Themethodtodetermineαandβ,andthusfinallythesensitivityfunctiong(Q),fromre-
cession analysis (Brutsaert and Nieber, 1977) was proposed byKirchner(2009) and is based
on discharge observations. Recession analysis can be an interesting alternative to auto- matic calibration, for example in regions where information on occurrence of precipita- tion is more accurate than the amounts, which is relevant for poorly gauged basins. Fig- ure 3.2 shows the recession analysis procedure which has been employed with different calibration periods.
During recession, i.e. excluding rainfall (P) and snowmelt (M), and during night time
(which makes it possible to neglect evapotranspiration), the sensitivity function can be expressed as (see Eq. 2.7):
g(Q) = dQ dS =αQ β ≈ −1 Q dQ dt |P,M,ET <<Q. (3.1)
In order to use Equation 3.1 to determine the sensitivity function parameters, the ob- served data has to be selected carefully. In this study, two different criteria for data se- lection have been used:
1. Dry and dark periods, too warm for snow cover to be present. 2. Dry and dark periods, too cold for snow melt to occur.
3.2. Methods
For an exact description of the criteria per set, seeTeuling et al.(2010). The strict criteria
for data selection only leave part of the data available for recession analysis. From the se- lected data (shown in the left panel of Figure 3.2),g(Q)is determined (middle panel of
Figure 3.2) with Equation 3.1. Because of measurement errors and a possible difference between catchment representation and reality, there is some scatter in the selected data. Therefore, thedatahasbeenbinnedaccordingtothemethodusedbyTeuling et al.(2010).
If 30 or less data points were selected for the two sets together,αandβwere fitted di-
rectly on all data points larger than zero (negative data points could exist due to small increases in discharge, e.g. due to the diurnal cycle in evapotranspiration or due to mea- surement errors), with a minimum of three data points. If more than 30 data points were selected, but only two bins or less,αandβwere also determined on all individual data
points above zero. If more than 30 data points and more than two bins were available,α
andβwere fitted on the binned data. These criteria are especially important during re-
cession analysis for a monthly time period, because the number of selected data points per month can be limited. Theα(intercept) andβ(slope) were determined by plotting
the logarithm ofg(Q)versus the logarithm ofQand fitting a straight line log(α) +β·
log(Q)(right panel in Figure 3.2), which is the classical approach of recession analysis
as described byBrutsaert and Nieber(1977) and referred to as ‘logRA’ in the sequel of this
Chapter. In addition,αandβwere determined on linear axes for bothg(Q)andQ, fit-
ting a power law. This method is further referred to as ‘linRA’.
Boussinesq equation for sloping aquifers
Boussinesq(1877)derivedanequationdescribinggroundwaterflowinanunconfinedaqui-
fer overlying an impermeable layer. An important assumption of this approach is the Dupuit-Forchheimer approximation, assuming that groundwater moves parallel to the impermeable layer in an unconfined aquifer and that groundwater discharge is propor- tional to the thickness of the saturated aquifer (Troch et al., 2013). With this assumption,
5/11/79 25/11/79 0 0.5 1 1.5 Q [mm h − 1] Obs.discharge Set 1 (Warm) Set 2 (Cold) 10−2 100 10−3 10−2 10−1 100 Q [mm h−1] Sensitivity function g(Q) [h − 1] 10−2 100 10−3 10−2 10−1 100 Q [mm h−1] Sensitivity function g(Q) [h − 1]
Figure 3.2: The process of recession analysis for a randomly selected month. Left panel: data are selected according to different criteria. Middle panel: sensitivity functiong(Q)is determined for each of the selected data points, based on Equation 3.1. Right panel: the points are binned, and a line is fitted through the data,αis represented by the intercept of the line,βby the slope of the line.
Boussinesq(1877) adapted Darcy's law and combined it with the continuity equation. An
important property of the Boussinesq equation (or its linearisations) is that analytical solutions of the equation can be expressed in the form of Equation 2.5 (Rupp and Selker,
2006), which gives the opportunity to estimate the two simple dynamical system param- eters in this study with the use of Boussinesq theory. The advantage of using a physically- based method such as the Boussinesq equation is that it does not need time consuming and costly discharge measurements. Disadvantage is that effective values for subsurface parameters at the catchment scale have to be estimated. It represents the whole catch- ment as a single hillslope draining to a channel, as shown in Figure 3.3, not taking into account heterogeneity in the topography or subsurface of the catchment. It is therefore a strong simplification of reality, especially in a catchment like the Rietholzbach with com- plex topography.
