Improved Monthly Frequency Method Based on Copula Functions for Studying Ecological Flow in the Hailang River Basin, Northeast China

Abstract

Climate change has intensified extreme hydrological events in cold regions, threatening the stability of river ecosystems. The traditional monthly frequency method for calculating ecological flow assumes equal guarantee rates across all months, overlooking the complex nonlinear dependencies between interannual and intermonthly flows. This approach may result in flow values for certain months during low-flow years exceeding those of corresponding months in high-flow years, failing to align with actual hydrological patterns. This study integrates Copula functions with the monthly frequency method to establish an improved ecological flow calculation framework, accurately characterizing the statistical correlation between interannual and intermonthly flow variability. The Hailang River basin in Northeast China was selected as the study area. First, the SWAT model was employed to simulate natural runoff processes from 1956 to 1965. The calibration phase demonstrated excellent performance (R² = 0.84, NSE = 0.83), and the validation phase also met standards (R² = 0.82, NSE = 0.81). The improved method selected optimal Copula functions for each month through rigorous statistical tests (AIC, BIC, RMSE, and K-S test), establishing joint probability distributions for annual and monthly average flows. The results indicate that different Copula types better align with monthly hydrological seasonal characteristics: Gaussian Copula suits February, May, and July; t-Copula suits August; Clayton Copula from September to December; Gumbel Copula for January, March, April, and June. Through conditional probability relationships (P(X₀≥x₀, 90%) = 0.9), the monthly guarantee rate range determined by the improved method spans 81.83% to 90.08%, significantly outperforming the uniform 90% guarantee rate employed by traditional methods. Verification using the Tennant method confirmed that ecological flows throughout the year met “excellent” or higher standards. Ecological flows exhibited pronounced seasonal variation, ranging from 6.2 m³/s during winter to spring to 96.93 m³/s during summer to autumn, providing scientific basis for basin-scale ecological water management. This study establishes a reliable methodological framework for ecological flow management in cold-region rivers.

1. Introduction

In recent years, due to the proliferation of watershed development and water resource utilization, natural runoff processes have undergone substantial changes. Human activities have altered the flow patterns of numerous rivers worldwide, resulting in adverse effects on biodiversity, water quality, and ecological processes [1]. Advanced remote sensing technologies, including high-resolution Digital Elevation Models (DEMs) and satellite-based land use monitoring, have become essential tools for quantifying hydrological alterations across large watersheds. These technologies enable comprehensive assessment of flow regime changes and their ecological implications [2]. Climate change has amplified and protracted regional extreme rainfall and drought events, diminishing the value of river ecosystem services. Concurrently, the proliferation of human activities, including dam construction, urbanization, and substantial water abstraction, has exerted multifaceted impacts on river ecosystems [3]. Recent studies have indicated that river ecosystems are exhibiting signs of degradation in both structure and ecological function due to climate change and human activities [4]. This phenomenon necessitates the development of robust, sustainable water resource management frameworks [5]. In the context of accelerating environmental change, ecological flow assessment has emerged as a critical tool for maintaining the integrity of river ecosystems [6]. A substantial body of research has validated the efficacy of appropriate ecological flow regulation as a means to safeguard ecosystem stability, foster biodiversity, and facilitate sustainable water resource development [7,8].

