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**Hydrology and Earth System Sciences**
An interactive open-access journal of the European Geosciences Union

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**Research article**
28 Feb 2018

**Research article** | 28 Feb 2018

Multiple causes of nonstationarity in the Weihe annual low-flow series

^{1}State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, P.R. China^{2}Department of Geosciences, University of Oslo, P.O. Box 1022 Blindern, 0315 Oslo, Norway

^{1}State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, P.R. China^{2}Department of Geosciences, University of Oslo, P.O. Box 1022 Blindern, 0315 Oslo, Norway

**Correspondence**: Lihua Xiong (xionglh@whu.edu.cn)

**Correspondence**: Lihua Xiong (xionglh@whu.edu.cn)

Abstract

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Under the background of global climate change and local
anthropogenic activities, multiple driving forces have introduced various
nonstationary components into low-flow series. This has led to a high demand
on low-flow frequency analysis that considers nonstationary conditions for
modeling. In this study, through a nonstationary frequency analysis framework
with the generalized linear model (GLM) to consider time-varying distribution
parameters, the multiple explanatory variables were incorporated to explain
the variation in low-flow distribution parameters. These variables are
comprised of the three indices of human activities (HAs; i.e., population,
POP; irrigation area, IAR; and gross domestic product, GDP)
and the eight measuring indices of the climate and catchment conditions
(i.e., total precipitation *P*, mean frequency of precipitation events
*λ*, temperature *T*, potential evapotranspiration (EP), climate
aridity index AI_{EP}, base-flow index (BFI), recession constant *K*
and the recession-related aridity index AI_{K}). This framework was
applied to model the annual minimum flow series of both Huaxian and Xianyang
gauging stations in the Weihe River, China (also known as the Wei He River). The results from stepwise
regression for the optimal explanatory variables show that the variables
related to irrigation, recession, temperature and precipitation play an
important role in modeling. Specifically, analysis of annual minimum
30-day flow in Huaxian shows that the nonstationary distribution
model with any one of all explanatory variables is better than the one
without explanatory variables, the nonstationary gamma distribution model
with four optimal variables is the best model and AI_{K} is of the
highest relative importance among these four variables, followed by IAR,
BFI and AI_{EP}. We conclude that the incorporation of multiple indices
related to low-flow generation permits tracing various driving forces. The
established link in nonstationary analysis will be beneficial to analyze
future occurrences of low-flow extremes in similar areas.

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Xiong, B., Xiong, L., Chen, J., Xu, C.-Y., and Li, L.: Multiple causes of nonstationarity in the Weihe annual low-flow series, Hydrol. Earth Syst. Sci., 22, 1525–1542, https://doi.org/10.5194/hess-22-1525-2018, 2018.

1 Introduction

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Low flow is defined as the flow of water in a stream during prolonged dry weather (WMO, 2009). Yu et al. (2014) quantitatively described a low-flow event as a segment of hydrograph during a period of dry weather with discharge values below a preset (relatively small) threshold. According to WMO (2009), annual minimum flows averaged over several days can be used to measure low flows. During low-flow periods, the magnitude of river flow will greatly restrict its various functions (e.g., providing water supply for production and living, diluting waste water, ensuring navigation, meeting ecological water requirement). Therefore, the investigation of the magnitude and frequency of low flows is of primary importance for engineering design and water resources management (Smakhtin, 2001). In recent years, low flows, as an important part of river flow regime, have been attracting an increasing attention of hydrologists and ecologists in the context of the significant impacts of climate change and human activities (HAs; Bradford and Heinonen, 2008; Du et al., 2015; Kam and Sheffield, 2015; Kormos et al., 2016; Liu et al., 2015; Sadri et al., 2016). In general, under the impact of a changing environment, combinations of multiple factors, such as precipitation change, temperature change, irrigation area (IAR) change and construction of reservoirs, can drive various patterns of streamflow changes (Liu et al., 2017; Tang et al., 2016). Unfortunately, when subjected to a variety of influencing forces, low flow is more vulnerable than high flow or mean flow. Therefore, it is a pretty important issue in hydrology to identify low-flow changes, track multiple driving factors and quantify their contributions from the perspective of hydrological frequency analysis.

In hydrological analysis and design, conventional frequency analysis estimates the statistics of a hydrological time series based on recorded data with the stationary hypothesis which means that this series is “free of trends, shifts or periodicity (cyclicity)” (Salas, 1993). However, global warming and human forces have changed climate and catchment conditions in some regions. Time-varying climate and catchment conditions (TCCCs) can affect all aspects of the flow regime, i.e., changing the frequency and magnitude of floods, altering flow seasonality and modifying the characteristics of low flows. The hypothesis of stationarity has been suspected (Milly et al., 2008). If this problematic method is still used, the frequency analysis may lead to high estimation error in hydrological design. Therefore, considerable literature has introduced the concept of hydrologic nonstationarity into analysis of various hydrological variables, such as annual runoff (Arora, 2002; Jiang et al., 2015a; Xiong et al., 2014; Yang and Yang, 2013), flood (Gilroy and Mccuen, 2012; Kwon et al., 2008; Yan et al., 2017; Zhang et al., 2015), low flow (Du et al., 2015; Jiang et al., 2015b; Liu et al., 2015), precipitation (Gu et al., 2017; Mondal and Mujumdar, 2015; Villarini et al., 2010) and so on. Compared with the literature on annual runoff, floods and precipitation, the literature on the nonstationary analysis of low flow is relatively limited.

