Introduction
Daily streamflows are often represented by flow duration curves (FDCs), which
illustrate the frequency with which flows are equaled or exceeded. FDCs have
important applications, including water allocation, wastewater management,
hydropower assessments, sediment transport, protection of ecosystem health,
and the generation of time series of daily streamflows (Archfield and Vogel,
2010; Castellarin et al., 2013; Smatkin, 2001; Vogel and Fennessey, 1995).
Broad regions of the world have insufficient records of streamflow and,
despite a decade of work focused on such ungaged and partially gaged basins,
accurate prediction of streamflow in these locations remains a challenge
(Sivapalan et al., 2003; Hrachowitz et al., 2013). Identification of a
probability distribution of daily streamflows would be instrumental to the
prediction of flows in ungaged basins. The goal of this study is to assess
whether a single probability distribution can adequately approximate the
distribution of daily streamflows, as represented by a period-of-record FDC
(FDCPOR), which reflects the long-term or steady-state hydrologic
regime at a site. This assessment is performed at the sub-continental scale
to enable consideration of a broad range of hydrologic conditions that may be
experienced in practice.
Methods to predict the FDCPOR in ungaged basins generally fall
into one of two categories: process-based or statistical. For an extensive
review of these methods, refer to chap. 7 in the book Runoff prediction in ungaged basins (Castellarin et al., 2013). Process-based
models are an increasingly popular method of estimating FDCs at ungaged
basins because they offer the ability to relate physical watershed
characteristics to streamflow regimes. While promising for regions without
any streamflow data, process-based FDCPOR models require numerous
assumptions regarding runoff and climate mechanisms (Basso et al., 2015;
Botter et al., 2008; Doulatyari et al., 2005; Müller and Thompson, 2016;
Schaefli et al., 2013; Yokoo and Sivapalan, 2011).
Historically, most studies predicting FDCPOR at ungaged sites have used
statistical methods, such as regression and index-flow methods, due to their
parsimony and relative ease of use in operational hydrology (Castellarin
et al., 2013). Yet, daily streamflow observations exhibit a very high degree
of serial correlation, seasonality, and other complexities and are thus
neither independent nor identically distributed. Klemeš (2000) warned
that ignoring these complexities can be problematic, particularly if the
FDCPOR is used to extrapolate upper tails of the distribution.
Furthermore, the fact that daily streamflows often range over many orders of
magnitude presents a considerable challenge to the identification of an
appropriate distribution. Although multiple parameters are needed to
describe the complex distribution of daily streamflows, it is also important
that the model be parsimonious, because each additional parameter can hinder
estimation, parameter identifiability, and interpretation (Castellarin et
al., 2007).
Despite these challenges, there is a relatively large literature which has
sought to approximate the distribution of daily streamflow with a single
probability distribution for practical purposes. The main motivations have
been estimation of FDCs at ungaged sites (Castellarin et al., 2004, 2007;
Fennessey and Vogel, 1990; Li et al., 2010; Mendicino and Senatore, 2013;
Rianna et al., 2011; Viola et al., 2011) or estimation of time series of
daily streamflow at ungaged sites (Fennessey, 1994; Smatkin and Masse, 2000;
Archfield and Vogel, 2010). To estimate FDCs at ungaged sites, regional
regression models of distribution parameters can be used when basin
characteristic data are available at both ungaged sites and gaged sites in
the region. A number of distributions have been proposed to describe daily
streamflow. Li et al. (2010) found that the three-parameter lognormal
distribution (LN) adequately represented FDCPOR for
southeastern Australia. In Italy, both the four-parameter kappa (KAP) and the
generalized Pareto (GPA), a special case of KAP, have been used to describe
FDCPOR in index-flow studies (Castellarin et al., 2004, 2007;
Mendicino and Senatore, 2013). Similarly, both GPA and KAP were found to
provide a good approximation for FDCPOR in the northeastern
United States (US) (Archfield, 2009; Fennessey, 1994; Vogel and Fennessey,
1993). However, Archfield (2009) highlighted challenges in fitting both KAP
and GPA to tails of the FDCPOR, noting that these fitted
distributions often exhibit lower bounds that can result in the generation of
negative flows. Multiple authors have noted that a complex distribution with
at least four parameters is needed to approximate the probability
distribution of daily streamflows (Archfield, 2009; Castellarin et al., 2004;
LeBoutillier and Waylen, 1993).
