# Impact-Detection Algorithm That Uses Point Clouds as Topographic Inputs for 3D Rockfall Simulations

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}area to calculate the slope of a 1 m

^{2}pixel on flat terrain, and this area increases to 18 m

^{2}and 52 m

^{2}for slopes of 60° and 80°, respectively.

#### 1.1. Point Cloud Characteristics Related to the Impact-Detection Algorithm

#### 1.1.1. Acquisition of Point Clouds

^{2}on flat terrain. However, steep slopes suffer from a density diminution similar to that of gridded DTMs (Figure 2), thereby decreasing the usefulness of airborne laser scans to generate detailed elevation models of near-vertical slopes [37].

^{2}) because the acquisition is carried perpendicular to the slope [38]. Occlusion and shadowing effects can affect this type of data, such as on horizontal surfaces [30].

#### 1.1.2. Point Cloud Preparation

## 2. Impact-Detection Algorithm

_{1}(m)), intermediate (d

_{2}(m)) and shortest (d

_{3}(m)) diameters are given by the respective lengths of the box sides. When searching for a contact with the point cloud, the falling rock is simplified to a sphere with a diameter equal to d

_{1}. Compared with approaches that use a terrain grid or mesh, the area above the surface in which the falling rock can evolve is not clearly defined by point clouds. Therefore, the rock cannot be simplified to a single point without dimensions, as is the case with lumped mass approaches. Thus, the space between the points must be sufficiently fine compared to the size of the rock to prevent the latter from passing through the surface. Points free of artifacts and evenly distributed with a spacing of approximately a quarter of d

_{1}are generally sufficient.

_{tree}(m)) and diameters at breast height (DBH (cm)) equivalent to those of the trees (Figure 7). The points of these lines should then contain the information needed to evaluate the local radius of the stem (r

_{stem}(m)) for any impact height with the ground (ΔH

_{ground}(m)) as follows:

_{1}) to create some freefall so that it does not stop at the first iteration. The initial translational velocities (v (m·s

^{−1})) can be attributed to the rock, but it is unnecessary if the rock’s initial location is slightly offset from the surface. Initially, the rotation velocities are usually zero.

_{1}. The drag force (F

_{D}(N)) is assumed constant on these short distance increments. It is determined using Rayleigh’s law with an air density (ρ (kg·m

^{3})) of 1.2, a rock reference surface (A (m

^{2})) of an ellipse with diameters d

_{1}–d

_{3}and a drag coefficient (C

_{D}) of 0.9:

_{1}) from the center of the rock or shorter than ½ d

_{1}+ r

_{stem}for impacts against the trees. If the answer is no, then an impact did not occur, the location and velocities of the particles are saved (step 7), and a verification is performed if the stopping thresholds are met (step 8, Figure 6). If the stopping thresholds are not met, then the process starts back at step 2 (Figure 6). On the other hand, if points occur inside the rock, then an impact occurs, which means that the freefall displacement was too large and the rock partly passed through the terrain surface. The previous freefall distance must be rewound (with negative time steps) until the rock is tangent to the surface, which is performed in step 4 (Figure 6 and Figure 9).

_{1}+ r

_{stem}if the last point corresponds to a tree). If this last point is not part of a tree, then the impact position is further refined by isolating the local points of the terrain in the footprint of the impacting rock, e.g., the points included in a radius of ½ d

_{1}around the last tangent point. The plane is then adjusted to them, such as by using the eigenvectors of the covariance matrix of the point’s distances to their centroid (see the “PlanePCA” method in [43]). If the plane corresponds well to the points (e.g., with a RMS < 0.04 d

_{1}) and the terrain is locally smooth, then the impacting rock is displaced by reverse freefall until it is tangent to the plane and the perceived orientation of the terrain is then given by the orientation of the plane. If the plane is inconsistent with the points or if only three points were considered for the fit, then the perceived orientation of the terrain is given by the normalized vector starting from the impacted point and pointing toward the center of mass of the rock (Figure 5). The characteristics of the impacted surface are evaluated in step 5 to pursue with a rebound model for rockfall simulations.

## 3. Application Cases

#### 3.1. Quantifying the Terrain Surface Roughness Perceived by Falling Rocks of Different Sizes

#### 3.1.1. Methodology

_{1}) at 0.3, 1, 3 and 10 m long.

#### 3.1.2. Results and Discussion

^{2}) of 0.95. The range covered by the distributions widens as the terrain roughness increases for both vertical and lateral perceived angles.

#### 3.1.3. Partial Conclusions

#### 3.2. Performing 3D Rockfall Simulations on Detailed Terrain Models: A Proof of Concept

#### 3.2.1. Rebound Model against the Ground

$I$ | $=$ | impacting rock moment of inertia | (kg·m^{2}); |

${\omega}_{1}$ | $=$ | initial rotational velocity | (rad·s^{−1}); |

$m$ | $=$ | impacting rock mass | (kg); |

${v}_{t1}$ | $=$ | initial tangent velocity | (m·s^{−1}); |

${f}_{friction}$ | $=$ | friction damping function; | |

${S}_{f}$ | $=$ | scaling damping factor; | |

${v}_{n1}$ | $=$ | initial normal velocity | (m·s^{−1}); |

${f}_{inelastic}$ | $=$ | inelastic damping function; | |

${\omega}_{2}$ | $=$ | final rotational velocity | (rad·s^{−1}); |

${v}_{t2}$ | $=$ | final tangent velocity | (m·s^{−1}); |

${v}_{n2}$ | $=$ | final normal velocity | (m·s^{−1}). |

#### 3.2.2. Rebound Model against Tree Stems

${v}_{1}$ | $=$ | initial translational velocity | (m·s^{−1}); |

${E}_{dtree}$ | $=$ | stem capacity to dissipate translational kinetic energy | (J); |

