Landslides threaten the safety of vehicles on highways. When analyzing the risk of a landslide hitting moving vehicles, the spacing between vehicles and the types of vehicles on the highway can be highly uncertain and have often been omitted in previous studies. Using a highway slope in Hong Kong as a case study, this paper presents a method for assessing the risk of moving vehicles being hit by a rainfall-induced landslide; this method also allows for the possible number of different types of vehicles hit by the landslide to be investigated. In this case study, the annual failure probability of the slope is analyzed based on historical slope failure data from Hong Kong. The spatial impact of the landslide is evaluated based on an empirical run-out prediction model. The consequences of the landslide are assessed using probabilistic modeling of the traffic, which can consider uncertainties in the vehicle spacing, vehicle types and slope failure time. Using the suggested method, the expected annual number of vehicles and people hit by the landslide can be conveniently calculated. This method can also be used to derive the cumulative frequency–number of fatalities curve for societal risk assessment. Using the suggested method, the effect of factors like the annual failure probability of the slope and the density of vehicles on the risk level of the slope can be conveniently assessed. The method described in this paper can provide a new guideline for highway slope design in terms of managing the risk of landslides hitting moving vehicles.

Encompassing a total land area of about 1100 km

There are many uncertainties in the assessment of the hazard posed by landslides to moving vehicles, such as the occurrence of the landslide, the spatial impact of the landslide, the number of vehicles hit by the landslide and the type of vehicles hit by the landslide. Risk assessment is a framework in which both the uncertainties and the consequences of a hazard can be addressed, and it has been increasingly used for landslide risk management (e.g., Lessing et al., 1983; Fell, 1994; Dai et al., 2002; Remondo et al., 2008; Erener, 2012; Vega and Hidalgo, 2016). Indeed, landslide risk assessment has been accepted as an effective tool for the planning of land use in Hong Kong. Nevertheless, risk assessment for moving vehicles affected by landslides is special because the elements at risk are highly mobile. In the past, many studies have been conducted on the individual risk associated with landslides, which is often measured by the annual probability that a person who frequently uses the highway is killed by a landslide (e.g., Bunce et al., 1997; Fell et al., 2005; Dorren et al., 2009; Michoud et al., 2012; Macciotta et al., 2015, 2017). Several studies have also examined the societal risk of vehicles being hit be landslides, in which the societal risk is measured in terms of the annual probability that at least one fatality occurs in 1 year (e.g., Budetta, 2004; Peila and Guardini, 2008; Pierson, 2012; Ferlisi et al., 2012; Corominas and Mavrouli, 2013; Macciotta et al., 2019). These studies have provided both useful insights and practical tools for the analysis and management of landslide/rockfall hazards. Nevertheless, it has commonly been assumed that traffic is uniformly distributed in time and space and that each vehicle is the same length (the mean length of all vehicles; e.g., Hungr et al., 1999; Nicolet et al., 2016). In reality, there is randomness associated with the spacing of vehicles on a highway. If such uncertainties are ignored, the resulting uncertainty associated with the number of vehicles hit by a landslide cannot be considered in the risk assessment process. Moreover, there might be various types of vehicles on a highway, and different types of vehicles may have different lengths and significantly different passenger capacities. If the difference between the different types of vehicles is ignored, it might be hard to estimate the number of people (passengers) hit by a landslide, which is also an important aspect of risk assessment.

Using Kennedy Road in Wan Chai, Hong Kong, as a case study, this paper aims
to present a new method for assessing the risk of moving vehicles being hit by a
rainfall-induced landslide; this method also allows for the possible number of different types
of vehicles hit by the landslide to be investigated. In general,
the quantitative analysis of vehicles endangered by landslides includes three
scenarios: (1) a moving vehicle is impacted by falling material, (2) a moving vehicle impacts falling material on the highway, and (3) a line of
stationary vehicles is impacted by falling material (Bunce et al., 1997).
In this study, our focus is on the risk assessment of moving vehicles being
impacted by a falling landslide. The structure of this paper is as follows.
Firstly, the annual failure probability of the slope is calculated based on
historical data from Hong Kong. Then, the spatial impact of the landslide is
analyzed based on the run-out distance analysis. Thereafter, the consequences
of the landslide are analyzed via a probabilistic model of traffic. Finally,
the annual expected numbers of vehicles and people hit by the
landslide are calculated, and the development of an