Given the complexity of daily streamflow, some studies have focused on only a
portion of the FDCPOR, such as flows below the median (Fennessey
and Vogel, 1990) or above the mean (Segura et al., 2013). Others have studied
the distribution of streamflow by season. For eight rivers across the US,
Bowers et al. (2012) developed a method to identify wet and dry season FDCs
and found discharge data in wet seasons to be well approximated by a
lognormal distribution, but dry season flows sometimes better fit with a
power law distribution. The study also illustrated the challenges of
conducting comprehensive seasonal analyses; findings varied across rivers and
depended upon season, suggesting that seasonal analysis of this kind is often
site-specific. A couple of papers have documented attempts to fit a
probability distribution to a mean annual FDC or a median annual FDC
(FDCMED), two types of hypothetical FDCs that express the
likelihood of daily streamflow being exceeded during a typical year
(Fennessey, 1994; LeBoutillier and Wayland, 1993). The FDCMED,
introduced by Vogel and Fennessey (1994), has a number of applications, from
ecology to hydropower (Lang et al., 2004; Müller et al., 2014; Kroll et
al., 2015). FDCMED are increasingly common and enable the
computation of tolerance or uncertainty intervals along with associated
hypothesis tests for flow alteration (see Kroll et al., 2015).
To address the practical goal of estimating FDCs, this study aims to
determine whether or not an existing probability distribution is capable of
approximating the distribution of daily streamflow for nearly 400 perennial
rivers with near-natural streamflow conditions across the conterminous US.
Differences in the performance of hypothesized probability distributions in
approximating FDCPOR are compared across physiographic regions of the
US to illustrate where these methods might be most successful. In addition,
this study also considers the ability of a single probability distribution
to represent the FDCMED.
The paper is organized as follows. First, the method to construct an
FDCPOR is described and the goodness-of-fit (GOF) metrics and study
region are introduced. The results are then presented, including L-moment
ratio diagrams and quantitative GOF comparisons among the fitted probability
distributions. These GOF results are then compared by physiographic region
within the US and the FDCMED results are shown. Finally, the conclusion
summarizes study findings and provides directions for future research.
Methods
FDC estimation
An empirical FDCPOR is constructed by ranking daily streamflows from
all recorded years and plotting them against an estimate of their exceedance
probability, known as a plotting position (Vogel and Fennessey,
1994). An FDC is defined as the complement of the cumulative distribution
function:
1-FQq,whereFQ(q)=PQ≤q,
where q represents observed streamflow and
FQ(q) is the empirical cumulative
distribution function of observed streamflow. The first step in constructing
an FDCPOR is to rank the flows, q, in ascending order. For leap years,
flows from 29 February were removed to maintain consistent sample sizes
across years. To obtain the probability with which each flow is exceeded,
the Weibull plotting position was used, as it provides an unbiased estimate
of exceedance probability, regardless of the underlying probability
distribution of the ranked observations (Vogel and Fennessey, 1994):
PQ>q=1-i365n+1,
where i represents the rank and n represents the number of years of record.
Vogel and Fennessey (1994) review several alternative nonparametric plotting
positions for constructing empirical FDCs at a gaged site, some of which are
preferred for smaller samples. The Weibull plotting position is selected
here given the large sample sizes considered (at least 40 years of daily
data leading to sample sizes greater than 40×365=14600).
Map of the conterminous United States showing physiographic regions
and the streamgages included in the study. Boxplots on the lower left show
the range of drainage areas and record lengths represented by study
streamgages.
