# Semiautomated Statistical Discontinuity Analyses from Scanline Data of Fractured Rock Masses
## 2 Methods and Theoretical Background
### Theoretical Background  
#### 2.2.1 Determination of Discontinuity Sets  
The first step of the statistical discontinuity analyses focuses on the determination of the discontinuity sets of the rock mass. Given that several commercial or freeware-based software products with implemented clustering algorithms are available, a new script was not developed in this study. Almost all of these products allow the import and export of tabulated data, such as the scanline logging sheet presented herein. Some of these products are able to define discontinuity sets by advanced statistical methods. [PERSON] and [PERSON] [24] proposed a discontinuity set clustering algorithm based on the K-means algorithm for clustering of the normal vectors of all joints, with the possibility of manual adaptation of individual discontinuity assignment subsequently. This method requires the information of the number of discontinuity sets as a prior input. [PERSON] [25] developed an expert-supervised method for grouping of orientation data that requires the visual assignment of initial values for the mean orientation of one to a maximum of seven clusters. The fully automatic clustering method of [PERSON] et al. [26] is based on vector quantization. Similar to other methods, the number of clusters has to be defined in advance.  
However, all of these methods assign discontinuities to discontinuity sets and calculate their mean solely on the statistical distribution of the measured orientation data. Information regarding the genesis of the discontinuity set is not considered, but can be helpful for the discrimination of discontinuity sets [27]. Overlapping clusters of discontinuity sets are common and often related to multiphase fracturing processes, indicating that within a single discontinuity set not all the discontinuities are of the same age. For example, a comparison of the linear discontinuity frequency of the main joint set in a granitic rock mass measured at ground surface and in an investigation drift 550 to 1250 m below the surface show a decrease in frequency by a factor of 3.6 [28]. The depth dependence of joint density indicates that additional discontinuity formation processes near the ground surface took place, most likely due to stress release by gravitational unloading. According to [PERSON] et al. [29], the complex regional and local stress field and its alteration over time may lead to the formation of a joint continua that is characterized by a coaxial angular continuum of fracture planes, enclosing a maximum angle of 45\({}^{\circ}\) and including extension and hybrid shear fractures (i.e., failure in the shear extension-fracture transition). Under such conditions, defining a well-developed and delineable discontinuity set is difficult, as a large variation of orientation measurements and overlaps with other sets occur. This results in discontinuity orientation clusters with a highly dispersed scatter around a mean. From the rock mechanical aspect, it is useful to discriminate the discontinuity sets not only on the basis of the statistical distribution of the orientation but also on structural-geological criteria related to the formation processes (i.e., sedimentary bedding, foliation planes, stress-induced joint and shear fractures), which may help delimiting sets. Numerous studies show that a maximum of 3-5 discontinuity sets are sufficient in many cases to characterize a rock mass (e.g., [28; 30]). A high number of sets may be obtained by the unfavorable delimitation of structural homogeneous areas and should be avoided, because it can lead to a too low number of discontinuities for each set for the subsequent analyses.