# Preliminary Modeling of Rockfall Runout: Definition of the Input Parameters for the QGIS Plugin QPROTO

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The QPROTO Plugin

_{0}and y

_{0}are the coordinates of the source point S, H(x

_{0},y

_{0}) the elevation of the source point, h

_{p}(x,y) the elevation of the topographic surface at point P(x,y) and ϕ

_{p}the energy angle. In Figure 2, the heights defined at the location of point P(x,y) are linked to the energy balance of the falling block. H(x

_{0},y

_{0}) represents the total energy content of the block which is defined by the energy conservation principle as the sum of potential h

_{p}(x,y), kinetic h

_{k}(x,y) and friction or dissipative contributions (${h}_{d}\left(x,y\right)=\overline{PS}\cdot tan{\varphi}_{p}$).

_{k}(x,y) at each point of the cone adopting the following relationship (Equation (2)):

## 3. Estimation of the QPROTO Input Parameters: The Proposed Methodology

- Creation of simplified synthetic slopes and definition of their geometrical and physical characteristics;
- Execution of a statistically representative number of detailed 3D rockfall numerical simulations on the synthetic slopes in different conditions (presence of forest, block volume and shape etc.), on the basis of the parametric variation of some variables;
- Statistical interpretation of the results in terms of energy angle and lateral angle;
- Correlation of the cone angles with the considered variables;
- Interpolation of the results and definition of charts for the estimation of the cone angles;
- Validation of the results through their application to real study cases.

#### 3.1. Synthetic Slopes

#### 3.2. Parametric Analyses

^{3}was introduced and the following five classes of volume were assumed for the parametrical study: 0.1, 0.5, 1.0, 5.0 and 10.0 m

^{3}. Finally, both spherical and cubic shapes were taken into account in the Rockyfor3D simulations. In Table 4, a recap of the main characteristics of the blocks adopted in this work is reported. Finally, the probabilistic nature of Rockyfor 3D [27] allowed to perform 20,000 launches for each analysis in order to acquire a statistically representative number of simulations, necessary for a reliable statistical interpretation of the results.

#### 3.3. Statistical Interpretation of the Results

## 4. Results

#### 4.1. Influence of the Slope Characteristics

#### 4.1.1. Slope Inclination

^{2}value in both cases.

#### 4.1.2. Forest Coverage

- Figure 7a shows the results on a 30° inclined slope, where most of the simulated blocks are distributed on the transit zone in the presence of dense forest, according to the mitigation effect of trees. The energy angle ${\varphi}_{p,2\%}$ increases from 26° without trees to 31.1° in presence of trees, while the lateral angle ${\alpha}_{98\%}$ increases from 5.2° to 18.4°. Such values are a consequence of the large number of blocks deposited just below the source and will be discussed later (Section 5);
- Figure 7b shows the scenario with a 45° inclined slope, where the effect of trees is minor and blocks are distributed on the whole slope (transit and accumulation zone) in the presence of dense forest; the energy angle ${\varphi}_{p,2\%}$ increases from 33.9° to 37.6°, while the lateral angle α
_{98%}increases from 9.6° to 11.3°. A lower number of blocks have accumulated just below the source and their effect on the statistical result is lower; - Figure 7c shows the scenario with a 60° inclined slope, where the effect of trees is at a minimum. Due to the steep inclination of the slope, the majority of blocks reaches the foot, and the extension of the accumulation zone is similar to that obtained without trees; the energy angle ${\varphi}_{p,2\%}$ increases from 44.2° to 46°, while the lateral angle α
_{98%}decreases from 12.4° to 9.9°.

#### 4.2. Influence of Block Characteristics

#### 4.2.1. Block Volume

- For ${\omega}_{2}=$ 30° (Figure 9a), only blocks of 0.1 m
^{3}volume stop above the foot of the slope, while larger volumes reach the stopping zone. Energy angles ${\varphi}_{p,2\%}$ are in the range [34.3°, 21.8°] with increasing block volume; - For ${\omega}_{2}=$ 45° (Figure 9b), all volumes reach the foot of the slope and are deposited in the stopping zone. Energy angles ${\varphi}_{p,2\%}$ are in the range [40.6°, 29.1°] with increasing block volume;
- For ${\omega}_{2}=$ 60° (Figure 9c), the map of deposits has the same configurations of the 45° slope. Energy angles ${\varphi}_{p,2\%}$ are in the range [50.6°, 38.3°] with increasing block volume.

