Ensemble probability distribution of annual runoff for the past 70 years in two main watersheds of China

Water resources in China are unevenly distributed in both time and space, and per capita water resources are limited (Jiang 2009; Niu et al. 2022; Wu et al. 2023). With the rapid development of social economy, the contradiction of water supply, ecology, and other water use between upstream and downstream has been increasingly prominent in many areas. In recent years, water allocation has been carried out on a yearly basis in the basin to solve the water problem. The long-term variation of runoff demonstrates significant uncertainty and can be regarded as a stochastic process due to its pronounced long-term auto-correlation. This behavior, known as long-term persistence, is evident across paleoclimatic, climatic, and annual scales (Pizarro et al. 2022; Guo et al. 2024). Therefore, the probability distribution of future runoff is often used as the basis for water resources planning and allocation and flood control regulation. Due to the uneven spatial distribution of rainfall-runoff, the probability distribution of annual runoff may be different in different regions of a watershed, especially in large basins. For the water allocation of the whole basin, regional water transfer coordination should consider the different probability distributions of annual runoff in each region of the basin, that is, the probability combination problem.

The commonly used methods for analyzing the probability combination of runoff or the joint probability of different elements in different regions (rivers) include multivariate normal probability distribution, which has been used to analyze the probability combination of flood peak and flood volume (Morán-Vásquez et al. 2022, 2023), multivariate joint probability distribution derived from the marginal probability of the logarithmic normal distribution of runoff (flood) in different regions (rivers) (Hangshing & Dabral 2018; Liu et al. 2018; Latif & Mustafa 2020; Zhong et al. 2021), and two-dimensional normal distribution, which is used to calculate the coincidence probability of basin design flood (Chen et al. 2012; Jianping et al. 2018), the multivariate joint probability distribution of annual runoff in different watersheds derived by converting runoff into one-dimensional variables (Wang et al. 2009; Jiang et al. 2017), and the copula function applied to the multiple joint probability distribution of the hydrological analysis (Peng et al. 2017; Guo et al. 2018; Lilienthal et al. 2018; Liu et al. 2020; Xiang et al. 2020; Latif & Mustafa 2021; Wen et al. 2022). Koutsoyiannis et al. (2008) conducted a comprehensive comparison of stochastic and deterministic methods for medium-range flow prediction in the Nile River. These methods mostly solve the runoff (flood) problems encountered by different regions (rivers) or different joint probability elements.

However, the most common problem of the probability combination of annual runoff in basin-wide water allocation is the corresponding situation of runoff in different regions of the basin, that is, the probability combination of total runoff and runoff in each region of the basin when the runoff with design probability occurs at the control section of the basin. In the conventional design flood, there were two methods to deal with this problem: the typical year method and the regional composition method with the same probability (Jing et al. 2020; Xiong et al. 2020). In the typical year method, several representative years with unfavorable runoff conditions for regulation are selected from the measured data. The corresponding design runoff for each region is then calculated based on the runoff composition proportion of each region and the design runoff at the basin control section. This method assumes that the selected representative years adequately reflect the overall hydrological behavior of the basin, including the concept of concentration time (Giandotti 1934; Grimaldi et al. 2012). The regional composition method, on the other hand, assumes that each sub-region within the basin has the same probability of flood occurrence as the control section of the entire basin. Using this assumption, the corresponding runoff for each sub-region is calculated based on the water balance principle. This method provides a detailed analysis by considering the contributions and hydrological characteristics of each sub-region. Both methods are comparatively reliable and suitable for flood–prevention design, as they address different aspects of hydrological analysis and management.

The runoff at the basin control section is an aggregation of the runoff from different regions within the basin. In recent years, numerous studies have focused on the spatial correlation among flows in sub-basins. For instance, Grimaldi et al. (2012) explored the stochastic generation of spatially coherent river discharge peaks for continental event-based flood risk assessment. This study highlights the importance of understanding and predicting the spatial distribution of river discharge during large-scale flood events. Additionally, our research provides new insights and methods by analyzing the spatial correlation among sub-basins, further enhancing the understanding of complex hydrological processes within basins. At the same time, there is also a correlation between the runoff of the basin control section and the runoff of each region in the basin (Yang et al. 2022). Therefore, when the runoff of a basin control section is a specified design value, the runoff of each region in the basin may be randomly distributed in a relatively small range. Under this condition, the mean square deviation of the probability distribution is less than the corresponding value of the measured regional runoff data series in the basin (Fischer & Schumann 2021). This paper studies the probability combination problem of annual runoff in a river basin, which is the most commonly used index in water resources planning and allocation (Cai et al. 2021). This paper attempts to calculate the probability distribution of annual runoff of each region in the basin under the condition that the design value of the basin control section is given.

