WATER RESOURCES RESEARCH, VOL. 48, W09544, doi:10.1029/2011WR011543, 2012
Can time domain and source area tracers reduce uncertainty in
rainfall-runoff models in larger heterogeneous catchments?
R. Capell,1 D. Tetzlaff,1 and C. Soulsby1
Received 31 October 2011; revised 16 July 2012; accepted 2 August 2012; published 25 September 2012.
[1] We used both time domain (deuterium) and source area (alkalinity) tracers to reduce
uncertainty in simple conceptual rainfall-runoff models applied to a larger (749 km2)
heterogeneous catchment with both upland and lowland headwaters. Stepwise, tracer-aided
model development resulted in different model structures for the uplands and lowlands as
representative elementary watersheds (REWs). These were differentiated by the
parameterization of a nonlinear overland flow mechanism in the former, and the
incorporation of high soil moisture storage capacity in the latter. Use of tracers and
recession characteristics also helped to reduce parameter uncertainty and provided models
that could simulate flows and tracer responses reasonably well over a full hydrological year.
However, it was apparent that other processes (e.g., more complex mixing, fractionation,
etc.) would need to be parameterized to explain the full variation in isotope dynamics. It is
also evident that the information content of tracer data declines as the intensity of sampling
decreases, particularly in the lowlands. The models of the REWs were coupled to provide
plausible simulations of the up-scaled flow and tracer response at the outfall of the 749 km2
catchment, though the usefulness of source area tracers decreased markedly at this larger
scale. Whereas the approach provides a step toward simple models that are likely to give the
‘‘right answer for the right reasons,’’ further improvements appear to require increased
parameterization and/or higher-resolution tracer data.
Citation: Capell, R., D. Tetzlaff, and C. Soulsby (2012), Can time domain and source area tracers reduce uncertainty in rainfall-runoff
models in larger heterogeneous catchments?, Water Resour. Res., 48, W09544, doi:10.1029/2011WR011543.
1.
Introduction
[2] Recent progress in rainfall-runoff modeling in small
experimental catchments has seen the use of time domain
(e.g., isotopes) or source area (e.g., alkalinity) tracers to
constrain the selection of appropriate model structures and
parameter ranges [Beven, 2012]. It is uncommon for both
time domain and source area tracers to be used concurrently in such tracer-aided models, and rarely have they
been applied to larger mesoscale catchments (>100 km2).
Yet this up-scaling is important, as it is commensurate with
the scales where models are needed to inform management
decisions concerning climate change impacts, flood alleviation, sustainable allocation of water resources, and protection of riverine ecology [Reed et al., 2006]. Unfortunately,
spatial variability of catchment characteristics and associated
hydroclimatic drivers make it challenging to optimize appropriate model structures at such larger scales and the potential
for tracers in assisting this process is largely unexplored
[Soulsby et al., 2008]. Contrasting geographic regions usu-
1
Northern Rivers Institute, School of Geosciences, University of
Aberdeen, Aberdeen, UK.
Corresponding author: R. Capell, School of Geosciences, University of
Aberdeen, Elphinstone Rd., Aberdeen AB24 3UF, Scotland, UK.
(rene.capell@gmail.com)
©2012. American Geophysical Union. All Rights Reserved.
0043-1397/12/2011WR011543
ally require different model structures. Although these may
be achieved by using more spatially distributed models, this
necessitates either high-resolution data or a level of parameterization, which makes model identification based on rainfall-runoff relationships virtually impossible [Beven, 2001].
On the other hand, increased heterogeneity of larger catchments can inhibit simple lumped modeling approaches,
particularly where marked landscape changes result in different dominant runoff generation processes [Beven, 2000].
Increased parameterization can offset this shortcoming
[Fenicia et al., 2008a]. However, increased complexity in
modeling approaches makes it more difficult to use additional process-based ‘‘soft’’ data, such as tracers, to aid
model evaluation [Seibert and McDonnell, 2002]. Thus,
there remain strong arguments for retaining low-parameter
approaches, which allow additional opportunities to assess
whether models provide the ‘‘right answers for the right reasons’’ [Kirchner, 2006]. This is particularly important if
calibrated models are to be used as tools to evaluate the
effects of environmental change on hydrological function
[Beven, 2007; Dunn et al., 2008b].
[3] The uncertainties associated with extrapolating pointscale empirical observations increase in larger-scale catchments, where routine hydrometric and meteorological data are
usually available, but process-based information is usually restricted. Here, environmental tracers can present versatile and
comparatively cheap tools for adding information to hydrometric data and detecting the dominant influences on the integrated runoff response by interpolating the likely processes
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governing the dynamics in a top-down manner [e.g., Klemes,
1983; Kendall and Coplen, 2001; Soulsby et al., 2006b].
Tracers can thus provide empirical insights into the dynamics
of catchment systems that have the potential to be useful in
modeling [e.g., Seibert et al., 2003; Birkel et al., 2010]. Information provided by both geochemical and isotopic tracer
responses in catchment runoff models can help to constrain
viable parameter sets and more fundamentally increase
confidence in the choice of model structure [e.g., Uhlenbrook et al., 2004; McGuire et al., 2007; Birkel et al.,
2010; McMillan et al., 2011, 2012].
[4] There is a long history of using tracers in modeling
studies. For example, Robson et al. [1992] used acid neutralizing capacity as the source area tracer of the geographic provenance of water in the Plynlimon catchments
and compared this with the equivalent partitioning of water
with an application of TOPMODEL. In the same catchment, Neal et al. [1988] used chloride as a conservative
time domain tracer to demonstrate that, whereas the Birkenes model could simulate rainfall-runoff relationships, it
failed to simulate stream chloride concentrations, indicating that storage changes and mixing processes were poorly
represented in the model. Despite this, it is only recently
that both ‘‘source area’’ and ‘‘time domain’’ tracers have
been used in conjunction in dynamic rainfall-runoff models
to constrain model parameters and structures in a consistent
way [e.g., Birkel et. al., 2011a]. Moreover, most other modeling applications of geographic source area or time domain
tracers either to constrain uncertainty or as a basis for model
rejection have focused on smaller (typically <10 km2) experimental catchments [e.g., Fenicia et al., 2008b; Uhlenbrook
et al., 2002; Page et al., 2007; Fenicia et al., 2010]. The
utility of tracers in aiding modeling studies at larger scales is
thus an area of significant research potential.
[5] The usefulness of identifying hydrologically similar
areas, or representative elementary watersheds (REW), is
well established as a basis for hydrological modeling at
larger scales [e.g., Wood et al., 1988], and recent work has
suggested that marked differences in catchment characteristics can provide important classification tools that can aid
this disaggregation [e.g., Buttle, 2006; Wagener et al.,
2007; Carillo et al., 2011; Sawicz et al., 2011]. Very recent
approaches have used such connections between key landscape controls and dominant runoff generation processes to
delineate hydrologically similar areas [Savenije, 2010;
Gharari et al., 2011; Ali et al., 2012]. However, different
landscape controls for delineating dominant runoff generation processes may be needed at different spatial scales.
For example, soil cover has been identified as the first-order
control on runoff generation processes at the hillslope [e.g.,
Scherrer and Naef, 2003] and small catchment scale
[Schmocker-Fackel et al., 2007]. Tetzlaff et al. [2007] also
showed that generalized hydrological properties of soil types
can serve as predictive landscape controls on runoff dynamics even in larger catchments (>10 km2), using the U.K. Hydrology of Soil Types (HOST) classification [Boorman
et al., 1995]. However, in larger river systems that encompass different geological provinces, bedrock characteristics
have been shown to be a dominant landscape control on runoff generation processes [e.g., Didszun and Uhlenbrook,
2008; Fröhlich et al., 2008; Gleeson and Manning, 2008;
Tetzlaff and Soulsby, 2008; Tetzlaff et al., 2011]. Studies
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that seek to integrate tracers into modeling processes at
larger catchment scales thus need to identify the first-order
landscape controls to delineate the different REWs.
