Geomorphic signatures on Brutsaert base flow recession analysis


  • Raphaël Mutzner,

    Corresponding author
    1. School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
    • Corresponding author: R. Mutzner, School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, Station 2, CH-1015, Lausanne, Switzerland. (

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  • Enrico Bertuzzo,

    1. School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
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  • Paolo Tarolli,

    1. Department of Land, Environment, Agriculture and Forestry, University of Padua, Padua, Italy
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  • Steven V. Weijs,

    1. School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
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  • Ludovico Nicotina,

    1. School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
    2. Risk Management Solutions Ltd., London, UK
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  • Serena Ceola,

    1. School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
    2. Department DICAM, University of Bologna, Bologna, Italy
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  • Nevena Tomasic,

    1. School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
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  • Ignacio Rodriguez-Iturbe,

    1. Department of Civil and Environmental Engineering, Princeton University, Princeton, New Jersey, USA
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  • Marc B. Parlange,

    1. School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
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  • Andrea Rinaldo

    1. School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
    2. Dipartimento ICEA, University of Padua, Padua, Italy
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[1] This paper addresses the signatures of catchment geomorphology on base flow recession curves. Its relevance relates to the implied predictability of base flow features, which are central to catchment-scale transport processes and to ecohydrological function. Moving from the classical recession curve analysis method, originally applied in the Finger Lakes Region of New York, a large set of recession curves has been analyzed from Swiss streamflow data. For these catchments, digital elevation models have been precisely analyzed and a method aimed at the geomorphic origins of recession curves has been applied to the Swiss data set. The method links river network morphology, epitomized by time-varying distribution of contributing channel sites, with the classic parameterization of recession events. This is done by assimilating two scaling exponents, β and bG, with |dQ/dt| ∝ Qβ where Q is at-a-station gauged flow rate and N(l) ∝ inline image where l is the downstream distance from the channel heads receding in time, N(l) is the number of draining channel reaches located at distance l from their heads, and G(l) is the total drainage network length at a distance greater or equal to l, the active drainage network. We find that the method provides good results in catchments where drainage density can be regarded as spatially constant. A correction to the method is proposed which accounts for arbitrary local drainage densities affecting the local drainage inflow per unit channel length. Such corrections properly vanish when the drainage density become spatially constant. Overall, definite geomorphic signatures are recognizable for recession curves, with notable theoretical and practical implications.

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:

display math(1)

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 qL, the total discharge Q(t) can be expressed as:

display math(2)

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/dtN·c). Equation (2) can be differentiated in time:

display math(3)

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.

Figure 1.

Idealized example of a recession described by the geomorphological conceptual models. The Active Drainage Network (ADN) is represented in blue solid lines and the dry part of the ADN is represented in red solid lines. The sources (here N = 14) are represented in red solid circles, the blue shaded area represents the fraction of the basin drained by the ADN (AT).

[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.

2. Study Areas and Available Data

[6] We analyze 26 catchments located in Switzerland (Figure 2) presenting different sizes, relief, slope, and soil properties. The watersheds are all characterized by relatively little anthropogenic influence on the streamflow behavior. The hourly averaged streamflow data of the 26 gauging stations are obtained from the Swiss Federal Office for the Environment. The rainfall data is obtained through SwissMetNet, a network of automatic weather stations operated by MeteoSwiss. All the stations are measuring the precipitation within a maximum radius of 10 km from the corresponding discharge gauging station. The DEM of the 26 basins were extracted from a 25 m resolution DEM of Switzerland provided by the Swiss Federal Office of Topography with standard commercial geographic information system (GIS) software. Table 1 summarizes the principal characteristics of the 26 catchments.

Figure 2.

Map of the experimental catchments used in this study. Some geomorphologic characteristics and description of the available data are listed in Table 1.