Ecological flow, also termed environmental flow, represents the minimum water quantity required to sustain aquatic ecosystems and maintain the natural ecological functions of rivers, lakes, and wetlands [9]. More specifically, ecological flow encompasses the flow regime characteristics, including magnitude, duration, timing, frequency, and the rate of change, which are essential for supporting native aquatic species, riparian vegetation, and overall ecosystem health [10]. Within the context of watershed ecology, ecological flow requirements vary spatially across different sub-basins due to variations in channel morphology, habitat diversity, and species composition. This spatial variability necessitates a comprehensive understanding of the interconnected hydrological and ecological processes that operate at multiple spatial and temporal scales. Ecological flow exhibits a threshold value; flows that exceed or fall below this threshold have the potential to disrupt ecological balance. Therefore, the establishment of reasonable threshold values is essential for facilitating the rational utilization of water resources and maintaining ecological stability [11]. According to the “Specifications for Calculating Water Requirements for River and Lake Ecological Environments” (SL/T 712-2021 [12]), the basic ecological flow rates for rivers, lakes, and wetlands are determined as the fundamental water quantity conditions to achieve ecological protection objectives and ensure the stability of ecological functions. According to surveys, there are over 200 ecological flow analysis methods worldwide, primarily categorized into four major types: hydrological methods, hydraulic methods, habitat simulation methods, and holistic methods [13]. Hydrological methods, which form the foundation of this study, rely primarily on statistical analysis of historical flow records to derive ecological flow recommendations. These methods assume that natural flow variability provides a template for maintaining ecological integrity and include approaches such as the Tennant method, historical flow curve method, variable range method, minimum monthly average flow method, monthly minimum ecological flow method, and monthly frequency method [14]. Unlike habitat simulation methods that require detailed biological data, or holistic methods that integrate multiple assessment approaches, hydrological methods offer the advantage of computational efficiency and broad applicability. This makes them particularly suitable for data-limited environments typical of many cold-region watersheds. The monthly frequency method, which serves as the core methodological foundation for this study, is based on the hydrological conditions of the river and ecological objectives, calculating the ecological flow rate for each month by selecting an appropriate guarantee rate. This approach recognizes that ecological water demands vary seasonally, requiring different flow thresholds for different months. However, traditional applications of this method treat annual average flow and monthly average flow independently, failing to account for the statistical dependencies between interannual flow variations and monthly flow patterns within specific year types. This independence assumption can lead to unrealistic scenarios where calculated monthly ecological flows during dry years exceed those during wet years for the same calendar month. Such outcomes violate fundamental hydrological principles and potentially under-protect ecosystems during critical low-flow periods. Dai et al. [15] employed the monthly frequency method to categorize river flow into three stages: low-flow, normal-flow, and high-flow periods. They selected three distinct combinations of guaranteed rates to calculate the ecological flow process. The Tennant method [16] is a frequently employed technique for the study of ecological water demand in rivers and lakes, with the capacity to expeditiously evaluate the adequacy of ecological flow in a watershed. Srivastava and Maity [17] employed the Tennant method to ascertain river ecological flow by dividing annual average flow into seven distinct grade standards. Ali and Hasan [18] compared various ecological flow calculation methods and identified a method suitable for the Gorai River that can capture the temporal variations of monthly average flow rates across different seasons. In recent years, there has been an increasing emphasis on enhancing the scientific rigor and applicability of ecological flow analysis. One approach that has been adopted is the incorporation of hydrological models into ecological flow studies. The SWAT model, in particular, has gained prominence due to its capacity to simulate the water cycle process within a watershed, encompassing precipitation, evaporation, soil infiltration, surface runoff, and groundwater recharge [19]. The SWAT model’s effectiveness is significantly enhanced by the integration of high-quality remote sensing datasets. These include 30 m resolution Digital Elevation Models (DEMs) for topographic analysis and satellite-derived land use/land cover data for spatial parameterization. The SWAT model facilitates the acquisition of long-term runoff data for each sub-basin. Integrating the SWAT model with hydrological methods enables a comprehensive analysis of ecological flow changes across the entire watershed [20]. However, when calculating river ecological flow using the monthly frequency method in hydrological methods, the difference in the guarantee rate between the annual average flow and the monthly average flow of the river channel is not considered. This methodological limitation arises because traditional approaches apply a fixed guarantee rate to each month’s flow series independently, without considering whether these monthly flows occur within dry, normal, or wet years. Consequently, the monthly runoff during the low-flow year may not correspond to the months with low flows, potentially resulting in ecological flow calculation results that do not meet ecological requirements. Therefore, it is necessary to improve the monthly frequency method for calculating ecological flow. This will allow for the accommodation of the difference in guarantee rates between annual average flow and monthly average flow. The result will be more reasonable ecological flow calculation results. Copula functions are a powerful joint distribution modeling tool. They can overcome the limitations of traditional statistical models in describing variable correlations. They can also accurately characterize the nonlinear and heterogeneous dependencies between annual and monthly average runoff [21]. Copula functions are currently widely applied in hydrology, particularly in drought-risk-related studies, and have been demonstrated to offer significant advantages in multivariate hydrological extremes analysis [22]. Shiau pioneered the introduction of Copula functions into drought characteristic analysis, establishing a theoretical foundation for drought risk assessment by constructing joint distributions of drought duration and intensity [23]. Wen et al. established a precipitation-runoff correlation analysis framework based on Bayesian model averaging and Archimedean Copula functions, substantially enhancing the precision of multivariate hydrological analysis [24]. However, most studies employ a single type of Copula function for analysis, failing to adequately account for seasonal variations in the correlation structures between different hydrological variables. Furthermore, in ecological flow calculations, the traditional monthly frequency method overlooks the complex relationship between annual mean flow and monthly mean flow guarantee rates, potentially leading to results that deviate from actual hydrological conditions. By applying Copula functions to improve the monthly frequency method, a joint probability distribution between annual average flow and monthly average flow can be constructed. This approach maintains the correlation between the two variables while accounting for variations in flow guarantee rates across months.