[6] The aim of this study is to explore the utility of both
time domain and source area tracers in the larger, nested
catchments of the North Esk in Scotland, which has two
distinct hydrogeomorphologic/hydroclimatic provinces.
This builds on previous work where the spatial and temporal dynamics of runoff in the catchment were examined
using a range of environmental tracers [Capell et al., 2011,
2012]. Catchment geology together with landscape evolution history (glaciation, soils, topography), were shown to
be the most important factors controlling large-scale runoff
dynamics and hydrochemistry. This results in shallow responsive soils overlying low permeability bedrock in the
wetter, cooler uplands giving a rapid runoff response. In
contrast, deeper, more freely draining soils facilitate
recharge of sandstone aquifers in the drier, warmer lowlands, increase baseflows, and attenuate storm hydrographs
[Capell et al., 2011, 2012]. Based on these previous findings, the specific research objectives are: (1) to develop parsimonious tracer-aided rainfall-runoff models based on
landscape heterogeneity with a simple division into two
major hydrological units representing the uplands and lowlands; (2) to identify appropriate parameter ranges and
model structures representing dominant runoff processes in
these different landscapes using tracer constraints (with alkalinity as a source area tracer and deuterium as a time domain
tracer); and (3) to combine the upland and lowland models
to simulate flow and tracer dynamics at a larger catchment
scale, using this as a learning framework to explore the
potential and problems associated with such tracer-aided
upscaling in more complex, heterogeneous landscapes.
2.
Study Sites and Data
[7] The 749 km2 North Esk catchment in northeast Scotland combines contrasting lowland and upland landscape
characteristics. An abrupt change between landscape units
is caused by a major geological discontinuity that separates
the northern mountains of Scotland from the lowlands, and
results in a distinct geomorphologic separation in the study
catchment (Figure 1 and Table 1). Along with the North
Esk outfall, two of its subcatchments were investigated :
the 56 km2 Water of Mark (upland catchment) and the
131 km2 Luther Water (lowland catchment).
[8] A detailed description of the study area is given by
Capell et al. [2011]. In brief, the mountainous western parts
of the North Esk—including the Water of Mark—have a
maximum elevation of 932 m a.s.l. The eastern part of the
North Esk with the Luther Water catchment drains the lowlands. The upland geology is dominated by granitic and
metamorphic bedrock, with low permeability apart from
fractures [cf. Haria and Shand, 2004]. In contrast, the sedimentary bedrock of the lowlands, predominantly sandstones and conglomerates, date from the lower Devonian
and form an important regional aquifer system. This aquifer
is actively recharged and chlorofluorocarbon (CFC) dating
resulted in estimated average groundwater ages of up to
Dochartaigh et al., 2006].
40 years [O
[9] The distribution of soils is strongly influenced by geology and topography. The uplands are dominated by peats
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Figure 1. (a) Catchment topography with simplified stream network and sampling sites. Subcatchments outlined (b) bedrock geology and quaternary sediments. Scotland overview map with location of
the study catchment inset.
and peaty soils on higher elevation plateaus and upper hillslopes and mineral podzols on lower slopes. Lowland
soils are predominantly brown forest soils and mineral podzols. The HOST mapping scheme [Boorman et al., 1995]
was used in previous studies to classify dominant flow
paths based on soil cover, which proved to be an effective
predictor for regionalization at the catchment scale
[Soulsby et al., 2006a; Tetzlaff et al., 2009]. This scheme
distinguishes hydrologically responsive soils with dominantly lateral flow paths and little recharge to depth, from
more freely draining soils with dominantly vertical flow
paths that recharge groundwater. Under the prevailing cool
and wet climate conditions, responsive soils are usually at,
or near to, saturation, and saturation excess overland and
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Table 1. Summary of Physical and Hydrological Catchment
Properties for the Three Studied Catchments
Uplands
River name
Area (km2)
Lowlands
Outlet
Water of Mark Luther Water North Esk
56
131
749
Topography
600
257
932
11.2
66
146
24
518
5.8
51
310
12
932
8.5
78
Bedrock Geology
–
78.0
22.0
79.8
17.5
2.7
37.7
55.3
7.0
10.9
53.9
–
3.6
21.8
14.5
85.4
25.3
23.8
7.5
9.4
19.9
46.5
53.4
Hydrometry (October 2004–October 2009)
Qmean (mm d1)
1.4
0.9
1
Q95 (mm d )
0.54
0.27
1
Q5 (mm d )
8.7
4.0
Mean annual sum (mm)
1066
502
1.3
0.37
6.4
752
Precipitation (mm)
October 2008–October 2009
1256
1165
Mean elevation (m)
Min. elevation (m)
Max. elevation (m)
Mean slope ( )
Max. slope ( )
Sedimentary (%)
Metamorphic (%)
Igneous (%)
Dominant Soils (HOST)
Peaty podzol (%)
25.6
Brown forest soil (gleying) (%)
–
Eroded peat (%)
31.0
Blanket peat (%)
34.0
Humus iron podzol (%)
9.0
Sum responsive soils (%)
97.3
Sum freely draining soils (%)
2.8
1161
near surface lateral flows dominate storm runoff generation
and produce flashy storm hydrographs (Figure 2a). Freely
draining soils dominate in areas that facilitate significant
groundwater recharge and large volumes of aquifer storage,
which produces longer mean transit times and attenuates
the runoff response (Figure 2b). Responsive soils cover
97% of the upland catchment (Water of Mark), whereas
freely draining soils (85%) dominate the lowland catchment (Luther Water). The North Esk catchment integrates
both landscape types resulting in 53% freely draining and
47% responsive soils, respectively, and the streamflow
response reflects this (Figure 2c).
[10] Five years of daily hydrometric and climatic data
(October 2004 to September 2009), which included one
year with weekly tracer data (October 2008 to September
2009), were used. Discharge data were provided by the
Scottish Environmental Protection Agency (SEPA). Precipitation data were available from the British Atmospheric
Data Centre (BADC) at a daily resolution from nine gauges
in and around the catchment (maximum distance : 10 km).
For the lowland areas, a gradient-inverse-distance-squared
(GIDS) interpolation [Hrachowitz et al., 2009] was used to
estimate daily catchment-wide precipitation. In the upland
areas, interpolation results were unreliable because of a
lack of measurements at higher elevations. Therefore, station data was used with a simple altitudinal correction to
ensure reasonable water balance. Given the size of the
study catchments and the subdued terrain in the lowland
areas, regional groundwater flows may be an additional
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source of error in the catchment water balance, though the
geological structure of the aquifer (a syncline with a NE–
SW axis) suggests such errors are probably small. Daily air
temperatures from two sites in the North Esk covering the
upland–lowland elevation gradient were available through
the BADC. The average of both stations was used as catchment temperature at the outlet. Daily actual evapotranspiration ET (mm), estimated with a Penman-Monteith equation
adjusted to aerodynamic and canopy roughness characteristics, was available from the U.K. Environmental Change
Network (ECN) at Glensaugh, in the northeast of the catchment. To account for higher temperatures and land use differences in the lowland areas, annual ETa estimates from
the Met Office Rainfall and Evaporation Calculation System (MORECS) were used to correct the evapotranspiration for lowland areas.
