Many studies have shown that downstream flood regimes have been significantly altered by upstream reservoir operation. Reservoir effects on the downstream flow regime are normally performed by comparing the pre-dam and post-dam frequencies of certain streamflow indicators, such as floods and droughts. In this study, a rainfall–reservoir composite index (RRCI) is developed to precisely quantify reservoir impacts on downstream flood frequency under a framework of a covariate-based nonstationary flood frequency analysis using the Bayesian inference method. The RRCI is derived from a combination of both a reservoir index (RI) for measuring the effects of reservoir storage capacity and a rainfall index. More precisely, the OR joint (the type of possible joint events based on the OR operator) exceedance probability (OR-JEP) of certain scheduling-related variables selected out of five variables that describe the multiday antecedent rainfall input (MARI) is used to measure the effects of antecedent rainfall on reservoir operation. Then, the RI-dependent or RRCI-dependent distribution parameters and five distributions, the gamma, Weibull, lognormal, Gumbel, and generalized extreme value, are used to analyze the annual maximum daily flow (AMDF) of the Ankang, Huangjiagang, and Huangzhuang gauging stations of the Han River, China. A phenomenon is observed in which although most of the floods that peak downstream of reservoirs have been reduced in magnitude by upstream reservoirs, some relatively large flood events have still occurred, such as at the Huangzhuang station in 1983. The results of nonstationary flood frequency analysis show that, in comparison to the RI, the RRCI that combines both the RI and the OR-JEP resulted in a much better explanation for such phenomena of flood occurrences downstream of reservoirs. A Bayesian inference of the 100-year return level of the AMDF shows that the optimal RRCI-dependent distribution, compared to the RI-dependent one, results in relatively smaller estimated values. However, exceptions exist due to some low OR-JEP values. In addition, it provides a smaller uncertainty range. This study highlights the necessity of including antecedent rainfall effects, in addition to the effects of reservoir storage capacity, on reservoir operation to assess the reservoir effects on downstream flood frequency. This analysis can provide a more comprehensive approach for downstream flood risk management under the impacts of reservoirs.
River floods are generated by various complex nonlinear processes involving physical factors, including “hydrological pre-conditions (e.g., soil saturation, snow cover), meteorological conditions (e.g., amount, intensity, and the spatial and temporal distribution of rainfall), runoff generation processes, and river routing (e.g., superposition of flood waves in the main river and its tributaries)” (Wyżga et al., 2016). In general, without reservoirs, the downstream flood extremes of most rain-dominated basins are primarily related to extreme rainfall events in the drainage area. However, with reservoirs, the downstream flood regimes should be totally different due to upstream flood control scheduling. In the literature, the significant hydrological alterations caused by reservoirs have been demonstrated in the many areas of the world. Graf (1999) showed that dams have more significant effects on streamflow in America than global climate change. Benito and Thorndycraft (2005) reported various significant changes across the United States in pre-dam and post-dam hydrologic regimes (e.g., minimum and maximum flows over different durations). Batalla et al. (2004) demonstrated an evident reservoir-induced hydrologic alteration in northeastern Spain. Yang et al. (2008) demonstrated the spatial variability in hydrological regimes alterations caused by the reservoirs in the middle and lower Yellow River in China. Mei et al. (2015) found that the Three Gorges Dam, the largest dam in the world, has significantly changed downstream hydrological regimes. In recent years, the cause–effect mechanisms of downstream flood peak reductions were also investigated by some researchers (Ayalew et al., 2013, 2015; Volpi et al., 2018). For example, Volpi et al. (2018) suggested that for a single reservoir, the downstream flood peak reduction was primarily dependent on its position along the river, its spillway, and its storage capacity based on a parsimonious instantaneous unit-hydrograph-based model. These studies have revealed that it is crucial to assess the impacts of reservoirs on downstream flood regimes for the success of downstream flood risk management.
Flood frequency analysis is the most common technique used by hydrologists to gain knowledge of flood regimes. In conventional or stationary frequency analyses, a basic hypothesis is that hydrologic time series maintains stationarity; i.e., it is “free of trends, shifts, or periodicity (cyclicity)” (Salas, 1993). However, in many cases, observations of changes in flood regimes have demonstrated that this strict assumption is invalid (Kwon et al., 2008; Milly et al., 2008). Nonstationarity in downstream flood regimes of dams makes frequency analyses more complicated. Actually, the frequency of downstream floods of dams is closely related to upstream flood operations. In recent years, there have been many attempts to link flood-generating mechanisms and reservoir operations to the frequency of downstream floods (Gilroy and Mccuen, 2012; Goel et al., 1997; Lee et al., 2017; Liang et al., 2018; Su and Chen, 2019; Yan et al., 2017).
