Excess nutrients in surface waters, such as phosphorus (P) from agriculture,
result in poor water quality, with adverse effects on ecological health and
costs for remediation. However, understanding and prediction of P transfers
in catchments have been limited by inadequate data and over-parameterised
models with high uncertainty. We show that, with high temporal resolution
data, we are able to identify simple dynamic models that capture the P load
dynamics in three contrasting agricultural catchments in the UK. For a flashy
catchment, a linear, second-order (two pathways) model for discharge gave
high simulation efficiencies for short-term storm sequences and was useful in
highlighting uncertainties in out-of-bank flows. A model with non-linear
rainfall input was appropriate for predicting seasonal or annual cumulative P
loads where antecedent conditions affected the catchment response. For
second-order models, the time constant for the fast pathway varied between 2
and 15 h for all three catchments and for both discharge and P,
confirming that high temporal resolution data are necessary to capture the
dynamic responses in small catchments (10–50 km
The quality of both surface waters and groundwater is under increasing pressure from numerous sources, including intensive agricultural practices, water abstraction, climate change, and changes in food production and housing provisions to cope with population growth (Carpenter and Bennett, 2011). Sediment and nutrient concentrations and loads are of concern to water utility companies and to environmental regulators who are striving to meet stringent water quality standards. However, accurate estimation of loads requires accurate, high temporal resolution measurements of both discharge and nutrient concentrations (Johnes, 2007) and should include quantification of observational uncertainties (McMillan et al., 2012). Sediment and nitrogen are frequently and relatively easily measured in situ. In contrast, phosphorus (P) concentrations for water quality assessments are typically measured by manual or automatic sampling followed by laboratory analysis, often at monthly resolution, which do not capture the dynamic nature of P concentrations, and result in biased estimates of P load (Cassidy and Jordan, 2011). Phosphorus concentration in rivers and streams is controlled by many factors, including rainfall, runoff, point sources, diffuse inputs, and in-stream P retention and processing. Some of these factors, particularly for small catchments, change at timescales of minutes to hours, and thus the dynamics of P concentration and load need to be studied at similar timescales. In this study, hourly time series of rainfall, runoff and P concentrations are used to help understand hydrological transport pathways of P for three contrasting agricultural catchments across the UK.
There is a wide range of complexity in hydrological and water quality models,
applicable on a range of scales and for different purposes. In most models
there is a balance between practical simplifications and model complexity,
which depends on catchment size and knowledge (or lack thereof) of the hydrological
processes, data availability and computing power. Some of the less complex
models for diffuse pollution include export coefficient models (Johnes, 1996)
and the phosphorus indicators tool (PIT) (Heathwaite et al., 2003; Liu et
al., 2005). The most complex water quality models are idealised,
process-based representations of our best understanding of reality, with a
highly complex, fixed structure and many parameters, for which there is often
little or no site-specific data (Dean et al., 2009). These models often
include a component for sediment-bound P, where the sediment transfer is
based on a form of the universal soil loss equation (USLE), which is a
semi-empirical model known to perform poorly (Evans and Boardman, 2016).