${v}_{2}$ | $=$ | final translational velocity | (m·s^{−1}). |

${E}_{dissM}$ | $=$ | maximum stem capacity to dissipate kinetic energy | (J); |

$d{E}_{vert}$ | $=$ | ratio of energy dissipated due to the impact height on the stem; | |

$d{E}_{hor}$ | $=$ | ratio of energy dissipated due to the impact lateral offset on the stem; | |

$d{E}_{\theta i}$ | $=$ | ratio of energy dissipated due to the incident angle. |

_{azimuth}(°)) is empirically defined based on 286 observed impacts on tree stems (mean DBH of 31 cm with a Std Dev of 21 cm) by falling rocks (mean volume of 0.49 m

^{3}with a Std Dev of 0.3 m

^{3}) at an on-site average velocity of 8.2 m·s

^{−1}[49] (Table 1):

^{3}rock should not be strongly deviated by a small tree.

#### 3.2.3. Terrain Models and Simulation Parameters

- A set of five artificial slopes with different roughnesses.
- A natural cliff with a scree slope composed of large boulders.
- A steep alpine terrain with a mature forest and an artificial fence.

_{n}and R

_{t}values of approximately 0.35 and 0.85, respectively, with a Std Dev of 0.04 and range between 0 and 1). Four spherical rock sizes were simulated on this site, with d

_{1}of 0.3, 1, 3 and 10 m with masses of 38 kg, 1414 kg, 38 t and 1414 t, respectively. The starting height was offset by ½ d

_{1}to ensure that all rock sizes would have the same initial freefall height.

_{1}between 10 and 15 m. The 3D terrain model of the site was created from ALS data combined with numerous TLS acquisitions from a Teledyne Optech Ilris 3D scanner and an SfM model from pictures acquired from the ground. Its point density was reduced based on spacing between points of approximately 0.25 m, thus giving a total of 1.8 million points. The damping coefficients used for that site were identical to those of the first site. The simulated rocks for this site have an ellipsoid shape (for inertia only) with d

_{1}, d

_{2}and d

_{3}of 1.9, 1.3 and 0.9 m, respectively, and a mass of 3143 kg.

^{−2}). Its point density was reduced so that the spacing between the points of approximately 0.46 m, thus giving a total of 1.9 million points. The damping coefficients used for that site were adjusted to correspond to typical talus cover (random normally distributed R

_{n}and R

_{t}values of approximately 0.32 and 0.83 with Std Dev of 0.04 and bounded between 0 and 1). The trees were identified using the FINT tool [57], which compared the terrain model with the surface model generated from the ALS data. Their central position was identified by finding the local maxima, and their DBHs were deduced from the tree heights (default DBH function in FINT [57]):

_{1}, d

_{2}and d

_{3}of 1.9, 1.3 and 0.9 m, respectively, and a mass of 3143 kg.

#### 3.2.4. Rockfall Proof of Concept Results and Brief Discussion

^{−1}) after falling 70 m from the steep cliff. Again, the impacts against the detailed terrain in the point cloud are correctly detected by the presented algorithm. Here, the trajectories deviate due to the large boulders and surface roughness, and some reach longer runouts when following paths with smaller roughness.

^{−1}, which is approximately half of the maximum velocities reached at the second site. This finding can be explained by the gentler slopes at the last site compared to the 70 m high cliff of the second site where the rocks were mostly in an undisturbed freefalling phase. The impacts against the artificial fence were properly detected by the algorithm, with the trajectories being intercepted (Figure 19b). From there, the capacity of the fence could be evaluated with advance rock–barrier interaction modeling (e.g., Coulibaly et al. [58]) or by combining the algorithm with a metamodel (e.g., Toe et al. [59]) using different parameters, e.g., the energy, angle and position at the impact. A quick evaluation of the capacity can also simply be performed by verifying that the impact energy and geometry are inside the certified limits of the fence.

## 4. Validations

#### 4.1. Validation 1: Ground Interaction

#### 4.1.1. Validation Method

^{2}were simulated with each software and the proposed algorithm.

#### 4.1.2. Results and Discussion

#### 4.1.3. Conclusions of Validation 1

#### 4.2. Validation 2: Tree Stem Interaction

#### 4.2.1. Validation Method

_{1}, d

_{2}and d

_{3}dimensions of 1.0, 0.8 and 0.5 m, respectively, a mass of 565 kg and a moment of inertia of 46 kg·m

^{2}. The d

_{2}–d

_{3}dimensions are only used to calculate the moment of inertia, with the mass distributed based on an ellipsoid. The shape is otherwise not considered in any other way with such a rebound model, as in the model of Pfeiffer and Bowen [6].

#### 4.2.2. Results and Discussion

_{1}and DBH.