Previous studies have meaningfully increased the knowledge about reservoir-induced nonstationarity of downstream hydrological extreme frequencies (Ayalew et al., 2013; López and Francés, 2013; Liang et al., 2018; Magilligan and Nislow, 2005; Su and Chen, 2019; Wang et al., 2017; Zhang et al., 2015). There are two main approaches to incorporate reservoir effects into flood frequency analyses: the hydrological model simulation approach and the nonstationary frequency modeling approach. In the first approach, the regulated flood time series can be simulated using three model components: the stochastic rainfall generator, the rainfall-runoff model, and the reservoir flood operation module, which includes the reservoir storage capacity, the size of release structures, and the operation rules. The continuous simulation method can explicitly account for the reservoir effects on floods in the hypothetical case. However, it is difficult to apply this approach to a majority of real cases (Volpi et al., 2018) because the simplifying assumptions of this approach are only satisfied in a few of basins with single small reservoirs. Furthermore, even if the basins meet the simplifying assumptions, the detailed information required in this approach is likely unavailable. Thus, our attention is focused on the second method, the nonstationary frequency modeling approach. Nonstationary distribution models have been widely used to deal with the nonstationarity of extreme value series. In nonstationary distribution models, the distribution parameters are expressed as the functions of covariates to determine the conditional distributions of extreme value series. According to extreme value theory, the maximum series can generally be described using the generalized extreme value (GEV) distribution. Thus, previous studies (El Adlouni et al., 2007; Ouarda and El-Adlouni, 2011) have used the nonstationary generalized extreme value distribution to describe the nonstationary maximum series. Scarf (1992) modeled the changes in the location and scale parameters of the GEV over time using the power function relationship. Coles (2001) introduced several time-dependent structures (e.g., trend, quadratic, and change point) into the location, scale, and shape parameters of the GEV. El Adlouni et al. (2007) provided a general nonstationary GEV model with an improved parameter estimate method. In recent years, “generalized additive models for location, scale, and shape” (GAMLSSs) have been widely used in nonstationary hydrological frequency analyses (Du et al., 2015; Jiang et al., 2014; López and Francés, 2013; Rigby and Stasinopoulos, 2005; Villarini et al., 2009). The GAMLSS provides various candidate distributions for frequency analysis, such as Weibull, gamma, Gumbel, and lognormal distributions. However, the GEV has rarely been involved in the candidate distributions of GAMLSSs. In terms of a parameter estimation method for the nonstationary distribution model, the maximum likelihood (ML) method is the most common parameter estimate method. However, the ML method for a nonstationary distribution model can lead to very high quantile estimator variances when using numerical techniques to solve the likelihood function when using a small sample (El Adlouni et al., 2007). El Adlouni et al. (2007) developed the generalized maximum likelihood (GML) method and demonstrated that the GML method had better performance than the ML method in all their cases. Ouarda and El-Adlouni (2011) introduced the Bayesian nonstationary frequency analysis. The Bayesian inference can obtain multiple estimates, forming a posterior distribution of model parameters. Thus, the Bayesian method is able to conveniently describe the uncertainty of flood estimates associated with the uncertainty of model parameters.
In the nonstationary frequency modeling approach, a dimensionless reservoir index (RI) was proposed by López and Francés (2013) as an indicator of reservoir effects, and it generally is used as a covariate for the expression of the distribution parameters (e.g., location parameter; Jiang et al., 2014; López and Francés, 2013). Liang et al. (2018) modified the reservoir index by replacing the mean annual runoff in the expression of the RI with the annual runoff. Therefore, the modified reservoir index can reflect the impact of reservoirs on downstream flood extremes under various total inflow conditions each year. However, the precision and accuracy in the quantitative analysis of the reservoir effects on downstream floods need to be further improved. In fact, the effects of reservoirs may be closely related not only to the static reservoir storage capacity but also to the dynamic reservoir operations associated with multiple characteristics, such as the peak, the intensity, and the total volume of the multiday antecedent rainfall input (MARI) and not just annual runoff.
Therefore, the aim of the study is to develop an indicator, referred to as the rainfall–reservoir composite index (RRCI), that combines the effects of reservoir storage capacity and the MARI on reservoir operation. This indicator is then used as a covariate to assess the reservoir effects on the downstream flood frequency. The specific objectives of this study are (1) to develop the RRCI, (2) to compare the RRCI with the RI using a covariate-based nonstationary flood frequency analysis, and (3) to obtain the downstream flood estimation and its uncertainty based on the optimal nonstationary distribution using the Bayesian inference.
