On Channel Network Fractal Properties: A Case of Study of the Hutt River Basin, New Zealand

The paper considers river networks as three‐dimensional self‐affine fractal objects. The Hutt River basin (New Zealand∥ was selected for detailed analysis on the basis of a digital elevation model (DEM). To characterize network properties quantitatively we used three scaling exponents in the relationships l ∝ ℒ υl , w ℒ υw , and h ∝ ℒ υh where l , w, h are some characteristic longitudinal, transversal, and vertical scales of a channel network; ℒ is the total length of channel network in three‐dimensional space; and υ l , υ w , and υ h are the self‐affine scaling exponents. We determined υ l , υ w , and υ h using L p ∝ A β , ℒ p ∝ A ϵ , and S ∝ A −θ , where L p is the length of the projection of the longest river channel on the horizontal plane, ℒ p is the total length of channel network projection on the horizontal plane, A is the catchment area, and S is the local slope. An approximate relationship υ h ≈ υ l − θ(υ l + υ w ) is derived which connects the main scaling exponents. For two New Zealand rivers, we found υ l =0.60 and υ w =0.40. On the basis of simple considerations, we estimated a range of possible values of υ h from 0.1 to 0.5 with 0.2 for the case study. The slope‐area‐elevation relation introduced by Willgoose [1994] was applied to interpret data concerning υ h . The influence of threshold area (TA) values on the scaling properties of channel networks is shown to be small, and double scaling relationships are suggested for connecting the physical scaling of channel networks with scaling caused by threshold effect.