Table 1. Summary of Geomorphological Characteristics, Discharge, and Precipitation Data for the 27 Watershedsa
Number of BasinNameGeomorphologyDischarge DataPrecipitation Data
Basin Surface (km2)Mean Basin Altitude (m)Max Basin Altitude (m)Mean Basin Slope (m/km)Glacier Ext. (%)Coordinates Gauging Station X,YStation Altitude (m)Measure StartNameCoordinates Precipitation Station X,YDistance from Discharge Station (km)
  1. a

    The coordinates are given in the Swiss Coordinate System.

1Murg-Wängi78650103525.50714,105261,7204661 Jan 1974Eschlikon715,095258,2603.60
2Gürbe-Belp, Mülimatt117849217652.20604,810192,6805221 Jan 1974Belp605,140193,8051.17
3Goldach Bleiche49.8833125139.30753,190261,5903991 Jan 1974Arbon749,840263,1503.70
4Necker-Mogelsberg- Aachsäge88.2959153235.60727,110247,2906061 Jan 1974Mogelsberg728,220247,0101.14
5Venoge-Ecublens, Les bois231700143120.80532,040154,1603831 Jan 1979Marcelin527,060152,1605.37
6Murg-Walliswil207637111918.60629,340233,5554191 Jun 1980Wynau626,400233,8502.95
7Sellenbodenbach-Neuenkirch10.561583836.50658,530218,2905151 Sep 1990Sempach657,010220,9403.05
9Sitter-Appenzell74.212522500139.60.08749,040244,2207691 Jan 1974Appenzell747,735244,4751.33
10Sense-Thörishaus-Sensematt3521068219025.40593,350193,0205531 Jan 1977Schwarzenburg5931501847208.30
11Melera-Melera1.051419172871.10726,988114,6709441 Jan 1974Bellinzona721,060116,8006.30
12Rotenbach-Schweinsberg1.6514541628151.30587,980170,59012751 Aug 1995Sangernboden593,220173,9056.20
13Riale di Calneggia-Pontit24199629212310684,970135,9708901 Jan 1974Bosco-681,160130,0257.06
14Areuse-Boudry3771060160727.30554,350199,9404441 Jan 1983Combe-551,220201,4503.48
15Ilfis-Langnau1881051209230.90627,320198,6006851 Apr 1989Kurzeneialp630,575207,0109.02
16Verzasca-Campiol1851661286472.40708,420122,2904901 Aug 1989Cimetta704,370117,5156.26
17Alp-Einsiedeln46.41155189940.80698,640223,0208401 Feb 1991Altmatt695,420220,7703.93
18Massa-Blatten bei Naters19529454195183.265.9643,700137,29014461 Jan 1974Brig640,570129,0808.79
19Krummbach-Klusmatten19.822763268121.83644,500119,42017951 Jan 1974Simplon-647,570116,1104.51
20Rhone-Gletsch38.92719362274.252.2670,810157,20017611 Jan 1974Grimsel668,583158,2152.45
21Grosstalbach – Isenthal43.918202952211.19.3685,500196,0507671 Jan 1974Isenthal685,460196,1100.07
22Alpbach-Erstfel- Bodenberg20.62200319841427.7688,560185,12010221 Jan 1974Altdorf690,174193,5588.59
23Ova da Cluozza-Zernez26.9236831651802.2804,930174,83015091 Jan 1974Zernez802,720175,2302.25
24Dischmabach – Davos43.323723146129.42.1786,220183,37016681 Jan 1974Dischma786,600182,9900.54
25Poschiavin-La Rösa14.12283303291.30.35802,120132,01018601 Jan 1974Cavaglia799,850138,4704.21
26Rein da Sumvitg,Encardens21.824503168165.56.7718,810167,69014901 Jan 1977Vrin727,220168,6708.47
27Val Ferret, Ferret20.42422.832063852.1574,93283,77911731 May 2008Val FerretVariousVariousVarious

