## ABSTRACT

Instream ecological flow (IEF) provides a flow reference value for maintaining the stability of river ecosystems and enriching biodiversity. Existing methods for determining the IEF often consider only one of the intra-annual runoff distribution characteristics or the inter-annual runoff distribution characteristics. In this study, a copula-based ecological runoff determination method is proposed to consider both annual and monthly runoff magnitudes. The marginal distribution functions of annual and monthly runoff are first determined using the maximum likelihood method. Then the copula function is used to construct the joint distribution function of annual and monthly runoff series. Using the flow duration curve method, the IEF under different annual and monthly runoff exceedance probabilities is calculated. The probability values are combined with the initial IEF values to determine the final IEF process. A case study is conducted using real-world data from the Tongtian River in China. The results show that the total annual basic ecological water demands of the Tongtian River in abundant, flat, and dry water years are 138, 102 and 72.2 billion m^{3}. The proposed method effectively avoids the disturbance of extreme runoff.

## HIGHLIGHTS

An innovative ecological runoff determination method is proposed to consider both annual and monthly runoff magnitudes.

Copula functions are used to construct the joint distribution function of annual and monthly runoff series.

A case study is conducted using real-world data of the Tongtian River in China.

## INTRODUCTION

Instream ecological flow (IEF) is the appropriate flow required to maintain the health of river ecosystems and ensure human survival and development (Poff 2018). Globally, typical cases abound of unmet ecological water needs leading to prominent ecological problems or ecological disasters in regional economies, societies and environments (Absalon *et al.* 2023; Xu *et al.* 2023).

The determination of IEFs in rivers is a prerequisite and basis for water resource utilization and protection. In the 1970s, a class of methods for determining IEF based on the analysis of historical hydrological data was proposed. Tennant (1976) studied the relationship between river flow and river habitat on a number of rivers in the Midwestern United States. The year was divided into two periods, normal water use and fish spawning, based on the timing of fish spawning, and corresponding IEF ranges were set for each.

Following the hydrological methods, a number of hydrodynamic methods and habitat modeling methods emerged in the 1980s. Hydraulic methods determine the IEF by establishing the relationship between hydraulic elements such as flow velocity, water depth, wetted perimeter, river width, and river flow. Commonly used methods include the wetted perimeter method (Sedghi-Asl & Poursalehan 2024) and the R2Cross (Oikonomou *et al.* 2021). The advantage of the hydraulic methods is that their data can be obtained by simple field tests. However, such methods do not reflect seasonal variation factors and are therefore not usually used for seasonal rivers but can provide a hydraulic basis for other methods. The habitat modeling method uses eco-hydrological relationships to determine the IEF of a river, for example, by establishing the relationship between the area response of the habitats of target aquatic ecosystem species (fish, etc.) under different flow rates to determine the IEF. The most representative methodology is the Instream Flow Incremental Methodology, which was proposed in the United States in the early 1980s. This methodology is used for assessing the impacts of water resource development, utilization, and management on aquatic organisms and off-channel ecosystems. Its main purpose is to address water resource management issues (Petts 2009). However, the quantitative biological information required by this method is difficult to obtain and focuses only on the conservation of some specific species rather than the whole river ecosystem. Thus, the application of this method is somewhat limited. In addition, the weighted usable area method (Park *et al.* 2023) and River2D (Li *et al.* 2024) are typical habitat-modeling methods.

In order to maintain the integrity of river ecosystems, scholars have begun to analyze ecological water demand in terms of the holistic nature of ecosystems, to take biological information into account when calculating IEFs, and to establish a habitat approach to the relationship between river hydrological–hydrodynamic parameters and biological growth states. Mathews & Bao (1991) developed a method in Texas that combines regional hydrological characteristics with different biological characteristics to determine IEFs by means of monthly flows at certain assurance rates. With the development of theories such as the river continuum theory and the flood pulse theory, holistic methods that emphasize the overall flow demand of the ecosystem have gradually been born since the late 1990s. The building block methodology (BBM) effectively maintains the health of river ecosystems by dividing the river flow and setting the corresponding flow based on four conditions (Hughes 1999). The method emphasizes that the river is an integrated ecosystem and is based on maintaining the natural condition of the river's water. It studies the relationship between the river flow, sediment transport, riverbed shape, and riparian zone communities and also observes changes in the size of the flow and the corresponding changes in the river ecosystem over a long period of time. The whole process of determining IEFs requires the participation of experts from different disciplines, and the procedure is complex and difficult to use.

On the basis of determining the IEF, further research has been carried out on the use of water resources and ecosystem protection, including research on the relationship between river IEF and fish habitat, research on the relationship between river IEF, aquatic organisms, and water quality, and research on the optimized allocation of reservoir scheduling taking into account the IEF. In recent years, the study of ecosystem water demand has begun to evolve into a combination of modeling and remote-sensing methods. For example, Chimtengo *et al.* (2014) analyzed the application of traditional water allocation schemes in the Rivirivi River Basin in Malawi, as well as the changes in river flow regimes before and after dam construction and the reasons for this. Remote-sensing techniques were used to analyze land-use changes and extrapolate their potential impact on water resources, and modeling techniques were used to estimate ecosystem water demand. The determination of IEF is the basis of these studies, and the development of scientifically sound IEF targets for rivers is of great significance.

