# Definitions and Concepts for Quantitative Rockfall Hazard and Risk Analysis

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## Abstract

**:**

## 1. Introduction

## 2. The Rockfall Process

#### 2.1. Failure

**rock compartment**with the surface by which it was in contact with the cliff. The detachment (or release) results from a

**failure**process by which the rock compartment begins to move. This failure usually occurs by a sliding or a toppling mechanism, but tensile, bending, and buckling failures may also play a role. According to the principle of the Varnes classification, a composite term can be used to describe both the failure mechanism and the transport process (examples: rock topple–rock fall, rockslide–rock fall, rockslide–rock avalanche). The mechanical analysis of slide and topple was described, for example, by [11,12]. The term “rock compartment” is preferred to “rock mass”, which refers to a larger rock volume in rock engineering. Most failures are controlled by the discontinuities of the rock mass, but non-structurally-controlled failures can occur when the rock mass strength is exceeded, and result in the movement of a rock compartment that slides on the rest of the rock mass.

**scar**or the

**source area**or the

**release area**of the rockfall. The deposited material is called the

**deposit**or

**debris**. When the deposit forms a continuous layer, it is known as talus. Its slope is usually lower than 40°.

#### 2.2. Fragmentation

**fragmentation**of a rock compartment results from the

**disaggregation**that occurs during the failure process (Figure 2) and the

**breakage**(dynamic fragmentation) that occurs during the impacts of the rock fragments between each other or on the substrate. The fragmentation of a potentially unstable rock compartment should be considered in the hazard analysis [15].

#### 2.3. Rockfall Modelling

**single block methods**, [16]) or by including a fragmentation law [17,18]. Rock mass falls can be analyzed by modelling each rock fragment (

**multi-block methods**, including discrete element methods [19]) or as a

**granular flow**remaining in a fairly continuous contact with the flow bed [9]. The methods that model explicitly the rock fragments may or not model the breakage. They also may or not model the interaction with the vegetation [3]. The simplest methods allow a lot of simulations to be run in a probabilistic approach.

**reach angle**that express the energy loss during the propagation. The

**energy line**represents the energy (divided by the weight) of a rock mass as a function of the horizontal displacement along the rockfall path. Theoretically, it starts from the gravity center of the rock mass before its detachment. The energy consists of potential energy and kinetic energy. The energy line decreases along the rockfall path due to the energy that is lost when the rock fragments rebound, roll, or slide on the ground. Thus, its inclination reflects the energy loss per unit of horizontal displacement. When a rock mass reaches the intersection of the energy line with the topography, the energy consists only of its potential energy and then it stops.

**maximal reach angle**. Authors in [22] suggest an alternative angle, the

**shadow angle**, which is defined from the apex of the talus slope to the outer margin of the rockfall shadow. The rockfall shadow is the area downslope of the talus slope, which is covered discontinuously by scattered large boulders that have rolled or bounced beyond the base of the talus. This alternative approach supposes a profile such that most of the kinetic energy is absorbed during the first impact on the talus slope. The effect of rockfall activity is integrated over time by considering the longest boulder run-out in a given rockfall shadow. Note that the reach angle and the shadow angle as defined originally are related respectively to a rockfall event and a rockfall area and represent a minimum of the angles that can be measured for individual boulders. The statistical analysis of the

**individual boulder reach (or shadow) angles**allows a probabilistic analysis of the propagation [23].

## 3. Rockfall Hazard

#### 3.1. Frequency and Probability

#### 3.2. Localized Hazard

_{A}, the expression failure probability [15], onset probability [3,29], rockfall probability [2], and release probability have been used. For P

_{B}, the expressions propagation probability [15,30,31], transit or impact probability [29], reach probability [32], and travel probability [33] have been used. For P

_{C}, the expressions impact probability [22,34], passage probability [29], reach probability [35], occurrence probability [3], and rockfall probability [36] have been used. As

**passing**through a point for a rock compartment needs to be

**released**(or fail) and to

**propagate**down to (or

**reach**) this point, we suggest using the following expressions:

**failure**probability or

**release**probability for event A;

**propagation**probability or

**reach**probability for event B; and

**passage**probability for event C. Note that the expressions probability of occurrence and onset probability are too general, because the occurrence or onset may refer to different events (failure, passage, impact).

**impact probability**to describe the probability of an element at risk to be impacted by a rockfall. This probability is different from the passage probability if the element at risk is mobile, as its exposure is included.

_{p}is the passage probability, P

_{f}is the failure probability, and P

_{r}is the reach probability.

_{ri}that at least one fragment reaches a point, given that a single block i has been released. The probability P

_{r}that at least one fragment reaches a point, given that the whole compartment has fallen is as follows:

_{ri}of the blocks are small, P

_{r}can be approximated by the following:

_{ri}, P

_{r}can be approximated by [15] the following:

_{pi}, the passage probability P

_{p}is given by the following:

#### 3.3. Diffuse Hazard

**passage temporal frequency**that is obtained from a

**rockfall inventory**(or data base) covering a known period. This inventory may identify rockfall events having occurred in a given area or rock fragments deposited in an area of interest. As the frequency is strongly dependent on the volume, an inventory should include the volume of each event or fragment. Note that the concept of temporal frequency is not suitable to describe a localized hazard, because the release of a given rock compartment occurs once only. However, in some cases, a big localized compartment that has a given failure probability can be viewed also as a homogenous area where falls of smaller blocks represent a diffuse hazard.

