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∝Lvl, w∝Lvw, and h∝Lvh 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 vl, vw, and vh are the self-affine scaling exponents. We determined vl, vw, and vh using Lp∝A&bgr;, Lp∝Aϵ, and S∝A-&thgr;, where Lp is the length of the projection of the longest river channel on the horizontal plane, Lp 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 vh≈vl-&thgr;(vl+vw) is derived which connects the main scaling exponents. For two New Zealand rivers, we found vl=0.60 and vw=0.40. On the basis of simple considerations, we estimated a range of possible values of vh 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 vh. 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. ¿ American Geophysical Union 1996