To date, more than 200 IEF determination methods have been developed, among which hydrological methods have been the most widely used due to their advantages of low data-requirements, simplicity of operation, and low cost (Chen *et al.* 2023). However, existing hydrological methods tend to consider only one of the intra-annual runoff distribution characteristics or inter-annual runoff distribution characteristics, while ignoring the other. For example, the Tennant method qualitatively characterizes habitat into eight classes, from very poor to maximum, based on a specific percentage of mean annual flow (Srivastava & Maity 2023). However, this method does not characterize monthly-scale changes in IEFs and applies to large, year-round rivers and not to seasonal rivers (Wang *et al.* 2022). The variable monthly flow (VMF) method divides the year into abundant, flat, and dry periods and determines the IEF by assigning the percentage of mean monthly flow. The VMF method reflects the intra-annual variation of IEF demand in rivers, especially enhancing the protection of river ecosystems during dry periods, but does not take into account inter-annual variation in runoff. There is an urgent need to propose an innovative IEF determination approach to account for inter- and intra-annual variability in river runoff.

Therefore, this paper proposes a copula-based IEF determination method that takes into account inter- and intra-annual runoff variability from the ecological objective of maintaining the natural hydrological condition of the river. The marginal distribution functions of annual and monthly runoff are first determined using the maximum likelihood method. Then the copula function is used to construct the joint distribution function of annual and monthly runoff series. Using the flow duration curve (FDC) method, the IEF under different annual and monthly runoff exceedance probabilities is calculated. The probability values are combined with the initial IEF values to determine the final IEF process. A case study is conducted using real-world data of the Tongtian River in China. The proposed approach provides flow reference values for maintaining the structural stability of river ecosystems and enriching biodiversity.

## MARGINAL DISTRIBUTION FUNCTION CONSTRUCTION

The Pearson Type III (P-III), Log-Normal (LOGN), Generalized Extreme Value (GEV), Weibull, and Logistic are univariate probability distributions that are commonly used in hydrological frequency analysis. These distributions are used as candidate marginal distributions, and parameter estimation is performed using maximum likelihood estimation (Wang *et al.* 2023) for the annual runoff series and the monthly runoff series for each month.

A goodness-of-fit test is then performed on the annual runoff series and the monthly runoff series. The Kolmogorov–Smirnov (K–S) test is used to test the degree of fit. The candidate distribution with the highest degree of fit is selected as the marginal function distribution of the annual flow series and monthly flow series for each month.

## COPULA FUNCTION CONSTRUCTION

A copula function can construct the joint distribution of multi-variables and analyze the correlation between variables. There are a number of copula function types. The most widely used in the field of hydrology and water resources are the Archimedean copula, meta-elliptic copula and empirical copula. The commonly used copula functions are used as candidate joint distribution functions for annual and monthly runoff series. Parameter estimation and goodness-of-fit tests are then performed. The root mean square error (RMSE) is used to analyze and determine the optimized copula function.

### Copula function types

#### Archimedean copula

Archimedean copula function structure is simple. The constructed joint distribution function form is diverse and adaptable. It thus occupies an important position in practical applications. As an example, several copula function forms commonly used in Archimedean copula functions are introduced for two-dimensional distributions:

#### Meta-elliptic copula

The meta-elliptic copula is derived from the elliptical distribution and is an extension of the multidimensional normal distribution. It can fit the multivariate extreme value distribution and non-normal distribution better. As an example, the common form of a two-dimensional distribution is as follows:

#### Empirical copula

*X*and

*Y*as and , respectively, then the empirical copula function of the sample is as follows:where ;

*I*is a function, when and otherwise.

### Parameter estimation

Once the marginal distributions of annual and monthly runoff are determined, copula functions are used to connect the marginal distributions of each variable in order to construct the joint distribution function. The maximum likelihood method is employed to estimate the respective parameter values of the candidate copula functions separately.

### Selection of copula function

*t*copula, are utilized to construct the joint probability distribution of annual and monthly flow series. The closeness of the constructed joint distribution to the empirical copula is compared based on the RMSE. The optimized copula function is determined as a joint probability distribution function for the annual and monthly runoff series with the objective of minimizing the RMSE between the theoretical and empirical joint probability distributions. The equations for the RMSE of the five alternative copula functions are as follows:where , , ; , ,, are the RMSE of the Gumbel–Hougaard copula, Frank copula, Clayton copula, Gaussian copula, and Student's

*t*copula.

