## 2 Theoretical Background  
The IBSD of a natural or man-made fractured rock slope is influenced by (a) the orientation and number of fracture sets, (b) the frequency and spacing, and (c) the persistence (size) of the fractures. Natural meso-scale fractures (i.e., joints) are often non-persistent and therefore characterized by finite fracture trace lengths or sizes. According to [5], persistence is defined as the areal extent or size of a fracture within a plane. In 2D, persistence can be estimated by observing the discontinuity trace lengths on the surface of rock exposures. Non-persistent fractures often show terminations within the intact rock which geometrically lead to intact rock bridges of variable extend. These rock bridges affect the overall rock mass strength, given that the intact rock is usually characterized by higher strengths and thus has a major impact on the stability of rock masses [15], but it is nearly impossible to map them at in situ conditions. It is obvious that rock mass failure occurs much easier if fractures are fully persistent and rock blocks are already formed as if pre-existing intact rock bridges have to fail either suddenly or in the long-term by sub-critical crack growth [11; 15; 16]. In many cases, the block area \(A_{0}\) [m\({}^{2}\)] or block volume \(V_{0}\) [m\({}^{3}\)] is estimated either on the basis of numerical DFN modelling or analytically by determining the mean normal set spacing of three major fracture sets (\(s_{1}\), \(s_{2}\), and \(s_{3}\) in [m]) and the angle between these fracture sets (\(\gamma_{11}\), \(\gamma_{13}\), and \(\gamma_{23}\) in [\({}^{\circ}\)]). According to [14], the mean block area and block volume assuming persistent fractures is given by:  
\[A_{0}=\frac{s_{1}\cdot s_{2}}{\sin(\gamma_{12})} \tag{1}\]  
\[V_{0}=\frac{s_{1}\cdot s_{2}\cdot s_{3}}{\sin(\gamma_{12})\cdot\sin(\gamma_{13 })\cdot\sin(\gamma_{23})} \tag{2}\]  
This common analytical approach based on the assumption of persistent fractures cause an underestimation of the block areas/volumes of non-persistent fracture sets (Figure 1). This could have adverse impacts for rockfall hazard assessments, such as underestimated run out length and block energies. In [13], the authors validated the supposition of [17], suggesting that the mean block area and block volume of rock masses with non-persistent fracture sets is related to the fracture persistence according to,  
\[A_{b}=\frac{s_{1}\cdot s_{2}}{\sin(\gamma_{12})\cdot\sqrt{p_{1}\cdot p_{2}}} \tag{3}\]  
\[V_{b}=\frac{s_{1}\cdot s_{2}\cdot s_{3}}{\sin(\gamma_{12})\cdot\sin(\gamma_{13 })\cdot\sin(\gamma_{23})\cdot\sqrt[3]{p_{1}\cdot p_{2}\cdot p_{3}}} \tag{4}\]  
where \(A_{b}\) and \(V_{b}\) are the block area and block volume assuming non-persistent fractures, respectively, and \(p_{1}\), \(p_{2}\), and \(p_{3}\) are persistence factors in the range between 0 and 1. Subscripts 1 to 3 represent the assigned number to fracture sets. Assuming a Veneziano fracture network model [18; 19], i.e., Poisson line and Poisson plane processes, the persistence factor is given as the ratio between the accumulated fracture trace length in a sampling plane to the total characteristic length of the rock mass under consideration (i.e., the sampling plane). Alternative approaches to estimating the persistence factor have been presented by, e.g., [14; 19], but they are all based on ratios between fracture and intact rock bridge geometries, which is a ratio that is usually very difficult to measure during practical campaigns in the field. Validation of these equations by [13] based on stochastic analyses using numerical fracture network simulation tools implemented in the discrete element codes UDEC and 3 DEC (Itasca), respectively, suggest that simulated block sizes only differ by \(<\)1% and around 4% from the Equations (3) and (4), respectively. However, the persistence factors \(p_{1}\), \(p_{2}\) and \(p_{3}\) require a ratio between the sum of the trace lengths of the joint sections in a plane and the length of the intact rock bridges in between.