Introduction
The flow duration curve (FDC) provides a compact summary of the variability
of daily streamflow by indicating what proportion of the flow regime exceeds
a given flow rate. FDCs have considerable practical relevance, particularly
in supporting decisions that are affected by the availability and reliability
of surface water. Common applications of FDCs include the design and
management of hydropower infrastructure
e.g.,, the determination of
environmental flow standards for ecosystem protection
e.g.,, the allocation of water resources for
consumptive uses e.g.,, or the prediction of
streamflow time series in ungauged or poorly gauged catchments
e.g.,.
Despite their utility, empirical FDCs are unavailable for many basins,
primarily because they require extensive on-site observations of daily
streamflow . Globally, the majority of catchments remain
ungauged (or the gauge data that exist are subject to significant quality
assurance and data availability constraints). Furthermore, the global number
of stream gauges continues to decline because of ongoing budgetary
constraints faced by water monitoring agencies
. Therefore FDCs must typically be
estimated in data-scarce areas. The most widely used techniques for FDC
estimation are simple, graphical methods. Such empirical methods are easy to
implement but often rely on overly simplistic assumptions that lead to
substantial prediction errors. For instance, in Nepal, the regionalization
method prescribed in official design manuals
e.g., relies on one in
situ observation of streamflow during the dry season to scale
standardized regional indices for monthly flows. The procedure neglects the
inter-annual variability of low flows, which leads to important biases in the
predicted flow distributions (see Sect. S1 of the Supplement). Even in gauged
catchments, FDCs constructed from historical observations may not represent
current flow conditions well, because flow regimes are impacted by climate
change and anthropogenic alterations of the catchments
e.g.,. Predicting streamflow
in ungauged basins, particularly in the context of environmental change,
remains both a fundamental necessity for water managers and a major research
challenge .
Recent efforts to predict FDCs in ungauged catchments focus on statistical
approaches that predict the flow distribution based on the catchment's
similarity to nearby, gauged watersheds . Index flow
approaches, which regionalize specific index flows (typically the mean flow),
and use those indices to rescale empirical FDCs from similar catchments, are
particularly popular
e.g.,.
While differing in methodological details, all index flow approaches assume
that FDCs do not vary within homogeneous regions, except by a scaling factor.
Because they do not assume any specific runoff-generating process,
statistical methods are versatile. They have been successfully applied
globally to predict FDCs in a variety of climates and catchment types
. However, methods are also insensitive to the
diversity of controls on the shape of the FDC exerted by climate processes
and catchment characteristics. This may affect their reliability under
non-stationary conditions . Finally, the
calibration of statistical methods relies on extensive streamflow
observations from a large number of representative and well-characterized
catchments e.g.,.
Their performance is therefore sensitive to the spatial density of available
gauges , and their reliability in regions where
streamflow data are truly scarce is uncertain.
Stochastic, process-based models that mechanistically link the drivers,
state, and response of the system are a promising avenue to address these
issues. In these models, basic assumptions about the stochastic structure
of rainfall and the (deterministic) response of catchments allow the analytic
derivation of streamflow probability density functions (PDFs). (Note that
because the FDC can be obtained directly by transforming the PDF, a
predictive technique that yields the streamflow PDF will also allow the FDC
to be estimated.) show that runoff follows a gamma
distribution if catchments behave as a linear reservoir, forced by
stochastic rainfall that follows a marked Poisson process. The resulting
gamma distribution depends on two parameters that are determined by the
recession characteristics of the catchment, and by the frequency and
intensity of effective rain. This process-based approach to the streamflow
PDF has been extended to include the fast flow component of streamflow
, non-linearities in subsurface storage–runoff
relationships , the effects of short-term snowmelt
, and the carryover of subsurface storage between
seasons in seasonally dry climates . Although the
stochastic framework allows the effects of changes in climate or landscape to
be independently modeled, it relies on strong simplifying assumptions about
the spatial homogeneity of catchments. These assumptions make the existing
process models less versatile than statistical methods. Nonetheless, the
approach has low calibration requirements because it relies on a small number
of parameters, which can be determined using rainfall, climate, and
geomorphological characteristics of the catchments
. This information is increasingly available
in ungauged basins, thanks to remote-sensing technologies, even when
ground-based measurements are sparse.
Process-based models successfully reproduce streamflow PDFs in numerous
gauged catchments worldwide
, including Nepal
. Yet their predictive performance in ungauged
basins remains largely unassessed, particularly in regions where the local
gauge density is globally representative (as opposed to densely monitored
catchments in developed countries such as, e.g., France and Austria in
). For lower gauge densities, it is unclear whether
the advantages of the process-based approaches, which are derived from an
explicit representation of flow-generating processes, are outweighted by the
limitations imposed by the restrictive assumptions underlying these methods
– and whether this trade-off is altered by non-stationarity in climate
drivers.
(a) Global histogram of the approximate spatial density of streamflow
gauges by nation, represented by the sample of 8540 gauges indexed by the Global
Runoff Data Center for 146 countries . With a density of 1.6 gauges
per 10 000 km2, Nepal falls close to the mode of the global distribution.
(b) Location of the rain gauges, streamflow gauges, and
corresponding Nepalese
catchments used in the analysis.
Using Nepal as a test case, this study compares the process-based and
statistical approaches on the basis of (i) their ability to predict FDCs in
ungauged basins, (ii) their sensitivity to data scarcity, represented both by
the spatial density of the stream gauge network and by the temporal extent
(length) of the available streamflow records, and (iii) their ability to
accommodate changes in the rainfall regime.