[7] We also analyze the Val Ferret catchment (catchment 27 in Table 1), an experimental site we have monitored since 2008 [see Simoni et al., 2011; Tobin et al., 2013]. Streamflow data are available every 5 min and obtained through water level measurements and a rating curve that was derived using the salt-dilution method on a yearly basis for the period 2008–2012. Recently, Weijs et al. [2013] decreased the error on the rating curve by combining the water level with the natural electrical conductivity of stream water which appears particularly useful for Alpine watersheds. The rainfall data for the Val Ferret catchment is obtained along with other forcing parameters by a wireless network of up to 26 small meteorological stations deployed on representative sites of the catchment [Simoni et al., 2011]. For this catchment, we used both a 5 m resolution light detection and ranging (LiDAR)-derived DEM and the 25 m resolution DEM provided by the Swiss Federal Office of Topography. Moreover, the actual channel network of the Val Ferret catchment has been surveyed by Differential Global Positioning System during an extensive field campaign in 2011.

3. Methods

[8] In order to compute the recession exponents, measured discharge has been averaged from hourly values to daily values in order to filter out the diurnal contributions of snow or icemelt to the streamflow daily periodicity. Precipitation has been integrated over the same period and used to define recession events as 6 consecutive days without precipitation. Moreover, following Biswal and Marani [2010], we only considered events with peak discharge larger than the average discharge in order to enhance the geomorphic signature on the base flow recession and to insure that the whole catchment is active at the onset of recession—unlike the original method from Brutsaert and Nieber [1977] where the focus was on the groundwater hydraulics signature. The discharge variations and values have been computed as inline image and inline image following Brutsaert and Nieber [1977], where Δt is the time step of 1 day used in the analysis. Recession events with nonmonotonically decreasing discharge values were discarded from the analysis (i.e., events containing days with dQ/dt ≥ 0 were removed from the analysis). In the log-log plot, the single –dQ/dt versus Q curves tend to be shifted depending on the maximum peak discharge, antecedent soil moisture and evapotranspiration. According to Biswal and Marani [2010], Biswal and Nagesh Kumar [2012], and Shaw and Riha [2012], a value of the exponent β is obtained for every recession event by fitting a linear model in the log-log space using ordinary least squares. In the following, we will refer to the recession exponent β of a catchment as the median of the exponent β's frequency distribution. This event-based approach differs from the previous studies linking the analyzed recession discharges based on a one-to-one relationship between the amount of water stored in the catchment and the discharge occurring during recessions.

[9] The variables N(t) and G(t), instrumental for the geomorphological analysis, can be obtained from the analysis of DEMs by extracting the channel network on the basis of standard topographic threshold methods [O'Callaghan and Mark, 1984; Tarboton et al., 1991] or slope-dependent or topographic curvature-dependent support areas mimicking different channel initiation processes [Montgomery and Dietrich, 1988, 1992; Montgomery and Foufoula-Georgiou, 1993; Sofia et al., 2011; Tarolli and Dalla Fontana, 2009], the latter being capable of rendering spatially heterogeneous drainage densities. As done in Biswal and Marani [2010], we use a simple flow accumulation threshold as channel network extraction method to standardize procedures, except for the case where ground truthing was available (for catchment 27, see section 2). After a certain time t, the number of sources N(t) is determined by the number of reaches located at distance l = c(t − to) from their farthest upstream initial source (Figure 1). Under the assumption that the ADN varies quickly, such that the term in dqL/dt can be neglected, one has dQ/dt ∝ −N(t) from equation (3) and Q ∝ G(t) from equation (2). Inserting these two relations in equation (1), one finally has

display math(4)

where bG = β if the geomorphological exponent correctly captures the exponent obtained from the analysis of the recession curves.