#### 4.2.3. Conclusions of Validation 2

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Bühler, Y.; Christen, M.; Glover, J.; Christen, M.; Bartelt, P. Signifiance of digital elevation model resolution for numerical rockfall simulation. In Proceedings of the 3rd International Symposium Rock Slope Stability, Lyon, France, 15–17 November 2016; pp. 101–102. [Google Scholar]
- Noël, F.; Jaboyedoff, M.; Cloutier, C.; Mayers, M.; Locat, J. The effect of slope roughness on 3D rockfall simulation results. In Proceedings of the RocExs 2017—6th Interdisciplinary Workshop on Rockfall Protection, Barcelona, Spain, 22–24 May 2017; Corominas, J., Moya, J., Janeras, M., Eds.; International Center for Numerical Methods in Engineering (CIMNE): Barcelona, Spain, 2017; pp. 31–34. [Google Scholar]
- Agliardi, F.; Crosta, G.B. High resolution three-dimensional numerical modelling of rockfalls. Int. J. Rock Mech. Min. Sci.
**2003**, 40, 455–471. [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; Lollino, G., Giordan, D., Crosta, G.B., Corominas, J., Azzam, R., Wasowski, J., Sciarra, N., Eds.; Springer International Publishing: Cham, Switzerland, 2015; Volume 2, pp. 43–58. [Google Scholar] [CrossRef] [Green Version]
- Noël, F. Cartographie Semi-Automatisée des Chutes de Pierres Le Long D’Infrastructures Linéaires. Master’s Thesis, Université Laval, Québec, QC, Canada, 2016. [Google Scholar]
- Pfeiffer, T.J.; Bowen, T.D. Computer Simulation of Rockfalls. Environ. Eng. Geosci.
**1989**, 26, 135–146. [Google Scholar] [CrossRef] - Wyllie, D.C. Rock Fall Engineering: Development and Calibration of an Improved Model for Analysis of Rock Fall Hazards on Highways and Railways. Ph.D. Thesis, University of British Columbia, Vancouver, BC, Canada, 2014. [Google Scholar]
- Spadari, M.; Giacomini, A.; Buzzi, O.; Fityus, S.; Giani, G.P. In situ rockfall testing in New South Wales, Australia. Int. J. Rock Mech. Min. Sci.
**2012**, 49, 84–93. [Google Scholar] [CrossRef] - Spadari, M.; Kardani, M.; de Carteret, R.; Giacomini, A.; Buzzi, O.; Fityus, S.; Sloan, S. Statistical evaluation of rockfall energy ranges for different geological settings of New South Wales, Australia. Eng. Geol.
**2013**, 158, 57–65. [Google Scholar] [CrossRef] - Buzzi, O.; Giacomini, A.; Spadari, M. Laboratory Investigation on High Values of Restitution Coefficients. Rock Mech. Rock Eng.
**2012**, 45. [Google Scholar] [CrossRef] - Wang, Y.; Jiang, W.; Cheng, S.; Song, P.; Mao, C. Effects of the impact angle on the coefficient of restitution in rockfall analysis based on a medium-scale laboratory test. Nat. Hazards Earth Syst. Sci.
**2018**, 18, 3045–3061. [Google Scholar] [CrossRef] [Green Version] - Asteriou, P. Effect of Impact Angle and Rotational Motion of Spherical Blocks on the Coefficients of Restitution for Rockfalls. Geotech. Geol. Eng.
**2019**, 37, 2523–2533. [Google Scholar] [CrossRef] - Asteriou, P.; Saroglou, H.; Tsiambaos, G. Geotechnical and kinematic parameters affecting the coefficients of restitution for rock fall analysis. Int. J. Rock Mech. Min. Sci.
**2012**, 54, 103–113. [Google Scholar] [CrossRef] - Chau, K.T.; Wong, R.H.C.; Wu, J.J. Coefficient of restitution and rotational motions of rockfall impacts. Int. J. Rock Mech. Min. Sci.
**2002**, 39, 69–77. [Google Scholar] [CrossRef] - Rocscience Inc. RocFall v8.0 2020. Available online: https://www.rocscience.com/software/rocfall (accessed on 15 February 2021).
- Pfeiffer, T.J.; Higgins, J.D. Rockfall hazard analysis using the Colorado rockfall simulation program. Transp. Res. Rec.
**1990**, 1288, 117–126. [Google Scholar] - Jones, C.L.; Higgins, J.D.; Andrew, R.D. MI-66 Colorado Rockfall Simulation Program; Version 4.0; Colorado Geological Survey, Division of Minerals and Geology, Department of Natural Resources: Denver, CO, USA, March 2000; 127p, Available online: https://coloradogeologicalsurvey.org/publications/colorado-rockfall-simulation-program (accessed on 24 April 2021).
- Berger, F.; Dorren, L. Objective Comparison of Rockfall Models using Real Size Experimental Data. In Disaster Mitigation Debris Flows Slope Failures and Landslides, Proceedings of the INTERPRAEVENT International Symposium, Niigata, Japan, 25–29 September 2006; Universal Academy Press: Niigata, Japan, 2006; pp. 245–252. [Google Scholar]
- Bourrier, F.; Berger, F.; Tardif, P.; Dorren, L.; Hungr, O. Rockfall rebound: Comparison of detailed field experiments and alternative modelling approaches. Earth Surf. Processes Landf.
**2012**, 37, 656–665. [Google Scholar] [CrossRef] - Garcia, B. Analyse des Mécanismes D’Interaction Entre un Bloc Rocheux et un Versant de Propagation: Application à L’Ingénierie. Ph.D. Thesis, Université Grenoble Alpes (ComUE), Gières, France, 2019. [Google Scholar]
- Bourrier, F.; Toe, D.; Garcia, B.; Baroth, J.; Lambert, S. Experimental investigations on complex block propagation for the assessment of propagation models quality. Landslides