Nepal provides an ideal setting to compare the two approaches, for four
reasons. First, the country is representative of global availability of
streamflow data, as measured by the density of its stream gauge network
(Fig. a). Second, methods drawn from both statistical and
process-based approaches have been developed and validated in Nepal. Here we
compare the stochastic–dynamic framework developed in
with the index flow model described in
. Third, flow generation processes in Nepalese
Himalayan catchments are complex, particularly with respect to the spatial
and temporal properties of precipitation. Rainfall derives from the Indian
summer monsoon and is strongly affected by topography. As a result, local
rainfall is temporally autocorrelated, spatially heterogeneous, and highly
seasonal. There is also significant carryover of groundwater storage between
the wet and dry seasons, so that dry-season discharge reflects the features
of the antecedent wet season. These characteristics violate many of the
assumptions that underlie the process-based method. The analysis in Nepal is
therefore likely to provide a conservative estimate of the potential
performance of the process-based method in ungauged basins. Finally,
developing reliable methods for FDC prediction in Nepal represents an
opportunity for “use-inspired science” .
Nepal has an enormous untapped hydropower potential and is in dire need of
electrical power, particularly in rural areas. A reliable method to estimate
FDCs in ungauged catchments would be a valuable tool to support the
development of micro-hydropower, a sustainable technology for rural
electrification .
Section describes the two models and the procedures used to
estimate their parameters from streamflow and rainfall observations.
Section presents the results of the comparative analysis in
Nepal. Section examines the key sources of errors for both
models and discusses implications for both “Prediction in ungauged
catchments” (PUB) and “Predictions under change” (PUC) beyond Nepal.
Methods
Compared approaches
Process-based model
The process-based approach models daily streamflow as a random variable.
Subject to strong simplifying assumptions about rainfall stochasticity and
runoff generation, the streamflow PDF can be analytically derived. During the
wet season, daily rainfall is represented as a stationary marked Poisson
process with exponentially distributed depths. Assuming linear
evapotranspiration losses, showed that effective
rain, that is, the portion of the total rainfall that contributes to
streamflow generation, also follows a stationary marked Poisson process. For
a spatially homogenous catchment with an exponentially distributed response
time (i.e., a catchment that behaves as a linear reservoir), this effective
rainfall will produce gamma-distributed streamflow. The parameters of the
gamma distribution are derived from the frequency (λP) and
mean depth (αP) of rainfall, and from the recession constant
(k) of the catchment. If rainfall in the dry season is sufficiently minimal
that effective rainfall does not contribute to runoff generation, then
dry-season streamflow represents only the discharge of groundwater stored
during the previous wet season. This discharge is modeled as a single
seasonal recession with stochastic initial conditions that depend on the
wet-season properties. Because groundwater is not replenished during the dry
season, the water table is subject to a large transient drawdown, resulting
in a non-linear discharge behavior and a power-law relation between recession rate and
discharge . We showed in
that the distributions of streamflow, and
therefore the FDC, in seasonally dry climates that meet the assumptions
above, can be expressed analytically as a function of seven independent
parameters: the frequency (λP) and mean intensity
(αP) of wet-season rainfall, maximum daily evapotranspiration
during the wet season (ET), the water storage capacity of the soil in the
root zone (SSC), the (linear) wet-season recession constant (k), the
duration of the dry season (Td), and the exponent of the power-law
recession during the dry season (b). The model admits an additional input
parameter, the scale a of the power-law seasonal recession, which we showed
in can be expressed as a function of k, b,
λP, and αP. The formal derivation of the model
is summarized in Appendix .
The model was successfully validated in a variety of regions with seasonally
dry climates worldwide, including Nepal, where observed FDCs were predicted
in 24 gauged catchments with a median Nash–Sutcliffe coefficient of 0.90 on
log-transformed flow quantiles . The approach
successfully reproduced both the rain-driven distribution of flows during the
wet season and the release of stored monsoon water during the dry-season
recession. In this study, we assess the operational performance of the
process-based approach as a tool to predict streamflow in ungauged
catchments. Therefore, we do not further attempt to attribute model errors to
parameters versus the model structure in the results presented in
Sect. , since in practice these errors are confounded in any
real application. The relative significance of these two error sources is
nonetheless discussed in Sect. .
In ungauged catchments, the process-based model is implemented as follows.
Three of the seven parameters of the model (Td, λP,
αP) are rainfall characteristics that can be estimated in
ungauged basins using meteorological observations. Recession parameters (k
and b) describe aquifer properties that are challenging to observe at the
catchment scale. They can be estimated using observed streamflow time series
in nearby gauged basins and subsequently interpolated from nearby gauges
using the geostatistical approach described in ,
which accounts for the topology of the stream network. The last two
parameters (ET) and (SSC) describe catchment-scale soil moisture dynamics
that are arduous to determine empirically. Previous applications of the model
relied on reasonable values of ET and SSC, based on land use, soil, and
climate characteristics of the catchment
e.g.,. Alternatively,
runoff coefficients can be used to directly relate rainfall statistics to
streamflow increments . Runoff coefficients
describe the ratio of mean discharge to mean precipitation, and can be
predicted in ungauged basins using water balance models and meteorological
observations. This approach circumvents the need to estimate ET and SSC, but
the accuracy of predicted runoff coefficients in ungauged catchments is
critically dependent on the type of water balance model used and on the
availability of appropriate calibration data
. Instead, this study follows the former
procedure and uses reasonable estimates of ET and SSC for Nepal.