This study employs the scientific scenario method to simulate runoff, thereby circumventing the limitations of conventional approaches that exclusively concentrate on calculating ecological flow at river outlet cross-sections. Given the close hydraulic connections between watersheds, conducting ecological flow distribution across the entire watershed enables a more comprehensive consideration of water resource allocation within the watershed. Conventional component survey methods often encounter challenges in acquiring long-term, daily-scale runoff process data. Physically based distributed modeling enables precise reconstruction of runoff processes, providing more detailed and accurate data for ecological flow assessment and research. Objectives of this study are to (1) construct a natural runoff model for the Hailang River basin (1956–1965) using the SWAT model integrated with high-resolution remote sensing datasets to provide robust data support for ecological flow calculations; (2) develop an improved monthly frequency method using optimal Copula functions to construct joint probability distributions between annual and monthly average flows, with month-specific Copula types selected through rigorous statistical testing; (3) establish conditional probability relationships for monthly flows during low-flow years using P(X₀≥x₀, 90%) = 0.9 for dry year identification and 90% conditional probability for monthly flow determination; and (4) validate the improved methodology using the Tennant method to establish ecological flow grade standards and applies this method to calculate ecological flow for the entire Hailang River basin. This comprehensive approach provides a scientific basis for optimizing water resource allocation and formulating reasonable flow ecological scheduling plans for cold-region river basins. The methodology has broader applicability to similar watersheds facing challenges from climate change and anthropogenic impacts.

2. Overview of the Study Area

The Hailang River is located in the southeastern part of Heilongjiang Province, within the jurisdiction of Hailin City. A negligible portion of its upper reaches is situated within Jilin Province. The topography of the Hailang River basin exhibits an elevation gradient from west to east, with an average elevation of 500 m above sea level. The river’s main stem measures 231 km in length, and the total basin area is 5251 square kilometers [25]. The Hailang River basin is subject to the influence of the Pacific monsoon and the Siberian high-pressure system, resulting in short, hot, and humid summers and long, cold winters. The annual flood season, characterized by high precipitation, extends from May to September. In contrast, the non-flood season, marked by low precipitation, commences in October and persists until April of the subsequent year. The long-term average temperature hovers around 3 °C, with July and August being the hottest months, each averaging above 20 °C [26]. The extreme maximum temperature recorded is 35.8 °C. January and February are the coldest months, averaging between −15 °C and −19 °C, with the lowest temperature reaching −38.8 °C. The first day when the daily average temperature consistently exceeds 10 °C occurs in mid-April, with the last such day in mid-October, spanning approximately 180 days. The accumulated temperature reaches 2489 °C, and the maximum frost depth is 2 m. This exhibits the typical characteristics of a temperate monsoon climate. Precipitation is influenced by orographic lifting, with an increase from the lower to the upper reaches. The long-term average precipitation distribution within the basin exhibits significant variability, with the upper reaches receiving an average of 1110 millimeters, while the lower reaches receive only 530 millimeters [27]. The multi-year average annual runoff depth in this watershed ranges between 150 mm and 500 mm. The annual variation in runoff volume is considerable, primarily governed by rainfall intensity, with abundant water flow in summer and reduced flow in winter. The distribution of flow throughout the year is highly uneven, with the majority concentrated during the summer and autumn seasons. The geographical area under study is delineated in Figure 1.

3. Data and Methods

3.1. Data Source and Processing

3.1.1. Digital Elevation Model (DEM)

The digital elevation data were obtained from the Spatial Geographic Data Cloud (http://www.gscloud.cn, accessed on 21 January 2025) platform [28], specifically the GDEMV3 digital elevation data with a resolution of 30 m. This dataset encompasses the entire geographical area of the study region, with corresponding spatial resolution, thereby effectively reflecting the slope variations of the ground surface. The GDEMV3 DEM data used in this study are non-generalized, maintaining the original 30 m spatial resolution without terrain smoothing or simplification processing, which adequately captures the topographic variations required for hydrological modeling. The 30 m resolution of the digital elevation data is adequate for the construction and analysis of the SWAT model [29]. DEM processing and watershed delineation were performed using ArcGIS version 10.8 with Spatial Analyst and 3D Analyst extensions. The watershed boundary extraction and stream network delineation utilized the ArcSWAT interface version 2012.10_4.19.