[10] In this work, we propose a revised approach of the conceptual model developed in Biswal and Marani [2010]. We assume that the directly contributing discharge Q is drawn not from an unconfined aquifer like in the traditional way but in our case by the unchanneled area draining directly in the ADN:

display math(5)

where ADN is the Active Drainage Network at time t, AT is the area draining directly in the ADN, ai is the directly contributing area at site i and qi is the discharge per pixel at site i, see Figure 1. The second approximation derives from assuming q constant as done in Biswal and Marani [2010]. As such we assume that the variation of the contributing discharge is proportional to the rate of change in directly contributing area therein and hence on local drainage density in the sense of Tucker et al. [2001]:

display math(6)

where c is the speed at which the sources recede, assumed constant in space and time as in a negative traveling wave. The approach is thus based on the computation of the direct drainage areas of the ADN. Moreover, the change in direct drainage area is given by the area AN draining directly in the sources of the ADN at time t:

display math(7)

[11] Combining equations (1), (5), (6), and (7), we obtain similarly to equation (4):

display math(8)

where bA = β if the geomorphological exponent obtained with the new method matches the exponent obtained from the analysis of the recession curves. In this study, we propose to compute the geomorphological exponents bG of equation (4) and bA of equation (8) for the 27 watersheds considered and to compare them with the values of the recession exponent β obtained from the recession analysis. Then, we study the differences between the two methods in terms of mean catchment altitude, aiming at improving our understanding of the geomorphological origin of the recession curves.

4. Results

[12] Examples of three catchments (basins 15, 25, and 27, see Table 1 for geomorphological characteristics), where the two different methods have been applied are shown in Figure 3. The examples present different cases and results characteristic of the correction we propose in this work. The channel network has been suitably extracted from the DEM using a flow accumulation threshold of 100 pixels and is here color coded in blue in Figure 3a. Different flow accumulation threshold values and another channel network extraction method (depending on a slope-area threshold) have been used without appreciable changes in the following results. For the study of the Val Ferret catchment (watershed 27 in Table 1), the monitored network (see section 2) has been used as the basis for the determination of the ADN. The distribution of the distance from any unchanneled site to its nearest stream channel following the steepest path has been studied in relevant subcatchments of the watersheds. It appears that the hillslope distance to the nearest channel and therefore the local drainage density [Tucker et al., 2001] cannot be considered constant in most high mountain catchments and especially in the case of the Val Ferret catchment where the real monitored network has been used.

Figure 3.

Example of the results obtained for watershed 15 (column 1), watershed 24 (column 2), and watershed 27 (column 3), see basins characteristics in Table 1. The different lines correspond to (a) the channel network obtained with the area threshold method in blue and in black and white the distance L to the nearest channel following the steepest path (except for watershed 27, column 3 where the real, monitored network has been used), (b) the cloud of points in gray obtained from the recession analysis with some events and their fit represented in different colors. Note that only few events are presented in colors for better visibility, (c) number of sources N(l) versus ADN length G(l) from which we obtain the geomorphologic exponent bG, and (d) total area draining directly in all sources of the ADN AN versus total area draining directly in the ADN AT from which we obtain the geomorphologic exponent bA.

[13] The recession slope analysis has been carried out for each of the 27 basins by fitting each recession event separately with least squares and by computing the frequency distribution of the β values. Some examples of individual fits are color coded in Figure 3b, along with the cloud of points obtained for all the events (gray dots). The solid black line represents the fit of all the events. As expected, the exponent β of equation (1) obtained with the global fit is less than the average of the coefficients fitted on single events. The results of the recession slope analysis are presented in Table 2. In the following, the values of a catchment's median exponent β are used as a comparative basis for the scaling exponents bG and bA of the different geomorphological models. Note that, in general, the standard deviations of the exponent β are relatively large due to the inferences of fast responses of the catchments to precipitation or glacier melts. The low extreme (β = 1.06, basin 20, see in Table 2) corresponds to a largely glaciated, high-altitude catchment whereas the high extreme (β = 6.24, basin 6, Table 2) corresponds to a highly urbanized catchment. Both cases have been discarded in the following calculations.