Statistical model
The statistical approach is entirely driven by observation data and does not
assume any specific runoff generation process. Instead, it identifies and
exploits statistical correlations that may occur between streamflow observed
at existing gauges and the geology, topography, and climate of the
corresponding catchments. The index flow model used in this study was
developed by to regionalize FDCs in Nepal to
assess the potential for small hydropower development. The model is based on
local flow indices for mean (Qm=E[Q]) and low
(q95=Q95/Qm, where Q95 is the 95th streamflow
percentile) flows, and uses a non-parametric approach to represent the shape
of the FDC. Empirical FDCs from available gauges are normalized by
Qm and pooled into equally sized groups based on the q95 index
of the gauge. A standardized curve is determined for each group by taking the
average of the normalized flows corresponding to each duration, in order to
represent the average catchment response in the group. The chosen statistical
approach is considerably less complex than many alternative state-of-the-art
methods using multiple (often non-linear) equations to relate multiple flow
quantiles to a variety of observed covariates (see
, for a review). However,
is, to our knowledge, the most recent
statistical method specifically developed and validated in the study region.
The approach is parsimonious and adapted to situations, where in situ
observations of catchment characteristics are scarce. The method is therefore
representative of the level of complexity of statistical approaches likely to
be implemented in developing countries for practical hydrological engineering
purposes.
Catchment characteristics. Median values and interquartile distances
(IQD) are given for the whole sample of 25 gauges. The table also presents
characteristics of the Chepe Kohla watershed considered in the analysis as a
case study.
Streamflow
Topography
Climate
Recession
Qm
q95
Ny
A
zm
zM
Py
Tmons
λP
αP
AR
CV
ET
k
b
All gauges
Median
76.1
0.14
22
1355
481
5209
1952
99
0.71
18.8
0.29
0.92
2.5
0.17
2.38
Min
7.3
0.06
10
130
116
1913
1260
88
0.54
12.1
0.09
0.61
0.40
0.07
1.99
Max
1462.4
0.25
41
32 817
1641
8369
4030
152
0.91
33.0
0.51
1.53
3.27
0.32
2.99
Chepe Kohla
23.0
0.14
31
277
475
4711
3050
100
0.84
26.5
0.09
1.03
2.1
0.20
2.41
Qm is mean annual flow in m3 s-1; q95 is
the 95th flow percentile normalized by Qm; Ny indicates
the number of observation years; A is the catchment area in km2;
zm and zM are, respectively, the minimum and maximum
elevation of the basin's meters; Py is mean precipitation in
mm yr-1; Tmons is the estimated duration of the monsoon in
days; λP is rainfall frequency during the monsoon (in
d-1); αP is mean rainfall intensity in mm d-1; AR
is the first-order autocorrelation coefficient of rainfall occurrence
(AR =0 if rainfall follows a Poissonian process), CV is the coefficient of
variation of rainfall intensity on rainy days (CV =1 if depths are
exponentially distributed); ET (mm d-1) is the reference
evapotranspiration during the rainy season ; k
is the linear recession constant estimated during the monsoon (in d-1)
and b is the non-linear exponent of the seasonal recession. A soil moisture
capacity of 16 mm is assumed throughout the country
.
Predictions in ungauged catchments are obtained by first using linear
regressions to predict Qm and q95. Although the original
method calls for a stepwise multiple regression approach to determine
regression covariates inductively, we used the regression models obtained in
: Qm is regressed against annual
rainfall (Ry) and gauge elevation (zmin) as a proxy for
evapotranspiration, and q95 is regressed against the ratios of catchment
area occupied by each of the considered geological units. The two regressions
loosely represent the long-term water balance and short-term response of the
catchment. The predicted low-flow index is then used to determine the
standardized FDC shape, which is finally multiplied by the predicted mean
flow to obtain the FDC. An important assumption, inherent to the linear
regression models, is that the dependent variable (here Qm and
q95) is not spatially correlated when controlling for the considered
covariates. This assumption is reasonable in Nepal, where the typical
distance between stream gauges is much larger than the correlation scale of
runoff . In more densely gauged areas (or if runoff
is correlated over larger distances), streamflow observations at neighboring
or flow-connected gauges are likely to be correlated. In these regions,
accounting for the effect of distance and stream network topology when
interpolating flow indices (e.g., using TopREML )
will improve predictions.
Study region and data
The two methods were evaluated using observed streamflow data from 25
Nepalese catchments mapped in Fig. b. The gauges in this data set
have at
least 10 years of daily streamflow records. They were checked for
consistency, using double mass plots and bias: we
discarded non-glaciated catchments that had a precipitation deficit
in their long-term water balance. Watersheds were
delineated using the ASTER GDEM v2 digital elevation model
. The study watersheds are located in central Nepal but
cover a wide variety of catchment sizes, elevation ranges, precipitation
characteristics, and geological units (Table ).
We focused on the Chepe Kohla catchment in central Nepal (Fig. b,
insert) as a case study for analyses requiring resampling (Sect. )
or simulation (Sect. ) of streamflow time series. The Chepe Kohla
watershed has a long (by Nepalese standards) record of daily streamflow
observations (31 years) and is representative of the full sample of gauges in
terms of topography and recession behavior (Table ). The
catchment is also small (i.e., close to spatially homogenous), and local
rainfall is well approximated by a marked Poisson process (first-order
autocorrelation coefficient of rainfall occurrence (AR): 0.09; coefficient of
variation of rainfall depths (CV): 1.09), echoing the underlying assumptions
of the process-based model.
Rainfall characteristics over the sampled catchments were obtained from 178
precipitation gauges
, also mapped
in Fig. b. The average duration of the dry season (Td)
was estimated at each precipitation gauge by fitting a step function to the
corresponding rainfall time series , and wet-season
precipitation records were used to compute the frequency and mean intensity
of rainfall (λP and αP). Rainfall
characteristics were then aggregated at the catchment level by assuming that
the rain process aggregates linearly within the basins. For rainfall
occurrence, we assumed that the duration between rain events caused by two
consecutive storms can be estimated as the average of the inter-arrival times
measured at the rain gauges within the catchment. This allows us to compute
catchment-level rainfall frequency as
λP=1Ng∑iNg1λP(i)-1,
where λP(i) designates rainfall frequency observed at gauge i
and Ng the number of rain gauges within the catchment. Similarly, the
catchment-level duration between rainy seasons is assumed to be the average
of the durations observed within the catchment:
Td=1Ng∑iNgTd(i).