To quantify the effects of reservoirs on the frequency of the annual maximum daily flow (AMDF) series downstream of reservoirs, a three-step framework (Fig. 1), termed the covariate-based flood frequency analysis using the RRCI as a covariate, was established. In this section, the methods of this framework are introduced. First, a RI is defined by additionally considering the effects of reservoir sediment deposition on the storage capacity. Second, the RRCI is developed by combining the RI and a rainfall index. Next, the C-vine copula model is used to construct and calculate the rainfall index. Finally, the nonstationary distribution models that utilize the Bayesian estimation are explained.
Flowchart of the nonstationary covariate-based flood frequency analysis using the rainfall–reservoir composite index (RRCI).
Intuitively, the larger the reservoir capacity relative to the flow of a
downstream gauging station, the greater the possible effects of the
reservoir on the streamflow regime. To quantify reservoir-induced
alterations to the downstream streamflow regime, Batalla et al. (2004)
proposed an impounded runoff index (IRI), which is a ratio of reservoir
capacity (
In addition to the reservoir capacity, the MARI, which is an event of continuous multiday multivariate
rainfall that forms the inflow event that will be regulated by the reservoir
system to become the downstream extreme flow, is a key constraint for
scheduling the reservoir system. In this study, to add the antecedent
rainfall effects into the new indicator of reservoir effects, five variables
were used to describe the MARI: the maximum
A new index is proposed in this study, the RRCI, to more comprehensively assess the effects of
reservoirs on floods by incorporating the effects of the MARI. This index is
defined as
Relationship in Eq. (2).
In this subsection, a C-vine Copula model for the construction of the
continuous
Decomposition of a C-vine copula using four variables and three trees (denoted by T1, T2, and T3).
The covariate-based extreme frequency analysis has been widely used (Villarini et al., 2009; Ouarda and El-Adlouni, 2011; López and Francés, 2013; Xiong et al., 2018). According to these studies, five distributions, namely the gamma (GA), Weibull (WEI), lognormal (LOGNO), Gumbel (GU), and the generalized extreme value (GEV) distribution, were used as candidate distributions in this study. In addition, their density functions, the corresponding moments, and the used link functions are shown in Table 1. In the following, the nonstationary distribution models based on Bayesian estimation are developed for a covariate-based flood frequency analysis.
Summary of the probability density functions, the corresponding moments, and the used link functions for nonstationary flood frequency analysis.
Suppose that the flood variable,
Seven nonstationary scenarios for the formulas of the two
distribution parameters (i.e.,
In the following, the Bayesian inference is introduced. The
GEV_S23 (representing the nonstationary GEV distribution with
the S23 scenario) model was used as an example, and the model parameter
vector
The procedure of model selection can identify which of the five
distributions is optimal and which of the seven nonstationary scenarios is
optimal. If all the distribution parameters are identified as constants
(S0), this process will be a stationary frequency analysis. To select the
optimal model, the Schwarz Bayesian criterion (SBC; Schwarz, 1978) for each
fitted model object is calculated by the following:
The Han River (Fig. 4), with the coordinates of 30
Geographic location of the reservoirs, gauging stations, and rainfall stations along the Han River.
Information of the five major reservoirs in the Han River basin.
The assessment analysis of reservoir effects on flood frequency utilized
streamflow data, reservoir data, and rainfall data. The AMDF series was extracted from the daily streamflow records of the
three gauges in the Han River basin; namely the Ankang (AK) station
with a drainage area of 38 600 km
To confirm the impact of reservoirs on the AMDF
in the study area, the mean and standard deviation of the AMDF before and
after the construction of the two large reservoirs, the Danjiangkou
Reservoir (1967) upstream of the HJG and HZ stations and the Ankang Reservoir (1992) upstream of the AK, HJG, and HZ stations, were compared.
According to Table 4, the mean and standard deviation of the AMDF of the AK,
HJG, and HZ stations were significantly reduced. By using the HJG station as
an example, the mean of the AMDF (1992–2013) is 4139 m
Change in the mean and standard deviation of the AMDF after the construction of the two large reservoirs (Danjiangkou Reservoir, completed by 1967, and the Ankang Reservoir, finished by 1992).