**2020**, 18, 1–16. [Google Scholar] [CrossRef] - Caviezel, A.; Demmel, S.E.; Ringenbach, A.; Bühler, Y.; Lu, G.; Christen, M.; Dinneen, C.E.; Eberhard, L.A.; Von Rickenbach, D.; Bartelt, P. Reconstruction of four-dimensional rockfall trajectories using remote sensing and rock-based accelerometers and gyroscopes. Earth Surf. Dyn.
**2019**, 7, 199–210. [Google Scholar] [CrossRef] [Green Version] - Lambert, S.; Bourrier, F.; Toe, D. Improving three-dimensional rockfall trajectory simulation codes for assessing the efficiency of protective embankments. Int. J. Rock Mech. Min. Sci.
**2013**, 60, 26–36. [Google Scholar] [CrossRef] - Guzzetti, F.; Crosta, G.B.; Detti, R.; Agliardi, F. STONE: A computer program for the three-dimensional simulation of rock-falls. Comput. Geosci.
**2002**, 28, 1079–1093. [Google Scholar] [CrossRef] - Lan, H.; Derek Martin, C.; Lim, C.H. RockFall analyst: A GIS extension for three-dimensional and spatially distributed rockfall hazard modeling. Comput. Geosci.
**2007**, 33, 262–279. [Google Scholar] [CrossRef] - Dorren, L. Rockyfor3D (v5.2) Revealed—Transparent Description of the Complete 3D Rockfall Model; EcorisQ Paper: Geneva Switzerland, 2015; p. 31. [Google Scholar]
- Matasci, B.; Stock, G.M.; Jaboyedoff, M.; Carrea, D.; Collins, B.D.; Guérin, A.; Ravanel, L. Assessing rockfall susceptibility in steep and overhanging slopes using three-dimensional analysis of failure mechanisms. Landslides
**2018**, 15, 859–878. [Google Scholar] [CrossRef] - Riquelme, A.J.; Tomás, R.; Abellán, A. Characterization of rock slopes through slope mass rating using 3D point clouds. Int. J. Rock Mech. Min. Sci.
**2016**, 84, 1–12. [Google Scholar] [CrossRef] [Green Version] - Assali, P.; Grussenmeyer, P.; Villemin, T.; Pollet, N.; Viguier, F. Surveying and modeling of rock discontinuities by terrestrial laser scanning and photogrammetry: Semi-automatic approaches for linear outcrop inspection. J. Struct. Geol.
**2014**, 66, 102–114. [Google Scholar] [CrossRef] - Slob, S. Automated Rock Mass Characterisation Using 3-D Terrestrial Laser Scanning. Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 2010. [Google Scholar]
- Lato, M.J.; Diederichs, M.S.; Hutchinson, D.J.; Harrap, R. Evaluating roadside rockmasses for rockfall hazards using LiDAR data: Optimizing data collection and processing protocols. Nat. Hazards
**2012**, 60, 831–864. [Google Scholar] [CrossRef] - Guerin, A.; Stock, G.M.; Radue, M.J.; Jaboyedoff, M.; Collins, B.D.; Matasci, B.; Avdievitch, N.; Derron, M.-H. Quantifying 40 years of rockfall activity in Yosemite Valley with historical Structure-from-Motion photogrammetry and terrestrial laser scanning. Geomorphology
**2020**, 356, 107069. [Google Scholar] [CrossRef] - Gauthier, D.; Hutchinson, D.J.; Lato, M.; Edwards, T.; Bunce, C.; Wood, D.F. On the precision, accuracy, and utility of oblique aerial photogrammetry (OAP) for rock slope monitoring and assessment. In Proceedings of the Conférence Canadienne de Géotechnique GEOQuébec 2015, Québec, QC, Canada, 20–23 September 2015. [Google Scholar]
- Bell, A.D.F.; Mckinley, J.M.; Hughes, D.A.B.; Hendry, M.; Macciotta, R. Spatial and temporal analyses using Terrestrial LiDAR for monitoring of landslides to determine key slope instability thresholds: Examples from Northern Ireland and Canada. In Proceedings of the 6th Canadian GeoHazards Conference GeoHazards 6, Kingston, ON, Canada, 15–18 June 2014; p. 10. [Google Scholar]
- Kromer, R.; Abellán, A.; Hutchinson, D.; Lato, M.; Edwards, T.; Jaboyedoff, M. A 4D Filtering and Calibration Technique for Small-Scale Point Cloud Change Detection with a Terrestrial Laser Scanner. Remote Sens.
**2015**, 7, 13029–13052. [Google Scholar] [CrossRef] [Green Version] - Noël, F.; Cloutier, C.; Turmel, D.; Locat, J. Using point clouds as topography input for 3D rockfall modeling. In Landslides and Engineered Slopes: Experience, Theory and Practice; CRC Press: Napoli, Italy, 2016; pp. 1531–1535. [Google Scholar] [CrossRef]
- Abellán, A.; Oppikofer, T.; Jaboyedoff, M.; Rosser, N.J.; Lim, M.; Lato, M.J. Terrestrial laser scanning of rock slope instabilities. Earth Surf. Processes Landf.
**2014**, 39, 80–97. [Google Scholar] [CrossRef] - Jaboyedoff, M.; Oppikofer, T.; Abellán, A.; Derron, M.-H.; Loye, A.; Metzger, R.; Pedrazzini, A. Use of LIDAR in landslide investigations: A review. Nat. Hazards
**2012**, 61, 5–28. [Google Scholar] [CrossRef] [Green Version] - Girardeau-Montaut, D. Détection de Changement Sur des Données Géométriques Tridimensionnelles. Ph.D. Thesis, Télécom ParisTech, Paris, France, 2006. [Google Scholar]
- Rusu, R.B.; Cousins, S. 3D is here: Point Cloud Library (PCL). In Proceedings of the 2011 IEEE International Conference on Robotics and Automation, Shanghai, China, 9–13 May 2011; pp. 1–4. [Google Scholar] [CrossRef] [Green Version]