Table 2. Summary of the Results Obtained From the Recession Analysis (Exponent β), From the Model Comparing N Versus G (Geomorphological Exponent bG) and From the Model Comparing AN Versus AT (Geomorphological Exponent bA) and Their Respective Residuals Compared to β
 Number of EventsCoefficient β for All Events TogetherMedian Coefficient β for Separated EventsMean Coefficient β for Separated EventsStandard Deviation of βbG (N Versus G)Residuals Median β-bGbA (AN versus AT)Residuals Median β-bA
  1. a

    Results obtained with the 25 m resolution DEM.

  2. b

    Results obtained with the 5 m resolution DEM.

1301.752.582.700.981.70 ± 0.010.871.56 ± 0.031.02
2362. ± 0.011.591.99 ± 0.031.29
3301.662.162.120.651.75 ± 0.010.411.83 ± 0.070.34
4231.241.961.830.512.21 ± 0.01−0.252.82 ± 0.06−0.86
5591.732.542.801.061.89 ± 0.010.641.95 ± 0.030.59
6252.556.246.964.522.45 ± 0.023.782.52 ± 0.093.71
7191.032.543.262.281.80 ± 0.020.741.76 ± 0.110.78
8191.732.352.260.872.21 ± ± 0.07−0.49
9141.882.892.610.872.32 ± 0.020.562.49 ± 0.060.40
10251.812.912.861.291.99 ± 0.010.912.64 ± 0.020.27
11561.823.343.912.010.79 ± 0.092.561.12 ± 0.152.22
12131.481.201.771.031.10 ± ± 0.16−0.35
13361. ± 0.010.791.44 ± 0.080.76
14441.802.282.551.232.03 ± ± 0.020.11
15201.572.572.551.332.19 ± 0.010.382.37 ± 0.030.20
16231.872.392.701.051.60 ± 0.010.792.96 ± 0.04−0.57
17101.511.291.700.781.59 ± 0.01−0.302.04 ± 0.05−0.75
18170.742.042.492.071.98 ± ± 0.03−0.83
19121.422.863.011.231.36 ± 0.021.501.14 ± 0.08−0.34
20130.891.062.652.941.88 ± 0.01−0.812.99 ± 0.120.29
21251.732.733.633.161.94 ± 0.020.782.03 ± 0.060.70
22140.591.311.320.591.02 ± ± 0.13−0.84
23141.533.203.481.612.12 ± ± 0.25−0.34
24181.752.662.951.491.37 ± ± 0.070.29
25401.793.904.293.181.37 ± 0.022.531.29 ± 0.092.62
26111.722.822.811.791.62 ± ± 0.091.31
27121.053.475.514.591.48 ± 0.02a1.99a4.11 ± 0.15a−0.64a
1.53 ± 0.01b1.94b3.40 ± 0.04b0.07b

[14] The comparison of the two models is shown in Figure 3c, as per the method developed in Biswal and Marani [2010], we show the number of sources N plotted versus the total length G of the ADN. As postulated by the original method, the number of sources decreases or stays constant in time, resulting in a monotonically decreasing function N(G). The plots result in a piecewise constant function at low values of the network length since low-order channels have dried out already so that the ADN stems mainly from high-order streams. In order to better estimate the geomorphological parameters bG and bA, the data have been fitted only up to when 80% of the initial ADN has receded in order to avoid the last part where N(G) is piecewise constant. In Figure 3d, according to the proposed revision of the original method, we show the area AN draining directly in the sources of the ADN versus the total area AT draining in the ADN. In the three plots, the cloud of points is quite noisy at the end of the recession (that is, for the smallest total areas) which is a signature of the watershed geometrical attributes and of uneven local drainage densities. All the values of the exponents bG and bA are presented in Table 2. The two exponents bG and bA are similar in the case of catchment 15 (first column in Figure 3), a catchment with fairly constant hillslope to channel distance. This supports our ansatz that the different methods provide indistinguishable results in cases where drainage density can be regarded as relatively uniform in space. In the case of catchment 24 (second column in Figure 3) and especially in the case of catchment 27 (third column in Figure 3), major differences arise from the two methods. Both catchments exhibit very variable patterns in the hillslope distance to the nearest channel. Remarkably, however, it appears that a scaling relation between AN and AT can still be found and that allows a fair determination of the scaling exponent bA. In the case of catchment 27, the exponents bG obtained with the two DEM of different resolution were very similar but relatively different for the exponent bA, see Table 2. The values computed with the 5 m resolution DEM have been used in the following for the catchment 27.