^{3}, the effect of trees seems less important. The influence of trees can also be observed as a function of the slope inclination angle. From Figure 9, it is possible to notice that:

- For ${\omega}_{2}=$ 30°, energy angles ${\varphi}_{p,2\%}$ are in the range of [36.7°, 23.2°]. The increment with respect to the previous scenario (without forest) is a function of the block volume and decreases with increasing volume;
- For a ${\omega}_{2}=$ 45°, energy angles ${\varphi}_{p,2\%}$ are in the range of [44.5°, 29.8°] with the same increment with respect to the previous scenario (without forest);
- For a ${\omega}_{2}=$ 60°, energy angles ${\varphi}_{p,2\%}$ are in the range of [53.3°, 39.6°] with a constant increment of 2° with respect to the previous scenario (without forest).

#### 4.2.2. Block Shape

- For ${\omega}_{2}=$ 30°, energy angles ${\varphi}_{p,2\%}$ are in the range [41.4°, 25°] with increasing block volume in the absence of trees; in the presence of trees, they vary in the range of [42.4°, 26°]
- For ${\omega}_{2}=$ 45°, energy angle ${\varphi}_{p,2\%}$ are in the range [43.4°, 30.3°] with increasing block volume in the absence of trees, in the presence of trees, they vary in the range of [44.6°, 32°]
- For ${\omega}_{2}=$ 60° energy angle ${\varphi}_{p,2\%}$ are in the range [53.1°, 37.5°] with increasing block volume in the absence of trees, in the presence of trees, they vary in the range of [55.2°, 39.2°].

## 5. Discussion

^{3}scenarios. Figure 11a refers to slopes without trees while Figure 11b is related to dense forest simulations. Intermediate forest density can be associated to interpolated values between the two sets of curves.

- Computed energy angles can reproduce the mitigation effect of dense forests located on the transit zone of the slope;
- The growth of a block volume causes the reduction of the energy angle and provide larger invasion areas;
- The growth of slope angle ${\omega}_{2}$ causes the increase of the energy angle and the reduction of the invasion area.

## 6. Validation and Examples

^{3}) destroyed a flexible net barrier whose capacity is estimated around 800 kJ and hit three buildings. On average, the slope involved in this event is inclined of about 45°. A back-analysis of the event was first conducted by means of Rockyfor3D. A pre-event DTM with a resolution of 5 m was used, and 5 m

^{3}blocks with a rock matrix density of about 2600 kg/m

^{3}were simulated from the detachment niches, defined on the basis of event reports and the available digital cartography.

^{3}and a dense forest coverage (400 trees/ha). The energy angle ${\varphi}_{p}$ = 34° can be extracted from Figure 11b. A value of $\alpha $ = 10° was finally assumed to completely define the visibility cones. Seven points were selected inside the source area and the attributes listed in Table 1 are associated to each of them. Finally, since it is a simulation of a real event, a constant propensity index DI = 1 was assumed. The result of the analysis is shown in Figure 14b, again in terms of the mean kinetic energy. The comparison between Figure 14a,b shows that QPROTO results are overestimated both in terms of invasion area and mean energy, involving a portion of the hamlet on the right side of the damaged buildings. It is worth noting that the analysis with Rockyfor3D considers the flexible net barrier of 800 kJ capacity, which played a role in the rockfall event, decelerating blocks and reducing their kinetic energy. However, this role seems weak, as shown in Figure 14a, since the capacity of the barrier is low with respect to the kinetic energy of the blocks. At the current state of the work, it is not possible to consider how the presence of protective works on the slope can influence the energy line, and thus, its angle. Therefore, the preliminary QPROTO analysis does not consider this effect, and therefore, leads to a conservative result. On the basis of these observations, it is possible to conclude that the results obtained are acceptable and encouraging for future implementations.

^{3}. The fallen blocks partially hit and damaged the existing embankment located along the provincial road, and some boulders arrived on the provincial road, reaching the foot of the slope (Figure 15). Such blocks damaged the provincial road, destroyed two isolated buildings and compromised the safety of inhabitants (Figure 16b).

^{3}, which was the average observed volume at the foot of the slope according to the event report prepared by ARPA Piemonte [40].