- Brodu, N.; Lague, D. 3D Terrestrial lidar data classification of complex natural scenes using a multi-scale dimensionality criterion: Applications in geomorphology. ISPRS J. Photogramm. Remote Sens.
**2012**, 68. [Google Scholar] [CrossRef] [Green Version] - Jaboyedoff, M.; Metzger, R.; Oppikofer, T.; Couture, R.; Derron, M.H.; Locat, J.; Turmel, D. New insight techniques to analyze rock-slope relief using DEM and 3D-imaging cloud points: COLTOP-3D software. In Rock Mechanics: Meeting Society’s Challenges and Demands; American Rock Mechanics Association: Vancouver, BC, Canada, 2007; Volume 1, pp. 61–68. [Google Scholar]
- Klasing, K.; Althoff, D.; Wollherr, D.; Buss, M. Comparison of surface normal estimation methods for range sensing applications. In Proceedings of the 2009 IEEE International Conference on Robotics and Automation, Kobe, Japan, 12–17 May 2009; pp. 3206–3211. [Google Scholar] [CrossRef]
- Azimi, C.; Desvarreux, P.; Giraud, A.; Martin-Cocher, J. Méthode de Calcul de la Dynamique des Chutes de Blocs—Application à l’étude du Versant de la Montagne de La Pale (Vercors). Bulletin de Liaison des Laboratoires des Ponts et Chaussées. 1982, Volume 122, pp. 93–102. Available online: https://trid.trb.org/view/1041460 (accessed on 14 February 2021).
- Gischig, V.S.; Hungr, O.; Mitchell, A.; Bourrier, F. Pierre3D: A 3D stochastic rockfall simulator based on random ground roughness and hyperbolic restitution factors. Can. Geotech. J.
**2015**, 14, 1–14. [Google Scholar] [CrossRef] - Asteriou, P.; Tsiambaos, G. Effect of impact velocity, block mass and hardness on the coefficients of restitution for rockfall analysis. Int. J. Rock Mech. Min. Sci.
**2018**, 106, 41–50. [Google Scholar] [CrossRef] - Lu, G.; Caviezel, A.; Christen, M.; Demmel, S.E.; Ringenbach, A.; Bühler, Y.; Dinneen, C.E.; Gerber, W.; Bartelt, P. Modelling rockfall impact with scarring in compactable soils. Landslides
**2019**, 16, 2353–2367. [Google Scholar] [CrossRef] - Yan, P.; Zhang, J.; Kong, X.; Fang, Q. Numerical simulation of rockfall trajectory with consideration of arbitrary shapes of falling rocks and terrain. Comput. Geotech.
**2020**, 122, 103511. [Google Scholar] [CrossRef] - Dorren, L.K.A.; Berger, F.; le Hir, C.; Mermin, E.; Tardif, P. Mechanisms, effects and management implications of rockfall in forests. For. Ecol. Manag.
**2005**, 215, 183–195. [Google Scholar] [CrossRef] - Dorren, L.K.A.; Berger, F.; Putters, U.S. Real-size experiments and 3-D simulation of rockfall on forested and non-forested slopes. Nat. Hazards Earth Syst. Sci.
**2006**, 6, 145–153. [Google Scholar] [CrossRef] [Green Version] - Dorren, L.K.A.; Berger, F. Stem breakage of trees and energy dissipation during rockfall impacts. Tree Physiol.
**2006**, 26, 63–71. [Google Scholar] [CrossRef] - Jonsson, M.J.O. Energy Absorption of Trees in a Rockfall Protection Forest. Ph.D. Thesis, ETH Zürich, Zürich, Switzerland, 2007. [Google Scholar] [CrossRef]
- Noël, F.; Wyser, E.; Jaboyedoff, M.; Derron, M.-H.; Cloutier, C.; Turmel, D.; Locat, J. Real-size rockfall experiment: How different rockfall simulation impact models perform when confronted with reality? In Proceedings of the Geohazards 7 Engineering Resiliency in a Changing Climate, Canmore, AB, Canada, 3–6 June 2018; Volume 26, p. 8. [Google Scholar]
- Ferrari, F.; Thoeni, K.; Giacomini, A.; Lambert, C. A rapid approach to estimate the rockfall energies and distances at the base of rock cliffs. Georisk Assess. Manag. Risk Eng. Syst. Geohazards
**2016**, 10, 179–199. [Google Scholar] [CrossRef] - Ji, Z.-M.; Chen, Z.-J.; Niu, Q.-H.; Wang, T.-J.; Song, H.; Wang, T.-H. Laboratory study on the influencing factors and their control for the coefficient of restitution during rockfall impacts. Landslides
**2019**, 16, 1939–1963. [Google Scholar] [CrossRef] - Ye, Y.; Zeng, Y.; Thoeni, K.; Giacomini, A. An Experimental and Theoretical Study of the Normal Coefficient of Restitution for Marble Spheres. Rock Mech. Rock Eng.
**2019**, 52, 1705–1722. [Google Scholar] [CrossRef] - Dorren, L. FINT—Find Individual Trees User Manual; EcorisQ: Genève, Switzerland, 2014. [Google Scholar]
- Coulibaly, J.B.; Chanut, M.-A.; Lambert, S.; Nicot, F. Toward a Generic Computational Approach for Flexible Rockfall Barrier Modeling. Rock Mech. Rock Eng.
**2019**, 52, 4475–4496. [Google Scholar] [CrossRef] - Toe, D.; Mentani, A.; Govoni, L.; Bourrier, F.; Gottardi, G.; Lambert, S. Introducing Meta-models for a More Efficient Hazard Mitigation Strategy with Rockfall Protection Barriers. Rock Mech. Rock Eng.
**2018**, 51, 1097–1109. [Google Scholar] [CrossRef] [Green Version] - Dorren, L.K.A. Rockfall and Protection Forests—Models, Experiments and Reality; Universität für Bodenkultur: Wien, Österreich, 2008. [Google Scholar]
- Zevenbergen, L.W.; Thorne, C.R. Quantitative analysis of land surface topography. Earth Surf. Processes Landf.
**1987**, 12, 47–56. [Google Scholar] [CrossRef]

