ORIGINAL PAPER
Uncertainty assessment of a process-based integrated catchmentmodel of phosphorus
Sarah Dean Æ Jim Freer Æ Keith Beven ÆAndrew J. Wade Æ Dan Butterfield
Published online: 7 October 2008
� Springer-Verlag 2008
Abstract Despite the many models developed for phos-
phorus concentration prediction at differing spatial and
temporal scales, there has been little effort to quantify
uncertainty in their predictions. Model prediction uncer-
tainty quantification is desirable, for informed decision-
making in river-systems management. An uncertainty
analysis of the process-based model, integrated catchment
model of phosphorus (INCA-P), within the generalised
likelihood uncertainty estimation (GLUE) framework is
presented. The framework is applied to the Lugg catchment
(1,077 km2), a River Wye tributary, on the England–Wales
border. Daily discharge and monthly phosphorus (total
reactive and total), for a limited number of reaches, are
used to initially assess uncertainty and sensitivity of 44
model parameters, identified as being most important for
discharge and phosphorus predictions. This study demon-
strates that parameter homogeneity assumptions (spatial
heterogeneity is treated as land use type fractional areas)
can achieve higher model fits, than a previous expertly
calibrated parameter set. The model is capable of repro-
ducing the hydrology, but a threshold Nash-Sutcliffe
co-efficient of determination (E or R2) of 0.3 is not
achieved when simulating observed total phosphorus (TP)
data in the upland reaches or total reactive phosphorus
(TRP) in any reach. Despite this, the model reproduces the
general dynamics of TP and TRP, in point source domi-
nated lower reaches. This paper discusses why this
application of INCA-P fails to find any parameter sets,
which simultaneously describe all observed data accept-
ably. The discussion focuses on uncertainty of readily
available input data, and whether such process-based
models should be used when there isn’t sufficient data to
support the many parameters.
Keywords INCA-P � GLUE � Uncertainty estimation �Phosphorus models � Diffuse agricultural pollution �Water quality modelling
1 Introduction
The water framework directive (WFD) legislation was
introduced in December 2000 (EC 2000/60/EC) to improve
the chemical and ecological status of European freshwater,
transitional and coastal waters. This legislation is funda-
mentally a new approach, as it considers the catchment as a
whole. Every catchment designated as being at significant
risk (e.g. from diffuse pollution, acidification) requires a
river basin management plan (RBMP). The WFD super-
sedes seven previous pieces of legislation and will address
all aspects of water quality as well as factors within
catchments that could negatively impact on water quality
and associated ecology, the first time this important link
has been enshrined in a legislative framework. The impli-
cations for terrestrial and aquatic habitats are great, and
necessitate the furthering of current knowledge of pressures
and impacts. Consequently, there is a real need to develop
S. Dean � J. Freer � K. Beven
Lancaster Environment Centre, Lancaster University,
Lancaster LA1 4YQ, UK
A. J. Wade � D. Butterfield
Aquatic Environments Research Centre,
Department of Geography, University of Reading,
Reading RG6 6AB, UK
Present Address:J. Freer (&)
School of Geographical Sciences, University of Bristol,
University Road, Bristol BS8 1SS, UK
e-mail: [email protected]
123
Stoch Environ Res Risk Assess (2009) 23:991–1010
DOI 10.1007/s00477-008-0273-z
tools to predict surface and groundwater responses, and
their associated dependent habitats, to both anthropogenic
pressures and remediation projects. Liu et al. (2005) state
that water quality models are a necessity in such catchment
management because of their ability to apply current
knowledge to predict water quality in response to different
scenarios (e.g. such as the likely consequences of different
farming practices).
Dynamic, process-based models of pollutant sources and
catchment dynamics are necessarily complex because they
attempt to describe all factors and processes so that the
relative importance of these may be understood and
investigated in response to environmental change. Fully
distributed process-based models are the most complex
form of environmental model as they attempt to model
every process deemed important for every location in a
catchment, generally on a grid-basis. However, it is gen-
erally accepted that a compromise is required between
available data, process representation, and runtime speed,
when using the model within a Monte-Carlo based sensi-
tivity or uncertainty analysis, to develop a pragmatic
approach (Langan et al. 1997). The key sources and pro-
cesses controlling nutrient water quality characteristics are
well established, but the understanding of how the sources
and processes vary in time and space is still limited. This is
due to the heterogeneity of environmental factors which
define source-areas and control process rates and delivery
from the land to the stream network, such as land use, soil
type, moisture and temperature, and flow-routing. Often the
data available to develop and apply predictive models are
generally insufficient, even for small research catchments.
Thus, while it is useful to develop models based on process
understanding, they will always necessarily be simplifica-
tions of reality. These simplifying assumptions are a source
of uncertainty in a model, and the robustness of any model
application will be dependent upon the validity of the
assumptions made. It is important, therefore, that the
uncertainties in model predictions are well understood.
Therefore, estimating prediction uncertainties in water
quality modelling is becoming increasingly appreciated
(Krueger et al. 2007; Page et al. 2004, 2005; Radwan et al.
2004; Rode et al. 2007; Singh et al. 2007; van Griensven
and Meixner 2006).
Parameter values in process-based models are often
difficult to measure or estimate, especially at the scale
required by the model, i.e. the effective model unit scale
which is the scale at which processes are represented in the
model. This necessitates the use of effective parameter
values to compensate for the underlying variability in
processes, site characteristics and limits in the model’s
process representations (Beven 1996, 2002, 2006). In
general, there are no techniques available for measuring
effective parameter values; so they are estimated through
calibration. Process-based models often suffer from over-
parameterisation, where the model parameters cannot be
identified with certainty from the information content of
the available observed data. This often leads to poorly
constrained parameter values resulting in many different
parameter sets producing acceptable fits of the observed
data, termed equifinality (Beven 2006). This also means
that any ‘optimum’ parameter set will not be robust (i.e.
may change) against a different period of calibration data
or errors in observations. Attempts are being made to
overcome this, for example through the use of high-fre-
quency data (Arnscheidt et al. 2007; Jordan et al. 2005,
2007; Kirchner et al. 2004); soft-data (e.g. Rankinen et al.
2006), and the assessment of internal measurements so that
models are tested against observations made at points
within a catchment rather than just at the catchment outlet
(e.g. Freer et al. 2004; Gallart et al. 2007).
Sensitivity and uncertainty analyses provide model users
with information regarding the effect of model parameters
and input data on the resultant model prediction. Sensi-
tivity analysis is particularly concerned with identifying the
parameters that are most influential in the model simula-
tions, whereas an uncertainty analysis is used to estimate
the error in a model output given uncertainty in the model
structure, parameters and input data. The generalised
likelihood uncertainty estimation (GLUE) technique
(Beven and Binley 1992) is a framework for evaluating a
model, given an acceptance of the equifinality concept.
GLUE provides information on the uncertainties in a
model’s predictions. This information helps model-users to
understand the confidence they can have in a model’s
prediction. Beven (2006, 2008) provides a full background
on the GLUE methodology. GLUE has been applied to
many different environmental models, including sediment
and geochemical models (Beven and Binley 1992; Brazier
et al. 2000; Freer et al. 1996, 2003, 2004; Zak and Beven
1999b). Here, GLUE is applied to the integrated catchment
model of phosphorus (INCA-P), in an application to the
Lugg Catchment, UK.
The INCA-P is a physically based, highly parameterised
model, requiring a variety of input variables, some of
which are often poorly known and impossible to measure,
and so require calibration. To date, there has been no
published uncertainty analysis performed on INCA-P,
although the in-stream part of the model structure has been
subject to both sensitivity and uncertainty analysis (Wade
et al. 2001, 2002b, c). There have also been numerous
assessments of the ability of the integrated catchment
model of nitrogen (INCA-N) to predict nitrogen dynamics
at the catchment scale (Granlund et al. 2004; Limbrick
et al. 2000; McIntyre et al. 2005; Raat et al. 2004;
Rankinen et al. 2006). The model developers envisage that
INCA-P could be used as part of a model hierarchy in
992 Stoch Environ Res Risk Assess (2009) 23:991–1010
123
which steady-state models, e.g. Phosphorus Indicator Tool
(Heathwaite et al. 2003) or Export Coefficient Model
(Johnes 1996), would produce a summary of the national
export of phosphorus and INCA-P would provide a detailed
assessment of particular catchments which are of particular
concern or interest (Wade et al. 2004).
