WATER RESOURCES RESEARCH, VOL. 47, W03515, doi:10.1029/2010WR009851, 2011
Identification of nonlinearity in rainfall-flow response using
data-based mechanistic modeling
Neil McIntyre,1 Peter Young,2,3 Barbara Orellana,1 Miles Marshall,1,4 Brian Reynolds,4 and
Howard Wheater1,5
Received 4 August 2010; revised 9 November 2010; accepted 29 December 2010; published 12 March 2011.
[1] Data-based mechanistic (DBM) modeling is an established approach to time series
model identification and estimation, which seeks model structures and parameters that are
both statistically optimal and consistent with plausible mechanistic interpretations of the
system. This paper describes the application of the DBM method to 10 min, relatively high
precision, rainfall-flow data, including observations of both surface flow and subsurface
flow. For a generally wet winter period, the preferred surface flow model is nonlinear in
flow generation and linear in routing, while the preferred subsurface flow model is linear in
flow generation and nonlinear in routing. These models have mechanistic interpretations in
terms of mass balance, hydrodynamics, and conceptual flow pathways. The four-parameter
surface and subsurface flow models explain 91% and 96% of the variance of the
corresponding observations. Other plausible models were identified but were less
parsimonious or were more reliant on prior perceptions. For a wet summer validation
period, the models performed as well as in the calibration period; however, when a long dry
spell was included, the performance deteriorated. It is speculated that this is because of
complex wetting-drying dynamics and potential nonstationarity of the soil properties that
are not sufficiently revealed in the available data. Conceptual models informed by the DBM
results matched the DBM model performance for subsurface flow but gave poorer
performance for the more complex surface flow responses. It is concluded that the DBM
method can identify nonlinearity in both flow generation and routing and provide
conceptual insights that can go beyond prior expectations.
Citation: McIntyre, N., P. Young, B. Orellana, M. Marshall, B. Reynolds, and H. Wheater (2011), Identification of nonlinearity in
rainfall-flow response using data-based mechanistic modeling, Water Resour. Res., 47, W03515, doi:10.1029/2010WR009851.
1.
Introduction
[2] A general challenge in the field of hydrological modeling is model structure identification [Dunn et al., 2008].
There are several practical criteria which may be applied
for selecting a model structure, for example, a model which
needs no more than the available data and produces the
required output; a model with which the modeler is familiar and which has a history of working well; a model
which, after estimation of parameter values, seems to perform satisfactorily on the catchment under investigation or
on an analog catchment ; or a model of suitable complexity
so that uncertainty in parameter values is minimized. In
practice, for any application, normally only one or perhaps
a few readily available model structures are considered
1
Department of Civil and Environmental Engineering, Imperial College
London, London, UK.
2
Lancaster Environment Centre, Lancaster University, Lancaster, UK.
3
Fenner School of Environment and Society, Australian National University, Canberra, ACT, Australia.
4
Centre for Ecology and Hydrology, Environment Centre Wales, Bangor,
UK.
5
National Hydrology Research Centre, University of Saskatchewan,
Saskatoon, Saskatchewan, Canada.
Copyright 2011 by the American Geophysical Union.
0043-1397/11/2010WR009851
without using a rigorous selection process, and consequently, many potentially better models are overlooked. It
is commonly noted that ideally, the search for the optimal
model structure should be approached more systematically,
and the development of identification procedures has
received significant attention from hydrological modelers
[e.g., Wagener et al., 2003; Young, 2003; Lee et al., 2005;
Kuczera et al., 2006; Lin and Beck, 2007; Liu and Gupta,
2007; Son and Sivapalan, 2007; Clark et al., 2008; Fenicia et al., 2008; Bulygina and Gupta, 2009; Reichert and
Mieleitner, 2009].
[3] A common model identification procedure is based on
developing prior hypotheses about hydrological processes in
the catchment under study, conceptualizing these into a numerical model, and testing the performance (usually against
measurements of flow from the catchment outlet) and, if performance is unacceptable, then rejecting and renewing the hypothesis, proposing a new model, and so on. This may be
called the ‘‘hypothetico-deductive’’ approach [Young, 2003].
A feature of this approach is that the prior range of potentially
relevant processes may be wide, leading to comprehensive,
highly parameterized, highly uncertain, ‘‘physics-based’’
models [Beven and Freer, 2001]. While this type of model
has a role in speculative scenario analysis [e.g., Jackson
et al., 2008], it does not easily allow for the identification of
the dominant modes of response that are most valuable for
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developing conceptual understanding and for uncovering
behavior that was unexpected a priori.
[4] An alternative procedure for model structure identification is the inductive, data-based mechanistic (DBM)
approach, which has developed over many years from its
original conception [Young, 1978] to the first use of the
name DBM by Young and Lees [1993]. Its initial application
to rainfall-flow modeling [e.g., Young, 1974, 1993; Whitehead et al., 1976; Young and Beven, 1994] has led to continued use for this purpose over a number of years [e.g., Young,
2001, 2003; Young et al., 2007; Taylor et al., 2007].
[5] The DBM approach aims to impose minimal prior
constraints on the model and to evolve the model through
the assimilation and analysis of measured data. By relying
as much as possible on information in the measurements,
such an ideology minimizes the subjectivity in hypothesis
making, and by seeking the simplest possible model structures supported by the data (i.e., parsimony), the approach
aims to minimize the risk of accepting wrong, overparameterized models and seeks to isolate the few dominant modes
of response that dictate the dynamic behavior of the system
[Young and Ratto, 2009, 2011]. Recognizing that any set of
hydrological measurements gives an incomplete picture of
the physical system, a range of possible response modes
may well be omitted with such an approach, which is dependent on the information content in the data. It follows
that a data-based model should not be used to extrapolate
far beyond the realm of the measurements on which it is
based. However, increased confidence in predictions and
increased applicability of the model for exploring scenarios
may be justified if the model has physical characteristics
that relate well to physical processes being investigated, despite its largely empirical origin.
[6] The DBM approach aims to identify statistically
optimal yet physically plausible models. It is normally
based on statistical identification of linear or nonlinear
transfer functions and the subsequent decomposition of
these transfer functions into models that may have a physical interpretation, using algorithms available in the CAPTAIN Toolbox for Matlab (the fully functional CAPTAIN
Toolbox for Matlab can be downloaded from http://
www.es.lancs.ac.uk/cres/captain/ and is available for use,
without a license, for 3 months). Only models that are considered to perform well, are statistically well defined and
parsimonious, and have an acceptable physical interpretation are accepted. The identified structure may then be used
in a number of ways: for example, the assessment of conceptual models [Ratto et al., 2007], real-time forecasting
[Young, 2002; Romanowicz et al., 2006], identification of
spatial signals in response [McIntyre and Marshall, 2010],
and providing links with reduced-order models that are
used in the ‘‘emulation’’ of associated physically based
models [Young and Ratto, 2009, 2011].
