# An Open-Source Algorithm for 3D ROck Slope Kinematic Analysis (ROKA)
## 1 Introduction  
Discontinuities have a great influence on rock slope stability because, along their surfaces, the cohesion of rock can be very small or null, and therefore, they can induce different Modes of Failure (MOFs). Rock slope stability can be analyzed using different approaches, from simple geometrical methods (e.g., kinematic analysis) to complex numerical methods (e.g., Limit Equilibrium (LE); Finite Element Method (FEM); Block Element Method (BEM); Discrete Element Method (DEM; FEM-DEM; etc.)). The Kinematic Analysis (KA) is certainly the simplest and quickest technique because it is purely geometric. It evaluates the possible MOFs, analyzing the angular relationships between slope surface and discontinuities [1] without considering the forces responsible for the MOF [2]. In any case, before carrying out more sophisticated analyses by numerical methods, it is often useful to proceed with an evaluation of the potential mechanisms of kinematic instability affecting a slope. Several studies have been realized to determine the possible MOFs using stereographic projection-based techniques [3, 4, 5, 6, 7, 8, 9, 10]. Among them, the Markland test [4] and its refinement proposed by [PERSON] [6] is one of the most used methods. The main pitfalls based on the principle of the traditional Markland test are as follows [11]: (a) the assumption of a uniform slope angle and slope azimuth, (b) the possible presence of non-tightly clustered discontinuity data, and (c) the possible non-uniform spatial distribution of discontinuity orientations and sizes. The assumption of a uniform slope angle and slope azimuth could negatively influence the KA of a natural slope with more than one oriented surface (multi-faced slope) [12]. As emphasized by [11], the traditional KA considers only the mean and most representative orientation of the identified discontinuity sets, due to the large number measures. This assumption makes the method qualitative and, anyhow, it is applicable only to tight discontinuity clusters. Moreover, the authors emphasize that the discontinuities orientation and size could change onto the rock slope surface. Therefore, some discontinuities with a specific orientation or dimension could be present only onto limited portions of the multi-faced rock slope. [13] show that the size of the discontinuities, not considered in the KA, has a crucial role in the possible formation of unstable intersections. These limitations are principally related to a stereographic approach, where the KA is performed using only the mean attitudes of the rock slope and of the discontinuity sets. Today, the computer technology permits overcoming these limits: the problem of the consideration of uniform slope angle, slope azimuth, and friction angle could be easily resolved performing the KA with the different orientations of the rock slope, using the method proposed by [12] or, as suggested by several authors (e.g., [11; 14; 15; 16; 17]) performing a sensitivity analysis using different KAs calculated for each possible slope orientation; the problem of the presence of non-tightly clustered discontinuity data could be easily resolved considering all the orientations of the discontinuities (not only the representative) in order to consider all the orientation variability [11]. Notwithstanding, the spatial distribution of discontinuity orientations and sizes still remains a limitation because no proposed method considers the 3D location of the discontinuity (a certain set of discontinuities can be present only in a specific portion of the rock slope) and its dimension (small discontinuities normally have less influence on stability than larger ones). This limit is principally related to techniques of data acquisition: with a traditional field survey, it is not possible to map all the discontinuities affecting a rock slope due to the inaccessibility of some portions of it, and it can be really difficult to define exactly the 3D position of the discontinuity measured by hand in the field.