^{3}and a block mass of 2600 kg were assumed as representative of the phenomenon with reference to all the homogeneous areas because not enough data were available to obtain a more detailed characterization. Since the aim of the analysis is to test the methodology for the estimation of the QPROTO parameters, the detachment propensity DI was set equal to 1 within each homogeneous area. This means that all the computed parameters have the same probability of occurrence within any point of the propagation area, and the maps “count” and “susceptibility” are equal. In other words, the susceptibility within the invasion zone only depends on the number of source points and on the cone parameters ${\varphi}_{p}$ and $\alpha .$

## 7. Conclusions

- For unitary blocks, a linear variation of both energy and lateral angles can be assumed as a function of slope inclination if non-forested slopes are considered and all the simulated blocks reach the stopping zone (i.e., the foot of the slope);
- The effect of trees consists of reducing the extension of the runout area and, thus, in increasing the energy angle: a linear equation can be assumed as a function of the slope inclination in the case of ${\varphi}_{p}$. In the presence of trees, unitary blocks, in many cases, do not reach the foot of the slope and stop at the transit zone. Therefore, an overestimation of the lateral spreading can be observed as a consequence of the statistical processing of the data. Here, some preliminary results concerning the estimation of the lateral angle are reported, and more detailed analyses will be carried out in future works;
- The influence of trees is also a function of the slope inclination: it is higher for less steep slopes and gradually decreases when the slope inclination increases. The linear variation of the energy angle with slope inclination is, therefore, less pronounced in the case of densely forested slopes;
- Runout distance is strongly influenced by block volume: smaller volumes reach the closest distances on the slope, while larger blocks run farther. The variation of the energy angle with reference to block volume is not linear because it depends on the possibility that blocks reach the foot of the slope or stop in the transit zone;
- Trees can influence the path of smaller blocks, reducing the extension of the runout area. This effect reduces when block volume increases;
- Spherical and cubic blocks show similar behaviors. Considering the quick nature of the plugin QPROTO, a mean value of the energy angles can be considered as representative of generic isometric blocks.

- The elevation of the source point with respect to the foot of the slope (and consequently the length of the transit zone) was kept constant in all the parametric analyses. Since it affects the length of boulder paths, it is supposed to strongly influence the results of the analyses and the cone angles. Further investigations on this topic will be carried out in the future in order to extend the parametric analyses to different elevation classes of the source points;
- Block slenderness has not yet been taken into account because of the difficulty in interpreting the results provided by Rockyfor3D. This software is based on a hybrid model and simulates the block impacts (with the soil or with a tree) using equivalent spheres, whose dimensions are a function of the three initial dimensions of the block (height, width and thickness). This could affect the results in terms of runout area, and many uncertainties can be related to these changes of shape. To simulate the real behavior of blocks of elongated or slab shape, the use of a fully 3D rigid body model could be helpful and will be evaluated in the future;
- The forest coverage adopted in this work, and described in Section 3, has medium protective capacity against rockfall. To generalize the obtained results, in the future, a real sensitivity study will be carried out in order to consider different tree diameters, forest density, types of trees etc. Some interesting suggestions can be derived from previous studies on the protective role of forests, such as the one performed in the framework of the RockTheAlps project [41];
- In this work, a very smooth slope and elastic geomaterials were adopted in order to produce precautionary results. In case of small scales and preliminary rockfall analyses, very rough information concerning soil characteristics are usually available and using precautionary values could be the better solution;
- The entire work is based on trajectographic analyses carried out on artificially created simplified slopes, to control the input variable with high levels of detail. The obtained results are in good agreement both with real cases and literature data but can be unrealistic if the slope is characterized by strong orientation gradient, which can deviate blocks from the original dip direction. Currently, the problem can be faced only through a high number of source points, in order to obtain a wider invasion area by the overlapping of differently oriented cones. A GIS study of slope morphometric features (concavity, convexity, curvature etc.) using well-documented real cases could help in introducing this aspect in the evaluation of the orientation and the lateral aperture of the cone. This will be investigated in the future development of the research. Moreover, some laboratory experiments carried out with scaled slopes could be useful to further validate the obtained results.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Spatial definition of the visibility cone as it is defined in QPROTO: ${\varphi}_{p}$ = energy angle in the vertical plane, $\alpha =$ lateral angle in the horizontal plane and θ = dip direction (modified from [16]).

