Determination of ecological flow based on probability distributions of annual and monthly flows

The rapid development of the social economy, behind the growth of various economic indicators, is also accompanied by the destruction of the ecological environment (Shi et al. 2020; Ikram et al. 2021). In the process of continuously transforming nature, human beings have increased the development of water resources, ignoring the river ecology's demand for water (Cheng et al. 2023). Water is vital to the virtuous cycle of the entire ecological environment, and meeting the ecological flow in water resource management is the most basic condition for protecting the ecological environment.

Ecological flow studies began in the 1940s. At that time, the United States Fish and Wildlife Service (USFWS) initiated a study of ecological water requirements in response to concerns about the impacts of dam construction and water development on fish habitat. Quantitative and process-based studies of ecological flows continued over the next two decades, with work focusing on the relationships between instream flows and fish, macrofauna, and plants (Sajedipour et al. 2017). After the 1970s, the concept of minimum acceptable flow was introduced to effectively address river navigation and pollution problems (Tharme 2003). In the early 1980s, a comprehensive restructuring of watershed development and management goals was initiated in the United States, which was the beginning of ecological and environmental water demand allocation studies. However, the concept of ecological flow or ecological water demand was not explicitly mentioned at that time (Wang et al. 2022; Chen et al. 2023). Since then, ecological water demand has gradually become the focus of global attention as studies continue to deepen. At the same time, the concern of ecological flow studies is no longer limited to the instream ecosystems and has started to expand to ecosystems beyond the rivers. Gleick (1998) emphasized that continued population growth as well as inadequate access to and poor management of freshwater resources will exacerbate all types of ecological problems. The seven principles of sustainable development were further discussed, including the basic amount of water needed to maintain human health and ecosystems. Whipple et al. (1999) argued that the planning and development of water resources need to give more consideration to the needs and management of the ecological environment. It was also pointed out that in the process of development and utilization, it is necessary to coordinate the balance between economic water demand and ecological environment water demand. Acreman & Dunbar (2004) pointed out that to meet human needs to protect and rebuild ecosystems, a number of organizations defined ecological flow as the flow required to achieve desired ecological goals and reviewed the methods used to calculate ecological flow at that time.

Methods for determining ecological flows in rivers are improving with ongoing studies. More than 200 methods for ecological flow determination are available. These methods include four major categories, namely, hydrological methods, hydraulic methods, physical habitat modeling methods, and holistic methods. Hydrological methods are used to derive ecological flows in rivers based on historical flow information. The main methods are the Tennant method, the 7Q10 method (Ames 2006), the variable monthly flow (VMF) method, the flow–duration curve (FDC) method (Liang et al. 2015), and the range of variability approach (RVA) (Lin et al. 2016). The hydraulic methods determine the required ecological flow of a river based on the hydraulic parameters of the channel (e.g. width, depth, hydraulic radius, and flow velocity). The representative methods are the wetted perimeter method and the R2-CROSS method (Liu et al. 2011). The hydraulic 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. Physical habitat modeling methods combine changes in river flow with species-specific habitat conditions to analyze the water demand of river ecosystems. The most representative method is the Instream Flow Incremental Methodology. However, it focuses only on the conservation of some specific species rather than the river ecosystem as a whole, so the application of this method is somewhat limited (Stewart et al. 2005). The most representative holistic approach is the Building Block Methodology. This method emphasizes that the river is a complete ecosystem and is based on maintaining the natural conditions of the river, observing changes in the size of the flow and the corresponding changes in the river ecosystem over a long period. The whole process requires the participation of experts from different disciplines, and the procedures are more complex and difficult to use (O'Keeffe et al. 2019).