Previous hydrological literature on frequency analysis of nonstationary hydrological series mainly focuses on two aspects: development of the nonstationary method and exploration of covariates reflecting changing environments. Strupczewski et al. (2001) presented the method of time-varying moment which assumes that the hydrological variable of interest obeys a certain distribution type, but its moments change over time. The method of time-varying moment was modified to be the method of time-varying parameter values for the distribution representative of hydrologic data (Richard et al., 2002). Villarini et al. (2009) presented this method using the generalized additive models for location, scale and shape parameters (GAMLSS; Rigby and Stasinopoulos, 2005), a flexible framework to assess nonstationary time series. The time-varying parameter method can be extended to the physical covariate analysis by replacing time with any other physical covariates (Jiang et al., 2015b; Kwon et al., 2008; López and Francés, 2013; Liu et al., 2015; Villarini and Strong, 2014). For example, Jiang et al. (2015b) used reservoir index as an explanatory variable based on the time-varying copula method for bivariate frequency analysis of nonstationary low-flow series in Hanjiang River, China. Du et al. (2015) took precipitation and air temperature as the explanatory variables to explain the inter-annual variability in low flows of the Weihe River, China (also known as the Wei He River). Liu et al. (2015) took the sea surface temperature in the Nino3 region, the Pacific Decadal Oscillation, the sunspot number (3 years ahead), the winter areal temperature and precipitation as the candidate explanatory variables to explain the inter-annual variability in low flows of Yichang station, China. Kam and Sheffield (2015) ascribed the increasing inter-annual variability of low flows over the eastern United States to the North Atlantic Oscillation and Pacific North America.

To our knowledge, compared with the nonstationary flood frequency analysis, the studies on the nonstationary frequency analysis of low-flow series are not very extensive because of incomplete knowledge of low-flow generation (Smakhtin, 2001). Most of these studies explain nonstationarity of low-flow series only by using climatic indicators or a single indicator of human activity. However, the indicators of catchment conditions (e.g., recession rate) related to physical hydrological processes have seldom been attached in nonstationary modeling of low-flow series. This lack of linking with hydrological processes makes it impossible to accurately quantify the contributions of influencing factors for the nonstationarity of low-flow series, and such a scientific demand for tracing the sources of nonstationarity of low-flow series and qualifying their contributions motivated the present study. The knowledge of low-flow generation has been increased by efforts of hydrologists, which can help develop physical covariates to address nonstationarity. Low flows generally originate from groundwater or other delayed outflows (Smakhtin, 2001; Tallaksen, 1995). Their generation relates to both an extended dry weather period (leading to a climatic water deficit) and complex hydrological processes which determine how these deficits propagate through the vegetation, soil and groundwater system to streamflow (WMO, 2009). Thus, not only climate condition drivers (e.g., potential evaporation exceeds precipitation), but also catchment condition drivers (e.g., the faster hydrologic response rate to precipitation) can cause low flows.

The significant factors such as precipitation, temperature,
evapotranspiration (EP), streamflow recession, large-scale teleconnections and
human forces may play important roles in influencing low-flow generation
(Botter et al., 2013; Giuntoli et al., 2013; Gottschalk et al., 2013; Kormos
et al., 2016; Sadri et al., 2016). Gottschalk et al. (2013) presented a
derived low-flow probability distribution function with climate and catchment
characteristics parameters (i.e., the mean length of dry spells
*λ*^{−1} and recession constant of streamflow *K* ) as its distribution
parameters. Botter et al. (2013) derived a measurable index
(${\mathit{\lambda}}^{-\mathrm{1}}/K)$ which can be used for discriminating erratic river
flow regimes from persistent river flow regimes. Recently, Van Loon and
Laaha (2015) used climate and catchment characteristics (e.g., the duration of
dry spells in precipitation and the base-flow index, BFI) to explain the duration
and deficit of the hydrological drought event and offered a further understanding
of low-flow generation. These studies indicated that climate and catchment
conditions play an important role in producing low flows.

The goal of this study is to trace origins of nonstationarity in low flows
through developing a nonstationary low-flow frequency analysis framework with
the consideration of the time-varying climate and catchment conditions and human activity. In this framework, the climate and catchment
conditions are quantified using the eight indices, i.e., meteorological
variables (total precipitation *P*, mean frequency of precipitation events
*λ*, temperature *T* and potential evapotranspiration), basin
storage characteristics (base-flow index, recession constant *K*) and
aridity indexes (climate aridity index AI_{EP}, the recession-related
aridity index AI_{K}). The specific objectives of this study are
(1) to find the most important index to explain the nonstationarity
of low-flow series, (2) to determine the best subset of TCCCs indices
and/or human activity indices (i.e., population, POP; irrigation area; and gross domestic product, GDP) for the final model through
the stepwise selection method to identify nonstationary mode of low-flow series
and (3) to quantify the contribution of selected explanatory
variables to the nonstationarity.

This paper is organized as follows. Section 2 describes the methods. The Weihe River basin and available data sets used in this study are described in Sect. 3, followed by a presentation of the results and discussion in Sect. 4. Section 5 summarizes the main conclusions.

2 Methodology

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The flowchart of how to organize the nonstationary low-flow frequency analysis framework is shown in Fig. 1. The whole process is divided into three steps. The first step is the preliminary analysis, including the graphical presentation of both explanatory variables and low-flow series, the statistical test for nonstationarity, and the correlations between each explanatory variable and each low-flow series. The second step is the single covariate analysis for the most important explanatory variable. The third step is the multiple covariate analysis for the optimal combination. We use a low-flow frequency analysis model and stepwise regression method to accomplish the last two steps. In the following subsections, first, the low-flow frequency analysis model is constructed based on the nonstationary probability distributions method, in which distribution parameters serving as response variables can vary as functions of explanatory variables. Second, the distribution types used to build the nonstationary model are outlined. Then, the candidate explanatory variables related to the time-varying climate and catchment conditions and human activity are clarified. Finally, estimation of model parameters and selection of models are illustrated.