The probability distribution of annual runoff of each region in a given basin is derived from the two-dimensional joint probability distribution of two related random variables. The functions used for joint probability distribution are mainly the logarithmic normal distribution, exponential distribution, Gumbel function, and P-III function (Nerantzaki & Papalexiou 2022). The logarithmic normal distribution is a function that has been proved to be suitable for runoff probability distribution by long-term practices (Vivekanandan & Srishailam 2021; Lee et al. 2022; Scala et al. 2022). Previous research has extensively explored the dynamics of water resources, particularly in the context of annual runoff and its probability distribution (Herman et al. 2020; Zhu et al. 2020; Gao et al. 2021; Hu et al. 2021; Zhou et al. 2022). These studies laid the groundwork for understanding the intricacies of water allocation in large basins. However, as the challenges of water distribution grow more complex with increasing demand and climatic uncertainties, there is a pressing need for refined methodologies that consider both spatial and temporal variations. The present study builds on this foundation by offering a nuanced analysis of the Songhua River and Yangtze River basins, integrating the probability distribution of annual runoff across different regions of the basin. Such an approach not only provides a comprehensive understanding of the basins' water dynamics but also offers practical insights for policymakers and stakeholders in water resource management.

In comparison, small basins are more particularly susceptible to the same weather and rainfall events, while the correlation of annual runoff in different regions in the basin is generally better than that in large basins. Therefore, the Songhua River and Yangtze River basins with large drainage areas are selected as the study areas in this paper. The Songhua River basin is located at 41°42′–51°48′ north latitude, with a drainage area of 556,800 km², a temperate continental monsoon climate, and an average annual precipitation of 525 mm. Jiamusi hydrological station, with a drainage area of 528,277 km², is the outlet of the basin, and basically controls the runoff of the whole basin. The Yangtze River basin is located at 24°30′–35°45′ north latitude, with a drainage area of 1,800,000 km², a temperate to subtropical continental monsoon climate, and an average annual precipitation of 1,067 mm. Datong hydrological station, with a drainage area of 1,705,383 km², is the outlet of the basin, and basically controls the runoff of the whole basin.

The data series of the Songhua River basin from the year 1954 to 2023 and the Yangtze River basin from the year 1950 to 2023 are used. The hydrological year of the Songhua River basin from May 1 to April 30 of the next year, and the hydrological year of the Yangtze River basin from April 1 to March 31 of the next year are adopted. The annual mean discharge is used to represent the annual runoff.

The control nodes are Jiangqiao, Harbin, and Jiamusi hydrological stations in the Songhua River basin, and Cuntan, Yichang, Hankou, and Datong hydrological stations in the Yangtze River basin. The annual mean discharge series of the region between two stations is obtained by subtracting the annual mean discharge of the downstream station from the annual mean discharge of the upstream station. The statistical characteristic values of annual runoff in the three regions and control sections of the Songhua River and the four regions and control sections of the Yangtze River are shown in Table 1.

The probability combination relationship of annual runoff in the basin is analyzed using the annual mean discharge of the hydrological year to represent the annual runoff. The control section of the basin represents the runoff of the whole basin, and the annual runoff of the control section is an important indicator for the planning and allocation of water resources in the basin. Therefore, the probability combination relationship between the annual runoff of each region in the basin and the annual runoff of the basin control section is established. According to the correlation between the annual runoff of each region in the basin and the annual runoff of the basin control section, the probability distribution of annual runoff of each region in the basin is deduced when the annual runoff of the basin control section is a given design value.

The analysis starts from the correlation between the annual runoff of each region in the basin and the control section of the basin. The runoff of the basin control section is collected by the runoff in each region of the basin. Runoff in different regions of the basin is often formed by rainfall in the same weather system. Therefore, the annual runoff of each region in a basin should have a certain correlation with the control section of the basin.

For the two data series X and Y with linear correlation, the linear regression equation can be established through the least square method.

