }}_0^1{x}^{\alpha -1}{\left(1-x\right)}^{\beta -1}dx$$

(11)

$$\alpha =\overline{x}\left\{\left[\overline{x}\left(1-\overline{x}\right)/s^2\right]-1\right\}$$

(12)

$$\beta =\left(1-\overline{x}\right)\left\{\left[\overline{x}\left(1-\overline{x}\right)/s^2\right]-1\right\}$$

(13)

where B is the Beta distribution, which takes values between 0 and 1; α and β are the parameters of the Beta distribution; $\overline{x}$ is the mean or expected value; and σ is the standard deviation.

2.6. Risk Estimation

The values per m² 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:

$${\left(P|l\right)}_i=e\left({\displaystyle \sum }{N}_{xj}{V}_{xj}\right)$$

(14)

$$E\left[L|l\right]={\displaystyle \sum }_i^N{\left(L|l\right)}_i{F}_i$$

(15)

where (P|l)_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[L|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:

$$(L|l) p{x}_j={\displaystyle \frac{E\left[L|l\right]}{N_{px}}}$$

(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]):

$$Lp{x}_{ji }=Fp{x}_j{l}_i \left(L|l\right)p{x}_j$$

(17)

where Lpxji is the loss in pixel j given precipitation event i; Fpxjli 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. To facilitate the spatial analysis calculations, all layers were rasterized into 10 × 10 m pixels.

The total value of losses L for each precipitation event i was obtained by adding the loss values at the pixel level with the following formula.

$$L|Even{t}_i={\displaystyle \sum }_{j=i}^n{L}_{pxji}$$

(18)

The loss values for each return period and their annual exceedance probabilities were integrated through a loss exceedance curve, following which the area under the curve was calculated to estimate the expected annual loss for the entire study area, using the following formula (adapted from Cardona et al. [46]):

$$AAL={\displaystyle \sum }_{i=n}^n\left(L|Even{t}_i\right)F_a\left(Even{t}_i\right),$$

(19)

where AAL is the average annual loss, (L|Event_i) is the value of the loss for precipitation event_i, and F_a (Event_i) is the annual frequency of precipitation event i.

Estimation of the AAL as the area under the exceedance curve was performed by applying the interval sum method [49,50]. The AAL is a key metric for risk analysis and is calculated in catastrophic risk models for different hazards such as hurricane, floods, and earthquakes. However, it had not been previously used in landslide disaster risk studies, due to difficulties in estimating the losses associated with different return periods. The methodology developed in the present study overcomes these limitations.

The proposed risk assessment is focused on quantifying direct losses or damages caused by slow-moving translational and rotational landslides. This may result in underestimation of the total risk associated with those hazards, as it does not consider social, environmental, or indirect economic losses. It is important to note that, as the risk is associated with slow-moving landslides, the likelihood of injuries and loss of human lives will be practically non-existent. Additionally, the areas exposed to slow-moving landslides are housing areas and, so, the economic activity is minimal, limited to small micro-enterprises located within homes. The interruptions of communication routes and their economic impact are also expected to be limited, as road closures due to landslides will be very localized events. Furthermore, the damage approach aligns with other widely disseminated catastrophic risk analysis models, such as Hazus [51,52] or CAPRA [46], contributing to the methodological comparability of the model.

3. Results

3.1. Spatial Probability

Figure 6 shows the results of modeling the spatial probability of SF < 1 with the PISA-m program.

Table 1. Average and standard deviation of the SF < 1 probability (between 0 and 1) for the different layers.

Layer	$\overline{\mathit{X}}$	σ
Colluviums and residual soils	0.37	0.26
Validated landslide inventory	0.44	0.25
Colluvium of basalts, andesites, and rhyolites	0.42	0.28
Residual soil from Valle de Ángeles group	0.4	0.23
Colluviums and residual soils	0.3	0.22

summarizes the average probability that SF < 1 for the different types of colluviums and residual soils and the inventory of translational and rotational landslides.