Generally, a nonstationary frequency analysis model can be established based
on the time-varying distribution parameters method (Du et al., 2015;
López and Francés, 2013; Liu et al., 2015; Richard et al., 2002;
Villarini and Strong, 2014). For the nonstationary probability distribution
*f*_{Y}(*Y*_{t}|*θ*^{t}), let
*Y*_{t} be a random variable at time $t(t=\mathrm{1},\mathrm{2},\mathrm{\dots},N)$ and vector
${\mathit{\theta}}^{t}=[{\mathit{\theta}}_{\mathrm{1}}^{t},{\mathit{\theta}}_{\mathrm{2}}^{t},\mathrm{\dots},{\mathit{\theta}}_{m}^{t}]$
be the time-varying parameters. The number of parameters *m* in hydrological
frequency analysis is generally limited to three or less. The function
relationship between the *k*th parameter ${\mathit{\theta}}_{k}^{t}$ and the
multiple explanatory variables is expressed as follows:

$$\begin{array}{}\text{(1)}& {\displaystyle}{g}_{k}\left({\mathit{\theta}}_{k}^{t}\right)={h}_{k}\left({x}_{\mathrm{1}}^{t},{x}_{\mathrm{2}}^{t},\mathrm{\dots},{x}_{n}^{t}\right),\end{array}$$

where ${x}_{\mathrm{1}}^{t},{x}_{\mathrm{2}}^{t},\mathrm{\dots},{x}_{n}^{t}$ are explanatory variables, *n*
is the number of explanatory variables, *g*_{k}(⋅) is the link
function which ensures the compliance with restrictions on the sample space
and is usually set to natural logarithm for the given negative predictions
and *h*_{k}(⋅) is the function for nonstationary modeling. The
generalized linear model theory (GLM; Dobson and Barnett, 2012) is used to build
function relationships between distribution parameters and their explanatory
variables. In GLMs, the response relationship can be generally expressed as

$$\begin{array}{}\text{(2)}& {\displaystyle}{g}_{k}\left({\mathit{\theta}}_{k}^{t}\right)={\mathit{\alpha}}_{\mathrm{0}k}+\sum _{i=\mathrm{1}}{\mathit{\alpha}}_{ik}{x}_{i}^{t},\end{array}$$

where ${\mathit{\alpha}}_{ik}(i=\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{\dots},n,k=\mathrm{1},\mathrm{\dots},m)$ are the GLM parameters.

In order to compare the nonstationary models constructed by various combinations of explanatory variables, Eq. (2) is modified in this study using the dimensionless method for the standard GLM parameters. The value of ${\mathit{\theta}}_{k}^{t}$ could be assumed to be equal to its mean (${\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{k})$ when all explanatory variables are equal to their mean (${\stackrel{\mathrm{\u203e}}{x}}_{i})$, i.e.,

$$\begin{array}{}\text{(3)}& {\displaystyle}{\mathit{\theta}}_{k}^{t}\left({x}_{\mathrm{1}}^{t}={\stackrel{\mathrm{\u203e}}{x}}_{\mathrm{1}},{x}_{\mathrm{2}}^{t}={\stackrel{\mathrm{\u203e}}{x}}_{\mathrm{2}},\mathrm{\dots},{x}_{n}^{t}={\stackrel{\mathrm{\u203e}}{x}}_{n}\right)={\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{k}.\end{array}$$

Equation (2) is then modified as

$$\begin{array}{}\text{(4)}& {\displaystyle}\begin{array}{l}{g}_{k}\left({\displaystyle \frac{{\mathit{\theta}}_{k}^{t}}{{\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{k}}}\right)={\mathit{\beta}}_{\mathrm{0}k}+\sum _{i=\mathrm{1}}^{i=n}{\mathit{\beta}}_{ik}{z}_{i}^{t}\\ {z}_{i}^{t}={\displaystyle \frac{{x}_{i}^{t}-{\stackrel{\mathrm{\u203e}}{x}}_{i}}{{s}_{i}}},i=\mathrm{1},\mathrm{2},\mathrm{\dots},n\\ {\mathit{\beta}}_{\mathrm{0}k}={g}_{k}\left({\displaystyle \frac{{\mathit{\theta}}_{k}^{t}}{{\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{k}}}\left|{\mathit{\theta}}_{k}^{t}={\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{k}\right.\right)={g}_{k}\left(\mathrm{1}\right),\end{array}\end{array}$$

where ${z}_{i}^{t}$ is the normalized explanatory variable, *s*_{i} is the
standard deviation of ${x}_{i}^{t}$ and ${\mathit{\beta}}_{ik}(i=\mathrm{1},\mathrm{2},\mathrm{\dots},n,k=\mathrm{1},\mathrm{\dots},m)$ are the standard GLM parameters. Letting the link
function *g*_{k}(⋅) be the natural logarithmic function ln (⋅)
and ${\mathit{\theta}}_{l}^{t}$ be the distribution parameter in $[{\mathit{\theta}}_{\mathrm{1}}^{t},{\mathit{\theta}}_{\mathrm{2}}^{t},\mathrm{\dots},{\mathit{\theta}}_{m}^{t}]$ with the most significant change, the
degree of nonstationarity in low-flow series can be defined as $\mathrm{ln}\left({\mathit{\theta}}_{l}^{t}\right)-\mathrm{ln}\left({\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{l}\right)$. Then, the contribution ${c}_{i}^{t}$
of each explanatory variable ${x}_{i}^{t}$ to $\mathrm{ln}\left({\mathit{\theta}}_{l}^{t}\right)-\mathrm{ln}\left({\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{l}\right)$ could be defined as

$$\begin{array}{}\text{(5)}& {\displaystyle}{c}_{i}^{t}={\mathit{\beta}}_{il}{\displaystyle \frac{{x}_{i}^{t}-{\stackrel{\mathrm{\u203e}}{x}}_{i}}{{s}_{i}}}.\end{array}$$

We need to select the form of probability distribution *f*_{Y}(⋅) to
determine what type of nonstationary frequency curves will be produced.
Various probability distributions have been compared or suggested in modeling
of low-flow series (Du et al., 2015; Hewa et al., 2007; Liu et al., 2015;
Matalas, 1963; Smakhtin, 2001). An extensive overview of distribution
functions for low flow is given in Tallaksen et al. (2004). Following these
recommendations, we consider five distributions, i.e., Pearson type III (PIII),
gamma (GA), Weibull (WEI), lognormal (LOGNO) and generalized extreme value
(GEV) as candidates in this study (Table 1). In the case of Pearson type III
distribution, considering that the parameter *θ*_{3} of
Pearson type III as lower bound should approach zero and the parameter *θ*_{3} of GEV is quite sensitive and difficult to be estimated, we
assume them to be constant in this study.