3.1.2. Hydrological Data

The hydrological data were obtained from the Changtingzi Hydrological Station, encompassing monthly flow measurements from 1956 to 2016. These data are essential for simulating hydrological processes and analyzing ecological flow in the watershed. The monthly flow data from the Changtingzi Hydrological Station reflect the hydrological characteristics and changes of the watershed. The monthly flow data provide a robust foundation for calibrating and validating the SWAT model. The geographical location of the hydrological station is indicated in Figure 1.

3.1.3. Meteorological Data

The selected meteorological data are daily climate observation data from three basic meteorological stations near the watershed. These data were obtained from the National Meteorological Administration (http://data.cma.cn/, accessed on 11 January 2025) platform [30]. The stations are Hailin Station (54,092), Ning’an Station (54,098), and Mudanjiang Station (54,094). The data were processed using the SWAT weather software (V 2012). Missing data were interpolated to construct the SWAT model meteorological database. This database was then imported into the model. The locations of the meteorological stations are delineated in Figure 1.

3.1.4. Land Use

Land use data are critical to the SWAT model [31]. The 30 m land use dataset was obtained from Yang Jie and Huang Xin’s study, “30-m annual land cover and its dynamics in China from 1990 to 2019” [32].

3.1.5. Soil Type

The soil data were obtained from the Harmonized World Soil Database (HWSD) [33] of the National Qinghai–Tibet Plateau Science Data Center (https://data.tpdc.ac.cn/, accessed on 21 January 2025) platform [34]. The HWSD provides comprehensive soil property information with global coverage, containing soil parameters essential for hydrological modeling. The soil land use types present within the Hailang River basin were retrieved, and the relevant parameters were calculated using the soil water characteristic calculation software SPAW Hydrology version 6.02.75 and statistical analysis methods [35]. The SPAW software was used to derive key soil hydraulic properties including soil texture, bulk density, available water capacity, and saturated hydraulic conductivity. Subsequently, these parameters were integrated with parameters from the global soil database to construct a comprehensive soil database for the Hailang River basin. The soil types present within the study area were categorized into 14 classes based on the FAO soil classification system, providing the necessary spatial and attribute data for SWAT model parameterization.

3.2. Hydrological Change Point Detection

The Mann–Whitney–Pettitt method [36,37] is based on rank statistics and detects change points by computing statistical measures at each possible division point within the time series to identify locations where rank differences are most pronounced. The software is well suited for the analysis of continuous data in the context of climate change, and it is capable of detecting both single and multiple change points. The Mann–Whitney–Pettitt analysis was implemented using MATLAB R2021b with the Statistics and Machine Learning Toolbox. The pettitt.test function was employed for single change point detection, while ordered clustering analysis was performed using custom algorithms based on variance minimization principles.

$$U_t={\sum }_{i=1}^{t-1}{\sum }_{j=t}^{n}sgn\left(x_j-x_i\right)$$

(1)

In the formula, ${\mathrm{U}}_{\mathrm{t}}$ is a statistic employed for the identification of change points. These points are determined through the execution of specific calculations on time series data. The maximum absolute value of t corresponds to the potential change point. $t$ represents a particular point in time within the time series, and i and j denote data point indices. The conditions for these indices are as follows: 1 < i < t − 1 and t < j < n [38].

Ordered clustering is a classification method based on data similarity [39]. The algorithm performs a calculation of the sum of the squares of the deviations between the various segments. This calculation is used to achieve the division of the data into categories, with the objective being to maximize the differences between categories and minimize the differences within categories.

$$V_{\tau }={\sum }_{i=1}^{\tau }{\left(x_i-{\overline{x}}_{\tau }\right)}^2$$

(2)

$$V_{n-\tau }={\sum }_{i=\tau +1}^n{\left(x_i-{\overline{x}}_{n-\tau }\right)}^2$$

(3)

$$S=\underset{2\le \tau \le n-1}{min}{S}_0(\tau )=\underset{2\le \tau \le n-1}{min}\left(V_{\tau }+V_{n-\tau }\right)$$

(4)

In the given equation, ${\overline{\mathrm{x}}}_{\mathsf{\tau }};{\overline{x}}_{n-\tau }$ represents the mean values of the two sequences, prior to and following the specified point, respectively. These values are also known as the cluster centers of the two sequences. Min indicates the minimum value. When S takes the minimum value in Equation (4), τ is the optimal bisection point and can be inferred as the change point [40].