[11] Weekly stream tracer samples were analyzed for
stable water isotopes (2H), alkalinity, and major ion concentrations. 2H was analyzed using a laser spectrometer
(DLT 100, Los Gatos Research) at a precision of 62 %
applying five standards for calibration. The 2H signatures
are presented as (%) concentration derived from Vienna
standard mean ocean water (VSMOW). 2H concentrations
in precipitation were available from two sites located north
and south of the North Esk basin [Birkel et al., 2011b;
Speed et al., 2010] and weekly averages were used as precipitation 2H input signature for modeling purposes. The
2H precipitation signatures of the subcatchments were further corrected to average stream signatures, which showed
a close correlation to mean catchment elevation [Capell
et al., 2012]. Stream water alkalinities were determined by
acidimetric titration [Neal, 2001]. The collected stream
water samples were also analyzed for major ion content
using inductively coupled plasma mass spectrometry and ion
chromatography. Stream water alkalinities and 2H signatures were used in the development and evaluation of simple
conceptual models, as the alkalinity summarized much of
the variation of other ions [Capell et al., 2011, 2012].
3.
Model Development
3.1. Translating Perceptual Catchment Understanding
Into Conceptual Rainfall-Runoff Models
[12] Simple, lumped water balance models were developed to simulate daily rainfall-runoff dynamics in the contrasting landscapes (as REWs) of the North Esk. The model
structures were extended in a stepwise fashion to determine
the lowest level of complexity necessary to capture the
main processes driving runoff generation following a ‘‘topdown’’ approach. The selected model components and configurations are summarized in Table 2; this shows the water
balance and tracer model equations ordered by the flow of
water through the model, as well as model parameters used
in each component and the performance measures for each
model in the upland and lowland catchments. In addition to
rainfall-runoff data, tracer data helped conceptualize the
dominant runoff generation processes in relation to different landscape types.
[13] In all tested model configurations, a degree-day
snowmelt routine was implemented to account for snow
accumulation and melt events during the winter season
(Table 2, equation (1)). The snow module is similar to the
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Figure 2. Rainfall-runoff response in (a) upland Water of Mark, (b) lowland Luther Water, and (c)
outlet of the North Esk catchment.
HBV model snow routine [Bergström, 1975]: precipitation
is accumulated as snow if temperature T ( C) falls below a
threshold value Tthresh ( C), usually close to 0 C, and melt
water is released from the snow pack according to a factor
Dmelt (mm C1 d1), forming effective precipitation input
Peff to the runoff generation modules. A proportion of the
meltwater is retained in the snow pack and allowed to
refreeze if temperatures fall below Tthresh. To minimize
model parameterization, the maximum volume of retained
water is fixed at 10% of the snow pack and the maximum
refreezing volume at 5% of resident meltwater. These values are based on previous studies using the HBV model
[Seibert, 2000].
[14] For the runoff generation routines in both the
uplands and lowlands, model development started from a
simple linear reservoir model (Model 1 in Table 2), with
discharge Q being proportional to the active catchment
storage S and the runoff coefficient k (Table 2, equation
(8)). This initial structure was iteratively tested against
observed hydrometric data and then extended with the aim
of better capturing the observed discharge dynamics, which
were evaluated with a range of performance measures.
Besides single linear storage concepts, different model configurations with two runoff-generation storage components
were also evaluated. Both included a linear base flow storage (Table 2, equation (8)); however, for the generation of
fast runoff components (Q1) from shallow storages (S1), single linear (Model 2) and nonlinear storages (Model 3) were
alternatively implemented (Table 2, equations (4)–(6)).
Evapotranspiration ET is subtracted sequentially starting with
evaporation from the snow pack (if present), followed by the
resident shallow storage volumes. In the configuration using a
soil storage Ss, which explicitly conceptualizes shallow water
held by the soil matrix, this serves as a reservoir for ET.
[15] The models were further extended to assess whether
conceptualization of assumed dominant runoff processes
indicated from previous tracers studies and the catchment
soil (HOST classes) and geology characteristics could
improve their performance. Thus, in Model 4, an additional
soil storage Ss, accounting for the predominance of deeper
soils that mainly contribute to groundwater recharge in the
lowlands, was implemented (Table 2, equation (2)). Precipitation only enters the runoff-generating storage components if the soil storage capacity Ss,max (mm) is exceeded.
Previous tracer studies inferred such a mechanism with
high concentrations of geochemical tracers, even during
peak flows, suggesting a strong influence of pre-event
water in these areas [Capell et al., 2011]. The assumption
of linear shallow storage, however, turned out to be insufficient to capture observed peak flow dynamics. These peak
flows were most likely dominated by fast translatory flow
through near surface storages aided by agricultural drains
[Birkel et al., 2011b]. These lowland shallow runoff sources were, therefore, conceptualized as the nonlinear shallow storage implemented in Model 3 (Table 2, equation
(5)) and retained in Model 4. A final model (Model 5)
included a conceptualization of saturation overland flow
(SOF) (Table 2, equation (3)) to explore the potential
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(5)
(7)
(8)
Q1 ¼ kS S1 (mm d1)(solved analytically)
S2 ðtÞ ¼ S2 ðt 1Þ Q2 ðt 1Þ þ R t (mm d1)
Q2 ¼ kG S2 (mm d1) (solved numerically)
Alkalinity Concentration Equilibrium Routine
yðtÞ ¼ ðyðt 1Þ y0 Þe þ ð1 e Þ þ y0 (eq l1)
Linear shallow storage runoff
Baseflow storage
Linear baseflow storage runoff
Bounded exponential growth model
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(14)
pS2 (mm)
4
5
0.28
0.27
0.29
0.19
0.28
0.32
0.49
0.36
0.24
0.00
0.33
0.35
0.36
0.35
0.40
0.43
0.45
0.44
0.38
0.04
0.44
0.38
0.41
0.45
0.43
0.59
0.50
0.60
0.51
0.04
0.46
0.59
0.53
0.53
0.44
0.71
0.68
0.72
0.63
0.44
SOF only present in the uplands model
Implemented as passive mixing volumes
in each storage component
0.53
0.63
0.55
0.64
0.60
0.62
0.62
0.66
0.44
0.19
Peff will be updated sequentially in soil buffer
if snow melt is generated
Implemented in shallow and baseflow storages,
with y0 representing the input concentration
of Peff and R in S1 and S2, respectively
3
a
Peff and ET were updated sequentially while passing through the model routines. Models 4 (lowlands) and 5 (uplands) are the configurations chosen for the study catchments. The equations are presented in the
order of flow through the model structures, corresponding to the conceptual model structures (Figure 3).
Model Maximum Performance With Uplands Data for the Calibration Period (October 2008–September 2009)
Model Maximum Performance With Lowlands Data for the Calibration Period (October 2008–September 2009)
(15)
(13)
pS1 (mm)
(12)
(11)
(10)
(9)
(6)
2
1
Model Number
CAPELL ET AL.: CAN TRACERS REDUCE UNCERTAINTY IN RAINFALL-RUNOFF MODELS?