[7] This paper describes the application of the DBM
modeling approach to identify nonlinearity in the rainfallflow relationship and the subsequent attempt to infer a conceptual model on the basis of the DBM modeling results.
As opposed to previous DBM applications to rainfall-flow
modeling, which have used hourly to daily data, this study
uses 10 min resolution measurements of rainfall and of
flow in two parallel flow pathways. It is proposed that
because of the small spatial scale in this case, the rainfall
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measurements represent the catchment average rainfall
more precisely than typical DBM applications where there
is significant noise in the rainfall inputs associated with
spatial sampling error. With this ‘‘low-noise’’ rainfall,
along with dual-pathway flow data, we are able to extract
more information about nonlinear processes than has previously been achieved using DBM modeling, including internal routing nonlinearity, rather than just nonlinearity in the
effective rainfall input.
2.
DBM Modeling
2.1. Linear Transfer Function
[8] The general form of a single input discrete time linear transfer function is as follows:
qk ¼
^
b0 þ b1 z1 þ b2 z2 þ þ bm zm
xk=t ;
1 þ a1 z1 þ a2 z2 þ þ an zn
ð1Þ
where x is the measured model input, ^q is the model output,
a ¼ [a1 a2 an]T and b ¼ [b0 b1 bm]T are parameter vectors, t is the uniform sampling interval used in the model
(¼ tk tk1), k is the time step number (sampling index),
zr is the backward shift operator (i.e., zr xk ¼ xkr), is
the time delay between the onsets of x and ^q (restricted to
being an integer multiple of the sampling interval t when
using a discrete time model), and m þ 1 and n are the number of numerator and denominator terms, respectively.
Within the DBM context, the parameters of equation (1)
are usually estimated using the refined instrumental variable methods [Young, 1984 and references therein]
(enlarged and revised version in text, in press, 2001) and
available in the CAPTAIN Toolbox. In the context of
DBM rainfall-flow modeling, where ^q is the estimated
streamflow, it is common for the measured input x to be
defined as the estimated effective rainfall ^uk , in which case
equation (1) acts as a flow-routing function. Here and for
the rest of the paper, we use the term ‘‘flow routing’’ to
describe the movement of effective rainfall through the
subsurface and/or surface of a catchment to a specified outlet location. While the generation of effective rainfall and
its movement as flow through a catchment are not distinct
processes, they are commonly treated this way in rainfallflow models [Beven, 2001], and this is a fundamental
assumption used in the DBM modeling described below.
[9] Equation (1) defines a family of model structures
depending on how many terms are included and the time
delay (i.e., the values of m, n, and ). Naturally, we seek a
model structure that gives a good fit between modeled
and observed response (flow in our context), as measured,
for example, using the coefficient of determination (RT2)
applied to the simulated model residuals [Pedregal et al.,
2007, equation 6.54]. RT2 is equivalent here to the NashSutcliffe efficiency, which is commonly used in hydrological modeling. To ensure reasonable parsimony, it is common to also evaluate a criterion that moderates the measure
of fit with a measure of parameter uncertainty, for example,
the well-known Akaike information criterion [Akaike,
1974] or Young’s information criterion (YIC) [Young,
1990]. The final model selection criterion is that the identified transfer function should be decomposable into a conceptually plausible system of hydrological storages and
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potentially also pathways that bypass the storage [e.g.,
Young and Lees, 1993; Young, 2005].
[10] The simplest form of equation (1) that is commonly
used in hydrological modeling is
^
qk ¼
b0
^
uk=t ;
1 þ a1 z1
T¼
t
:
lnða1 Þ
ð3Þ
[11] A steady state gain parameter G can also be derived
from equation (2) in the form
G¼
b0
;
1 þ a1
ð4Þ
where in the case where q^ and ^u are measured in the same
units, G > 1 would imply that the system is gaining water
from somewhere other than effective rainfall and G < 1
would imply a loss of water. Another transfer function
commonly used in hydrological applications is the secondorder process
^qk ¼
b0 þ b1 z1
uk=t ;
^
1 þ a1 z1 þ a2 z2
ð5Þ
which if the roots of the denominator polynomial 1 þ a1z1
þ a2z2 are real, can be decomposed, for example, into two
first-order systems in parallel or series or as a feedback connection, where the parallel connection is normally the most
physically appropriate in the present context. These types of
model structures are commonly identified as suitable when
using daily resolution data [e.g., Young, 1993; Young and
Beven, 1994; Lees, 2000; Young, 2003]. More complex
transfer functions have been identified using hourly resolution data: for example, Young et al. [1997] and Young
[1998, 2008a] identify models equivalent to two stores plus
a instantaneous bypass flow, while Young [2010, 2011] identifies a third-order model with three stores and a bypass flow.
2.2. Nonlinearity and State-Dependent Parameter
Estimation
[12] Effective rainfall is usually estimated from measured rainfall using a nonlinear model. Various conceptual
soil moisture models are available and may be applied prior
to the linear transfer function identification [e.g., Jakeman
et al., 1990; Young, 2003; Chappell et al., 2006]. However, in keeping with the data-based ethos of the DBM
approach, it is more common to employ an empirical model
described by
^
uk ¼ f ð yÞrk ;
inally, for example, Young [1993] and Young and Beven
[1994] identified a power law function of observed flow q
as the wetness index [see also Young et al., 1997; Lees,
2000, Young, 2002; Romanowicz et al., 2006; McIntyre
and Marshall, 2010]
ð2Þ
which is a discrete time solution to a single linear store. A
time constant (or residence time) representing the storage
effects can be derived by conversion to continuous time
using the well-known expression [e.g., Young, 1984]
ð6Þ
where f(y) is an empirical nonlinear function of a chosen
catchment wetness index y and r is measured rainfall. Orig-
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^
uk ¼ cqk rk :
ð7Þ
[13] Here is a parameter that needs to be identified as
part of the model fitting procedure, with high values signifying a highly nonlinear response and a zero value being
equivalent to a linear effective rainfall (or loss) model. The
coefficient c is not identified but can be selected by the
modeler, for example, to impose the constraint ^u r or to
ensure that the total effective rainfall is equal in volume to
the resulting flow. While the use of observed flow as a wetness index restricts the predictive use of the model to forecasting applications [e.g., Romanowicz et al., 2006] and
precludes the direct application of the model to long-term
predictive simulation, the objective here is identification of
the form of nonlinearity, which can later inform the development of such a predictive simulation model.
