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, wℒ^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.