Finally, the precipitation depth received on any given day by a catchment is
assumed to be the average of the precipitation depths observed by individual
rain gauges. It follows that the aggregated mean rainfall intensity can be
expressed as
αP=λP-11Ng∑iNgλP(i)αP(i).
If no precipitation station is located within the catchment, rainfall
characteristics observed at the rain station closest to the catchment
centroid were considered. Although aggregating rainfall time series
before computing their statistics would better account for spatial
correlation in rainfall, aggregating rainfall statistics instead
allows for non-overlapping observation periods (assuming rainfall is
stationary). This is important in the context of Nepal, where rain gauges are
scarce with sporadic observations. Unfortunately, the low density of rain
gauges within the considered basins prevents a formal treatment of spatial
correlation when aggregating frequencies. However, in a previous study
we observed large spatial correlation ranges on
rainfall occurrence in Nepal (125 km during the monsoon). Under these
conditions the selected method stands out as the most parsimonious approach
to utilize multiple, yet sparse, rainfall observations.
Recession characteristics were estimated using streamflow observations as
described in . We computed wet-season recession
constants (k) by regressing the logarithm of streamflow against time for
each period of consecutively decreasing streamflow during the wet season. The
recession constant was then obtained by taking the median value of the
regression coefficients of recessions lasting more than 4 days. The power-law
exponent of dry-season recessions (b) was obtained by fitting a non-linear
recession curve
Q(t)=(Q01-b-a(1-b)t)11-b
to base flow, which was computed from observed streamflow time series using
the Lyne–Hollick algorithm . The last streamflow
peak of the wet season was taken as initial flow condition Q0, and we used
a stochastic optimization algorithm (simulated annealing,
) to minimize least square fitting errors. In
ungauged catchments, the scale exponent of the seasonal recession was
approximated as
a≈λ-re-rm-1αQ⋅(m+1),
where r=1-b; m is the ratio between the frequency λ of effective
rain events and the linear recession constant k, and αQ is the
average depth of effective rain events (see Appendix ).
Potential evapotranspiration was approximated by applying the empirical
relation estimated by for Nepal during the rainy
season (July–September):
ET≈4.0-0.0008⋅zmean,
where ET is given in mm d-1 and zmean is the average
elevation of the catchment in meters. The formula provides daily average
evapotranspiration estimates for each month. It accounts for elevation but
assumes a spatially homogenous elevation gradient. A uniform soil moisture
capacity of 50 mm was assumed throughout the country, based on empirical
observations reported in . By neglecting local
variation in soil characteristics, this produces conservative estimates of
the performance of the process-based model in ungauged basins.
Comparative analyses
Predictions in ungauged basins
We used three cross-validation techniques to evaluate the predictive ability
of both methods in ungauged basins. Firstly, a leave-one-out analysis was
carried out to assess predictive performances in a realistic situation, where
FDCs are predicted in Nepal using all streamflow gauges available in the
region. Secondly, we examined the sensitivity of the methods to decreasing
data availability by reducing the number of gauges available to calibrate the
models. Finally, we performed a similar data-degradation procedure, but in
this case we reduced the number of daily streamflow observations, while
holding the number of gauges constant. This final analysis accounts for the
challenges posed by recent or temporary installation of stream gauges, which
introduce uncertainties into the estimation of model parameters due to the
short streamflow records used. These errors can propagate through the model
and affect the prediction of FDCs.
Numerical simulation analysis to assess predictions under change.
Future rainfall characteristics (frequency λP, mean intensity
αP, auto-correlation coefficient AR and coefficient of
variation CV) are determined according to expected changes in rain regimes in
Nepal (see Sect. ) and fed into a stochastic rainfall generator.
The resulting 1000 years of synthetic daily rainfall values
(PSynth(t)) are fed into a rainfall–runoff model that simulates
the processes described in Sect. . The rainfall–runoff model uses
current recession, soil, and evapotranspiration conditions observed at the
Chepe Kohla catchment. The resulting 1000 years of synthetic daily flow
values (QSynth(t)) are then reordered to construct an empirical
synthetic (future) FDC, which was compared (in terms of the Nash–Sutcliffe
coefficient) to modeled FDCs predicted by the statistical and process-based
models. The process-based model admits current recession conditions
but future estimates for rainfall frequency (λP) and
mean intensity (αP). Note that unlike the numerically
generated empirical FDC, the process-based model assumes Poissonian rainfall
with exponentially distributed depths, that is, CV =1 and AR =0.
Current low-flow characteristics (q95) are fed into the statistical
model, as well as the current or future (i.e., computed from synthetic
streamflow time series) mean flow, depending on the extent to which mean
rainfall is an unbiased predictor of mean flow (Cases 1 and 2 described in
Sect. ).
In a leave-one-out analysis, one gauge is “left out” of the data set, and
streamflow is predicted at the “missing” location using observations from
the remaining gauges. The predicted FDC is then compared to observations from
the omitted gauge. The resulting error between observation and prediction
yields the prediction performance of the method at that catchment if it was
not gauged. Repeating the procedure for all gauges offers an approximation to
the overall prediction error of the method. To measure this error, we
constructed error duration curves , where the
relative prediction error at each flow quantile is plotted against the
corresponding duration. Error duration curves allow the partitioning of
prediction errors across flow quantiles to be visualized. General prediction
performances (across all durations) at individual gauges were also determined
using the Nash–Sutcliffe coefficient (NSC) on log streamflow quantiles
:
NSC=1∑t=1364lnQt(emp)-lnQt(mod)2∑t=1364lnQt(emp)-ElnQt(emp)2,
where Qt(emp) and Qt(mod) are the empirical and
modeled streamflow quantiles of duration t.