**Figure 2.**3D sketch of the cone generation within the QPROTO plugin. The cone originates from the S(x

_{0},y

_{0}) point and is sectioned by a vertical plane passing through the generic topographic point P(x,y). The light grey shows the volume of the cone, which is limited by the topographic surface and the plane defined by the energy line (orange line). The horizontal dark green line refers to the total energy of the block.

**Figure 3.**Sketch of the synthetic slope adopted in the present work with indication of the main zones and dimensions. The energy line connects the source point S to the i-th stopping point P

_{i}. The energy angle ${\varphi}_{p}$ is defined in the vertical plane as the angle enclosing between the horizontal plane (black lines) and the energy line. The lateral angle $\alpha $ is defined by the projection of the energy line on the horizontal plane and the dip direction of the source point.

**Figure 4.**Examples of the typical histograms and cumulative distribution functions (CDFs) of (

**a**) the energy angle ${\varphi}_{p}$ and (

**b**) the lateral angle $\alpha $ obtained in the case of 5 m

^{3}spherical block, 0 trees/ha, and ${\omega}_{2}={30}^{\xb0}$.

**Figure 5.**Example of the frequency distribution obtained for the set of 20,000 launches described in Figure 4: (

**a**) three-dimensional histogram; (

**b**) two-dimensional view of the same histogram. The magenta vertical and horizontal lines highlight the value of ${\varphi}_{p,2\%}$ and ${\alpha}_{98\%}$, respectively.

**Figure 6.**Linear variation of angles ${\varphi}_{p}$ and $\alpha $ as a function of slope inclination ${\omega}_{2}$. Spherical blocks of unitary volume and absence of forest are considered here.

**Figure 7.**Maps of deposited blocks in the case of a spherical block with a unitary volume. Upper and lower limits of the transit zone (zone 2) are indicated in black dashed lines. The blue area refers to the scenario with 0 trees/ha, while the red one refers to the scenario with 400 trees/ha. Slope inclination ${\omega}_{2}$: (

**a**) 30°, (

**b**) 45° and (

**c**) 60°.

**Figure 8.**Effect of forest on the variation of angles ${\varphi}_{p}$ and $\alpha $ as a function of slope inclination ${\omega}_{2}$. Spherical blocks of unitary volume are considered here.

**Figure 9.**Results of the parametrical simulations for (

**a**) ${\omega}_{2}$ = 30°, (

**b**) 45° and (

**c**) 60° slopes. Energy angle values are given as a function of block volumes and forest coverage.

**Figure 10.**Mean values of the energy angle on account of isometric cubic/spherical blocks, as a function of block volume, forest coverage and slope inclination.

**Figure 11.**Resulting chart for the estimation of the energy angle ${\varphi}_{p}$ as a function of slope inclination and block volume. Contour black lines refer to the same energy angle conditions: (

**a**) non-forested slope; (

**b**) forested slope (400 trees/ha).

**Figure 12.**Overestimated runout zone assuming ${\alpha}_{98\%}$ for the cone in the case a significant number of blocks stops just below the source. Reference is made to the scenario described in Figure 7a.

**Figure 14.**Results of the back analyses for the 2011 rockfall event in Cels-Morlière hamlet (rockfall sources in yellow, barrier in dashed white line, impacted buildings in red). (

**a**) Rockyfor3D mean kinetic energy; (

**b**) QPROTO mean kinetic energy.

**Figure 15.**Map of the 2010 event. The red line highlights the runout area of the phenomenon and the red triangles define the location of the observed fallen blocks. The embankment (yellow line), located along the provincial road, was impacted by a cluster of boulders in the southern part of the runout zone.

**Figure 16.**(

**a**) Back analysis of the 2010 event; results in terms of mean energy (kJ). Runout area is shown with a blue line, source points with blue dots, provincial road no. 216 in white and impacted buildings in grey. Orthophoto 2010 Regione Piemonte. (

**b**) Effects of the May 2010 event on the Melezet site: at the top, blocks that stopped along the provincial road no. 216 (no direct damage to the buildings); at the bottom, a wood building that was seriously damaged (Bld.3 in Figure 16a).

**Figure 17.**Homogeneous source areas with indication of the energy angle ${\varphi}_{p}$ associated to each one. Runout area of 2010 event in magenta. Orthophoto AGEA 2018.