**Figure 1.**Surface roughness is frequently lost when representing a site with a digital terrain model. On the lower image, most of the surface roughness is absent on the triangulated mesh from a gridded DTM raster with a 2 m cell size when compared to the same view with a 1 cm spaced 3D point cloud (upper image).

**Figure 2.**Comparison of rasterized and point cloud terrain models. On the gridded terrain surface, the point density is not homogenous (

**a**) and the near vertical face is underrepresented; however, in the 3D point cloud, the points can be evenly distributed (

**b**).

**Figure 3.**Strong biases affect the source distribution on the gridded DTM (raster). Here, the 45° steep slope has 128 pixels used as rockfall sources while the 85° slope only has 8 pixels that can be rockfall sources.

**Figure 4.**Smoothing of the terrain surface resulting from the surface orientation calculation using a plane fitting method with different radii (r) on point clouds (

**a**,

**b**) compared to the important smoothing of rasterized data (

**c**). The vegetation was removed from the point cloud and raster to compute the orientation and is shown here as a scale. The bottom-right tree is 9 m high.

**Figure 5.**Distinction between the terrain orientation and the orientation perceived by rocks moving from left to right when encountering a local bump. Neglecting any scarring, the local terrain orientation does not change; however, the perceived orientation depends on the size of the rocks. For the smallest rock, all incident velocities can be decomposed as a normal component to the terrain. For the medium-sized rock, the normal velocity component is larger than the tangential velocity component. The normal velocity component is smaller than the tangent velocity component for the largest impacting rock. The smoothing effect of a low-resolution terrain model can also be illustrated by doubling the width of the local bump and reducing its height by half.

**Figure 8.**Algorithm steps 2 and 3. Freefalling positions are shown in gradations of gray. Impact verification is performed for each position. An impact is detected when there are points inside the falling rock. The point spacing must be smaller than the width of the minimum area covered and generally provides the optimum results at ¼ of d

_{1}.

**Figure 9.**Algorithm step 4. The impact’s location is adjusted by reverting the freefall preceding step 3 (Figure 6c) until the rock is tangent to the surface. In this figure, the perceived surface is the best fitted plane of the points identified in a radius of ½ d

_{1}from the last point to be tangent to the rock when reverting the freefall. If the fit is bad or if a tree is impacted, then the last tangent point is simply used as the local surface.

**Figure 10.**Measurement of the perceived deviation on the difference of DTM (DoD), represented here by the layer “Detailed terrain sample” combined with a flat “Coarse terrain”. The perceived deviation varies depending on the applied incident angle because some part of the terrain cannot be reached at shallow impact angles. In this 2D representation, the deviation measured is called vertical deviation. The lateral deviation is measured in the vertical plane perpendicular to the vertical profile of this figure (deviating toward the reader, or the opposite).

**Figure 11.**Preparation of the DTM difference (DoD) is performed by subtracting the elevation from the coarse terrain model (on which rockfall simulation might be performed) from the elevation of the detailed terrain sample. This approach extracts the roughness of the detailed terrain model and removes the global terrain orientations present on the coarse model.

**Figure 12.**Six selected sites for their characteristic surface roughness encountered on rockfall-prone terrain. The central section of each 3D scene was isolated to extract its surface roughness. Here, each scene is shown with the 3D SfM model textured and with contours of the foreground elements highlighted using the Eye Dome Lighting filter.

**Figure 13.**Vertical and lateral perceived deviation angles for the six tested sites with four different impacting rock sizes (more than 20 million detected impacts summarized here). The left column shows the surface’s shaded top views of the sites. The middle column shows the vertical perceived deviation angles. The right column shows the lateral perceived deviation angles. The vertical axis of each graph corresponds to the centiles. The results for the 0.3, 1.0, 3.0 and 10 m wide rocks are represented with black, blue, green, and red lines, respectively.