NSE
log(NSE)
R2
Alkaline R2
2H R2
NSE
log(NSE)
R2
Alkaline R2
2H R2
Runoff signatures from SOF
QSOF ðtÞ ¼ Peff ðtÞ (%)
Peff ðtÞPeff ðtÞ þ S1 ðt 1Þ S1 ðt 1Þ þ pS1
Runoff signatures from shallow storage Q1 ðtÞ ¼
(%)
Peff ðtÞ þ S
1 ðt 1Þ þ pS1
S1 ðtÞR þ S2 ðt 1Þ S2 ðt 1Þ þ pS2
Runoff signatures from baseflow storageQ2 ðtÞ ¼
(%)
R þ S2 ðt 1Þ þ pS2
QSOF ðtÞQSOF ðtÞ þ Q1 ðtÞQ1 ðtÞ þ Q2 ðtÞQ2 ðtÞ
Combined runoff signatures
QðtÞ ¼
(%)
QSOF þ Q1 ðtÞ þ Q2 ðtÞ
Peff ðtÞ ¼
Isotope Mixing Model Routine
PðtÞPðtÞ þ SNOWðt 1ÞSNOWðt 1Þ
(%)
Peff ðtÞ þ SNOWðt 1Þ
PðtÞPðtÞ þ Peff ðt 1ÞSs ðt 1Þ
Peff signatures from soil buffer storage Peff ðtÞ ¼
(%)
Peff ðtÞ þ Ss ðt 1Þ
Peff signatures in snow pack
y0 (meq l1) (meq l1) (d1)
kG (d1)
kS (d1)
kS (d1) aS (-)
(3)
(4)
Shallow storage
SOF runoff
Soil buffer storage
(2)
(1)
Q1 ¼ kS S11þaS (mm d1)
Tthresh ( C) Dmelt
(mm C1 d1)
Parameter
Nonlinear shallow storage runoff
Snow pack retained if T < Tthresh , melt computed with: Peff ðtÞ ¼ Dmelt ðT Tthresh Þ
Model Equation
Runoff Generation, Storage Routine
Ss,max (mm)
Ss ðtÞ ¼ Ss ðt 1Þ þ Peff ðtÞ ETðtÞ t (mm d1) Peff ¼ P ðSs;max Ss Þ=t (mm d1)
1þaSOF
1
S1
cSOF ¼ 1 1 Smax
(-), QSOF ¼ cSOF Peff (mm d )
aSOF (-) Smax (mm)
1
S1 ðtÞ ¼ S1 ðt 1Þ Q1 ðt 1Þ þ ðPeff ðtÞ QSOF Þ ETðtÞ R t (mm d )
R (mm d1)
Day-degree snow routine
Module
Equation
Number
Table 2. Model Equations and Configurations Used in the Stepwise Development Process, and Performance Measures for Model Evaluationa
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CAPELL ET AL.: CAN TRACERS REDUCE UNCERTAINTY IN RAINFALL-RUNOFF MODELS?
importance of shallow lateral flows relating to the high proportion of responsive, thin soils in these areas. SOF runoff
is generated from Peff (mm) depending on the volume in
the shallow storage S1 (mm). The fraction of Peff contributing to SOF, cSOF (-), is determined as a power function
with the maximum storage parameter Smax (mm) and exponent aSOF (-). This gives the SOF runoff QSOF (mm) at each
time step. The remaining fraction of Peff is routed to a linear shallow runoff store S1.
[16] In the upland areas, previous tracer-based investigations revealed a dynamic contribution of surface or nearsurface lateral flow paths to stormflow but also a base flow
component with fairly long turnover times [Capell et al.,
2011, 2012]. The model development utilized this information as substantially under-estimated peak flows, or base
flow depletion to zero, were simulated using for standard
linear and nonlinear storage model structures (Models 1–3).
This is reflected in poor overall performances of the rainfallrunoff modules for these model configurations (Table 2) and
necessitated the integration of a separate, saturation-dependent runoff generation parameter (Table 2, equation (3)) [Seibert et al., 2003; Savenije, 2010]. This also improved the
simulations of alkalinity and deuterium. The overall efficiency statistics for flow in the best models were still quite
low, but this mainly reflected a crude approximation of
snowmelt in topographically complex catchments like the
Water of Mark [Kling and Nachtnebel, 2009].
[17] Similarly, whereas the standard model configurations
(Models 1–3) performed better for the lowland catchment
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(Table 2), the incorporation of the soil storage parameter
(Table 2, equation (2))—which was also informed by previous observations of tracer dynamics [Capell et al., 2011,
2012]—resulted in much better performance statistics for
flow simulations. This store was needed to filter precipitation
inputs and required to provide a threshold before runoff could
be generated by upper and lower storages for storm runoff and
base flow generation. The parameter also improved alkalinity
predictions significantly and was essential in getting the deuterium variations close to measured values. The main failure in
the flow simulations was the prediction of runoff peaks during
the rewetting period at the end of the summer; however, it is
likely that rainfall measurement errors in summer convective
events are a reason for this, as well as model errors.
[18] Figure 3 shows a conceptual diagram of the final
model structures accepted for lowland (Model 4) and
upland areas (Model 5). The models differentiate components representing near-surface storages and groundwater
from deeper sources with longer flow paths. As discussed
above, these were selected based on model performance
evaluation, which is consistent with the landscape characteristics and the perceptual models developed from the a priori
tracer analyses. For modeling at the catchment outlet, the two
model structures were coupled and precipitation distributed
according to the proportion of uplands and lowlands. The parameter sets for the coupled model were restricted to ranges
identified as behavioral in the subcatchment models. The models were implemented using explicit forward approximation.
An exception was the use of an analytical solution for the
Figure 3. Conceptual model structures for uplands (a) and lowlands (b). Both structures incorporate a
simple snow accumulation and melt module, active below a threshold air temperature (TT), and a linear
base flow storage (S2). The upland model conceptualizes saturation excess overland flow (QSOF) as a
function of the linear shallow storage (S1), whereas the lowland model features an additional soil storage
(Ss) and a nonlinear shallow storage (S1). Both model structures use additional passive storage volumes
for isotope tracer mixing (pS1 and pS2), which are not connected to the runoff generation.
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shallow linear storage Q1 (Table 2, equation (6)), as it was
assumed this might improve simulations of processes where
rapid subdaily runoff dynamics were observed. Although subsequent testing indicated that this did not result in significant
improvements in the modeled runoff response, the analytical
solution was retained for the final model.
[19] A recession analysis of both upland and lowland
catchments revealed little temporal variability in behavior
for the low-flow end of the discharge recessions, which
underpins the justification for two-storage model structures
[Capell et al., 2012]. Master recession curves [Lamb and
Beven, 1997; Fenicia et al., 2006] were constructed using
5 years of discharge data selecting all recession events with
a minimum length of 6 days (Figure 4). In both catchments,
exponential decay functions were fitted with coefficients of
determination R2 of 0.99 (uplands) and 0.96 (lowlands),
respectively. Based on these, the low-flow behavior in both
model structures was conceptualized with linear groundwater storages S2 (Table 2, equations (7) and (8)) generating base flow from deeper groundwater sources. The
regression parameters derived from the master recession
curves were used in the model structures to constrain acceptable parameter ranges for the base flow storage coefficients kG (d1) in both landscapes (Table 3). In both
models the recharge from the shallow storage S1 into S2 is
also conceptualized as recharge parameter R (mm), assuming a constant recharge rate. Base flow runoff from these
storages is computed identically to the shallow storage in
the upland model (Table 2, equation (4)).
3.2. Tracer Incorporation
[20] Previous analysis of tracer dynamics [Capell et al.,
2011, 2012] helped identify important runoff generation
mechanisms at larger scales and guide their conceptualization into simplified lumped models. To further evaluate
appropriateness of the structures and identify feasible parameter ranges, tracer transport modules were also integrated into the modeling. A recession function between the
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geographic source area tracer alkalinity and the timing of
the discharge recession was implemented to simulate stream
water alkalinities and for model evaluation [Table 2, equation (9)]. The alkalinity recession characteristics were used
to constrain parameter ranges, in the same way as the discharge recession parameters. Alkalinity is a proxy for acid
neutralization capacity (ANC) and has been widely used as
a source area tracer in Scottish upland catchments [e.g.,
Soulsby et al., 2003; Tetzlaff et al., 2008; Birkel et al.,
2011b]. Alkalinity correlates with the concentration of
weathering-derived solutes, thus is highest during low flows
representing water from sources with longer contact times
with minerogenic materials in the subsoil and aquifers.
