## 3 Considering the ALS Workflow and Its Impact on the Efficacy of Data Products
### Data Processing
#### 3.2.5 Surface Model Generation
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Method**} & **Exact/Smooth** & **Weighting** & **Allows for** \\  & **Interpolator** & **Method** & **Extrapolation** \\ \hline Triangulation Linear Interpolation & Exact & None & No \\ Nearest Neighbor & Assigned & None & No \\ Natural Neighbor & Smooth & Area & No \\ Moving Average & Smooth & None & Yes \\ Local Polynomial & Smooth & Least Squares & Yes \\ Minimum Curvature & Smooth & None & Yes \\ Inverse Distance to a Power & Smooth/Exact & Distance & Yes \\ Kriging & Smooth/Exact & Variogram [58] & Yes \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Characteristics of main surface interpolation methods. Figure 20: Creation of surface elevation model via interpolation. irregular network (TIN) to a regular spaced raster. This is accomplished by using bilinear interpolation to determine grid node elevations from the triangular facets. The result is a surface model that looks just like a TIN but with a higher number of data points supporting it. The minimum curvature method produces the smoothest possible surface with the minimum amount of curvature while trying to be as close as possible to the observed data (not an exact interpolator). The inverse distance weighting (IDW) algorithm will consider all data points around a grid node to obtain its interpolated value, but will weigh the data according to the distance of the data point to the grid node. The closer the data point the higher the weight and thus the higher the contribution to the interpolated grid node value. IDW can be either an exact or smoothing interpolator depending on how the weighting function is set. In the case of Figure 21 the DEM generated with IDW was set with a power of two which makes the IDW a smoother interpolator as is evident in the figure. Kriging [58] is perhaps the most powerful interpolator with properties of being an exact interpolator that preserves the observation values if they correspond to the grid nodes and a smooth and optimal predictor that recognizes trends suggested by sparse data. For instance kriging will consider a set of high points to be a ridge rather than a set of isolated peaks. The difference in the results is more evident when the density of the points used is sparse, as it will be discussed in Section 4.3 where the ground points used to produce the DEMs were systematically decimated to assess the erosion of features with a lower number of ground returns. Figure 21: Shaded relief map of DEMs generated with different interpolation methods **(a)** Kriging; **(b)** Inverse distance weighting to a power (IDW); **(c)** Triangulation with linear interpolation; **(d)** Minimum Curvature. Methods to access the absolute accuracy of an elevation raster and other ALS datasets are covered in Section 4.1.2. It will be explained that the methods require an independent elevation dataset or reference dataset acquired with higher accuracy. However, at this point it is useful to point out that there are methods that can be used to assess the internal accuracy of an interpolation method used for the production of the elevation raster. These methods are based on cross validation [59, 60] which consists of selecting a random sample of the data points to be interpolated and one at a time a single sample is removed from the dataset and then surface interpolation is performed without this data point. The difference in elevation between the data point and the interpolated model is then computed. This process is repeated for all the random sample points and the root mean square error (RMSE) of elevation is computed. The lower the RMSE the higher the internal accuracy or cross validation of the elevation raster. In Figure 21 the cross validation RMSE for each interpolation method considering 1000 random points is provided.