## 2 Materials and Methods
### Ground-Based Photogrammetry and Structural Mapping  
Ground-based photogrammetry at the study areas were performed with digital single-lens reflex (DSLR) cameras with sensor size of 35.9 mm by 24 mm and 36.3 million pixels. The lens used at the Tornado Mountain site had a 200 mm fixed focal length and the lens used for Site S042 had a 150 mm fixed focal length. The distance between the camera stations and the target slope area varied between approximately 500 m and 600 m at the Tornado Mountain site, and between approximately 200 m and 600 m at Site S042. This corresponds to ground pixel sizes between 0.7 cm and 2 cm.  
Photographs were processed in ADAM Technology's photogrammetry software suite [38]. This software package builds a digital surface model based on overlapping sets of photos using photogrammetric principles, with tools to (automatically or manually) identify exposed discontinuity faces and map their orientations. In this study, orientations were mapped manually for all observable discontinuity faces. The software also has a tool that allows for measuring discontinuity spacing.  
Discontinuity orientations obtained from the virtual structural mapping were input into Roccience's Dips software package [39]. Dips provides stereographic projection plots of discontinuity orientations and allows for defining discontinuity sets with manual intervention. The software further provides histograms and exports the mapped orientations corresponding to each discontinuity set defined.  
The discontinuity orientations for each set were imported into Wolfram's Mathematica software package [40]. This software is capable of working with large databases, conducts complex explicit  
Figure 1: Method for calculating fragmented rockfall volume distribution from remote sensing, virtual structural mapping and discrete fracture networks.  
mathematical and numerical calculations, and provides advanced visualization of data and results. Histograms with the probability densities of set orientations were fitted to probability density functions (PDFs). The number of observations required for an 80% confidence is estimated at 41 according to [41], and up to 271 for a 90% confidence. The number of observations achieved per set in this study varied between 18 and 75. This corresponded to limitations in aerial extents of the outcrop of interest (Tornado Mountain rockfall source) or limitations about observable discontinuities. This limitation necessitated the use of engineering judgment when defining the PDF for each set of discontinuities. PDFs were truncated at the minimum and maximum values measured to avoid unrealistic values at the distribution tails. Normal distributions were assumed for the PDFs following observations in [41], with the exception of one set which fit a LogNormal distribution. These PDFs were adopted as the mathematical representation of discontinuity orientation variability for each discontinuity set.  
Pearson's \(\chi^{2}\) tests were adopted to evaluate the fit of these distributions to the orientation data. The statistic value (\(\chi^{2}\)) in this test is obtained through Equation (1), where \(n\) is the number of bins in the histogram, \(X_{i}\) is the observed data, and \(E_{i}\) is the expected value based on the theoretical distribution.  
\[\chi^{2}=\sum_{i=1}^{n}\frac{(X_{i}-E_{i})^{2}}{E_{i}} \tag{1}\]  
The null hypothesis is that the data are a subset obtained from the theoretical distribution. The value of \(\chi^{2}\) obtained through Equation (1) is mapped to the \(\chi^{2}\) Distribution to obtain the probability that this or a larger difference will be observed between the data and the distribution. If the probability is higher than the criteria adopted (commonly 0.05), the hypothesis is not rejected [42].