## 2 Methodology
### Hole Filling  
Data gaps (holes) are inevitable in any laser scan data because of obstructions that block the line of sight to the objects of interest. In particular, for this study, scanning needed to be completed relatively close (approximately 20-40 m) from the cliff face so that the equipment could be set up adjacent to the highway for accessibility. This close proximity to the cliff results in several portions of each scan being oblique to the surface, which results in more holes from topographic obstructions compared to scanning from vantage points with a better view further back from the cliff surface [27]. (It is noted that this problem can be somewhat mitigated by spacing scans closer; however, the increased number of setups reduces the amount of coverage possible in a given time window). Passing vehicles also resulted in several holes on the cliff surface in each scan. Holes in the model were filled by generating a thin plate spline (TPS) from the cell centroid points in cells surrounding the holes (Figure 5). Because the TPS technique is computationally inefficient across large point cloud datasets, a TPS is computed for a small window of the dataset at a time. However, note that large holes require a larger window size to be filled. To avoid poor fits in areas with sparse data, 50% of the cells in the window were required to have data in order for holes within that window to be filled. Also, to minimize artifacts and edge effects, holes are only filled in the center cells of the window. For purposes of this study, the window size is defined as the number of pixels included in each direction outward from the center pixel that are included for the hole filling. For example, with a window size of 5 (11 \(\times\)11 cells), all cells with data in the 11 \(\times\)11 cells were used to compute the TPS, but cells with holes were only filled in the center 6 \(\times\)6 cells (Figure 5). Note that the interpolated points are not used in the next overlapping window to compute the TPS. Figure 5: Schematic illustrating the hole filling process using thin plate spline (TPS) interpolation. The computation of TPS was adopted from [48, 49]. The TPS can be calculated from the centroid points for each cell within the search window through a linear equation system based on minimizing bending energy in the TPS. A regularization parameter (\(\beta\geq 0\)) can control how tightly the TPS fits the input points. The solution of this linear equation system results in local influence weights (\(w\)) and coefficients (\(a\)) that can be used in the following interpolation equation to determine the z' coordinate for the cells without data:  
\[z^{\prime}(x^{\prime},y^{\prime})=a_{1}+a_{2}x^{\prime}+a_{3}y^{\prime}+\sum \limits_{i=1}^{M}w_{i}U\left(\left|P^{*}xi\right.\right.\left.P^{*}y_{i} \right|-\left|x^{\prime}\right.\left.y^{\prime}\right|\right) \tag{10}\]  
where x' and y' are the coordinates at the center of the cell without data in the transformed coordinate system. U is a weighting function based on the distance between the cell without data and each of the points in the subdatset that are used to compute the TPS. \[U(\mathrm{r})=\left\{ \begin{array}{cc}r^{2}\log(r)&r>0\\ 0&r=0\end{array} \right. \tag{11}\]  
Figure 6 shows an example of hole filling in a surface model generated at Site A using the approach described above. In this example, both large and small holes can be filled across even larger data gaps by increasing the window size for visualization purposes; however, as with any technique, interpolation errors across these extents could generate significant problems for volumetric estimates, particularly on gradually eroding slopes. Hence, some judgment is necessary for implementation. Figure 6: Example of hole filling for the 2014 surface model for Site A using a grid cell size of 0.05 m and window size of 20 (_i.e._, TPS search window of 41 \(\times\) 41 cells (2 \(\times\) 2 m) but hole fill size of 21 \(\times\) 21 cells (1 \(\times\) 1 m)).