The effect of the number of calibration gauges was assessed using a jackknife
cross-validation analysis . At each
of 10 000 iterations, a selected fraction of the available gauges was
randomly sampled (without replacement) and used to predict the FDC at one
(randomly selected) remaining gauge. Prediction accuracies for flow duration
curves (given by the NSC) and uncertainties on the spatial interpolation of
model parameters were reported for each iteration. The procedure was repeated
for decreasing numbers of selected “training” gauges.
The available streamflow data did not allow a direct evaluation of the
effects of time-series length through cross-validation, because such an
analysis requires substantial overlaps in the monitoring periods of all
gauges. Therefore we focused the final analysis on the Chepe Kohla catchment,
which has the longest observation record in our data set. We evaluated the
effect of the length of the available observation records on parameter
estimation, and propagated the ensuing uncertainty in the parameters to the
FDCs predicted by each model. To do this, we selected a fixed number of full
years of streamflow observations, estimated the parameters, predicted the FDC
using these parameters, and compared the results to the empirical FDC
obtained from the full observation record. The procedure was repeated 10 000
times. The estimation errors in the model parameters and the resulting FDC
prediction performances (NSC) were recorded as a function of the number of
sampled years. This analysis is not intended to describe the models' ability
to predict FDCs at catchments with short observation records: in this case,
constructing an empirical FDC using the available (however short) observation
record is likely to be the best course of action
. Instead, the analysis is intended to
simulate the effect of short observation records on FDC prediction at nearby,
ungauged catchments. The underlying assumptions behind this analysis
are that (i) the error associated with interpolation is independent of the
flow record length, and (ii) the Chepe Kohla catchment is representative of
Nepalese basins.
Sensitivity of models to changes in the precipitation regime.
(a) Empirical and simulated flow duration curves at Chepe Kohla. The
simulated FDC obtained from the stochastic rainfall generator and the bucket
watershed model (solid) reproduce the empirical FDC constructed from the
observed streamflow well (grey dots). Rainfall changes expected in Nepal
(αP/αP,0=1.2,
λP/λP,0=0.98) do not have a substantial
influence on the simulated flow distribution (dashed). αP and
λP designate the mean depth and frequency of wet-season
rainfall, respectively. (b) Sensitivities to relative changes in
rainfall frequency and intensity over the Chepe Kohla catchment. The
performance of the process-based model is not affected by rainfall changes
(dotted). The sensitivity of the statistical model depends on its ability to
predict changes in mean flow from annual rainfall. The model is highly
sensitive to rain changes if average streamflow cannot be predicted (dashed),
and is robust to moderate changes if average flow is perfectly predicted
(solid). (c) The linear regression of the statistical model
underestimates annual flows at the Chepe Kohla when using a cross-sectional
sample (25 gauges) to estimate the local relation between average rainfall
and average runoff.
Predictions under change
We used numerical simulations to assess the ability of both models to predict
streamflow when subject to changing rainfall regimes, as described in Fig. .
Synthetic streamflow time series were generated by coupling the stochastic
rainfall generator described in to a rainfall–runoff
model. The generated wet-season rainfall is a first-order Markov process
(i.e., rainfall occurrence on a given day is correlated with rainfall
occurrence on the previous day) with gamma-distributed rainfall intensities,
and as such produces a rainfall record that explicitly violates the
assumptions under-pinning the process-based model. The duration of the rainy
season was assumed constant, and no rainfall was generated during the dry
season. Wet-season streamflow was simulated by feeding synthetic rainfall
into a linear reservoir (with a recession constant k) with linear
evapotranspiration losses, as in . Dry-season
discharge was obtained by simulating non-linear seasonal recessions of
duration Td starting at randomly selected runoff peaks in the
(previously generated) wet-season streamflow. These assumptions are close to
the observed reality in Nepal, as seen in Fig. a, where the FDC
constructed from the simulated streamflow is a close approximation to the
empirical FDC in the Chepe Kohla watershed. We translated the effect of
shifts in precipitation regimes into changed streamflow for the Chepe Kohla
catchment by considering a range of future combinations for rainfall
frequencies and intensities. In line with what is expected in Nepal
, we considered negative
changes in the frequency and positive changes in the mean daily rainfall
depth. We neglected changes in soil moisture capacity, evapotranspiration,
rainfall autocorrelation, and the duration of the
rainy season. These parameters are explicit in the process-based model, so we
expect differences in the sensitivity of the process-based and statistical
models to climate change to be underestimated by this procedure. For each
rainfall scenario, we evaluated the performance of the models in a changing
climate by generating 1000 years of daily streamflow using future rainfall
frequencies and intensities.
Flow duration curve prediction performance in ungauged basins. The
error duration curves of the leave-one-out cross-validation analysis using
the process-based and statistical models are presented in panels (a)
and (b), respectively. Relative errors are plotted on a log scale in
order to allow the graphs to be balanced on the y axis: a relative
prediction error of 2 (the model predicts double the observed value) is at
the same distance from y=1 (perfect prediction) as a relative error of
1/2 (the model predicts half the observed value). Durations are plotted on
the x axis, with x=0 and x=1 for the highest and lowest flow quantiles,
respectively. Panel (c) shows box plots of Nash–Sutcliffe
coefficients computed from log-transformed flow quantiles.
We compared the synthetic FDCs to model predictions that were made with
future rainfall statistics but contemporary recession and
low-flow parameters (Fig. ). The statistical method in
uses a linear regression over a cross-sectional
sample of observations to predict mean flow based on mean rainfall and
altitude. The regression may fail to capture a variety of unobserved
characteristics affecting both rainfall and streamflow (e.g., local
topographic features), and hence may not capture the causal relation between
the two variables. The extent of this bias cannot be quantified a priori, so
we considered two extreme cases: infinite and zero bias. The infinite bias
case (Case 1 in Fig. ) represents the case where no effective
relationship can be determined between rainfall and mean flow. The best
estimator of future mean flow is then the current flow condition.