We look for variables ${x}_{\mathrm{1}}^{t},{x}_{\mathrm{2}}^{t},\mathrm{\dots},{x}_{n}^{t}$ that can
explain parts of the variations in distribution parameters *θ*^{t}. From the perspective of low-flow generation, the dependency between
low-flow regime and both climate and catchment conditions has been presented
by previous studies (Botter et al., 2013; Gottschalk et al., 2013; Van Loon
and Laaha, 2015). We focus on eight measuring indices: precipitation, mean
frequency of precipitation events, temperature, potential
evapotranspiration, climate aridity index, base-flow index, recession
constant and recession-related aridity index. These indices were chosen to
incorporate time-varying climate and catchment conditions in
nonstationary modeling of low-flow frequency and serve as candidate
explanatory variables. Climate variables (i.e., precipitation, mean frequency
of precipitation events, temperature, potential evapotranspiration and
climate aridity index) are related to both water supply source and water
loss and are therefore selected as candidate variables. It has been shown
that the base-flow index and recession constant reflect the storage and release
capability of the catchments (Van Loon and Laaha, 2015). The recession-related
aridity index reflects both the water supply and storage capability (Botter
et al., 2013). In addition to TCCCs indices, the three indices of human
activity (irrigation area, population and gross domestic product) are
related to water withdrawal loss for agricultural, domestic and industrial
purposes and are therefore included. The detailed reasons for selecting all
indices are summarized in Table 3. The values of them at each year could be
estimated from hydro-meteorological data and human activity data. Annual
precipitation (*P*) and temperature (*T*) are calculated directly by
meteorological data. The remaining TCCCs indices need to be estimated
indirectly. Detailed estimation procedures are shown in the following
subsections.

Annual mean frequency of precipitation events is defined as an index to represent the intensity of precipitation recharge to the streamflow:

$$\begin{array}{}\text{(6)}& {\displaystyle}\mathit{\lambda}={\displaystyle \frac{\mathrm{1}}{W}}\sum _{w=\mathrm{1}}^{w=W}{\displaystyle \frac{{N}_{w}\left(A\right)}{{t}_{r}}},\end{array}$$

where *N*_{w}(A) is the number of daily rainfall events *A*
(with values more than the threshold 0.5 mm) in *w*th windows with a
length *t*_{r}; *W* is the number of windows.

The ratio of annual potential evaporation to precipitation, commonly known as the climate aridity index, has been used to assess the impacts of climate change on annual runoff (Arora, 2002; Jiang et al., 2015a). The climate aridity index largely reflects the climatic regimes in a region and determines runoff rates (Arora, 2002). Therefore, we choose the annual climate aridity index as a measure of time-varying climate and catchment conditions and estimate its value in a whole region using

$$\begin{array}{}\text{(7)}& {\displaystyle}{\mathrm{AI}}_{\mathrm{EP}}={\displaystyle \frac{\mathrm{EP}}{P}},\end{array}$$

where *P* is annual areal precipitation (mm) and EP is annual areal potential
evapotranspiration (mm). The Hargreaves equation
(Hargreaves and Samani, 1985) is applied to calculate EP using the R package
“Evapotranspiration” (Guo, 2014).

The base-flow index (BFI) is defined as the ratio of base flow to total flow. This index has been applied to quantify catchment conditions (e.g., soil, geology and storage-related descriptors) to explain hydrological drought severity (Van Loon and Laaha, 2015). We also choose annual base-flow index as a measure of TCCCs. BFI is estimated using a hydrograph separation procedure in the R package “lfstat” (Koffler and Laaha, 2013).

The recession constant is an important catchment characteristic index measuring the timescale of the hydrological response and reflecting water retention ability in the upstream catchment (Botter et al., 2013). Various estimation methods have been developed to extract recession segments and to parameterize characteristic recession behavior of a catchment (Hall, 1968; Sawaske and Freyberg, 2014; Tallaksen, 1995).

In this study, annual recession analysis (ARA) is performed to obtain the annual
streamflow recession constant (*K*). In ARA, the linearized
Dupuit–Boussinesq
equation is used to parameterize characteristic recession behavior of a
catchment and is written as

$$\begin{array}{}\text{(8)}& {\displaystyle}-{\displaystyle \frac{\mathrm{d}{Q}_{t}}{\mathrm{d}t}}={\displaystyle \frac{\mathrm{1}}{K}}{Q}_{t},\end{array}$$

where *Q*_{t} is the value at time *t*. Equation (8) is investigated by plotting
data points $\frac{\mathrm{d}{Q}_{t}}{\mathrm{d}t}$ against *Q*_{t} of all extracted recession
segments from hydrographs at each year. The criteria of recession segment
extraction are based on the Manual on Low-flow Estimation and Prediction
(WMO, 2009). Then, the annual recession rate (*K*^{−1}) is estimated as the
slope of the fitted straight line of these data points with the least-squares method.
We calculated *K* using the R package “lfstat” (Koffler and Laaha, 2013).

In this study, the recession-related aridity index is defined as the ratio of
recession rate (*K*^{−1}) to mean precipitation frequency (*λ*),
denoted as

$$\begin{array}{}\text{(9)}& {\displaystyle}{\mathrm{AI}}_{K}={\displaystyle \frac{{K}^{-\mathrm{1}}}{\mathit{\lambda}}}.\end{array}$$

This ratio plays an important role in controlling the river flow regime (Botter
et al., 2013; Gottschalk et al., 2013) and serves as an indicator measuring
the recession-related aridity degree of the streamflow in the river channel. For
example, the faster recession process or lower precipitation frequency may lead
to increased runoff loss or decreased precipitation supply. Consequently,
the higher the value AI_{K} is, the more likely low-flow events occur,
and vice versa.

