Understanding the projection performance of hydrological models under
contrasting climatic conditions supports robust decision making, which
highlights the need to adopt time-varying parameters in hydrological
modeling to reduce performance degradation. Many existing studies model
the time-varying parameters as functions of physically based covariates;
however, a major challenge remains in finding effective information to
control the large uncertainties that are linked to the additional parameters
within the functions. This paper formulated the time-varying parameters for
a lumped hydrological model as explicit functions of temporal covariates and
used a hierarchical Bayesian (HB) framework to incorporate the spatial
coherence of adjacent catchments to improve the robustness of the projection
performance. Four modeling scenarios with different spatial coherence
schemes and one scenario with a stationary scheme for model parameters
were used to explore the transferability of hydrological models under
contrasting climatic conditions. Three spatially adjacent catchments in
southeast Australia were selected as case studies to examine the validity of
the proposed method. Results showed that (1) the time-varying function
improved the model performance but also amplified the projection uncertainty
compared with the stationary setting of model parameters, (2) the proposed HB
method successfully reduced the projection uncertainty and improved the
robustness of model performance, and (3) model parameters calibrated over
dry years were not suitable for predicting runoff over wet years because of
a large degradation in projection performance. This study improves our
understanding of the spatial coherence of time-varying parameters, which
will help improve the projection performance under differing climatic
conditions.
Introduction
Long-term streamflow projection is an important part of effective water
resources planning because it can predict future scarcity in water supply
and help prevent floods. Streamflow projections typically involve the
following: (i) calibrating hydrological model parameters with partial
historical observations (e.g., precipitation, evaporation, and streamflow);
(ii) projecting streamflow under periods that are outside of those for model
calibration; and (iii) evaluating the model projection performance with
certain criteria. One of the most basic assumptions of this process – that
the calibrated model parameters are stationary and can be applied to predict
catchment behaviors in the near future, has been widely questioned (Brigode et al., 2013; Broderick et al., 2016; Chiew et al., 2009, 2014; Ciais et al., 2005; Clarke, 2007; Cook et al., 2004;
Coron et al., 2012; Deng et al., 2016; Merz et al., 2011; Moore and
Wondzell, 2005; Moradkhani et al., 2005, 2012; Pathiraja
et al., 2016, 2018; Patil and Stieglitz, 2015; Westra et
al., 2014; Xiong et al., 2019; Zhang et al., 2018).
Many previous studies have explored the transferability of stationary
parameters to periods with different climatic conditions. They have
concluded that hydrological model parameters are sensitive to the climatic
conditions of the calibration period (Chiew et al., 2009, 2014; Coron et al., 2012; Merz et al., 2011; Renard et al., 2011; Seiller et
al., 2012; Vaze et al., 2010). For instance, Merz et al. (2011)
calibrated model parameters using six consecutive 5-year periods between
1976 and 2006 for 273 catchments in Austria and found that the calibrated
parameters representing snow and soil moisture processes showed a
significant trend in the study area. Other studies have found that
degradation in model performance was directly related to the difference in
precipitation between the calibration and verification periods (Coron et
al., 2012; Vaze et al., 2010). One proposal for managing this problem is to
calibrate model parameters in periods with similar climatic conditions to
the near future, but future streamflow observations are unavailable. Thus,
it is still necessary to reduce the magnitude of performance loss and
improve the robustness of the projection performance using calibrated
parameters based on the historical records, even though the climatic
conditions in the future may be dissimilar to those used for model
calibration.
Several recent studies have found that hydrological models with time-varying
parameters exhibited a significant improvement in their projection performance
compared with those using the stationary parameters (Deng et al., 2016, 2018; Westra et al., 2014). The functional method is one of the most
promising ways to model time-varying parameters and shows its excellence in
improving the model projection performance (Guo et al., 2017; Westra et
al., 2014; Wright et al., 2015). This method models the time-varying
parameter(s) as the function(s) of physically based covariates (e.g.,
temporal covariate and Normalized Difference Vegetation Index). Generally,
the hydrological model is run with various assumed functions, and the best
functional forms of time-varying parameters can be obtained by comparing the
evaluation criteria. However, a major challenge for the application of the
functional method remains in finding effective information to control the
large uncertainties that are linked to the additional parameters describing
these regression functions.
The similarity of adjacent catchments has been verified, along with the validity of
controlling the estimation uncertainty of model parameters (Bracken et
al., 2018; Cha et al., 2016; Cooley et al., 2007; Lima and Lall, 2009;
Najafi and Moradkhani, 2014; Sun and Lall, 2015; Sun et al., 2015; Yan and
Moradkhani, 2015). The level of similarity of different catchments is known
as spatial coherence. For instance, Sun and Lall (2015) used the
spatial coherence of trends in annual maximum precipitation in the United
States and successfully reduced the parameter estimation uncertainty in
their on-site frequency analysis. In general, there are three methods to
consider the spatial coherence between different catchments in parameter
estimation. The first one is no pooling, which means every catchment is
modeled independently, and all parameters are catchment-specific. The second
one is complete pooling, which means all parameters are considered to be
common across all catchments. The third and last one is the hierarchical Bayesian
(HB) framework, also known as partial pooling, which means some parameters
are allowed to vary by catchments and some parameters are assumed to drown
from a common hyper-distribution across the region that consists of
different catchments. In these three approaches, the HB framework has been
proven to be the most efficient method to incorporate the spatial coherence to
reduce the estimation uncertainty because it has the advantage of shrinking
the local parameter toward the common regional mean and including an
estimation of its variance or covariance across the catchments (Bracken
et al., 2018; Sun and Lall, 2015; Sun et al., 2015). In the field of
hydrological modeling, most preceding studies were focused on no-pooling models that neglect the spatial coherence between catchments (Heuvelmans et al., 2006; Lebecherel et al., 2016; Merz and Bloschl,
2004; Oudin et al., 2008; Singh et al., 2012; Tegegne and Kim, 2018; Xu et
al., 2018); little attention has been paid to the HB framework. Thus, we
want to fill this gap and explore the applicability of the spatial coherence
through the HB framework in hydrological modeling with the time-varying
parameters.