Conversely, if regression coefficients perfectly describe the effect of
annual rainfall on average flow (Case 2 in Fig. ), then the
future flow conditions can be perfectly estimated using the (known) future
annual rainfall. We modeled this situation by estimating Qm
directly from the (simulated) future flow conditions. While the two
cases differed in the determination of mean flow (Qm), the low-flow
parameter (q95) was determined from current flow conditions in both
cases. In , q95 is normalized by
Qm and represents recession behavior, which is assumed independent
of rainfall. The process-based predictions were obtained by inserting future
rainfall statistics and contemporary recession constants into the analytical
FDC equation described in Appendix A. The two models were compared by
plotting prediction performances (NSC) against the relative change in the
frequency and intensity of synthetic rainfall.
Although the recession assumptions of the process-based model are taken to
generate the synthetic streamflow used as a control, we believe that the
analysis is not biased against the statistical approach for three reasons.
Firstly, the only parameter of the statistical approach that is influenced by
rainfall (Qm) is also computed from synthetic streamflow (Case 2 in
Fig. ). Secondly, although based on identical recession
assumptions, the process-based model and the synthetic streamflow generator
are driven by different stochastic rainfall processes (i.e., Poisson and
Markov, respectively). Lastly and most importantly, empirical observations
reveal that synthetic streamflow distributions generated under
contemporaneous rainfall conditions reproduce closely FDCs constructed from
gauge records (Fig. a), showing that the underlying recession
assumptions are, in fact, representative of runoff processes actually
occurring in Nepal.
Results
Prediction in ungauged basins
Results from the leave-one-out cross-validation analysis are presented in
Fig. and show that both methods perform similarly in the
prediction of FDCs in ungauged basins. Error duration curves
(Fig. a and b) show comparable streamflow prediction
uncertainties: 75 % of the predicted flow quantiles are between half and
double the observed streamflow for both models, although the low flows in the
process-based model display an increasing upwards bias (Fig. b).
Considering the Nash–Sutcliffe coefficients computed at the individual basin
level, the mean and median performances are again comparable for both models,
but the accuracy of the statistical model predictions is more variable across
sites than the process model predictions, as indicated by the larger spread
of the Nash–Sutcliffe coefficients (Fig. c).
Sensitivity of models to data scarcity.
(a) Cross-validation analysis showing the sensitivity of both models
to a decreasing number of calibration gauges. (b) Resampling
analysis of streamflow observations in the Chepe Kohla (N=10 000)
catchment showing the effect of the number of observation years. In
panels (a) and (b), the effects on FDC prediction
performances (top) are shown by plotting the ratio of calibration gauges
sampled (or the number of observation years) against the relative Nash–Sutcliffe coefficient (with the NSC for the full set of available data as
reference). The plot shows the median value for all iterations, and the error
bars indicate the interquartile (25–75 %) range. The prediction
uncertainties of model parameters (bottom) are given in absolute values of
relative prediction errors.
Figure a (top) shows prediction performances of both models as the
number of streamflow gauges available for predictions decreases, and
indicates that the performance of both models is relatively insensitive to
the gauge density, until it declines to less than approximately 0.6 gauges
per 10 000 km2. For such situations, which represent discarding of more
than half the available gauges in Nepal, the statistical model performance
declines rapidly compared to the process-based model. Prediction performances
are strongly affected by uncertainties on the interpolation of model
parameters, as seen in Fig. a (bottom). Interpolation uncertainties
are generally larger for the flow indices of the statistical model
(Qm and q95) than for the recession parameters of the
process-based model (k and b). This explains the larger spread in
prediction performances of the former (Fig. c and error bars in
Fig. a (top)). The parameter uncertainties are also relatively
insensitive to the total gauge density until about 60 % of the originally
available gauges are discarded. At this point, the uncertainties associated
with estimation of the flow indices increase significantly, while the
process-based model parameters remain more reasonably estimated.
When considering short observation windows, parameter uncertainties also
drive the performance of the models. Figure b (top) shows the
prediction performance of both models at the Chepe Khola watershed, as the
number of observation years used to estimate the model parameters is reduced.
In this case, the statistical model outperforms the process-based model when
less than 10 years of streamflow observations are available. The parameter
uncertainties associated with the short time-series estimates
(Fig. b, bottom) suggest that a longer time series of streamflow
observations is needed to accurately estimate the wet-season recession
parameter (k), resulting in the lower performance of the process-based
model for short streamflow records.
Prediction under change
Simulation results presented in Fig. b show both models' ability
to predict a simulated future flow duration curve of the Chepe River under a range of different possible
changes in rainfall regimes. In all simulations, parameters describing the
hydrological response of the basin (k, b, and q95) are determined
using current flow conditions, and evapotranspiration is assumed constant.