[14] Alternative nonlinear models for the estimation of
effective rainfall may be prespecified, their parameters
optimized, and their performances intercompared [e.g.,
Ochieng and Otieno, 2009]; however, prespecification is
somewhat unsatisfactory given the goal of minimizing prior
assumptions about model structure. Instead, the nature of
the nonlinearity can be examined using nonparametric
state-dependent parameter estimation [Young, 2000, 2003].
Combining equations (1) and (6), the full rainfall-flow
model then takes the form
^
qk ¼
b0 þ b1 z1 þ b2 z2 þ þ bm zm
f ð yÞrk=t ;
1 þ a1 z1 þ a2 z2 þ þ an zn
ð8Þ
or, equivalently, each parameter in the b vector can be considered to be a function of y, i.e.,
^
qk ¼
b0 ð yÞ þ b1 ð yÞz1 þ b2 ð yÞz2 þ þ bm ð yÞzm
rk=t :
1 þ a1 z1 þ a2 z2 þ þ an zn
ð9Þ
[15] The nature of the functions bi(y) (i ¼ 1, m) can be
explored by estimating the vector b as time variable using
state-dependent parameter estimation techniques [see
Young, 2000, 2003]. Plotting bi against y then shows the nature of any significant relationship that can then be parameterized. For example, in the hydrological context, using y ¼
qk, various authors have justified the parameterized power
law of equation (7) in this way [e.g., Lees, 2000; Ratto et
al., 2007; Young, 2008a]. Equation (9) may be generalized
further to allow parameters ai, i ¼ 1, 2, . . . , n, to also be
dependent on qk, which may signify internal nonlinearity in
the routing model and time variable recession characteristics. Although this kind of nonlinearity exists in theory and
nonlinear flow-routing models have been used in DBM and
other applications [e.g., Moore and Bell, 2002; Romanowicz
et al., 2006; Young et al., 2006; Segond et al., 2007], the
dominant nonlinearity has traditionally been linked to flow
generation via an ‘‘effective rainfall’’ input nonlinearity,
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rather than any form of flow-routing nonlinearity [Wheater
et al., 1993; Beven, 2001]. Indeed, in the majority of flowrouting applications, combinations of linear stores or other
simple unit hydrograph functions are employed. Also, prior
to the case study in section 3, it seems that the DBM
approach had not been used to help distinguish between the
two sources of nonlinearity in the rainfall-flow response.
3.
Case Study
[16] The case study aims to investigate whether with
minimal prior hypotheses but good quality, high-resolution,
supporting data the DBM method can be used to identify
the nonlinear structure of a rainfall-flow model.
3.1. Case Study Data
[17] The plot under study (Figure 1) is within the Pontbren experimental catchment in Powys, mid-Wales, United
Kingdom (see Marshall et al. [2009] for a location plan).
The plot is part of a field used for sheep grazing and is agriculturally improved, meaning it has been underdrained by a
network of tile drains and at various unknown dates has
been plowed, fertilized, and reseeded with more productive
grasses and clover. Soils are silty clay loams of the Cegin
and Sannan series, with a distinct upper layer of approximately 30 cm depth overlying a relatively impermeable
lower layer [Marshall et al., 2009]. For example, median
sampled saturated hydraulic conductivities for the upper
and lower layers were 1.67 and 0.018 m/d, respectively
[Wheater et al., 2008]. Tile drains are located at around
750 mm beneath the surface. There is some evidence of
mole activity that together with other bioturbation effects
and soil structural change, could potentially create a macropore network. The climate is wet, with average annual rainfall in the field equal to 1449 mm (as measured between 1
April 2007 and 31 March 2009). Within the field is an
instrumented plot (Figure 1) with a quite well defined but
not isolated catchment area. The surface water catchment
of the plot has an area of 0.0044 km2, with average surface
slope of 12.5%, while the tile drain network implies an approximate subsurface catchment area of 0.0036 km 2.
[18] The plot was monitored between July 2005 and July
2009, although there are significant gaps in the data series
Figure 1.
Features of the case study plot.
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[McIntyre and Marshall, 2010]. Measurements included
rainfall, overland flow, subsurface drain flow, soil water
pressure at a network of sites at multiple depths, and
groundwater levels [Marshall et al., 2009]. The rainfall is
monitored at 10 min resolution by a tipping bucket gauge
with a nearby storage gauge (there was no significant difference in volumes between the two). The surface flow is
intercepted by a gutter at the bottom edge of the plot, which
transmits flow to a 45 V notch weir box in series with a
tipping bucket to verify measurements. The flow from the
network of tile drains collects in one main tile drain, the
outlet of which is connected to a 45 V notch weir box,
which is also in series with a tipping bucket. The surface
and drain flows are measured at 1 min intervals, and the average flows over 5 min intervals are logged. Review of the
time series flow data (e.g., Figure 2) illustrates that the flow
from the plot is dominated by subsurface drain flow, with
surface flow relatively intermittent and short-lived. For
example, from November 2006 to October 2007, 94% of
the recorded flow volume was drain flow. Flow peaks are
also usually dominated by drain flow, although in wet conditions the peak surface flow can exceed the peak drain
flow. Both pathways exhibit relatively flashy responses,
with sudden bursts of rainfall in wet periods causing the
drain flow to peak typically within 40 min of peak rainfall
and the surface flow within 20 min. The response of the
groundwater is, in general, a slow seasonal signal, without
notable response to individual wet periods, which is
assumed to mean that little water infiltrates though the relatively impermeable subsoil.
[19] While we consider the flow data to be a relatively
accurate record of what is collected in the weir boxes (relative, that is, to flow data typically used in catchment-scale
rainfall-flow studies), it cannot easily be argued that the
measurements accurately represent the total plot outflow.
First, it may be assumed that the surface flow collection
gutter did not always intercept all the surface flow because
of intermittent interference from animals, and during the
largest recorded flow event (17 – 18 January 2007) the surface flow downpipe (connecting the gutter to the weir box)
was observed to be partly blocked by storm debris. Second,
the bounds of this plot were defined by topography and the
identifiable subsurface drainage network, and there may
have been significant unmeasured outflow particularly from
throughflow. And third, although the water table variability
is generally small and slow [Marshall et al., 2009], there is
likely to be some unmeasured groundwater recharge. A
water balance calculation for a winter period, assuming no
net storage (as supported by tensiometer measurements)
and that evaporation losses are equal to potential evaporation, suggests that 18% of the outflow is unmeasured. In
summer, in periods of drain flow less than 0.02 L/s, some
diurnal variations of up to approximately 0.001 L/s can be
observed (e.g., Figure 6), with peaks at around 4:00 A.M.
every day. The same signal appears in the soil water pressure measurements; hence, it is presumed to be related to
the diurnal cycle of transpiration.
