# An Open-Source Algorithm for 3D ROck Slope Kinematic Analysis (ROKA)
## 3 Cases of Study
### Case B: Subvertical Rock Cliff of Quartzites--Ornea (CN, Italy), Ligurian Alps  
The second study rock slope is a sub-vertical cliff that is composed mainly by quartzites and is located along the right side of the Tanaro Valley, in the Ligurian Alps (44.147\({}^{\circ}\) latitude, 7.919\({}^{\circ}\) longitude), close to the village of Ormea (CN, Italy). The rock cliff is \(\approx\)80 m high and \(\approx\)100 m wide, and it is characterized by frequent instability events (rock falls) that imperil some houses and infrastructures below it (road and bridge). For this reason, it was studied using the unmanned aerial vehicle-based digital photogrammetry. The dataset of the slope was already presented and validated in [23]. The 3D point cloud has 98 million points, with a point density of 1000 pts/m\({}^{2}\) and a mean planimetric and vertical accuracy of 3.3 and 0.9 cm. One thousand thirty-six discontinuities were manually mapped (Figure 14). Figure 13: Rendering of the 3D model indicating the results of the ROKA Kinematic Analysis (KA) of the possible Mode of Failure (MOF). The color scale represents the critical value of the discontinuity planes and intersections that could activate (**a**) planar failure, (**b**) flexural toppling, (**c**) wedge failure, and (**d**) direct toppling (see Section 2.5 for the critical value definition). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Total Number of Critical Discontinuities** & **Wedge Failure** & **Direct Toppling** \\ \hline
4349 & 1214 & 153 \\
**Percentual** & 28\% & 4\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: According to the ROKA Kinematic Analysis, the table shows the number of discontinuities intersections that could activate wedge failure and direct toppling modes of failure. Three main sets of discontinuity (Figure 15) were identified: the bedding and the K2 and K3 fracture sets. The traditional KA was performed using the orientations of mapped discontinuities. A mean attitude of the rock slope of 300\({}^{\circ}\)/75\({}^{\circ}\) (dip direction/dip) indicates that all the considered MOFs can be activated (Figure 16). Figure 14: (**a**) 3D discontinuities manually mapped onto the Digital Outcrop Model of the vertical rock cliff of Ormea [23] and (**b**) a detail of the preceding image. Al the discontinuities are represented by Baecher’s discs [33]. Figure 15: Lower hemisphere projection and contour (Fisher distribution per 1% area) of the poles of the 1036 mapped discontinuities. The orientations of the three recognized sets’ mean pole and plane values are represented by the red dots and lines, respectively, and the orientation is expressed in dip direction and dip. In particular, the results reported in Tables 6 and 7 show that the most probable MOF is the wedge failure, with 17% of all the possible discontinuity intersections that could activate it. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Discontinuity Set** &
\begin{tabular}{c} **Total Number of** \\ **Discontinuities** \\ \end{tabular} & **Planar Failure** & **Flexural Toppling** \\ \hline K2 & 334 & 28 & 8\% & 21 & 6\% \\ Random & 340 & 24 & 7\% & 22 & 6.5\% \\ All & 1036 & 52 & 5\% & 43 & 4\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: According to the traditional Kinematic Analysis, the table shows the number of discontinuities that could activate planar and flexural toppling modes of failure. The bedding and K3 fracture set are not reported because their planes orientation does not suggest this possible failure. Figure 16: Traditional Kinematic Analysis (KA) of **(a)** planar failure, **(b)** flexural toppling, **(c)** wedge failure, and **(d)** direct toppling Modes of Failures (MOFs) performed onto a lower hemisphere and equal angle projection considering a mean slope attitude of 300\({}^{\circ}\)/75\({}^{\circ}\) (dip direction/dip). In the colored areas fall all the discontinuities and their intersections that could activate a specific MOF.