The above existing ecological flow determinations are based on the actual hydrological and ecological conditions of the river, and it is difficult to have a universal approach. Hydrological methods are usually the preferred method when hydrological data are available. For example, Zhang et al. (2019) comprehensively considered the 7Q10, RVA (hydrological method), and River2D (habitat simulation method) to determine the integrated ecological water demand threshold of the study area and constructed three kinds of ecological scheduling models based on this value to realize the quantitative analysis of the relationship between power generation and ecology. The results of the study showed that power generation and ecological demand are mutually constrained and conflicting. Shadkam et al. (2016) calculated the ecological flow requirements for the world's second largest hypersaline lake based on the VMF method and estimated that 3.7 billion m³ of water per year is required to protect Urmia Lake. However, all these hydrological methods analyze the hydrological status of rivers only from an intra- or inter-annual perspective. In fact, the hydrological processes of rivers have inter- and intra-annual variations, which are not adequately taken into account by the existing methods. This paper aims to propose an ecological flow determination method based on the joint probability distribution of annual and monthly flows to take into account the abundance and drought encounters of annual and monthly flows, to compensate for the deficiencies of the existing hydrological methods, and to better maintain the ecological objectives of the natural hydrological conditions of the river. The contribution of this paper includes the following: (1) marginal distributions of annual and monthly flows are fitted, and joint probability distributions of annual and monthly flows are constructed using Copula functions. (2) Based on Bayesian inference, the conditional probabilities of different monthly flow sizes under different annual flow size conditions are calculated. (3) Combining the conditional probability with the FDC method, an innovative ecological flow determination method is proposed to overcome the deficiencies of the existing hydrological methods.

Univariate probability distributions (e.g. Pearson type III (P-III), Lognormal, Generalized Extreme Value, Weibull, and Logistic) commonly used in the discipline of hydrology and water resources are first adopted as the candidate marginal distribution functions of annual and monthly flows. The candidate marginal distribution functions are used to fit the annual and monthly flow series, and their parameters are determined using the great likelihood estimation method. Hypothesis testing is then used to assess the significance level of the constructed marginal distributions of annual and monthly flows to ensure that the theoretical probability distributions represent the actual annual and monthly flows well. The hypothesis testing methods that can be used include the Kolmogorov–Smirnov (K–S) test, Anderson–Darling (A–D) test, and t-test. The correlation between the annual and monthly marginal distributions is finally analyzed and used to confirm the feasibility of establishing a joint probability distribution function, which can be evaluated using the Spearman correlation coefficient, Pearson correlation coefficient, and Kendall's correlation coefficient.

Copula functions are currently the main method used in hydrology and water resources to create joint probability distributions for multiple variables. De Michele & Salvadori (2003) used a Copula function to establish the joint probability distribution function of rainfall duration and rainfall intensity, which opened up the application of Copula functions in the field of hydrology and water resources. Favre et al. (2004) discussed the application of Copula functions in multivariate modeling and explored the relationship between flood peaks and flood volumes. Shiau (2006) defined the intensity and duration of drought events, and the recurrence period of drought events was obtained by optimized fitting of the marginal and Copula functions. Zhang & Singh (2007) derived trivariate rainfall frequency distributions using the Gumbel–Hougaard Copula, which does not assume the rainfall variables to be independent, normal, or have the same type of marginal distributions. The trivariate distribution was then employed to determine joint conditional return periods and was tested using rainfall data from the Amite River Basin in Louisiana. Karmakar & Simonovic (2009) used a Copula function to jointly analyze different characteristic variables of floods. Wang et al. (2010) applied a two-dimensional Archimedean Copula function to analyze the flood frequency at the confluence of rivers. The commonly used Copula functions in the field of hydrology and water resources are the Archimedean Copula, Meta-elliptic Copula, and Empirical Copula.

The Archimedean Copula function structure is simple; the constructed joint distribution function form is diverse and adaptable, and thus it 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:

• (1) Gumbel–Hougaard Copula

  

  (1)

  where is a parameter; and obey a uniform distribution between [0,1].
• (2) Frank Copula

  

  (2)
• (3) Clayton (Cook–Johnson) Copula

  

  (3)

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:

• (1) Gaussian Copula

  

  (4)

  where Φ is the multivariate normal distribution function and ρ is the linear correlation coefficient.
• (2) Student-t Copula

  

  (5)

  where ϑ is the functional degree of freedom.