**Figure 18.**QPROTO results for the site of Melezet. Source areas are shown in light brown, local viability in pink, provincial road no. 216 in magenta, bridges in black. (

**a**) Mean energy in kJ, (

**b**) susceptibility. Orthophoto AGEA 2018.

No. | Attribute | Description |
---|---|---|

$0$ | ID | Identification number of the source point |

$1$ | ELEVATION | Height of the source point a.s.l. (m) |

2 | ASPECT | Dip direction θ of the slope in the source point (°) |

3 | ENERGY ANGLE | Energy line angle ${\varphi}_{p}$ of the cone with apex in the source point (°) |

4 | LATERAL ANGLE | Lateral angle $\alpha $ of the cone with apex in the source point (°) |

5 | VISIBILITY DISTANCE | Distance to which the analysis can be extended (m) |

6 | DETACHMENT PROPENSITY | Detachment index DI of each source point (-) |

7 | BOULDER MASS | Mass of the boulder (kg) |

Type | Description | |
---|---|---|

count | Raster map | Number of source points that can view each cell |

susceptibility | Raster map | Weighted view frequency |

v_min | Raster map | Minimum computed block velocity |

v_mean | Raster map | Mean computed block velocity |

v_max | Raster map | Maximum computed block velocity |

e_min | Raster map | Minimum computed kinetic energy |

e_min | Raster map | Mean computed kinetic energy |

e_max | Raster map | Maximum computed kinetic energy |

w_en | Raster map | Maximum weighted kinetic energy |

w_tot_en | Raster map | Total weighted kinetic energy |

Finalpoints | Shape file | Details on points located in the runout zone |

_Log.txt | Log file | Report of the computed analysis |

Characteristics | Zone 1 | Zone 2 | Zone 3 |
---|---|---|---|

$H$ (m) | 50 | 230 | 20 |

$\omega $ (°) | 70.0 | 30.0, 45.0, 60.0 | 1.5 |

L (m) | 500 | 500 | 500 |

Soil type (-) | 4 | 4 | 1 |

Roughness (m) | 0.05 | 0.05 | 0.05 |

Forest coverage (trees/ha) | 0 | 0, 400 | 0 |

Characteristic | Values |
---|---|

Density (kg m^{−3}) | 2500 |

Shape (-) | Cubic, spherical |

Volume (m^{3}) | 0.1, 0.5, 1.0, 5.0, 10.0 |

Attribute | Value |
---|---|

ID | Progressive number |

Elevation (m) | From the analysis of the DTM |

Aspect θ (°) | From the analysis of the DTM |

Energy angle ${\varphi}_{p}$ (°) | Form the proposed methodology |

Detachment propensity DI (-) | 1 |

Block mass (kg) | 2600 |

**Table 6.**Mean slope orientation and energy angle for each homogeneous area, obtained from the chart in Figure 11b, with the assumption of a dense forest coverage and boulder volume V = 1 m

^{3}.

Homogeneous Area | Mean Slope Inclination (°) | Energy Angle ${\mathit{\varphi}}_{\mathit{p}}$ (°) |
---|---|---|

0 | 34.0 | 35.0 |

1 | 40.0 | 37.0 |

2 | 36.0 | 36.0 |

3 | 35.5 | 36.0 |

4 | 38.0 | 37.0 |

5 | 34,4 | 35.0 |

6 | 35.0 | 35.0 |

7 | 32.0 | 34.0 |

8 | 33.0 | 34.0 |

9 | 41.4 | 38.0 |

10 | 39.4 | 37.0 |

11 | 38.6 | 37.0 |

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## Share and Cite

**MDPI and ACS Style**

Castelli, M.; Torsello, G.; Vallero, G.
Preliminary Modeling of Rockfall Runout: Definition of the Input Parameters for the QGIS Plugin QPROTO. *Geosciences* **2021**, *11*, 88.
https://doi.org/10.3390/geosciences11020088

**AMA Style**

Castelli M, Torsello G, Vallero G.
Preliminary Modeling of Rockfall Runout: Definition of the Input Parameters for the QGIS Plugin QPROTO. *Geosciences*. 2021; 11(2):88.
https://doi.org/10.3390/geosciences11020088

**Chicago/Turabian Style**

Castelli, Marta, Giulia Torsello, and Gianmarco Vallero.
2021. "Preliminary Modeling of Rockfall Runout: Definition of the Input Parameters for the QGIS Plugin QPROTO" *Geosciences* 11, no. 2: 88.
https://doi.org/10.3390/geosciences11020088