Selection of candidate distributions
As an initial assessment, L-moment ratio diagrams were used to narrow the
pool of potential candidate probability distributions. L-moments are linear
combinations of probability-weighted moments (Hosking and Wallis, 1997).
Estimates of L-moment ratios exhibit substantially less bias than moment
ratio estimators and are resistant to the influence of data outliers
(Hosking and Wallis, 1997). The advantages of using L-moment diagrams in
distribution identification are described in Vogel and Fennessey (1993) and
Hosking and Wallis (1997). L-moments can be directly related to ordinary
product moments of a probability distribution.
Theoretical relationships between L-moment ratios have been determined for a
wide class of probability distributions (Hosking and Wallis, 1997). These
relations can be plotted on an L-moment ratio diagram with L-moment ratios
estimated from the daily streamflows to provide a visual method of comparing
various probability distributions to observed data. Vogel and Fennessey (1993) demonstrate that L-moment ratio diagrams are often superior to
ordinary moment ratio diagrams, especially for extremely long records of
highly skewed samples of daily streamflow, as is the focus of this study.
Even when parent distributions are complex, L-moment ratio diagrams are
useful in identifying simpler distributions that fit the observed data
sufficiently well (Stedinger et al., 1993). For a description of the
theory of L-moments, see Hosking (1990).
Goodness-of-fit evaluation
To evaluate the suitability of a model to reproduce observations, a measure
of the standardized mean square error commonly referred to as Nash–Sutcliffe
efficiency (NSE) is used. The estimator of NSE for a streamgage site is
NSE=1-∑x=1X(Qx-Qxpred)2∑x=1X(Qx-Qx‾)2,
where Qx represents observed flow at quantile x,
Qxpred represents predicted flow at quantile x,
Qx‾ represents the mean value of the observed flows, and X
represents the total number of daily flows. NSE values range from -∞
to a maximum of 1, which here would indicate that the estimated flow
quantiles matched observed flow quantiles exactly. Because NSE is heavily
influenced by the highest flows, NSE is computed based on the natural
logarithms of the flows and is referred to as LNSE.
Theoretical L-skew and L-kurtosis ratios of three- and
four-parameter distributions compared to empirical L-skew and L-kurtosis
ratios from (a) daily streamflows at 420 US sites, (b)
flows simulated from three-parameter generalized Pareto, (c) flows
simulated from three-parameter lognormal, and (d) flows simulated
from four-parameter kappa distributions.
Part of the reason why FDCPOR are so widely used in practice is that
they provide a graphical illustration of the complete relationship between
the magnitude and frequency of streamflow. Examples of poor, good, and very
good fits by candidate distributions to FDCPOR are presented to
illustrate LNSE values visually. Lastly, error duration curves are given for
each candidate distribution to illustrate how error is distributed across
exceedance probabilities. Error is measured by calculating the ratio of
predicted quantiles of flow to observed ranked flows for each site.
Results
Graphical identification of candidate distributions
To identify candidate probability distributions, theoretical L-moment
ratios are compared to sample L-moment ratios in Fig. 2a. Four-parameter
KAP is represented by the shaded area below the generalized logistic curve
and above the theoretical L-moment ratio limits. The lower bound of the
five-parameter Wakeby (WAK) distribution is also plotted as a curve. Sample
estimates of L-moment ratios computed from empirical FDCPOR at
study sites are shown as points. Empirical L-moment ratios mostly fall
below the generalized logistic and generalized extreme value curves and above
the Pearson type III and WAK lower bound curves (Fig. 2a). The points are
clustered around the three-parameter GPA and LN curves; thus, these two
distributions are identified as possible parent distributions. The empirical
L-moment ratios are also consistent with both KAP and WAK distributions.
(a) Boxplots showing the range of streamgage
Nash–Sutcliffe efficiencies for natural logarithms of daily streamflows
(LNSE) based on hypothesized generalized Pareto (GPA), lognormal (LN), and
kappa (KAP) distributions with points omitted from the plots listed in
brackets, and example streamgage sites with (b) very good fits (LNSE
above 0.99), (c) good fits (LNSE between 0.93 and 0.99), and
(d) poor fits (LNSEs below 0.93).