The objectives of this paper were to (1) verify the effect of the
time-varying model parameter scheme on model projection performance and
uncertainty analysis compared with stationary model parameters, (2) verify
the projection performance of a scheme that considers the spatial coherence of adjacent
catchments through the HB framework compared with spatial incoherence, and
(3) compare the model projection performance for different climatic transfer
schemes.
The rest of the paper is organized as follows. Section 2 outlines the
methodology employed in this study including differential split-sample test
(DSST) for segmenting the historical series, the hydrological model, and the
two-level HB framework for incorporating spatial coherence from adjacent
catchments. Section 3 presents the information on the study area and data.
The results and discussion are described in Sect. 4. Section 5 summarizes
the main conclusions of the study.
Methodology
The methodology is outlined by a flowchart in Fig. 1, and is summarized as
follows:
A temporal parameter transfer scheme is implemented (described in Sect. 2.1) using a classic DSST procedure in which the available data are
divided into wet and dry years.
A daily conceptual rainfall–runoff model is used (outlined in Sect. 2.2).
A two-level HB framework is used to incorporate spatial coherence in
hydrological modeling (described in Sect. 2.3). The process layer (first
level) of the framework models the temporal variation in the model
parameters using a time-varying function, while the prior layer (second
level) models the spatial coherence of the regression parameters in the
time-varying function. Four modeling scenarios with different spatial
coherence schemes and one scenario with a stationary scheme for the model
parameters are used to evaluate the transferability of hydrological models
under contrasting climatic conditions.
Likelihood function and parameter estimation methods are applied
(outlined in Sect. 2.4).
The criteria are used to evaluate the model performance for various
model scenarios (described in Sect. 2.5).
Flow chart of the methodology for integrating inputs from
spatially coherent catchments and temporal variation of model parameters
into a hydrological model under contrasting climatic conditions (wet and dry
years).
Differential split sampling test
To verify the projection performance of the rainfall–runoff model under
contrasting climatic conditions (wet and dry years), a classic DSST using
annual rainfall records was adopted.
Two separate tasks were needed to develop the DSST method into a working
system. The first step was to define “dry years”. The method to define the
dry years is adopted from Saft et al. (2015), which is a rigorous
identification method that treats autocorrelation in the regression
residuals, undertakes global significance testing, and defines the start and
end of the droughts individually for each catchment. Saft et al. (2015)
tested several algorithms for dry-year delineation, which considered
different combinations of dry run length, dry run anomaly, and various
boundary criteria and found that the identification results of dry years by
one of the algorithms showed marginal dependence on the algorithm and the
main results were robust to different algorithms. The detailed processes
could be found on Saft et al. (2015) and are also generalized as follows.
First, the annual rainfall data were calculated relative to the annual
mean, and the anomaly series was divided by the mean annual rainfall and
smoothed with a 3-year moving window. Second, the first year of the
drought remained the start of the first 3 years of the negative anomaly period.
Third, the exact end date of the dry years was determined through analysis
of the unsmoothed anomaly data from the last negative 3-year anomaly. The
end year was identified as the last year of this 3 year period unless (i)
there was a year with a positive anomaly >15 % of the mean, in
which case the end year is set to the year prior to that year, (ii)
the last 2 years have slightly positive anomalies (but each <15 % of the mean), in which case the end year is set to the first year of
positive anomaly, or (iii) to ensure that the dry years are sufficiently long
and severe, in the subsequent analysis, the authors use dry years with the
following characteristics: length ≥7 years; mean dry years
anomaly <-5 %.
In the second step, the wet years were defined as the complement of the dry
years in the historical records. A similar approach to define the dry and
wet years was used by Fowler et al. (2016).
In the DSST method, the model parameters calibrated in the wet years were
evaluated in the dry years, and vice versa. In addition, criteria (i.e,
NSEsqrt, BIAS, DIC, MaxF, and MinF, illustrated in Sect. 2.5) were
used to evaluate the performance of the calibrated parameters for different
transfer schemes.
The rainfall–runoff model
The hydrological model used in this study is the GR4J (modèle du
Génie Rural à 4 paramètres Journalier), which is a lumped
conceptual rainfall–runoff model (Perrin et al., 2003). The original
version of the GR4J model (Fig. 2) comprised four parameters (Perrin et
al., 2003): production store capacity (θ1 mm), groundwater
exchange coefficient (θ2 mm), 1-day-ahead maximum capacity of
the routing store (θ3 mm), and the time base of the unit
hydrograph (θ4 d). More details on the GR4J model can be
found in Perrin et al. (2003).
Schematic diagram of the GR4J rainfall–runoff model adopted by
Perrin et al. (2003). In the figure, P and E refer to precipitation and
evapotranspiration, respectively; En and Pn denote net
precipitation and net evapotranspiration, respectively; Ps refers to the
part of precipitation that fills the production store (i.e., S). The
production store is determined as a function of the water level S in the
production store. θ1, θ2, θ3, and
θ4 denote model parameters. The Perc refers to the percolation
leakage that is a function of production store S and parameter
θ1. The Pr refers to the total quantity of water that reaches the
routing functions. UH1 and UH2 denote two-unit hydrographs. Q1
and Q9 refer the corresponding output of the unit hydrographs,
respectively; F indicates the groundwater exchange term; R is the level in
the routing store. Qr refers to the outflow of the routing store,
Qd is a function of water exchange, and Q refers to the total
streamflow.