[15] The geomorphological exponents of the two conceptual models, bG and bA, have been compared to the recession exponents β of all the basins in Figures 4a and 4b, respectively. The horizontal uncertainty bars represent the standard deviation of the recession exponents β computed on a single event basis and the solid black line represents the one-to-one relation. The upper plot is visibly more scattered than the lower plot where the points are more aligned along the one to one line. More formally, a simple frequency distribution of the difference between the geomorphological exponents bG and bA and the median recession exponents β is presented in the insets of Figures 4a and 4b, respectively. With a mean residual of 0.20, it appears clearly that, on average, the exponent bA obtained with the new method matches better the exponent β of the recession analysis than the exponent bG obtained with the first method (mean residual of 0.67). The improvement of the new method is even more remarkable when the residuals are classified along mean basin altitudes, see color coding in Figure 5. For low-altitude basins (mean altitude below 1000 m, first line in green in Figure 5), the mean residuals decrease between the first and the second method. For the watersheds at medium mean altitudes (mean altitude between 1000 and 2000 m in red in Figure 5) and especially for watersheds at high mean altitudes (mean altitude above 2000 m in blue in Figure 5), the new method improves the results with a larger decrease in the mean residual. Assuming increasing spatial complexity of the channel network with mean basin altitude, our results suggest that the proposed revised method provides better results in catchments where local drainage density is naturally heterogeneous reflecting the variety of channel initiation processes (i.e., proglacial, high-altitude catchments). In the case of low-altitude catchments, where local drainage density tends to be uniform in space, the two methods provide similar results.

Figure 4.

Recession exponent β versus geomorphological exponent for (a) bG with the N versus G method and (b) bA with the AN versus AT method. The insets in Figures 4a and 4b correspond to a simple frequency distribution of the residuals β − bG and β − bA, respectively.

Figure 5.

Histogram of the residuals between the recession exponent β and the geomorphological exponents (a, c, and e) bG (left histograms) and (b, d, and f) bA (right histograms) classified along mean basin altitude, from top to bottom in green (first row, Figures 5a and 5b) for basins at mean altitude below 1000 m, in red (Figures 5c and 5d) for basins at mean altitude between 1000 and 2000 m and in blue (Figures 5e and 5f) for basins at mean altitude above 2000 m.

5. Discussion

[16] We find that the discrepancies arising between the exponent β obtained from the analysis of the recession curves and the geomorphological parameter bG obtained from the original method proposed in Biswal and Marani [2010] are relatively small for low-altitude basins but larger for higher altitude basins. Compared to the watersheds studied in Biswal and Marani [2010], the watersheds chosen in this study are probably less suited to the original conditions envisioned by Brutsaert and Nieber [1977]: they are relatively smaller, structurally inconsistent with the conceptual model of simple drainage of an unconfined aquifer and generally exhibiting faster responses to rainfall impulses especially due to steep hillslopes. Often, as highlighted in the physical features of Table 1, the catchment response shows signatures of snow or ice melt, both leading to larger uncertainties in the evaluation of the recession exponent β. Moreover, such discrepancy is very large in the case of an urbanized watershed (basin 6 in Tables 1 and 2), confirming that the method is mostly suited to watersheds with little anthropogenic influence. However, the discrepancies between bG and β cannot be explained only by the uncertainty in the exponent β, and some assumptions made by the method must be relaxed when hydrologic and geomorphic conditions required are not met.

