## 4 Results
### Site S042  
Orientation data from virtual discontinuity mapping was provided in [43]. Orientation data was plotted in a stereographic projection as shown in Figure 8. The density contours were used to manually define five joint sets (J1 through J5 in Figure 8). The projection is an Equal Area projection consisting of 207 discontinuity entries. Discontinuity J1 was the most well defined (clear clustering--see Figure 8 in red contours) followed by discontinuities J2 and J3. Discontinuities J4 and J5 were not as well defined, probably due to the low frequency of their traces. Histograms of joint set orientation (dip and dip direction) are shown in Figure 9 for joint sets J1 through J3. These histograms correspond to the probability densities for each bin (bins are 2\({}^{\circ}\) wide). As opposed to the Tornado Mountain case study, PDFs for the orientations have been defined based on the statistical descriptors of the data (mean and standard deviation), truncated at the observed minimum and maximum measurements, and assuming the normality observed for large datasets of discontinuity orientations in [41]. The dip measurements of J2 were fitted to a lognormal distribution to better represent the shape of the data. The distributions were truncated at the maximum and minimum values to avoid unrealistic values at the tails of the distributions. The distributions adopted, as well as their mean, standard deviation, and minimum and maximum values, are presented in Figure 9. The probability values from the Pearson's \(\chi^{2}\) tests for goodness of fit are shown in Table 2. These probabilities are all significantly higher than the critical value adopted (0.05); therefore, it is highly probable that the distributions are an adequate representation of the true orientation distributions. Figure 7: Fragmented rockfall volume cumulative relative distribution as calculated by the persistent DFN method and field surveys, for all DFN calculated volumes (**a**) and for DFN volumes truncated in accordance with the range of surveyed volumes (**b**). Tornado Mountain. The histograms for joint sets J4 and J5 are shown in Figure 10. The number of measurements obtained for J4 and J5 were 6 and 4, respectively. These two joint sets were not considered for the method any further due to their low frequency. Range of spacing values for joint set J1 through J3 are reported in [43] and shown in Table 3. Unfortunately, detailed measurements are not reported and only the minimum, maximum and mean values were available. This study adopted triangular distributions for the spacing between the minimum and maximum values reported and with modes equal to the mean values reported. This last decision was made to try and capture any skewness in the data reflected by the mean values. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Joint Set** & **Dip Dist. Fit** & **Dip Direction Dist. Fit** \\ \hline
1 & 0.78 & 0.27 \\
2 & 0.13 & 0.55 \\
3 & 0.21 & 0.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Pearson’s \(\chi^{2}\) probability values for the goodness of fit tests for measured orientation PDF adopted. Values over 0.05 are considered an adequate fit. Figure 8: Virtually mapped discontinuity density contours in stereographic projection for Site S042 and manually identified joint sets. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Joint Set** & **Min** & **Mean** & **Max** & **Distribution Adopted** \\ \hline
1 & 0.1 & 0.5 & 2.4 & Triangular, mode = 0.5 \\
2 & 0.1 & 0.3 & 0.4 & Triangular, mode = 0.3 \\
3 & 0.1 & 1.0 & 2.8 & Triangular, mode = 1.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Range of spacing values for joint sets J1 through J3. One iteration of the 10 m \(\times\) 10 m \(\times\) 10 m persistent DFN built based on the distributions in Tables 2 and 3 is shown in Figure 11a. The blocks defined by this DFN are shown in Figure 11b, randomly colorized. The realizations resulted in 6077 blocks.