The GR4J model is a parsimonious but efficient model. The model has been
used successfully across a wide range of hydro-climatic conditions across
the world, including the crash testing of model performance under
contrasting climatic conditions (Coron et al., 2012), and the simulation
of runoff for revisiting the deficiency in insufficient model calibration (Fowler et al., 2016). For example, Fowler et al. (2016) verified that
conceptual rainfall–runoff models were more capable under changing climatic
conditions than previously thought. These characteristics make the GR4J
particularly suitable as a starting point for implementing modifications
and/or improving predictive ability under changing climatic conditions.
The HB framework for the time-varying model parameter
In this study, various versions were constructed for evaluating the
projection capabilities of models for contrasting climatic conditions (wet
and dry years), and for considering the temporal variation and spatial
coherence of parameter θ1.
Process layer: temporal variation of the model parameter
As described in the literature (Pan et al., 2019; Perrin et al., 2003;
Renard et al., 2011; Westra et al., 2014), parameter θ1, which
represents the primary storage of water in the catchment, is the most
sensitive parameter in the GR4J model structure, and the stochastic
variations of this parameter have the largest impact on model projection
performance (Renard et al., 2011; Westra et al., 2014). In addition, the
temporal variation in the catchment storage capacity was physically
interpretable. Periodic variations in the production store capacity θ1 can be induced by the periodicity in precipitation (Pan et al., 2018) and in seasonal vegetation growth and
senescence. In the present study, θ1 was constructed to account
for the periodical variation that had a significant impact on the
extensionality of the model. The periodical variation in catchment storage
capacity θ1 is described by a sine function, using amplitude and
frequency.
Thus, for any catchment c, the full temporal regression function for θ1 at the process layer is as follows:
Process layer:θ1(c,t)=α(c)+β(c)sinω(c)t,
where α, β, ω are regression parameters
for the specific DSST method; α signifies the intercept;
β,ω represents the amplitude and frequency
of the sine function, respectively; and t is the time step. According to the
definition of the GR4J model (Perrin et al., 2003), the value of θ1 must be a positive value. If model parameter θ1 is
constant then β=0 and α>0 suffice in Eq. (1). Meanwhile, the
value of ω becomes irrelevant. Thus, the resulting model simplifies
to a stationary hydrological model.
Prior layer: spatial coherence of regression parameters
For a heterogeneous region that is distinctly nonuniform in climatic and
geologic conditions, different catchments within the region typically have
different catchment storage capacities and different values of production
store capacity θ1. For a homogeneous region prescribed by
similar climatic and geologic conditions in each part, the production store
capacity (in Eq. 1) is expected to be the same among different catchments of
the region. The model could be improved by considering spatial input, i.e.,
the spatial coherence of parameters across adjacent catchments (Chen et
al., 2014; Lima et al., 2016; Merz and Bloschl, 2004; Oudin et al., 2008;
Patil and Stieglitz, 2015; Renard et al., 2011; Sun et al., 2014).
In this study, independent Gaussian prior distributions were used for the
amplitude β and frequency ω at the prior layer to include the
potential spatial coherence. Their equations are as follows:
Prior layer:β(c)=Nμ2,σ22,ω(c)=Nμ3,σ32,
where μ2, μ3, σ2, and σ3 are
hyper-parameters, and N. represents the hyper-distribution,
i.e., a Gaussian distribution. Independent Gaussian distributions were
assumed for the amplitude β and frequency ω that were used to
model spatial coherence based on practical considerations. The prior layer
of the HB framework aims to describe the variation of β,ω in space by means of a Gaussian spatial process in which
the mean value depends on covariates describing regional characteristics.
Amplitude β and frequency ω are the most important parameters
in the regression function and can reflect the spatial connection of
the variation and cyclicity of catchment production storage capacity among
catchments. The Gaussian distribution is one of the widely used
distributions for describing the prior layer within the HB framework and has
been applied in many previous studies, such as Sun et al. (2015) and
Chen et al. (2014). In addition, the Gaussian
distributions were introduced to describe the spatial coherence of β and ω because there are still uncountable factors that may have impacts
on the spatial coherence between adjacent catchments, which might make the
coherence tend to converge a central value (but with finite variance) and
obey the central limit theorem.
Modeling scenarios
Five modeling scenarios (Table 1) were carried out to assess the effect of
the spatial coherence on the time-varying function. Different levels of spatial
coherence of β,ω were assumed in scenarios 1
to 4, while in scenario 5 parameter θ1 was set to be constant to
provide a comparison. It should be noted that the estimates for spatially
coherent regression parameters would be shared by different catchments while
other quantities would be regarded as catchment-specific variables. For
example, amplitude β is spatially linked in scenario 1, i.e.,
βc=Nμ2,σ22, which
means that the estimates of β are shared by all catchments.
Meanwhile, regression parameters ω1-1, ω1-2, and
ω1-3 are used as independent variables to represent the
frequency of model parameter θ1 in different catchments. The
number of unknown quantities in different scenarios are as follows: 15
in scenarios 1 and 2, 13 in scenario 3, and 18 in scenario 4. The
prior ranges of all unknown quantities (including model parameters θ2, θ3, and θ4; regression parameters α,
β, and ω; and hyper-parameters μ2, σ2,
μ3, and σ3) in different scenarios and both DSST schemes
could be found in Table S1 in the Supplement. It should be noted that
in a specific scenario, some unknown quantities might not exist. For
example, μ3 and σ3 did not exist in scenario 1 while
μ2 and σ2 did not exist in scenario 2.
