# Introducing Uncertainty in Risk Calculation along Roads Using a Simple Stochastic Approach

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## Abstract

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## 1. Introduction

_{0}during a period Δt is given by

_{r}is the temporal frequency of rupture for a given period within a given perimeter and f

_{r}is the probability of rupture associated with a given magnitude or volume (here, λ = λ

_{r}× f

_{r}). PS is a spatial weight if the exact location is not known, Pp is the probability of propagation calculated from the rockfall source at a given location, Exp is the exposure, and E corresponds to the value or unit of the object at risk and V is its vulnerability.

## 2. Model Data

_{0}= 100 events reach the road per year for volumes greater than V

_{0}= 0.001 m

^{3}, and they are distributed according to a cumulative power with the observed b equal to 0.434 and a = N

_{0}× V

_{0}

^{b}= 4.99 (Figure 1):

^{3}) affect only one of the lanes, and these smaller volumes do not necessarily affect a car passing over them; however, for volumes above 100 m

^{3}, the road section is fully covered by the rockfall over a width D and Pp = 1. Exposure is calculated according to D, which increases roughly with the cubic root of the volume. The average vehicle length L

_{v}is 5.4 m, and 5000 vehicles travel per day. Here, only fatal accidents for at least one occupant are counted; therefore, vulnerability is equal to lethality, injuries are not considered, and E is implicitly set to 1. As an example, using the values chosen by [5] for the class of blocks from 0.1 to 1 m

^{3}, we obtain (Table 1):

_{v}is the speed of a vehicle and N

_{v}is the number of vehicles per year. The sum of all classes up to 10

^{5}m

^{3}indicates an average annual frequency of fatal accidents of 0.106, i.e., approximately one accident every 10 years (Table 1). By using the upper bounds of the classes, the risk is increased when compared to that obtained through the use of the average of the classes. The following paragraph attempts to overcome this problem by introducing simulations, which allow uncertainty to be incorporated in the model, and the values of the vulnerability or probability of death and the probability of impact are modified according to functions instead of with the discrete sets of values used by [5] (Figure 2).

## 3. Introducing Uncertainty into Risk Calculation

^{5}m

^{3}) and minimum (10

^{−3}m

^{3}) volumes of the distribution function, respectively. Let F

_{max}= 4.99 × 0.001

^{0.434}= 100 and F

_{min}= 4.99 × 100,000

^{0.434}= 0.0337. Starting from the power-law cumulative distribution, it is quite easily inverted and thus, we can draw values at random in an equiprobable way between F

_{min}and F

_{max}such that the simulated frequency is given by

_{v}, and N

_{v}. We do not randomize L

_{v}because the length of the zone affecting the passengers is not easy to estimate and does not change much; the goal is also to be coherent with [5]. The limits are chosen based on so-called “expert knowledge” while assuming reasonable ranges for values centered on the average value.

## 4. Results

^{6}simulations for one year with a number of annual rockfalls distributed according to Figure 3, we obtain an average frequency of 0.059 events per year, i.e., one event every 16.8 years (Table 3). The median is 0.047, i.e., a longer time than that obtained by [5] separates the potential accidents. The fact that we are no longer working with classes reduces the average frequency (it is divided by almost half). The so-called exceedance curves indicate that there is a 95% chance that there are less than 45.6 years between two events (Figure 4). The probability of having an event within less than 7.3 years is 5%, which is not negligible.

## 5. Discussion and Conclusions

_{0}= 130, we also fit the data (Figure 1) by maximizing the frequency; the average return period is 12.9 years (median 15.3) (Table 3, case B). By using a random number of occupants (1 or 2), we obtain that T = 11.2 years (median 14.1; case C), and if both are used, the result is 8.6 years (median 10.4; case D). This shows that a reasonable hypothesis can lead to agreement with the observed data. It also shows that for models A to D, the exceedance probability of 5% of the return period ranges from 3.9 and 7.3 years. It is also noteworthy that the centered 95% confidence level ranges for the return period decrease with hazard and occupant increases; for cases A to D, the results are 51.2 to 6.1 years (range 45.1), 35.4 to 5.1 years (30.3), 34.8 to 3.9 years (30.9), and 24.1 to 3.3 years (20.8), respectively.