The results show that explicitly modeling rainfall–runoff processes allows
the process-based model to accommodate the effects of the changing
precipitation regime. In contrast, the performance of the statistical model
is affected to various degrees by shifts in rainfall
regimes, depending on how the model translates changes in annual
precipitation to changes in average flows. If these shifts are perfectly
represented by the model, then prediction errors arise solely from changes in
the shape of the FDC, and the process and statistical models perform
similarly in the Chepe Kohla watershed across the full range of considered
rainfall scenarios (Fig. b, dashed curve). If, however, average
(future) streamflows cannot be reliably predicted from the predicted changes
in annual rainfall, the statistical model does not accommodate flow regime
changes at all. In this case, future FDCs are modeled using current
streamflow observations, and the ensuing prediction errors can be substantial
(Fig. b, dotted curve). The simulated cases provide upper and
lower bounds for the actual performance of the statistical model in future
rainfall regimes. We evaluated the model's ability to predict Qm by
using cross-sectional data (i.e., average streamflow and annual rainfall from
the 25 catchments) to estimate the linear relation between Qm and
annual rainfall Ry. Applied to the Chepe Kohla watershed, the
estimated regression coefficients allowed the annual streamflow to be
estimated from annual precipitation with a bias of -13 % and a
coefficient of determination of R2=0.57 (Fig. c). Regardless,
prediction errors remained negligible for both bounds (NSC > 0.95) for
the range of changes actually anticipated in Nepal (e.g.,
ΔλP/λP≈0.98 and
ΔαP/αP≈1.20 for the 2⋅CO2 scenario – ).
Discussion
Predictions in ungauged basins
The analysis suggests that both statistical and process-based methods to
estimate FDCs in ungauged basins perform comparably in Nepal, over a wide
range of gauge densities and observation durations. Yet prediction
performances varied significantly between the models as data became
increasingly sparse. The statistical method is more sensitive to spatially
sparse data, which degrades the interpolation accuracy of Qm. In
contrast, the estimation method for recession parameters makes the
process-based approach more sensitive to temporally restricted observations,
which reduce the accuracy with which recession parameters can be estimated.
This suggests that the performance of the two models in ungauged basins is
affected by different sources of uncertainty. In this section, we investigate
the source of prediction error in each method and discuss the implications
for their application in ungauged basins beyond Nepal.
Sources of uncertainty
The statistical model relies on two assumptions about the correlations of
observed data. The first assumption is that catchments with similar low-flow
indices (q95) have identical hydrological responses, and therefore
identical FDC shapes. Second, the model assumes that the flow indices
(Qm and q95) at ungauged catchments can be best predicted
using linear regressions against observable covariates (annual rainfall,
elevation, and geology). The latter assumption does not hold if the flow
indices are spatially auto-correlated, or if the posited linear relations are
spatially heterogeneous or, in fact, non-linear. Furthermore, “omitted
variable” biases will arise if an unobserved
variable is correlated with both a covariate and a flow index. For instance,
local topographic features may affect both the annual rainfall and the
average streamflow in mountainous regions. Violation of the second assumption
leads to substantial uncertainty in the interpolation of the flow indices in
Nepal and drives the prediction errors of the statistical approach, as shown
in Sect. S2 of the Supplement.
While the performance of the process-based model is also driven by parameter
estimation uncertainties, these errors arise from simplifying assumptions
about local hydrological processes (rather than uncertainties from their
statistical interpolation from neighboring gauges). Additional
cross-validation analyses (shown in Sect. S2 of the Supplement) suggest that
uncertainties caused by the aggregation of observed point-rainfall statistics
at the catchment level drive prediction errors of high-flow quantiles. While
increasingly accurate remote sensing rainfall data will progressively allow
such spatial heterogeneities to be resolved, current precipitation products
(e.g., TRMM 3B42) remain substantially biased in mountainous regions like
Nepal, where they do not outperform available rain gauges in predicting the
frequency and intensity of areal rainfall . A second
source of error arises from the simplifying assumptions made about streamflow
recession that do not hold perfectly in the observed catchments. Because they
describe the same watershed, the wet and dry recession parameters are assumed
to be physically related. In , the scale
parameter of the non-linear seasonal recession (a) is expressed as an
explicit function of the two recession parameters (k and b) for
sufficiently short recession times, where power-law recessions can be
approximated by exponential functions. We show in the Supplement (Sect. S2)
that, although this approach provides more accurate estimates of a than
would be obtained through spatial interpolation, estimation uncertainties
remain, propagate through the model, and result in prediction errors during
the dry season.
Applicability beyond Nepal
This study compares two specific methods on their ability to predict FDCs in
the particular context of ungauged Nepalese basins. Results are thus not
necessarily representative of the relative performance of process-based and
statistical methods in general, particularly in regions where abundant field
data allow more advanced statistical approaches to be implemented. Yet
fundamentally, the statistical model relies on observed correlations rather
than assumptions about hydrologic mechanisms. Because FDC shapes are modeled
non-parametrically, the approach is applicable to regions with highly
variable catchment responses. However, prediction performance in ungauged
basins is constrained by interpolation errors in the mean flow. This makes
the method unsuitable for regions where the local determinants of mean flow
(i.e., rainfall, evapotranspiration, glacial melt) cannot be accurately
monitored at the catchment level. In contrast, a key advantage of the
process-based model is its ability to exploit characteristics of the
stochastic structure of rainfall that can be estimated from daily rainfall
observations. The model is appropriate for regions where the spatial
heterogeneity of runoff is driven by rainfall, and where the frequency and
intensity of rainfall depths at the catchment level can be readily estimated
(i.e., small catchments with numerous rain gauges, or places where satellite
observations provide a good representation of rainfall statistics). Unlike
rainfall, recession behavior arises from lumped and complex interactions
between climate, vegetation, and groundwater processes that typically cannot
be monitored in a spatially explicit manner. The process-based model is
therefore inappropriate for regions where the hydrologic response of the
catchment is the main source of runoff heterogeneity, or where the assumed
recession behavior (in particular the relation between a, k, and b)
does not occur.
Conveniently, the appropriate implementation contexts for both methods appear
to be complementary, and the optimal method in a given region is determined
by the driving source of runoff heterogeneity in the catchments. Ultimately,
the performance of both methods is constrained by their ability to estimate
their parameters in ungauged basins. This relation is apparent in Fig. ,
where drops in prediction performances correspond to increases in
the estimation uncertainty of model parameters. Under these conditions, the
performance of each method is driven by the ability of the available
observations to capture the variability of the model parameters. When
interpolated from neighboring gauges, uncertainties are governed by the
interplay between the layout of the gauges and the spatial correlation range
of the considered model parameter. When estimated from short observation
records, accuracy is determined by the extent to which the available record
is representative of the temporal variability of the parameter. These
interactions between data availability and runoff variability are inherently
local and will affect the determination of the most appropriate method for
any given region.