Different spatial coherence scenarios for amplitude β and
frequency ω in the time-varying functional form of model parameter
θ1. To explore the performance of spatial coherence within the
time-varying function, different levels of spatial coherence for amplitude
β and frequency ω were assumed for the first three scenarios;
in contrast, no spatial coherence is assumed in scenario 4, and a temporally
stable θ1 is assumed in scenario 5.
CategoryScenarioβωConstraintsTime-varying1Parameter β is region-relatedParameter ω is catchment-specificθ1=α(c)+β(c)sin[ω(c)t],while β(c)=N(μ2,σ22)Spatial2Parameter β is catchment-specificParameter ω is region-relatedθ1=α(c)+β(c)sin[ω(c)t],coherencewhile ω(c)=N(μ3, σ32)θ1=α(c)+β(c)sin[ω(c)t],3Parameter β is region-relatedParameter ω is region-relatedwhile β(c)=N(μ2, σ22)and ω(c)=N(μ3, σ32)No spatial4Parameter β is catchment-specificParameter ω is catchment-specificθ1=α(c)+β(c)sin[ω(c)t]Time-invariantcoherence5No parameters β or ωθ1 is stationary
NB: θ1 represents the production storage capacity of the
catchment; β is the slope describing long-term change during the
modeling period, and ω is the amplitude of the sine function
describing its seasonal variation during the modeling period; μ2,
σ2, μ3, and σ3 are hyper-parameters.
Estimation and projection
The objective function and parameter inference methods were used to derive
the posterior distribution of all unknown quantities, as illustrated below.
Objective function
For a specific catchment, the model parameters were calibrated to minimize
the following objective function, which was adopted from Coron et al. (2012):
3εcθ1,θ2,θ3,θ4=-RMSEQ1+1+BIAS,4whereRMSEQ=1T∑t=1TQsimt-Qobst2
and RMSEQ refers to the root-mean-square error, in
which Qsim is derived by the adopted hydrological model. T represents the
number of the time series while t is the time step.
Coron et al. (2012) showed that this objective function performed well.
In this function, the combination of RMSEQ and
BIAS (Eq. 7) gives weight to dynamic representation as well as the water balance.
Using square-root-transformed flows to compute the RMSE reduces the
influence of high flows during the calibration period and provides a good
compromise between alternative criteria.
In the case of multiple catchments, the objective function of the HB
framework was the product of Eq. (3) and the conditional probability of spatial
coherence of regression parameters fN. It was written as follows:
Scenario 1:Λ=∏c=1Cεcθ1(t,c),θ2(c),θ3(c),θ4(c)|α(c),β,ω(c)⋅fN(β|μ2,σ2);Scenario 2:Λ=∏c=1Cεcθ1t,c,θ2c,θ3c,θ4c|αc,βc,ω⋅fNω|μ3,σ3;Scenario 3:Λ=∏c=1Cεcθ1t,c,θ2c,θ3c,θ4c|αc,β,ω⋅∏n=12fNβ,ω|μ2,σ2,μ3,σ3;Scenario 4:Λ=∏c=1Cεcθ1t,c,θ2c,θ3c,θ4c;Scenario 5:Λ=∏c=1Cεcθ1c,θ2c,θ3c,θ4c.
Here, the number of catchments in the region is represented by C, and the
Gaussian spatial function between regression parameters β and ω and
hyper-parameters μ2, μ3, σ2, and σ3
are denoted by fN(). N refers to the Gaussian distribution and n
represents the number of regression parameters that are spatially coherent.
Inference
The uniform distribution is used as the prior distribution for
hyper-parameters and spatially irrelevant parameters. Meanwhile, spatially
relevant parameters are sampled from the Gaussian distributions. Because the
prior distribution has no impact on the final evaluation of different
scenarios, the prior distributions are not presented in Eq. (5). The likelihood
functions defined in Eqs. (3) and (5) pose a computational challenge because
their dimensionality grows (primarily related to the number of
catchment-specific parameters) with the number of catchments considered. The
unknown quantities, including model parameters (θ2, θ3, and θ4), regression parameters α, β, and
ω, and hyper-parameters μ2, σ2, μ3, and
σ3 (if present), are sampled and estimated simultaneously using
the Shuffled Complex Evolution Metropolis (SCEM-UA) sampling method (Ajami et al., 2007; Vrugt et al., 2003, 2009). The SCEM-UA
sampling method is a widely used Markov Chain–Monte Carlo algorithm for
simulating the posterior probability distribution of parameters that are
conditional on the current choice of parameters and data. When compared with
traditional Metropolis–Hasting samplers, the SCEM-UA algorithm more
efficiently reduces the number of model simulations needed to infer the
posterior distribution of parameters (Ajami et al., 2007; Duan et al.,
2007; Liu et al., 2014; Liu and Gupta, 2007; Vrugt et al., 2003).
Convergence is assessed by evolving three parallel chains with 30 000 random
samples, the posterior distributions of parameters are evaluated by the
Gelman–Rubin convergence value, and it is confirmed that the convergence value
is smaller than the threshold 1.2 (Gelman et al., 2013).
Model performance criteria
Five criteria were used to assess the projection performance during the
verification periods.