Nowadays, many different forms of the Boussinesq equation are in use, see for example the overview given inRupp and Selker(2006). For the Boussinesq equation adapted for
sloping aquifers, relevant in this study,Rupp and Selker(2006) used numerical simula-
tions to empirically derive an analytical solution to the non-linear Boussinesq equation for the late part of the recession. Assumptions of the applied Boussinesq equation are that no water flows through the bottom and water divide boundaries (see Figure 3.3), and that the water level in the stream is constant.Rupp and Selker(2006) present the fol-
lowing equations forαandβin terms of physical parameters: α= (n+ 1) 2 (n+ 0.01)ϕA· 2k DLsinφ (n+ 1)Dn n+11 ·(10−3A)β, (3.2) β= n n+ 1. (3.3) Φ L h0
Figure 3.3: Water table profile for the applied Boussinesq model, which is adapted for sloping aquifers. Zero flux boundaries at the right hand side and at the bottom are assumed. Parameters
3.3. Results and discussion
Table 3.1: Parameter values for the Boussinesq equation
Unit Value Description
k m h−1 range: Saturated hydraulic conductivity
6.5·10−5 Rietholzbach average (König and Lang, 1994)
0.1170 Nagelfluh (König and Lang, 1994)
0.0360 Lowest estimate (Lehner and Seneviratne, 2010)
1.8 Highest estimate (Lehner and Seneviratne, 2010)
L m 2500 Channel length
ϕ - 0.1;0.2;0.25;0.3;0.35 Drainable porosity
A m2 3.31·106 Catchment area
φ ◦ 14.5 Average slope
D m 1 Average depth of the soil
n - 0;1;10;20;30;40;50;60;70 Saturated hydraulic conductivity profile parameter
Withϕ(-) the drainable porosity,A(m2) the catchment area,kD(m h-1) the saturated hydraulic conductivity at depthD,L(m) the length of the aquifer,φ(-) the slope of the
aquifer, andD(m) the depth of the aquifer. Then-parameter determines the saturated
hydraulic conductivity profile, with the following equation (Rupp and Selker, 2006):
k(z) =kD(z/D)n. (3.4)
Forn=0, no variation with depth is assumed,n=1 indicates a linear profile with saturated
hydraulic conductivity decreasing with depth,n >1 leads to power law profiles. Note
that forn=0,β=0 (thus leading to a linear reservoir). Experience from earlier studies (e.g.
Kirchner,2009;Teuling et al.,2010;Brauer et al., 2013)showedthatβis hardlyever equalto
zero, so a range of possible values fornwas used. ThekDin Equation 3.2 and 3.4 differs from the averagek¯in the area (see e.g.König and Lang, 1994). Integration of Equation 3.4
leads to the following relationship:
kD= (n+ 1)·¯k. (3.5)
The parameter values that have been used for Equation (3.2) and (3.3) are given in Table 3.1. For the saturated hydraulic conductivity¯k, the drainable porosityϕ, and the satu-
rated hydraulic conductivity profilen, exact values were unknown, hence a range of val-
ues was used (Table 3.1). This lead to 180 different parameter sets. For each set the model efficiencies, NSE(Q) and NSE(logQ), over the full validation period were determined.
3.3
Results and discussion
First the results from the automatic calibration procedure are discussed, followed by the results obtained with recession analysis and Boussinesq theory. Finally, a comparison between the three methods is made. The discussed model efficiencies are based on a common validation period covering the full data length, which was 32 years, so that all reported NSE values refer to the same 32 year period and can be compared directly. No
split sample validation was employed. Hence, there is at least one month overlap be- tween the calibration and the validation period. The overlap between calibration and validation period increases further with longer calibration periods: for the calibration period of 32 years, the calibration and validation period are identical. Since the analy- sis focusses on much shorter periods than the full validation period, the effects of this approach on the results are believed to be small.