[20] The data period used for the DBM model identification is 10 November 2006 to 25 January 2007. This period
is chosen because the relevant variables have been continuously monitored during this period and there are many
events of varied magnitude and duration but without
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Figure 2. Model results in the winter calibration period using the three parallel linear stores (equation
(10)). The corresponding results using the preferred model is in Figure S3.
snowfall occurrence. Marshall et al. [2009] noted that large
cracks appeared in the plot’s soil during the summer of
2006 and that this affected the flow response for several
months afterward. This will be considered when interpreting results. Although this winter period is described here as
‘‘wet,’’ it contained relatively dry spells, including a 15 day
period with almost zero rainfall, and soil water pressure
measurements show that the plot was not continually saturated [Marshall et al., 2009]. A second, drier period of 9
May 2007 to 30 July 2007 is used for validation. Rainfallflow data are shown in Figure 2 for the winter calibration
period (see Figure 6 in section 3.7 for the summer calibration period).
3.2. DBM Model Identification With Time-Invariant
Parameters
[21] The DBM methods were implemented using the discrete time transfer function identification modules within
the CAPTAIN toolbox [Pedregal et al., 2007; Taylor et al.,
2007]. Continuous time transfer function modeling is an alternative approach, also available within CAPTAIN, which
can provide better defined and less biased parameter estimates in cases where fairly rapidly sampled data are available. This is possible in this case, but the discrete time
results yielded similar inferences and are reported here.
The sampling interval used is the smallest possible with the
available rain gauge data, t ¼ 10 min. The simplified
refined instrumental variable [see Young, 2008b] option of
the RIVBJ routine within the CAPTAIN Toolbox is used
for transfer function parameter estimation.
[22] The model identification strategy begins with a relatively straightforward and conventional DBM model identification exercise and then progressively explores avenues
aimed at extracting more information from the data. Hence,
to begin with, the aggregated surface and subsurface flow
is used as the observed flow, and a linear flow-routing
model is serially connected with a conventional nonlinear
flow generation model, equation (7). Subsequently, models
are fitted to the data from the two individual flow pathways
to look for new information. Information about nonlinearity
is then explored using state-dependent parameter estimation. It is assumed that any losses from the plot, including
possible unmeasured components of inflow or outflow discussed previously, and any error in the estimated catchment
area, are accounted for implicitly in the parameters of the
model. This may affect only the volume scaling, for example, parameter c in equation (7), although if inflow or outflow volumes are related to the wetness of the plot, it is
expected to also affect the nonlinearity parameter .
[23] Note that the unit of flow used in all the modeling is
mm/t, where t ¼ 600 s. Therefore, the aggregated flow
is not a single measurable variable; rather, it corresponds
to Qs/As 600 þ Qd/Ad 600, where Q is flow in L/s, A is
catchment area in m 2, and the subscripts s and d refer to the
surface and subsurface catchments, respectively. For the
first stage of analysis, the observed surface and drain flow
are aggregated in this way for the purpose of fitting the
model. The nonlinearity parameter is optimized by maximizing RT2 for each considered transfer function structure.
This included all combinations of models within the ranges
m ¼ 1 – 3 and n ¼ 1 – 3, giving nine model structures in
total. The performance and optimized parameters of two of
the tested structures are given in Table 1 (models 1 and 2).
Model 2, with two parallel stores, gives the best RT2 value
(0.92), and this model is about the same as model 1, a single store, in terms of YIC (11.6 compared to 11.9). YIC
is a logarithmic measure, so that a more negative value,
coupled with a comparatively high RT2, indicates a better
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12.2
10.4
0.87
0.92
0.91
0.92
0.93
0.93
0.88
0.91
0.96
0.96
11.9
11.6
12.4
8.0
Here s is the measured surface flow, and d is the measured drain flow.
Young’s information criterion (YIC) is not calculated for the nonlinear models because the parameter covariance is unknown.
Nonlinear one-store routing models.
c
a
b
0.54
0.44
1.04
0.93
10
10
0.80
0.80
1.07
21.8
3.72
1.37
0.45
0.25
0.28
3.65
0.47
q¼sþd
q¼sþd
s
s
s
s
d
d
d
d
1
2
3
4
5
6
7
8
9
10
1, 1, 0
2, 2, 0
1, 1, 0
2, 2, 0
1, 1, 0c
1, 1, 0c
1, 1, 0
2, 2, 0
1, 1, 1c
1, 1, 1c
1.65
1.92
1.02
0.99
0.73
0.68
1.28
2.64
0.52
0.62
0.37
0.44
0.82
0.81
0.70
0.68
0.27
0.57
0.06
12.5
2.3
0.97
0.35
0.27
0.51
0.65
0.69
split to T1
T2 (h)
T1 (h)
c
Model
Structure
n, m, Modeled
Variablea
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identified model: for a more detailed explanation; see
Young [1990] and Appendix 3 of Young [2001]. The identification of two parallel stores as a suitable model for this
plot is neither new nor surprising: the same result was
found by McIntyre and Marshall [2010] in their broader
analysis of the Pontbren streamflow data, and it is known a
priori from the measurements that there are dual flow pathways. Nevertheless, this result provides a benchmark to
assess whether or not the model complexity can be developed systematically by applying the DBM analysis to the
separated surface and drain flow components of flow.
