# Semiautomated Statistical Discontinuity Analyses from Scanline Data of Fractured Rock Masses
## 2 Methods and Theoretical Background
### Theoretical Background
#### 2.2.2 Determination of the Total Spacing and Normal Set Spacing  
Discontinuity spacing is a basic parameter that affects the size of the blocks and thus rock mass strength, deformability, and hydraulic conductivity. [PERSON] [14] differentiated three types of discontinuity spacing: (i) total spacing, (ii) set spacing, and (iii) normal set spacing. Total spacing is the spacing determined between a pair of immediately adjacent discontinuities, measured along a defined scanline. Set spacing is the spacing between a pair of immediately adjacent discontinuities from a particular discontinuity set, measured along a defined scanline. Normal set spacing is the spacing that is determined along a scanline oriented parallel to the mean normal to the set. Whereas the former two types of spacing parameters are scanline direction-dependent, normal set spacing is a general rock mass parameter. The normal set spacing \(X_{n}\) of a set is calculated on basis of themeasured distance between two discontinuities of the same set along the scanline \(X_{d}\) and the angle \(\delta\) between the orientation of the normal vector of the discontinuity and the scanline orientation [14; 31]:  
\[X_{n}=X_{d}\cos\delta \tag{1}\]  
where \(\delta\) is the acute angle between the discontinuity set normal and the sampling line and may be expressed as:  
\[\delta=|\cos(\alpha_{n}-\alpha_{s})\cos\beta_{n}\cos\beta_{s}+\sin\beta_{n} \sin\beta_{s}| \tag{2}\]  
where \(\alpha_{n}/\beta_{n}\) are the trend/plunge of the normal set vector and \(\alpha_{s}/\beta_{s}\) are the trend/plunge of the scanline. A limitation of the maximum angle between scanline and the orientation of the normal vector of the set can be set to avoid extreme normal set spacing values [32]. The mean normal set spacing \(\overline{X}\) is determined by:  
\[\overline{X}=\sum_{i=1}^{n}X_{i}/n \tag{3}\]  
where \(X_{i}\) is the \(i\)-th spacing value and \(n\) the total number of spacing values.  
The discontinuity spacing values from scanlines can be visualized by histograms, which indicate if the intersection points of traces along the scanline are evenly spaced, clustered, or randomly distributed. [PERSON] and [PERSON] [21] proposed studying the pattern of spacing values by applying density probability functions. Based on several case studies, they observed that randomly located intersection points of traces along a scanline produce a negative exponential probability density distribution of spacing values:  
\[f(x)=\lambda e^{-\lambda x} \tag{4}\]  
where \(f(x)\) is the probability density \(\lambda\) is the mean linear discontinuity frequency, and \(x\) is the spacing value. Furthermore, they showed that the superimposition of different density distributions, such as negative, uniform, or normal distribution, will also lead to a negative exponential density distribution. Hence, for many situations, especially when the formation of a set is related to multiphase fracturing processes, it is likely that the measured distribution will coincide with a negative exponential form. However, in some cases, other density probability functions, for example, the log-normal distribution or the Weibull distribution may better represent the measured data set [28; 33].