Several suitable candidate Copula functions are selected based on the marginal distribution of annual and monthly flows, and the parameters of the candidate Copula functions are estimated using the great likelihood estimation method. The best Copula function is selected as the joint distribution function of annual and monthly flows according to the degree of goodness of fit. Graphical evaluation can be used to visualize the degree of fit. The theoretical and empirical probability values are plotted on a scatter plot, and the theoretical distribution is more representative of the actual distribution if the points are more evenly distributed around the 45° line. When the theoretical and empirical probability distributions of different candidate Copula functions are fitted similarly and cannot be visually compared by graphical evaluation, the RMSE will be an effective method of quantitative analysis. The optimal Copula function is selected as the joint probability distribution function for the annual and monthly flow series with the criterion of minimizing the RMSE.

The annual and monthly flow series are grouped into high, medium, and low groups according to runoff magnitude. Each monthly flow series is divided into high, medium, and low groups, bounded by 25 and 75% of the exceedance probability of each monthly flow. Similarly, the annual flow series is divided into high, medium, and low groups bounded by 25 and 75% of the annual flow exceedance probability in preparation for calculating the final ecological flow. It is worth noting that in applying the methodological framework proposed in this study, other more appropriate methods for grouping annual and monthly flow series can be used depending on the actual situation.

The FDC-based method is used to calculate the initial ecological flow and is a widely used hydrological method for determining ecological flows in rivers. The FDC is defined as the percentage of time that a given flow rate is equaled or exceeded during a specified time period (Vogel & Fennessey 1995). The vertical coordinate is the average daily, monthly, or annual flow rate, and the horizontal coordinate is the probability or frequency that the actual flow rate exceeds the flow rate corresponding to the vertical coordinate. Different types of FDCs represent the link between the magnitude of runoff and the frequency of occurrence on a daily, monthly, or annual scale; can adequately reflect characteristics such as the magnitude and frequency of flow; and are widely used to evaluate changes in hydrologic conditions. Vogel et al. (2007) further proposed the dimensionless river ecological indicators Ecosurplus and Ecodeficit based on FDC, which directly reflect the river ecological flow surplus and deficit generated by flow changes. Various ecological flow determination methods have been developed since the Ecosurplus and Ecodeficit were proposed. In this study, the flow series of each month under each monthly flow grouping is sorted by flow magnitude to create an FDC. The flow value corresponding to the 90% exceedance probability is then used as the initial ecological flow for the month in this grouping (high, medium, or low).

In this case, the flow observation data from Pingshan Station, the controlling hydrological station of the Jinsha River, was selected. The Jinsha River is home to a variety of endemic and rare fish species, such as round-mouthed copper fish, Chinese sturgeon, white alligator, Dabry's sturgeon, rosy barb, and more than 60 other species. The Pingshan Station is downstream of the Wudongde–Baihetan–Xiluodu–Xiangjiaba cascade reservoir group, and its flow data can better reflect the runoff change characteristics and hydrological laws of this cascade reservoir group. The construction and operation of the reservoirs on the Jinsha River have altered the natural state of runoff, changing the patterns of annual and monthly flow and impacting the ecological environment. The proposed ecological flow determination method was utilized to formulate a more scientific and reasonable ecological runoff flow, which can be used as a red line for the development of Jinsha River's water resources and hydroelectric energy resources, and to better protect natural species of flora and fauna.

We organized the flow observations at Pingshan Station from 1940 to 2011 into monthly and annual flow series. It has been shown that the runoff of the Jinsha River obeys the P-III distribution (Huang et al. 2018; Jia et al. 2020). Therefore, we used the P-III distribution as the marginal distribution function to fit the annual flow series and the flow series of each month at the Pingshan Station. The parameters of the P-III distribution were estimated by the great likelihood estimation method, and the results of the parameter estimation are shown in Table 1. The K–S test and A–D test were used to test the fitting degree of the marginal distribution function, and the results show that the marginal distributions of the annual flow and the monthly flow have a significance level of more than 95%.