The objectives of this study are:
1. to assess the model uncertainty in the application of
the INCA-P model to the Lugg catchment, using the
GLUE uncertainty framework;
2. to investigate parameter uncertainty and how these
uncertainties impact upon the model predictions;
3. to consider whether, allowing for the lack and quality
of data available for calibration, such complex models
are suitable for assessing phosphorus dynamics at the
catchment scale; and
4. to consider the suitability of INCA-P as a potential
phosphorus modelling tool for implementing the WFD.
2 Materials and methods
2.1 The GLUE framework
A full account of the GLUE methodology and rationale can
be found in Beven and Binley (1992), Beven and Freer
(2001), and Beven (2006, 2008). What follows is a brief
description.
GLUE is an extension of the regional sensitivity analysis
(RSA) proposed by Spear and Hornberger (1980). For any
given set of observed data and particular choice of per-
formance measure, there will be an optimal model structure
and parameter set that best describes that dataset. However,
RSA and GLUE recognise that there may be many models
(structures and parameter sets) that give acceptable results,
and equally, given a different or additional set of calibra-
tion data, or a different performance measure, the optimal
model is likely to be different. As GLUE considers
parameter sets as opposed to the individual parameters,
interactions between parameters in providing a good (or
bad) fit is implicitly accounted for. GLUE differs from
RSA in the way it treats acceptable model simulations.
They both use a performance measure threshold to define
acceptable models but, whereas RSA treats all acceptable
models equally in looking at global sensitivities, GLUE
calculates a likelihood weight for each simulation by
evaluating the performance of the simulation in compari-
son with observed data and then uses those weights to
evaluate uncertainties in predicted model outputs over all
the simulations considered acceptable.
GLUE requires a number of prior decisions to be made,
which should always be reported:
1. which parameters to vary;
2. which model structure(s) to consider;
3. the ranges within which the parameter values should
be varied;
4. the likelihood weights to be used in assessing the
performance of a model simulation; and
5. a procedure for creating the uncertainty prediction
bounds.
Likelihood weights are calculated for all acceptable
model simulations. The choice of performance measure is
important; it should reflect our best understanding of the
possible errors in the observed data (see discussion in
Beven 2006). However, in certain cases, such as this study,
an understanding of the possible combined errors in the
modelling application (input and output observation error
combined with model structure error) are not known and,
therefore, the choice of measure is necessarily subjective
but must provide a relative measure of model performance.
The resultant likelihood weights are defined as 0 for
unacceptable model simulations and should increase in
value as model simulations improve. The performance
measures used in this study are:
R2 ¼ 1� SSE
SSTwhen R2 [ 0 ð1Þ
1=RMSE = 1
, ffiffiffiffiffiffiffiffiffiSSE
n
rð2Þ
where
SST ¼Xn
i¼1
wiðyi � �yÞ2 ð3Þ
SSE ¼Xn
i¼1
wiðyi � yiÞ2 ð4Þ
where i is the current time step, n is the number of data
points, SST is the sum of squares of the observations (yi)
around the observed mean ð�yÞ; SSE is the sum of squared
model errors with yi being the simulated value, RMSE is
the root mean squared error, and R2 is a coefficient of
determination; it is a comparison between the ability of the
model to simulate the observed data and using the mean of
the observed data as a predictor of the observed data. R2
usually ranges between 0 and 1, but when SSE is greater
than SST, the mean of the data would be a better predictor,
and a negative R2 value is obtained. 1/RMSE is also a SSE
based measure of fit, but always takes positive values. Both
these coefficients are biased towards fitting high values of
discharge and concentration because they are based on
squared errors. Wade et al. (2004) used R2 in a previous
assessment of INCA-P with which we compare our simu-
lations, and this measure has also been commonly used in
other uncertainty analyses (McIntyre et al. 2005; McIntyre
Stoch Environ Res Risk Assess (2009) 23:991–1010 993
123
and Wheater 2004; Rankinen et al. 2006). Therefore, R2
was the performance measure used to determine whether a
model simulation was acceptable. However, due to the total
phosphorus (TP) R2 results being almost all negative (see
later discussion), 1/RMSE was used to evaluate the simu-
lations in terms of sensitivity and prediction bounds as the
use of negative R2 values would not provide an appropri-
ately weighted performance measure. However, R2 is still
referred to when making references to Wade et al. (2004).
2.2 The INCA-P model
INCA-P was developed to investigate transport and reten-
tion of phosphorus in terrestrial and aquatic environments,
and to quantify the impacts of phosphorus loads on in-
stream macrophyte biomass. A full description of INCA-P,
including process equations used, appeared in Wade et al.
(2002a) and an initial application is reported in Wade et al.
(2007). Only those parts relevant to this study are described
herein. INCA-P is a dynamic, semi-distributed, process-
based model which predicts discharge and concentrations
of suspended sediment, soluble reactive phosphorus (SRP)
and TP concentrations in stream water by tracking dis-
charge and phosphorus through the soil and groundwater to
the main channel. The model also simulates the following
processes: bed sediment resuspension, suspended sediment
deposition, the growth effect of phosphorus on macro-
phytes and epiphytes, and the feedback of the growth on
phosphorus concentrations in the stream water.
Briefly, the INCA-P model consists of three
components:
• A land-phase hydrological model: this calculates dis-
charge through different pathways (direct, soil and
groundwater) and their stores. Discharge and phospho-
rus is controlled through this component.
• A land-phase phosphorus model: this component deals
with the various phosphorus stores in soil and ground-
water, and phosphorus transformations.
• An in-stream phosphorus model: this component sim-
ulates the phosphorus processes operating once it
reaches the stream—dilution and transformations, as
well as the concomitant algal, epiphyte and macrophyte
growth responses.
INCA-P is based on a simple mixing model approach,
whereby conceptually the water (and any phosphorus being
transported) is mixed from the different land uses (up to six
user-defined classes) within each reach and then routed
along the main stream. The in-stream model is based on the
Kennet model (Wade et al. 2001) which simulates in-
stream phosphorus and macrophyte/epiphyte dynamics
(Wade et al. 2002a). Phosphorus sources can include fer-
tiliser, plant residue, slurry, animal waste and wastewater.
Typically, when INCA-P is applied to a catchment, the
sensitive parameter values are calibrated by comparing
model output to observed discharge, suspended sediment
(SRP) and TP measured along the main channel. The user
guide (Butterfield et al. 2004) contains the calibration
guidelines. A parameter set derived by the model devel-
opers, using expert calibration (i.e. manual changing of
parameters) of INCA-P to the Lugg catchment, is used as a
benchmark for comparing the results obtained from this
GLUE analysis (Wade et al. 2004, 2007).
INCA-P has been used recently on behalf of the Envi-
ronment Agency (EA), UK, to look at the effectiveness of
eutrophication control through reductions in phosphorus
(Wade et al. 2004, 2007). It was applied to three catch-
ments in the UK, including the Lugg Catchment. The
report concluded that the representation of discharge, sus-
pended sediment and phosphorus ‘‘appeared reasonable’’
(Wade et al. 2004). It was noted that there was a clear need
for detailed point source data and a more general assess-
ment of the relative contributions of point and diffuse
sources, for which the current monitoring networks of the
EA are unfortunately not yet sufficient.