The model parameters including ${\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{k}(k=\mathrm{1},\mathrm{2},\mathrm{\dots},m)$ and ${\mathit{\beta}}_{ik}(i=\mathrm{1},\mathrm{2},\mathrm{\dots},n,k=\mathrm{1},\mathrm{\dots},m)$ need to be estimated. ${\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{k}(k=\mathrm{1},\mathrm{2},\mathrm{\dots},m)$ are estimated from outputs of stationary frequency analysis through the maximum likelihood method. We have

$$\begin{array}{}\text{(10)}& {\displaystyle}L\left({\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{\mathrm{1}},{\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{\mathrm{2}},\mathrm{\dots},{\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{m}\right)=\sum _{t=\mathrm{1}}^{t=N}\mathrm{ln}\left[{f}_{Y}\left({y}_{t}\left|{\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{\mathrm{1}},{\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{\mathrm{2}},\mathrm{\dots},{\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{m}\right.\right)\right],\end{array}$$

where *y*_{t} is observed low flow at time *t* and *N* is the number of samples.
The parameters ${\mathit{\beta}}_{ik}(i=\mathrm{1},\mathrm{2},\mathrm{\dots},n,k=\mathrm{1},\mathrm{\dots},m)$ are estimated
through the maximum likelihood method to produce nonstationary low-flow
frequency curves:

$$\begin{array}{ll}\text{(11)}& {\displaystyle}& {\displaystyle}L\left(\begin{array}{l}{\mathit{\beta}}_{\mathrm{11}},\mathrm{\dots},{\mathit{\beta}}_{n\mathrm{1}}\\ \mathrm{\dots}\\ {\mathit{\beta}}_{\mathrm{1}m},\mathrm{\dots},{\mathit{\beta}}_{nm}\end{array}\right)=\sum _{t=\mathrm{1}}^{t=N}\mathrm{ln}{\displaystyle}& {\displaystyle}\left\{{f}_{Y}\left({y}_{t}\left|{\mathit{\theta}}_{\mathrm{1}}^{t}\left({z}_{\mathrm{1}}^{t},\mathrm{\dots},{z}_{n}^{t}\left|{\mathit{\beta}}_{\mathrm{11}},\mathrm{\dots},{\mathit{\beta}}_{n\mathrm{1}}\right.\right),\mathrm{\dots},{\mathit{\theta}}_{m}^{t}\left({z}_{\mathrm{1}}^{t},\mathrm{\dots},{z}_{n}^{t}\left|{\mathit{\beta}}_{\mathrm{1}m},\mathrm{\dots},{\mathit{\beta}}_{nm}\right.\right)\right.\right)\right\}.\end{array}$$

The residuals (normalized randomized quintile residuals) are used to test the goodness of fit of fitted model objects (Dunn and Symth, 1996):

$$\begin{array}{}\text{(12)}& {\displaystyle}{\widehat{r}}_{t}={\mathrm{\Phi}}^{-\mathrm{1}}\left({F}_{Y}\left({y}_{t}\left|{\widehat{\mathit{\theta}}}^{t}\right.\right)\right),\end{array}$$

where *F*_{Y}(⋅) is the cumulative distribution of *y*_{t} and ${\mathrm{\Phi}}^{-\mathrm{1}}\left(\cdot \right)$ is the inverse function of the standard
normal distribution. The distribution of the true residuals ${\widehat{r}}_{t}$
converges to standard normal if the fitted model is correct. A worm plot
(Buuren and Fredriks, 2001) is used to check whether ${\widehat{r}}_{t}$ have a
standard normal distribution.

Model selection contains the selection of the type of probability
distribution and the selection of the explanatory variables to explain the
response variables (i.e., distribution parameters *θ*_{1} and *θ*_{2}). In order to obtain the final optimal model, the selection of the
explanatory variables for *θ*_{1} and *θ*_{2} is conducted by
stepwise selection strategies (Stasinopoulos and Rigby, 2007; Venables,
2002): i.e., select a best subset of candidate explanatory variables for
*θ*_{1} using a forward approach (which starts with no explanatory
variable in the model and tests the addition of each explanatory variable
using a chosen model fit criterion); given this subset for *θ*_{1}
select another subset for *θ*_{2} (forward). The stepwise selection
strategies can get a series of stepwise models with different numbers of
explanatory variables, as shown in Fig. 1. In order to detect how the number
of explanatory variables influences the performance of the model for
describing nonstationarity, we investigate the eight types of stepwise
models as shown in Table 2: the zero-covariate model or stationary model
(M0), the time covariate model (M1), the single physical covariate model M2
(single TCCCs covariate model M2a or single HA covariate model M2b), two
TCCCs covariates model (M3), the optimal TCCCs covariates model (M4), the
optimal HA covariates model (M5) and the final model (M6). The model fit
criterion is based on the Akaike's information criterion (AIC; Akaike, 1974) as
shown by the following

$$\begin{array}{}\text{(13)}& {\displaystyle}\mathrm{AIC}=-\mathrm{2}\mathrm{ML}+\mathrm{2}\mathrm{df},\end{array}$$

where ML is the log-likelihood in Eq. (11) and df is the number of degrees of freedom. The model with the lower AIC value was considered better.

3 Study area and data

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The Weihe River, located in the southeast of the northwest Loess Plateau, is
the largest tributary of the Yellow River, China. The Weihe River has a
drainage area of 134 766 km^{2}, covering the coordinates of
$\mathrm{33}{}^{\circ}{\mathrm{42}}^{\prime}$–$\mathrm{37}{}^{\circ}{\mathrm{20}}^{\prime}$ N, $\mathrm{104}{}^{\circ}{\mathrm{18}}^{\prime}$–$\mathrm{110}{}^{\circ}{\mathrm{37}}^{\prime}$ E
(Fig. 2). This catchment generally has
a semi-arid climate, with extensive continental monsoonal influence. Average
annual precipitation of the whole area over the period 1954–2009 is about
540 mm and has a wide range (400–1000 mm) in various regions. Under the
significant impacts of climate change and human activities in the Weihe River
basin in recent decades, the hydrological regime of the river has changed
over time (Du et al., 2015; Jiang et al., 2015a; Xiong et al., 2015).

