Landslides spatial probability

Landslides spatial probability is based on the concept of landslides susceptibility and refers to the classification, area, volume, and distribution in space of events that occurred or may occur within a designated study area. The methods for assessing landslides susceptibility are classified into qualitative and quantitative approaches. Qualitative susceptibility assessment is carried out based on a compilation of landslides data, relying on geomorphological maps, parameters, and indices, and it requires the analysis of experts. These methods are widely used (^{Fell et al., 2008}; ^{Van Westen et al., 2008}; ^{Guzzetti et al., 2012}; ^{Sheng et al., 2020}). In the quantification of landslides susceptibility, prominent methods include data processing (statistical) techniques and deterministic analysis. Data-driven analyses involve the statistical evaluation of combined factors that have produced landslides in the past, resulting in quantitative predictions for areas with similar conditions where landslides have not occurred. These methods assume that the conditions that led to landslides in the past will continue to do so in the future, and as a result, they are widely used (^{Guzzetti et al., 2005}; ^{Brenning, 2005}; ^{Catani et al., 2013}; ^{Reichenbach et al., 2018}; ^{Abhirup et al., 2020}). Deterministic methods model and simulate landslide processes using models based on slope stability analysis (^{Silva et al., 2008}; ^{Oliva & Gallardo, 2018}; ^{Oliva-González et al., 2019}; ^{Mavrouli et al., 2019}; Quiau et al., 2019; Parra et al., 2022). These methods are applicable only when there is enough topographic, geological, and hydrological data available to develop an adequate geotechnical database for the study area.

Landslide temporal probability

Although landslides are events that do not follow a particular probability model, their frequency of occurrence can be estimated using different methods. When detailed multitemporal landslide inventory maps are available, the temporal probability of landslides can be estimated through the analysis of the empirical recurrence of slope failures (^{Guzzetti et al., 2006}), using statistical or heuristic methods. The resulting classes are then associated with the temporal probability of the triggering events (^{Van Westen et al., 2011}). Some authors have established, through expert judgment, relationships between the factor of safety to the landslide and the annual probability of slope failures (^{Silva et al., 2008}).

When there is no multitemporal inventory of landslides or maps linking landslides with their triggering events, the temporal probability can be evaluated using probabilistic models that consider the occurrence of independent random events in the time (^{Crovelli, 2000}; ^{Guzzetti et al., 2006}; ^{Sheng et al., 2020}).

Probability of landslide size

The probability of landslide size is calculated by observing the relationship between the volume of landslides and their cumulative frequency, using the information obtained from the multitemporal landslide inventories that exist for the study area. To determine the probability densities, the double Pareto and inverse Gamma distributions are widely employed (^{Stark & Hovius, 2001}; ^{Malamud et al., 2004}; ^{Guzzetti et al., 2006}) but some authors have used distributions obtained through the use of the ordinary least squares method (^{Sheng et al., 2020}).

Landslides in Tijuana urban área

On average, in the urban area of Tijuana, Mexico, more than two landslides occur per year. These phenomena produce direct and indirect adverse effects on physical infrastructure, lifelines, and people, causing chaos, suffering, and a sense of helplessness wthin the affected communities, as well as large material losses in various economic sectors (Figure 1). Due to their frequency and magnitude, these events themselves constitute emergencies, the effective management of which poses growing challenges for authorities. Furthermore, they negatively impact the socio-economic structure of the municipality.

Source: authors

Figure 1. ) Some urban landslides occurred in Tijuana. Note. (a Lomas Conjunto Residencial Landslide, 2009; (b) Laderas de Monterrey Landslide, 2010; (c) Liberal Lomas del Rubí Landslide, 2018; (d) Cumbres del Rubí Landslide, 2019. 

From 1990 to September 2020, more than seventy landslides have been registered in the urban area of Tijuana. These incidents have resulted in the complete or partial destruction of over a thousand buildings, mainly homes, along with extensive damage to essential infrastructure such as road networks, electricity, drinking water, and drainage (Table 1).

Table 1 Summary of landslides that occurred in Tijuana urban area from January 1990 to September 2020, and their impacts on homes. 

Source: authors.

The Tijuana city was built on terrain with complex geological, geomorphological, and hydrological characteristics, situated within a region of great seismic activity. These factors were not adequately considered during urban planning, leading to a rapid and haphazard expansion of the city. Consequently, the city spread into hillside areas potentially unstable, as well as soils and fillings affected by previous landslides (^{Oliva et al., 2019}). The main objective of this study is to assess the hazard landslides in a neighborhood of the Tijuana city, using deterministic methods based on slope stability models and probabilistic methods based on landslides that occurred in nearby sites with similar characteristics.