[24] For each of the two flow components the same array
of nine different transfer function structures was tested. At
this stage, we maintained the nonlinear model of equation
(7) for each flow component and, again, optimized for
each tested linear transfer function. Selected results are
included in Table 1. For the surface flow component, a single linear store with ¼ 0 and ¼ 0:82 (model 3 in Table
1) is selected as the best compromise between performance
and parsimony, and for the drain flow, a parallel store
model ¼ 0 and ¼ 0:57 (model 8) is preferred. Superimposing these two models gives the transfer function
4
6
4
6
5
4
4
6
6
4
0.71
0.81
0.76
0.84
0.95
RT2
YICb
Gain G
Number of
Parameters N
Time Lag
(min)
Nonlinear Routing Parameters
Linear Routing Parameters
Flow Generation
Parameters
Table 1. Optimized Performance and Parameter Values for Model Defined by Equation (7) Combined With Different Transfer Functions for the Winter Period
Performance
MCINTYRE ET AL.: NONLINEARITY IN RAINFALL-FLOW RESPONSE
^
qk ¼
0:282
0:041
0:003
^
^
u
ud;k ;
þ
þ
s;k
1 0:628z1
1 0:885z1 1 0:992z1
ð10aÞ
^
us;k ¼ 1:018s0:82
k rk ;
ð10bÞ
^
ud;k ¼ 2:641dk0:57 rk ;
ð10cÞ
where s is the measured surface flow, d is the measured
drain flow, ^us is the estimated effective rainfall going to
surface flow, and ^ud is the estimated effective rainfall going
to drain flow. Equation (10a) is equivalent to three parallel
flow pathways [e.g., Young et al., 1997; Young, 2005,
2008a], but unlike in previous models, the split between the
three pathways is not constant because of the presence of
parallel nonlinear effective rainfall models (equations
(10b) and (10c)). The aggregated surface and drain flow
time series obtained from equation (10) is shown in Figure
2, along with aggregated observed flow, rainfall, and estimated effective rainfall. In Figure 2, the flow is log transformed to exaggerate the low flow errors. The RT2 value for
the aggregated flow is 0.93. Although this is only 0.01 more
than the original two parallel store models and therefore,
arguably, a small if not insignificant performance improvement, the more complex transfer function facilitates a simulation which can distinguish between surface and subsurface
pathways. This distinction is potentially important, for
example, if the DBM results are used to develop models for
assessing impacts of soil and drainage management.
[25] A feature of the result in Figure 2 is that the two
highest flow peaks (on the 13 December and the 18 January) are modeled well only because the estimated effective
rainfall is significantly higher than (almost twice the volume of) the observed rainfall. This is mathematically possible because the constraint ^uk rk is not imposed upon the
nonlinear model, equation (7). In general, the occurrence of
^uk > rk may be considered physically reasonable because if
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Figure 3. Model performances for four example events (a – d) using equation (10) and (e – h) using
equation (16).
rainfall measured from one or more rain gauges is treated
as an estimate of catchment average rainfall, it is expected
to contain significant spatial sampling errors. And, of
course, the DBM model is inherently stochastic, so that the
estimated uncertainty has to be taken into account in any
evaluation of the model characteristics. In the case study,
however, errors arising from sampling the catchment area
are presumed to be small because the plot covers an area of
only around 65 m 65 m, with the rain gauge located
almost centrally within it, and comparisons with storage
gauge readings show no volume bias (although both may
suffer from undercatch). Furthermore, closer examination
of the hydrographs in Figure 2 (e.g., Figure 3) shows that
the accumulated volume of flow over each of the two largest events (12 – 13 December and 17 – 18 January) is overestimated by the model despite the good fit to peaks,
supporting the view that ^uk has been overestimated. Adding
to concerns about the model of equation (10), Figure 2
illustrates persistent underestimation of low flows. Therefore, despite its impressive performance in terms of RT2,
we do not accept the model of equation (10) on the grounds
that the generated effective rainfall is not physically realistic and there is a low flow bias. Instead, we go on to more
critically assess the nonlinear component of the model.
3.3. State-Dependent Parameter Analysis for Drain
Flow
[26] We proceed now to examine whether the nonlinear
component of the drain flow model can be improved using
state-dependent parameter estimation. To facilitate clear
signals of state dependence, the simplest of the applicable
transfer function structures, equivalent to one single store,
is preferred as a starting point [Young, 2003]. Adapting
equation (2) to this form, we obtain
d^k ¼
b0 ðdk Þ
rk=t :
1 þ a1 ðdk Þz1
ð11Þ
[27] Both the a1 and b0 parameters were simultaneously
identified as state-dependent parameters, using the estimation procedure described by Pedregal et al. [2007, chapter 5]
and Young et al. [2001], assuming for now that ¼ 0. The
default settings in the CAPTAIN Toolbox SDP function
were used, except that the code for the number of iterations
used to estimate the noise variance ratio smoothing parameter was set to 2 on the recommendation of Pedregal et al.
[2007, p. 98] and the integrated random walk [see Pedregal
et al. 2007, p. 19] was used to ensure a smooth estimate
of the state-dependent parameters. As well as using the
observed drain flow (d) as the state on which the parameters
are dependent, as specified in equation (11), the observed
surface flow (s), the aggregated flow (s þ d), and the plot average soil water tension (p) were also considered, with the
aim of identifying the most representative catchment wetness index (y) for use in equation (6). Although state dependence was present in all cases (the results are shown in the
auxiliary material in Figure S1), by far, the least noisy dependence was achieved using d as the state, and these results
are presented in detail in Figure 4 and subsequent analysis.1
1
Auxiliary materials are available in the HTML. doi:10.1029/
2010WR009851.
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Figure 4. Model parameters as functions of flow in the winter period: (a) parameter b0 in the drain
flow model, (b) parameter a1 in the drain flow model, (c) parameter b0 in the surface flow model, and (d)
parameter a1 in the surface flow model.
[28] Figures 4a and 4b show the relationships b0(d) and
a1(d) for drain flow in the winter period. Figure 4a supports
the power law function, equation (7), previously assumed,
although the result may also arise from dependency of a1
on the value of b0, as explained below. The state dependence of a1 for drain flow is illustrated in Figure 4b, showing
that the time constant T decreases as flow increases. This
result may be interpreted in two ways: the state dependency
of T illustrated by Figure 4b may be because a single drain
flow store has been used to represent a dual-pathway drain
flow system (this is easily confirmed by applying the single-store state-dependent parameter analysis to the outputs
of a parallel store model); alternatively, or additionally, the
result may be interpreted as a kinematic routing effect.
3.4. Proposed Mechanistic Interpretations of the State
Dependence
[29] The kinematic wave model [e.g., Beven, 2001, p.
177] requires that v ¼ q ; where v is celerity, q is flow,
and and are parameters related to the physical properties of the flow path. Celerity may also be treated as equal
to a representative length x divided by a representative
time of travel, which in this case, is the time constant T
since there is no time delay ; hence, v ¼ x = T . Using the
approximation to equation (3), T ¼ t =ð1 þ a1 Þ; parameter a1 can then itself be expressed in the form
a1 ðdk Þ ¼ dk 1 ;
either ¼ 0 or ¼ 10 min. This theoretically derived
expression is consistent with the form of the empirical
relationship in Figure 4b, where < 1. Therefore, the nonlinear kinematic routing is accepted as a plausible model,
and equation (11) becomes
d^k ¼
b0 ðdk Þ
rk=t :
1 þ dk 1 z1
ð13Þ
[30] If the gain parameter G is forced to be constant,
from equation (4), b0 ðdk Þ ¼ dk G; and the drain flow
model would become
d^k ¼
dk
Grk=t :
1 þ dk 1 z1
ð14Þ
[31] An alternative explanation for the result in Figure
4a can now be seen: a1 may vary because of kinematic
routing effects, and hence, b0 must vary in order to maintain a constant steady state gain between rainfall and flow.