Table 1

P-III distribution parameter estimation results

Flow series .	P-III distribution parameter .
	.	α .	1/β .
Annual	2,070.054	10.406	241.560
January	683.750	20.661	47.802
February	1,084.283	4.251	83.852
March	422.204	1.929	497.434
April	1,139.823	2.644	151.599
May	−986.109	51.020	64.426
June	1,803.358	5.536	567.403
July	2,004.268	9.183	810.698
August	3,084.967	4.726	1492.066
September	−16,960.288	100.000	269.369
October	3,001.163	5.050	695.772
November	1,583.662	8.163	230.373
December	727.527	22.676	64.168

Flow series .	P-III distribution parameter .
	.	α .	1/β .
Annual	2,070.054	10.406	241.560
January	683.750	20.661	47.802
February	1,084.283	4.251	83.852
March	422.204	1.929	497.434
April	1,139.823	2.644	151.599
May	−986.109	51.020	64.426
June	1,803.358	5.536	567.403
July	2,004.268	9.183	810.698
August	3,084.967	4.726	1492.066
September	−16,960.288	100.000	269.369
October	3,001.163	5.050	695.772
November	1,583.662	8.163	230.373
December	727.527	22.676	64.168

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The correlation between the monthly and annual flows was tested using the Pearson correlation coefficient, and the results are shown in Table 2. As shown in the table, the correlation coefficients of the January–June monthly flow and annual flow are all less than 0.5, the correlation is poor, and the synchronization between the monthly and annual flow is poor. The correlation coefficients of the July–December monthly flow and annual flow are greater than 0.5, and there is a greater possibility that the monthly and annual flows are at the same frequency, but we cannot completely ignore the possibility of asynchronization. Therefore, it is necessary to establish the joint probability distribution of monthly and annual flows to determine the ecological flow for the Jinsha River.

We selected two-dimensional Clayton Copula, Frank Copula, Gumbel–Hougaard Copula, Gaussian Copula, and Student-t Copula functions as candidate joint distribution functions of the monthly and annual flows. The parameters of each candidate joint distribution function were estimated using the great likelihood estimation method, and the parameter estimation results are shown in Table 3.

Table 3

Parameter estimation results for candidate joint distribution functions

Annual runoff .	Clayton .	Frank .	Gumbel–Hougaard .	Gaussian .	Student-t .
	.	.	.	.	.	.
January	0.141	0.949	1.132	0.127	0.173	3.810
February	0.177	1.259	1.170	0.181	0.205	26.514
March	0.168	1.527	1.254	0.133	0.235	6.106
April	0.107	1.319	1.157	0.164	0.204	14.205
May	0.153	1.248	1.188	0.229	0.256	1.124 × 107
June	0.482	3.505	1.508	0.468	0.487	35.289
July	1.258	5.012	1.917	0.697	0.715	4.669 × 106
August	1.779	8.072	2.409	0.819	0.830	1.298 × 107
September	1.889	6.111	2.071	0.743	0.758	5.885
October	1.779	5.254	1.746	0.678	0.694	1.088 × 106
November	2.072	7.208	2.173	0.793	0.807	1.480 × 107
December	3.047	9.708	2.797	0.873	0.882	4.669 × 106