Prediction under change
Expected shifts in the frequency and intensity of monsoon rainfall over Nepal
only have a marginal impact on the streamflow distributions in the Chepe
Kohla catchment, as shown by the numerical simulation presented in
Fig. a (dashed curve). Consequently, changes in rainfall regime
do not appear to affect the performance of either model (Fig. b),
unless they are significantly larger than expected. Climate change may
nonetheless affect flow predictions elsewhere. It is therefore helpful to
consider the conditions under which FDCs can be reliably predicted in a
changing climate.
Although rainfall stationarity is an inherent assumption of the process-based
approach, climate change can be incorporated by updating the relevant
parameters to their future value to predict the (pseudo-)stationary future
state of the system. The method accounts for otherwise confounding changes in
the frequency and intensity of rainfall, which are expected in Nepal. By
explicitly accounting for soil moisture dynamics and recession behavior, the
model emulates the (causal) effect of rainfall on streamflow. As a result,
the method reliably predicts the distribution of future streamflow, provided
that governing flow generation processes are in line with the basic
assumptions listed in Sect. .
In contrast, the statistical model is solely based on observed correlations,
leading to two important sources of errors for predictions under change.
First, the model only accommodates rainfall changes to the extent that the
estimated statistical relation between rainfall and runoff is representative
of local runoff coefficients. The model will not reliably predict future
streamflows if runoff coefficients are strongly spatially heterogeneous, or
if the cross-sectional sample of gauges fails to capture important processes
governing mean flow. This source of uncertainty appears to be significant in
Nepal, as illustrated by the substantial bias in annual flow predictions in
Fig. c. Secondly, the statistical model only considers the effect
of average rainfall on average flow: the effect of rainfall
distribution on streamflow
distribution is ignored. As a result, the model cannot predict
changes in the shape of FDCs that are brought about by changing rainfall. The
prediction performance of the statistical approach is therefore determined by
the resilience of the flow regime, that is, the extent to which streamflow
distribution is affected by shifting rain signals
: the method will perform poorly in catchments
with non-resilient flow regimes. The Monte Carlo analysis presented in the
Supplement (Sects. S3 and S4) shows that streamflow resilience in seasonally
dry catchments depends on two distinct seasonal effects: a “direct” effect
driven by the ratio between λP and k during the wet season,
and an “indirect” effect during the dry season, when resilience is
determined by the interplay between Q0 (i.e., wet-season rainfall) and
b. In seasonally dry climates, we expect the statistical method to be most
reliable in regions where wet seasons are short with limited total rainfall
but persistent flow regimes, and where the recession behavior during the dry
season is close to linear.
Lastly, a key assumption in this study is that catchment response (in terms
of low-flow or recession characteristics) is independent of climate. It is
possible that shifts in climate have an effect on catchment response by
affecting the partitioning of effective rainfall between storage and runoff.
Although not quantitatively assessed in this study, we expect that this
effect would negatively affect the performance of both approaches.
Conclusions
Stochastic, process-based models predicted the FDCs for ungauged catchments
in Nepal well, with a performance that was comparable to that of statistical
models. It suggests that in regions with globally representative gauge
densities, and under seasonally dry climates, the advantages of the
statistical approaches relative to stochastic models noted in previous
analyses may not apply. Fundamentally, the
performances of both approaches are strongly affected by the method chosen to
estimate model parameters in ungauged basins, so this conclusion comes with
the caveat that this study cannot be interpreted as a general benchmark to
compare these approaches at a global level. Although we believe that the
selected models are appropriate to compare process-based and statistical
approaches for practical PUB application in Nepal, their relative performance
may be different in other regions, where more abundant information on
catchment characteristics allow more complex (and presumably more accurate)
regionalization approaches to be applied. Thus, substantial research remains
to be done to compare these approaches in other parts of the world, where
locally appropriate methods should be carefully considered.
Nonetheless, this study finds a complementarity between the different sources
of uncertainty in the stochastic and statistical methods. This suggests that
model selection should be driven by a consideration of the main drivers of
heterogeneity in any study catchment: process-based models are advisable if
climate is likely to be the main source of runoff heterogeneity. Conversely,
statistical methods are more appropriate for regions with substantially
different recession behaviors across catchments. These distinctions provide a
potentially robust basis for model selection in any given application.
The results also suggest that the sensitivity of statistical approaches to
changes in rainfall statistics is dependent on the “resilience” of the flow
regime as defined by . Overall, the process-based
models are more reliable in projecting FDCs into new rainfall regimes. This
is particularly true for catchments characterized by a strong wet-season
runoff and a rapid, strongly non-linear hydrologic response, because their
flow regime is particularly vulnerable to rainfall changes, making the
assumptions of the statistical model inappropriate.
The excellent performance of both process-based and statistical models for
the FDC and PDF in ungauged basins suggests that extending probabilistic
analyses in such basins to also include flow-derived variables such as
hydropower capacity or ecological responses
may be feasible. While these prospects are
enticing, we note that a model's ability to predict an FDC with high fidelity
is not necessarily indicative of prediction performances on all derived
stochastic properties. For instance, demonstrate that the
crossing properties of streamflow can be very poorly estimated by stochastic
process-based models, even in applications where the same models predict the
PDF of flow well. Further exploration of the potential opportunities and
limitations afforded by use of probabilistic models in ungauged basins offers
a promising avenue for future study.