Therefore, the state-dependent analysis results for the drain
flow lead to three alternative propositions: (1) there are
two parallel linear routing stores and nonlinear flow generation, (2) there is one nonlinear routing store as in equation
(14) with linear flow generation, or (3) there is a mixture of
the nonlinear flow generation and nonlinear routing which,
assuming a single store, may be expressed as
ð12Þ
where ; t; and x have been combined into one parameter with units (mm/tÞ1 . The same form of a1(dk) is
observed when fixing ¼ 10 min; hence, (12) is valid for
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d^k ¼
dk
G^
ud;k=t ;
1 þ dk 1 z1
^
ud;k ¼ cdk rk :
ð15aÞ
ð15bÞ
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MCINTYRE ET AL.: NONLINEARITY IN RAINFALL-FLOW RESPONSE
3.5. Assessment of the Proposed Mechanistic
Interpretations
[32] The first proposal, that there are two parallel linear
routing stores and nonlinear flow generation, has already
been assessed (the drain flow model in equation (10)) and
was discounted. The second and third propositions are now
explored by optimizing the parameters of the mixed model,
equation (15). If the optimum value of is not significantly
different from zero, then the second proposition is supported, while if it is significantly greater than zero, then the
third proposition is supported. Prior to optimizing the parameters, the observed drain flow dk in equation (15a) is
replaced by simulated drain flow at the preceding time step
d^k1 . This prevents feedback effects that cause instability
in the validation period results (described in section 3.8).
Parameters G, , and are optimized to the drain flow
observations using Matlab’s nonlinear solver ‘‘fminsearch’’
with RT2 as the criterion. The time delay parameter is also
optimized by trial and error, and the c parameter is fixed, as
in the previously optimized models, so that the volume of
effective rainfall equals the volume of observed flow. The
optimized parameter values are shown in Table 1 (model
9). The optimized value of (0.06) implies that overall,
there is less flow generated during wetter periods than in
drier periods; however, this value is not considered to be
significantly different from zero, and so our second proposition is supported : a linear drain flow generation model.
[33] Another feature of the optimized equation (15) is
that G is close to unity. This is perhaps unsurprising, as the
value of c has been fixed so that the volume of effective
rainfall is equal to the value of observed flow, so that, in
principle, there should be no need for a gain parameter.
However, in practice, G 1 is a nontrivial result because it
signifies that consistency between observed and simulated
drain flow volumes coincides with the RT2 optimal model
(this consistency was not present when using linear routing,
as indicated by the values of G in Table 1). Fixing ¼ 0
and G ¼ 1 and reoptimizing ; ; and yield the values in
Table 1 (model 10) with associated RT2 ¼ 0.96. This is a
remarkably good performance for a four-parameter (c, ; ;
and ) model. Furthermore, unlike the majority of DBM
rainfall-flow models, this model avoids the use of observed
flow as a surrogate measure of wetness and therefore may
be used directly as a simulation model for prediction.
[34] However, model 10 is questionable for at least three
reasons. First, recalling that the plot soil was not continually saturated, a linear flow generation model is contrary to
the general experience that flow generation under unsaturated soil conditions is a nonlinear process. A second
related issue is that the value ¼ 0:93 is higher than
expected a priori: calibrated values for catchment-scale
models typically range from 0.2 to 0.7 [e.g., Wittenberg,
1993; Segond et al., 2007], and theoretical analysis for
unconfined saturated subsurface flow gives a value of 0.5
[Moore and Bell, 2002]. These two issues point to the possibility of interaction between and within the optimization, so that is lower than expected and is higher than
expected. However, plotting the response of RT2 against the
parameters of equation (15) shows unambiguously that
value of is close to zero and that is close to 0.93. This
plot is included in the auxiliary material in Figure S2.
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[35] The remaining question about model 10 is whether a
dual-pathway drain flow model, where one or both pathways are nonlinear in the form of equation (15), might perform better. Indeed, when a state-dependent analysis was
applied to the disaggregated fast flow component of drain
flow df (i.e., the observed drain flow minus the modeled
slow drain flow from equation (10)), it did indicate nonlinearity of the form evident in Figure 4b. To answer the
question, therefore, a nonlinear fast drain flow model was
optimized conditional on the linear slow drain flow model
specified within equation (10).
[36] With the RT2 value remaining at 0.96, the addition
of the slow flow store was not justified. Furthermore, a
response surface analysis of the type shown in Figure S2
indicated poor identifiability, especially for the slow flow
b0 parameter. On the other hand, an attraction of this dualpathway drain flow model is the more reasonable values (in
terms of prior expectations) obtained for and (0.18 and
0.65, respectively). In other words, there seems to be a
more reasonable split of nonlinearity between the flow generation component and the fast flow-routing component.
While at this stage in the analysis a dual-pathway model
cannot be ruled out as an alternative description of the drain
flow system, precedence is given to the more parsimonious
single-store model.
3.6. Surface Flow Model
[37] For the surface flow model, using surface flow as
the state (rather than drain flow, aggregated flow, or soil
water pressure) produced the clearest signals of state dependence in the parameters. Figure 4c shows a surprising
reduction in b0 at the highest values of surface flow, and
Figure 4d shows a1 increasing with increasing flow, particularly at the upper range of flows. This unexpected state dependence is largely because the surface runoff coefficient
for the largest storm event (17 – 18 January 2007) was considerably less than those for the medium size events, and its
time constant was larger. This is consistent with the field
observation that the surface flow downpipe (see section
3.1) failed to collect and efficiently transmit the full surface
flow on that occasion. If a nonlinear model of the form of
equation (15) is pursued for the surface flow, despite the
lack of clear evidence in Figure 4, then the optimized parameter values are as shown in Table 1 (model 5 when G is
optimized and model 6 when G is fixed to 1.0).
