# Semiautomated Statistical Discontinuity Analyses from Scanline Data of Fractured Rock Masses
## 2 Methods and Theoretical Background
### Theoretical Background
#### 2.2.4 Determination of Trace Lengths and Termination  
A systematic mapping of entire discontinuity surfaces is difficult, since fully exposed surfaces are rare and termination characteristics are mostly not visible in 3D. Advanced remote sensing methods based on UAV and terrestrial laser scanning (TLS) provide some improvement of the sampling situation, since areal sampling of large rock faces is performed and discontinuities that are aligned subparallel to the rock face are scanned preferably. However, discontinuities exposing as traces are underestimated by these methods. Scanline mapping can reduce this problem, but has to account for other sampling biases, resulting from (i) the tendency to intersect preferentially the longer traces, (ii) trimming of short traces, i.e., the difficulty in measuring discontinuity traces with a length below a certain threshold value, and (iii) curtailment of long traces [14; 15]. Some authors assume that discontinuities from a given set can be represented by circular disks where their diameters are distributed according to some probability density distribution. In order to avoid developing a model for the discontinuity shape, some authors have decided to treat the problem entirely in 2D [14; 15; 34]. Such 2D approaches are based on probability density distributions of traces, e.g., negative exponential, uniform, or triangular [14; 15], whereas others are distribution-independent [16]. Herein, four different approaches to estimate the mean full trace length \(\mu_{L}\) of a rock face for a set are presented, according to [14]. Approach I is related to considerations of [14; 15] and is based on the assumption of a negative exponential distribution of full trace lengths for a given discontinuity set over an entire rock face with the mean \(\mu_{L}\). Hence, the probability density distribution \(g(l)\) with its mean \(\mu_{SL}\) of the full trace lengths intersected by the scanline is given by:  
\[g(l)=\frac{l\ e^{-\left(l/\mu_{L}\right)}}{\left(\mu_{L}\right)^{2}} \tag{8}\]  
\[\mu_{SL}=2\ \mu_{L} \tag{9}\]  
Equation (9) indicates that the mean full trace length of the entire face is equal to half of the mean full trace length of the scanline intersected traces. Approach II determines the mean of the full trace length distribution by considering the numerical proportion of discontinuities that have a semi-trace length less than the user-specified curtailment value \(c\)[14; 15]. Given that the total sample size of semi-trace lengths for a given discontinuity set is \(n\) and a certain number \(r\) of these have a semi-trace length less than \(c\), then the mean of the full trace length \(\mu_{L}\) can be calculated by:  
\[\mu_{L}=\frac{c}{-ln\left(1-\frac{r}{n}\right)} \tag{10}\]  
Approach III is independent of the type of probability distribution and is based on a histogram of semi-trace lengths, i.e., number of values \(N(l))\) and its intercept with the vertical axis at the limit \(l=0\) of the horizontal axis [14]. Therefore, a curve has to be fitted to the histogram and extrapolated to \(l=0\). Taking this interception value \(N(l)\) the arbitrary defined class interval \(\Delta\) and the total number of data \(n\) the mean \(\mu_{L}\) of the full trace length is expressed by:  
\[\mu_{L}=\frac{n\ \Delta}{N(l)} \tag{11}\]  
It has to be mentioned that the result is sensitive to the quality of the histogram and the assumed shape of the best-fit curve. In order to avoid too subjective an estimation of the mean trace length, we implemented a 6 th-order polynomial function to fit a curve to the histogram. Approach IV is based on [PERSON] [35] and accounts for various linear sampling biases, such as curtailment and edge effects. His approach does not require any information aboutthe probability density distribution and considers the measured full trace lengths as well as the termination characteristics.