Results generated by such process-based models are often highly uncertain,
due to the uncertainty in both the model parameters and the model structure
(Parker et al., 2013; Jackson-Blake et al., 2015). A review of pollutant loss
studies using one process-based model, the soil water assessment tool (SWAT),
revealed that most applications used a monthly time step for calibration,
with few applications using a daily time step and none using a sub-daily time
step. Model fit for total P (TP) concentration, measured by the Nash–Sutcliffe
efficiency, often exceeded 0.5 but could be as low as
Location and topography of study catchments. Newby Beck, Eden,
Cumbria: location
Hydrological models are subject to uncertainties in structure, parameters and measurement data (both input and output observations) (Krueger et al., 2010), and understanding the errors in measurement data is a prerequisite to better understanding of the other uncertainties in modelling (McMillan et al., 2012). Young et al. (1996) recommended constructing models that capture the dominant modes of a system, with as few tuneable parameters as possible. Transfer function models, whose structure and parameters are determined by the information in the data, are considered to be among the most parsimonious for rainfall–flow relationships (McGuire and McDonnell, 2006; Young, 2003). Data-based mechanistic (DBM) modelling, which uses time-series data and fits a range of transfer functions, allows the structure of the model to be determined by the information in the monitoring data. There will still be structural errors in a DBM model, as it tries to represent a continuum of flow pathways with just the dominant modes, but this simplification will be determined by the information in the data rather than being pre-selected. This assists in getting the right answers for the right reasons (Kirchner, 2006). In contrast, there is a danger in process-based models that one might fit quite different model structures or parameter sets to the available data, i.e. the equifinality problem (Beven, 2006; Beven and Freer, 2001). An optimal DBM model and associated parameters are identified using statistical measures, but a model is only accepted if it has a plausible physical explanation (Young, 1998, 2003; Young and Beven, 1994; Young et al., 2004). With the increasing availability of high temporal resolution datasets for additional variables alongside stream discharge (Bieroza and Heathwaite, 2015; Bowes et al., 2015; Halliday et al., 2015; Outram et al., 2014), this technique has been used effectively for relating rainfall to hydrogen ion concentration in rivers (Jones and Chappell, 2014), and rainfall to dissolved organic carbon (Jones et al., 2014).
The aim of this study was to investigate, for the first time, whether simple
dynamic models of P load could be identified to help understand the
hydrological P processes within three contrasting agricultural catchments in
the UK that represent a range of climate, topography, soil and farming
types. Specifically, the objectives were as follows:
to identify rainfall–runoff models for each catchment, from hourly time
series data collected over 3 years; to develop models of P load exported from each catchment, using hourly time-series data of P concentrations measured with in situ bank-side
analysers; to improve understanding of the dominant modes of catchment response through
comparison of rainfall–runoff and rainfall–TP load models for each catchment.
If successful, this would be the first time that DBM modelling has been applied to high-resolution phosphorus data in catchment science.
Three rural catchments with different temperate climate, topography and farm
types were monitored at high temporal resolution as part of the UK
Demonstration Test Catchments (DTC) programme (Lloyd et al., 2016a, b; Outram
et al., 2014; McGonigle et al., 2014). These were Newby Beck at Newby, Eden
catchment, Cumbria (54.59
Rainfall was measured at 15 min resolution at three sites in each of the Newby Beck and Blackwater catchments (Outram et al., 2014; Perks et al., 2015) and summed to give hourly totals. The hourly totals from the different rain gauges were combined by areal weighting to give an hourly time series for the catchment. For the Wylye catchment, only daily rainfall was available for sites within the catchment, so raw tipping bucket data were obtained for several sites outside the catchment and analysed to produce an hourly time series which was considered most representative of the rainfall in the catchment. Further details of the rainfall analysis for the Wylye catchment are given in Sect. S1 in the Supplement.
River water level was measured at 15 min resolution in the three catchments,
with rating curves developed for discharge estimation (Outram et al., 2014;
Perks et al., 2015; Lloyd et al., 2016b). TP concentration
was determined in situ at 30 min intervals with a Hach Lange combined
Sigmatax sampling module and Phosphax analyser using acid digestion and
colorimetry (Jordan et al., 2007, 2013; Perks et al., 2015). Total P loads
for each hour were determined by multiplying discharge (averaged to 30 min
resolution) by TP concentration for each 30 min and summing to give hourly
totals:
Transfer function models relating the input (here, a time series of rainfall,
The identification algorithm always includes a noise model; by default this assumes normally distributed, uncorrelated errors, but an auto-regressive moving average (ARMA) structure can be specified. The Gaussian noise model still results in asymptotically unbiased parameter estimates, but not necessarily the most statistically efficient (close to minimum variance) (Taylor et al., 2007). In this study, models up to third order were considered initially, but higher order models showed no advantage, so only models up to second order were considered in subsequent evaluations. Full models (input–output (I-O) plus ARMA structured residual noise) were assessed initially and overall they did not produce better results in all cases; therefore, in order to keep a consistent approach for all catchments, structured noise models were not specified in later model identification. In addition, transfer function models with a structured noise component generally do not improve longer-term predictions of processes which are I-O dominated. The residuals structure was not strong enough for a structured noise model to improve the model fit consistently. If there was a strong structure in the residuals, it would suggest that something was being missed in the DBM system representation. The time delay constants were estimated from the data at the same time as the model structures.