Linear correlation between the five MARI variables and the AMDF
for
Figure 5 presents the linear correlation between the five MARI variables
(i.e., the maximum,
To obtain the annual values of the RRCI, the RI was estimated first. The RI was affected by the loss of the reservoir capacity but not to a great extent (Fig. S2). This happened because the main reservoirs (Danjiangkou and Ankang reservoirs) had a small loss rate of no more than 15 % (Table S1 and Fig. S1).
Correlation coefficients between the RRCI and the AMDF. The values in bold font are the highest absolute values of Pearson, Kendall, or Spearman correlation coefficient for the station.
The C-vine copula model was applied to calculate the OR-JEP of the
scheduling-related MARI variables. In the modeling of the univariate
marginal, the marginals of the intensity (
Results of the copula models for scheduling-related rainfall variables.
Table 6 shows the results of the copula modeling of the scheduling-related
variables using the aid of the R package “VineCopula”
(
Variation of the RI and the RRCI for
A nonstationary flood frequency analysis using the RRCI or the RI as the
covariate was performed to investigate how the reservoirs affected the
downstream flood frequency. A summary of results of fitting the
nonstationary models to the flood data is shown in Table 7. Based on the
SBC, the lowest values indicate that the best models for the AK, HJG, and HZ
stations are the nonstationary WEI distribution with S23, the nonstationary
GA distribution with S21, and the nonstationary WEI distribution with S21,
respectively, hereafter referred to as WEI_S23,
GA_S21, and WEI_S21, respectively. Note that
for any one of the five distributions (GA, WEI, LOGNO, GU, and GEV), the
RRCI-dependent scenario had a lower SBC value than the RI-dependent scenario
for each gauging station. Furthermore, for the RI-dependent and
RRCI-dependent scenarios, using the HZ station as an example, the optimal
formulas of the two distribution parameters, WEI_S11 WEI_S21
It was found that in Eqs. (13) and (14), there were negative estimates
of
Summary of the results of the nonstationary flood distribution models. The value in bold font is the lowest SBC value for the station.
Figure 7 compares the stationary scenario (S0), the RI-dependent scenario (S1), and the RRCI-dependent scenario (S2) of the same optimal distributions that explain all the flood values and the several largest flood values for each station. The Q–Q plots (Fig. 7a1–c1) show that overall, the RRCI-dependent scenario more adequately captured the entire empirical quantiles (particularly the smallest and largest empirical quantiles) than the two other scenarios for each station. Furthermore, as shown in Fig. 7a2–c2, for the seven largest floods (observed) of each station, the RRCI-dependent scenario produced lower quantile residuals than the two other scenarios.
Comparison of the stationary (S0), the RI-dependent (S1), and the
RRCI-dependent (S2) scenarios of the same optimal distributions for
Performance of
Figure 8 shows the performance of the best models: WEI_S23
for the AK station, GA_S21 for the HJG station, and
WEI_S21 for the HZ station. The points in the worm plots in
Fig. 8 are within the 95 % confidence interval, indicating that the
selected models are reasonable. In addition, according to the centile-curve
plots in Fig. 8, the AMDF series is well fitted by the best models.
Undoubtedly, with the incorporation of the effects of the MARI, the
RRCI-dependent scenario captured the presence of nonstationarity in the
downstream flood frequency well. The case of the HZ station was used for the
analysis (Fig. 8c1). After the construction of the Danjiangkou Reservoir (1967), due to reservoir operation, most of the values of the AMDF had been
reduced in magnitude by this reservoir. However, some relatively large flood
events still occurred several times, such as 25 600 m
Statistical inference of the 100-year return levels with a 95 %
uncertainty interval using the optimal RI-dependent and the RRCI-dependent
distributions:
The 100-year return levels at a 95 % credible interval from WEI_S23 and WEI_S13 for the AK station, GA_S21 and GA_S11 for the HJG station, and WEI_S21 and WEI_S11 for the HZ station are presented in Fig. 9. For each station, compared to the optimal RI-dependent distribution, the optimal RRCI-dependent distribution provided a lower 100-year return level. However, exceptions existed. In addition, after the construction of the main reservoir, the uncertainty range of the AK station was larger than that of the HJG and HZ stations. A possible explanation for the larger uncertainty range was that the sample size (1993–2015) of the regulated floods at the AK station was smaller. Furthermore, the dependent relationship between the RRCI and the AMDF at the AK station was weaker.
