## 1. Introduction

[2] Groundwater is the main contributor of a river catchment's base flow whose predictability during recession events is of crucial importance for water resource management. Recession curves have been widely studied in the past and their characteristics used to establish basin-scale parameters (see *Tallaksen* [1995] for a review). In particular, *Brutsaert and Nieber* [1977] analyzed daily discharge values of six basins in the Finger Lakes region of the northeastern US and proposed an analytical tool to characterize the recession flow based on the description of the discharge change rate *dQ*/*dt* as a function of the discharge *Q*. Unlike many nonlinear recession flow models, this method avoids the knowledge of the precise beginning of the recession event which can be difficult to evaluate due to the continuous nature of streamflow measurements. The main feature of their method is the comparison of the observations with analytical solutions of the Boussinesq equation for an unconfined rectangular aquifer under particular boundary conditions. Two exact solutions of the Boussinesq equation [*Boussinesq*, 1904; *Polubarinova-Kochina*, 1962] and an approximated linearized solution [*Boussinesq*, 1903] can be expressed in the form:

where *β* and *k* are constants depending on the flow regime considered. In order to avoid contributions from relatively fast subsurface flow, overland flow, and evapotranspiration, *Brutsaert and Nieber* [1977] recommended the use of the lower envelope of the point cloud in the ln(*−dQ*/*dt*) versus ln *Q* plot, corresponding to the slowest recession rate. Based on their study, they identified two typical values of *β*, describing the rate of decline in streamflow recessions: ∼1.5 for low *Q* (long-term response) and ∼3 for high *Q* (short-term response). Moreover, some parameters of the watershed such as the saturated hydraulic conductivity stemming from the Boussinesq equation have been computed from the intercept of the ln(*−dQ*/*dt*) versus ln *Q* plot.

[3] The method has been widely applied to estimate basin-scale parameters in relatively natural areas [*Brutsaert and Lopez*, 1998; *Brutsaert and Sugita*, 2008; *Brutsaert and Hiyama*, 2012; *Eng and Brutsaert*, 1999; *Malvicini et al*., 2005; *Mendoza et al*., 2003; *Parlange et al*., 2001; *Szilagyi et al*., 1998; *Troch et al*., 1993; *Vogel and Kroll*, 1992; *Zecharias and Brutsaert*, 1988] and in engineered catchments [*Rupp et al*., 2004; *Wang and Cai*, 2010], to formulate base flow in a watershed model [*Szilagyi and Parlange*, 1999], in order to separate the base flow [*Szilagyi and Parlange*, 1998] or to assess long-term groundwater storage changes [*Brutsaert*, 2008]. *Rupp and Selker* [2006a, 2006b] showed some limitations of the method, e.g., in the case of sloping aquifers. Recession curve studies moving from *Brutsaert and Nieber*'s [1977] work have proved of central importance in a broad range of topics ranging from comprehensive water resource management to studies on fluvial biodiversity, catchment-scale transport, ecohydrology, and the so-called old-water paradox [*Bertuzzo et al*., 2007; *Botter et al*., 2007a, 2007b, 2009, 2010; *Ceola et al*., 2010; *Harman et al*., 2009; *Kirchner*, 2009; *Kondolf et al*., 1987; *Palmroth et al*., 2010; *Rinaldo et al*., 1995a, 1995b, 2006, 2011; *Rodriguez-Iturbe et al*., 2009; *Tague and Grant*, 2004; *Thompson and Katul*, 2012; *Wittenberg*, 1999; *Zaliapin et al*., 2010].

