Are channel networks statistically self-similar?

Scaling properties of both field-mapped and threshold-delineated channel networks were studied by applying the box-counting method to three drainage basins in the western United States. This method involves (1) examination of power-law relations between the box size, epsilon , and the number of boxes, N, that intersect channel segments across a range of box sizes appropriate for the method and then (2) examining the standardized residuals for the least squares linear regressions of log N vs. log epsilon used to calculate a fractal dimension (D). For each channel network, the slope of the log N vs. log epsilon relation varies from 1 at small length scales to 2 at large length scales, a range that defines the limits to the applicability of the box-counting method. At length scales below which this slope equals 1, the plots simply record the linear aspect of streams; the length scale defining an upper limit to the application of the box-counting method corresponds to a box size large enough to intersect a channel in each box. Although a fractal dimension may be meaningfully defined only between these upper and lower length scales, neither the field-mapped nor the artificially delineated networks that we examined exhibit discrete fractal dimensions within this range. Instead, the slope of the log-log plot systematically varied with box size. The consistent lack of log-linear plots for the networks that we examined violates a fundamental requirement for fractal geometry and contrasts with general assertions about the fractal nature of river networks. A strong correlation between mean source-area size and the length scale above which the slope of plots implies D = 2 indicates that, although channel networks are not statistically self-similar, they are space filling at length scales greater than twice the mean source-basin length.