In the Weihe basin, the impacts of agricultural irrigation on runoff have
been found to be significant (Jiang et al., 2015a; Lin et al., 2012). Lin et
al. (2012) mentioned that the annual runoff of the Weihe River was
significantly affected by irrigation diversion of the Baoji Gorge irrigation
district. The irrigated area of the Baoji Gorge irrigation district increased over time
since the founding of P.R. China in 1949, and, due to one influential
irrigation system project in that area, it became more than twice as large
as the original irrigation area since 1971. Jiang et
al. (2015a) demonstrated that, in the Weihe basin, irrigated area, as
compared with the other indices, e.g., population, gross domestic product and
cultivated land area, was a more suitable human explanatory variable for
explaining the time-varying behavior of annual runoff. With the above
background, it is important to consider the effects of human activities that
mainly originate from irrigation diversion, especially for studying low-flow
series in this basin. The estimations of annual recession rate (*K*^{−1}) by
the daily streamflow data are expected to incorporate the information of
impacts of water diversions on the low flows in the river channel.

We used daily streamflow records (1954–2009) provided by the Hydrology Bureau
of the Yellow River Conservancy Commission from both Huaxian station (with a
drainage area of 106 500 km^{2}) and Xianyang station (with a drainage area
of 46 480 km^{2}). Low-flow extreme events were selected from the daily
streamflow series using the widely used annual minimum series method (WMO,
2009). AM_{n} is the annual minimum *n*-day flow during hydrological year
beginning on 1 March. Consequently, AM_{1},
AM_{7}, AM_{15} and AM_{30} are selected as low-flow
extreme events in this study. The original measure unit of streamflow data
(m^{3} s^{−1}) is converted to
10^{−4} m^{3} s^{−1} km^{−2} for convenience of
comparison of results between the Huaxian and Xianyang gauging stations

We downloaded daily total precipitation and daily mean air temperature
records for 19 meteorological stations over the basin from the National
Climate Center of the China Meteorological Administration (source:
http://www.cma.gov.cn/). The areal average daily series of both
variables above Huaxian and Xianyang stations are calculated using the
Thiessen polygon method (Szolgayova et al., 2014; Thiessen, 1911). The
annual average temperature (*T*) and annual total precipitation (*P*) over
the period 1954–2009 are calculated for each catchment.

Human activity data (i.e., gross domestic product, population and irrigation area) were taken from annals of statistics provided by the Shaanxi Provincial Bureau of Statistics (http://www.shaanxitj.gov.cn/) and Gansu Provincial Bureau of Statistics (source: http://www.gstj.gov.cn/).

4 Results and discussion

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The graphical representation and statistical test provide a preliminary analysis
for low-flow nonstationarity. The graphical representations of time-series
data help visualize the trends of related variables (i.e., low flow, TCCCs and
HA variables), the density distributions of TCCCs variables, and the
correlations between low-flow variables and these explanatory variables. In
Fig. 3, four annual minimum streamflow series (AM_{1}, AM_{7}, AM_{15} and AM_{30}) in both Huaxian and Xianyang gauging stations show overall
decreasing trends, as indicated by the fitted (dashed) trend lines. Compared
with Huaxian, Xianyang has a larger runoff modulus (the flow per square
kilometer) and a larger decrease in annual minimum streamflow series. For
example, the decline slope of AM_{30} is −0.0725
(10^{−4} m^{3} s^{−1} km^{−2} yr^{−1}) in Huaxian station while Xianyang
station it is −0.1338 (10^{−4} m^{3} s^{−1} km^{−2} yr^{−1}).

Figure 4 shows the kernel density estimations and time processes of TCCCs
variables for both Huaxian (H) and Xianyang (X) stations. The results show
that these variables have different variation patterns. For example, the
mean frequency of precipitation events (*λ*) has a decreasing trend,
while temperature (*T*) has an increasing trend. As presented by Fig. 5,
three HA variables have a significant upward trend, especially the
irrigation area which is increased greatly after about 1970,
suggesting that the impact of human activities in this basin has increased
over time.

The significance of trends in the four annual minimum streamflow series and
TCCCs variables is tested by the Mann–Kendall trend test (Kendall, 1975;
Mann, 1945; Yue et al., 2002), and the change points in these series are
detected by Pettitt's test (Pettitt, 1979). The results in Table 4 show
that, in both Huaxian and Xianyang stations, the decreasing trends in all the
four low-flow series (AM_{1}, AM_{7}, AM_{15} and AM_{30}) and two
explanatory variables (*λ* and *P*), as well as the increasing trends in *T*,
EP, and AI_{EP} are significant at the 0.05 level (Table 4), but BFI
shows no significant trends. However, *K* and AI_{K} had significantly
decreasing trends only in Huaxian station (*p* value <0.05). The results of
change-point detection show that all low-flow series are located at 1968–1971
(*p* value <0.05) except AM_{30} at Xianyang station whose change point is
located at 1993 (*p* value <0.05); for the eight candidate explanatory
variables, the change points of the variables related to temperature
(*T*, EP, AI_{EP}) in both stations are located at 1990–1993
(*p* value <0.05), the change points of the variables related to precipitation
(*λ*, *P*) in both stations are close at 1984–1990 (*p* value ≤0.186)
and the change points of the variables related to streamflow recession
(*K*, AI_{K}) in Huaxian station are located at 1968–1971
(*p* value <0.05). However, BFI in both stations and *K* and AI_{K} in Xianyang station show no significant change points.

A preliminary attribution analysis is performed using the Pearson correlation
matrix to investigate the relations between the annual minimum series and
eight candidate explanatory variables. Figure 6 indicates that there are
significant linear correlations between the four minimum low-flow series
(AM_{1}, AM_{7}, AM_{15} and AM_{30}) and all the explanatory
variables except GDP have the absolute values of Pearson correlation
coefficients larger than 0.27 (*p* value <0.05). These potential physical
causes of nonstationarity in low flows are further considered by
establishing the low-flow nonstationary model with TCCCs and HA variables in the following
section.