The first criterion was NSEsqrt, known as the arithmetic square
root of the Nash–Sutcliffe efficiency (Coron et al., 2012; Moriasi et al.,
2007; Nash and Sutcliffe, 1970). When compared with the classic NSE,
NSEsqrt gives an intermediate, more balanced picture of the overall
hydrograph fit because it can reduce the influence of high flow. It is
expressed as follows:NSEsqrt=1-∑t=1TQobst-Qsimt2∑t=1TQobst-Q‾obs2,where Qsimt and Qobst represent
the simulated and observed daily streamflow values for the tth day,
respectively, Q‾obs is the mean of the observed daily
streamflow for the calculation interval, and T refers to the length of the
calculation period.
The second criterion is the BIAS, one of the most popular indexes to
reflect the deviation degree between the modeled runoff and observations,
and this is also a part of the objective function Eq. (3).BIAS=∑t=1TQsimt-Qobst∑t=1TQobst
The third criterion is the deviance information criterion (DIC), which
was defined by Spiegelhalter et al. (2002). It is a widely used and
popular measure designed for Bayesian model comparison and is a Bayesian
alternative to the standard Akaike information criterion. The DIC value for
a Bayesian scenario is obtained as follows:DIC=-2logpqθ^Bayes,ξ+2pDIC,where pDIC is the effective number of parameters, defined aspDIC=2logpqθ^Bayes,ξ-1S∑s=1Slogpqθs,ξ,where p refers to probability, q represents the observations of streamflow,
and ξ denotes the time series of model input, e.g., rainfall and
potential evapotranspiration. Posterior mean θ^Bayes=Expectθq,ξ, and
s=1, … , S means the sequence number of the simulated
parameter set θs by the adopted SCEM-UA algorithm. According to Spiegelhalter et al. (2002), scenarios with smaller DIC would be
preferred to scenarios with larger DIC.
The fourth and fifth criteria are the mean annual maximum flow (MaxF,
mm d-1) and mean annual minimum flow (MinF, mm d-1), which are used to qualify
the performance of the high flows and low flows. These criteria are
self-explanatory and have been used in many studies to assess the magnitude
of maximum and minimum levels of flows (Ekstrom et al., 2018). The
scenarios with the least absolute variation between the modeled values and
the observed values are recognized as the best scenarios.
Study area and data
To evaluate the model performance, we used daily precipitation (mm d-1),
potential evapotranspiration (mm d-1), and streamflow (mm d-1) time series
records for three unregulated and unimpaired catchments in southeastern
Australia, taken from the national dataset of Australia (Zhang et
al., 2013), covering 1976–2011. The streams were unregulated: they were not
subject to dam or reservoir regulations, which can reduce the impact of
human activity. The observed streamflow record contained at least 11 835
daily observations (equivalent to record integrity of greater than 90 %)
for 1976–2011, with acceptable data quality. The first complete year of
data was used for model warm-up to reduce the impact of the initial soil
moisture conditions during the calibration period.
The attributes of the southeastern Australian catchments are shown in
Table 2 and Fig. 3. The IDs of these catchments are 225219 (Glencairn station on
the Macalister River: mean annual rainfall, potential evapotranspiration,
and runoff are 1106, 1184, and 368 mm, respectively), 405219 (Dohertys
station on the Goulburn River: mean annual rainfall, potential
evapotranspiration, and runoff are 1171, 1196, and 420 mm,
respectively), and 405264 (D/S of Frenchman Ck Jun station on the Big River:
mean annual rainfall, potential evapotranspiration, and runoff are 1408,
1160, and 465 mm, respectively). As shown in Fig. 3, these catchments
are adjacent to each other. All catchments experienced a severe multiyear
drought around the end of the millennium. Saft et al. (2015) identified
that the rainfall–runoff relationship in these catchments was altered during
the long-term drought.
Comparison of catchments attributes in terms of mean annual
rainfall (mm), mean annual evaporation (mm), and mean annual runoff (mm) for
1976–2011.
CatchmentsRiverObservationsObservationsMeanMean annualMeanIDnamestartendannualpotential evapo-annualrainfalltranspirationrunoff225219Macalister1 January 197630 December 201111061184368405219Goulburn1 January 197630 December 201111711196420405264Big1 January 197630 December 201114081160465
Locations of study catchments in Victoria, Australia. The
catchment IDs are 225219 (Macalister River catchment), 405219 (Goulburn
River catchment), and 405264 (Big River catchment).
Results and discussion
Results from the DSST were used to assess the model projection performance
for five scenarios under contrasting climatic conditions. First, a DSST was
conducted in each catchment to divide original records into wet and dry
years. Then, the projection performance for the five scenarios and
associated parameter uncertainties were evaluated using the criteria
described above.
Dry years identification
As illustrated in Table 3 and Fig. 4, the drought definition method
identified that the three catchments had similar dry-year characteristics,
with the same drought start (1997) and end (2009) points. The length of dry
years for the studied catchments is the same, 13 years. The mean dry years'
anomaly was more severe in the Macalister catchment (225219), with an
11.70 % reduction in the mean dry years' anomaly while the other two
catchments experienced reductions of 11.16 % (405219) and 11.14 %
(405264).
Drought identification results for the catchments.
NB: R1 and R2 refer to the runoff coefficient during the wet and
dry years, respectively.
The identified dry years in all catchments. The annual anomaly is
defined as a percentage of the mean annual rainfall.
In terms of changes in rainfall, on average catchments had an 11 %
reduction from the wet years to the dry years (Table 3). Meanwhile, these
catchments experienced a 26.3 % decrease in runoff during the dry years,
which is much more severe than the reduction in rainfall. The similar
findings can be derived out from the comparison of runoff coefficients of
different periods; that is, all catchments experienced a decrease in its
runoff coefficients during the dry years.