Lowest alkalinities occur during high flows when sources
like lateral flow from peats and thin acidic soils with short
contact times dominate the runoff. Thus, alkalinities can
vary substantially between different landscape types and it
is a useful tracer for detecting the temporal and spatial variation of runoff sources. In the North Esk, stream alkalinities
ranged from 29 meq l1 to 552 meq l1 in the upland catchment and 670 meq l1 to 1547 meq l1 in the lowland catchment (Table 4). The higher stormflow alkalinities at the
lowland site are indicative of the dominance of subsurface
flow paths. Previous analysis of long-term data showed
strong correlation of alkalinity and runoff in upland rivers
[Tetzlaff et al., 2008]. Reasonable power law regressions
representing flow-concentration relationships were found in
the 1 year weekly data for up- and lowlands (R2 of 0.72 and
0.62, respectively) (Figure 5). These were used similarly to
the master recession curves for discharge, to establish
stream water alkalinity recession curves for the study period
(October 2008 to September 2009) (Figure 5). A bounded
exponential growth model was found to be a good approximation for the temporal dynamics of stream alkalinities y(t)
(meq l1). A regression model of the form:
yðtÞ ¼ y0 þ ð1 et Þ
Figure 4. Master recession curves using 5 years daily discharge time series. (a) Upland catchment
using 27 recession events, and (b) lowland catchment using 26 recession events. The exponential regression functions were fitted to the well-confined lower end of the curves, below the dotted horizontal lines.
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Table 3. Regression Parameters of Empirically Derived Alkalinity Relationships Including Standard Errors, Passive Mixing Volumes
Necessary to Fit the 2H Stream Signatures, and Model Performance Measures
Parameters/
Performances
Upland
Lowland
Coupled Model
at Outlet
kG (d1)
y0 (meq l1)
(meq l1)
(d1)
pS1 (mm)
pS2 (mm)
NSE
VE
log(NSE)
R2
Alkaline R2
2H R2
NSE
0.037 6 0.01
119 6 6
386 6 20
0.07 6 0.01
0
>100
0.53 (0.45)
0.57 (0.5)
0.63 (0.5)
0.55 (0.45)
0.64 (0.3)
0.60 (0.4)
0.46
0.027 6 0.01
950 6 12
346 6 36
0.08 6 0.02
>100
>200
0.71 (0.6)
0.73 (0.6)
0.68 (0.6)
0.72 (0.6)
0.63 (0.3)
0.44 (0.15)
0.62
–
–
–
–
–
–
0.74
0.70
0.74
0.74
0.59
0.57
0.68/0.68
Empirically derived baseflow storage coefficient
Alkalinity equilibrium model
Deuterium transport model
Model maximum performance (performance thresholds)
Cross-evaluation (upland/lowland, outlet with both)
Table 4. Summary Statistics for Deuterium and Alkalinity for the Sampling Year 2008 to 2009
Alkalinity (meq l1)
Uplands
Lowlands
Outlet
2H (%)
Precipitation
Uplands
Lowlands
Outlet
n
Mean
Minimum
Maximum
Standard
Deviation
95th
Percentile
5th
Percentile
51
49
51
235
1123
573
29
670
253
552
1547
745
109
184
132
376
1329
733
44
784
298
51a
51
49
51
50.3
57.0
52.8
55.2
94.3
73.2
58.3
67.1
12.6
49.6
48.5
45.7
18.8
4.3
2.0
3.4
84.2
63.0
55.6
58.7
25.0
53.6
49.9
51.3
a
Volume-weighted composite samples from two stations around the catchment.
Figure 5. Empirical recession functions between stream water alkalinity and time during flow recession periods (period October 2008 to September 2009). (a) Upland catchment using seven recession
events, and (b) lowland catchment using eight recession events. Inset: concentration–discharge relationship of weekly samples. A bounded exponentially growth model was used for fitting the alkalinity recessions, a power law model for the concentration–discharge relationships.
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was fitted, with y0 (meq l1) being the initial alkalinity as
precipitation enters the subsurface, (meq l1) being the
long-term equilibrium alkalinity, and (d1) giving the
coefficient of increment toward equilibrium. In the hydrological model, a daily analytical solution of the bounded
exponential growth was implemented using the same parameters (equation (9) in Table 2).
[21] At each time step, alkalinity of Peff entering the subsurface storages is initialized as y0, as the storage volumes
are updated according to equation (7) in Table 2 and mixed
with incoming Peff and recharge R. The exponential growth
model (Figure 5) approximates alkalinity recession dynamics very well (R2 of 0.99 (uplands) and 0.96 (lowlands)).
However, some of the alkalinity variation is masked by the
estimation of daily alkalinities using the concentration discharge relationship and the subsequent loss of variability
found in the measured weekly samples. Nevertheless, the
samples covered a wide range of high and low flows—
therefore, viewed as representative of the overall dynamics—and the obtained parameters were then used in the
model structure to constrain parameter ranges (Table 3).
[22] The model was extended to facilitate the mixing and
routing of the stable isotope 2H as a time domain tracer
through the storages between precipitation entering and discharge leaving the system. This was used to estimate the
total storage volumes of the catchment models. 2H dynamics are characterized by strong intra-annual variability in
precipitation (standard deviation of 18.8 % for the year).
This is attenuated in the stream water signatures caused by
internal mixing processes (Table 4). Variability of 2H in
stream water varies spatially, with stronger attenuation
where larger mixing volumes and longer transit times are
involved in runoff generation. In the lowland catchment,
the attenuation of 2H in stream water (2.0 % std. dev.) is
much stronger than in the uplands (4.3 % std. dev.) indicating the influence of larger subsurface storage volumes.
These dynamics were used as an independent tool to evaluate
the storage mixing volumes necessary to be able to simulate
the isotope response in the stream. Passive (i.e., not contributing to runoff generation) mixing volumes pS1 and pS2, freely
parameterized, were assigned to the linear and nonlinear storages (Table 2, equations (13) and (14); Figure 3) to incorporate the mixing of 2H signatures with modeled stored water
volumes.
[23] The mixing model routine is defined in equations
(10) to (15) of Table 2. The 2H signatures of modeled storage and runoff components in are denoted in deviations
from VSMOW (%): P (in precipitation, model input variable), Ss (in soil buffer storage), Peff (effective precipitation from snow pack and soil buffer), S1 (in shallow
storage), S2 (in base flow storage), QSOF (in SOF runoff),
Q1 (in shallow storage runoff), Q2 (in base flow storage
runoff), and Q (modeled runoff). All storage component
and internal flow signatures were recalculated iteratively in
each time step using the volumetric model components,
adding the passive mixing volumes for shallow and base
flow storages. The isotope mixing equations were solved
numerically in the model code. For this study, the 2H was
considered to be conservative; no fractionation processes
were included and instantaneous and complete mixing of
isotope tracers in each storage compartment was assumed
[Dunn et al., 2010; Birkel et al., 2011b]. The model
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storages were initialized as being empty and a 3 year
warm-up period was run to fill and equilibrate the model
storage volumes. The 2H signatures of the model storages
(Ss, S1, S2) were initialized with the average 2H signature
measured in stream water during the 1 year tracer survey, a
method commonly used in transit time modeling [e.g., Hrachowitz et al., 2011].
4. Model Calibration and Parameter
Identifiability
[24] Model performance was assessed using multiobjective, multicriteria evaluation based on Monte Carlo (MC)
random sampling (106 realizations) during the calibration
year 2008–2009. The performance of the runoff dynamics
was evaluated using four objective evaluation criteria: the
Nash-Sutcliffe efficiency NSE [Nash and Sutcliffe, 1970]
together with the NSE of log-scaled runoff, log(NSE), the
coefficient of determination R2, and the volumetric efficiency (VE) [Criss and Winston, 2008], which results in an
unbiased evaluation of volumetric errors as runoff volume
deviations are not weighted depending on the flow conditions (Table 2). The performance of the alkalinity and isotope tracer dynamics were assessed using the coefficient of
determination R2, thus focusing on the model performance
of the simulated temporal dynamic and accounting for systematic concentration biases (Table 2).