Study area and data

The study area encompasses the Laderas de Monterrey neighborhood (32 ° 30’25.29 "N 117 ° 1’27.23"W), located in the center of the Tijuana urban area. The urbanized area of the neighborhood occupies an area of approximately 62,000 m2 and was built in the 1990s on land characterized by its irregular topography. The neighborhood’s buildings are individual homes averaging two levels in height and more than 150 m2 of the living area each. On May 3rd of 2010, a landslide occurred in this neighborhood destroying a total of eleven homes and causing significant damage to another eight. In addition, the road network and public services were seriously affected by the destruction of three streets and damage to the drinking water, electricity, and drainage networks (Figure 2).

Source: authors

Figure 2. (a) Laderas de Monterrey neighborhood before and after the landslide. Note. Location of the neighborhood in the urban area; (b) neighborhood in 2006; (c) neighborhood in 2019; (d) main escarp, (e) body and (f) foot of the landslide. 

The study area was selected because it was affected by the 2010 landslide and because it currently shows signs of geotechnical and structural pathologies which are evidence of terrain instability and warn of the possibility of new landslides, increasing concerns about the dangers that these phenomena pose for at least 46 homes and the more than 163 residents of the neighborhood. From the analysis and interpretation of the results of different studies and investigations carried out in the area from 2010 to date, data, parameters, and information related to the topography, geomorphology, geology, geotechnics, and hydrology of the study area were obtained.

Topography and geomorphology

The study area is located on a hill cut by erosion channels, indicative of the land’s susceptibility to physical and morphological transformations. The neighborhood has been established on a hillside with an average slope of 39 % that is part of the tributary micro-basins of the La Piedrera creek. Figure 3 shows the geomorphology of the site and the digital terrain elevation model obtained using UVA system (^{Escalante, 2016}).

Source: authors

Figure 3 Topography and geomorphology of the study area. 

Geology and geotechnics

At the base of the hillside, there is the typical lithology of the Otay Geological Formation, made up of medium-grained and light-gray color tobaceous sandstones; while in the upper part of the outcrop, the lithology is that of the San Diego Geological Formation, which is characterized by clayey sands with intercalations of boulders, gravel, and thin layers of clay. Recent geotechnical and geophysical studies indicate the presence of significant amounts of soft, low-strength materials, products of the collapse and removal of the natural terrain due to the landslide in 2010 and as a result of the deterioration of the surface soils and fillings due to weathering (Figure 4).

Table 2 shows the geotechnical parameters of the terrain, obtained with field and laboratory tests.

Source: authors

Figure 4 General stratigraphy of the study area. 

Table 2 Properties of the predominant soils in the study area. 

Source: authors.

Seismic activity

In the study area, as in the entire city of Tijuana, the tectonic framework is characterized by the faults that cross the peninsular mountain ranges whose activity directly affects the urban area (^{Hungr, 1997}; ^{Cruz-Castillo, 2002}). Studies carried out indicate that one of the triggers for the 2010 landslide was the 7.2 magnitude earthquake that occurred on April 4 of that year (a month earlier), with its epicenter in the Mexicali Valley, 159 km from Tijuana (^{Oliva et al., 2014}).

Hydrology and hydrogeology

The rains in the region are scarce and there are no permanent flows of surface water. However, small flows of underground water were detected by employing onshore geotechnical investigation surveys and geophysical exploration. The flows originate in the upper part of the hillside and roughly follow the path of the bed of an old creek that was filled during the construction of the neighborhood.

Procedure to assess risk

The probability of landslides size was not considered in the calculation of the risk because there is no detailed inventory of landslides that occurred in the area where the neighborhood is located. However, records of landslides that occurred in nearby sites, as well as the topographic, geomorphological, geological, and hydrological studies carried out, provide sufficient data and geotechnical parameters to assess the risk using deterministic methods based on slope stability models. The general scheme of the methodology used is presented in Figure 5.

Source: authors

Figure 5 Scheme of the methodology used. 

Spatial probability

The spatial probability was analyzed from the concept of susceptibility to landslides and was calculated using slope stability analysis methods, through which potential landslide surfaces and their corresponding factors of safety were obtained. The stability analyses were carried out in four profiles of the study area (Figure 6), which were selected considering the geometry of the terrain surface, the presence of groundwater flows and anomalies in the subsoil, as well as the geotechnical and structural pathologies associated with processes of terrain instability.