## DETERMINATION OF IEF

### Grouping of annual and monthly runoff series

Annual and monthly runoff series are grouped according to the probability of runoff volume exceedance. Based on the inter- and intra-annual variability characteristics of the study river, the annual runoff series is divided into *p* groups and the monthly runoff series into *q* groups. For example, annual and monthly runoff series can be divided into abundant water year group, flat water year group, and dry water year group based on the boundaries of 25% and 75% exceedance probability. If river runoff is highly variable, the annual and monthly runoff series can be divided into more groups for more detailed characterization, e.g., into ten groups bounded by 10%, 20%, … , 90% probability of exceedance. According to the actual hydrological characteristics of the study river, the annual runoff group *p* and the monthly runoff group *q* may not be equal.

### Calculation of initial IEF

The FDC method (Guo *et al.* 2021) is used to determine the initial IEF. Specifically, the FDC for each month within each monthly runoff group is constructed, and the FDC method is applied to take the flow corresponding to the 90% (or other) exceeding probability of the FDC as the basic IEF, so as to determine the IEF for each month within each group.

### Calculation of conditional probability

*m*= 1, 2, … , 12, denoting 12 months;

*mg*= 1, 2, … ,

*q*, denoting monthly-scale groupings;

*yg*= 1, 2, … ,

*p*, denoting yearly-scale groups and is the probability that annual runoff is in the

*yg*th group.

### IEF considering annual and monthly runoff variability

*m*th month under the

*yg*th annual runoff group and is the initial IEF of the

*m*th month under the

*mg*th monthly runoff group.

## CASE STUDY

The proposed method was applied to the Tongtian River in China to determine its IEF.

### Study area

Located in the hinterland of the Qinghai-Tibet Plateau, Sanjiangyuan is the birthplace of the Yangtze, Yellow and Lancang Rivers, and is known as the ‘Water Tower of China’. The ecosystem of the Sanjiangyuan region is very fragile due to the special geographic environment and climatic conditions (Sun *et al.* 2023). The Tongtian River is the uppermost main river of the Sanjiangyuan region between 90°E– 98°E and 32°N–36°N (as shown in Figure 1). It is also one of the water sources for the South-to-North Water Diversion Western Route Project. The Tongtian River has abundant water resources, with a multi-year average flow of 424 m^{3}/s (Zhang *et al.* 2021a). Its main stream is 769 km long with a large drop, and the river has a natural drop of 953 m. The river is also rich in water energy resources, with a theoretical reserve of 2,668 MW. According to the Tongtian River hydropower development plan, eight hydropower stations will be constructed to form a cascade hydropower system, with a total installed capacity of 2,838 MW and an annual power output of 12,832 million kWh (Zhang *et al.* 2021b). Hydropower development needs to give full consideration to the ecological environment protection of the Tongtian River, and ensuring its IEF demand is the most basic requirement. In this paper, the Tongtian River is used as a study case to determine its IEF using the proposed method, which provides a flow reference value for maintaining the stability of the river ecosystem and enriching biodiversity.

### Marginal distribution function construction

#### Marginal distribution function

Runoff data for the Tongtian River from 1957 to 2010 were organized into monthly runoff and annual runoff series. Candidate distribution functions (i.e., P-III, LOGN, GEV, Weibull, and Logistic) were used to fit the annual and monthly runoff series of the Tongtian River. The parameter values of each candidate marginal function were determined using the maximum likelihood estimation method, and the five probability distribution functions were tested using the K–S test. The results of the test are listed in Table 1. The *D*-value is the statistic of the K–S test. The *P*-value, in contrast to fixed significance-level *α* values, is calculated based on the test statistic and denotes the threshold value of the significance level in the sense that the null hypothesis will be accepted for all *α* values less than the *P*-value. For example, if *P* = 0.90, the null hypothesis will be accepted at all significance levels less than *P* (i.e., 0.75 and 0.80) and rejected at higher levels, including 0.95 and 0.99. Therefore, the *P*-value can be used as an indicator to select the best-fitting marginal distribution function. For each runoff series, the distribution function with the largest *P*-value among the candidate distribution functions was selected as the marginal distribution function for that runoff series. The GEV was selected as the marginal distribution function for the annual runoff series and also for the March, and August runoff series. The LOGN was selected as the marginal distribution function for the February, April, May, June, July, October, and November runoff series. The Weibull was selected as the marginal distribution function for the January, September, and December runoff series.