The highest spatial probability value corresponded to the landslide inventory, with a value of 0.44, followed by the colluvium of basalts, andesites, and rhyolites (0.42). The average probability value for all colluvium and residual soil cover was 0.37.

As a result of the review of the model with expert criteria based on geomorphological analysis, 14 potential landslides, classified by the model as stable, were identified. These landslides, totaling 2,861,296 m², were manually incorporated into the spatial probability model. They represent 7% of the total area of colluviums and residual soils. It was assumed that the probability of these landslide polygons was equal to that of the escarpment surrounding them (Figure 7).

3.2. Temporal Probability

A total of 52 landslide episodes in neighborhoods located on the layer of colluviums and residual soils were identified, with 70% of the episodes characterized by one or two movements.

Based on the antecedent rainfall indicators, the return period correlated with each number of landslides was estimated. In total, values were obtained for six return periods. The temporal probability (Pt) of a landslide being triggered in any pixel of the colluvium and residual soil layer was estimated for each return period; see the results provided in

Table 2. RP: return period. NLS: N⁰ of landslides. AD (pix): N⁰ of pixels in the landslides. Pt: temporal probability at the pixel level.

RP: (Years)	NLS	AD (pix) (Tai)	$\mathit{P}\mathit{t} \left(\frac{\mathit{T}{\mathit{a}}_{\mathit{i}}}{\mathit{T}\mathit{s}}\right)$
125	29	42,664	0.109
6	20	29,423	0.075
4	13	19,125	0.049
2	9	13,240	0.034
1.3	3	4413	0.011
1	1	1471	0.004

.

3.3. Vulnerability to Translational and Rotational Slides

The total number of dwellings estimated in the three landslides with surveys was 805. Using Equation (10) and considering the 323 available surveys, a sampling error of 4.22% was calculated. The population N was considered to be the total number of houses present on the three active landslides at the time of the survey implementation.

Using the values of the surveys, the mean and the standard deviation of the MDR were estimated and the parameters of the Beta function were obtained for each of the types of housing considered. The results are summarized in

Table 3. Average, standard deviation, and parameters of the Beta function for each type of housing.

House Type	$\overline{\mathit{X}}$	σ	α	β
Precarious brick housing	0.53	0.43	0.18	0.16
Wooden housing	0.47	0.38	0.35	0.40
Popular brick housing	0.42	0.34	0.46	0.63
Block housing	0.35	0.34	0.33	0.62

.

3.4. Risk Estimation

Table 4. Values exposed to landslides by type of neighborhood.

Neighborhood Type	Area (ha)	Exposed Houses 2020	Exposed Population 2020	Exposed Population 2020 (%)	Housing Area (103 m2)	m2 Value (USD) 1	Exposed Value (Million USD) 1	Exposed Value (%)
Residential	43.7	1197	3755	2	305.8	1246.0	381	30
Middle class	56.2	3170	11,638	7	337.4	940.5	317	25
Popular	132.7	9431	39,316	24	929.1	357.5	332	26
Precarious	606.6	24,072	109,566	67	4852.9	46.3	224	18
TOTAL	839.3	37,870	164,275	100	6425		1255	100

¹ USD values from 2020.

summarizes the results of the population projected for 2020 and the economic value exposed to landslides in the study area. The projected total population for 2020 in the MCD is 1,345,117 inhabitants and 328,879 households. The number of inhabitants and houses exposed to landslides represent 12% and 11.5% of the total population and housing, respectively. The total value of MCD homes exposed to landslides is estimated at USD 1255 million, representing 4.2% of the USD 30,195 million total housing stock replacement value for 2020.

The expected value of the economic loss conditioned to a landslide occurrence for each type of neighborhood is summarized in

Table 5. Relationship between exposed value and expected loss by type of neighborhood.

