# Evaluating the Drainage Density Characteristics on Climate and Drainage Area Using LiDAR Data
## 2 Data and Methodology
### Valley Network Extraction  
In this study, the valley network is extracted using a curvature-based method that was proposed and explained in the previous study conducted by [PERSON] et al. (2015, 2016) [17; 21]. Curvature-based methods for valley or channel network delineation usually require filtering the noise and insignificant features to obtain a robust curvature grid. The Perona-Malik filter used in this study is a nonlinear diffusive filter that efficiently smooths DEMs while preserving significant features such as valleys [19]. This filter has one parameter called \"time of forward diffusion,\" which indicates the number of iterations for filtering the DEM as a numerical representation of derivatives for smoothing [21]. The time of forward diffusion denoted by \(T_{F}\) is set to 50 in the study. From the filtered DEM, the curvature is calculated using Equation (1) [54]:  
\[\kappa=\frac{z_{xx}z^{y}-2z_{xy}z_{x}z_{y}+z_{yy}z_{x}{}^{2}}{\left(z_{x}{}^{2} +z_{y}{}^{2}\right)\sqrt{1+z_{x}{}^{2}+z_{y}{}^{2}}} \tag{1}\]  
where \(\kappa\) is the curvature, and \(z_{x}\) and \(z_{xx}\) (\(z_{y}\) and \(z_{yy}\)) denote the first and second derivatives of elevation (\(z\)) with respect to \(x\) (\(y\)). \(z_{xy}\) is the first derivative of \(z_{x}\) with respect to \(y\). Figure 3 shows the curvature extracted from the original and filtered DEMs overlaid with contour lines in the Isleta Drain watershed located in New Mexico. Figure 3: One-meter contour lines and the curvature grid for (**a**) the original DEM and (**b**) the smoothed DEM after applying Perona-Malik filter with the time of forward diffusion set to 50 (\(T_{F}=50\)) on the Isleta Drain watershed located in New Mexico. Figure 2: (**a**) The relationship (\(PE=176E_{P}/P^{-1.22}\)) between \(PE\) index and climate aridity index (\(E_{P}/P\)) and (**b**) drainage density (\(D_{d}\)) versus \(E_{P}/P\) from Melton [32] and Madduma Bandara [38]. Valley or drainage network extraction is based on the curvature analysis using positive and negative curvature as indicators of the significance of convergence or divergence. Valleys are defined as convergent surfaces which are associated with positive curvature. Ridges are the segments with negative curvature (i.e., divergent surface), which are typically located between the valleys as the signature of flow separation lines between tributaries. In order to differentiate valleys, a positive curvature threshold (\(\kappa_{\
u}\)) is automatically derived which is used to cluster the landscape into convergent (\(\kappa>\kappa_{\
u}\)), divergent (\(\kappa<-\kappa_{\
u}\)), and insignificant (\(|\kappa|\leq\kappa_{\
u}\)) pixels. \(\kappa_{\
u}\) is calculated through connected component analysis on the curvature grid. Figure 4 shows the number of connected components (\(N_{cc}\)) for any given curvature threshold, denoted by \(\kappa_{T}\) in the Isleta Drain watershed. A connected component is a set of connected pixels which are all either convergent or divergent (Figure 4a). In order to compute \(N_{cc}\) for each value of \(\kappa_{T}\), the territory is clustered into convergent (curvature greater than \(\kappa_{T}\)) and divergent (curvature less than \(-\kappa_{T}\)) pixels, and then the connected component is counted using a binary labeling algorithm [55]. As shown in Figure 4b, decreasing \(\kappa_{T}\) initially leads to more connected components since there are more pixels labeled as convergent or divergent. However, at a certain point, the number of connected components drops due to the merging process. In other words, the existing components start to merge together resulting in less \(N_{cc}\). The peak of the connected component curve is considered as the curvature threshold for valley extraction since it produces the most separated clusters in the landscape and efficiently identifies the local optimums in the curvature grid. Given the obtained \(\kappa_{\
u}\), the initial valley skeleton is generated by imposing \(\kappa>\kappa_{\
u}\) to the curvature grid.