The study slope is located on Kennedy Road in the Wan Chai District of Hong Kong,
as shown in Fig. 1. Wan Chai is one of the most traditional cultural areas
in Hong Kong and attracts many international tourists every year. Kennedy Road is a major road in this area with three lanes, and it
links with the Queen's Road in Wan Chai (TDHK, 2018). On 8 May 1992, the
slope failed during intense rainfall and hit a car traveling along
Kennedy Road, killing the driver (GEO, 1996). The slope is an old cut
slope formed in 1967–1968 and was covered by trees before the landslide event occurred. Figure 2 shows a typical cross section of the slope
and the landslide event. As shown in this figure, rainfall
infiltration triggered the failure of the soil mass below the retaining wall,
and the sliding mass hit the vehicle. The height of the slope (

According to TDHK (2018), vehicle traffic in Hong Kong is composed of private buses, non-franchised public buses, franchised buses, taxis, private cars, public light buses, private light buses, goods vehicles, special-purpose vehicles, government vehicles and motor cycles. The percentage of each type of vehicle with respect to the total numbers of vehicles is shown in Table 1 along with the typical length of each type of vehicle and its passenger capacity. The purpose of this case study is to analyze the annual risk of different types of vehicles being hit by a landslide if the slope fails again due to rainfall.

Location of the landslide studied in this paper.

Typical cross section of the slope and the landslide studied in this paper.

Plan view of the landslide studied in this paper.

Percentage contribution of the vehicle type to the total number of vehicles, and the length and passenger capacity of different vehicle types in Hong Kong.

There are multiple types of vehicles on highways. In a landslide critical
area of a road, the longer the vehicle, the greater the probability that
it will be hit by a landslide. Figure 4 shows the event tree model employed in
this study to assess the risk of a rainfall-induced landslide hitting type

Event tree of the annual risk evaluation of type

Based on the event tree shown in Fig. 4, the annual probability of

Let

Let

The total expected number of people hit by the landslide considering
all types of vehicles (

Equation (2) can be extended to estimate the expected monetary losses with respect to vehicles hit by a landslide when information regarding the price of different types of vehicles is available. Nevertheless, during the analysis of the risk of vehicles hit by landslides, the social impact, which can be better measured by the number of vehicles than the cost of the vehicles, is often more important than the economic losses. Hence, the risk of vehicles being hit by landslides is not measured in terms of monetary losses in this study.

Previously, the individual risk has often been used to measure the threat of a
landslide to a moving vehicle, which provides information about the
probability of a frequent user of the highway being killed by a landslide.
However, decision-makers may also be interested in the annual
expected numbers of vehicles/people hit by landslides, which can
be obtained using the method suggested in this paper. As will be shown later
in the case study, the above framework can be easily extended to calculate
the

As indicated by Eq. (1), the key points for assessing the annual risk associated with
type

The estimation of the annual landslide probability or the landslide susceptibility is fundamental in landslide hazard assessment. As almost all slope failures in Hong Kong are caused by rainfall infiltration (e.g., Lumb, 1975; Brand, 1984; Finlay et al., 1999), assessing the annual probability of rainfall-induced landslides is important. In general, there are two types of methods for evaluating the likelihood of slope failure: physically based methods involving slope stability analysis (e.g., Christian et al., 1994; Fenton and Griffiths, 2005; Huang et al., 2010) and empirical methods involving the statistical analysis of historical slope failure data (e.g., Chau et al., 2004; Zang and Tang, 2009). Currently, landslide probability analyses via slope stability analyses mainly focus on the likelihood of slope failure for a given rainfall. In reality, the occurrence of landslides in a given year is highly uncertain. Currently, the method for calculating the annual failure probability of a landslide using physically based models considering rainfall uncertainty is still not well established; hence, statistical methods are adopted in this study to estimate the annual landslide probability.

In Hong Kong, the failure of a slope is highly correlated with the 24 h
rainfall,

In Zhang and Tang (2009), the conditional failure probability of a slope for
a given type of rainfall is provided. To calculate the annual failure
probability of a slope, the uncertainty associated with the rainfall should
be analyzed. In this study, the uncertainty associated with rainfall can be
represented by the uncertainty associated with

Histogram and fitted PDF of the yearly maximum

CDF of the yearly maximum

In this study, the spatial impact of the landslide is characterized by the
landslide width and the run-out distance of the landslide. Let