Type of Neighborhood	E [L|l] (MUSD)	% EV
Residential	137	36
Middle class	139	44
Popular	146	44
Precarious	113	51
TOTAL	535	43

. As a result of the housing vulnerability (characterized by the MDR), the relationship between the expected loss and the exposed value is lower for residential neighborhoods (36%) and higher for precarious neighborhoods (51%), with the same value for residential and middle-class neighborhoods (44%).

Figure 8 shows the expected value of the economic losses at the pixel level. The residential neighborhoods located to the north of the study area (on the slopes of El Picacho hill) show the highest probable loss value at the pixel level, ranging between USD 25,188 and 1,044,164.

To estimate the probable loss for each return period (125, 6, 4, 3, 2, and 1 year), the spatial and temporal probability (value Pt in Table 2) and expected value of the economic losses maps were multiplied. The result of the estimation of probable losses for the 125-year return period is shown in Figure 9.

The estimated total value of losses was determined for each return period. The loss exceedance curve (Figure 10) was constructed using the total value of losses and the return period expressed as annual exceedance probabilities. The AAL was calculated as the area under the loss exceedance curve following the interval method, which is summarized in

Table 6. Summary of the values used to estimate the average annual loss (AAL) using the interval method. LI: lower limit. LS: upper limit. VI: interval value. Q: probability.

LI 1	LS 1	VI 1	LI 2	LS 2	P 2	AAL 1
0.78	2.1	1.3	1	0.77	0.23	0.23
2.1	6.6	3.7	0.77	0.5	0.27	1.01
6.6	9.6	7.9	0.5	0.25	0.25	1.99
9.6	14.6	11.8	0.25	0.16	0.08	0.99
14.6	21.3	17.6	0.16	0.008	0.16	2.79
21.3	n/a	21.3	0.008	0	0.008	0.17
				TOTAL AAL 1		7.26

¹ USD, ² probability values.

. The value obtained for the AAL was USD 7.26 million.

4. Discussion

While probabilistic models have been widely utilized for disaster risk estimation, their use requires making several assumptions and managing uncertainties regarding the hazards, the homogeneity of the exposed assets, and their vulnerability. For slow-moving landslides, there is a lack of vulnerability data on exposed assets and difficulty in estimating the intensity of landslides and their temporal probability, making disaster risk studies associated to these hazards particularly scarce. However, a conceptual model was developed in this study to analyze the disaster risk associated with slow landslides, which overcomes many of the identified limitations.

Additionally, using a physical estimation model for spatial probability estimation provides significant advantages over statistical models. Physical models do not rely on the principle of actualism, allowing for the identification of potentially unstable areas which may not be revealed by a statistical model. They also do not depend on an inventory of slope movements for development, contributing to the independence of the temporal and spatial probabilities. Moreover, using a physical model helps to overcome one of the main limitations of landslide susceptibility statistical models: the use of variables that are widely available but may not have the highest explanatory capacity [35].

The main assumptions made for this model included the following: (i) the soils were completely saturated; (ii) the data of the events recorded in the press represent the events that occurred in the study area; and (iii) the soil parameters follow a normal distribution.

Assuming that the soils were completely saturated, it is important to achieve independence between temporal and spatial probability. This means that the spatial probability model should not include variables that change over time. It is reasonable to assume that, for this movement to occur, there must be a significant level of saturation in the soil, as it is the main triggering factor. However, the assumption of total saturation could lead to an overestimation of spatial probability.

On the other hand, the data used to estimate temporal probability may contribute to an underestimation. Databases based on press reports, such as the one in this case, have a bias toward slope movements that caused damage and do not include other events that might have occurred in sparsely populated areas [34].

In relation to the assumption of the FOSM method that the different variables follow a normal distribution, this is always assumed when applying this probabilistic technique.