The scatter of points around the GPA or LN distribution curves could, in
theory, be due to sampling variability. However, given a sufficiently long
record, empirical L-moment ratios would be expected to fall directly on the
theoretical curves if the probability distribution of daily streamflow truly
arose from one of these distributions. The very large sample sizes here
suggest this is unlikely; nevertheless, synthetic daily streamflows were
generated to test this hypothesis. The method of L-moments (Hosking and
Wallis, 1997) was used to estimate distribution parameters from the ranked
observed daily streamflows (the empirical FDCPOR) for each study
gage. Distribution parameters were found to be inconsistent with KAP at 35
sites (9 %) and with WAK at 244 sites (61 %). Because WAK could not
be fit at over half of the study gages, a finding encountered previously for
New England (Archfield, 2009), WAK was removed from further consideration.
Based on distribution parameters for GPA, LN, and KAP, data of the same
record length as the daily streamflow observations at a given site were
simulated and L-moment ratios computed. These synthetic L-moment ratios
are plotted in Fig. 2b–d. As expected given the very large samples, the
synthetic L-moment ratios for GPA and LN fall on the empirical curves
representing these distributions. Thus, the scatter in L-moment ratios does
not appear to be due to sampling variability, but rather reflects the
complexity of the true distribution(s) from which daily streamflows arise.
Compared to GPA and LN, simulated L-moment ratios from KAP (Fig. 2d) appear
more consistent with the L-moment ratios estimated from empirical FDCs
(Fig. 2a). Thus, KAP appears to provide the best fit among the probability
distributions considered. Because there are benefits to having fewer
parameters in practice and because some gages do have L-moment ratios
consistent with theoretical GPA and LN L-moment ratios, GPA and LN
hypotheses are retained for future analyses.
National goodness-of-fit comparisons
In this section, additional measures of the GOF of the GPA, LN, and KAP
models for approximation of FDCPOR are considered. One
complication involves the generation of negative streamflows, which can occur
when the fitted lower bound of a distribution is less than zero. Negative
streamflows were predicted at 98 sites for GPA, 159 sites for LN, and 40
sites for KAP. Other studies have also encountered problems with the
generation of negative streamflow (Archfield, 2009; Castellarin et al.,
2007). To prevent these infeasible negative flow predictions, distributions
were constrained to ensure a theoretical lower bound of zero at study sites
for which negative flows would otherwise be generated. Both the GPA and LN
distributions include parameters representing theoretical lower bounds
(Hosking and Wallis, 1997). Constraining both of these lower bound parameters
to zero was relatively simple as it is equivalent to fitting two-parameter
versions of three-parameter GPA and LN distributions. For KAP, the lower
bound is a function of all four parameters, so enforcing a theoretical lower
bound requires solving for the four parameters simultaneously while
constraining the lower bound. The same approach as that used by Castellarin et al. (2007)
in constraining the KAP lower bound to zero was followed here. Following this
procedure, KAP parameters were infeasible based upon site L-moment ratios
at 42 sites (11 %).
Figure 3a gives boxplots showing the range of values of LNSE across sites
corresponding to the GPA, LN, and KAP hypotheses. To ensure fair comparison
across the three distributions, only LNSE values for sites for which KAP
could be estimated (356 sites) are shown, though the figure appears nearly
identical when the additional 42 sites are included for GPA and LN. KAP shows
the highest GOF, which is not surprising given that the distribution includes
an additional parameter compared to GPA and LN. Both GPA and LN also have
quite high values of LNSE (note that the y-axis ranges from 0.8 to 1). To
illustrate how these LNSE values translate into GOF, example
FDCPOR are given for three sites with varying GOF (Fig. 3b–d).