[25] Model calibration was based on MC sampling of parameter sets. Initially, MC sample sets were computed
applying random sampling from wide parameter ranges
using a uniform distribution (Table 3, Table 5). The model
performances derived from these MC samples were used as
an objective measure during the iterative testing of model
configurations. Following the selection of the upland and
lowland model structures based on the evaluation criteria
and prior knowledge, a stepwise integration of multicriteria
evaluation measures [Khu et al., 2008] was then used to
evaluate the performance of the parameter combinations of
the two models using: (1) the runoff dynamics, (2) the
additional tracer performances, and (3) the parameter
ranges from the independently derived recession functions
for base flow storage coefficients and the alkalinity equilibrium model (Table 5). Efficiency performance thresholds
were subjectively established for all four rainfall-runoff criteria to identify behavioral parameter sets with acceptable
scores for each. The critical threshold value was set as 0.6
for all criteria in the lowlands, and as 0.5 (VE, log(NSE))
and 0.45 (NSE, R2) corresponding to the weaker maximum
performances of the upland model (Table 4). These probably reflect the complex nature of the Water of Mark catchment with marked environmental gradients and topographic
complexity (Figure 1), which resulted in large uncertainties
over precipitation inputs and other hydroclimatic drivers
including snowmelt. The resulting parameter sets were further constrained by introducing additional information to the
performance evaluation procedure, seeking to reduce the
subjectivity in the selection of behavioral parameter sets
[Seibert and McDonnell, 2002]. For this, tracer performance
criteria and the parameter ranges established by the alkalinity-recession functions were used. Tracer model performance was evaluated using threshold criteria (Table 3).
Behavioral parameter ranges for the storage coefficients kG
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Table 5. Model Parameters for Upland and Lowland Model Structures With Parameter Ranges for Initial Monte Carlo Simulations and
Accepted Performance Envelopes Under Rainfall-Runoff Performance and Additional Tracer Performance Constraintsa
Upland
Lowland
Initial
Parameter
Sampling
Range
Parameter
Range,
Rainfall-Runoff
Constrained
Parameter
Range,
Constrained
by Recession
Functions
Initial
Parameter
Sampling
Range
Parameter
Range,
Rainfall-Runoff
Constrained
Parameter
Range,
Constrained
by Recession
Functions
Tthresh ( C)
Dmelt (mm
1 1
C d )
2–3
0.1–7
1.2–3
1.1–7
Snow Module
1.3–3
1.5–6.6
2–2
0.1–7
2–2
0.1–7
1.9–1.65
0.12–5.4
Smax (mm)
aSOF (-)
aS (-)
Ss,max (mm)
kS (d1)
R (mm d1)
kG (d21)
10–400
0–5
0.001–0.5
1–10
0.0001–0.15
47–400
0.0003–5
n.a.
n.a.
0.09–0.35
1.9–10
0.001–0.05
0.09–0.32
2.3–10
0.001–0.04
0–1
1–1000
0.0001–0.4
0.1–10
0.00002–0.3
n.a.
n.a.
0.001–0.99
1–1000
0.001–0.25
1.1–9.9
0.004–0.049
0.14–0.83
8–970
0.01–0.15
3.5–9.0
0.004–0.017
y0 (meq l21)
a (meq l-1)
b (d21)
25–200
100–700
0.02–0.11
25–200
100–700
0.02–0.11
Alkalinity Module
107–131
346–424
0.05–0.09
500–1500
150–750
0.04–0.12
500–1500
150–750
0.04–0.12
926–974
274–418
0.04–0.11
0–500
10–1000
0–500
10–1000
Deuterium Storage Module
0–498
84–997
1–1000
10–5000
3–997
10–2250
134–997
168–2100
Model
Parameter
pS1 (mm)
pS2 (mm)
Runoff Generation Module
116–400
0.0003–5
a
Parameters that are constrained by empirical recession functions are highlighted in bold.
and the alkalinity model parameters y0, , and were constrained using the empirically derived regression parameters
with boundary estimates based on the regression standard
errors (Table 3). The resulting parameter space is given in
Table 5.
[26] The rainfall-runoff performance of the final model
structures was considered reasonable with maximum NSEs
of 0.71 in the lowlands and 0.53 in the uplands (Table 2).
Evaluation MC runs in the hydrological year preceding the
sampling period (October 2007–September 2008), yielded
similar maximum performances with the identified parameter ranges. A cross-evaluation, forcing the uplands and lowlands model structures with data from the other respective
subcatchment decreased the model performances (maximum NSE: 0.62 in the lowlands and 0.44 in the uplands)
and confirmed the importance of the specific model structures for different catchment landscapes (Table 3). The
ability of both models to simulate the observed tracer dynamics was also regarded as reasonable [maximum R2 of
0.64 and 0.63 (alkalinity), 0.44 and 0.60 (2H) in lowlands
and uplands, respectively]. The weaker performances of the
modeled tracer dynamics were accepted given the simplifying conceptualizations necessary to integrate the tracers
into the lumped model schemes and the lower information
content of the measured data because of the lower sampling
density of the weekly tracer survey.
[27] Parameter identifiability under increasing constraints was assessed using cumulated efficiencies of the
sampled parameter space [Hornberger and Spear, 1981].
For the rainfall-runoff performance, an average efficiency
score E was calculated from the averaged scores of NSE,
log(NSE), and R2. For alkalinity and 2H model performances, R2 scores were used directly:
E¼
NSE þ log ðNSEÞ þ R2
3
[28] The cumulative efficiencies were calculated as the
accumulated, normalized difference between the efficiency
score and the chosen threshold values. They represent empirical cumulative distribution functions of the efficiency
score. Figures 6 and 7 indicate increasing parameter identifiability with added constraints for the majority of the
model components for the uplands and lowlands, respectively. Steep segments of the cumulative curves represent
high-efficiency scores for the parameter value given on the
x axis, whereas plateaus indicate poor efficiency scores.
[29] In the uplands (Figure 6), the multicriteria calibration
increases the identifiability of the model parameters. Alkalinity parameters are only constrained after the introduction
of the recession function, but this, in turn, constrains other
parameters; notably, recharge R and the nonlinearity factor
for the generation of SOF runoff, aSOF. The parameterization
of the 2H mixing model is generally poorly constrained in
the upland catchment, the occurrence of two-steep segments
in the shallow passive storage Sp, however, indicates a bimodal distribution with highest performances at small passive storages, reflecting strong short-term dynamics in the
2H response, and at large passive storages consistent with a
longer-termed damped response.
[30] In the lowland catchment (Figure 7), the snow
module parameter Tthresh and Dmelt show no sensitivity to
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Figure 6. Cumulated efficiency C (-) of parameter ranges under increasing constraints (sensitivity
analysis) for the upland subcatchment. Bold lines show the full sampled parameter space, thin lines
include a runoff performance threshold, and dashed lines include constraints derived from empirical
recession functions.
performance constraints as snow is much less significant
here compared to the uplands. After constraining the rainfall-runoff criteria the identifiability of base flow coefficient kG and the recharge R increases noticeably. The
constraints added by the recession functions enhances the
identifiability of kG and R. Additionally, the identifiability
of maximum soil storage Ss,max is greatly increased below
200 mm, probably because of short term variability
observed in the stream water 2H signature. The 2H model
shows a bimodal distribution for pS1, comparable to the
uplands. However, slow performance increase at low values
for pS1 indicates that below 100 mm passive storage the
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Figure 7. Cumulated efficiency C of parameter ranges under increasing constraints (sensitivity analysis) for the lowlands subcatchment. Bold lines show the full sampled parameter space, thin lines include
a runoff performance threshold, and dashed lines include constrains derived from empirical recession
functions.
2H is not sufficiently damped. Thus, minimum transit times
even at high flows are longer than in the uplands where the
2H model performs equally well with smaller passive
storages.