In general, the run-out distance of a landslide depends on factors like the
slope geometry, the soil profile, and geotechnical, hydraulic and
rheological properties of sliding mass. The methods to investigate the
run-out distance of a landslide can be divided into two categories (Hungr et
al., 2005): (1) analytical or numerical methods based on the physical laws
of solid and fluid dynamics (Scheidegger, 1973), which are usually solved
numerically (e.g., Hungr and McDougall, 2009; Luo et al., 2019), and (2) empirical methods based on field observations and geometric correlations
(e.g., Dai and Lee, 2002; Budetta and Riso, 2004). The use of the
physically based methods requires detailed information on the ground
condition as well as the geotechnical and hydraulic properties of the soils.
In contrast, empirical methods based on geometry of the landslide are
generally simple and relatively easy to use (e.g., Finlay et al., 1999; Dai
et al., 2002). In this study, empirical methods are adopted due to the lack of
information on the geotechnical and hydraulic conditions of the slope. In
particular, the following empirical equation is used (Corominas, 1996):

For the slope shown in Fig. 2, the height is 25 m, i.e.,

Based on Fig. 3, the landslide scar area is estimated to be 450 m

PDF of the run-out distance of the landslide studied in this paper.

As shown in Fig. 2, the horizontal distance from the crest of the landslide
scar to the side of Kennedy Road closest to the slope (

In this study, the width of the landslide is assumed to be equal to the width
of the slope, i.e.,

Let

Mean rates of different types of vehicles during the

In general, the presence of vehicles also depends on the time of day.
For example, Table 2 shows the

Number of vehicles passing a given cross section of the road per hour (

Let

Probability distribution of the number of private cars hit by
the landslide studied in this paper during different periods when the
spatial impact is

Equation (14) provides a probabilistic model of the number of vehicles hit by the
landslide, which can consider the uncertainties of vehicle spacing, vehicle
type and slope failure time. For example, Fig. 9a, b and c show the
probability distributions of the number of private cars hit by the
landslide during the morning peak, the normal period and the evening peak when the
spatial impact is

In reality, the slope can fail during any period of a day. Based on the
total probability theorem, the probability that

As an example, Fig. 9d shows the probability distribution of the number
of private cars hit by the landslide considering the uncertainty of
the failure time when the spatial impact is

In the above analyses, equations for evaluating

Annual expected number of elements,

Introducing the passenger capacity of each type of vehicle into Eq. (3), the
expected number of people hit by the landslide associated with each
type of vehicle can be computed, and the results are shown in Fig. 10b. As
can be seen from this figure, the expected number of people hit by
the landslide in private cars is highest with a value of

Society is less tolerant of events in which a large number of lives are
lost in a single event compared with the same number of lives being lost over a large
number of separate events – this can be measured via societal risk
(Cascini et al., 2008). In Hong Kong, societal risk is measured using the

Estimated annual frequency of

Figure 11 shows the relationships between the number of people hit by the landslide and the annual probability that such an event occurs for different vehicle types. As can be seen from this figure, the risk associated with type 5 vehicles (private cars) is the greatest and is unacceptable; the risk associated with type 1 vehicles (private buses), type 9 vehicles (special-purpose vehicles) and type 10 vehicles (government vehicles) is in the acceptable region; and the risk associated with the rest of the vehicle types is in the ALARP region. Indeed, the person hit by the landslide on 8 May 1992 was in a private car.

As the flow of all vehicles on the highway is modeled as a Poisson process,
the flow of people on the highway considering all vehicle type can also
be modeled as Poisson process with a mean rate of

In the above analysis, the annual failure probability of the slope only
represents the failure probability of an average slope in Hong Kong. To
investigate the effect of the failure probability of the slope, Fig. 12
shows how the annual expected number of vehicles and people hit by
the landslide for all types of vehicles changes with the annual failure
probability of the slope. As can be seen from this figure, the expected
number of vehicles hit by the landslide increases linearly as the annual
failure probability of the slope increases. When the failure probability of
the slope increase from

Impact of the annual failure probability of the slope on the annual expected number of elements hit by the landslide.

Impact of the annual failure probability of the slope on the annual societal risk.

The density of vehicles may vary from one road to another. To investigate
the effect of the density of vehicles, the annual expected number of vehicles
and people hit by the landslide and the annual societal risk for all
types of vehicles are investigated when the density of vehicles on the
highway increases from 0 to 300 vehicles per kilometer, and the results are
shown in Figs. 14 and 15, respectively. As can be seen from Fig. 14,
there is a linear increasing trend in the expected number of vehicles and
people impacted as the density of vehicles increases. When the density of vehicles is 300 vehicles per kilometer, the expected number of elements impacted can reach

Impact of the density of vehicles on the annual expected number of elements hit by the landslide.

Impact of the density of vehicles on the annual societal risk. The numbers in the legend refer to the number of vehicles per kilometer.