3.3. SWAT Model

The SWAT model, a semi-distributed watershed hydrological model, divides the entire study watershed into multiple sub-watersheds and hydrological response units based on physical process principles [41]. The watershed hydrological process is divided into two stages: the land runoff stage and the river convergence stage. Multiple key modules are integrated. The SWAT model version 2012.664 was implemented through the ArcSWAT interface version 2012.10_4.19. Model calibration and validation were conducted using SWAT-CUP version 5.1.6.2, specifically employing the SUFI-2 algorithm for parameter optimization and uncertainty analysis.

The surface runoff module calculation formula is as follows:

$$S{W}_t=S{W}_0+{\sum }_{i=1}^t{\left(P_d-Q_s-E_a-W_s-Q_g\right)}_i$$

(5)

In the aforementioned equation, $S{W}_t$ denotes the final soil moisture content (mm); t denotes the calculation period (days); $S{W}_0$ denotes the initial soil moisture content; $P_d$, $Q_s$, $E_a$, and $Q_g$ denote the daily rainfall, daily surface runoff, daily evapotranspiration (mm), and return flow ($Q_g$), respectively; and $W_s$ denotes the daily lateral seepage and infiltration in the soil profile (mm) [42].

The Nash efficiency coefficient, NSE, and the coefficient of determination, R², are calculated as follows [43]:

$$R^2={\left\{{\displaystyle \frac{{\sum }_{i=1}^{n^{\prime }}\left(Q_{S,i}-{\dot{Q}}_S\right)\left(Q_{M,i}-{\dot{Q}}_M\right)}{{\left[{\sum }_{i=1}^{n^{\prime }}{\left(Q_{S,i}-{\dot{Q}}_S\right)}^2{\sum }_{i=1}^{n^{\prime }}{\left(Q_{M,i}-{\dot{Q}}_M\right)}^2\right]}^{0.5}}}\right\}}^2$$

(6)

$$NSE=1-\left[{\displaystyle \frac{{\sum }_{i=1}^n{\left(Q_{S,i}-Q_{M,i}\right)}^2}{{\sum }_{i=1}^n{\left(Q_{S,i}-{\dot{Q}}_S\right)}^2}}\right]$$

(7)

In this study, the measured flow rate of the ith segment is denoted as $Q_{S,i}$, the average value of the measured flow is denoted as ${\dot{Q}}_S$, the simulated flow rate of the ith segment is denoted as $Q_{M,i}$, the average value of the simulated flow rate is denoted as ${\dot{Q}}_M$, n′ denotes the length of the measured time series, and n denotes the number of samples.

3.4. Ecological Flow Calculation

3.4.1. Monthly Frequency Method

The monthly frequency method is based on the hydrological conditions of rivers and ecological and environmental objectives. This approach selects a reasonable guarantee rate to derive the calculation method for ecological flow rates for each month of the year. Guo’s assumptions regarding the monthly flow guarantee rate are as follows: 80% during the low-flow period and 50% during the normal-flow period. For the flood period, the annual average flow is used directly as the ecological flow [44]. The following approach is adopted in this study: the monthly average flow corresponding to a 90% guarantee rate is used as the basic ecological flow for the corresponding month, forming the values for different time periods within the year.

3.4.2. Tennant Method

The Tennant method utilizes the long-term average flow rate as a basis, multiplying the average by the percentage of river flow required by the aquatic ecosystem to determine the environmental flow rate of the river [45]. When river flow reaches 10% of the long-term average flow, it can meet the basic short-term survival needs of most aquatic organisms; when river flow reaches 30% of the long-term average flow, it provides a suitable living environment for most aquatic organisms [16]. This method is characterized by its simplicity and efficacy, particularly in temperate climates and perennial rivers. The determination of this index is based on the division of wet and dry seasons and the ranking of long-term average natural monthly flow. The ecological flow classification method is delineated in

Table 1. Flow percentage of the Tennant method.

In-Stream Ecological Conditions Corresponding to Different Flow Percentages	Percentage of Natural Flow, %
	Non-Flood Period	Flood Period
Optimum range	60~100	60~100
Outstanding	40	60
Excellent	30	50
Good	20	40
Fair	10	30
Poor	10	10

[16].