Continuous-time and discrete-time model structures are described below (from Ockenden et al., 2017). The parameter estimates in both continuous-time models and discrete-time models are formulaically related (Table S3).
A second-order discrete linear transfer function, denoted by [2, 2,
In continuous time, a transfer function model with time delay
This method of model identification requires high temporal resolution data that capture the dynamic response to the driving input; therefore, it cannot work if input data (in this case, rainfall) are missing, and does not perform well if too much output data (in this case, discharge or TP load) are missing or not showing a response. For the Newby Beck catchment, linear models were identified for short storm sequences up to 1 month, and were considered applicable to periods of similar conditions. These short-term models had a simple linear structure and very few parameters (five for a second-order model). As this paper is evaluating a methodology, successful modelling on different timescales can be used as validation of the approach. Models were not identified for short periods for Blackwater and Wylye, as the presence of a much slower pathway (with a time constant of the same order as the length of the identification period) did not allow model parameter estimates to be sufficiently constrained over such short periods.
For longer time series, when seasonal change and antecedent wetness are
expected to have an impact on the response, linear models were improved by
inclusion of the rainfall–runoff non-linearity (Beven, 2012) based on
the storage state of the catchment, for which the discharge is used as a
proxy, i.e.
Model fit was assessed according to model bias, to evaluate systematic over-
or under-prediction of the model, and to
The DBM technique involves the simplified representation of complex systems, based on the information in the data (Young, 1998, 2001; Young et al., 2004). In practice, this means identifying models over a range of orders, and choosing the most appropriate model order. Generally the simplest (lowest order) model which balances model fit without over-parameterisation is chosen. The chosen models often have a very simple structure, which will certainly not be a true representation of all the processes, but may model the data adequately. This structural error is accepted as part of the DBM technique in order to reveal the dominant modes of response.
The instrumental variable algorithms (RIVCBJ and RIVBJ) allow unbiased estimation of the model parameters and their covariance matrices. Monte Carlo sampling within the parameter space determined by the covariance matrices allows for uncertainty in derived quantities, such as time constants, to be calculated. In general with DBM modelling, very little of the total uncertainty is due to the parameters, partly because there are so few of them and because the linear-dynamic part of the process that the model describes is well-defined. Note that in the case of transfer function models of the hydrograph, the models do not directly reflect the transport of water in the system since the hydrograph represents the integrated effects of celerities in the system rather than flow velocities (McDonnell and Beven, 2014).
A review of measurement data uncertainty is presented by McMillan et al. (2012), including uncertainties in rainfall observations. For all three catchments in this study, input data (rainfall) was based on three rain gauges in or near each catchment. This only gives a catchment rainfall estimate, which is affected by the non-homogeneity of the rainfall field and the rainfall regime, and therefore some of the mismatch between model fit and observations (for any modelling technique) may be attributed to uncertainties in the rainfall input.
Time series of hourly rainfall, runoff and total phosphorus (TP)
concentration at the three Demonstration Test Catchments: rainfall and
runoff
A rigorous treatment of the uncertainties in high-frequency nutrient data and
its subsequent impact on loads is given by Lloyd et al. (2016b). For Newby
Beck, where stage–discharge gaugings were available, the discharge
uncertainty was estimated using the method of McMillan and Westerberg (2015),
fitting multiple plausible rating curves and weighting with a likelihood
function. This method accounts for a mix of systematic and random measurement
errors. The uncertainty of the phosphorus concentration measurements was
estimated by comparing the time series from the bank-side analyser with the
laboratory spot samples taken for ground-truthing (Lloyd et al., 2016b),
fitting multiple regression curves and weightings according to McMillan and
Westerberg (2015). The time series of discharge and TP concentration, with
their uncertainty distributions, were then combined by resampling to give the
measurement data uncertainties on the TP loads. For the Wylye, discharge
measurement uncertainties were estimated using a standard deviation of
10 %, the maximum value calculated by Lloyd et al. (2016b) for the
gauging site at Brixton Deverill using the method of Coxon et al. (2015).