_{v}, and N

_{v}are not simple, but it seems that reasonable choices provide reliable results. Haimes (2015) [16] also showed that performing the risk calculation using a probabilistic approach reduced the risk compared to the average value. This type of approach is likely to be developed in landslide risk assessments, such as those of propagation models, by also introducing variability. This is a way to introduce the catastrophe model [14] into landslide risk assessments.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Wieczorek, G.F.; Nishenko, S.P.; Varnes, D.J. Analysis of rock falls in the Yosemite Valley, California. In Proceedings of the 35th US Symposium on Rock Mechanics (USRMS), Reno, NV, USA, 5–7 June 1995; pp. 85–89. [Google Scholar]
- Hovius, N.; Allen, P.A.; Stark, C.P. Sediment flux from a mountain belt derived by landslide mapping. Geology
**1997**, 25, 231–234. [Google Scholar] [CrossRef] [Green Version] - Dussauge, C.; Grasso, J.-R.; Helmstetter, A. Statistical analysis of rockfall volume distributions: Implications for rockfall dynamics. J. Geophys. Res. Solid Earth
**2003**, 108. [Google Scholar] [CrossRef] [Green Version] - Hantz, D. Quantitative assessment of diffuse rock fall hazard along a cliff foot. Nat. Hazards Earth Syst. Sci.
**2011**, 11, 1303–1309. [Google Scholar] [CrossRef] [Green Version] - Hungr, O.; Evans, S.G.; Hazzard, J. Magnitude and frequency of rock falls and rock slides along the main transportation corridors of southwestern British Columbia. Can. Geotech. J.
**1999**, 36, 224–238. [Google Scholar] [CrossRef] - Hoek, E. Practical Rock Engineering. Available online: https://www.rocscience.com/learning/hoeks-corner/course-notes-books (accessed on 16 March 2021).
- Corominas, J.; van Westen, C.; Frattini, P.; Cascini, L.; Malet, J.P.; Fotopoulou, S.; Catani, F.; Van Den Eeckhaut, M.; Mavrouli, O.; Agliardi, F.; et al. Recommendations for the quantitative analysis of landslide risk. Bull. Eng. Geol. Environ.
**2014**, 73, 209–263. [Google Scholar] [CrossRef] - Wyllie, D.C. Rock Slope Engineering: Civil Applications, 5th ed.; CRC Press: Boca Raton, FL, USA; London, UK; New York, NY, USA, 2018; p. 568. [Google Scholar]
- Dai, F.C.; Lee, C.F.; Ngai, Y.Y. Landslide risk assessment and management: An overview. Eng. Geol.
**2002**, 64, 65–87. [Google Scholar] [CrossRef] - Nadim, F. Tools and Strategies for Dealing with Uncertainty in Geotechnics. In Probabilistic Methods in Geotechnical Engineering; Griffiths, D.V., Fenton, G.A., Eds.; Springer: Vienna, Austria, 2007; pp. 71–95. [Google Scholar]
- Wang, X.; Frattini, P.; Crosta, G.B.; Zhang, L.; Agliardi, F.; Lari, S.; Yang, Z. Uncertainty assessment in quantitative rockfall risk assessment. Landslides
**2014**, 11, 711–722. [Google Scholar] [CrossRef] - Crosta, G.B.; Agliardi, F.; Frattini, P.; Lari, S. Key Issues in Rock Fall Modeling, Hazard and Risk Assessment for Rockfall Protection. In Engineering Geology for Society and Territory; Springer: Cham, Switzerland, 2015; Volume 2, pp. 43–58. [Google Scholar]
- Macciotta, R.; Martin, C.D.; Morgenstern, N.R.; Cruden, D.M. Quantitative risk assessment of slope hazards along a section of railway in the Canadian Cordillera—A methodology considering the uncertainty in the results. Landslides
**2016**, 13, 115–127. [Google Scholar] [CrossRef] - Mitchell-Wallace, K.; Jones, M.; Hillier, J.; Foote, M. Natural Catastrophe Risk Management and Modelling—A Practitioner’s Guide; Wiley-Blackwell: Hoboken, NJ, USA, 2017; 536p. [Google Scholar]
- Nicolet, P.; Foresti, L.; Caspar, O.; Jaboyedoff, M. Shallow landslide’s stochastic risk modelling based on the precipitation event of August 2005 in Switzerland: Results and implications. Nat. Hazards Earth Syst. Sci.
**2013**, 13, 3169–3184. [Google Scholar] [CrossRef] [Green Version] - Farvacque, M.; Eckert, N.; Bourrier, F.; Corona, C.; Lopez-Saez, J.; Toe, D. Quantile-based individual risk measures for rockfall-prone areas. Int. J. Disaster Risk Reduct.
**2020**, 53, 101932. [Google Scholar] [CrossRef] - Haimes, Y.Y. Risk Modeling, Assessment, and Management, 4th ed.; Wiley: Hoboken, NJ, USA, 2015; 720p. [Google Scholar]

**Figure 1.**Cumulative frequency distribution as a function of the magnitude (volume) of 390 events along 75 km of Highway 99 in British Columbia; this includes the adjustment proposed by [5] for 100 events per year, as well as a second curve when this value is modified to 130 events per year (modified from [5]).

**Figure 2.**Model for the probability of impact or spread (Pp) and a vulnerability (V) curve created from data from [5] to make the functions continuous.

**Figure 3.**Cumulative distribution of the number of events per year for the 10

^{6}simulations, based on a Poisson distribution using an average parameter of 100.