The primary sources of uncertainty for estimating the risk of slow-moving disasters include (i) random uncertainty related to the intensity of movements and the response of the house, as well as epistemological uncertainty due to the limited amount of data available to characterize the potential damage that landslides could cause on dwellings across the study area, and (ii) epistemological uncertainty due to the lack of knowledge regarding the input parameters for the Infinite Slope model and its applicability to the characterization of landslides in the study area. As part of the risk modeling design, various techniques have been proposed to address these uncertainties.

The random uncertainty associated with the intensity of the hazard and the response of infrastructures and the epistemological uncertainty linked to the generalization of a limited number of home damage surveys to all homes in the study area are common issues faced when using probabilistic disaster risk models for small areas such as cities or watersheds. In these models, vulnerability functions are built with a limited amount of empirical data or limited results from physical models. Due to the limited availability of data, it is not possible for these data to be statistically representative. To handle this source of uncertainty, the data are used to parameterize a statistical function that characterizes vulnerability and damage as a random variable, in a similar way to the method used in this study.

Regarding the epistemological uncertainty related to the use of a limited number of soil trial data to characterize the mechanical properties in the study area, probabilistic techniques were applied to manage the uncertainty associated with these data. Additionally, the soil mapping was a key input in the reduction in uncertainty associated with the mechanical properties, as it contributes to the definition of homogenous units. In a previous study, modeling was conducted on some pilot landslides to reduce the epistemological uncertainty related to the suitability of the infinite slope model to represent the landslides in the study area. This analysis revealed that the model has biases, particularly underestimating the landslide extension in low-slope areas. This bias was addressed by applying geomorphological methods to incorporate these low-slope areas into areas characterized by high landslide probability.

Validation of a probable loss estimate is complex, as it represents future values. In this context, the use of historical data for probabilistic risk validation has some limitations [42]. One of the main limitations is that probabilistic modeling represents future losses, which can exceed historic losses. Furthermore, in the study area, there is no information on historical losses attributed to landslides. The DesInventar database does not include loss information, and the existing national damage assessments do not discriminate losses associated with landslides. In a previous study, the significance of the colluvium and residual soil layer in explaining the spatial distribution of the landslide inventory was validated, obtaining an AUC of 88.64%. However, the subject of the present study is the estimation of probable losses and, so, this is the indicator that should be validated. One option to estimate the accuracy of the results could be to compare them with the probable losses obtained for the same study area using a different method [42]. However, to date, the present study is the only one providing an estimate of landslide risk made in the MCD. On the other hand, there are no widely accepted methods for landslide risk estimation that can be considered as a standard to validate the results of the present study [14,15,16].

A flood risk assessment was carried out in 2015 for the same study area [21]. As the estimation of flood risk is based on reliable and widely accepted methods [53], the comparison of the results provides a benchmark for validating the findings of the present study.

The results of the flood risk assessment were projected for 2020, obtaining an AAL of USD 6.34 million, with an exposed population of 92,531 people. The risk expressed as AAL was similar for both landslides and floods (USD 7.26 million and USD 6.34 million, respectively), although it was 14.5% higher for landslides. This difference in AAL can be attributed to the higher exposed population for landslides, which is 77% greater than that of floods.

In another study [54], which used physical methods to estimate risk for debris flows, a loss of EUR 3.11 million was calculated for an exposed value of homes, amounting to EUR 14.9 million over a 100-year return period, representing 20% of the exposed value. In the current study, the loss values for a return period of 125 years (USD 21.3 million) represent 2% of the exposed value (USD 1255 million), which is reasonable given that slow-moving landslides are far less destructive than debris flows.

A complete external validation of the risk estimation results was not possible due to inherent validation difficulties and data limitations in the study area. However, the reasonability of the results was confirmed by consulting external sources of information.