#### 3.3.1. Frequency

**emporal frequency**is the number of occurrences per unit of time. It can be divided by an area (cliff area for example) or a length (cliff length or road length for example), giving a

**spatial–temporal frequency**. When the qualifiers “temporal” and “spatial–temporal” are not used, confusion is possible with the term “frequency” used in the statistical sense, which does not refer to time.

- -
- -
**Fragment release frequency:**the number of rock fragments that detach from a given source area, per unit of time (and per unit of area for the spatial–temporal frequency). Ref. [34] proposed a method to derive the fragment release frequency from the failure frequency.- -
**Event passage frequency:**the number of rock fall events that pass through a given location, per unit of time (and per unit of length for the spatial–temporal frequency). In other words, it is the number of rock fall events, at least one fragment of which passes through the given location. The spatial–temporal passage frequency allows one to derive the passage frequency at any location according to its width, measured perpendicularly to the movement direction [22].- -
**Fragment passage frequency:**the number of rock fragments that pass through a given area or location, per unit of time (and per unit of length for the spatial–temporal frequency).- -

#### 3.3.2. Rockfall Event Inventory

^{3}that occurred in different regions during the post-glacial period and calculated spatial–temporal frequencies. One of the oldest known terrestrial rockfalls on Earth was dated to 1.2 billion years [51], but it is difficult to assess rockfall frequency for periods older than the post-glacial period.

#### 3.3.3. Rockfall Fragment Inventory

#### 3.3.4. Volume–Frequency Relation

**hazard curve**[56]. Ref. [29] considered also the fly height, introducing a rockfall hazard vector, the components of which are the passage frequency, the maximum velocity, and the maximum fly height.

#### 3.3.5. Derivation of the Passage Frequency from the Release Frequency through Propagation Analysis

_{fc}, the length T

_{s}of the simulated period is the number of rockfalls simulated per source cell N

_{sc}divided by λ

_{fc}.

_{bc}) divided by the length of the simulated period (T

_{s}) gives the fragment passage frequency per cell.

_{bc}≥ 1 (N

_{sc}

_{1}) divided by the length of the simulated period (T

_{s}) gives the event passage frequency per cell.

_{sc}

_{1}/N

_{sc}is the probability of propagation in a cell, given that a rockfall start from each source cell. Note that the frequencies per cell depend on the cell size and must be divided by the cell width to derive a usable spatial–temporal passage frequency.

_{f}for a given source area comprising N

_{c}cells, T

_{s}can be written as follows:

_{b}is the number of blocks passing through a line of interest, and N

_{s}

_{1}is the number of simulations with N

_{b}≥ 1. These passage frequencies can also be determined considering a minimal value of the fragment volume or energy or a minimal value of the energy of all blocks passing through the cell. The maximum kinetic energy or the maximum passing height of each block passing through a cell can be calculated, and their distribution functions can be displayed.

_{c}= 2). The failure frequency per cell (λ

_{fc}) is one event per year. Two simulations are carried out for the two cells (N

_{sc}= 2), simulating a 2-year period (T

_{s}= 2 years). Each rockfall event produces three blocks. Another presentation of diffuse rockfall hazard assessment is given in this Special Issue by [60].

## 4. Rockfall Risk

**risk**given by [1] is: “A measure of the probability and severity of an adverse effect to health, property or the environment. Risk is often estimated by the product of probability of a phenomenon of a given magnitude times the consequences. However, a more general interpretation of risk involves a comparison of the probability and consequences in a non-product form. For Quantitative Risk Assessment the use of the landslide intensity is recommended.” In addition to the product of probability times the consequences, the risk can be described by the annual probability of different levels of loss [1,61].

**Individual risk to life**(or individual human risk): “The annual probability that a particular life will be lost”.**Societal risk to life**(or societal human risk): “The risk of multiple fatalities or injuries in society as a whole”, which can be expressed as the annual number of deaths.**Non-human societal risk**concerns “financial, environmental, and other losses”. The elements at risk can be “buildings and engineering works, economic activities, public services utilities, infrastructure and environmental features in the area potentially affected by landslides”.

#### 4.1. Localized Hazard

#### 4.1.1. Trivial Case of a Unique Block without Fragmentation

_{p}is the passage probability for the period of interest of a location whose width is the width of the element at risk plus the width of a rock fragment if a lumped mass model is used for rockfall modelling.

_{t}is the

**temporal probability**of the element at risk. This probability is 1 for a static element at risk. For a moving element at risk (person or vehicle), it is the proportion of the time when it is in the considered location. For a vehicle moving perpendicularly to the trajectory of the block, the exposure is [57,62]

_{m}is the length of the vehicle, V

_{m}is its velocity, W

_{b}is the width of the block, and F

_{v}is the passage frequency of the vehicle considered. If the societal risk is considered, F

_{v}is the traffic (number of vehicles per time unit).