2.3 Evaluation catchment
A full description of the River Lugg catchment can be
found in (Wade et al. 2004, 2007). Data was supplied by
the model developers at the University of Reading as an
example of a catchment for which they felt INCA-P pro-
vided a good representation of observed discharge and
phosphorus dynamics. The River Lugg was chosen as a
study area because it is known to have very high
(approximately 1 mg l-1) stream water phosphorus con-
centrations (measured as TP) in its lower reaches. Such is
the concern about its water quality that, in 1994, the Lugg
was designated a ‘eutrophic sensitive’ area under the Urban
Wastewater Treatment Directive.
The River Lugg is a tributary of the Wye; its source is in
Powys. The catchment area of the entire Lugg is 1,077 km2,
and the catchment area to the lowest gauged point is
885 km2; the long-term annual mean rainfall (1961–1990)
is 977 mm in the north of the catchment and 877 mm in the
south. The headwaters drain Silurian rocks, and the
impermeable bedrock is covered by extensive deposits of
gravel in the valleys. The geology of the lower Lugg is
predominantly Devonian Old Red Sandstone. The upland
area consists of a mix of forestry, grazing and arable land,
whereas the lowland is dominated by arable cultivation
(http://www.environment-agency.gov.uk/hiflows). Many
point sources of pollution have been identified, including
sewage effluent, trade effluent and those from private
outlets (Wade et al. 2004, 2007). The most important con-
cerning downstream water quality are the Welsh Water
994 Stoch Environ Res Risk Assess (2009) 23:991–1010
123
sewage works at Leominster discharging into Reach 10 and
the trade effluent discharged from Cadbury’s chocolate
factory into Reach 12.
Discharge and water quality in the Lugg catchment are
monitored by the EA (Welsh office). The discharge is
recorded every 15 min; water quality is monitored by grab
sampling approximately monthly. This resolution of water
quality sampling is typical for the EA national monitoring
network. The catchment is divided into 22 sub-catchments
for the INCA-P application. There are 3 sub-catchments
with observed discharge data: Byton, Butts Bridge and
Lugwardine; 10 sub-catchments with suspended sediment
(not evaluated in this paper) and TRP data; and 8 sub-
catchments with TP observations (Fig. 1). The INCA-P
model provides predictions of SRP as opposed to TRP
(where the same analysis is performed on unfiltered sam-
ples). This discrepancy is discussed in Sects. 3 and 4.
2.3.1 Model set-up
The model set-up is given in detail elsewhere (Wade et al.
2004); what follows is a summary of input datasets, choice
of parameters to be varied and parameter ranges. INCA-P
hydrological input data requirements are: hydrologically
effective rainfall (HER), actual precipitation, air tempera-
ture, and soil moisture deficit (SMD). In the UK, output
from the meteorological office rainfall and evapotranspi-
ration calculation system (MORECS) is typically used to
derive the hydrologically effective rainfall and SMD based
on measurements of temperature, bright sunshine hours,
wind and humidity. Single site daily MORECS time series
data for hydrologically effective rainfall, SMD and air
temperature were licensed from the Meteorological Office.
The MORECS time series are based on data from two
weather stations, one representing the north of the catch-
ment (Llandrindod Wells) and the other the south (Madley)
(Table 1). HER is an expression of excess (or effective)
rainfall (Hough and Jones 1997); it is the water that pen-
etrates the soil after interception and evaporation have been
taken into account. The MORECS model also produces a
time series of SMD, which is used to control the rates of
phosphorus transformations in INCA-P.
Wade et al. (2004, 2007) applied the Llandrindod data to
the north of the catchment (Reaches 1–9) and the Madley
data to the south of the catchment (Reaches 10–22); the
implications of this are discussed in Sects. 3 and 4. The
average annual (1961–1990) rainfall recorded for the
three discharge stations at Byton (Reach 4), Butts Bridge
(Reach 8) and Lugwardine (Reach 19) are 977, 877 and
814 mm, respectively (http://www.environment-agency.
gov.uk/hiflows). Assuming that the rainfall for the six
year period considered in this study was well represented
by these averages, the estimated HER is approximately 66,
73 and 33% of rainfall at each gauging station, respec-
tively. This difference results from the MORECS method
of calculating HER. This is because, in part, the SMD
between the two weather station areas are very similar
during the winter months when both are close to zero, but
Fig. 1 The River Lugg and its tributaries: the location of the discharge gauges, and nutrient monitoring sites (Wade et al. 2004). NB: Suspended
solids and TRP are also measured at all of the TP sites
Stoch Environ Res Risk Assess (2009) 23:991–1010 995
123
quite different during the summer, with Madley experi-
encing a greater deficit.
INCA-P has a large number of parameters: 47 land use
dependent parameters, 20 reach-dependent parameters, and
25 independent parameters. If this catchment was treated
heterogeneously throughout the six land uses and 22
reaches, that would equate to 747 parameters requiring
identification. Some simplification in terms of the number
of parameters considered is therefore necessary. Irvine
et al. (2005) observed that, although complex physics-
based models can provide better causal relationships in
predicting the impacts of measures to implement the WFD,
it is generally not possible to obtain all the data required to
develop and apply them. McIntyre et al. (2005) performed
two investigations into the sensitivity of INCA-N. The first
investigation considered the catchment to be homogenous,
as done in the present study, and the second heterogeneous.
They concluded that the sensitivity analysis conducted
using a heterogeneous parameter set returned little infor-
mation not already uncovered through the homogenous
parameter set analysis, although they did find that, in
general, the optimum parameter values were reach-depen-
dent (heterogeneous). Using reach-dependent parameters
dramatically improved nitrogen simulation, but discharge
was unaffected and ammonium simulation was actually
worse.
In this study, a pragmatic strategy for homogeneous
parameterisation across the catchment is implemented to
reduce the computation time required to sample such a
large model space. The success with which the model
space is sampled is discussed in Sect. 3. The assumption is
made that each sub-catchment and each land use could be
described by the same set of INCA-P parameters, as there
is not enough information for each individual sub-catch-
ment regarding characteristics that may lead to differences
in hydrological or nutrient transport and transformation, to
justify treating them differently. Many of the parameters
which are kept constant are related to the initialisation of
the model. These are set at values supplied with the model
in order to be directly comparable with the results from
Wade et al. (2004, 2007) and any parameters which vary
between sub-catchments or river reaches are kept as such.