**Figure 4.**Simulation results. (

**a**) 10 realizations of the volume distributions; (

**b**) histogram of the simulated fatal accident frequencies; (

**c**) exceedance curve or probability that the frequency is greater than a given value; (

**d**) probability that the accident return period is smaller than a given value.

**Table 1.**Details of the risk calculations for different classes recalculated from [5], but with D = Vol

^{1/3}(see Supplementary Materials).

Volume | 4.99 × Vol^{−0.434} | λ_{r} × f_{r} | D~Vol^{(1/3)} | Exp | Pp | V | H × Pp × Exp × V | 1/R |
---|---|---|---|---|---|---|---|---|

(m^{3}) | (#/yr) | (#/yr) | (m) | (-) | (-) | (-) | (-) | (yr) |

0.001 | 100.000 | |||||||

0.010 | 36.813 | 63.187 | 0.2 | 0.0146 | 0.1 | 0.05 | 0.005 | 217.0 |

0.100 | 13.552 | 23.261 | 0.5 | 0.0154 | 0.2 | 0.1 | 0.007 | 139.9 |

1.0 | 4.989 | 8.563 | 1 | 0.0167 | 0.4 | 0.2 | 0.011 | 87.6 |

10 | 1.837 | 3.152 | 2 | 0.0193 | 0.6 | 0.5 | 0.018 | 54.9 |

100 | 0.676 | 1.160 | 5 | 0.0271 | 0.8 | 0.8 | 0.020 | 49.7 |

1000 | 0.249 | 0.427 | 10 | 0.0401 | 1.0 | 1.0 | 0.017 | 58.4 |

10,000 | 0.092 | 0.157 | 30 | 0.0922 | 1.0 | 1.0 | 0.014 | 69.0 |

>10,000 | 0.092 | 50 | 0.1443 | 1.0 | 1.0 | 0.013 | 75.7 | |

Total | 0.106 | 9.4 |

Variables | Units (Remarks) | Minimum | Maximum |
---|---|---|---|

Debris width D | m | D/2 | 3D/2 |

Vehicle speed v_{v} | km/h | 57.5 | 102.5 |

Number of vehicles N_{v} | Vehicles/day | 4500 | 5500 |

Probability of impact or propagation at the vehicle location Pp | (-) (Integrated in the calculation; one order of magnitude of volume variability) | log_{10}(V(d)) − 0.5 | log_{10}(V(d)) + 0.5 |

Vulnerability V(lethality) | idem | idem | idem |

**Table 3.**Characteristics of the exceedance curves in Figure 4 for the first two columns and for two other scenarios obtained by changing the number of occupants in the car and the average total number of rockfalls per year.

Thresholds | Frequency | Return Period T [Year] | |||
---|---|---|---|---|---|

Case | A | A | B | C | D |

(events/year) | 1 occ. N_{0} = 100 | 1 occ. N_{0} = 130 | 1–2 occ. N_{0} = 100 | 1–2 occ. N_{0} = 130 | |

Average | 0.059 | 16.8 | 13.0 | 11.2 | 8.6 |

Minimum (max. T) | 0.010 | 103.3 | 80.9 | 75.4 | 48.6 |

97.50% | 0.020 | 51.2 | 35.4 | 34.8 | 24.1 |

95% | 0.022 | 45.6 | 31.8 | 31.0 | 21.6 |

Median | 0.047 | 21.1 | 15.5 | 14.2 | 10.5 |

5% | 0.137 | 7.3 | 6.0 | 4.8 | 3.9 |

2.5 | 0.165 | 6.1 | 5.1 | 3.9 | 3.3 |

Maximum (Min. T) | 0.603 | 1.7 | 1.6 | 1.2 | 1.0 |

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**MDPI and ACS Style**

Jaboyedoff, M.; Choanji, T.; Derron, M.-H.; Fei, L.; Gutierrez, A.; Loiotine, L.; Noel, F.; Sun, C.; Wyser, E.; Wolff, C.
Introducing Uncertainty in Risk Calculation along Roads Using a Simple Stochastic Approach. *Geosciences* **2021**, *11*, 143.
https://doi.org/10.3390/geosciences11030143

**AMA Style**

Jaboyedoff M, Choanji T, Derron M-H, Fei L, Gutierrez A, Loiotine L, Noel F, Sun C, Wyser E, Wolff C.
Introducing Uncertainty in Risk Calculation along Roads Using a Simple Stochastic Approach. *Geosciences*. 2021; 11(3):143.
https://doi.org/10.3390/geosciences11030143

**Chicago/Turabian Style**

Jaboyedoff, Michel, Tiggi Choanji, Marc-Henri Derron, Li Fei, Amalia Gutierrez, Lidia Loiotine, François Noel, Chunwei Sun, Emmanuel Wyser, and Charlotte Wolff.
2021. "Introducing Uncertainty in Risk Calculation along Roads Using a Simple Stochastic Approach" *Geosciences* 11, no. 3: 143.
https://doi.org/10.3390/geosciences11030143