Runoff series . | P-III . | LOGN . | GEV . | Logistic . | Weibull . | |||||
---|---|---|---|---|---|---|---|---|---|---|

D
. | P
. | D
. | P
. | D
. | P
. | D
. | P
. | D
. | P
. | |

Annual | 0.071 | 0.931 | 0.067 | 0.953 | 0.066 | 0.961 | 0.104 | 0.570 | 0.073 | 0.913 |

Jan. | 0.088 | 0.769 | 0.074 | 0.908 | 0.067 | 0.953 | 0.074 | 0.904 | 0.066 | 0.960 |

Feb. | 0.083 | 0.821 | 0.053 | 0.996 | 0.053 | 0.996 | 0.090 | 0.739 | 0.056 | 0.993 |

Mar. | 0.087 | 0.771 | 0.087 | 0.775 | 0.084 | 0.808 | 0.110 | 0.496 | 0.093 | 0.703 |

Apr. | 0.096 | 0.663 | 0.095 | 0.679 | 0.100 | 0.614 | 0.122 | 0.370 | 0.113 | 0.460 |

May | 0.066 | 0.960 | 0.065 | 0.967 | 0.067 | 0.954 | 0.142 | 0.204 | 0.072 | 0.920 |

Jun. | 0.064 | 0.968 | 0.057 | 0.991 | 0.057 | 0.991 | 0.115 | 0.436 | 0.080 | 0.854 |

Jul. | 0.076 | 0.888 | 0.065 | 0.983 | 0.063 | 0.975 | 0.132 | 0.282 | 0.080 | 0.855 |

Aug. | 0.068 | 0.948 | 0.069 | 0.944 | 0.063 | 0.976 | 0.083 | 0.824 | 0.073 | 0.073 |

Sep. | 0.091 | 0.724 | 0.095 | 0.679 | 0.089 | 0.758 | 0.127 | 0.320 | 0.086 | 0.784 |

Oct. | 0.089 | 0.755 | 0.085 | 0.799 | 0.088 | 0.766 | 0.087 | 0.779 | 0.101 | 0.101 |

Nov. | 0.067 | 0.954 | 0.060 | 0.985 | 0.064 | 0.969 | 0.113 | 0.459 | 0.081 | 0.843 |

Dec. | 0.101 | 0.604 | 0.082 | 0.832 | 0.083 | 0.824 | 0.114 | 0.451 | 0.081 | 0.844 |

Runoff series . | P-III . | LOGN . | GEV . | Logistic . | Weibull . | |||||
---|---|---|---|---|---|---|---|---|---|---|

D
. | P
. | D
. | P
. | D
. | P
. | D
. | P
. | D
. | P
. | |

Annual | 0.071 | 0.931 | 0.067 | 0.953 | 0.066 | 0.961 | 0.104 | 0.570 | 0.073 | 0.913 |

Jan. | 0.088 | 0.769 | 0.074 | 0.908 | 0.067 | 0.953 | 0.074 | 0.904 | 0.066 | 0.960 |

Feb. | 0.083 | 0.821 | 0.053 | 0.996 | 0.053 | 0.996 | 0.090 | 0.739 | 0.056 | 0.993 |

Mar. | 0.087 | 0.771 | 0.087 | 0.775 | 0.084 | 0.808 | 0.110 | 0.496 | 0.093 | 0.703 |

Apr. | 0.096 | 0.663 | 0.095 | 0.679 | 0.100 | 0.614 | 0.122 | 0.370 | 0.113 | 0.460 |

May | 0.066 | 0.960 | 0.065 | 0.967 | 0.067 | 0.954 | 0.142 | 0.204 | 0.072 | 0.920 |

Jun. | 0.064 | 0.968 | 0.057 | 0.991 | 0.057 | 0.991 | 0.115 | 0.436 | 0.080 | 0.854 |

Jul. | 0.076 | 0.888 | 0.065 | 0.983 | 0.063 | 0.975 | 0.132 | 0.282 | 0.080 | 0.855 |

Aug. | 0.068 | 0.948 | 0.069 | 0.944 | 0.063 | 0.976 | 0.083 | 0.824 | 0.073 | 0.073 |

Sep. | 0.091 | 0.724 | 0.095 | 0.679 | 0.089 | 0.758 | 0.127 | 0.320 | 0.086 | 0.784 |

Oct. | 0.089 | 0.755 | 0.085 | 0.799 | 0.088 | 0.766 | 0.087 | 0.779 | 0.101 | 0.101 |

Nov. | 0.067 | 0.954 | 0.060 | 0.985 | 0.064 | 0.969 | 0.113 | 0.459 | 0.081 | 0.843 |

Dec. | 0.101 | 0.604 | 0.082 | 0.832 | 0.083 | 0.824 | 0.114 | 0.451 | 0.081 | 0.844 |

*Note:* The maximum *P*-value and the corresponding *D*-value for the annual runoff and for each monthly runoff are highlighted in bold, indicating that the function is the best marginal distribution function.

#### Annual and monthly runoff series correlation

After determining the respective marginal distribution functions of the annual and monthly runoff series, the correlation between the two variables was tested by calculating Kendall's rank correlation coefficient, and the results are shown in Table 2. It can be seen that the correlation coefficients between January–June runoff and annual runoff are all less than 0.5, the correlation between the monthly runoff series and the annual runoff series is poor, and the synchronization between annual runoff and monthly runoff is poor. The correlation coefficient between July–December runoff and annual runoff is greater than 0.5, which is a relatively high correlation, and it is more likely that the annual runoff is in the same frequency as the monthly runoff, but it is not possible to completely ignore the possibility of asynchrony. In general, the co-frequency between monthly and annual runoff varies from month to month. It is necessary to analyze each month separately, and both the co-frequency and cross-frequency of the annual and monthly abundance, flatness, and dryness need to be taken into account. Therefore, it is necessary to construct the joint probability distribution of annual and monthly runoff to analyze the ecological water demand of rivers.

