## 1. Introduction

[2] The dependencies between channel properties and river flows have been observed for a long time, and empirically described by the notion of hydraulic geometry (HG). Hydraulic geometry was first introduced in the pioneering work of *Leopold and Maddock* [1953] and refers to the power laws relating the channel width *W*, mean depth *D*, and mean velocity *V* to discharge *Q*: *W* = *aQ*^{b}, *D* = *cQ*^{f}, *V* = *kQ*^{m}. (Hereafter the width *W*, mean depth *D*, mean velocity *V*, cross-sectional area *C*_{A} and discharge *Q* are referred to as HG factors). These relationships have been observed to hold either for different discharges at an individual cross section (hereafter called at-station HG), or for different downstream locations related through some characteristic discharge of constant frequency of exceedance (hereafter denoted as downstream HG). Figure 1 (reproduced from *Leopold and Maddock* [1953]) illustrates the idea for one HG factor, the velocity.

[3] For at-station HG, the single power law relationships are widely used although some deviations from a single power law have been reported in the literature, either as a change in the exponent in the log-log plot of velocity and discharge with increasing discharge, or in general as loss of log-log linearity when discharge increases [e.g., *Richards*, 1976; *Wong and Laurenson*, 1984; *Bates*, 1990; *Pilgrim*, 1976]. In contrast, the log-log linearity in downstream HG has been supported by many empirical [*Carlston*, 1969; *Park*, 1977] and theoretical [*Parker*, 1979; *Huang et al.*, 2002] studies. Many empirical models [e.g., see *Rhoads*, 1991] consider the proportionality coefficient and the exponents of hydraulic geometry as functions of some specific discharge and the grain size of bed and/or bank material. Since both grain size and discharge generally depend on the contributing area (scale), it is clear that the magnitude of the power law exponents will also depend on the contributing area and frequency of occurrence of a specific discharge.

[4] Analytical derivation of scale-frequency-dependent HG based on first principles is not known to the best of authors knowledge, and this is probably because the at-station and downstream hydraulic geometries represent two different (although mutually dependent) processes, which “live” in different timescales. On the other hand, approaching HG from a statistical point of view seems fruitful, especially given that a rigorous mathematical-statistical framework exists, that of “multiscaling scale invariance”, within which processes whose spatial variability changes with scale and frequency can be concisely described. In essence, multiscaling scale invariance implies that the probability distribution function of a random field indexed by scale, can be appropriately rescaled via a random or nonrandom function which depends on scale only. The multiscaling framework has been successfully employed to provide a theoretical basis for the empirically observed scale-dependent behavior of flood peaks in the context of regional quantile analysis [e.g., *Gupta and Waymire*, 1990; *Gupta et al.*, 1994; *Gupta and Dawdy*, 1995]. Other applications of this concept have appeared in atmospheric turbulence, rain and clouds (e.g., see *Schertzer and Lovejoy* [1987] for an early reference), in river networks [e.g., *Gupta and Waymire*, 1989], and in solute transport [*Sposito and Jury*, 1988]. In this paper, it is shown that the multiscaling framework can provide a theoretical basis for interpreting and modeling the scale and frequency dependent relations between channel morphometry and discharge (known as HG).

[5] This paper is structured as follows. In section 2, empirical evidence is provided (based on 85 stations in Oklahoma and Kansas for basins ranging from 10 to above 10000 km^{2}) that at-station HG depends systematically on scale and downstream HG depends on the frequency of the characteristic discharge. In section 3, a multiscaling model is proposed for channel cross-sectional area and discharge and is used to derive generalized at-station and downstream HG for cross-sectional area and velocity. The parameters of the generalized at-station HG are analytical functions of the multiscaling model parameters and the contributing area (scale). Analytical derivation of the parameters of the downstream HG is not possible and these have been computed numerically. The theoretically derived HG (both at-site and downstream) is compared to the empirical HG with good agreement, supporting thus the proposed generalized model. In section 4, the hydrologic response of a hypothetical catchment has been computed based on geomorphologic nonlinear reservoirs in network model and assuming classical versus generalized HG. This comparison is revealing and highlights important implications of the scale-dependent (and thus spatially heterogeneous) HG on the nonlinearity of hydrologic response. Finally, based on the observation that a single power law relationship of velocity versus discharge may not hold for a wide range of discharges (it often breaks for discharges close and above bank-full), and also that there is a considerable spread in the log-log linear relationships of HG (pointing to the fact that they have to be seen as stochastic and not deterministic relationships), an extension of the lognormal multiscaling framework to a bivariate mixed lognormal multiscaling framework is proposed in section 5. A preliminary analysis shows indeed that this extended framework has the potential to capture the scale frequency dependence of HG for composite log-log linear relationships.