# An Open-Source Algorithm for 3D ROck Slope Kinematic Analysis (ROKA)
## 3 Cases of Study
### Case A: Northern Apennines--Calcareous Turbidites
Figure 10: 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 181\({}^{\circ}\)/56\({}^{\circ}\) (dip direction/dip), a friction angle of 30\({}^{\circ}\), and the lateral limits in \(\pm\) 20\({}^{\circ}\). In the colored areas fall all the orientations of the critical discontinuities and their intersections. Figure 9: Lower hemisphere projection and contour (Fisher distribution per 1% area) of the 1919 mapped discontinuities poles. The orientation (dip direction and dip) of the mean poles and planes of the five recognized sets are represented by the red dots and lines, respectively. The traditional KAs was performed considering a mean slope orientation of 181\({}^{\circ}\)/56\({}^{\circ}\) (dip direction/dip), a friction angle of 30\({}^{\circ}\), and lateral limits of 20\({}^{\circ}\). The principal results show the following:  
* Part of the K4 discontinuity set and some random discontinuities could act as a sliding surface (Figure 10a);
* Few K5 discontinuities and random fractures could activate the flexural toppling MOF (Figure 10b);
* Most of the orientations of the intersections between K2 and bedding and those between all the fracture sets (K2-K3, K2-K4, K2-K5, K3-K4, K3-K5 and K4-K5) could induce wedge failure (Figure 10c);
* The orientation of the intersection between K4 fractures and bedding could activate the direct toppling MOF (Figure 10d). The number of the critical discontinuities and intersections calculated considering all the discontinuities as infinite planes are reported in Tables 2-4. In this study, due to the small dimensions of the analyzed portion of the road cut, a 10-cm scan radius was used to obtain a representative \"local\" orientation of the slope, highlighting all the small changes. Subsequently, the algorithm tested whether the discontinuities and their intersections intersect the moving circular scan-plane (Figure 12), representing the slope's local geometry. If the intersection test is positive, ROKA algorithm performs the Kinematic Analysis (KA) of the possible MOFs (planar failure, flexural toppling, wedge failure, and direct toppling--Figure 13) considering the local orientation of the slope and the attitude of the detected discontinuity planes and intersections. Figure 11: The “local” orientation of the 3D model calculated from the normal vectors of the points that fall inside the moving scan-sphere, with a radius of 10 cm, is visualized as (**a**) dip and (**b**) dip direction. In (**c**), the areas in yellow represent portions of the slope that have an overhanging attitude. Figure 12: Rendering of the 3D model of the slope indicating (**a**) the number of the intersections between the scan-plane and discontinuities for every point of the cloud where the scan-plane (representing the local geometry of the slope) is centered, and (**b**) the number of the intersections between the scan-plane and the discontinuity intersections. The middle of the outcrop shows no intersection because it is highly vegetated (see Figure 6) and, therefore, no visible discontinuities were mapped (Figure 8). In Figure 13 (see percentage values), a classification of the different critical discontinuities, based on the length of their intersection with the slope (see Section 2.5), is reported. In this way, the most unstable portions of the slope are highlighted, allowing eventually to focus the further detailed stability analyses (e.g., field-based roughness, aperture, filling, seepage, wall strength characterization and definition). The results of algorithm KA are reported in Tables 4 and 5. The comparison of the results of the kinematic analyses performed by the traditional method and ROKA algorithm will be discussed further in the Discussion section.