[17] Our proposed correction of the method indeed accounts for uneven local drainage density, because it assumes that the local contributing discharge per unit length of receding ADN is limited by the local hillslope to channel distance and its directly contributing area. This is typically occurring in high altitude, proglacial dominated catchments where channel initiation processes are most diverse and unchanneled distances may vary from tens to thousands of meters. The inclusion aims at improving our understanding of the geomorphological origin of the recession curves, as noted in section 3. The system is described by the evolution in time of AT, the area draining directly into the ADN relative to the area AN draining into the sources of the ADN, leading to an empirical relation inline image. Our results suggest that the new geomorphological parameter bA is closer to the exponent β, resulting in a decrease of the residuals between the two exponents (Figure 4). In particular, the correction is substantial in the Val Ferret catchment (number 27) where the real, monitored network has been used for the calculations. For this high-altitude catchment, endowed with highly uneven local drainage density, the residuals between the exponents β − bG and β − bA decrease from 1.99 to −0.64 and from 1.94 to 0.07 between the two methods for the 25 m resolution and the 5 m resolution DEM, respectively. In other particular cases where the local drainage density is more even, the two methods produce very similar results (see catchments 3, 5, or 7) and the two methods give comparable results as expected from the fact that the approaches tend to collapse into the same formulation.

[18] We acknowledge several sources of uncertainty introduced in the model, chiefly through the resolution of the DEM and through the area threshold method adopted for the channel network extraction (which is known to fail in complex terrain where heterogeneities of channel initiation processes are major). In the Val Ferret catchment, the uncertainty in the channel network extraction is dramatically decreased by an accurate field channel network monitoring resulting in improved results for the method proposed here. However, the time series available for measured streamflows for this watershed is less compared to the 26 others leading to a greater uncertainty in the exponent β. Field campaigns in other small watersheds aimed at monitoring channel initiations would probably improve the performance and the reliability of this conceptual model. In the Val Ferret catchment, our results were not affected by the DEM resolution since the geomorphological parameter bA was closer to β than bG to β in both DEM resolution cases. Therefore, the uncertainty introduced by the DEM resolution might only alter the accuracy of the predicted exponent but not the essence of our method. We also acknowledge larger uncertainties in the estimation of the parameter bA due to the larger scatter in the AN versus AT plots which is the mark of uneven local drainage densities. We note, however, that the error in the estimate of the parameter bA is still small compared to the error in the exponent β. Finally, we note that our study pinpoints that the most critical assumption of the conceptual model lies in neglecting the term dq/dt in equation (3) with respect to the change in the ADN geometry. Further studies are thus needed in order to combine the results of the method here formulated with a possibly geomorphically controlled integration, modeling the speed of the negative traveling wave of active stream switch offs.

6. Concluding Remarks

[19] Catchment recession curves bear the signatures of geomorphology. Two conceptual models based on empirical relations obtained solely from the analysis of DEMs have been compared. Both models describe the impact of geomorphology on the recession curves of the stream network, described in the first model by the evolution of the number of sources in the active channel network versus the total active channel network length, and in the latter by the area draining in the sources of the receding network versus the total area draining directly in the network. From the analysis of 27 catchments relatively unaffected by anthropogenic influence, our results suggest that the two models give similar results in the cases where local drainage density is approximately constant. In the cases of spatially uneven local drainage density, the first model does not hold and the new model presented here improves the results for high-altitude basins. In general, we suggest that this conceptual model might be useful to estimate the low flow regime of natural ungauged basins by predicting its features solely from information remotely acquired and objectively manipulated through DEM data.


[20] The authors are grateful to the Swiss National Science Foundation for financial support (grant 200021_134982/1 and 200021-124930/1), to ERC Advanced Grant RINEC 226712 and to the NCCR-MICS and CCES fundings. The authors wish to thank the Commune d'Orsières for providing logistic support for the field campaigns in the Val Ferret catchment.