Figure 7 presents the AIC values of the four types of models (M0, M1, M2a and
M2b) fitted for the low-flow series (AM_{1}, AM_{7}, AM_{15} and AM_{30}). Some interesting results are shown as follows. First,
nonstationary models (M1, M2a and M2b) have lower AIC values than stationary
model (M0), which suggests that nonstationary models are worth considering.
Second, for Huaxian station, irrespective of the chosen explanatory
variables, the distribution type plays an important role in modeling
nonstationary low-flow series. For example, PIII, GA and WEI distributions in
AM_{15} and AM_{30} cases have lower AIC values than LOGNO and GEV
distributions. However, for Xianyang, choosing a suitable explanatory
variable may be more important than choosing a distribution type. For
example, variables *t*, *P*, *T*, AI_{EP}, POP and IAR in most cases
have lower AIC values than the other explanatory variables. Finally, in
Huaxian, the lowest AIC values for modeling AM_{1}, AM_{7}, AM_{15} and AM_{30} are found in GEV_M2b_IAR,
LOGNO_M2b_IAR, PIII_M2a_AI_{K} and GA_M2a_AI_{K},
respectively, while in Xianyang the lowest AIC values for
modeling AM_{1}, AM_{7}, AM_{15} and AM_{30} are found in GEV_M2b_IAR,
GEV_M2b_IAR, PIII_M2b_IAR and GEV_M2b_IAR, respectively. These
results indicated that for explaining nonstationarity of low flow in Huaxian
station, IAR is the most dominant HA variable and AI_{K} is the most
dominant TCCCs variable, while in Xianyang the most dominant HA variable is
IAR and the most dominant TCCCs
variables causing nonstationarity in AM_{1}, AM_{7}, AM_{15} and
AM_{30} are *K*, AI_{EP}, AI_{EP} and *T*,
respectively.

Figure 8 shows the diagnostic assessment of the GA_M2 model
(with the optimal explanatory variable) for AM_{30} in both Huaxian and
Xianyang stations. The centile curve plots of GA_M2 (Fig. 8a and b)
show the observed values of AM_{30}, the estimated median and the
areas between the 5th and 95th centile. Figure 8a shows the response
relationship between AM_{30} and AI_{K} in Huaxian: the increase in
AI_{K} means the smaller magnitude of low-flow events because a high value
of AI_{K} (faster stream recession or fewer rainy days) may lead to faster
water loss or less supply. In Fig. 8b, the higher values of IAR means the
smaller magnitude of low-flow events, which suggests that IAR plays an
important role in driving low-flow generation in Xianyang. Figure 8c and d
show that the worm points are within the 95 % confidence intervals, thereby
indicating a good model fit and a reasonable model construction.

Figure 9 shows the AIC values of the stationary model (M0), time covariate model
(M1) and physical covariate models (M2a, M2b, M3, M4, M5 and M6) for AM_{30}. As shown in Fig. 9, M4 (nonstationary GA distribution with the optimal
TCCCs variables) has a good performance; after adding the HA variables, M6
with the lowest AIC value is attained; it can be found that the combination
of multiple TCCCs variables plays a major role in changing the low flows of
the Weihe River, but the influence of HA variables should not be ignored.

A summary of frequency analysis based on nonstationary GA distribution
AM_{30} is presented in Table 5. We choose to focus on M4, M5 and M6.
When only using TCCCs variables to model nonstationary low-flow frequency
distribution, the results of M4 show the optimal combination of explanatory
variables for all low-flow series contains more than three variables. For
example, for AM_{30} of Huaxian, the optimal combination of TCCCs
variables includes AI_{K}, BFI and AI_{EP}. When only HA variables
are used, the results of M5 show IAR is important to the low flows in this
area. And M4 has a better performance than M5. When using both TCCCs
variables and HA variables, the results of M6 show the optimal combination
contains multiple TCCCs variables and the irrigation area. For
Huaxian, the optimal combination of all explanatory variables is AI_{K},
IAR, BFI and AI_{EP}, while for Xianyang, the optimal combination is
IAR, AI_{EP} and BFI. We can also find that if two TCCCs variables
are highly correlated, they do not seem to be selected as the explanatory
variables at the same time. For example, in terms of air temperature (*T*),
evapotranspiration and the climate aridity index (AI_{EP}), only
one of them will appear in the optimal combination. This suggests that
multicollinearity problem in the multiple variables analysis can be reduced,
which will help obtain more reliable GLM parameters for contribution
analysis.

The diagnostic assessment of the GA_M6 model for AM_{30} at
two stations is presented by Fig. 10. The centile curve plots of
GA_M6 (Fig. 10a and b) show a more sophisticated
nonstationary modeling than GA_M2 (Fig. 8). When using
GA_M6 to model AM_{30} in Huaxian (Fig. 10a), similar to
GA_M2, the lower low flows are found to also correspond to
a higher value of AI_{K}, but GA_M6 is able to identify the
more complex variation patterns of low flows through the incorporation of
IAR, BFI and AI_{EP}. Figure 10c and d show that the data points of
worm plots of GA_M6 are almost within the 95 % confidence
intervals, thereby indicating an acceptable model fit and a reasonable model
construction.

Figure 11 presents the contribution of each selected explanatory variable to
$\mathrm{ln}\left({\mathit{\theta}}_{\mathrm{1}}^{t}\right)-\mathrm{ln}\left({\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{\mathrm{1}}\right)$ in the observation year based on GA_M6 for AM_{30} in
Huaxian and Xianyang. We can find that for Huaxian, the simulation value of
$\mathrm{ln}\left({\mathit{\theta}}_{\mathrm{1}}^{t}\right)$ frequently occurs below $\mathrm{ln}\left({\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{\mathrm{1}}\right)$ during the two periods of about 1970–1982
and 1993–2003, which is in accordance with the observed decrease in AM_{30} of Huaxian station during these periods. In the former period 1970–1982,
both AI_{K} and BFI contribute a lot of negative amount to $\mathrm{ln}\left({\mathit{\theta}}_{\mathrm{1}}^{t}\right)-\mathrm{ln}\left({\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{\mathrm{1}}\right)$,
whereas, during 1993–2003, the contribution of both AI_{K} and BFI
decreases significantly. However, IAR has almost equal negative contribution to
$\mathrm{ln}\left({\mathit{\theta}}_{\mathrm{1}}^{t}\right)-\mathrm{ln}\left({\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{\mathrm{1}}\right)$ in both periods. Unlike the former three variables, the significant
negative contribution of AI_{EP} is only found in 1993–2003. For AM_{30} of Xianyang, the contribution of IAR, AI_{EP} and BFI is similar to
that at Huaxian station in two periods; however, AI_{K} is not included in
the final model.

