## 2 Methodology
### Change Detection and Individual Failure Isolation  
When a surface model for each epoch (e.g., epoch 1 and epoch 2) is generated, an estimated change grid can be computed simply by differencing the z' values within each cell. \[\Delta_{j}=z_{\mathrm{J}}^{\prime}j-z_{\mathrm{I}}^{\prime}j \tag{12}\]  
where \(\Delta_{\mathrm{j}}\) is the 1D change in cell \(j\) orthogonal to the best fit plane of the data. Note that the baseline surface must be created using the same transformation as the new surface for this approach to work. Following creation of a change grid, the next step is to isolate individual failure clusters (_i.e._, rockfalls). The change grid itself contains some noise and can be somewhat difficult to interpret, especially for small failures. To improve the clustering of points, an averaging filter (e.g., [50]) is applied to smooth the change grid using a roving window (e.g., focal statistics) approach:  
\[\Delta_{\mathrm{j-ave}}=\mathrm{averagefilter}(\Delta_{\mathrm{j}},\varepsilon) \tag{13}\]  
where \(\varepsilon\) is the smoothing window size, in cells. A value of \(\varepsilon=2\) (\(5\times 5\) cell window) was used in this study. This filter reduces noise present in the change grid reducing spurious values. It also helps group smaller clusters that are adjacent to one another that otherwise might go undetected, especially when they are close to the threshold value. If a significant amount of geo-referencing errors relative to the change of interest are suspected, the average difference between all cells of the surface models can be subtracted from each cell to remove those potential biases. However, this adjustment should be used with caution because it can remove any indication of overall change at the site. Also, to account for potential geo-referencing errors, a significant change grid [51,52] is created where change below a threshold (\(\gamma\)) is ignored. \[\delta_{j}= \begin{bmatrix}-1&\Delta_{j-ave}\leq-\gamma&\text{\emph{Erosion}}\\ 0&-\gamma<\Delta_{j-ave}<\gamma&\text{\emph{No.Significant.Change}}\\ 1&\Delta_{j-ave}>\gamma&\text{\emph{Accretion}}\end{bmatrix}  \tag{14}\]  
A recommended value for \(\gamma\) is the estimated RMS geo-referencing error. For this study, we used \(\gamma=0.05\) m, which slightly overestimates the amount of error present in the scans. A smaller threshold, which would enable detection of smaller rockfall events, could be used if more labor-intensive geo-referencing techniques were implemented for small sites; however, given that the overarching study focuses on long segments of highway, such techniques are not practical and often result in propagated error of a similar or greater magnitude over such long distances [27]. A connected components algorithm [53,54] was then utilized on the significant change grid (\(\delta_{\mathrm{j}}\)) in order to isolate individual failure clusters and assign each cell to a cluster. For the erosional events, values of \(\delta_{\mathrm{j}}=0\) and \(1\) were considered as background. For accretion events, a second run was completed with values of \(\delta_{\mathrm{j}}=0\) and \(-1\) considered as background. The connected components' algorithm implements a two-pass method. The first pass iterates through the grid from left to right and top to bottom examining the four neighboring grid cells found to the left and above the current grid cell. A series of conditions are used to generate a preliminary cluster ID for each cell contingent on the values of the current and neighboring cells. Following the assignment of an ID for a given grid cell, the occurrence of any adjacent cells with differing IDs arerecorded in a lookup table for later use (second pass). The first pass results in all non-background cells receiving a label; however, contiguous regions might still contain multiple cluster IDs. After utilizing a convention that favors the minimum ID for a region, a second pass is conducted in which the lookup table is used to consolidate cluster IDs.