## 1. Introduction

[2] The spatial heterogeneity and process complexity of subsurface flow imply that any feasible hydrological model will necessarily involve substantial simplifications and generalizations. The essential question for hydrologists is which simplifications and generalizations are the right ones. Physically based rainfall-runoff models (see *Beven* [2001] for an overview) attempt to link catchment behavior with measurable properties of the landscape, but many properties controlling subsurface flow are only measurable at scales that are many orders of magnitude smaller than the catchment itself. Thus, although it seems obvious that catchment models should be “physically based,” it seems less obvious how those models should be based on physics. Many hydrologic models are based on an implicit premise that the microphysics in the subsurface will “scale up” such that the behavior at larger scales will be described by the same governing equations (e.g., Darcy's law, Richards' equation), with “effective” parameters that somehow subsume the heterogeneity of the subsurface [*Beven*, 1989]. It is currently unclear whether this upscaling premise is correct, or whether the effective large-scale governing equations for these heterogeneous systems are different in form, not just different in the parameters, from the equations that describe the small-scale physics [*Kirchner*, 2006].

[3] This observation raises the question of how we can identify the right constitutive equations to describe the macroscopic behavior of these complex heterogeneous systems. For decades, hydrologists have used characteristic curves to describe the macroscopic behavior of blocks of soil, recognizing that these empirical functions integrate across the complex and heterogeneous processes that govern water movement at the pore scale. Likewise, one can pose the question of whether there are “characteristic curves” at the scale of small catchments, that can usefully integrate over the complexity and heterogeneity of the landscape at all scales below, say, a few square kilometers. And if such “characteristic curves” are meaningful and useful at the scale of small catchments, can they also be measured at that scale?

[4] Since at least the time of *Horton* [1936, 1937, 1941], a major theme in catchment hydrology has been the interpretation of streamflow variations in terms of the drainage behavior of hillslope or channel storage elements [e.g., *Nash*, 1957; *Laurenson*, 1964; *Lambert*, 1969, 1972; *Mein et al.*, 1974; *Brutsaert and Nieber*, 1977; *Rodriguez-Iturbe and Valdes*, 1979; *van der Tak and Bras*, 1990; *Rinaldo et al.*, 1991], whose parameter values are typically calibrated to the observed hydrograph (see *Beven* [2001] and *Brutsaert* [2005] for an overview). In some cases, these parameters can be interpreted as reflecting basin-scale hydraulic properties [e.g., *Brutsaert and Nieber*, 1977; *Brutsaert and Lopez*, 1998], and in others they can be correlated with catchment geomorphic characteristics [e.g., *Nash*, 1959], facilitating hydrologic prediction in ungauged catchments. However, the form of the constitutive relationship (the storage-discharge function) must normally be known in advance.

[5] Here I show that, if the catchment can be represented by a single storage element in which discharge is a function of storage alone, the form of this storage-discharge function can be estimated from analysis of streamflow fluctuations. In contrast to conventional methods of recession analysis (see reviews by *Hall* [1968], *Tallaksen* [1995], and *Smakhtin* [2001], and references therein), this approach does not specify the functional form of the storage-discharge relationship a priori, instead determining it directly from data. (For further comparisons between previous work and the present approach, see section 15.1 below.) Using this approach, one can construct a first-order nonlinear differential equation linking precipitation, evapotranspiration, and discharge, with no need to account explicitly for changes in storage; these are instead inferred from the resulting changes in discharge. This single equation allows one to predict streamflow hydrographs from precipitation and evapotranspiration time series. It can also be inverted, allowing one to use streamflow fluctuations to infer precipitation and evapotranspiration rates at whole-catchment scale.