Rainfall conditions may affect the failure probability of a slope as well as the traffic density and, hence, affect the risk. In this case study, the effect of rainfall conditions on the annual failure probability of the slope is considered using Eq. (6), which is based on both the chances of different types of rainfall and the failure probability of the slope under different rainfall conditions. Traffic conditions may also vary with rainfall conditions. However, data on the impact of rainfall conditions on the traffic density are rarely available. In this study, the impact of rainfall conditions on the traffic flow is not considered in the risk assessment.

The method used for the case study consists of three components: the hazard probability model, the spatial impact assessment model and the consequence assessment model. The annual failure probability of the slope is calculated based on the statistical analysis of past failure data in Hong Kong. It represents the failure probability of an average slope in Hong Kong, which is a common assumption adopted in empirical methods. When the method is applied in another region, the failure probability should be estimated using data from the region under study. Alternatively, to reflect the effects of factors like slope geometry and local ground conditions on the slope failure probability, the failure probability can also be estimated using physically based methods. As mentioned previously, current physically based methods mainly focus on the failure probability of a slope during a given rainfall event. It is important to also examine how to incorporate the uncertainty of the rainfall condition into the slope failure probability evaluation in future studies.

In this study, the spatial impact is estimated based on an empirical run-out distance prediction equation formulated using data from different types of landslides from several countries. When applying the method suggested in this paper in another region, the empirical equation should be tested to establish whether it can better fit landslides in the region under study or whether the run-out distance based on empirical relationships developed in the region under study should be estimated. The spatial impact of the landslide may also be estimated using physically based models. In recent years, large deformation analysis methods have been increasingly used for run-out distance analysis. It should be noted that, during the run-out distance analysis, the uncertainties in the geological condition and soil properties should be considered. Currently, large deformation analysis is often carried out in a deterministic way. It is highly desirable to combine large deformation analysis with the reliability theory such that the spatial impact of the landslide can also be predicted probabilistically.

The consequence assessment model is generally applicable and can be used to assess the impact of landslides on moving vehicles in other regions. Therefore, after the hazard probability model and the spatial impact model are replaced with models suitable for application in another region, the suggested method in this paper can also be used to assess the risk of moving vehicles being hit by a rainfall-induced landslide in another region.

There are multiple scenarios in which a landslide can impact vehicles on a highway. The focus of this paper is on the impact of falling materials on moving vehicles. In future studies, it would also be worthwhile developing methods to evaluate the effect of the uncertainty of the number and types of vehicles on the risk assessment of the impact of a landslide on vehicles in other scenarios.

When assessing the risk of a landslide hitting moving vehicles, the number
and types of vehicles hit can be highly uncertain. Using a case
study in Hong Kong, this paper suggests a method for assessing the risk of
vehicles being hit by a rainfall-induced landslide explicitly considering
the above factors. The research findings from this study can be summarized
as follows:

Using the method suggested in this paper, the expected annual number of vehicles/people hit by the landslide as well as the cumulative frequency–number of fatalities curve can be calculated. These results can complement existing results from previous studies on the risk assessment of landslides hitting moving vehicles, which mainly focus on the individual risk of a landslide or societal risk assessment relying on the probability of the occurrence of at least one fatality per year.

As the length, density and the passenger capacity of different vehicles are different, the annual number of vehicles/people hit by landslides varies for different types of vehicles. The societal risk associated with different types of vehicles is also variable. Thus, it is important to consider the different types of vehicles in the traffic flow.

The method suggested in this paper can be used to examine the effect of factors like the annual failure probability of the slope and the density of the vehicles on the road on the risk of a landslide hitting moving vehicles. The proposed method can be potentially useful in determining the target annual failure probability of a slope considering the traffic conditions on a highway, which can be used as a new guideline for highway landslide risk management.

In this case study, the annual failure probability of the slope is evaluated based on a statistical model, and the spatial impact of the landslide is analyzed using an empirical equation. While these methods are easy to use, they cannot consider the effect of local geology and soil condition on the failure and post-failure behavior of the slope. Further studies are needed to explore physically based methods to predict the annual failure probability and run-out distance with explicit consideration of the uncertainties involved.

All research data in this paper have been presented directly, and their sources have also been illustrated clearly.

The authors contributed equally to the development of this work.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Advances in extreme value analysis and application to natural hazards”. It is not associated with a conference.

This research has been supported by the National Key Research and Development Program of China (grant nos. 2018YFC0809600 and 2018YFC0809601), the National Natural Science Foundation of China (grant no. 41672276) and the Key Innovation Team Program of MOST of China (grant no. 2016RA4059).

This paper was edited by Sylvie Parey and reviewed by Johnny Alexander Vega and one anonymous referee.