3.5. Copula Construction

The term “copula” was introduced by Sklar [46] in 1959 to denote multivariate joint distribution functions, with results uniformly distributed in [0, 1]. First, the marginal distributions of two or more correlated variables must be constructed. Then, the joint distribution of these marginal distributions is constructed using the Copula function. Common Copula functions used in hydrology include the Archimedean Copula, meta-elliptic Copula, and empirical Copula [47]. The model simulation accuracy is evaluated using four metrics: the coefficient of determination (R²), root mean square error (RMSE), the Akaike information criterion (AIC), and the Bayesian information criterion (BIC) [48]. These are determined through parameter estimation and goodness-of-fit tests. R² close to 1 indicates a high correlation between the simulation results and the actual values, while R² close to 0 indicates a low correlation [49]. In general, smaller values of RMSE, AIC, and BIC are indicative of a more accurate model. All Copula analyses were performed using R software version 4.3.0. Parameter estimation was conducted using the Copula package (version 1.1-0) with the fitCopula function implementing maximum likelihood estimation. Goodness-of-fit testing utilized the gofCopula function with Kolmogorov–Smirnov test statistics. Additional packages included VineCopula (v2.4.4) for advanced dependency modeling, ks (v1.14.0) for kernel density estimation, and boot (v1.3-28) for bootstrap resampling in uncertainty analysis.

3.5.1. Archimedean Copula

The Archimedean Copula [50] is a widely utilized tool in the fields of probability statistics and risk management, owing to its straightforward formulation and adaptable structure. The present study principally employs the following Archimedean Copula functions:

Gumbel Copula:

$$C\left(u_1,u_2\right)=exp\left\{-{\left[{\left(-ln u_1\right)}^{\theta }+{\left(-ln u_2\right)}^{\theta }\right]}^{1/\theta }\right\},\theta \in [1,\infty )$$

(8)

Frank Copula:

$$C\left(u_1,u_2\right)=-{\displaystyle \frac{1}{\theta }}ln\left[1+{\displaystyle \frac{\left(e^{-\theta u_1}-1\right)\left(e^{-\theta u_2}-1\right)}{e^{-\theta }-1}}\right],\theta \in R$$

(9)

Clayton Copula [51,52]:

$$C\left(u_1,u_2\right)={\left(u_1^{-\theta }+u_2^{-\theta }-1\right)}^{-1/\theta },\theta \in (0,\infty )$$

(10)

where θ represents the parameter; u₁ and u₂ follow a uniform distribution within [0, 1].

3.5.2. Elliptic Copula

Elliptic Copula [53] is characterized by its ability to model data with symmetric dependencies between variables, thus inheriting the fundamental properties of elliptical distributions. Elliptic Copula includes Gaussian Copula and Student’s t-Copula.

Gaussian Copula:

$$C\left(u_1,u_2\right)={\int }_{-\infty }^{{\Phi }^{-1}\left(u_1\right)}{\int }_{-\infty }^{{\Phi }^{-1}\left(u_2\right)}{\displaystyle \frac{1}{2\pi {\left(1-{\rho }^2\right)}^{1/2}}}exp\left(-{\displaystyle \frac{x^2-2\rho xy+y^2}{2\left(1-{\rho }^2\right)}}\right)dxdy$$

(11)

where u₁ and u₂ are marginal probability values, and x and y are integration variables, representing two random variables that follow a standard normal distribution.

Student’s t-Copula [54]:

$$C\left(u_1,u_2\right)={\int }_{-\infty }^{t_{\vartheta }^{-1}\left(u_1\right)}{\int }_{-\infty }^{t_{\vartheta }^{-1}\left(u_2\right)}{\displaystyle \frac{1}{2\pi {\left(1-{\rho }^2\right)}^{1/2}}}{\left[1+\left({\displaystyle \frac{x^2-2\rho xy+y^2}{\vartheta \left(1-{\rho }^2\right)}}\right)\right]}^{-(\vartheta +2)/2}dxdy$$

(12)

where ϑ represents degrees of freedom.