Annual runoff .	Clayton .	Frank .	Gumbel–Hougaard .	Gaussian .	Student-t .
	.	.	.	.	.	.
January	0.141	0.949	1.132	0.127	0.173	3.810
February	0.177	1.259	1.170	0.181	0.205	26.514
March	0.168	1.527	1.254	0.133	0.235	6.106
April	0.107	1.319	1.157	0.164	0.204	14.205
May	0.153	1.248	1.188	0.229	0.256	1.124 × 107
June	0.482	3.505	1.508	0.468	0.487	35.289
July	1.258	5.012	1.917	0.697	0.715	4.669 × 106
August	1.779	8.072	2.409	0.819	0.830	1.298 × 107
September	1.889	6.111	2.071	0.743	0.758	5.885
October	1.779	5.254	1.746	0.678	0.694	1.088 × 106
November	2.072	7.208	2.173	0.793	0.807	1.480 × 107
December	3.047	9.708	2.797	0.873	0.882	4.669 × 106

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We calculated the RMSE between the five candidate and empirical joint distributions, and the results are shown in Table 4. The optimal Copula function was selected as the joint distribution function of monthly and annual flows with the criterion of minimum RMSE, which is highlighted in bold with a gray background in the table. The joint distribution functions of January, July, September, November, and December monthly and annual flows were determined as Student-t Copula. The joint distribution functions of February and April monthly and annual flows were determined as Frank Copula. The joint distribution function of March monthly and annual flows was determined as Gaussian Copula. The joint distribution functions of May, June, and August monthly and annual flows were determined as Gumbel–Hougaard Copula. The joint distribution function of October monthly and annual flows was determined as Clayton Copula.

The FDC-based method was used to calculate the initial ecological flow. The monthly flow series was divided into high, medium, and low groups, bounded by 25 and 75% of the exceedance probability for each month. The flow series of each month under each monthly flow grouping was sorted by flow magnitude to create an FDC. The flow value corresponding to the 90% exceedance probability was then determined as the initial ecological flow for the month in this grouping. The initial ecological flow results are shown in Table 5.

Section 4.2 established the joint probability distribution of annual and monthly flows (see Figure 2), from which the joint probability values for different combinations of annual and monthly flows can be obtained. The conditional probability of each month's flow size (which group of high, medium, and low flows the monthly flow belongs to) was then calculated based on Bayesian inference under different annual flow scenarios (which group of high, medium, and low flows the annual flow belongs to). The results are shown in Table 6.

After obtaining the conditional probability values, combined with the initial ecological flows, the final ecological flow process that takes into account the intra- and inter-annual flow variations was finally calculated. The results are shown in Table 7. The total annual ecological water demand of the Jinsha River is 1.50, 1.25, and 1.04 × 10¹¹ m³ under annual flow scenarios of high, medium, and low flows, respectively, which provide a red line for the development of water resources and hydro-energy resources of the Jinsha River, as well as for the better protection of the natural plant and animal species.

Most hydropower stations serve multiple functions, including power generation, flood control, and water supply. Their primary goal is to pursue efficient utilization of water resources and maximize economic benefits. However, if the operational models of these hydropower stations do not adequately consider the ecological flow requirements of the downstream river reaches, it can cause varying degrees of damage to the river ecosystem. For example, the construction of dams obstructs rivers, leading to changes in hydrological conditions and biological habitats, which affect the spawning and reproduction of migratory fish, resulting in alterations in the abundance and structural characteristics of riverine biological communities. The calculated ecological flow can be used as a constraint in the optimization scheduling model of the reservoir group on the Jinsha River, or the level of assurance for ecological flow can be set as an objective in multi-objective optimization scheduling. This approach can achieve the consideration of ecological factors in the reservoir scheduling process, balancing economic and ecological benefits to maximize overall benefits.

The reason for this difference is that the traditional FDC method only considers the inter-annual variability of streamflow, whereas the proposed method takes both inter-annual and intra-annual variability into account. Combined with the conditional probabilities listed in Table 6, when the annual flow is in the high group, the probability that the monthly flow is in the medium group always occupies a certain proportion of the 12 months, and the probability in the months of January–May exceeds that in the high group, so the calculated ecological flow is smaller than that obtained by FDC. When the annual flow is in the medium group, the probability that the monthly flow is in the medium group is the largest, and both are greater than 50%; the probability that the monthly flow is in the high or low group is relatively small, so the calculated ecological flow is similar to that obtained by FDC. When the annual flow is in the low group, although the probability that the monthly flow is in the low group is larger, there are some months where the probability that the monthly flow is in the medium group is larger or second only to the probability of being in the low group, so the calculated ecological flow is larger than that obtained by FDC.