Model performance in five scenarios
As shown in Figs. 5a, 6a, and 7, the calibrated model parameters
yielded a good simulation performance over the calibrated periods for all
criteria. For example, the mean NSEsqrt score during the calibration
period across these catchments remained close to about 0.7 or slightly
higher, regardless of which scenario was chosen. However, when the same
parameter sets were verified by simulating streamflow over drier or wetter
years, the model performance was degraded, including both the robustness and
accuracy of projection performance. Furthermore, the magnitude of
performance loss increases along with the variation in rainfall between the
calibration and verification periods.
NSEsqrt for each of the five scenarios for each catchment
during (a) the calibration period (wet years) and (b) the verification
period (dry years). The white dots represent the median estimates of the
results.
NSEsqrt for each of the five scenarios for each catchment
during (a) the calibration period (dry years) and (b) the verification
period (wet years). The white dots represent the median estimates of the
results.
Long-term simulation BIAS of Qmedian for five scenarios in all
catchments. Simulation BIAS is plotted as a 10-year moving average, and
10-year moving average streamflows are plotted for reference. The three left-hand
graphs are calibrated in the wet years and then verified in the dry
years, while the opposite sequence applies to the right-hand graphs.
Figure 5 shows the NSEsqrt performance for calibration in wet years
and verification in the dry years for each scenario in all catchments. All
scenarios performed well in all catchments with the mean NSEsqrt
reaching 0.81 during the wet calibration period, and then all scenarios
experienced a slight decrease in performance (NSEsqrt=0.75) during
the dry verification period. Scenario 4 (time-varying parameters without
spatial inputs) or scenario 5 (temporally stable parameters) generally
performed better during the calibration period than the scenarios that
considered different levels of spatial coherence for the regression
parameters. During the verification period, the NSEsqrt rank order
changed (Fig. 5b). Scenario 4 had a higher median NSEsqrt performance
than scenario 5 in catchments 225219 and 405264. Although the median
estimate in scenario 4 was slightly inferior to the latter in catchment
405219, its distribution of the NSEsqrt performance was much more
positively biased from the median estimates than scenario 5. Furthermore,
the former reaches a higher NSEsqrt performance than the latter when
comparing the top NSEsqrt performance of these two scenarios. Thus, it
indicates the validity of the time-varying scheme for improving model
performance. However, the introduction of additional regression parameters
(α, β, and ω) at the same time amplified the
model projection uncertainty in two of three catchments (405219 and 405264)
when comparing results from scenarios 4 and 5. Fortunately, the appropriate
adoption of spatial coherence alleviates this problem. In the DSST scheme of
calibrating in the wet years and verifying in the dry years, scenario 2
exhibited the smallest fluctuation range of NSEsqrt estimate in
catchments 405219 and 405264 and was the second-best scenario in catchment
225219. Conversely, scenario 3 exhibited the smallest fluctuation range of
NSEsqrt estimate in catchment 225219, and was the second-best scenario
in catchments 405219 and 405264. As for the median NSEsqrt estimate,
scenario 2 is the best scenario (which showed the best performance in
catchment 225219 and 405219 but was the fourth in catchment 405264),
followed by scenario 3 (which was the second-best scenario in catchments
405219 and 405264 and was the third in catchment 225219). In addition, the
highest median NSEsqrt performance in scenarios 4 and 5 during the
calibration period did not guarantee the same superior performance during
the verification period. This illustrates the deficiency of time-varying and
stationary schemes of model parameters when spatial inputs from adjacent
catchments are not considered.
Similarly, Fig. 6 illustrates the NSEsqrt performance for each
scenario in all catchments for calibration in the dry years and verification
in the wet years. All scenarios performed well for all catchments with the
mean NSEsqrt reaching 0.75 in the dry calibration period and 0.79 in
the wet verification period. As shown in Fig. 6, models experienced a
slight improvement in NSEsqrt performance when transferred from the dry
years to the wet years. However, the projection performance calibrated using
a contrasting climatic condition was inferior to the simulation performance
that was directly calibrated from the climatic condition, compared with
Figs. 5a and 6b, or 6a and 5b. For example, the
NSEsqrt performance in Fig. 6b is inferior to that in Fig. 5a.
By comparing scenarios in the calibration period, it was found that
scenarios 4 and 5 exhibited the highest performance in two of three
catchments (405219 and 405264), followed successively by scenario 3,
scenario 2, and scenario 1. During the verification period, the median
NSEsqrt performance in scenario 4 was 0.80 % higher than scenario 5;
however, the variation range in scenario 4 was 53 % wider than the latter.
These results demonstrate that the time-varying scheme (scenario 4) for
model parameters improved the median NSEsqrt performance but also
amplified the projection uncertainty compared with the results from the
stationary scheme (scenario 5) for model parameters. In the DSST scheme of
calibrating in the dry years and verifying in the wet years, scenario 3,
which considered both spatial coherence of β and ω between
different catchments, exhibited the highest median NSEsqrt for all
catchments, had the smallest fluctuation range in two catchments (225219 and
405264), and had the second least variation in catchment 40519
during the verification period. Conversely, scenario 2, the scenario with
the best median estimate performance during the verification period in
Fig. 5, is just the fourth in all five scenarios in this DSST scheme.
Compared with other model scenarios, the incorporation of spatial coherence
of both regression parameters in scenario 3 reduced the projection
uncertainty and improved the robustness of the model performance, with the
smallest fluctuation ranges in most options under the contrasting climatic
conditions. It indicates that the spatial setting of model parameters
between different catchments provided a clear input for reducing the
uncertainty of the model projection performance during the verification
period. In addition, it also should be noted that model parameters
calibrated over dry years, contrastively, were not suitable for predicting
runoff over wet years because of a larger degradation in projection
performance than the scheme with the adverse calibration–verification
direction.
