## 1. Introduction

[2] Downstream hydraulic geometry [after *Leopold and Maddock*, 1953] has become a standard tool to describe the variation with flow of stream channel geometry through the channel network. The equations are empirical, but significant attempts have been made to develop a theoretical basis for them [e.g., *Langbein and Leopold*, 1966; *Kirkby*, 1977; *Parker*, 1978, 1979; *Chang*, 1980; *Yang et al.*, 1981; *White et al.*, 1982; *Davies*, 1987; *Cao*, 1996; *Huang and Nanson*, 2000]. By their very construction they are incomplete statements with no strictly physical interpretation, but they can properly be interpreted as scaling relations between the “effective channel-forming flow” and the accommodation of that flow by the covariation of stream width, depth and velocity. The usual power law representation of them is, to this extent, appropriate.

[3] The hydraulic geometry arose by analogy with “regime relations” established early in the 20th century for the design of unlined irrigation canals. The earliest relations were deliberately empirical [*Kennedy*, 1895; *Lindley*, 1919], but they were succeeded by attempts to clothe the relations in the semblance of a theory (hence “regime theory” [see, e.g., *Lacey*, 1930; *Blench*, 1969]). These investigators recognized that a critical control over river channel form is exercised by the materials that form the bed and banks of the channel, but they were able to quantify the effect only by an index “silt factor” [*Lacey*, 1930; *Blench*, 1969] or by stratifying regime relations according to the character of bed and bank materials [*Lane*, 1957; *Simons and Albertson*, 1963]. This factor has mostly been ignored in considering the hydraulic geometry of rivers, although thoughtful investigators have introduced the related factor of vegetation-mediated bank strength [*Andrews*, 1984; *Hey and Thorne*, 1986; *Huang and Nanson*, 1998], again by index methods, while others have restricted investigations to a particular type of bounding sediment [*Bray*, 1973; *Neill*, 1973; *Charlton et al.*, 1978; *Andrews*, 1984; *Hey and Thorne*, 1986].

[4] The purpose of this paper is to explore the hydraulic geometry of selected river systems in terms of a rational regime theory, both as a test of the theory and in order to attempt to learn more about the covariation of terrain and hydrological conditions through a drainage system that leads to the familiar empirical equations for the variation of width, depth and velocity.