Wylye discharges were combined with a standard deviation of
0.11 mg L
Time-series data from each catchment (Fig. 2) indicated large contrasts in
the hydrological response of each study catchment, with Newby Beck (Eden)
showing a very flashy response to rainfall (Fig. 2a). Although a fast
response at certain times was also evident in the Blackwater (Wensum)
catchment (Fig. 2c) and the Wylye (Avon) catchment (Fig. 2e), there was also
a more pronounced seasonal response, particularly in the Wylye where a large
groundwater component could be observed in the winter periods. This indicates
the importance of both high-frequency data and a long-term record, to capture
both fast and slower dynamics adequately. The errors resulting from sampling
well below the catchment dynamics have been well documented elsewhere (e.g.
Johnes, 2007; Jones et al., 2012; Lloyd et al., 2016b; Moatar et al., 2013).
TP concentrations in all three study catchments revealed peaks that
corresponded with runoff, with maximum values of 1.0, 0.9 and
1.5 mg L
Observed rainfall, discharge, total phosphorus (TP) concentration and load for the period 1 October 2012–30 September 2013, for the three catchments.
Rainfall–runoff and rainfall-total phosphorus load (TP) models
identified for Newby Beck during the period 7 November–4 December 2015, with
estimations of discharge and TP load during Storm Desmond
(5–6 December 2015). CT linear
A summary of the observed total rainfall, runoff, mean concentration and TP load is given in Table 1 for the period 1 October 2012–30 September 2013 (the hydrological year with the most complete dataset). The lowest mean annual TP concentrations were observed in the Newby Beck catchment, but combined with the highest runoff this resulted in a high total annual TP load. Conversely, although mean annual TP concentration in the Blackwater was also higher than in Newby Beck, when combined with the lowest runoff, this resulted in the lowest total annual TP load. The rainfall–runoff ratio for Newby Beck (0.65) was much higher than for the Blackwater (0.31) or the Wylye (0.32), indicating a larger capacity for storage in the latter two catchments. Despite similarity in the rainfall–runoff ratio, total runoff in the Wylye was higher than the Blackwater because of the higher total rainfall.
Detailed analysis of the high-frequency data is not included here as it has already been published by several authors (e.g. Ockenden et al., 2016; Outram et al., 2014, including hysteresis analysis; Perks et al., 2015). Investigation of the relationships between TP concentration and streamflow indicated that, for all three catchments, the TP concentration was out of phase with the streamflow; distinct hysteresis loops (Figs. S1–S3), also observed by Outram et al. (2014), showed different TP concentrations on the rising stage of a storm hydrograph compared to the same stage on the falling hydrograph. This indicates that antecedent conditions and the storage state of the catchment are important in determining the response. In order to capture the effects of storage, dynamic models are required.
For short storm sequences up to about a month, when antecedent flows for events were rather similar, linear models were identified for the Newby Beck catchment. These were useful for infilling missing discharge or TP load data, or for highlighting and estimating uncertainties in discharge and TP load when extrapolation of the stage–discharge relationship was inappropriate. The model is only reliable for the conditions covered during the calibration period, but it may still be useful when there are known problems with a stage–discharge relationship (such as during extreme events). Indeed, the stage to discharge relationship is the weakest point of all the catchment models relying on stage measurements. Whilst it was still possible to identify linear models for short periods for the Blackwater and Wylye catchments, the parameter uncertainty for these models was large; the parameters cannot be well constrained when the (slow) time constant was of similar order to the period of identification. For this reason, linear models for short periods for the Blackwater and the Wylye were not considered useful.