The long-term variation in the AMDF series (Fig. 8) indicates that the
upstream reservoirs had evidently altered the downstream flood regimes. As
an example, since the completion of the Danjiangkou Reservoir in 1967, the
flood magnitude of the HZ station was evidently reduced overall. This is
consistent with the results of the effects of reservoirs on the hydrological
regime in this area found in previous studies (Cong et al., 2013; Guo et
al., 2008; Jiang et al., 2014; Lu et al., 2009). In this study, it was found
that there was a significant difference between downstream floods affected
by the same reservoir system (with the same RI value). In most cases,
relatively small downstream floods were obtained. However, it is of interest
to note that unexpected large downstream floods still occurred in a
few cases in spite of a large RI value. For example, most values of the
AMDF in the HZ station have been less 10 000 m
Summary of the rainfall information for the five largest floods after the construction (1967) of the Danjiangkou Reservoir in the HZ station.
With the combination of both the RI and the OR-JEP, the RRCI had a significant difference from RI (Fig. 6). With a few exceptions, the RRCI values were higher than the RI values. This indicates that the real reservoir impact may be underestimated by the RI in most cases. Moreover, the RI will also probably overestimate the real reservoir impact in a few cases because of not considering special rainfall events (i.e., the MARI with low values of the OR-JEP). The results of the covariate-based nonstationary flood frequency analysis (Table 7 and Figs. 7 and 8) demonstrate that, compared to the RI-dependent scenario, the RRCI-dependent scenario for the optimal nonstationary distribution more completely captured the presence of nonstationarity in the downstream flood frequency. Therefore, the RRCI might be a useful index for accessing the reservoir effects on downstream flood frequency.
Finally, the estimation errors of the OR-JEP should be noted. (1) Only those MARI samples that corresponded to the timing of the AMDF were included to estimate the OR-JEP. This means that some extreme MARI samples that corresponded to the non-maximum flow were not included, resulting in an estimation error for the OR-JEP. To reduce this error, it might be worth considering the use of the peaks-over-threshold sampling method. (2) The areal-averaged MARI was based on the records from 16 rainfall stations using the IDW method. The estimation error of the areal-averaged rainfall can be transferred to the OR-JEP estimation error. Additional rainfall site data and spatial distribution information were needed to reduce the OR-JEP estimation error. Nonetheless, the good performance of the downstream flood frequency model results demonstrated that the MARI samples still remained representative in this study.
Accurately assessing the impact of reservoirs on downstream floods is an important issue for flood risk management. In this study, to evaluate the effects of reservoirs on the downstream flood frequency of the Han River, the rainfall–reservoir composite index (RRCI) was derived from Eq. (2), which considers the combination of the reservoir index (RI) and the OR joint exceedance probability (OR-JEP) of scheduling-related rainfall variables. The main findings are summarized as follows. (1) The magnitude of the downstream flood events has been reduced by the reservoir system in the study area. However, the long-term variation in the observed AMDF series showed that despite the large reservoirs, unexpected large flood events still occurred several times, such as at the Huangzhuang station in 1983. One important cause of the unexpected large floods at the Huangzhuang station may have been related to the operation strategy of staged increases in the flood limit water level of the Danjiangkou Reservoir. (2) According to the results of the covariate-based nonstationary flood frequency analysis for each station, compared to the optimal RI-dependent distribution, the optimal RRCI-dependent distribution more completely captured the presence of nonstationarity in the downstream flood frequency. (3) Furthermore, in estimating the 100-year return level for each station, the optimal RRCI-dependent distribution provided a lower 100-year return level, but exceptions existed. In addition, it provided a smaller uncertainty range associated with the uncertainty of the model parameter.
Consequently, this study demonstrated the necessity of including the antecedent rainfall effects, in addition to the effects of reservoir storage capacity, on reservoir operation to assess the reservoir effects on downstream flood frequency. This study provides a comprehensive approach for downstream flood risk management under the impacts of reservoirs.
The data used in this paper can be requested by contacting the corresponding author Lihua Xiong at xionglh@whu.edu.cn.
The supplement related to this article is available online at:
BX and LX conceived and designed the study. LX, JX, and CYX provided funding. BX, LX, and CJ provided the methodology. BX and LX conducted case analysis. BX wrote the original draft. LX, JX, CYX, and TD reviewed and edited the paper. All authors read and approved the paper.
The authors declare that they have no conflict of interest.
We greatly appreciate the editor and the two reviewers for their insightful comments and constructive suggestions for improving the paper.
This research has been supported by the National Natural Science Foundation of China (grant nos. 41890822 and 51525902), the Research Council of Norway (FRINATEK project 274310), and the “Plan 111” Fund of the Ministry of Education of China (grant no. B18037).
This paper was edited by Jim Freer and reviewed by two anonymous referees.