# Semiautomated Statistical Discontinuity Analyses from Scanline Data of Fractured Rock Masses
## 4 Application Example
### Trace Length and Termination  
In Step V, the mean trace length of discontinuities for all data or each discontinuity set can be determined by four different approaches as per Section 2.2.4. These approaches are implemented in the m-file _trace_length_calculation.m_, providing an outcome of four mean trace length values. Before executing the m-file _trace_length_calculation.m_, the source data file has to be specified (e.g., for set 1, _trace_data_set1.dat_), containing six columns comprising the discontinuity number, scanline number, trace length above, trace length below, termination above, and termination below. In addition, a curtailment value \(c\) should be defined as proposed by approach II, and a histogram range and class interval should be defined for approach III, respectively.  
After executing the m-file, the four different mean trace length values based on approaches I to IV are determined and saved in a mat-file (e.g., for set 1, _trace_termination_set1.mat_). In addition, a termination index is calculated from semi-traces above the scanline (_variable termination_above_), from semi-traces below the scanline (_termination_below_), and from all measured semi-traces (_termination_all_) for each set. The representative termination index value is usually gained from all measured semi-traces; however, the values above and below are helpful to validate the impact of concealed discontinuities. Given that in most cases, subhorizontal scanlines are measured, it can be assumed that below the scanline, a larger number of discontinuities are concealed, resulting in an underestimation of the true termination index.  
Table 1 shows the results for the mean full trace length and termination analyses obtained by the m-file _trace_length_calculation.m_. The analyses indicate a remarkably large variation of the mean full trace length, depending on the applied approach. Approach I gave the smallest and approach III the highest values. Results of approaches II and IV are in between, with approach IV being smaller by 15% to 24%.  
In Step VI, the results of the trace length analyses are visualized as a histogram showing the semi-traces versus frequency. Functionally, the m-file _trace_length_histogram.m_ is similar to the m-file for plotting the spacing histogram. Figure 6 shows the histograms of all semi-traces (i.e., above and below of the scanlines) for sets 1, 2 and 3 based on the data of the case study. It is worth mentioning that the negative exponential function based on the mean trace length of Approach I maps the distribution of lengths quite well.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Applied Approach for Mean Trace Length [m] and** & **Set 1** & **Set 2** & **Set 3** \\
**Termination Index Calculation [\%].** & & & \\ \hline Approach I & 0.515 & 0.710 & 0.587 \\ Approach II & 0.770 & 1.160 & 0.969 \\ Approach III & 1.040 & 1.580 & 1.163 \\ Approach IV & 0.657 & 0.955 & 0.737 \\ Termination—above the scanline & 31.5 & 18.5 & 21.9 \\ Termination—below the scanline & 27.8 & 12.3 & 21.9 \\ Termination—all semi-traces & 29.7 & 15.4 & 21.9 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Results of trace length and termination index analyses for sets 1, 2, and 3.  
Figure 6: Frequency histogram versus semi-trace lengths with fitted negative exponential probability density distribution (blue line): (**a**) set 1 \(\mu_{L}\) = 0.515 m, (**b**) set 2 \(\mu_{L}\) = 0.710 m, and (**c**) set 3 \(\mu_{L}\) = 0.587 m.