The input data and initial values for phosphorus in the soil,
groundwater and in-stream reflect the historic applications
of phosphorus in the catchment. Land use types are also
differentiated by inputs of phosphorus which varies
throughout the catchment. The calibration guide in the
INCA-P user manual is used as a basis to identify param-
eters to vary, and the recommended ranges are used as a
starting point for the sampling ranges used in this GLUE
analysis. Where a range is not given in the guide, a range is
calculated using an order of magnitude variation above and
below the default value. Additional parameters, which are
thought to be interesting in terms of phosphorus dynamics,
are also considered in this analysis, including maximum
SMD and animal waste input to land. Initial ranges are
adjusted after a trial set of Monte Carlo simulations, as a
few parameters perform best on the edge of their range
whilst others only produce acceptable results in a narrow
band. The base flow index (BFI) range is selected differ-
ently, rather than sampling the recommended range of 0–1,
since Wade et al. (2004, 2007) already suggest the values
lie between 0.63 and 0.7 for the sub-catchments of the
River Lugg, the range 0.6–0.8 is used. Table 2 lists the 44
parameters being varied together with the minimum and
maximum values (the expert calibration values are inclu-
ded for comparison); where there is more than one value,
this indicates where land uses or reaches were parameter-
ised heterogeneously in Wade et al. (2004, 2007). Of the 44
parameters varied, 11 parameters are expertly calibrated
heterogeneously for each reach or land use, of those, three
related to hydrology and eight to phosphorus. The
remaining parameters are kept at default values, as speci-
fied in the user manual. In the studies by Wade et al. (2004,
2007), BFI was the only parameter relating to groundwater
that was calibrated as reach-dependent. The impact of
treating this parameter homogenously is explored in more
detail in Sects. 3 and 4. Parameters not sampled kept their
Table 1 Summary statistics for the MORECS time series relating to weather stations at Llandrindod Wells and Madley, including mean for
study period (1995–2000)
Total hydrologically effective rainfall (mm/year) Mean soil moisture deficit (mm/year) Mean air temperature (�C)
Llandrindod Wells Madley Llandrindod Wells Madley Llandrindod Wells Madley
1995 503.1 316.1 49.1 45.6 9.7 10.5
1996 391.1 205.6 36.6 51.3 8.6 9.2
1997 395.4 244.9 28.6 21.1 9.8 10.4
1998 819.5 273.1 8.5 45.7 9.7 10.3
1999 842.6 266.5 17.3 25.6 9.8 10.6
2000 902.7 311.4 11.6 42.0 9.4 10.2
1995–2000 642.3 269.6 25.3 38.5 9.5 10.2
996 Stoch Environ Res Risk Assess (2009) 23:991–1010
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Table 2 INCA-P parameter value ranges used to simulate the Lugg catchment (multiple values in the ‘expert-calibration’ column indicate that
there is more than one parameter to represent the six land uses or 22 reaches)
Parameter Expert calibration GLUE parameter range Unit
Initial direct runoff flow (applies to 6 land uses) 0.001 0 0.001 m3 s-1
Initial soil water flow (applies to 6 land uses) 0.005 0.001 0.01 m3 s-1
Initial groundwater flow (applies to 6 land uses) 0.008 0.003 0.015 m3 s-1
Initial soil water organic P (applies to 6 land uses) 0.1; 0.01 0 1 mg l-1
Initial soil water inorganic P (applies to 6 land uses) 0.1; 0.01 0 1 mg l-1
Initial groundwater organic P (applies to 6 land uses) 0.001 0 0.01 mg l-1
Initial groundwater inorganic P (applies to 6 land uses) 0.001 0 0.01 mg l-1
Initial direct runoff organic P (applies to 6 land uses) 0.025; 0.01 0 0.1 mg l-1
Initial direct runoff inorganic P (applies to 6 land uses) 0.025; 0.01 0 1 mg l-1
Initial firmly bound organic P (applies to 6 land uses) 0.01 0 0.01 mg l-1
Initial firmly bound inorganic P (applies to 6 land uses) 0.01 0 0.01 mg l-1
Initial soil water drainage volume (applies to 6 land uses) 5E?05 1E?05 1E?06 m3
Initial groundwater drainage volume (applies to 6 land uses) 1E?07 1E?06 1E?08 m3
Initial direct runoff drainage volume (applies to 6 land uses) 1,000 1,000 6E?07 m3
Organic P uptake (applies to 6 land uses) 0.08; 6; 0.18; 0 0 2 m day-1
Immobilisation rate (applies to 6 land uses) 0 0 0.1 m day-1
Mineralisation rate (applies to 6 lands uses) 0 0 0.25 m day-1
Firmly bound organic P input (applies to 6 land uses) 0.07; 0.01; 0.22; 0 0 5 m day-1
Firmly bound organic P output (applies to 6 land uses) 0; 0.1 0 5 m day-1
Inorganic P uptake (applies to 6 land uses) 0.08; 6;
0.18; 0
0 6 m day-1
Firmly bound inorganic P input (applies to 6 land uses) 0.07; 0.01; 0.22; 0 0 0.1 m day-1
Firmly bound inorganic P output (applies to 6 land uses) 0; 0.1 0 0.1 m day-1
Max. P uptake by plants (applies to 6 land uses) 100 0 105 kg P ha-1 year-1
Max. soil moisture deficit (applies to 6 land uses) 150 0 170 mm
Plant residue (applies to 6 land uses) 0 0 366 kg P ha-1 year-1
Animal waste (applies to 6 land uses) 0 0 366 kg P ha-1 year-1
Slurry (applies to 6 land uses) 0 0 162 kg P ha-1 year-1
Dirty water (applies to 6 land uses) 0 0 162 kg P ha-1 year-1
Inorganic P fertiliser (applies to 6 land uses) 0 5 80 kg P ha-1 year-1
Direct runoff time constant (applies to 6 land uses) 0.01 0.001 0.1 Day
Soil water time constant (applies to 6 land uses) 1 0.5 5 Day
Groundwater time constant (applies to 6 land uses) 70 10 200 Day
Max soil retention volume (applies to 6 land uses) 0.2 0.1 1 m
Initial flow 0.01 0 2 m3 s-1
Initial TP in the water column 0.01 0.01 0.05 mg l-1
Initial TP in the pore water 0.01 0.01 0.1 mg l-1
Sediment resuspension/settling rate 8 8 800 mm s m-3
P exchange (water column/pore water) 1,000 100 10,000 Day-1
Precipitation of P in water column 0 0 1 Day-1
Kd for bed sediment 0.6 0.0005 1 –
Kd for suspended sediment (applies to 22 reaches) 800 200 1E?06 dm3 kg-1
SUP proportion (applies to 22 reaches) 0.05 0.25 –
Base flow index (applies to 22 reaches) 0.65; 0.7; 0.63 0 1 –
Alphaa (applies to 22 reaches) 0 0 1 –
See (Wade et al. 2002a) for detailed parameter definitions. Other parameters required by the model that are kept fixed are given in Appendixa Alpha is the trigger for overland flow
Stoch Environ Res Risk Assess (2009) 23:991–1010 997
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reach and land use dependent values that were previously
identified by expert calibration (see Appendix); these
parameters are generally related to suspended sediment and
macrophyte/epiphyte growth.
A total of 200,000 simulations are made using Monte
Carlo random uniform sampling of the parameter ranges
shown in Table 2. It is recognised that 200,000 simulations
provide a limited sampling density when considering a 44-
dimensional parameter set; however, the results suggest
that the shape of the responses for individual parameters
are adequately defined (Figs. 2, 3). For each simulation
run, the performance measures (Eqs. 1, 2) are calculated
for each reach length shown in Fig. 1 relating to an
observed discharge or phosphorus dataset. The threshold of
acceptability is set at R2 C 0.6 for discharge and R2 C 0.3
for SRP and TP (the use of different thresholds ensures that
for both sets of observations some behavioural simulations
could be critically evaluated to assess the quality and/or
deficiencies of the best model predictions sampled).
3 Results
The GLUE framework is not a parameter optimisation
tool, since it is based on the use of multiple acceptable
models in estimating the prediction uncertainties generated
from Monte Carlo simulations using different parameter
sets. However, the results from the parameter set which
gained the combined highest R2 score, summing the R2
values for discharge, TRP and TP (with no weightings on
each type of information), are compared with those pre-
sented in Wade et al. (2004) using their expert calibration
parameter set. This results in the ‘best’ set from the Monte
Carlo simulations improving on this expert calibration for:
a) all flow reaches (Reaches 4, 8, and 19, having an R2 of
0.58, 0.36 and 0.48 for the expert calibration and for the
‘best’ parameter set these being 0.68, 0.64, and 0.65 R2,
respectively), and b) all TP simulations with an R2 [ 0
except for Reach 21 (Reaches 11, 12, and 21,
expert = 0.31, 0.32 and 0.47 R2 and from the ‘best’
parameters 0.37, 0.48, and 0.40 R2). Although this ‘best’
parameter set generally performs better than the expertly
calibrated parameter set, this too is unable to produce any
simulations that improve on fitting a mean of the observed
data (R2 = 0) for those upland catchments with observed
data for TP (Reaches 1, 3, 6, 9–10). Both sets of these
simulations are unable to reproduce the TRP with a
R2 C 0 for all reaches (Reaches 1, 3, 6, 9–12, 16, 19, 21).
This suggests treating the Lugg catchment homoge-
neously, as described, does not seem to have negatively
impacted the performance of the model when compared
with the heterogeneously calibration parameter set (with
spatially heterogeneous parameters).