Source: authors

Figure 6 Profiles used for slope stability analysis. 

The relationships used to estimate the spatial probability from the depth of the potential slip surface and its corresponding factor of safety (F.S) are presented in Table 3.

Table 3 Landslides spatial probability 

Source: modified from ^{Cuanalo et al., 2007}.

Temporal probability

Temporal probability was estimated considering that the failures in the slopes are independent random events in time. The Poisson model was adopted (^{Crovelli, 2000}; ^{Sheng et al., 2020}) which is expressed mathematically by equation (2).

In the equation, T is the return period, RI is the historical mean recurrence interval, t is the time interval of the landslide database, and N is the number of landslides recorded. In our case study, a database of landslides that occurred in an area of approximately 3.0 km2 was used, which has geomorphological, geological, geotechnical, and hydrological conditions similar to those existing in the Laderas de Monterrey neighborhood (Figure 7). According to database records, 10 landslides have occurred in this area in the last 20 years. Consequentlym landslides can be predicted at a rate of 0.5 per year and a mean recurrence interval (RI) of 2 years.

Source: authors.

Figure 7 Record of landslides that occurred in sites with similar conditions to the study area. 

Hazard assessment

The landslide hazard in the profiles of Figure 6 was evaluated based on spatial and temporal probabilities, using equation (3)

Landslide spatial probability assessment

Based on the results of the studies and available data, the geological-geotechnical models were elaborated to determine the susceptibility of the land to landslides in the selected profiles (Figure 8).

Source: authors

Figure 8 Geological-geotechnical models of the profiles used in slope stability analysis. 

Stability analyzes were performed using methods based on limit equilibrium, enabling the identification of the potential landslide surfaces along with their corresponding factors of safety (Table 4). The effect of seismicity activity on instability was accounted for by integrating the analysis the seismic accelerations of the ground, as outlined in the complementary technical standards of the state of Baja California (^{NTC, 2017}), as well as their amplification by site effect (^{Acosta-Chang et al., 2009}). Also, in the static condition, a 50 % saturation of the land was considered to evaluate the influence of the small underground water flows detected in the study area.

Table 4 Minimum factor of safety 

Source: authors.

Figure 9 shows the potential slip surfaces obtained in the most unfavorable scenarios using different limit equilibrium methods (^{SLOPE/W, 2012}). Likewise, Table 5 shows the spatial landslides probability obtained through slope stability analysis.

Source: authors

Figure 9 Areas of potential landslides and possible instabilities up and down slopes. 

Table 5 Landslides spatial probability in the study area. 

Source: authors.

Considering the pathologies and evidence of geotechnical instability existing in the study area and based on spatial probability values, landslide susceptibility was ordered into five classes: very low (0.00 - 0.20), low (0.20 - 0.40), moderate (0.40 - 0.60), high (0.60 - 0.80), and very high (0.80 - 1.00).

Landslide temporal probability estimation

The landslides temporal probability occurrence in the study area was estimated using equation (2), for return periods (T ) of 5, 10, 15, and 20 years, considering a mean recurrence interval (RI) of 2 years (Figure 10).

Source: authors

Figure 10 Landslides temporal probability in the study area. 

We have assumed that pass events serve as indicators of potential future occurrences. Therefore, utilizing the graph presented in Figure 10, it becomes possible to project the behavior of landslides in the studied area over the next 20 years.

Landslide hazard assessment

From the spatial and temporal probabilities, the landslides hazard in the study area was assessed, using equation (3). Table 6 and Figure 11 show the results.

Source: authors.

Figure 11 Landslide hazard for return periods of 5, 10, 15, and 20 years. 

The landslide hazard was divided into five classes, based on susceptibility and the temporal probability of occurrence in the study area: very low (0.00 0.15), low (0.15 0.30), moderate (0.30 0.45), high (0.45 0.60), and very high (0.60 - 0.75).

Under static conditions (without ground acceleration), the landslide hazard in profiles 1, 2, and 3 for return periods of 5, 10, 15, and 20 years is greater than 0.50, thus falling in the high and very high categories (Figure 11). In a return period equal to the mean recurrence interval that characterizes the area where the neighborhood is located (T = RI = 2), the hazard in profiles 1 and 2 is 0.55 (high), in profile 3 it is 0.39 (moderate) and in profile 4 it is 0.15 (low). In seismic conditions, with ground accelerations close to 0.30g, the hazard is very high throughout the study area for any return period. In these cases, large landslides will surely occur.

Table 6 Results of the quantitative assessment of the landslides hazard. 

Source: authors.