In order to obtain the runoff value of the specified design standard probability, the empirical probability calculated by Equation (1) also needs to adapt to the probability of the theoretical distribution function curve, such as the logarithmic normal distribution as well as the others.

According to the characteristics of the normal distribution, the common value of the logarithmic normal distribution, the value of P(x) = 50%, is the mean value of the lnX sequence (El). From hydrological probability analysis experience, it is known that the common value of the runoff sequence in China (P(x) = 50%), the runoff value e^El corresponding to El, is less than the mean value of the runoff sequence x.

If there is correlation between the lnX and lnY series, and the correlation coefficient R > 0, then the mean square deviation of the conditional probability distribution of lny is less than the mean square deviation of the lnY series, ⁠. If there is no correlation between the lnX and lnY series, that is, R = 0, then the mean and mean square deviation of the lny conditional probability distribution are equal to the mean and mean square deviation of the lnY series, respectively, that is, and ⁠. Therefore, the conditional probability distribution of the lny series is equivalent to the conventional probability distribution of the lny series.

Thus, under the condition that the annual runoff value of the basin control section is given, the mean square deviation of the logarithmic conditional probability distribution of regional annual runoff (⁠⁠) is equal to the square root of the mean difference between the data points and the regression line (Equation (5)) in the correlation diagram of the regional logarithmic annual runoff and the logarithmic annual runoff of the control section.

The above figures and tables show that there is a significant correlation between the annual mean discharge in each region of the two basins and their control sections. The correlation of the annual mean discharge series and logarithmic series between the three regions of the Songhua River basin and the Jiamusi control section is relatively high, and the correlation coefficients are 0.76–0.84. The correlations of the annual mean discharge series and logarithmic series between the four regions of the Yangtze River basin and the basin control section Datong are that the correlation between the two regions above Yichang is relatively low, with correlation coefficients of 0.56–0.58, while the correlation between the regions below Yichang is relatively high, with correlation coefficients of 0.82–0.87. The correlation coefficients of the annual mean discharge series and logarithmic series in each region of the two basins and their basin control sections are almost the same.

The mean value (⁠⁠) and mean square deviation (⁠⁠) of the logarithmic series of annual mean discharge in the three regions of the Songhua River, the mean value (⁠⁠) and mean square deviation (⁠⁠) of the logarithmic series of annual mean discharge in the Jiamusi control section, the mean value (⁠⁠) and mean square deviation (⁠⁠) of the logarithmic series of annual mean discharge in the four regions of the Yangtze River basin, and the mean value (⁠⁠) and mean square deviation (⁠⁠) of the logarithmic series of annual mean discharge in the Datong control section are shown in Table 3.

As shown in Figures 8 and 9, the gradient of cumulative probability distribution of annual mean discharge under various regional conditions is steeper than that of the conventional cumulative probability distribution of annual mean discharge, that is, the greater the correlation between the annual mean discharge of each region in the basin and the control section of the basin, the steeper the conditional cumulative probability distribution of regional annual mean discharge. In addition, as the cumulative probability value of annual mean discharge of a given basin control section increases (the annual mean discharge decreases), the conditional cumulative probability distribution curve moves in a smaller direction, while the shape of the distribution curve changes little. This means that when the annual mean discharge of the control section of the basin is a given value, that is, the design probability value, the annual mean discharge of each region in the basin will be randomly distributed within a relatively small range. With the decrease of the given annual mean discharge of the basin control section, the mean value of the logarithmic conditional cumulative probability distribution of annual mean discharge of each region in the basin decreases, and the distribution curve will move to the smaller side approximately.

The empirical cumulative probability of the virtual logarithmic annual mean discharge series (lnY*) in each region of the Songhua River and Yangtze River basins is calculated by using Equation (1). Figures 8 and 9 show the plot and the cumulative probability of the logarithmic conditional distribution. For the seven regions of the Songhua River and Yangtze River basins, the empirical cumulative probability of the virtual logarithmic annual mean discharge series and the logarithmic conditional cumulative probability of the logarithmic annual mean discharge are well fitted. To test the feasibility of using the control section to predict sub-basin interval discharge, we used a copula function to predict the discharge for the Yihan interval and compared it with the observed data (Supplementary Figure S2). The results showed a high degree of fit, with a Nash–Sutcliffe efficiency coefficient reaching 0.77 (Supplementary Figure S3).