Considering each reach independently, the GLUE
analysis is unable to identify any acceptable parameter sets
for the upland reaches when evaluated against the observed
TP data, and no acceptable parameter sets are found for any
reach for the TRP using the stipulated R2 criteria. This
Fig. 2 Relationship between
behavioural parameter value (x-
axis) and the performance
measure (y-axis) calculated
using observed flow data
998 Stoch Environ Res Risk Assess (2009) 23:991–1010
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suggests that, in this application, the process representa-
tions in INCA-P are better suited to the simulation of
reaches controlled by point inputs of phosphorus than
predicting the inputs from diffuse sources. In the River
Lugg, the mean TP concentration increases downstream
(see Fig. 7), suggesting additional phosphorus inputs from
sewage treatment works and trade effluents, especially
following the input from Leominster STW into Reach 10
(Wade et al. 2007).
Table 3 shows the number of acceptable parameter sets
remaining as each set of constraining data listed in column
one is used: when only the discharge at Reach 4 is con-
sidered, there are 119,371 acceptable runs; when discharge
at Reaches 4 and 8 are considered, there are 51,821
acceptable runs, and so forth. If all of the constraining data
were used (or indeed if any of the TRP data were used), no
acceptable simulations would remain, as shown in Table 3
(final row, second column). The third column shows how
many acceptable runs are achieved if only using one set of
constraining data at a time. Based on the stipulated R2
thresholds, INCA-P is incapable of producing acceptable
model simulations of the TP concentrations at all of the
reaches simultaneously using spatially lumped parameters.
However, if only the constraining data for discharge at
Reaches 4, 8 and 19 are used, the model is able to achieve a
R2 performance of at least 0.67. This result shows that
discharge can be represented adequately by assuming the
catchment to be uniform, although some improvement
might be gained by taking the difference in groundwater
inputs to the different reaches into account. However,
obtaining acceptable simulation results for TP and TRP
may well relate to the model structure and input data (see
discussion in later sections).
3.1 Sensitivity to individual parameters
Sensitivity is assessed by visual inspection of dotty plots,
which represent projections of the response surface onto
single parameter axes. While such plots do not show the
complex interactions between parameters in a high-
dimensional parameter-space, the vast majority of model
parameters show no visually detectable sensitivity across
the range evaluated. Clearly, this could be further evaluated
using global sensitivity methods (e.g. Campolongo et al.
Fig. 3 Relationship between
top 10% of parameter values
(behavioural and non-
behavioural) (x-axis) and the
performance measure (y-axis)
calculated using observed TP
data from Reach 21: a Firmly
Bound Organic Phosphorus
Input Rate; b Firmly Bound
Organic Phosphorus Output
Rate; c Firmly Bound Inorganic
Phosphorus Input Rate;
d Firmly Bound Inorganic
Phosphorus Output Rate; and
e Groundwater Time Constant.
Stars behavioural, dotsnon-behavioural
Table 3 The reduction in numbers of behavioural parameter sets for
different constraining data as each dataset is sequentially added
(second column) and for each reach considered independently (third
column)
Constraining dataa Cumulative acceptable
simulations
Acceptable simulations
per constraining dataset
Reach 4: flow 119,371 119,371
Reach 8: flow 51,821 96,332
Reach 19: flow 33,997 83,247
Reach 11: TP 0 1
Reach 12: TP 0 3
Reach 21: TP 0 41
a TP Simulations for River Reaches 1, 3, 6, 9, and 10 do not produce
any acceptable simulations and hence there are no cumulative
acceptable simulations
Stoch Environ Res Risk Assess (2009) 23:991–1010 999
123
2007; Jansen 1999), and this will be the focus of further
research. Figure 2 shows dotty plots only for parameters
showing sensitivity to the observed discharge data, and
Fig. 3 shows dotty plots for parameters showing sensitivity
to the TP observations. The performance measure used is
defined in Eq. 2.
3.1.1 Hydrology
The most sensitive hydrological parameters are initial
groundwater flow, soil water time constant, groundwater
time constant and BFI; this is consistent with the findings
of McIntyre et al. (2005) for INCA-N who, in addition,
identified initial soil water discharge as being sensitive
(BFI was not assessed in their study).
Some of the dotty plots shown reveal that the model
performance is still increasing at the edge of the sampled
range. The ranges sampled were initially identified in
Wade et al. (2004) or suggested in the INCA-P user guide
(Butterfield et al. 2004), although following the initial
sensitivity analysis some of these ranges are altered.
Unfortunately, in some cases, even these extended ranges
fail to show the complete pattern of parameter response.
However, it is beyond the project resources to extend the
ranges further and thus they will be assessed in more detail
in our future works.
Other than BFI, the response of Reach 8 is different to
the responses of the other two catchments (Fig. 2). The
groundwater time constant shows the biggest difference in
response by the three catchments. The difference seen at
Reach 8 may be caused by the driving meteorological data.
As previously explained, there are two meteorological
driving datasets derived from MORECS, one applied to the
upland catchments and one to the lowlands. Reach 8 is
approximately centrally located, and was assigned the input
hydrological data from the Llandrod Wells in the original
report, which was assumed representative of the upland
area of the catchment (Wade et al. 2004). The ratio
between observed discharge and HER should be approxi-
mately 1, the ratio between these data for the three
catchments are 1.09, 0.84 and 1.31 for Reaches 4, 8 and 19,
respectively. The ratios for Reaches 4 and 19 are both over
one, indicating that there is more discharge than HER in
the catchment, whereas Reach 8 shows the opposite.
Therefore, the similar responses of Reaches 4 and 19 and
the significant difference in Reach 8 are primarily con-
trolled by these HER differences. Reach 8 is routing excess
rainfall through the groundwater storage in order to pro-
duce the correct observed discharge (Fig. 2). We suggest
that heterogeneous parameters are therefore required,
although the possibility of having inadequate driving data
cannot be discounted. If the lowland rainfall data were
applied to Reaches 7–22, the ratio at Reach 8 would be
0.96. These findings illustrate the fundamental importance
of an accurate estimation of inputs to the system in
obtaining behavioural predictions (even if the perfect
model was available, see Beven (2006)).
3.1.2 Total reactive and total phosphorus
To assess the response of the model and identify why it
might be failing, the top 10% of results are plotted; these
include acceptable and unacceptable model outputs (stars-
acceptable, dots-unacceptable). As all the reaches show
similar responses (unlike the discharge), Fig. 3 shows only
the parameters at Reach 21 which appear visually sensitive.
The fact that all of the reaches produce similar responses
further confirms the acceptability of the approach of
treating phosphorus related parameters homogenously.
It is worth highlighting here that the model fits are poor
for total and SRP simulations, and that acceptable simu-
lations (R2 C 0.3) are only obtained for Reaches 11, 12 and
21 for TP (see Table 3).
The groundwater time constant shows parameter sensi-
tivity to both observed discharge and TP data (Fig. 3e). The
sensitivity of the groundwater time constant is indicative of
the reliance of TP concentration on the accuracy of the
hydrology component. This is likely due to the parameter
controlling the amount of water available to dilute diffuse
and point source inputs to stream. Any water quality model
can only be as good as the water quantity model driving it.
The firmly bound phosphorus output rates relate to the
rate at which phosphorus is released into the available soil
phosphorus store and the input rates relate to the rate at
which phosphorus is locked away into the firmly bound
phosphorus store. Figure 3 shows that the best simulations
occur when the output rate is approaching zero and the
input rate is approaching the maximum of the suggested
range. Consequently, over time, there is a net increase in
the amount of firmly bound phosphorus locked away in the
soil (being made unavailable). However, the amount of
phosphorus being added from fertiliser and manure on an
annual basis remains constant within each catchment. The
mass balance calculated within INCA-P at the end of the
simulation confirms that the amount of firmly bound
phosphorus stores in the soil has increased for all land uses
in all reaches. The phenomenon of catchments acting as a
sink for phosphorus has been observed in other UK
catchments (Neal et al. 2004b).
3.2 Calculation of indicative prediction bounds
The acceptable parameter sets from the GLUE analysis
(with R2 [ 0.6 for hydrology and R2 [ 0.3 for TP and
SRP) are used to produce 5% and 95% prediction bounds.