Table 2 shows results from rainfall–runoff and rainfall–TP load models identified for Newby Beck for a series of contiguous storms in November 2015, immediately preceding Storm Desmond (5–6 December 2015), which caused catastrophic flooding in Cumbria and Lancashire, UK. During Storm Desmond, Honister Pass in Cumbria received the highest 24 h rainfall on record (341 mm) and Thirlmere received the highest 48 h rainfall on record (405 mm). The storm was remarkable for the duration of sustained rainfall. At Newby Beck, 156 mm of rainfall was recorded in 36 h. Although the monitoring equipment was recording during Storm Desmond, the peak flows during the storm were out of bank for around 31 h (compared to less than 3.5 h during more typical storms), with anecdotal evidence that the gauging point was significantly bypassed, so these out of bank flows were highly uncertain. This measurement uncertainty is shown by the shaded bands in Fig. 3 (discharge model) and Fig. 4 (TP load model), which span the observed (calculated from stage) discharge and TP load. This is more visible in the zoomed-in periods for discharge (Fig. 3b) and TP load (Fig. 4b). Concentrations were assumed to be reasonably accurate, but discharge was likely underestimated, therefore TP loads were consequently underestimated too. Storm Desmond was not included in the model identification period. Using the models from the November period to simulate flows (Fig. 3) and TP load during Storm Desmond (Fig. 4) suggests that both discharge and TP load were underestimated. Time series and histograms of the residuals are given in Fig. S7 for discharge and Fig. S8 for TP load. The zoomed-in period for the TP load model (Fig. 4b) suggests that whilst the transfer function model got the timing of the load peak and the decay approximately right, the model generally started to respond before the observed load responded.
Although there are uncertainties associated with whether it is valid to extend the models identified above to an extreme event such as Storm Desmond, we believe that this highlights the possible underestimation in discharge and TP load during Storm Desmond and that the models in Table 2 might provide more realistic estimations of the true values.
Observed and modelled discharge per unit area
Observed and modelled total phosphorus (TP) load
Structure, response characteristics and model fit statistics of
rainfall–runoff and rainfall–TP load models for each catchment. Models were
calibrated on all or part of hydrological years 2012 and 2013 and validated
on all or part of hydrological year 2014.
Longer-term models, based on 2 years of hourly data, were identified for
each catchment. Model fits (
The dynamic response characteristics of time constant and percentage on each
flow pathway (for definitions see Table S4), determined after partial
fraction decomposition, can be compared between the study catchments for both
discrete and continuous-time models. The time constants are associated with
the dominant pathways and indicate how quickly each impulse response (of
water or TP mass) is depleted to 37 % (or fraction
The marginal distributions of the time constants and proportion of flow or TP
load (Table 3) were determined from 1000 to 10 000 Monte Carlo realisations
using the covariance of the parameter estimates. The parameter uncertainties
estimated within the DBM methodology were small, even for the response
characteristics of the TP load models, which had higher uncertainty than
rainfall–runoff models; TP load models had coefficients of variation of less
than 3 % for fast time constants, less than 6 % for slow time
constants and less than 2 % for proportions on pathways. For the
rainfall–runoff models, the time constant for the fast pathway was
2.9
First-order model between effective rainfall and total phosphorus
(TP) load at Newby Beck for the identification period
1 October 2011–30 September 2013. Continuous-time model with structure [1, 1,
1] (see Table 3);
For the rainfall–TP load models, at Newby Beck the best identified model was
a first-order model relating the effective rainfall (from the runoff model,
i.e. calculated one step at a time using the simulated discharge,
First-order model between effective rainfall and total phosphorus
(TP) load at Newby Beck, expanded from Fig. 5, for storm events in May
2012
Expanded sections of Fig. 5 are shown for storms in May 2012 (Fig. 6a) and November 2012 (Fig 6b). Time series of residuals and residuals against observed values are given for the discharge model in Fig. S9 and for the TP load model in Fig. S10. Although Fig. 5 illustrates several storms where the model underestimated the peak TP load, the model matched the shape and peak of the May 2012 storm quite well. However, once again the model started to respond to the rainfall before the observations showed a response. Figure 6b shows an example of a storm in which the TP load was underestimated by the model. The model parameter uncertainty was considerably smaller than the measurement data uncertainty. The model did not always lie within the bands indicated by the measurement data uncertainty, whereas the total model prediction uncertainty (including the residual uncertainty) would span most of the observations, indicating that the simple structure of the model does not capture all the dynamics, and that there are other sources of uncertainty (such as rainfall input) which are not quantified.