**Figure 14.**Vertical and lateral perceived deviation angles for the Saint-Eynard mid-scree site with four different impacting rock sizes (more than 60 million detected impacts summarized here). The left column shows how the rocks perceive the surface, with shaded top views of the sites. The middle column shows the vertical perceived deviation angles for the 18 incident angles (5° to 90° in 5° increments). The right column shows the lateral perceived deviation angles. The vertical axis of each graph corresponds to the centiles. The results for the 0.3, 1.0, 3.0 and 10 m wide impacting rocks are represented with black, blue, green and red lines, respectively.

**Figure 15.**Parameters for the impact model against tree stems. The lateral offset is measured from the center of the stem to the contact point with the impacting rock, perpendicular to the initial velocity in the horizontal plane.

**Figure 16.**Profile of the five artificial slopes with different roughnesses. The undulations only begin after 2 m horizontally to ensure that the first impact has the same geometry. The vertical part of the profile was not considered for the simulation; it is only shown here to demonstrate the freefalling height preceding the first impact. The XZ plane space is parallel to the trihedron respective red–blue axes.

**Figure 17.**Two real sites where the algorithm was tested to simulate rockfalls. (

**a**) Gros Bras Mountains located in Canada. The scree slope composed of many large rock fragments is clearly visible. (

**b**) Moderately steep alpine slope near La Verda–Rougemont in Switzerland. Impact marks and boulders from a 2019 rockfall event and older events are visible. The forest has stems of 57 cm DBH on average (Std Dev of 23 cm).

**Figure 18.**Centerline of the simulated rockfalls (10 per slope) for rocks of different sizes on the artificial slopes of the first site. They are colored based on their translational velocities. The translational velocities increase as the size of the falling rocks increases, which is highlighted on the histograms next to the scale bars with the white arrows. Additionally, the centerline of the trajectories is properly offset to the terrain in relation to the radius of the different rock.

**Figure 19.**Centerline of 20 selected simulated trajectories on the second site (Gros Bras Mount, Canada) in (

**a**) and 130 on the third site (La Verda–Rougemont, Switzerland) in (

**b**–

**d**). The trajectories are colored based on their translational velocities. The central line of the stems considered for the simulations on the third site is shown by the vertical brown lines. The center of mass of the rocks when they stopped is shown in (

**c**,

**d**) by the white squares.

**Figure 20.**Rifle rockfall test site used for the comparison. The different cells with attributed terrain materials are shown with vertical colored bands and numbers. The 10,000 trajectories simulated with each software and the presented algorithm (this paper) combined with the same impact model as the others are shown in magenta for RocFall 8, in graded colors for the proposed algorithm and in black for CRSP 4. The results of CRSP 4 were created using the original damping coefficients and artificial terrain roughness for the 13 cells found to reproduce the data from the original rockfall experiment performed at that site [6,16,17]. Surprisingly, they had to be modified to new adjusted values with RocFall 8 and the proposed algorithm to produce the results shown above, which match the CRSP 4 results.

**Figure 21.**Distribution of the simulated maximum translational kinetic energy obtained along the Rifle rockfall test site.

**Figure 22.**Distribution of the simulated maximum jumping (bouncing) heights obtained along the Rifle rockfall test site. To correspond to the lumped mass results, the trajectories simulated with the proposed algorithm that considers the size of the rock were shifted down by ½ d

_{1}.

**Figure 23.**Detected impacts against the tree stems for the simulation with the proposed algorithm seen in 3D and from above. The red points are located at the center of mass of the rock at the impact, and their distances from the center of the stems at a height of 1.3 m, thus corresponding to ½ (DBH + d

_{1}) based on Equation (1).

**Figure 24.**Results for the comparison on the alpine slope near La Verda–Rougemont in Switzerland. The left windows (

**a**,

**c**) show the reach probability results from the proposed algorithm compared to those from Rockyfor3D 5.2.15. The right windows (

**b**,

**d**) show the same results, but with tree stems considered. The colors are based on a log10 distribution to show the changes in order of magnitude instead of linearly, which enhances the subtle changes in the low reach probability values. The 10 m contour lines and hill shade base layers are from a 1 m cell size gridded DTM.

**Figure 25.**Results for the comparison on the alpine slope near La Verda–Rougemont in Switzerland. The left windows (

**a**,

**c**) show the maximum translational kinetic energy results from the proposed algorithm compared to those from Rockyfor3D 5.2.15. The right windows (

**b**,

**d**) show the same results but consider the tree stems. The 10 m contour lines and hill shade base layers are from a 1 m cell size gridded DTM.

**Figure 26.**Enlarged view of the simulated maximum translational kinetic energy close to the foot of the slope. The same scale as in the previous figure is used, with energies from 0 to 500 kJ shown in rainbow colors from violet to red.