The prediction bounds are calculated from the outputs of
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all the behavioural models and for each reach length,
weighted by a ‘likelihood’ value that is conditional on the
choices identified for the GLUE analysis for each param-
eter set. Values below the acceptability threshold are
assigned a likelihood of zero and values above the
threshold are assigned a likelihood proportional to 1/
RMSE, rescaled such that they sum to unity. These
rescaled weights are then applied to the model outputs at
each timestep to create uncertainty bounds representing the
5th and 95th quantiles of predicted values. These quantiles
are then plotted as prediction bounds with the observed
data. Only the water year of 1995/1996 is presented.
3.2.1 Hydrology
Figure 4 shows the observed and predicted bounds for
discharge for the water year 1995/1996 for Reaches 4, 8
and 19, respectively. It is worth noting that the GLUE
prediction limits are not confidence limits in a statistical
sense, but are rather derived from the likelihood weighted
distribution of outputs from all the behavioural models.
This means that deficiencies in the model performance are
not compensated by a statistical model of the errors; thus,
the observed dynamics not reproduced can be identified
(see Beven 2006 for a more complete discussion). Fig-
ure 4a shows that in Reach 4 the model under-predicts
discharge during the wet season, but seems to predict well
the discharge during the drier months of June to October.
Reach 8 (Fig. 4b) similarly shows a good prediction of
discharge in the dry season, but an over-prediction of peak
discharges and an under-prediction of the falling limb in
the wet season. Reach 19 (Fig. 4c) shows that the contri-
bution of groundwater during the drier months is over-
estimated, but the representation of the rising and falling
limbs in the wet season are visually good. The difference in
the ability of the model to accurately predict the observed
discharge during the drier months may be a result of using
the same BFI for all reaches. According to Wade et al.
(2004), the expertly calibrated BFI values for the three
reaches are 0.65, 0.7 and 0.63, respectively. However, the
EA hiflows database (http://www.environment-agency.gov.
uk/hiflows) gives the BFIHOST values [BFI derived using
the Hydrology of Soil Types (HOST) classification] as
0.593, 0.610 and 0.587, respectively. Although different,
they both agree in the ranking of the reaches, as do
the ‘peak’ of the distributions derived from the GLUE
simulations: 0.608, 0.609 and 0.603. It is suggested that
the use of a single BFI value has led to an over-prediction
in groundwater contribution during the drier months of
the year in Reach 19, based on the assumption that the
optimal homogeneous BFI value is higher than the true
BFI value of Reach 19, and lower than the true value at
Reach 8.
The difference in the ability of the model to accurately
predict the observed discharge during storm peaks may be
related to how the areas relating to each weather station are
divided. Reach 8 is very near the boundary between the
two respective areas, and the difference in HER between
the two areas is high. It is highly likely that the true HER at
Reach 8 is less than what is being used to drive the model
and hence the model over-predicts storm events in this
reach, whilst at Reach 4 the opposite appears to be true and
the model is under-predicting both storm peaks and
recession curves. The model at Reach 19 visually fits this
part of the observed hydrograph well. The difference in the
ability of the model to predict accurately the observed
discharge data during the falling limb of a storm event is
again related to using the same value for the BFI.
3.2.2 Phosphorus
Unlike the hydrology, there are very few model simulations
that reproduce the TP data acceptably and none that
reproduces the TRP data. As mentioned, the data being
used to assess the SRP model output is actually TRP. Wade
et al. (2007) demonstrate that INCA-P is capable of
reproducing the basic dynamics of the observed TRP, if not
the absolute values. However, even after the model output
has been adjusted (as described in Sect. 4.2) using a ratio
formulated between TRP and SRP, this study is still unable
to find any simulations which meet the fairly relaxed
threshold of R2 C 0.3.
All of the reaches show a similar response to changes in
parameter values in predicting TP, and hence only pre-
diction bounds for Reach 21 with the largest number of
behavioural parameter sets (41, see Table 3) are shown to
assess the model responses (Figs. 5, 6). In all reaches, there
are observations that are not well reproduced by the
behavioural models.
Figure 5 does clearly show, however, the width of the
prediction bounds increasing during the autumn, indicating
the increase in uncertainty of TP concentration during this
‘flushing’ period. It also shows how the model underesti-
mates the concentration of phosphorus during the early
autumn flush (August and September), but is well within
the wide prediction bounds at the peak of the autumn flush
in October. Figure 7 shows the temporal and spatial vari-
ation of the observed TP for multiple reaches. A pattern of
an autumn flush can be clearly seen in most of the reaches.
The pattern is less clear in Reaches 1 and 6, where con-
centrations are lowest. There is a small sewage treatment
works entering into Reach 3 which has a significant impact
on the TP concentration compared to the neighbouring
reaches with observed data. The effect of phosphorus
stripping commencing in the sewage treatment works in the
latter half of the time series is also very apparent.
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Fig. 4 1995/1996 observed flow with 5 and 95% GLUE prediction bounds for a Reach 4; b Reach 8; and c Reach 19
1002 Stoch Environ Res Risk Assess (2009) 23:991–1010
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4 Discussion
The results show that under the assumptions detailed, all
the simulations run using INCA-P can be rejected as
unacceptable representations of the observed data when
evaluated over all the available observed data sets. This is
not the first time that a GLUE study has rejected all models
(Freer et al. 2003; Zak and Beven 1999a). Assessing all
models as unacceptable then requires a consideration of
why this is the case in this application.
4.1 Ability of INCA-P to produce behavioural
simulations
The success of a model in producing an acceptable simu-
lation depends on four aspects: an adequate model
structure; adequate input data to drive the model; adequate
observations with which to evaluate model performance,
and the definition of ‘acceptable’ behaviour. The difficulty
in finding acceptable model runs for TP and TRP in this
application may be a result of all these causes. The data do
Fig. 5 Observed TP
concentrations with 5 and 95%
GLUE prediction bounds for all
41 behavioural models in Reach
21, for water year 1995/1996
Fig. 6 Observed TP concentrations with 5 and 95% GLUE prediction
bounds for all 41 behavioural models in Reach 21, for the entire study
period. For comparison, the expert calibration derived simulation (not
shown) is generally contained within the prediction bounds; it tends to
be slightly under the lower bound during periods of low concentration
Stoch Environ Res Risk Assess (2009) 23:991–1010 1003
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not allow us to speculate about whether some process
representations in INCA-P could be improved (although
see the next section), since the inputs with which the model
is being driven are so uncertain.
The model successfully produces the expected differ-
ence in response between a reach dominated by point
sources of phosphorus and a reach dominated by diffuse
sources (Wade et al. (2004, 2007)). Figure 8 shows the
observed and simulated data for two differing reaches. The
simulated data points are those produced from the best
(highest total R2 across all performance measures) model
run from the Monte Carlo realisations. Figure 8a shows a
weak positive relationship (between TP (Reach 1) and
discharge (Reach 4). Such a weak positive response is
indicative of an agricultural catchment, where rainfall
instigates the mobilisation and delivery of phosphorus
associated with soil particles to the channel (Jarvie et al.
2006; May et al. 2001; Wade et al. 2004). Jarvie et al.
Fig. 7 Available observed total
phosphorus data for the study
period for different river reaches
to show the significant
variability in observed TP
concentrations
Fig. 8 Differences in the
flow—TP concentration
between a diffuse (a) and a
point (b) dominated reach: aReach 4 flow plotted against
reach 1 TP concentration; bReach 19 flow plotted against
Reach 21 TP concentration.
Crosses best Monte Carlo
simulation, stars observed
1004 Stoch Environ Res Risk Assess (2009) 23:991–1010
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(2006) identified the same response in the Chitterne
catchment, but attributed the relationship to the interactions
with sewage soakaways rather than overland diffuse pol-
lution. The weakness of the signal is caused by the
complex interactions of factors that directly affect phos-
phorus mobilisation and transport from land to water.
