Statistical analysis of drainage density from digital terrain data

Drainage density (D_d), defined as the total length of channels per unit area, is a fundamental property of natural terrain that reflects local climate, relief, geology, and other factors. Accurate measurement of D_d is important for numerous geomorphic and hydrologic applications, yet it is a surprisingly difficult quantity to measure, particularly over large areas. Here, we develop a consistent and efficient method for generating maps of D_d using digital terrain data. The method relies on (i) measuring hillslope flow path distance at every unchanneled site within a basin, and (ii) analyzing this field as a random space function. As a consequence, we measure not only its mean (which is half the inverse of the traditional definition of drainage density) but also its variance, higher moments, and spatial correlation structure. This yields a theoretically sound tool for estimating spatial variability of drainage density. Averaging length-to-channel over an appropriate spatial scale also makes it possible to derive continuous maps of D_d and its spatial variations. We show that the autocorrelation length scale provides a natural and objective choice for spatial averaging. This mapping technique is applied to a region of highly variable D_d in the northern Apennines, Italy. We show that the method is capable of revealing large-scale patterns of variation in D_d that are correlated with lithology and relief. The method provides a new and more general way to quantitatively define and measure D_d, to test geomorphic models, and to incorporate D_d variations into regional-scale hydrologic models.