3.6. Establishment of Conditional and Joint Probability Relationships Using Improved Monthly Frequency Method

According to the definition of conditional probability, this study establishes the relationship between the conditional probability of monthly average flow in dry years and the joint probability. The guarantee rate for dry years is defined as P(X₀≥x₀, 90%) = 0.9, which serves as the criterion for identifying low-flow year conditions. Based on this framework, the relationship between the conditional probability of monthly average flow and the joint probability can be expressed as follows:

$$\begin{array}{c}P\left(X_m⩾x_{m, \mathit{Dry} \mathit{year} \mathit{design} \mathit{value} },X_0⩾x_{0.90\%}\right)=\\ P\left(X_m⩾x_{m, \mathit{Dry} \mathit{year}\mathit{ }\mathit{design} \mathit{value} }\mid X_0⩾x_{0.90\%}\right)P\left(X_0⩾x_{0.90\%}\right)\end{array}$$

(13)

The 90% conditional probability is to be set as the target conditional probability for the monthly average flow. The corresponding indicator is to be derived as the basic ecological flow design value. The $P\left(X_m⩾x_{m, \mathit{Dry} \mathit{year} \mathit{design} \mathit{value}}\right)$ is to be solved using the marginal distribution function to obtain the ecological flow for the corresponding month.

Statistical computations and probability calculations were performed using R version 4.3.0 with the stats package (v4.3.0). Data visualization was created using ggplot2 (v3.4.0), while additional data preprocessing was conducted using Microsoft Excel 2019.

The technical roadmap for this study is divided into four main sections: (1) Runoff Data Analysis and Change Point Detection, (2) SWAT Model Hydrological Simulation, (3) Marginal Distribution Construction and Copula Function Selection, and (4) Ecological Flow Calculation and Validation. The technical roadmap’s specifics are illustrated in Figure 2.

4. Results and Analysis

4.1. Change Point Detection Results

Given the nonparametric characteristics of hydrological time series, specialized statistical methods are required for change point detection. This study employed two analytical approaches: the Mann–Whitney–Pettitt method and ordered clustering analysis. These methods were used to identify single breakpoints in the measured flow time series of the Hailang River basin from 1956 to 2016 (Figure 3). At the α = 0.05 significance level, the standardized test identified 1966 as the significant breakpoint in the flow series. Historical analysis indicates that large-scale reservoir construction commenced in Heilongjiang Province during the mid-1960s. Various water storage projects within Hailin City were concentrated for completion and commissioning between 1965 and 1967, directly altering the natural river flow processes [55]. During the period 1956–1965, human activity exerted relatively minor influence. Flow data from this era adequately represent the natural hydrological conditions of the Hailang River basin, providing a reliable baseline for ecological flow calculations.

4.2. SWAT Model Simulation

The threshold value of 10,000 pixels was determined through a systematic analysis based on the relationship between spatial resolution, terrain characteristics, and hydrological modeling requirements. According to Oliveira and Paradella [56], the DEM spatial resolution directly influences topographic analysis results, with threshold selection being critical for maintaining the accuracy of derived hydrological parameters. For our 30 m DEM covering the 5251 km² Hailang River basin, a threshold of 10,000 pixels corresponds to approximately 9 km² minimum sub-basin area, which aligns with third-order stream networks typical for mesoscale hydrological analysis. This threshold selection is further validated by quality assessments demonstrating that data accuracy is compatible with 1:80,000 to 1:100,000 scale mapping requirements for undulated relief conditions similar to our study area [57,58]. The 9 km² minimum sub-basin area ensures that each hydrological response unit captures significant terrain and climatic variations while maintaining computational efficiency for the SWAT model calibration [59]. Sensitivity analysis conducted with thresholds of 5000 pixels, 10,000 pixels, and 15,000 pixels confirmed that 10,000 pixels provides optimal balance between spatial detail and model stability, avoiding over-segmentation that could introduce noise in ecological flow calculations. A sub-basin is defined as an area delineated by grid cells within the river network where the cumulative runoff volume exceeds the threshold defined by the catchment area. The quantity of sub-basins is contingent upon the catchment area threshold. In effect, as the threshold diminishes in magnitude, the number of sub-basins is concomitantly increased. Conversely, as the threshold amplifies in magnitude, the number of sub-basins is concomitantly diminished. It has been demonstrated that the magnitude of the threshold directly correlates with the number of sub-basins, a relationship that has been consistently demonstrated in watershed studies across different regions and basin types [60,61,62]. The delineation of hydrological response units is predicated on a comprehensive array of factors, including but not limited to the soil database, land use data, and relevant slope classifications. Land use data are an indispensable component of the model; however, acquiring land use data from 1956 to 1965 presents a significant challenge. Consequently, 1980 land use data are employed to simulate the natural runoff of the Hailang River basin. The classification of land use types encompasses seven categories: impervious surfaces, forest, water bodies, wetlands, cropland, grassland, and bare land (Figure 4). Soil types are categorized into 14 classes, whilst slope grades calculated via DEM are classified into four thresholds: 0°–8°, 8°–30°, 30°–60°, and >60° [63]. To account for the influence of topography on runoff in the study basin, the proportions of slope land use area and soil type were set at 5%, 10%, and 10%, respectively. We constructed a data index table for meteorological data and imported it into the model to complete the preliminary setup. The data were imported into SWAT-CUP and calibrated using the SUFI-2 calibration method. Subsequently, the calibration parameters were imported into the SWAT model and the model was rerun. The parameters shallow groundwater evaporation depth threshold (REVAPMN), groundwater inflow threshold to the main river channel (GWQMN), temperature lapse rate (TLAPS), plant absorption compensation coefficient (EPCO), groundwater re-evaporation coefficient (GW_REVAP), and groundwater runoff delay time (GW_DELAY) have a significant impact on the model.