[38] Model 6 has the same number of parameters as the
linear single-store model, while the RT2 value has improved
from 0.91 to 0.93. Also, the value ¼ 0:28 is comparable
with established values for overland flow; for example, the
classic Chezy friction equation for surface flow would give
¼ 0:33. This supports an argument that the DBM results
should be overruled in this case, given prior knowledge of
process nonlinearity and field measurement problems. However, in keeping with the DBM ethos that adding complexity
to the model on the basis of prior expectations should be
avoided wherever possible, the linear surface flow routing in
equation (10) is favored, although the nonlinear counterpart
cannot be ruled out as an alternative plausible description.
3.7. Preferred Aggregated Flow Model
[39] Combining the preferred surface and subsurface
models (models 3 and 10 in Table 1) leads to
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^qk ¼
MCINTYRE ET AL.: NONLINEARITY IN RAINFALL-FLOW RESPONSE
0:93
0:443d^k1
0:282
^
^
us;k þ
ud;k1 ; ð16aÞ
1
0:93
1 0:628z
1 þ 0:443d^k1
1 z1
^
ud;k ¼ 0:623rk ;
ð16bÞ
^
us;k ¼ 1:018s0:82
k rk :
ð16cÞ
[40] Equation (16) gives an RT2 value for the aggregated
flow of 0.97. To illustrate the performance in detail, four
example events are included in Figure 3, comparing results to
those obtained from the benchmark linear routing model,
equation (10). The performance over the full winter time series is shown in the auxiliary material in Figure S3. These
illustrations show that the overestimation of effective rainfall
during the largest events, 12 – 13 December and 17 – 18 January, arising from the assumption of equation (10) that all the
nonlinearity was in the flow generation has been solved. Figure S3 shows that the low flow bias has also been substantially solved, and the absence of a drain flow steady state gain
(i.e., G ¼ 1.0) signifies a more satisfactory mass balance than
is implicit in equation (10). Adoption of the alternative more
complex models (dual-pathway nonlinear drain flow routing
and/or nonlinear surface flow routing) produced the same or
only marginally better performances.
[41] Some properties of the model residuals obtained
from equation (16) are shown in Figure 5. Figures 5b
and 5d show some bias toward negative residuals at the
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medium range of rainfall and flows. This indicates some
systematic error in the drain flow recessions where, potentially, underestimation of medium flows is needed to
achieve good peak flow and base flow performance. The
bias persists if we use the dual-pathway nonlinear drain
flow routing, and so the bias cannot easily be interpreted as
lack of flow pathways. Figures 5b, 5c, and 5d also show
positive residuals during the large event of 17 – 18 January
2007, which was dominated by surface flow. These residuals are associated largely with timing errors (see Figure
3h); however, interpretation is complicated by the scope
for observation error in surface flow during this event, as
already noted. There is no evidence in Figures 3, 4, 5, or S3
that the swelling of the soil observed by Marshall et al.
[2009] following the dry summer of 2006 has altered the
flow response within this winter period.
3.8. A Summer Validation Period
[42] All the results presented so far are for the wet winter
period 10 November 2006 to 25 January 2007. Insight into
processes may be gained by closer analysis of the summer
‘‘validation’’ period of 9 May to 30 July. This period followed a dry April where observed flows fell to zero by the
beginning of May (for this reason 1 May was not used as
the start of the period; a zero initial condition is problematic because of the flow dependence of parameters). The
dry spell continued until 15 June with only two significant
flow events, and flows were sufficiently low that diurnal
transpiration signals of around 0.001 L/s became visible, as
Figure 5. Error analysis (equation (16), winter period). (a) The x axis and y axis are curtailed so that
the distribution tails are more visible: the actual peak value of relative frequency is 0.32 in the bin centered around an error of 0.0006 mm t1 . (b – d) The actual maximum errors are also shown.
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previously noted. The subsequent wet period lasted until
the end of July (Figure 6). Given the differences between
the winter and summer conditions, this is a difficult validation period because some mechanisms operative over the
summer period may not have been active over the winter,
so the results reported in this section should be considered
with this in mind.
[43] The model of equation (16), estimated using the winter period, performed well on the wet summer period (Figure 6, from 15 June), with an RT2 value for the aggregated
flow of 0.96, and if the parameters are reoptimized (as
before, for drain and surface flow models separately), their
values are close to those estimated for the winter period.
However, equation (16) performs less well if the whole 2
month summer period is included (the entire period in Figure 6), with an RT2 value for the aggregated flow of 0.83.
The loss in performance lies in both surface and drain flow
components. The error structure over the summer period is
dominated by overestimation of flow events within and immediately after the dry weather and underestimation of similarly sized events in the subsequent wetter period. If the
summer period is extended to include the dry weather in
August, this pattern repeats.
[44] However, reoptimizing the parameters within the
model structure of equation (16) (including in the drain
flow model), giving an optimal RT2 value of 0.91 for the
aggregated flow for the period shown in Figure 6, does not
recover the level of performance achieved in the wet periods. Plausible reasons for the apparent structural change in
the model during and after the dry periods are the increased
significance of the diurnal transpiration cycle, the presence
of the strong nonlinearity and hysteresis usually found in
drying-wetting cycles of clay rich soils [e.g., Topp, 1971],
the presence of infiltration excess flow (although there is no
observed evidence of this), changes in the properties of the
Figure 6.
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grass, and/or changes in the soil structure, for example, a
stronger role of macropores after dry periods. As noted,
Marshall et al. [2009] reported large cracks appeared in the
plot’s soil during the dry summer of 2006, and these
appeared to affect the groundwater dynamics. However,
there was no visible change to the soil structure or unusual
groundwater dynamics during the 2 month validation period. Furthermore, there was no evidence that the change in
response was related to the available potential evaporation
estimates. A state-dependent parameter estimation exercise
on the 2 month summer period produced results similar to
those for the winter period, except with a very noisy relationship between b0 and flow for the drain flow model. This
implies that influences other than rainfall and integrated antecedent wetness affected the variability of drain flow generation in the summer, potentially relating to the factors
suggested above. Examination of runoff and soil moisture
data over multiple experimental plots is currently underway
to help resolve this.
3.9. Wetness Index Simulation
[45] One motivation for the DBM analysis is to assist in
identification of a model that is capable of predicting flow
response to scenarios of future rainfall inputs. The routing
models identified here may be applied directly to prediction
and so can the wet period drain flow generation model. The
optimal surface flow and dry period drain flow generation
models, however, used observed flow as a surrogate wetness index. A possible solution is to use the flow estimated
at the previous time step; however, this led to unsatisfactory results for both drain and surface flows. Therefore, the
value of the DBM analysis for making predictions partly
rests in its potential for use in multistep-ahead forecasting
when the data exhibits time delays and to inform hypotheses about models that can simulate catchment wetness.
Model results in the summer validation period using equation (16).