Comparing the DIC results for both DSST schemes in Tables 4 and 5, the
best DIC value is achieved by scenario 3, which incorporates the spatial
coherence of both regression parameters and is the most complex scenario in
the comparison. This finding is consistent with the results obtained by using the
NSEsqrt criterion and showed the validity of the spatial coherence of
both regression parameters in ensuring the robustness of the hydrological
projection performance. In addition, when comparing the DIC results of scenarios
4 and 5, the setting of time-varying functions improved the DIC performance
in both DSST schemes. This finding also agreed with the results obtained by using the
NSEsqrt criterion and indicated the positive implications of the
time-varying model parameters on the projection performance.
Comparison of five scenarios in terms of the deviance information
criterion (DIC) when model parameters were calibrated in the wet years and
verified in the dry years.
Comparison of five scenarios in terms of the deviance information
criterion (DIC) when model parameters were calibrated in the dry years and
verified in the wet years.
Tables 6 and 7 illustrate the performance of high and low flows during the
verification period in terms of MaxF and MinF estimates for the median
projected streamflows in both DSST schemes. As shown in Table 7, for the
projection of the high-flow part, scenario 3 exhibits the best performance in
all catchments among five scenarios under the scheme of calibrating in the
dry years and verifying in the wet years. For the projection performance in
the other DSST scheme (Table 6), scenario 3 has the best projection
performance in the high-flow part in catchment 225219 and is the second-best
scenario in the other two catchments. It indicates that the incorporation of
spatial coherence of both amplitude β and frequency ω
successfully improves the projection performance in the high-flow part. As
for the projection of the low-flow part, the discrepancy between the results
of different scenarios and the observed low flows is not obvious (the
absolute differences between the observed values and modeled values are very
small). Furthermore, scenario 3 shows the best-projected performance in two
catchments (405219 and 405264) in the scheme of calibrating in dry years and
verifying in wet years, and it is the best scenario in catchment 405264 in the
scheme of calibrating in wet years and verifying in dry years. In addition,
scenario 3 is the second-best option in catchments 225219 and 405219 under
the scheme of calibrating in wet years and verifying in dry years. Combined
with the projection performance of both high and low flows, scenario 3
achieves its superior projection performance mainly by the improvement in
the prediction of high-flow parts.
Comparison of the projection performance of median flows during the
verification period associated with the mean annual maximum flow (MaxF,
mm d-1) and mean annual minimum flow (MinF, mm d-1) when model parameters were
calibrated in the wet years and verified in the dry years. The percentage
represents the percentage variation between the modeled value and the observed
value.
Note: (1) the data in 1976 have been used for model warm-up to reduce the impact of
the initial soil moisture conditions during the calibration period, and is
not counted in the table; (2) the scenarios with bold values are labeled as the best scenario for
projecting the streamflow during the verification periods, and the values
from these scenarios have the least absolute percentage difference with the
observed values.
Comparison of the projection performance of median flows during the
verification period associated with the mean annual maximum flow (MaxF,
mm d-1) and mean annual maximum flow (MinF, mm d-1) when model parameters were
calibrated in the dry years and verified in the wet years. The percentage
represents the percentage of variation between the modeled value and the observed
value.
Note: (1) The data in 1997 have been used for model warm-up to reduce the impact of
the initial soil moisture conditions during the calibration period, and is
not counted in the table; (2) The scenarios with bold values are labeled as the best scenario for
projecting the streamflow during the verification periods, and the values
from these scenarios have the least absolute percentage difference with the
observed values.
Figure 7 shows the BIAS estimates for the median of the posterior
distribution of model parameters for all modeling scenarios across all
catchments when transferability between the wet and dry years was examined.
Although BIAS was a component of the objective function (Eq. 3), the
10-year rolling average BIAS still deviated considerably from a value of 1
for all the scenarios in the two DSST schemes. The median estimates of the
posterior distribution in both scenarios performed well in the NSEsqrt
criterion for both periods. However, the median estimates did not ensure
unbiased simulations over the modeling period; one scenario with a higher
NSEsqrt criterion may have an altered BIAS during the modeling period.
The BIAS results in catchments 225219 and 405219 showed some similarity: all
scenarios tended to underestimate streamflow along the time sequence in both
DSST schemes. Conversely, all scenarios tended to overestimate the
streamflow in catchment 405264 in both schemes. By comparing the BIAS
performance for the five scenarios, it was observed that the spatial setting
of modeling scenarios generally tended to enlarge the BIAS in all
catchments, while the difference between scenarios 4 and 5 was very small.
Posterior distributions of the regression parameters (β
and ω) for the production storage capacity (θ1)
for the four model scenarios in each catchment when calibrated in the wet
years and verified in the dry years. The solid horizontal lines within the
violin plots denote the 25th and 75th percentiles of the posterior
distribution, while the white dots denote median estimates.
Posterior distributions of the regression parameters (β and
ω) for the production storage capacity (θ1) for the
four model scenarios in each catchment when calibrated in the dry years and
verified in the wet years. The solid horizontal lines within the violin
plots denote the 25th and 75th percentiles of the posterior
distribution, while the white dots denote median estimates.