It is important to note that there was substantial variability in how
FDCPOR appear across similar LNSE values, and these are only
three examples. First, in Fig. 3b, an empirical and fitted FDCPOR
with LNSE values above 0.99 for all three distributions is given. For this
example site located in Pennsylvania, nearly the entire FDCPOR is
captured except for the very lowest flows. A site with “good” fits, all
with LNSE values between 0.93 and 0.99, is shown in Fig. 3c. For this site
located in Michigan, GPA over-estimates the highest flows and under-estimates
the lowest flows. LN and KAP predict the upper tail well, but KAP has trouble
predicting the lower tail. Finally, Fig. 3d illustrates a site in Virginia
where all three distributions show poor fits (LNSE values below 0.93).
Error duration plots illustrating the range of errors (the ratio
of predicted quantiles of flow to observed ranked flows) across exceedance
probabilities for generalized Pareto, lognormal, and kappa hypotheses. Each
grey line represents the estimated relative error for a study streamgage and
the black horizontal line at one shows no error.
(a) By physiographic region, boxplots of streamgage
Nash–Sutcliffe efficiencies of natural logarithms of daily streamflows
(LNSE) based on hypothesized generalized Pareto (GPA), lognormal (LN), and
kappa (KAP) distributions with points outside the bounds of the plots listed
in brackets. Below the region name, the number of study gages located in that
region is listed as N. Boxplots are only given for regions with at least 20
study gages to facilitate a relatively fair comparison across regions.
(b) Maps of the conterminous United States illustrating streamgage
LNSE values for the GPA, LN, and KAP hypotheses.
To assess how the magnitude of errors varied across exceedance probabilities,
error duration curves are shown in Fig. 4 (similar to the error duration
plots given in Müller and Thompson, 2016). These plots illustrate how
error, the ratio of predicted quantiles of flow to observed ranked flows, is
distributed across the quantiles for GPA, LN, and KAP. Values of one indicate
no error and above one indicate that predicted flows are greater than
observed flows for a given quantile. Each grey line represents the error for
a study site. All three distributions dramatically over-predict the highest
flows for some sites, but the spread of error is highest for the lowest flows
(exceedance probabilities closer to one). These errors highlight the
challenge of having one distribution represent the tail behavior of both low
and high flows. While GPA and LN errors appear relatively comparable, the
spread of errors for KAP is generally smaller across all quantiles.
Goodness of fit by physiographic region
Perhaps the sites with poor fits to FDCPOR are primarily located
within certain regions of the US. Focusing on such a large study region
provides both a challenge and an opportunity to compare the GOF of candidate
distributions across regions within the US. Figure 5a shows boxplots of LNSE
by probability distribution for eight physiographic regions in the US (all of
the regions which included at least 20 study sites). Sample sizes are given,
as well as the number of sites within each region for which
FDCPOR could not be estimated with KAP. (This was a particular
problem in the Piedmont region, where only 8 of the 24 sites had feasible KAP
parameters.) These boxplots illustrate that there are some regions in the US
for which all three distributions provide a very good fit, such as the New
England, Appalachian, and Valley and Ridge regions. A three-parameter
distribution such as GPA or LN might be adequate to describe FDCs in these
regions, as Fennessey (1994) found to be the case for the mid-Atlantic
region. For most regions, KAP provides the best fit, which is not surprising
given that it has an additional parameter compared to GPA and LN. The
Cascade-Sierra mountains appear to be a particularly difficult region to
capture with these three candidate distributions, as none show very high GOF.
Maps of the US illustrating LNSE for GPA, LN, and KAP are given in Fig. 5b.
For GPA (left), nearly all “poor fits” (LNSE < 0.93) are at sites
in the western half of the country. Very good fits (LNSE > 0.99)
are found throughout the US, but are primarily clustered in New England and
the mid-Atlantic regions. For LN (middle map), more sites have LNSE values
above 0.99 compared to GPA, particularly in the eastern half of the country,
and there are fewer sites on the West Coast, with LNSE values below 0.93.