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[46] A range of simple models were tested, and the discussion here focuses on the one which gave the most satisfactory results. The DBM results in Figure 4 are closely
associated with models that replace the observed flow in
equation (7) with a simulated catchment wetness index
[Young, 2003]. This includes the models described by Jakeman et al. [1990] and the subsequent versions described by
Ye et al. [1995] and McIntyre and Al Qurashi [2009]. For
the winter period, attempts to optimize these models led to
the same conclusion as the DBM analysis : the generated
flow is not significantly related to variability of wetness,
and the linear effective rainfall model of equation (16b) is
preferred. For the summer period, a three-parameter version of the catchment wetness index model (which assumes
a first-order loss from the soil store and that effective rainfall is proportional to wetness raised to a power ) combined with the nonlinear routing store in equation (16a)
achieved a satisfactory performance over the full range of
summer flows, with RT2 ¼ 0.95. Arguably, such simple
drain flow generation models would not have been presumed to work well prior to a DBM analysis, and parameter
values would have been significantly biased if the routing
nonlinearity had not been separated out. However, given
the difficulty of addressing the hysteretic and nonstationary
behavior proposed in section 3.8, identifying a single predictive model to simulate flows over both the winter and
summer periods is more challenging and has not been
addressed in this work.
[47] Identifying a predictive simulation model of surface
flow also remains a challenge The best performance for surface flow in the winter period was RT2 ¼ 0.79 using a fiveparameter catchment wetness index model, which
accounted for losses using daily estimates of potential evaporation and included an infiltration excess parameter, along
with a linear routing store structure (as in model 3 in Table
1). Using a nonlinear surface-routing store (as in model 5)
improved the result only slightly, with RT2 ¼ 0.81. In the
summer period, the best model (the five-parameter catchment wetness index model with a single linear routing store)
achieved only RT2 ¼ 0.55 after optimization. It seems that
the surface flow generation processes are not controlled
solely by a lumped soil wetness index: the observed flow
was by far the best wetness index, as suggested in the previously cited DBM rainfall-flow modeling studies.
[48] It may be conceptually attractive to use the same
wetness index for the generation of flow for all pathways,
as is normal practice in conceptual modeling at both plot
and catchment scales [e.g., Lee et al., 2005; Krueger et al.,
2010]. The previous DBM results (Figure S1) indicate that
this may not be effective for the case study plot, and this is
confirmed by testing wetness index models and explicit soil
moisture accounting models, including those proposed for
this same plot prior to DBM analysis [Wheater et al.,
2008]. While it would not be reasonable to suggest that the
surface and subsurface flow processes are independent, the
DBM results illustrate that the separation is stronger than
presumed a priori.
4.
Conclusions
[49] This paper has demonstrated the identification and
estimation of a plot-scale rainfall-flow model using the
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data-based mechanistic (DBM) modeling approach, including the allocation of nonlinearity between the flow generation and routing components of the model. The data were
considered to be more accurate than typically used for rainfall-flow modeling because of the double measurement
systems used for flow and low errors in spatial rainfall estimates. Parsimonious models of two parallel pathways, drain
flow and surface flow, were identified independently, and
the models were superimposed into a model of total flow. In
winter and summer wet periods, a linear model was identified for the drain flow generation (i.e., a proportional loss
model), and there was stronger than expected nonlinearity
in drain flow routing. The wet period performances of the
surface flow and drain flow models (RT2 ¼ 0.91 and RT2 ¼
0.96) are remarkable for models with only four parameters,
and superimposing the models achieved an even better performance for aggregated flow (RT2 ¼ 0.97).
[50] Despite the good performance in relatively wet periods, the model was less successful when forced to alternate
between very dry and very wet periods, and it is speculated
that this is due to nonlinearity in soil moisture dynamics
and/or nonstationarity in the vegetation or soil structure,
for example, due to opening and closing of macropores, as
observed by Marshall et al. [2009]. The DBM method
could be used to explore the cause of this variability if suitable additional data were considered, for example, air temperature, which may be a primary cause of the structural
change in the soil. A conceptual model that was based on
the DBM modeling results and which matched the DBM
performance was found for drain flow but was elusive for
surface flow.
[51] The search for methods to assist in identification of
model structures has received considerable interest from
hydrologists, as illustrated by the papers cited in section 1.
Two features stand out from these and other papers: the
difficulty of distinguishing between candidate model structures given data and parameter uncertainty and, in the context of rainfall-flow modeling, the common result that all
the nonlinearity is included in the effective rainfall generation model. The latter feature is partly a consequence of the
former: the typical accuracy of rainfall-flow data may not
permit unambiguous identification of routing nonlinearity.
In addition, arguably, the linear routing is commonly used
simply for convenience because of the numerical efficiency
of the analytical solutions.
[52] In contrast to these previous studies, a significant aspect of the DBM modeling reported in this paper is the identification and estimation of nonlinear state dependency
within the model (i.e., in the feedback within the model,
rather than just in the effective rainfall nonlinearity at the
model input). Here the effect of state dependency is to
change the recession characteristics of the drain flow pathway as a nonlinear function of the soil wetness, revealing
that the associated time constant decreases nonlinearly as the
drain flow increases. This is the first DBM rainfall-flow modeling study that has clearly identified such characteristics.
Furthermore, the paper has shown that apportioning nonlinearity between flow generation and flow-routing components in this way has considerable rewards in terms of the
model performance and the physical plausibility of its internal functioning. For example, the identification of a nonlinear
loss model conditional on a linear routing model may corrupt
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MCINTYRE ET AL.: NONLINEARITY IN RAINFALL-FLOW RESPONSE
the internal realism of the model despite good flow performance. Having reached this conclusion using high-quality experimental data, future investigation is required using more
typical, catchment-scale, operational data sets.
[53] Acknowledgments. This research was partly funded by the UK
Flood Risk Management Research Consortium Phase 2, EPSRC grant EP/
F020511/1. The CAPTAIN toolbox (http://www.es.lancs.ac.uk/cres/captain) was used under license from Lancaster University. The Pontbren
experiment is possible due to the support of the Pontbren Group, Llyn Hir
landowners, and Coed Cymru.
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M. Marshall, N. McIntyre, B. Orellana, and H. Wheater, Department of
Civil and Environmental Engineering, Imperial College London, London
SW7 2AZ, UK. ([email protected])
B. Reynolds, Centre for Ecology and Hydrology, Environment Centre
Wales, Deiniol Road, Bangor LL57 2UW, UK.
P. Young, Lancaster Environment Centre, Lancaster University, Lancaster LA1 4YQ, UK.
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