A new class of probabilistic models of the width function, based on so-called iterated random pulse (IRP) processes, is proposed. IRP processes reproduce the main characteristics of empirical width functions (nonnegativity, nonstationarity, and power law decay of the spectrum) and require few and easily accessible parameters. IRP models are based on a simple conceptualization of the geometrical structure of river basins and exploit in a natural way the self-similarity of natural channel networks. A result that is derived from the IRP representation is that the exponent &agr; of Hack's law, L~A&agr;, and the exponent &bgr; of the power spectral density of the width function, S(&ohgr;)~|&ohgr;|-&bgr;, are related as &agr;=1/&bgr;. Empirical values of &bgr; are typically in the range 1.8--2.0 and are consistent with this theoretical result and the usual range of &agr;. ¿ 2000 American Geophysical Union