**Table 1.**Probabilities for deviation occurring after an impact against a stem (after Dorren et al. [49]).

Impact Configuration | Probability of Deviation |
---|---|

Frontal | |

$\raisebox{1ex}{$0$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\le \frac{{l}_{offset}}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.DBH}\le \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ | $\mathbb{P}\left(0\xb0\le \Delta {\theta}_{azimuth}\le 22.5\xb0\right)=0.44$ $\mathbb{P}\left(22.5\xb0<\Delta {\theta}_{azimuth}\le 67.5\xb0\right)=0.50$ $\mathbb{P}\left(67.5\xb0<\Delta {\theta}_{azimuth}\le 76\xb0\right)=0.06$ |

Lateral | |

$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.<\frac{{l}_{offset}}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.DBH}\le \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ | $\mathbb{P}\left(0\xb0\le \Delta {\theta}_{azimuth}\le 22.5\xb0\right)=0.11$ $\mathbb{P}\left(22.5\xb0<\Delta {\theta}_{azimuth}\le 67.5\xb0\right)=0.84$ $\mathbb{P}\left(67.5\xb0<\Delta {\theta}_{azimuth}\le 76\xb0\right)=0.05$ |

Scratch | |

$\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.<\frac{{l}_{offset}}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.DBH}\le \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ | $\mathbb{P}\left(0\xb0\le \Delta {\theta}_{azimuth}\le 22.5\xb0\right)=0.72$ $\mathbb{P}\left(22.5\xb0<\Delta {\theta}_{azimuth}\le 67.5\xb0\right)=0.24$ $\mathbb{P}\left(67.5\xb0<\Delta {\theta}_{azimuth}\le 76\xb0\right)=0.04$ |

Parameters | Site 1a | Site 1b | Site 1c | Site 1d | Site 2 | Site 3 |
---|---|---|---|---|---|---|

Rn | 0.35 | 0.35 | 0.35 | 0.35 | 0.35 | 0.32 |

Rn variability (Std Dev) | 0.04 | 0.04 | 0.04 | 0.04 | 0.04 | 0.04 |

Rt | 0.85 | 0.85 | 0.85 | 0.85 | 0.85 | 0.83 |

Rt variability (Std Dev) | 0.04 | 0.04 | 0.04 | 0.04 | 0.04 | 0.04 |

d1 (m) | 0.3 | 1 | 3 | 10 | 1.9 | 1.9 |

d2 (m) | 0.3 | 1 | 3 | 10 | 1.3 | 1.3 |

d3 (m) | 0.3 | 1 | 3 | 10 | 0.9 | 0.9 |

Mass (kg) | 38 | 1414 | 38 × 10^{3} | 1414 × 10^{3} | 3143 | 3143 |

Point spacing (m) | 0.02 | 0.02 | 0.02 | 0.02 | 0.25 | 0.46 |

Number of points (·10^{6}) | 10 | 10 | 10 | 10 | 1.8 | 1.9 |

**Table 3.**Damping coefficients and artificial roughnesses used for the Rifle rockfall test site. The first 13 values are from Jones et al. [17]. They were replaced by the adjusted parameters shown on the bottom line for a second round of simulations with RocFall 8 and the proposed algorithm to obtain matching results with CRSP 4.

Cell Number | $\mathbf{Roughness}\text{}\left({\mathbf{S}}_{\perp}\right)$ | θ_{max} | R_{n} | R_{t} |
---|---|---|---|---|

- | (m) | (°) | - | - |

1 | 0.12 | 11.3 | 0.25 | 0.82 |

2 | 0.18 | 16.7 | 0.32 | 0.84 |

3 | 0.24 | 21.8 | 0.32 | 0.84 |

4 | 0.61 | 45.0 | 0.32 | 0.84 |

5 | 0.24 | 21.8 | 0.3 | 0.84 |

6 | 0.24 | 21.8 | 0.3 | 0.84 |

7 | 0.24 | 21.8 | 0.3 | 0.83 |

8 | 0.30 | 26.6 | 0.33 | 0.82 |

9 | 0.24 | 21.8 | 0.33 | 0.82 |

10 | 0.43 | 35.0 | 0.34 | 0.84 |

11 | 0.37 | 31.0 | 0.34 | 0.84 |

12 | 0.24 | 21.8 | 0.34 | 0.84 |

13 | 0.12 | 11.3 | 0.34 | 0.85 |

Adjusted | 0.18 | 16.7 | 0.28 | 0.98 |

**Table 4.**Damping parameters based on slope thresholds used for the alpine slope near La Verda–Rougemont in Switzerland.

Slope | Soil Type | Rg70 | Rg20 | Rg10 |
---|---|---|---|---|

(°) | - | (m) | (m) | (m) |

0–2 | 3 | 0.03 | 0.03 | 0.03 |

2–7 | 4 | 0.05 | 0.05 | 0.10 |

7–25 | 4 | 0.05 | 0.10 | 0.20 |

25–45 | 5 | 0.00 | 0.05 | 0.10 |

45–90 | 6 | 0.00 | 0.00 | 0.00 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Noël, F.; Cloutier, C.; Jaboyedoff, M.; Locat, J.
Impact-Detection Algorithm That Uses Point Clouds as Topographic Inputs for 3D Rockfall Simulations. *Geosciences* **2021**, *11*, 188.
https://doi.org/10.3390/geosciences11050188

**AMA Style**

Noël F, Cloutier C, Jaboyedoff M, Locat J.
Impact-Detection Algorithm That Uses Point Clouds as Topographic Inputs for 3D Rockfall Simulations. *Geosciences*. 2021; 11(5):188.
https://doi.org/10.3390/geosciences11050188

**Chicago/Turabian Style**

Noël, François, Catherine Cloutier, Michel Jaboyedoff, and Jacques Locat.
2021. "Impact-Detection Algorithm That Uses Point Clouds as Topographic Inputs for 3D Rockfall Simulations" *Geosciences* 11, no. 5: 188.
https://doi.org/10.3390/geosciences11050188