Second-order model between effective rainfall and total phosphorus
(TP) load at Wylye for the identification period
1 October 2012–30 September 2013. Continuous-time model with structure [2,
2, 6] (see Table 3);
For the Wylye, the best identified TP load model was a second-order model
relating effective rainfall to TP load, with 42
Second-order model between effective rainfall and total phosphorus
(TP) load at Wylye for storm events in November 2012
Figure 8 shows expanded sections of the Wylye TP load model, including a large storm in which the load is underestimated (Fig. 8a) and two smaller storms where the model overestimated the loads (Fig. 8b). For the Wylye catchment, the measurement uncertainty was dominated by the uncertainty of the data from the TP sensor, rather than the uncertainty in the discharge (Lloyd et al., 2016b). However, some of the mismatch between model and observations here might also be attributable to uncertainty in rainfall input: in Fig. 8a there could be an underestimate in catchment rainfall not captured by the rain gauges; conversely, in Fig. 8b the rain gauges may have captured more than the catchment-average rainfall. Time series of residuals and residuals against observed values are given for the Wylye discharge model in Fig. S11 and for the TP load model in Fig. S12.
Second-order model between rainfall and total phosphorus (TP) load
at Blackwater for the identification period 26 October 2012–28 July 2013.
Continuous-time model with structure [2, 2, 4] (see Table 3);
The TP load model used for the Blackwater was a linear model relating
rainfall directly to TP load. The second-order TP model gave fast and slow
time constants of 12.5
Second-order model between rainfall and total phosphorus (TP) load
at Blackwater for storms in December 2012
The proportion of TP load exported on the fast pathway was considerably greater for all catchments than the corresponding proportion of water on the fast pathway, by a factor of approximately 2 for Newby Beck and Blackwater and approximately 5 for the Wylye. This suggests that on the fast water pathways, generally associated with shallower pathways such as shallow sub-surface flow, field drains and surface runoff, there is more release of TP than on deeper water pathways. This is consistent with soil profiles in agricultural areas, which generally show P concentrated on the surface and in the near-surface soil layers, with a decrease in P with depth (Heathwaite and Dils, 2000).
Validation of the TP model for Blackwater and Wylye was performed on a shorter period than for Newby Beck (half of the hydrological year 2013–2014) because of missing data (Table 3, Figs. S15–S18). The power law used to represent the rainfall–runoff non-linearity did not validate the data very well in the Blackwater catchment. Different representations of the rainfall–runoff linearity were also investigated, such as the Bedford Ouse Sub-Model (Chappell et al., 2006; Young, 2001; Young and Whitehead, 1977), in which the soil storage is related to an antecedent precipitation index. Although changes in the model non-linearity representation made minor differences to model fit, none of the model variants validated the data well for the Blackwater catchment. This suggests that there may be a different mechanism at work in the Blackwater catchment, in which a fast pathway only becomes active once the soil is fully saturated, or the groundwater level rises to a certain level (Outram et al., 2016). This could be due to the shallow slopes, which encourage infiltration rather than runoff. Alternatively, the response may be more dominated by point sources which are not as rainfall-driven, or sources such as sediment-laden runoff from impervious surfaces (roads or yards), which are rainfall-driven but do not behave in the same non-linear way as the runoff from soil.
In addition, the conditions experienced during the 2 years used for model identification may not be very similar to the validation period. From the data in Fig. 1c, the winter of 2011 and spring of 2012 showed much lower discharge than the same months in subsequent years. The groundwater recharge, which is shown as an increase in the baseflow in winter, was obvious for winter 2012–2013 and winter 2013–2014 for both the Blackwater (Fig. 2c) and the Wylye (Fig. 2e), but was not evident for either catchment for the winter of 2011–2012. Because of the slow time constants for these catchments, the dataset for model identification ideally needs to be longer than for the Newby Beck catchment, where the dynamics are much faster. This study suggests that the dataset used here was not long enough for the Blackwater catchment to capture an adequate range of conditions.
Advantages and limitations of the DBM modelling method for rainfall–TP load.