. | Jan. . | Feb. . | Mar. . | Apr. . | May . | Jun. . |
---|---|---|---|---|---|---|

Kendall's rank correlation coefficient | 0.147 | 0.116 | 0.153 | 0.094 | 0.226 | 0.470 |

. | Jul. . | Aug. . | Sep. . | Oct. . | Nov. . | Dec. . |

Kendall's rank correlation coefficient | 0.578 | 0.529 | 0.582 | 0.581 | 0.671 | 0.730 |

. | Jan. . | Feb. . | Mar. . | Apr. . | May . | Jun. . |
---|---|---|---|---|---|---|

Kendall's rank correlation coefficient | 0.147 | 0.116 | 0.153 | 0.094 | 0.226 | 0.470 |

. | Jul. . | Aug. . | Sep. . | Oct. . | Nov. . | Dec. . |

Kendall's rank correlation coefficient | 0.578 | 0.529 | 0.582 | 0.581 | 0.671 | 0.730 |

### Joint distribution function construction

#### Parameter estimation for candidate copula functions

The candidate two-dimensional copula functions (i.e., Gumbel–Hougaard copula, Frank copula, Clayton copula, Gaussian copula, and Student's *t* copula) were used to construct the joint probability distributions of the annual runoff series and the monthly runoff series of each month. The parameters of each joint distribution function were estimated using the maximum likelihood estimation method. The results of parameter estimation are listed in Table 3.

Month . | Gumbel–Hougaard . | Frank . | Clayton . | Gaussian . | Student's t. | |
---|---|---|---|---|---|---|

. | . | . | . | . | . | |

Jan. | 1.149 | 1.577 | 0.260 | 0.247 | 0.248 | 1.272 × 10^{7} |

Feb. | 1.117 | 1.137 | 0.186 | 0.192 | 0.194 | 4.222 × 10^{6} |

Mar. | 1.177 | 1.565 | 0.397 | 0.296 | 0.303 | 4.315 × 10^{6} |

Apr. | 1.166 | 1.115 | 0.221 | 0.245 | 0.172 | 5.297 |

May | 1.222 | 1.992 | 0.482 | 0.348 | 0.351 | 1.262 × 10^{7} |

Jun. | 1.790 | 5.153 | 1.170 | 0.689 | 0.691 | 30.354 |

Jul. | 2.232 | 7.628 | 1.758 | 0.804 | 0.810 | 1.284 × 10^{7} |

Aug. | 1.921 | 6.400 | 1.541 | 0.735 | 0.744 | 50.793 |

Sep. | 2.019 | 6.950 | 1.294 | 0.754 | 0.753 | 4.669 × 10^{6} |

Oct. | 2.483 | 7.714 | 1.484 | 0.802 | 0.800 | 7.333 |

Nov. | 2.817 | 9.852 | 1.765 | 0.863 | 0.865 | 1.418 × 10^{7} |

Dec. | 3.149 | 12.242 | 1.561 | 0.874 | 0.876 | 1.290 × 10^{7} |

Month . | Gumbel–Hougaard . | Frank . | Clayton . | Gaussian . | Student's t. | |
---|---|---|---|---|---|---|

. | . | . | . | . | . | |

Jan. | 1.149 | 1.577 | 0.260 | 0.247 | 0.248 | 1.272 × 10^{7} |

Feb. | 1.117 | 1.137 | 0.186 | 0.192 | 0.194 | 4.222 × 10^{6} |

Mar. | 1.177 | 1.565 | 0.397 | 0.296 | 0.303 | 4.315 × 10^{6} |

Apr. | 1.166 | 1.115 | 0.221 | 0.245 | 0.172 | 5.297 |

May | 1.222 | 1.992 | 0.482 | 0.348 | 0.351 | 1.262 × 10^{7} |

Jun. | 1.790 | 5.153 | 1.170 | 0.689 | 0.691 | 30.354 |

Jul. | 2.232 | 7.628 | 1.758 | 0.804 | 0.810 | 1.284 × 10^{7} |

Aug. | 1.921 | 6.400 | 1.541 | 0.735 | 0.744 | 50.793 |

Sep. | 2.019 | 6.950 | 1.294 | 0.754 | 0.753 | 4.669 × 10^{6} |

Oct. | 2.483 | 7.714 | 1.484 | 0.802 | 0.800 | 7.333 |

Nov. | 2.817 | 9.852 | 1.765 | 0.863 | 0.865 | 1.418 × 10^{7} |

Dec. | 3.149 | 12.242 | 1.561 | 0.874 | 0.876 | 1.290 × 10^{7} |

#### Selection of copula function

The RMSE values were used to select the optimized copula function from the candidate copula functions. This function serves as the joint distribution function for annual and monthly runoff. After determining the parameter values of the five copula functions, the RMSE between the theoretical joint probability values and the empirical joint probability values of the annual and the monthly runoff series was calculated. The results are listed in Table 4.