[31] For the catchment outlet, where outputs from the
contrasting REWs are integrated in the observed runoff dynamics, the two model structures were coupled and parameterized using the constrained parameter ranges identified
in the two subcatchments (Table 5). Percentage geology
was chosen to distinguish between the two main landscape
units. Precipitation input was then distributed between both
modeled regions based on the proportion of sedimentary
bedrock in the lowlands (38%) and metamorphic and igneous bedrocks in the uplands (62%). The distribution parameter was allowed to vary 65% but no identifiable performance
difference was detected. MC simulations (106 random samples) were used to evaluate the model performance. The performance of the coupled model with the a priori constrained
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parameter ranges (Table 5) is comparable (maximum NSE
0.74) to those of the models in up- and lowlands. The tracer
model performances are also reasonable (maximum R2 of
0.59 (alkalinity) and 0.57 (2H)). A cross-evaluation using
both model structures separately to model the outlet runoff
gave maximum NSE values of 0.68 for each, which is an acceptable if slightly poorer performance but does not account
for the known differences in runoff processes in the contrasting catchment parts.
5.
Modeled Hydrographs and Tracer Responses
[32] Figure 8 shows the precipitation, observed and final
modeled hydrographs and tracer responses in the up- and
lowland subcatchments and the integrating outlet for the
study period 2008–2009. The modeled envelopes are
defined by the 5th and 95th percentiles of simulation values
for all accepted parameter sets using the constraints
described above. Despite similar climate forcing, the
upland hydrograph shows a more rapid and intense runoff
response to storm events, and a steeper decline to base flow
(Figure 8a). In contrast, the lowland catchment hydrograph
shows a much more subdued response with smaller peak
flows and limited direct runoff reaction to smaller precipitation events, particularly in summer (Figure 8b). The outlet
catchment hydrograph visibly integrates both aspects; the
upland influence is evident in the runoff peaks, but these are
moderated in smaller events and more pronounced base flow
recession periods are evident (Figure 8c). Precipitation is
fairly evenly distributed throughout the year though the biggest storm events occur in autumn and winter. The intra-annual runoff dynamics follow the precipitation distribution,
and snow melt events are responsible for large winter and
spring floods in late December 2008 and mid-February 2009.
[33] The observed stream water alkalinity dynamics are
strongly influenced by dominant soils and geology. The range
varied between 400 meq l1 in the upland and 1000 meq l1
in the lowland catchment, but both with pronounced temporal
dynamics reflecting the active differentiation of geographic
sources (Figure 8). Again, the catchment outlet shows clear
mixing as it integrates the subcatchment responses. The 2H
signatures observed in the precipitation show strong temporal
variation, with 2H concentrations being most depleted in
winter and most enriched in summer. In comparison, the signature in runoff is strongly attenuated but shows a pronounced response to storm events in the uplands, which
propagates to the larger-scale catchment outlet, whereas the
lowlands dynamics are much more attenuated.
[34] The modeled runoff time series envelopes bracket
the observed dynamics quite well at most times in the
uplands. Peak flows in the uplands are sometimes overestimated in moderate-sized events and the timing and rate of
snowmelt are only crudely captured as the melt in December is delayed whereas the February melt proceeds too
quickly. Nevertheless, the timing of the onset of stormflow
and the recession into base flow are generally well matched
(Figure 8). This applies also for the lowland catchment,
where the modeled envelope neatly follows the recession
period in March to June 2009 particularly well. However, a
major precipitation event in July 2009 showed little
response in the observed runoff, which the model overestimates. This may reflect the model poorly capturing the
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rewetting process, or it may be that the precipitation event
is overestimated if a more localized convective summer
precipitation event was erroneously extrapolated to a larger
area (Figure 8b). The hydrograph dynamics of the larger
catchment outlet are captured well by the coupled model,
even though some more moderate flow fluctuations
observed in July and August 2009 are not entirely bracketed by the runoff envelopes.
[35] The modeled alkalinity envelopes capture the
observed dynamics in the up- and lowlands, though some
subtle variations are missed. (Figures 8a and 8b). For example, in the uplands the model’s failure to capture the length
of the February melt results in over estimation of alkalinities. Conversely, in the lowlands, low-flow alkalinity at the
start of the year is systematically under predicted, despite
good flow simulations, implying that the end-member was
not as well-defined, as assumed. However, model failures
are much more apparent at the larger scale of the catchment
outlet where the model systematically underestimates alkalinity concentrations, especially at low flows (Figure 8c).
This is a limitation that is mainly caused by the transfer of
constrained parameter ranges from the separately fitted
upland and lowland models to the coupled model. More
extensive tracer surveys indicate that some lowland tributaries in close proximity had higher alkalinities than those in
the Luther Water as a result of local geological differences,
which violates the mixing assumption implicit in the parameter transfer and points to limitations in using alkalinity, or
indeed other weathering-based tracers, in larger catchments.
[36] The results of the 2H models are ambiguous. On
the one hand, the precipitation signal is sufficiently damped
in all three catchments and the models capture the most
dynamic characteristics of the relationship between 2H in
precipitation and discharge at the more responsive upland
(Figure 8a) and whole catchment (Figure 8c) scales. On the
other hand, there is a notable systematic overestimation of
the short-term variation in the summer season in all models, but most notably in the lowlands, probably relating to
the selective depletion of the modeled shallow storage
components by evapotranspiration and subsequent runoff
generation of more enriched and less-well-mixed precipitation, when much more complex mixing processes are likely
to occur in reality. In the lowlands, the lack of dynamic
stream response provides limited variability for the model to
track, however, in the winter the model underestimates the
deuterium values, again implying that the precipitation signal is insufficiently well mixed in the model. However, this
is not evident at the large catchment scale, as the influence
of the upland headwaters provides a larger volume of water
where isotopic values are better predicted and also exhibits
variability that the integrated model is able to capture.
6.
Discussion and Wider Implications
[37] This study provided an opportunity to combine both
geographic source area tracers and time domain tracers to
aid the development of rainfall-runoff models in large
catchments with contrasting headwaters. Specific traceraided models were developed for REWs for upland and
lowland tributaries and then integrated to simulate flow and
tracer responses in the larger catchment. The integration of
such alternative sources of ‘‘soft’’ or ‘‘orthogonal’’ data is
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Figure 8. Observed and modeled time series of runoff, alkalinity, and 2H. Modeled time series as 5th to 95th percentile accepted model result envelopes :
(a) Water of Mark subcatchment and upland model, (b) Luther Water subcatchment with lowland model, and (c) integrating North Esk catchment using the
coupled outlet model.
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increasingly viewed as an efficient, viable means of constraining structural and parameter uncertainty to produce
more realistic process representation in hydrological models
[Seibert and McDonnell, 2002]. This is an area of considerable potential as the use of multiple model schemes like
FUSE [e.g., Clark et al., 2008, 2010; McMillan et al.,
2012] and FLEX is becoming possible and high-resolution
tracer data sets are becoming more readily available. These
approaches are consistent with the philosophy of using
models as learning tools for understanding catchment functioning [Beven, 2007; Dunn et al., 2008b]. Moreover, integrating additional data into models, while maintaining low
levels of parameterization, allows the capabilities and limitations of simple conceptualizations of hydrological systems
to be explored [Kirchner, 2006]. In larger catchments with
marked differences in landscape evolution history, tracers
can help to identify changes in dominant runoff generation
mechanisms in the main REWs and inform the choice of
model structures. Given the simplicity of the lumped
approach used, together with the size of the study catchments, the associated uncertainties in measurements and
catchment heterogeneity, the accepted model configurations
generally performed quite well. Some important insights
emerge from the work, which highlight some of the challenges and possible limitations to tracer-aided modeling at
these larger scales.