The benefits and limitations of the modelling method for TP load are summarised in Table 4. For catchments that exhibit rapidly changing dynamics, such as response to storm events, models calibrated with daily data will have large uncertainties associated with the parameters (and output) because the input data do not capture the high-frequency dynamics of processes such as P transfer. This study shows that simple transfer function models using data with sub-daily resolution can simulate the dynamics of TP load, with model fits at least as good as generally achieved with process-based models (Gassman et al., 2007; Moriasi et al., 2007) and with low parameter uncertainty. Full direct model comparisons are not currently possible, as the published results for process-based models used different catchments and data sets. It is still advisable to validate a fitted model using at least a split record test (Klemes, 1986). This highlights the importance of long and complete datasets with good time resolution for properly representing both flow and TP loads for such catchments. The high data demand of DBM models is noted in Table 4. Technology and monitoring methods are improving all the time so that high-frequency data are now more readily available (e.g. Jordan et al., 2007, 2005; Outram et al., 2014; Skeffington et al., 2015) This requirement for adequate datasets is often an obstacle in the use of the DBM modelling method, but as such datasets become more available, the method can be used to improve our understanding of catchments. We should embrace efforts to improve data coverage and ways to use it widely.
The models in Table 3 have been identified using a consistent method, to test how well this modelling method copes with the different characteristics of the three catchments. The method has been successfully applied to all the catchments, although less successfully for the Blackwater catchment. It is likely that the models could be improved if catchment-specific adjustments were made or used alongside other models in a hypothetico-inductive manner (Young, 2013). For instance, in the Blackwater catchment, the use of state-dependent parameters (Young, 1984) might be more successful to capture the rainfall–runoff non-linearity. This means that, rather than using the form of the non-linearity specified by Eq. (6), the parameters could be allowed to vary according to some other observed state. In addition, model fit might be improved by accounting for heteroscedasticity of residuals (shown in residual analysis, Figs. S9–S14), through transformation of data and residuals (e.g. Yang et al., 2007). Models for all catchments could be improved by having a longer dataset, to ensure, as far as possible, that environmental conditions during a future simulation period have already been experienced during the identification period.
The models showed a pattern of underestimation of high-level TP load events and, to a lesser extent, overestimation of lower level events (Figs. 10, 12 and 14). This was more apparent for TP load than for the discharge model (Figs. 9, 11 and 13), although in many cases this was within the limits of the uncertainty in the observed data. This suggests that, for the TP load model, the non-linearity may be rainfall, discharge or load-dependent to a greater extent than allowed for in the non-linearity of Eq. (6). This could be explored using state-dependent parameter estimation, on which the power law of Eq. (6) for the flow non-linearity was originally based (Young and Beven, 1994; Young, 1984). In addition, models with at least two terms in the numerator polynomial could provide more flexibility for a differencing effect, i.e. a consistent flushing effect with higher load occurring during the rising limb of the discharge peak. This mechanism is not represented in first-order models [1 1 del], as for Newby Beck, as it requires two terms of the numerator polynomial.
The use of process-based models is often justified on the basis that the inclusion of adequate process representations will lead to more robust estimation of the response to changing environmental conditions. This is the basis for arguing that process-based models are better suited for predicting the impacts of future change. However, they also involve a plethora of (often difficult to validate) assumptions in their model structures and parameters. In practice, parameters set during calibration are rarely changed to account for changes in the modelled processes under future conditions, although by calibrating models for conditions similar to the expected future conditions, it may be possible to incorporate non-stationary parameter values (Nijzink et al., 2016). This idea could be integrated into DBM models by choosing identification periods which are most likely to reflect the conditions of the simulation period or through the use of state-dependent parameters. Thus, whilst the data-based assumption of similar conditions may be questioned when limited periods have been used for identification, usually restricted by data availability, we argue that many of the factors contributing to catchment response will not have changed (e.g. catchment topography, soil type and geology) and that this assumption will in many circumstances be no more restrictive than the (different) assumptions made when using process-based models. Clearly, where the factors contributing to catchment response have obviously changed (such as if all septic tanks were upgraded or if farm budgeting reduced the additions of P), then simple transfer function models would not be able to predict the changes over time, whereas, in theory, process-based models might be able to account for such changes, albeit with much uncertainty (e.g. Dean et al., 2009; Yang et al., 2008). However, for rainfall-dominated responses, or responses to changes in rainfall patterns, simple transfer function models can provide valuable understanding of the dominant modes of a catchment, which, in turn, can be used to target management interventions.