The impacts of both human activities and climate change on low flows of the study area led to time-varying climate and catchment conditions. Nonstationary modeling for annual low-flow series using TCCCs variables and/or HA variables as explanatory variables is clearly different from either the stationary model (M0) or the time covariate model (M1). The result demonstrates that considering multiple drivers (e.g., the variability in catchment conditions), especially in such an artificially influenced river, is necessary for nonstationary modeling of annual low-flow series.

In this study area, nonstationary modeling considering TCCCs is supported by
the following facts and findings. For human activities, an important
milestone representative is the completion and operation of the irrigation
system on the plateau in the Baoji Gorge irrigation district since 1971 (Sect. 3.1).
Figure 5c shows the change in irrigation area in this basin. And the
change-point detection test in Sect. 4.1 shows that significant change
points of low-flow series occur exactly at around 1971. This result
demonstrates that changes in AM_{30} may involve a consequence of this
project. In addition to human activities, climate change also makes a
considerable contribution to nonstationarity of low flows, as suggested by
nonstationary modeling using TCCCs variables with stepwise analysis.
Actually, climate driving pattern may strengthen after nearly 1990, which is
indicated by change-point detection test of both annual mean temperature
(*T*) and annual precipitation (*P*) as well as the behavior of annual
low-flow series after nearly 1990. There are two faster recession periods, the
1970s and the 1990s, as shown in Fig. 4. The reasons for the faster recession
are likely to be related to the above-mentioned project (e.g., the increasing
diversion for irrigation) and climate change (e.g., the intensified
evaporation) but also could be human alterations on catchment properties,
such as vegetation cover change. In conclusion, the temporal variability in
irrigation area, air temperature, precipitation (the frequency and volume of
rain events) and streamflow recession should be the main driving factors of
generating low-flow regimes in this basin. Overall, the causes of
nonstationarity in the category for two gauging stations have no clear
difference but have some differences in the relative importance. As shown
in Table 5, when modeling the low-flow series of Huaxian using TCCCs
variables, the optimal model (M4) preferred the variables that are related to
recession process; however, for Xianyang, the preferred variables are
related to temperature. The reason for this may be that, as a downstream
station, Huaxian station suffers more intensive human activity, so that the
importance of temperature change to the low-flow change is reduced meanwhile
the importance of streamflow recession (related to the capability of water
storage) change is enhanced. Ignoring the negative impacts of the errors in
estimating annual recession constants (*K*) which are caused by insufficient
data points of extracted stream segments at some wet years may lead to the
propagation of high errors in annual recession analysis and accordingly
affect the quality of nonstationary frequency analysis when *K* is used as
an explanatory variable. Further study will give a more reliable estimation of
*K* through the improvement of annual recession analysis. In addition, it should be noted
that the population recorded in the annals of statistics may not be equal
to the actual population living in the catchment. If the population in
the annals is used as the explanatory variable, this difference may lead to
the uncertainty of model parameter estimations. Nonetheless, it is the best
population data so far and the explanatory variable POP is excluded in the
final model (M6).

5 Conclusion

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There is an increasing need to develop an effective nonstationary low-flow
frequency model to deal with nonstationarities caused by climate change and
time-varying anthropogenic activities. In this study, time-varying climate
and catchment conditions in the Weihe River basin were measured by
annual time series of the eight indices, i.e., total precipitation (*P*),
mean frequency of precipitation events (*λ*), temperature (*T*),
potential evapotranspiration, climate aridity index (AI_{EP}),
base-flow index, recession constant (*K*) and the recession-related
aridity index (AI_{K}). The nonstationary distribution model was
developed using these eight TCCCs indices and/or three HA indices as
candidate explanatory variables for frequency analysis of time-varying
annual low-flow series caused by multiple drivers. The main driving forces
of the decrease in low flows in the Weihe River include reduced
precipitation, warming climate, increasing irrigation area and faster
streamflow recession. Therefore, a complex deterioration mechanism resulting
from these factors demonstrates that, in this arid and semi-arid area, the
water resources could be vulnerable to adverse environmental changes, thus
portending increasing water shortages. The nonstationary low-flow model
considering TCCCs can provide the knowledge of low-flow generation mechanism
and give a more reliable design of low flows for infrastructure and water
supply.

Data availability

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Data availability.

The data used in this paper can be requested by contacting the corresponding author Lihua Xiong at xionglh@whu.edu.cn.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

The study was financially supported by the National Natural Science
Foundation of China (NSFC grants 51525902 and 51479139) and projects from
State Key Laboratory of Water Resources and Hydropower Engineering Science,
Wuhan University. We greatly appreciate the editor and three reviewers for
their insightful comments and constructive suggestions that helped us to
improve the manuscript.

Edited by:
Fuqiang Tian

Reviewed by: Xiaohong Chen, Dengfeng Liu,

and one anonymous referee

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Short summary

In changing environments, extreme low-flow events are expected to increase. Frequency analysis of low-flow events considering the impacts of changing environments has attracted increasing attention. This study developed a frequency analysis framework by applying 11 indices to trace the main causes of the change in the annual extreme low-flow events of the Weihe River. We showed that the fluctuation in annual low-flow series was affected by climate, streamflow recession and irrigation area.

In changing environments, extreme low-flow events are expected to increase. Frequency analysis...

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