These factors include antecedent conditions, soil P status,
farm management styles (fertiliser usage and livestock
management) and rainfall (quantity and intensity). Fig-
ure 8b shows a weak negative relationship between TP
concentration (Reach 21) and discharge (Reach 19). The
response shows that TP concentrations decrease with dis-
charge (dilution), which is indicative of point sources of
phosphorus (Neal et al. 2004a). The few higher points at
high discharges are from the beginning of the time series
before phosphorus stripping was introduced at Leominster
STW. The highest point was taken during the first event
following the first dry spell of the time series. This appli-
cation of INCA-P only finds acceptable model runs for
reaches dominated by point sources. The signal from a
point source is likely to be much higher than from a diffuse
source and so easier to identify against background varia-
tion and measurement error. Wade et al. (2004) state that
the poor performance in Reach 1 is due to the inability of
INCA-P to predict responses where source areas and flow
pathways are highly heterogeneous, and also note that work
is required to improve the TRP simulations, particularly
where concentrations are low and the result of diffuse
pollution.
Diffuse pollution from agriculture can be very difficult
to predict because of the many factors driving it, and the
relatively low concentrations typically measured are sub-
ject to higher percentage error and are closer to detection
limits of analytical techniques. Diffuse sources are also
intermittent and transient, dependent on factors for which
there is little or no data, e.g. the sporadic cleaning out of a
cow yard or variable phosphorus transport pathways
(Beven et al. 2005). These kinds of incidences, if they
coincide with the taking of sparse observations of stream
concentrations (typical of a national monitoring network),
can dominate a weak signal, but are extremely difficult to
account for.
4.2 Failure of INCA-P to produce any acceptable
TRP runs
In this study, INCA-P does not reproduce acceptably the
observations of TRP as defined by the R2 threshold of
explaining more than 30% of the observed variance. A
visual comparison of the simulated and observed stream
water TRP dynamics (Fig. 6) does show that, in the lower
reaches dominated by point sources, the general patterns of
dilution during winter and concentration during summer
and a general downward trend in concentration due to
phosphorus removal from Leominster STW and Cadburys
are reproduced. Because no behavioural simulations are
identified in this study, no further analysis is done to look
at prediction bounds or sensitivities, but it is interesting to
consider here why no behavioural simulations are found.
SRP concentrations in INCA-P are calculated from the TP
concentration. INCA-P uses a linear relationship to relate
SRP to TP, whereas the observed relationship is more
complex.
INCA-P simulates SRP but the available phosphorus
fraction from the EA measurements is TRP. Using a ratio
of 1.32 derived from Haygarth et al. (1997) to estimate the
TRP from the model output of SRP data, it is possible to
say that both the expert calibration and the ‘best’ MC run
are still considered unacceptable, although the R2 scores
(not shown) are marginally higher (see Fig. 9 for the
adjusted relationship). Figure 9 further shows that although
INCA-P predicts the main trend of the relationship between
TRP and TP, it is the outliers in the observed data in this
relationship that are the primary reason for the failure of
Fig. 9 Relationship between total reactive phosphorus (TRP) con-
centration and total phosphorus (TP) concentration for all reaches.
Crosses simulated (using expert calibration), stars observed. The plotshows that the observed TRP to TP relationship is variable, but the
model which initially predicts SRP calculates a linear relationship to
TP. In this case, the model output has been adjusted to better reflect
TRP using a ratio derived from Haygarth et al. (1997)
Stoch Environ Res Risk Assess (2009) 23:991–1010 1005
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the model to produce any acceptable simulations in the
GLUE simulations.
In developing INCA-P, data from the River Lambourn
shows a strong linear relationship between soluble unre-
active phosphorus and TP (Wade et al. 2002c). This
implies that soluble unreactive phosphorus can, therefore,
be represented as a constant fraction of TP. Wade et al.
(2002a, b, c) acknowledge that this is a simplification, and
have since added an additional parameter to allow the
proportion of TP existing as soluble unreactive phosphorus
to be varied by reach (which would allow the general trend
of the observations, but not the outliers, in Fig. 9 to be
more closely matched). However, they also state that the
soluble unreactive phosphorus proportion is not only spa-
tially variable but should also vary with discharge (Wade
et al. 2002a), so that the use of a constant proportion is still
a simplification. This raises an interesting problem in
model evaluation. If it is known a priori that the outliers in
Fig. 9 (or any other predicted variable) cannot be predicted
by the model, then should they be used in model evalua-
tion, should they be treated in terms of the ‘‘effective
observation error’’ discussed by Beven (2006), or should
some other explicit error model be used for the observa-
tions? These approaches remain to be tested in future
works.
4.3 Data limitations
The measured stream water phosphorus concentrations are
subject to high levels of uncertainty due to a number of
factors, as detailed in Jarvie et al. (2002): sampling
method, storage, and analytical techniques can all con-
tribute to an uncertain result. Some of the analytical
techniques used to determine fractions of phosphorus can
become unreliable when certain other compounds are also
present in the water sample. They also comment on how
much EA data, referred to as ‘orthophosphate as P’, is in
fact molybdate reactive phosphorus (or TRP) as the anal-
ysis is performed on unfiltered samples, and so will contain
some particulate phosphorus which is reactive to the
reagents.
There is also an issue of incommensurability between
what is modelled and the available observed phosphorus
data. Samples for phosphorus analysis are collected by the
EA as grab samples which are only truly representative of
that single point in time and space, whereas in common
with most models working on a daily timestep, this data is
assumed representative of a daily mean value. However,
there are many studies showing how phosphorus concen-
trations can vary with discharge at sub-daily timesteps
(Cooke and Prepas 1998; Johnes 2007; Kronvang and
Bruhn 1996; May et al. 2001; Pionke et al. 1996; Svendsen
et al. 1995); hence, the assumption that a grab sample is
representative of a daily mean could have strong implica-
tions for the ability of the model to represent the observed
data. As well as the issue of incommensurability of the
observed data, there is also an issue with the amount of
data available and its information content. The motivation
behind the sampling regime of the EA is driven by consent
verifications and EC Directives; they sample fairly infre-
quently at most locations (approximately monthly) and are
not concerned with obtaining samples representative of the
full discharge range. Hence, the data being used to calibrate
the parameter values has relatively low information content
regarding the full dynamics of phosphorus transport.
Despite this, a user should still expect the model to
reproduce the majority of available data, if the measure-
ment and commensurability issues are allowed for.
In this application of INCA-P, the hydrological driving
input data (HER, SMD and temperature) used are derived
from two single-site MORECS calculations. As mentioned
previously, there is a large difference in HER between the
two sites which is not seen in the average rainfall or dis-
charge observed in the Lugg catchment. The decision to
divide the catchment into upper and lower sections at
Reach 9 is inappropriate, and if the division occurred at
Reach 7 better fits may be found. It is also suggested that
the use of MORECS derived HER data as opposed to
rainfall data may be inappropriate, and that a better fit
might be obtained if more local HER data, i.e. from within
the catchment, is available and used.
The phosphorus-related driving data are derived from a
number of sources (Wade et al. 2004, 2007). MORECS
uses an estimate of the growing season for each land use to
calculate evapotranspiration. These estimates are used
within INCA-P. The land use and livestock numbers are
based on a 1 km2 grid provided by ADAS based on the
1995 agricultural census figures, which have been re-
mapped by ADAS to allow for undisclosed data. It is
assumed that the figures from 1995 would be representative
of the study period. Fertiliser practice has been generalised
to a regional level, the monthly load estimates were based
on a farm survey conducted in the River Ant catchment
(Johnes and Butterfield 2003). A 70:30 split between
organic and inorganic phosphorus is also assumed. As
noted in Wade et al. (2007), due to the Non-Disclosure Act
(whereby a single holding cannot be identified), the actual
amount of phosphorus input may be lower than reported.