Parameter uncertainty analysis
The uncertainty of the parameters was characterized by the posterior
distribution of the regression parameters and was derived by the MCMC
iteration. As mentioned in Sect. 2.3.2, amplitude β and frequency
ω were assumed to have different levels of spatial coherence in each
modeling scenario (Table 1); these scenarios in each DSST regime are
compared in Figs. 8 and 9. It should be mentioned that there was no
regression parameter in scenario 5. Solid lines in the violin plots
represent the 25th and 75th percentiles of the posterior
distribution. The white dots in the violin plot denote the median estimate
of the posterior distribution. In the upper plots in Figs. 8 and 9, it can
be clearly seen that the first three scenarios had a much smaller variation
interval than scenario 4 in terms of amplitude β, which denotes the
amplitude of the sine function. The catchment averages of both schemes of
the median estimates of β in the first three scenarios are 2.78,
-4.91, and 9.26 respectively, while that in the fourth scenario is much
larger, reaching -39.20. Scenario 3, which considered both spatial
coherence of amplitude β and frequency ω, has the narrowest
interval of β for all catchments, followed successively by scenario 1
(only considered the spatial coherence of the amplitude β), scenario
2 (only frequency ω was spatially coherent), and scenario 4 (no
regression parameter was spatially coherent). With regard to the regression
parameter ω, which denotes the frequency of the sine function (in
the lower figures of Figs. 8 and 9), its median estimates in both groups of four
scenarios differ slightly. As shown in Fig. 8, the catchment averages of
frequency ω for different scenarios are 0.24, 0.14, 0.15, and 0.18,
while those in Fig. 9 are 0.15, 0.26, 0.23, and 0.17 respectively. The
period T of the sine term could be derived based on the estimates of
ω by equation T=2π/ω. Thus, the mean periods T of model
parameter θ1 for different scenarios are 26.2, 46.3, 41.9, and
35.2 in Fig. 8, respectively. Similarly, the mean periods T are 42.9,
24.1, 27.4, and 38.0 in Fig. 9, respectively. In addition, we used the
Hilbert–Huang transform method (Huang et al., 1998) to identify the
potential periods of the series of several climate variables (including the
daily rainfall, daily potential evapotranspiration, daily maximum
temperature, and daily minimum temperature in the studied catchments). It was
found that these daily series have periods of 22.2–49.1 d.
Thus, we guess that the potential periods of these climate variables may be
the possible reasons for the periods of time-varying parameters. It also
should be mentioned that the adopted Hilbert spectrum method is one of the
most popular methods for analyzing nonlinear and nonstationary data.
Huang et al. (1999) indicated that this method is better than the Fourier
transform method and wavelet transform method in processing nonlinear and
nonstationary data.
In summary, by combining the results of parameter uncertainty estimation and
model projection performance evaluation, the incorporation of spatial
coherence successfully improved the robustness of the projection performance
in both DSST schemes by controlling the estimation uncertainty of amplitude
β.
Conclusions
In this study, a two-level HB framework was used to incorporate the spatial
coherence of adjacent catchments to improve the hydrological projection
performance of sensitive time-varying parameters for a lumped conceptual
rainfall–runoff model (GR4J) under contrasting climatic conditions. First,
a temporal parameter transfer scheme was implemented, using a DSST procedure
in which the available data were divided into wet and dry years. Then, the
model was calibrated in the wet years and evaluated in the dry years, and
vice versa. In the first level of the proposed HB framework, the most
sensitive parameter in the GR4J model, i.e., the production storage capacity
(θ1), was allowed to vary with time to account for the periodic
variation that had significant impacts on the extensionality of the model.
The periodic variation in catchment storage capacity was represented by a
sine function for θ1 (parameterized by amplitude and
frequency). In the second level, four modeling scenarios with different
spatial coherence schemes and one scenario with a stationary scheme of
catchment storage capacity were used to evaluate the transferability of
hydrological models under contrasting climatic conditions. Finally, the
proposed method was applied to three spatially adjacent, unregulated, and
unimpaired catchments in southeast Australia. The study concludes that (1)
the time-varying setting was valid in improving the model performance but
also extended the projection uncertainty in contrast to the stationary
setting, (2) the inclusion of spatial coherence successfully reduced the
projection uncertainty and improved the robustness of model performance, and
(3) a large performance degradation has been found in the DSST scheme with
its model parameters calibrated over dry years and verified in the wet
years. This study improves our understanding of the spatial coherence of
time-varying parameters, which will help improve the projection performance
under differing climatic conditions. However, there are several unsolved
problems that need to be addressed. First, the spatial setting of regression
parameters may expand the BIAS between the simulation and streamflow
observation with a single objective function; the potential physical
mechanism behind this result should be explored further. Second, this
study was confined to spatially coherent catchments that are similar in
climatic and hydrogeological conditions; further research is needed to
determine which factors have the most significant impacts on model
projection performance when considering obvious inputs from other
catchments.
Data availability
The precipitation, potential evapotranspiration, and streamflow data of the studied catchments in south-eastern Australia are taken from publicly available data (10.4225/08/58b5baad4fcc2, Zhang et al., 2013).
The supplement related to this article is available online at: https://doi.org/10.5194/hess-23-3405-2019-supplement.
Author contributions
All of the authors helped to conceive and design the analysis. ZP
and PL performed the analysis and wrote the paper. SG, JX,
JC, and LC contributed to the writing of the paper and made
comments.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
The numerical calculations were done on the supercomputing system in the Supercomputing Center of Wuhan University. The authors would like to thank the editor and anonymous reviewers for their comments, as well as Chong-Yu Xu in the University of Oslo for proofreading an earlier version of the paper, which helped improve the quality of the paper.
Financial support
This research has been supported by the National Key Research and Development Program (grant no. 2018YFC0407202), the National Natural Science Foundation of China (grant nos. 51861125102 and 51879193), the Natural Science Foundation of Hubei Province (grant no. 2017CFA015), and the Innovation Team in Key Field of the Ministry of Science and Technology (grant nos. 2018RA4014).
Review statement
This paper was edited by Fabrizio Fenicia and reviewed by two anonymous referees.
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