[38] As well as aiding the development of plausible
model structures, tracer data provided an effective means
for constraining model parameters for the groundwater subroutine (Table 5). The additional information content of the
alkalinity data was utilized by the simple alkalinity-recession function and the associated equilibrium conceptualization (Figure 5) used to estimate concentrations in the model
storage components, thus, providing additional constraints
on the dynamics of runoff sources.
[39] The use of 2H signatures, which are strongly dampened after passage through the catchment, highlighted the
incapacity of simple storage structures to sufficiently provide
mixing volumes especially during stormflow generation (see
2H model performances in Table 2). This led to the integration of buffering volumes and storm runoff mixing mechanisms into the model concepts. This, however, also highlights
the limitation of lumped storage conceptualizations as a computational basis for the modeling of larger catchments as their
structure necessitates simplifying assumptions on transport
and mixing processes in the subsurface. This was particularly
apparent in simulating the lowland deuterium response,
where the low signal-to-noise ratio provided limited tracer
variation in streams and a weak test of the models ‘‘goodness
of fit.’’ Moreover, the simple representation of ET losses,
mixing and drainage from the upper soil store particularly in
the lowlands seemed to result in the systematic underestimation of the damping of precipitation inputs [e.g., Dunn et al.,
2008a]. This led to the model over-estimating 2H values in
the summer and under-estimating them in the winter.
Improvements could result from inclusion of additional parameters to overcome the assumption of complete mixing,
which is unrealistic [e.g., Kirchner et al., 2000]. However,
this would move the models to greater complexity and possible increase problems of identifiability.
[40] Elucidating the nature of mixing in the main stores
represents a potentially fertile area in integrating empirical
W09544
data with catchment scale models [McNamara et al., 2011].
Recent regional studies in the Scottish Highlands have
shown that tracer-based assessment of the minimum catchment storage needed to cause tracer based damping often
implies storages (103 mm) two orders of magnitudes
greater than the storage changes caused by the annual water
balance (101 mm) [Soulsby et al., 2011]. In the case of
the present study, the different storage estimates identified
with the 2H integration into the landscape-specific models
are consistent with known characteristics of the lowlands
Dochartaigh
bedrock aquifer in the catchment lowlands [O
et al., 2006], whereas the upland bedrock is considered
largely impermeable and groundwater is restricted to fractures and shallow drifts [e.g., Soulsby et al., 2000, 2006b].
The mixing volumes of each storage component for the
2H mixing model (pS1 and pS2, Table 3) showed lower
performance limits, which allowed for an approximation of
the minimum subsurface mixing volume in each landscape.
The minimum storage estimates in the model components
generating runoff were identified as at least 100 mm passive mixing volume in the uplands model and at least 300
mm in the lowlands model. The volumes are an order of
magnitude greater than the dynamic model storages, and
are higher by at least a factor of two in the lowlands.
[41] In this study, we implemented the models using
fixed step forward approximation in almost all model components, the only exception was the linear shallow storage
(Table 2, equation (6)). It has recently been stressed how
numerical implementation can have a significant influence
on the stability of parameter performances, as well as the
size of internal fluxes and storage volumes, which can be
important, especially when comparing performances of different model structures [Kavetski and Clark, 2010]. The
chosen implementation can lead to instability of performance ranges and thus misleading acceptability of parameter
sets because of artifacts of numerical approximation, which
adds to uncertainty in the internal storage estimates and,
more generally, in the validity of the performance levels of
parameter ranges [Clark and Kavetski, 2010]. Although a
full exploration of these issues was beyond the scope of
this study, some testing in the most dynamic upland catchment indicated that similar simulations were given by both
explicit-forward and analytical solutions for the model
structure used here. Moreover, the acceptable tracer simulations provide additional reassurance that the uncertainties
in the internal fluxes are reasonably constrained.
[42] The combination of models for the upland and lowland REWs were able to simulate the flow and isotope
response quite well at the larger catchment scale. However,
also some of the limitations of using tracers at larger scales
became apparent. As with the lowlands, the low signal-tonoise ratio in the damped stream water isotope response, together with the weekly data, give only a weak basis to test
the models capabilities. More serious are the limitations of
alkalinity as a source area tracer at larger scales. The
observed stream alkalinity at the catchment integrates the
spatial end members of the total catchment area. Although
the investigated subcatchments covered around 25% of the
total catchment area, local geological and land use differences elsewhere in the lowlands resulted in inputs of more
alkaline water [cf. Capell et al., 2011], thus the end-member
space was not adequately sampled [cf. Hooper et al., 1990]
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CAPELL ET AL.: CAN TRACERS REDUCE UNCERTAINTY IN RAINFALL-RUNOFF MODELS?
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in the approach used. Although the large-scale alkalinity dynamics were crudely captured, the end-member values were
poorly constrained despite the broad envelope of the simulations (Figure 7). Furthermore, even though stream water alkalinity can be considered a conservative tracer at event
timescales [e.g., Soulsby et al., 2003], the alkalinity of geographical sources will change toward equilibration endpoints over longer time periods and on larger scales with
many spatially disconnected sources, the sensitivity of alkalinity as end-member mixing tracer becomes increasingly
limited. An alternative would be to use an increased number
of end members, and/or end members defined by multivariate characterization of whole ion chemistries [Capell et al.,
2011]. However, at larger scales, similar problems of multiple sources and nonconservative behavior mean that use
in simple model conceptualizations would be extremely
challenging.
7.
Conclusion
[43] Insights gained from this study have been instructive in learning about the conceptualizations needed for
modeling runoff generation and time domain and source
area tracer transport in contrasting REWs and their integration at larger catchment scales [Dunn et al., 2008b]. This,
in turn, informs future challenges in such efforts to improve
process-based models that retain low levels of parameterization and are based on relatively simple, integrated field
measurements. Unfortunately, it seems that improvements
are contingent upon either increased model complexity or
tracer data that is collected at a higher temporal and spatial
resolution. The former includes new parameterization to
account for processes affecting isotopic tracers (e.g., partial
mixing, fractionation, etc.), or more fundamentally better
representation of snowmelt in headwater areas. The latter
includes more intense stream water sampling on daily or
subdaily time steps; for isotope tracers, this is becoming
increasingly feasible and cost effective with new technological developments [Berman et al., 2009]. This would allow
a better test of modeling capabilities, especially where the
tracer damping results in reduced variability in weekly samples. In addition, measurements of tracer signals in hydrological stores that reflect the internal model variables (i.e.,
soil water, shallow and deeper groundwater, etc.) might
help to identify the limitations of the current model structure [Seibert et al., 2003; Birkel et al., 2011b]. This would
have the advantage of allowing tracer signals to be tracked
through different stores providing new model diagnostics
[cf. McMillan et al., 2012]. Such characterization is obviously more easily envisaged in smaller REWs, and the development of tracer-aided models at larger scales is likely to
remain challenging and frustrated by greater uncertainty.
Notation
P
Peff
ET
QSOF
Ss
catchment-wide precipitation, mm
effective precipitation, i.e., precipitation amount
after passage through snow and soil layers, mm
actual evapotranspiration, mm
modeled saturation excess overland flow runoff,
mm
modeled storage in soil layer, mm
S1
Q1
R
S2
Q2
W09544
modeled shallow storage, mm
modeled runoff from shallow storage, mm
modeled recharge from shallow to base flow storage (freely parameterizable), mm
modeled base flow storage, mm
modeled runoff from base flow storage, mm
[44] Acknowledgments. This work is supported by a University of
Aberdeen, College of Physical Sciences Ph.D. studentship. The authors
would like to acknowledge discharge data provision by the Scottish Environmental Protection Agency (SEPA) and precipitation data provision by
the Metoffice via the British Atmospheric Data Centre (BADC). The
authors are grateful to Christian Birkel for helpful comments on an earlier
draft of the manuscript. We are also grateful to Martyn Clark and three
anonymous reviewers for their comments in the peer revision process.
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