Month . | Gumbel–Hougaard . | Frank . | Clayton . | Gaussian . | Student's t
. |
---|---|---|---|---|---|

Jan. | 0.022 | 0.019 | 0.020 | 0.019 | 0.019 |

Feb. | 0.019 | 0.017 | 0.017 | 0.017 | 0.017 |

Mar. | 0.020 | 0.019 | 0.015 | 0.019 | 0.019 |

Apr. | 0.019 | 0.019 | 0.017 | 0.019 | 0.017 |

May | 0.022 | 0.019 | 0.021 | 0.019 | 0.019 |

Jun. | 0.020 | 0.021 | 0.028 | 0.018 | 0.018 |

Jul. | 0.021 | 0.023 | 0.028 | 0.019 | 0.019 |

Aug. | 0.025 | 0.019 | 0.022 | 0.018 | 0.017 |

Sep. | 0.026 | 0.017 | 0.037 | 0.020 | 0.020 |

Oct. | 0.011 | 0.011 | 0.031 | 0.009 | 0.009 |

Nov. | 0.010 | 0.011 | 0.040 | 0.010 | 0.010 |

Dec. | 0.015 | 0.014 | 0.054 | 0.016 | 0.015 |

Month . | Gumbel–Hougaard . | Frank . | Clayton . | Gaussian . | Student's t
. |
---|---|---|---|---|---|

Jan. | 0.022 | 0.019 | 0.020 | 0.019 | 0.019 |

Feb. | 0.019 | 0.017 | 0.017 | 0.017 | 0.017 |

Mar. | 0.020 | 0.019 | 0.015 | 0.019 | 0.019 |

Apr. | 0.019 | 0.019 | 0.017 | 0.019 | 0.017 |

May | 0.022 | 0.019 | 0.021 | 0.019 | 0.019 |

Jun. | 0.020 | 0.021 | 0.028 | 0.018 | 0.018 |

Jul. | 0.021 | 0.023 | 0.028 | 0.019 | 0.019 |

Aug. | 0.025 | 0.019 | 0.022 | 0.018 | 0.017 |

Sep. | 0.026 | 0.017 | 0.037 | 0.020 | 0.020 |

Oct. | 0.011 | 0.011 | 0.031 | 0.009 | 0.009 |

Nov. | 0.010 | 0.011 | 0.040 | 0.010 | 0.010 |

Dec. | 0.015 | 0.014 | 0.054 | 0.016 | 0.015 |

*Note:* The minimum RMSE for annual runoff and each monthly runoff is highlighted in bold with a gray background, indicating that this copula function is the best joint distribution function.

The optimized copula function was determined as a joint probability distribution function for the annual and monthly flow series to minimize the RMSE between the theoretical and empirical probability distributions. Based on the RMSE values, it was determined that the Frank copula would be selected as the joint distribution function of the annual runoff with the September and December monthly runoff, the Clayton copula as the joint distribution function of the annual runoff with the March and April monthly runoff, the Gaussian copula as the joint distribution function of the annual runoff with the January, June, and July monthly runoff, and Student's *t* copula as the joint distribution function of annual runoff with February, May, August, October, and November monthly runoff.

### Determination of IEF

In this case study, the annual and monthly runoff series of each month were categorized into the abundant water year/month group, the flat water year/month group, and the dry water year/month group with 25% and 75% exceedance probability, respectively. The IEFs were determined for these three groups.

#### Initial IEF

Exceeding probability (monthly runoff) . | Jan. . | Feb. . | Mar. . | Apr. . | May . | Jun. . |
---|---|---|---|---|---|---|

PM ≤ 25% | 73.6 | 74.2 | 81.9 | 170 | 322 | 784 |

25% < PM ≤ 75% | 60.2 | 57.9 | 67.0 | 138 | 210 | 438 |

75% < PM | 45.6 | 44.2 | 55.7 | 99.3 | 156 | 186 |

Exceeding probability (monthly runoff) . | Jul. . | Aug. . | Sep. . | Oct. . | Nov. . | Dec. . |

PM ≤ 25% | 1,430 | 1,220 | 1,170 | 537 | 204 | 90.9 |

25% < PM ≤ 75% | 776 | 727 | 657 | 349 | 145 | 69.7 |

75% < PM | 386 | 431 | 342 | 179 | 99.3 | 54.4 |

Exceeding probability (monthly runoff) . | Jan. . | Feb. . | Mar. . | Apr. . | May . | Jun. . |
---|---|---|---|---|---|---|