High temporal resolution data (hourly) of discharge and TP load have been used to identify simple transfer function models that capture the dynamics of rainfall–runoff and rainfall–phosphorus load in three diverse agricultural catchments. Linear models were identified for short storm sequences in the flashy Newby Beck catchment, when antecedent flows for events were similar. Models identified for November 2015 were used to simulate flows and TP loads in the devastating Storm Desmond (5–6 December 2015), supporting our belief that the discharge and TP load calculated from recorded data during this storm were considerably underestimated. In these circumstances, simple models could be useful to infill missing data or to highlight or estimate uncertainties in the recorded data. Linear models for short periods were not appropriate for the less flashy Blackwater and Wylye catchments when the slow time constant (for a second-order model) was similar in length to the time period of identification, making the parameter uncertainty large.
Longer-term models were identified for each of the three catchments based on 2 years of data. Comparison of rainfall–runoff and rainfall–TP load models for each catchment allowed a better understanding of the dominant modes of transport within each catchment, which was based on the time series data alone, rather than other (unmeasured) catchment parameters. In all three catchments, a higher proportion of the TP load was exported via a fast pathway than the corresponding proportion of water on the fast pathway. In agreement with soil profiles in agricultural areas, this suggested that there is more release of TP on fast (generally shallower) water pathways such as shallow sub-surface flow, field drains and surface runoff.
For successful simulations of future conditions, the models require long datasets to ensure that a full range of driving conditions has been included in the identification period. However, this study shows that simple transfer function models can be successful in modelling TP loads and explaining dominant transport modes. Transfer function models make good use of high-frequency data, require very few parameters with low uncertainty and allow physical interpretation based solely on the information in the data.
The data used in this study are openly available from Lancaster University data archive (Ockenden, 2017).
The DTC data are available from each DTC consortium until the archive is transferred to Defra (Department for Environment, Food & Rural Affairs) as the holding body.
Information about the following can be found in the Supplement:
Estimation of hourly rainfall time series for the Wylye catchment
(Sect. S1); Model assessment criteria (Sect. S2); Study catchment characteristics (Table S1); Notation (Table S2); Structure of models and relationship between
parameters from discrete-time and continuous-time models (Table S3); Definition of time constants, steady-state gains and fraction on each pathway
for discrete-time and continuous-time models (Table S4); Model structures and
parameters identified (Table S5); Hourly streamflow against total phosphorus
concentration for the Newby Beck catchment (Fig. S1), the Blackwater
catchment (Fig. S2) and the Wylye catchment (Fig. S3); Hourly streamflow
against total phosphorus load for the Newby Beck catchment (Fig. S4), the
Blackwater catchment (Fig. S5) and the Wylye catchment (Fig. S6); Time series
of residuals and histograms of residuals for short term model, Newby Beck
(Figs. S7–S8); Residual analysis, long-term models (Figs. S9–S14); Model validation (Figs. S15–S18).
MCO ran the DBM model and led the writing of the paper. WT assisted with DBM modelling. PMH was overall project lead with KJB, PJW, PDF and JZ also helping manage the project. All authors participated in interpretation of results and the writing and editing process. MCO, KJB, ALC, RE, PDF, KJF, KMH, MJH, RK, CJAM, MLV, CW, PJW, JGZ and PMH contributed to NUTCAT 2050; ALC, KMH, CB, SB, RJC, JEF and PMH are part of the DTC project.
Jim Freer is a member of the editorial board of Hydrology and Earth System Sciences.
This work was funded by the Natural Environment Research Council (NERC) as part of the NUTCAT 2050 project, grants NE/K002392/1, NE/K002430/1 and NE/K002406/1, and supported by the Joint UK BEIS/Defra Met Office Hadley Centre Climate Programme (GA01101). The authors are grateful to the UK Demonstration Test Catchments (DTC) research platform for provision of the field data (Defra projects WQ02010, WQ0211, WQ0212 and LM0304). Edited by: Christian Stamm Reviewed by: Sebastian Stoll and three anonymous referees