It can be seen from Equation (4) that since the annual runoff of each region in the basin is related to the annual runoff of the basin control section, the mean square deviation (⁠⁠) of the logarithmic conditional probability distribution of annual runoff of each region is smaller than the mean square deviation (⁠⁠) of the logarithmic series of the annual runoff of each region. By comparing the data in Tables 2, 3 and 6, it can be verified that the larger the correlation coefficient between them, the smaller the mean square deviation (⁠⁠) of conditional probability distribution of the regional annual runoff. Therefore, for a given annual runoff value of the basin control section, the range of the random distribution of the regional annual runoff within the basin becomes smaller, so the slope of the conditional probability curve becomes steeper. Similarly, because the annual runoff of each region in the basin is positively correlated with the basin control section, if the annual runoff value (⁠⁠) given by the basin control section decreases, the mean value (⁠⁠) of logarithmic conditional probability distribution of annual runoff of each region in the basin also decreases, and the greater the correlation coefficient, the more it decreases. It can be seen from the logarithmic distribution characteristics in Equation (3) that when the cumulative probability of the annual runoff of the basin control section is 50%, the value ⁠, therefore ⁠, that is, the mean value of the logarithmic conditional probability distribution of the regional annual runoff of the basin is equal to the mean value of its logarithmic series (see Tables 3–5). With the increase of the cumulative probability value of annual runoff of the basin control section (the annual runoff value x* decreases), the mean value (⁠⁠) of the logarithmic conditional probability distribution of annual runoff of each region in the basin gradually decreases (see Tables 4 and 5). Therefore, the annual runoff of the basin control section decreases, and the conditional cumulative probability distribution range (curve) of the logarithmic annual runoff of each region in the basin moves to a smaller direction. In addition, since the mean square deviation of the logarithmic conditional probability distribution of the regional annual runoff corresponds to the different annual runoff of basin control sections, the logarithmic conditional cumulative probability distribution curve of regional annual runoff within the basin shifts to the direction of low flow, and its shape is basically unchanged.

It is clear that by substituting the annual runoff of the basin control section into the annual runoff regression Equation (14), the sum of the annual runoff of each region in the basin is equivalent to the annual runoff of the control section of the basin.

Under the condition of the given annual runoff of the basin control section, the logarithmic conditional probability distribution of annual runoff of each region in the basin can be regarded as the probability distribution of the logarithmic annual runoff series. Solution of the given annual runoff of the basin control section on the regression line (Equation (5)) of the correlation diagram of the logarithmic annual runoff of the regional and basin control sections is taken as the mean value (⁠⁠) of this logarithmic annual runoff series. The difference between the data points on the regression line (Equation (5)) in the correlation diagram between the logarithmic annual runoff of each region and the basin control section will lead to the fluctuation of the logarithmic annual runoff series every year.

Overall, this paper comprehensively analyzes the correlation and probability combination between the annual runoff of controlled sections in the Songhua River and Yangtze River basins and the annual runoff in various areas of the basin. Details are as follows.

This study provides a comprehensive analysis of the annual runoff in the basin and its sub-basins, utilizing both the year method and the regional composition method. The integration of these methods allows for a detailed understanding of the temporal and spatial distribution of runoff, revealing the significant contributions of sub-basin flows to the overall basin runoff.

By fitting the entire empirical cumulative distribution function (CDF) using the log-normal distribution, our analysis captures the full range of variability in annual runoff. This approach provides a robust framework for understanding the probabilistic characteristics of runoff, which is crucial for effective water resource management and flood risk assessment.

The study confirms the presence of strong long-term auto-correlation in the runoff process, characterized by long-term persistence behavior. This finding aligns with previous studies and underscores the importance of considering temporal dependencies in hydrological modeling.

The authors appreciated the editor and anonymous reviewers for their constructive comments and suggestions on the revision of this paper. This work was supported by the National Key R&D Program of China [grant number 2021YFB3900604-04/05] and the National Natural Science Foundation of China [grant number 42271084].

F.S.: Formal analysis, Writing – original draft. G.W.: Methodology. S.N.: Software, Supervision. Y.T.: Funding acquisition, Resources. J.Y.: Data curation. H.L.: Software. X.X.: Writing – review and editing. M.Z.: Software, Visualization. Y.C.: Software. All authors have read and agreed to the published version of the manuscript.

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

The authors declare there is no conflict.