**vulnerability**or lethality of the element at risk or its degree of loss (a number between 0 and 1). It depends mainly of the energy of the block. The relation between the energy and the vulnerability can be called the vulnerability curve

^{35}.

#### 4.1.2. Case of a Rock Compartment with Fragmentation

_{p}(E

_{j}) is the probability that the total energy of the blocks passing through the considered location belongs to the interval (E

_{j},E

_{j}

_{+1}), and V(E

_{j}) is the vulnerability of the element at risk for this energy interval.

#### 4.1.3. Case of Several Rock Compartments

_{pij}is the probability that the compartment i reaches the considered location with an energy of class j, and P

_{t}is the temporal probability of the element at risk at the considered location. The risks corresponding to the different classes of energy cannot be added as in Equation (16), because the corresponding events are not exclusive (a compartment can produce an energy of class j and another one can produce an energy of a different class). The total risk is

#### 4.2. Diffuse Hazard

_{p}is the passage frequency of the events for which the total energy of all the blocks reaching the considered location belongs to the interval (E

_{j}, E

_{j}

_{+1}). Alternatively, volume intervals can also be considered. P

_{t}is the temporal probability of the element at risk to be in the considered location (Equation (17)). V(E

_{j}) is the vulnerability of the element at risk for this energy interval.

_{a}is the width of the area of interest. As the passage frequency refers to an area of interest that is much wider than the element at risk, the probability that a rockfall reaching this area impacts the element at risk must be introduced [59]. Refs. [57,59] named this probability the

**spatial probability**(of an event reaching the element at risk given that it occurs in the area of interest). It can be expressed as

_{e}is the width of an event in the area of interest. Note that [59] neglected the length of the element at risk. For a vehicle crossing the whole area of interest, the risk is then

_{t}P

_{s}can be called the

**temporal spatial probability**or the exposure.

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Rockfall with a volume bigger than 1000 m

^{3}occurred in 2008 in Saint-Paul de Varces (French Subalpine Ranges). Note steep talus slope and large boulder runouts. Photo S. Gominet (IRMa).

**Figure 3.**Detection of rockfalls with diachronic terrestrial laser scanning.

**Up**: 3D view of the cliff. Red and green points are detected rockfalls. Red perimeters are smooth surfaces without rockfalls.

**Down**: Photo of the cliff [46].

**Figure 4.**Examples of rockfall volume–release frequency relations.

**Left**: bedded Sequanian limestone (topographical inventory).

**Right**: massive Urgonian limestone (topographical inventory under 1000 m

^{3}, historical inventory over 1000 m

^{3}). The spatial–temporal frequency of rockfalls bigger than 1 m

^{3}is at about 10 times smaller in massive limestone than in bedded limestone. However, the frequency of rockfalls bigger than 1000 m

^{3}is higher in massive limestone.

**Table 1.**Example of calculation of human risk for a 195 m long trail section exposed to a diffuse rockfall hazard (modified from [44]).

Volume Class (m^{3}) | Rockfall Release Frequency (Events/Year) | Reach Probability | Temporal Spatial Probability | Vulnerability | Annual Risk (Human Life) |
---|---|---|---|---|---|

V < 0.05 | 16.32 | 0.119 | 0.010 | 0.5 | 9.9 × 10^{−3} |

0.05 < V < 0.5 | 0.25 | 0.328 | 0.019 | 0.9 | 1.4 × 10^{−3} |

0.5 < V < 5 | 3.3 × 10^{−2} | 0.590 | 0.022 | 1.0 | 4.3 × 10^{−4} |

5 < V < 50 | 4.3 × 10^{−3} | 0.765 | 0.066 | 1.0 | 2.2 × 10^{−4} |

50 < V < 500 | 5.7 × 10^{−4} | 0.832 | 0.124 | 1.0 | 5.9 × 10^{−5} |

V > 500 | 8 × 10^{−5} | 0.874 | 0.153 | 1.0 | 1.0 × 10^{−5} |

Total risk | 0.012 |

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**MDPI and ACS Style**

Hantz, D.; Corominas, J.; Crosta, G.B.; Jaboyedoff, M.
Definitions and Concepts for Quantitative Rockfall Hazard and Risk Analysis. *Geosciences* **2021**, *11*, 158.
https://doi.org/10.3390/geosciences11040158

**AMA Style**

Hantz D, Corominas J, Crosta GB, Jaboyedoff M.
Definitions and Concepts for Quantitative Rockfall Hazard and Risk Analysis. *Geosciences*. 2021; 11(4):158.
https://doi.org/10.3390/geosciences11040158

**Chicago/Turabian Style**

Hantz, Didier, Jordi Corominas, Giovanni B. Crosta, and Michel Jaboyedoff.
2021. "Definitions and Concepts for Quantitative Rockfall Hazard and Risk Analysis" *Geosciences* 11, no. 4: 158.
https://doi.org/10.3390/geosciences11040158