[4] Recently, several studies [*Biswal and Marani*, 2010; *Biswal and Nagesh Kumar*, 2012; *McMillan et al*., 2011; *Rupp et al*., 2009; *Shaw and Riha*, 2012] analyzed the recession events on a seasonal or event-based time scale and discussed their shifts in the ln(−*dQ*/*dt*) versus ln *Q* plot, linking them to different antecedent soil moisture and evapotranspiration happening over the season. In particular, *Biswal and Marani* [2010], *Biswal and Nagesh Kumar* [2012], and *Shaw and Riha* [2012] proposed to obtain the parameters *β* and *k* of equation (1) by fitting a linear model to every recession event in order to obtain a distribution of the parameters instead of fitting a line to the lower envelope of the point cloud. With this method, the authors found that the slopes of the individual recession curves, i.e., the parameter *β*, were in general larger than the one of the lower envelope, resulting in an underestimation of the streamflow decline rate when described by a unique *dQ*/*dt* − *Q* relationship. Moreover, *Biswal and Marani* [2010] proposed to link recession event parameterization to river morphology through a time-varying geometry of saturated channel sites. In particular, they developed a theory based on geomorphological arguments to link the exponent *β* of equation (1) to that characterizing an empirical relation resulting from the analysis of Digital Elevation Models (DEM). In this model, the variation of the discharge is linked to the direct drainage into a time-varying Active Drainage Network (ADN). When the recession hydrograph at an outlet is dominated by drainage of the unconfined aquifer as in *Brutsaert and Nieber* [1977], the ADN is the length of the channel network instantaneously intersecting it. The model relies on four main assumptions. First, the authors assume that the recession flow could be studied as a succession of steady flows since the time scale at which the discharge varies is much longer than the time scale of water propagating in the network. Second, by assuming a spatially constant discharge per unit length *q _{L}*, the total discharge

*Q*(

*t*) can be expressed as:

where *G*(*l*(*t*)) is the total length of the drainage network actively contributing at time *t* and *l*(*t*) is the distance between the actual source of the ADN and their location at the beginning of the recession event (Figure 1). Third, they assume that all sources of the ADN recede at the same speed *c = dl*/*dt*, constant in space and time such that the change in time of the network length is proportional to the number of sources *N* (*dG/dt = dG/dl*·*dl/dt* ∝ *N*·*c*). Equation (2) can be differentiated in time:

where the third assumption has been employed. The first term at right-hand side of equation (3) embeds the geomorphologic signature and the second the Brutsaert recession proper. Fourth, they studied the case where the variation in time of the ADN, assumed to be much larger than the variation in time of the discharge per unit length, dominates the second term in equation (3) which can thus be neglected. This work has clearly established that base flow recession curves bear the signatures of the geomorphological structure of the contributing river basin.

[5] However, the innovative method proposed by *Biswal and Marani* [2010] postulates constant drainage density, classically defined as the total length of stream channels divided by the area they drain [*Horton*, 1932, 1945] and properly described by a random space function endowed with spatial correlation [*Tucker et al*., 2001]. Other formulations have also been proposed [*Marani et al*., 2003] as it was shown that networks with the same Hortonian drainage density may embed rather different distributions of unchanneled pathways, and thus different extents of the actual density of the drainage network. Random functions are defined by the statistical properties of the length of the (steepest-descent) distance from any unchanneled site to the first occurrence of a stream channel [*Tucker et al*., 2001]. The consequences are far from obvious. In fact, the Hortonian definition applies reasonably well only in cases where locally the mean unchanneled lengths vary little from subcatchment to subcatchment, thus postulating that channel initiation processes are homogeneous—technically, whenever automatic network extractions assume it, like, e.g., in the case of constant support area (for a review see e.g. *Rodriguez-Iturbe and Rinaldo* [2001]). This is seldom the case in nature [e.g., *Montgomery and Dietrich*, 1988, 1992]. Typically, in proglacial catchments, mean unchanneled distances exhibit a broad range varying from tens of meters in shallow-soiled topographically concave source areas to a few km in deep moraines [e.g., *Montgomery and Foufoula-Georgiou*, 1993; *Tarolli and Dalla Fontana*, 2009]. Thus, one wonders whether the geomorphic framework proposed by *Biswal and Marani* [2010] for predicting the shape of Brutsaert recession curves can be suitably generalized to account for spatially uneven drainage densities. In practice, one needs to relax certain assumptions therein and check empirically whether geomorphic signatures could still be interpreted in such context, possibly improving the explanatory power of the original method and reducing to it in the limit case of constant drainage density. This is precisely what this paper addresses.