The most common approach to extract the channel network from digital elevation models is to specify a threshold area, S, which is the minimum area required to drain to a point within which a channel forms. This threshold area is usually specified arbitrarily, although it is recognized that different threshold areas will result in substantially different channel networks for the same basin. This paper studies the effects of S, which is also the scale of observation, on scaling and morphometric properties such as the number of sources and the areas of source basins and lateral basins. The Gardon basin, located in southern France, was extensively studied. The results indicate that morphometric properties vary considerably with S and that the spatial distribution of source basins can be considered as a fractal Sierpinski space. A simple model based on the procedure similar to the Sierpinski carpet construction is proposed to explain empirical relations between the number of source areas and the threshold, S. The model presents a new approach to estimate basin fractal properties and defines new indices to characterize the spatial distribution of first‐order basins. This methodology is useful for hydrologists and geomorphologists dealing with river networks and spatial patterns of various basin properties such as vegetation, soil, soil moisture, and human activities.