Finally, the map of KAP LNSE (right) illustrates that, of the 356 sites which
could be fit with KAP, the majority are well approximated by KAP, as
indicated by LNSE values above 0.99. However, a limitation of KAP is that it
could not be used to estimate FDCPOR at 42 sites in the study
region due to parameters inconsistent with KAP. Martinez and Gupta (2010)
found a relatively similar geographic pattern in GOF for a monthly water
balance model applied across the conterminous US. They attributed this
pattern in GOF to aridity, with worse model performance generally found at
water-limited sites.
Median annual flow duration curves
The FDCPOR reflects the steady-state or long-term behavior of the
frequency–magnitude relationship for streamflow. Alternatively, if flows in
a typical year are of interest, then median annual FDCs (FDCMED)
are useful (Vogel and Fennessey, 1994). Less dependent upon the specific
period of record than FDCPOR, FDCMED are increasingly
applied in practice when hydrologic conditions for a typical year are of
interest. For example, FDCMED have recently been used to predict
hydropower production (Mohor et al., 2015; Müller et al., 2014), evaluate
regional similarity between streams under different flow conditions (Patil
and Stieglitz, 2011), and characterize baseflow variability (Hamel et al.,
2015). FDCMED are also used to compare streamflow regimes in
different catchments (Hrachowitz et al., 2009), to assess before and after
watershed land-use changes (Kinoshita and Hogue, 2014), and to quantify fish
passage delays (Lang et al., 2004). More generally, FDCMED are
useful in testing hypotheses regarding any form of flow alteration (Kroll et
al., 2015).
(a) L-moment diagram with empirical L-moment ratios of
the median annual flow duration curves (FDCMED) estimated at
study streamgages; (b) boxplots of streamgage Nash–Sutcliffe
efficiencies for natural logarithms of FDCMED (LNSE) based on
hypothesized generalized Pareto (GPA), lognormal (LN), and kappa (KAP)
distributions with points outside the bounds of the plots listed in brackets;
(c) error duration plots for FDCMED illustrating the
range of error (the ratio of predicted quantiles of FDCMED to
empirical FDCMED) across exceedance probabilities for the GPA,
LN, and KAP hypotheses. Each grey line represents the estimated relative
error for a study streamgage and the black horizontal line at one shows no
error.
A few studies have attempted to fit a probability distribution to FDCs in a
typical year. LeBoutillier and Wayland (1993) found a five-parameter mixed
lognormal distribution to be superior to two- and three-parameter lognormal,
Gamma, and generalized extreme value distributions for mean annual FDCs of
four rivers in Canada. For the mid-Atlantic US, Fennessey (1994) identified
the GPA as a suitable distribution for both FDCPOR and
FDCMED, developed regional regression models to relate GPA model
parameters to basin characteristics, and then used those models to predict
FDCs at ungaged locations. FDCMED can also be estimated
seasonally, and seasonal FDCMED have been used to evaluate
impacts on ecological flow regimes (Gao et al., 2009; Lin et al., 2014; Vogel
et al., 2007).
The procedure for constructing an FDCMED is similar to the method
for constructing an FDCPOR, but rather than ranking all recorded
flows, flows are ranked within each calendar year, resulting in rankings of
1–365 for each year. Then, the median flow at each ranking is selected to
represent the given quantile within the FDCMED. The majority of
the FDCPOR and FDCMED curves are generally very
similar, only differing at the lowest and highest exceedance probabilities.
This is because the most extreme flows on record are included in
FDCPOR but are not included in FDCMED, as the median
estimator is insensitive to outliers. See Vogel and Fennessey (1994) for a
more detailed discussion of the relationship between FDCPOR and
FDCMED.
Figure 6a shows the relationship between empirical L-skew and L-kurtosis
for FDCMED at study sites. These L-moment ratios appear quite
similar to those found for FDCPOR (Fig. 2a), as do the LNSE
values for GOF by distribution shown in boxplots (Fig. 6b). As for
FDCPOR, distributions were constrained to ensure no negative
streamflows are predicted, and KAP appears to provide the best fit to
FDCMED. Figure 6c shows error duration curves for
FDCMED. The main difference between these plots and the error
duration plots for FDCPOR (Fig. 4) is that the errors are smaller at
the lowest and highest exceedances. This may be due to the fact that
FDCMED curves are generally quite similar to FDCPOR
but lack the most extreme high and low flows.