This may account for the large amount of phosphorus
accumulating within the firmly bound store. The data
available for the sewage work and trade effluent discharges
in this application is limited. For the smaller sites, an
average TP concentration and discharge is applied across
the entire study period, while the larger sites (Leominster
STW and Cadburys Factory) had monthly averaged values.
It was also decided in the original studies (Wade et al.
1006 Stoch Environ Res Risk Assess (2009) 23:991–1010
123
2004, 2007) that only those effluents being discharged
directly into the main channel would be included. Because
point sources on tributaries cannot be included in a single
model application. Wade et al. (2004, 2007) recognised
that this may be an over-simplification and further inves-
tigations to test the validity of this assumption are needed.
5 Conclusions
Despite the growing popularity of the INCA-P model, and
the suggested incorporation of it into the hierarchy of
models for use by the EA, there had been no peer-reviewed
consideration of the uncertainty in its predictions. This
paper has presented an uncertainty analysis of the INCA-P
model within the GLUE framework in an application to the
River Lugg (1,077 km2). It has improved understanding of
the performance of the INCA-P model through the con-
sideration of 200,000 Monte Carlo realisations of
parameter sets, and comparison of the model performance
using the best homogeneous parameter Monte Carlo run
and the expert calibrated heterogeneous parameter set.
Similar to the findings of McIntyre et al. (2005), in an
application of the nitrate version (INCA-N), it was found
that the most hydrologically sensitive parameters in the
Lugg application were groundwater-related. This was
expected due to the permeable nature of the catchment. A
large number of acceptable models of the hydrology were
found, although the prediction bounds over all behavioural
models showed some consistent departures from measured
discharges in the different reaches. The TP simulations
were sensitive to the groundwater parameters but also to
the parameters controlling the firmly bound phosphorus
stores. Although none of the models tried produced
acceptable models to all of the TP and TRP concentration
observations as defined by a R2 threshold of 0.3, it was
found that INCA-P was capable of reproducing the basic
difference between point and diffuse sources of pollution
and the seasonal and inter-annual patterns in the TP and
TRP concentrations in the lower reaches of the Lugg
system.
The failure of the model demonstrated here is in part a
result of limitations in the model structure, but also a result
of inadequate input data and measurement and commen-
surability errors in the observations with which the model
outputs were compared. Following a workshop held as part
of the Euro-limpacs project, INCA-P has since been
updated to better account for terrestrial sediment transport,
soil type, slope and particulate and soluble fractions of
phosphorus. Irrespective of changes to the model structure,
the application of any model depends on having adequate
input data in time and space, here in describing the vari-
ability in the hydrological conditions and flow pathways in
the different sub-catchments, in the spatial heterogeneity of
land use, and in the variability in inputs of phosphorus. In
addition, the data and knowledge are not available to
enable good prior estimates to be made of effective values
of so many parameters at the sub-catchment scales required
in such a model. Until these issues are confronted, the
performance of the INCA-P model and confidence in its
predictions will necessarily be limited.
5.1 Wider comment
The implementation of the WFD is driving a need for
models that can predict the impact of spatially distributed
phosphorus sources, changing flow-pathways and the likely
response to changes in climate, land-management and point
source inputs. INCA-P has the capability to make these
types of dynamic predictions, but this work has shown the
limited accuracy of the model predictions due to uncertainty
in the input data and in the representation of complex pro-
cesses by a simple model structure. As technologies for
remote sensing and high-frequency in-situ monitoring of
soil- and stream-waters become more reliable, widespread
and cheaper, there is then hope that new data will become
available that can be used to improve process representa-
tions and better constrain the uncertainty in predicting
phosphorus concentrations. It is generally the case, how-
ever, that improving process representations increases the
number of parameters that need to be estimated to run the
model. Thus, there will be an inevitable compromise
between improvements and uncertainty in applications to
those majority of sites that will not be data rich (Beven 2000,
2002). Krueger et al. (2007) have suggested that problems of
this type would require new modelling strategies within an
uncertainty learning framework for improving phosphorus
model development and evaluation. Importantly, we need to
further investigate what is the correct balance between the
process complexity of the models we develop with the
quality and quantity of observations we will ever have to
drive and evaluate models used to inform WFD policy
decisions at national scales of interest.
There is still much research to be done to build confi-
dence in this type of modelling. Ideally, model output
should be compared with long-term ([30 years) data sets
to identify if the model has been able to capture major
perturbations in a river system; this would provide some
confidence that a model was responsive to environmental
change. Such models can also be useful as ‘learning’ tools
for exploring the possible responses of a catchment to
environmental change and investigating the key factors and
processes operating. Such analysis may help to spark new
thoughts for experiments and monitoring with which to
assess the impacts of environmental change. The present
results suggest, however, that such studies need to estimate
Stoch Environ Res Risk Assess (2009) 23:991–1010 1007
123
the uncertainties involved in making such predictions and
explicitly consider the magnitude of such uncertainties in
drawing conclusions that might be later used in decision
making.
Acknowledgments This research was supported by the NERC
Grant Number NER/L/S/2001/00658. The contribution of AJ Wade
and D Butterfield was supported by the Environment Agency Science
Project P2-137
Appendix
See Table 4.
Table 4 Constant parameter values not adjusted as part of the Monte Carlo simulations (multiple values in the ‘expert calibration’ column
indicate that there is not just one parameter to represent all six land uses or all 22 reaches)
Parameter Expert calibration Units
Max. temperature difference (applies to 6 land uses) 0; 4.5 �C
Fertilisation addition start day (applies to 6 land uses) 0 Day
Fertiliser addition period (applies to 6 land uses) 0 Day
Plant growth start day (applies to 6 land uses) 1; 70; 77 Day
Plant growth period (applies to 6 land uses) 365; 160; 221 Day
Diffuse direct runoff suspended sediment input (applies to 6 land uses) 150 mg l-1
Diffuse boron input (applies to 6 land uses) 0 mg l-1
Direct runoff sustainable flow (applies to 6 land uses) 9,999 m3 s-1
Soil water sustainable flow (applies to 6 land uses) 9,999 m3 s-1
Groundwater sustainable flow (applies to 6 land uses) 9,999 m3 s-1
Mineralisation temperature threshold (applies to 6 land uses) 9,999 �C
Immobilisation temperature threshold (applies to 6 land uses) 9,999 �C
Diffuse soil water suspended sediment input (applies to 6 land uses) 80 mg l-1
Diffuse groundwater suspended sediment input (applies to 6 land uses) 10 mg l-1
Initial suspended sediment 5 mg l-1
Initial boron 0 mg l-1
Initial macrophyte mass 1 g C m-2
Initial epiphyte mass 0.01 g C m-2
Sediment grain size 100 lm
Sediment settled or resuspended 0.1 kg
Initial live phytoplankton 2.12 lg CHl ‘a’ l-1
Initial dead phytoplankton 1 lg CHl ‘a’ l-1
Proportion of P in epiphytes 0.0054 g P (g C)-1
Half-saturation of P for epiphyte growth 0.02 mg P l-1
Half-saturation of P for macrophyte growth 0.02 mg P l-1
Macrophyte self-shading constant 10 g C m-2
Proportion of P in macrophytes 0.0054 g P (g C)-1
Phytoplankton death rate 0 Day-1
Phytoplankton growth rate 0 Day-1
Half-saturation constant for phytoplankton growth (mg P l-1) 1 mg P l-1
Phytoplankton self-shading constant 1 g C m-2
Dead phytoplankton settling rate 0 Day-1
Flow a parameter (applies to 22 reaches) 0.04 n/a
Flow b parameter (applies to 22 reaches) 0.67 n/a
Macrophyte temperature dependency (applies to 22 reaches) 1.066 n/a
Epiphyte temperature dependency (applies to 22 reaches) 1.066 n/a
Phytoplankton temperature dependency (applies to 22 reaches) 1.066 n/a
Bed suspension potential (applies to 22 reaches) 0.1 n/a
Bulk sediment density (applies to 22 reaches) 2.65 kg m-3
1008 Stoch Environ Res Risk Assess (2009) 23:991–1010
123
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