When evaluating the expected loss data, it is essential to consider the varying impact these losses have, according to the socio-economic status of families. Research from around the world has demonstrated that, when individuals living in poverty experience disasters, the relative impact on wealth is two to three times greater compared to those not living in poverty [55]. Potential losses will have significantly different implications for precarious and residential neighborhoods, especially in a country such as Honduras, where there are high levels of inequality. In Honduras, the poorest 20% of the population only accounts for 5% of the income, while the richest 20% controls 55% of the income [56]. These inequalities should be taken into account when developing risk reduction interventions, which should prioritize the most vulnerable families.

In terms of the scope of the risk estimation results, it is important to consider that they reflect the risk in the 2020 exposure scenario. According to the Intergovernmental Panel for Climate Change [57], increased exposure and vulnerability are part of the main trends that determine the increase in the risk of climate disasters at a global level. Similarly, if measures are not taken, the risk values for slow-moving landslides estimated for the MDC for the year 2020 will increase with the growth of the population, as well as the proportion of the urban sprawl located on the colluvium and residual soils. With a projected year-on-year growth rate of 1.5% for 2020 [21], the population is expected to increase by roughly 25% over 15 years, increasing the landslide exposure along with the associated risk. Moreover, it is projected that the frequency and severity of landslides will increase with the intensification of extreme rainfall due to climate change [58]. Consequently, the results obtained from this analysis serve as a starting point for understanding the expected losses caused by slow-moving landslides in the study area, which will escalate together with increasing human exposure and the effects of climate change. Therefore, they will need to be updated to reflect changes in the urban configuration. Furthermore, the results do not consider the effects of climate change on the temporal probability of landslides, and these elements would need to be considered in future studies.

5. Conclusions

In this study, new methods were applied to estimate the hazard, vulnerability, and risk of rotational and translational landslides in the MCD while considering various sources of uncertainty. This study proposes a novel methodology that overcomes some of the limitations of previous studies and conceptual models and allows for a full qualitative estimation of the slow-moving landslide disaster risk.

The spatial probability was estimated using the infinite slope model, with a probabilistic analytical method. The probabilistic method was employed to manage the uncertainty associated with the input data, which can be a limitation when applying the infinite slope model. The average spatial probability results (expressed as the probability of SF < 1) were 0.37 in colluvium and residual soil cover and 0.44 in the validated inventory of landslides and residual soils.

Based on empirical data, probabilistic physical vulnerability functions were developed for the four existing single-floor housing types: wooden, precarious brick, popular brick, and block. The average damage (expressed as a percentage of the replacement value of the dwelling) was found to vary between 53% for precarious brick dwellings and 30% for block dwellings.

The conducted research yielded significant insights into the spatial and temporal patterns of direct potential losses due to landslides in the MCD area. By integrating the likelihood of landslides, the vulnerability of residences, and exposed values, we were able to quantify and geographically illustrate the probable economic losses linked to landslides in the MCD. The estimated AAL from landslides in the region was USD 7.26 million. Furthermore, the anticipated loss corresponding to an event with a recurrence interval of 125 years was estimated at USD 21.3 million. The proposed methodology and the vulnerability functions could be applied in future slow-moving landslide disaster risk estimation studies. As a result of the probabilistic methodology and the inherent limitations on validating the results, these values should be considered as a reference for the potential losses caused by landslides in the study area and not as exact values of the potential losses.

The application of the results of this work will contribute significantly to the reduction in landslide risks in the MCD by facilitating the strategic implementation of risk mitigation measures and their associated cost–benefit analyses. In areas with colluvial and residual soils that are currently uninhabited, restriction and control of the location of new housing and infrastructure is recommended. In inhabited areas on colluvium and residual soils, it is recommended that the indicators of possible landslides should be monitored and controlled, and also that funds should be invested in rainwater drainage to prevent the occurrence or reactivation of landslides. In addition, it would be highly recommended to raise awareness and educate the population living in areas with a high probability of landslide occurrence. These conceptual recommendations should be validated through rigorous cost–benefit analyses. The results of the present study can serve as an input for such estimations.