Discussion and conclusions
Due to the complexity associated with time series of daily streamflows, the
challenge set forth in this study – to identify a single probability
distribution that could approximate the distribution of daily flows – was an
ambitious one. Based upon multiple goodness-of-fit (GOF) assessments, three
candidate probability distributions were identified which can approximate
period-of-record (FDCPOR) and median annual (FDCMED)
flow duration curves at perennial, unregulated streamgage sites in much of
the conterminous United States (US). Previous work on this subject identified
the need for at least four parameters to describe the complex distribution of
daily streamflows; this study built off of earlier studies by investigating the
suitability of a probability distribution for streamflow at the
sub-continental scale across widely varying physiographic and hydroclimatic
settings. For these study streamgages, four-parameter kappa (KAP) was found
to provide a very good fit to the distribution of daily streamflows across
most of the US (at the 89 % of sites that had valid KAP parameters). A
special case of the KAP distribution, three-parameter generalized Pareto
(GPA), can provide an acceptable fit for certain regions of the US,
particularly New England, Appalachia, and the Valley and Ridge regions.
Compared to GPA, three-parameter lognormal (LN) was found to result in
predictions with better GOF, particularly in the Pacific Border and
Cascade-Sierra regions. To prevent the prediction of infeasible negative
streamflows, all three distributions required lower bound constraints for
some sites. More work is needed on parameter estimation that enforces the
conditions that streamflows be both non-negative and exceed
theoretical distributional lower bounds.
Few previous studies have sought to evaluate theoretical probability distributions for modeling FDC in a typical year, but the growing use of FDCMED suggests that these findings could have broad applications. Users of
FDCMED should be aware that the FDCMED only provides
information about the behavior of streamflow in a typical year; thus, it is
important to illustrate the entire family of annual FDCs which gave rise to
the computation of the FDCMED. To predict either
FDCPOR or FDCMED at ungaged sites, regional
regression models of distribution parameters can be developed based on the
relation between basin characteristics and distribution parameters at a set
of neighboring gages. Then, with knowledge of basin characteristics at the
ungaged site, the FDC can be estimated from distribution parameters predicted
by the regional regression model.
There are many limitations of this work. First, daily streamflows are not
independent and exhibit a high level of serial correlation. This correlation
will impact confidence intervals or any other form of uncertainty analysis
associated with modeled FDCs. Furthermore, daily streamflows exhibit
seasonality and are thus far from being identically distributed, which is
assumed whenever one attempts to fit a single distribution to a random
variable. The seasonality of daily streamflows suggests that distributional
analyses of this nature should be done at a seasonal level, as was recently
carried out on a broad scale for daily precipitation (see Papalexiou and
Koutsoyiannis, 2016). The definition of seasons, as well as the parent
distributions which can approximate streamflows within those seasons, has
been shown to vary across sites (Bowers et al., 2012). Given the large range
of hydroclimatic conditions affecting study streamgages, a seasonal analysis was beyond the scope
of this study, but future studies should consider the impact of seasonality
on the GOF of FDCs. In addition, this study included only perennial and
unregulated streams. While there is some existing literature on intermittent
regimes (Mendicino and Senatore, 2013; Pumo et al., 2014; Rianna et al.,
2011) and the impacts of human regulation on flow duration curves (Gao et
al., 2009; Kroll et al., 2015), additional research on these topics would
improve understanding of flows across a wider range of streams.
Daily streamflow varies over 4 or 5 orders of magnitude and is subject to
seasonality and serial correlation. When viewed though this lens, the finding
of any reasonable candidate distribution that provides some explanatory power
– such as those explored here – is somewhat remarkable. Future research on
intermittent sites, differences across seasons, lower bound constraints, and
additional distributional types, such as mixed distributions, can help to
improve prediction of daily streamflows at ungaged sites across the globe.