PM ≤ 25% | 73.6 | 74.2 | 81.9 | 170 | 322 | 784 |

25% < PM ≤ 75% | 60.2 | 57.9 | 67.0 | 138 | 210 | 438 |

75% < PM | 45.6 | 44.2 | 55.7 | 99.3 | 156 | 186 |

Exceeding probability (monthly runoff) . | Jul. . | Aug. . | Sep. . | Oct. . | Nov. . | Dec. . |

PM ≤ 25% | 1,430 | 1,220 | 1,170 | 537 | 204 | 90.9 |

25% < PM ≤ 75% | 776 | 727 | 657 | 349 | 145 | 69.7 |

75% < PM | 386 | 431 | 342 | 179 | 99.3 | 54.4 |

*Note:* PM indicates the exceeding probability of monthly runoff.

#### Joint probability and conditional probability

Bayesian inference was used to determine the conditional probability of the magnitude of each monthly runoff (to which group the monthly runoff belongs) under different annual runoff scenarios (to which group the annual runoff belongs). The results of the conditional probabilities are shown in Table 6.

. | PY ≤ 25% . | 25% < PY ≤75% . | 75% < PY . | ||||||
---|---|---|---|---|---|---|---|---|---|

Month . | PM ≤ 25% . | 25% < PM ≤75% . | 75% < PM . | PM ≤ 25% . | 25% < PM ≤75% . | 75% < PM . | PM ≤ 25% . | 25% < PM ≤75% . | 75% < PM . |

Jan. | 0.356 | 0.488 | 0.156 | 0.244 | 0.512 | 0.244 | 0.156 | 0.488 | 0.356 |

Feb. | 0.332 | 0.493 | 0.175 | 0.246 | 0.507 | 0.246 | 0.175 | 0.493 | 0.332 |

Mar. | 0.317 | 0.526 | 0.156 | 0.263 | 0.520 | 0.216 | 0.156 | 0.433 | 0.411 |

Apr. | 0.289 | 0.519 | 0.192 | 0.259 | 0.510 | 0.231 | 0.192 | 0.461 | 0.347 |

May | 0.405 | 0.476 | 0.119 | 0.238 | 0.524 | 0.238 | 0.119 | 0.476 | 0.405 |

Jun. | 0.594 | 0.382 | 0.024 | 0.191 | 0.618 | 0.191 | 0.024 | 0.382 | 0.594 |

Jul. | 0.680 | 0.314 | 0.006 | 0.157 | 0.686 | 0.157 | 0.006 | 0.314 | 0.680 |

Aug. | 0.634 | 0.351 | 0.015 | 0.176 | 0.649 | 0.176 | 0.015 | 0.351 | 0.634 |

Sep. | 0.655 | 0.333 | 0.012 | 0.166 | 0.667 | 0.166 | 0.012 | 0.333 | 0.655 |

Oct. | 0.682 | 0.305 | 0.013 | 0.153 | 0.695 | 0.153 | 0.013 | 0.305 | 0.682 |

Nov. | 0.735 | 0.264 | 0.001 | 0.132 | 0.736 | 0.132 | 0.001 | 0.264 | 0.735 |

Dec. | 0.462 | 0.499 | 0.040 | 0.249 | 0.603 | 0.147 | 0.040 | 0.294 | 0.666 |

. | PY ≤ 25% . | 25% < PY ≤75% . | 75% < PY . | ||||||
---|---|---|---|---|---|---|---|---|---|

Month . | PM ≤ 25% . | 25% < PM ≤75% . | 75% < PM . | PM ≤ 25% . | 25% < PM ≤75% . | 75% < PM . | PM ≤ 25% . | 25% < PM ≤75% . | 75% < PM . |

Jan. | 0.356 | 0.488 | 0.156 | 0.244 | 0.512 | 0.244 | 0.156 | 0.488 | 0.356 |

Feb. | 0.332 | 0.493 | 0.175 | 0.246 | 0.507 | 0.246 | 0.175 | 0.493 | 0.332 |

Mar. | 0.317 | 0.526 | 0.156 | 0.263 | 0.520 | 0.216 | 0.156 | 0.433 | 0.411 |

Apr. | 0.289 | 0.519 | 0.192 | 0.259 | 0.510 | 0.231 | 0.192 | 0.461 | 0.347 |

May | 0.405 | 0.476 | 0.119 | 0.238 | 0.524 | 0.238 | 0.119 | 0.476 | 0.405 |

Jun. | 0.594 | 0.382 | 0.024 | 0.191 | 0.618 | 0.191 | 0.024 | 0.382 | 0.594 |

Jul. | 0.680 | 0.314 | 0.006 | 0.157 | 0.686 | 0.157 | 0.006 | 0.314 | 0.680 |

Aug. | 0.634 | 0.351 | 0.015 | 0.176 | 0.649 | 0.176 | 0.015 | 0.351 | 0.634 |

Sep. | 0.655 | 0.333 | 0.012 | 0.166 | 0.667 | 0.166 | 0.012 | 0.333 | 0.655 |

Oct. | 0.682 | 0.305 | 0.013 | 0.153 | 0.695 | 0.153 | 0.013 | 0.305 | 0.682 |

Nov. | 0.735 | 0.264 | 0.001 | 0.132 | 0.736 | 0.132 | 0.001 | 0.264 | 0.735 |

Dec. | 0.462 | 0.499 | 0.040 | 0.249 | 0.603 | 0.147 | 0.040 | 0.294 | 0.666 |

*Note:* PY indicates the exceeding probability of annual runoff.