Most hydrologic methods (including traditional FDC methodologies) only consider intra-annual variability characteristics of runoff and rarely consider inter-annual variability characteristics of runoff. The proposed method utilizes Copula functions to construct joint probability distributions for annual and monthly flows and further calculates conditional probabilities for different monthly flow sizes under different annual flow conditions based on Bayesian inference. Thus, the ecological flow calculated by the proposed method takes into account the probability of abundance and drought encounters of annual and monthly flows. It is due to this feature that the ecological flow of a river determined by this method provides a more balanced representation of the actual ecological water demand of the river under different coming water situations than other hydrological methods. The proposed method also has good adaptability and flexibility. In this paper, the annual and monthly flows are divided into three high, medium, and low groups according to their size. Future studies can increase or decrease the grouping according to the actual needs of the study objects. It can even be divided into each year to reflect the different situations of incoming water in each year. The proposed method belongs to the category of hydrological methods, and like other hydrological approaches, it can determine ecological flow based solely on flow data. However, it also shares a common limitation of hydrological methods, which is the lack of biological knowledge support. Future research should also integrate considerations from biology, such as biological habitats.

The method proposed in this study essentially belongs to hydrological approaches, which are relatively simple and easy to implement but still lack support from biological knowledge. In the future, it will be necessary to further incorporate biological elements such as habitats. When calculating ecological flow, three different flow levels – high, medium, and low – were distinguished. When applying this method to other studies, groups can be formed based on actual needs.

In developing and utilizing water resources, maintaining ecological flow is the most basic requirement for protecting the ecological environment. In this paper, an innovative ecological flow determination method based on the joint probability distribution of annual and monthly flows was proposed to compensate for the deficiencies of the existing hydrological methods that inadequately consider the abundance and drought encounters of annual and monthly runoff. A case study was conducted in the Jinsha River of China. Several findings can be revealed as follows:

• (1) The P-III distribution fits the marginal distribution functions of monthly and annual flows well, with significance levels exceeding 95% by both the K–S test and the A–D test.

• (2) Student-t Copula was selected as the joint distribution function of monthly and annual flows for January, July, September, November, and December; Frank Copula as the joint distribution function of monthly and annual flows for February and April; Gaussian Copula as the joint distribution function of monthly and annual flows for March; Gumbel–Hougaard Copula as the joint distribution function of monthly and annual flows for May, June, and August; and Clayton Copula as the joint distribution function of monthly and annual flows for October.

• (3) Combining the conditional probability with the FDC-based method, the ecological flow process that takes into account the intra- and inter-annual variations was finally determined. The total annual ecological water demand of the Jinsha River is 1.50, 1.25, and 1.04 × 10¹¹ m³ under annual flow scenarios of high, medium, and low flows, respectively, which provide a red line for the development of water resources and hydro-energy resources of the Jinsha River, as well as for the better protection of the natural plant and animal species.

BF, XZ, and YZ conceptualized the article. XZ, FY, and YZ developed the methodology. XZ and FY rendered support in formal analysis and investigated the process. BF, XZ, FY, and YZ wrote and prepared the original draft. BF, XZ, FY, and YZ wrote the review and edited the article. BF and YZ rendered support in funding acquisition. XZ developed the resources. BF supervised the article.

This work was supported by the National Key Research and Development Program of China (Grant No. 2023YFC3206005), the Major Science and Technology Program of the Ministry of Water Resources of China (Grant No. SKS-2022034), the Special Research Fund of Nanjing Hydraulic Research Institute (Grant No. Y123001), and the National Natural Science Foundation of China (Grant No. 52209032).

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

The authors declare there is no conflict.