#### IEF considering annual and monthly runoff variability

After determining the conditional probability values as well as the initial ecological water demand, the IEF for each month under different annual flow groupings is finally calculated. The results are listed in Table 7. The IEF process was converted into ecological water demand. The total annual basic ecological water demands of the Tongtian River under abundant, flat, and dry water years are 138, 102 and 72.2 billion m^{3}.

Exceeding probability (annual runoff) . | Jan. . | Feb. . | Mar. . | Apr. . | May . | Jun. . |
---|---|---|---|---|---|---|

PY ≤ 25% | 62.7 | 60.9 | 69.9 | 139 | 249 | 638 |

25% < PY ≤ 75% | 59.9 | 58.5 | 68.5 | 137 | 224 | 456 |

75% < PY | 57.1 | 56.2 | 64.7 | 131 | 202 | 297 |

Exceeding probability (annual runoff) . | Jul. . | Aug. . | Sep. . | Oct. . | Nov. . | Dec. . |

PY ≤ 25% | 1,220 | 1,040 | 989 | 475 | 188 | 78.9 |

25% < PY ≤ 75% | 817 | 762 | 689 | 352 | 147 | 72.7 |

75% < PY | 514 | 547 | 456 | 236 | 112 | 60.3 |

Exceeding probability (annual runoff) . | Jan. . | Feb. . | Mar. . | Apr. . | May . | Jun. . |
---|---|---|---|---|---|---|

PY ≤ 25% | 62.7 | 60.9 | 69.9 | 139 | 249 | 638 |

25% < PY ≤ 75% | 59.9 | 58.5 | 68.5 | 137 | 224 | 456 |

75% < PY | 57.1 | 56.2 | 64.7 | 131 | 202 | 297 |

Exceeding probability (annual runoff) . | Jul. . | Aug. . | Sep. . | Oct. . | Nov. . | Dec. . |

PY ≤ 25% | 1,220 | 1,040 | 989 | 475 | 188 | 78.9 |

25% < PY ≤ 75% | 817 | 762 | 689 | 352 | 147 | 72.7 |

75% < PY | 514 | 547 | 456 | 236 | 112 | 60.3 |

## CONCLUSIONS

This paper proposed a copula-based IEF determination method that takes into account inter- and intra-annual runoff variability. A case study was conducted based on the proposed method using real-world data from the Tongtian River in China. Several findings can be revealed as follows:

(1) The values of the respective parameters of the candidate distribution functions for annual runoff and monthly runoff of each month were estimated by the maximum likelihood method, and the K–S test was utilized to evaluate the degree of fit and select the best marginal distribution function. The GEV was selected as the marginal distribution function for the annual runoff series and also for the March and August monthly runoff series. The LOGN was selected as the marginal distribution function for the February, April, May, June, July, October, and November monthly runoff series. The Weibull was selected as the marginal distribution function for the January, September, and December monthly runoff series.

(2) The candidate copula functions were compared by RMSE to finalize the best joint distribution function. It was determined that Frank copula was chosen as the joint distribution function of annual runoff with September and December monthly runoff, the Clayton copula as the joint distribution function of annual runoff with March and April monthly runoff, the Gaussian copula as the joint distribution function of annual runoff with January, June, and July monthly runoff, and Student's -

*t*copula as the joint distribution function of annual runoff with February, May, August, October, and November monthly runoff.(3) The initial IEFs were calculated for different annual and monthly runoff volumes’ exceedance probabilities using the FDC method. The probability values were combined with the initial IEF values to determine the final IEF process. The total annual basic ecological water demands of the Tongtian River under abundant, flat, and dry water years are 138, 102 and 72.2 billion m

^{3}.

## AUTHORS' CONTRIBUTIONS

Y.Z. and G.F. primarily led the conceptualization of the project. Methodology design and development were mainly done by X.L., W.Y., and Y.Z. X.L. and Y.Z. conducted formal analysis and investigation. The original draft of the manuscript was prepared by X.L., W.Y., and Y.Z. All authors contributed to the review and editing process, including X.L., W.Y., Y.Z., and G.F. Y.Z. and G.F. were responsible for securing funding for the project. Y.Z. provided the necessary resources for the research. G.F. supervised the entire project.

## FUNDING

This work is supported by the National Natural Science Foundation of China (Grant Nos 52209032 and 42102286).

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.