The natural runoff model (Figure 5) utilizes the period from 1956 to 1961 as the calibration period and the period from 1962 to 1965 as the validation period. The simulation results of the natural runoff model indicate that the simulation performance during the calibration period and validation period is satisfactory. The coefficient of determination R² for the calibration period is 0.84, and the Nash efficiency coefficient NSE is 0.83; for the validation period, the coefficient of determination R² is 0.82, and the Nash efficiency coefficient NSE is 0.81. The model simulation results are consistent with the actual flow data, indicating that the model can accurately simulate the spatiotemporal variation characteristics of natural runoff.

4.3. Copula Joint Distribution Analysis

This study utilizes a natural runoff model to simulate monthly flows in the Hailang River basin from 1987 to 2016 under natural conditions as input data. The 16th sub-basin, where the hydrological station is located, is selected as an example to calculate the joint distribution between annual average flow (AAF) and monthly average flow (MAF). The accuracy of the model simulation is evaluated using four indicators: the coefficient of determination (R²), root mean square error (RMSE), the Akaike information criterion (AIC), and the Bayesian information criterion (BIC). The optimal Copula function was selected (

Table 2. Optimal Copula function under the combination of annual average flow (AAF) and monthly average flow (MAF).

Period	Copula Function Type	AIC	BIC	RMSE	K-S
Jan.	Gaussian	1.4379	2.8391	0.060035	0.34203
	T	5.2298	9.4334	0.059861	0.34203
	Clayton	2	3.4012	0.060177	0.5372
	Frank	1.9346	3.3358	0.059751	0.34203
	Gumbel	0.060901	1.4621	0.059099	0.34203
Feb.	Gaussian	1.8187	3.2199	0.057566	0.76005
	T	5.7387	9.9423	0.0566	0.76005
	Clayton	2	3.4012	0.053389	0.76005
	Frank	1.9793	3.3805	0.055101	0.76005
	Gumbel	2	3.4012	0.053389	0.76005
Mar.	Gaussian	0.26288	1.6641	0.033664	0.93601
	T	4.2577	8.4613	0.034115	0.93601
	Clayton	1.6749	3.0761	0.041849	0.76005
	Frank	0.85045	2.2517	0.034812	0.76005
	Gumbel	−0.021686	1.3795	0.036723	0.76005
Apr.	Gaussian	−1.0618	0.33941	0.051321	0.93601
	T	2.9332	7.1368	0.051777	0.93601
	Clayton	1.6837	3.0849	0.064702	0.93601
	Frank	−1.183	0.21821	0.049554	0.93601
	Gumbel	−1.5988	−0.19759	0.051583	0.93601
May	Gaussian	−5.0474	−3.6462	0.037558	0.93601
	T	−1.0534	3.1502	0.037909	0.93601
	Clayton	−0.26009	1.1411	0.0509	0.76005
	Frank	−3.2721	−1.8709	0.038662	0.93601
	Gumbel	−4.7289	−3.3277	0.041725	0.93601
Jun.	Gaussian	−4.268	−2.8668	0.060751	0.5372
	T	−0.77274	3.4309	0.061394	0.5372
	Clayton	0.91608	2.3173	0.075018	0.76005
	Frank	−3.4419	−2.0407	0.058922	0.5372
	Gumbel	−6.1334	−4.7322	0.061759	0.76005
Jul.	Gaussian	−25.092	−23.691	0.056971	0.76005
	T	−21.097	−16.894	0.057233	0.76005
	Clayton	−15.459	−14.057	0.07139	0.5372
	Frank	−24.66	−23.259	0.052756	0.76005
	Gumbel	−21.596	−20.195	0.059493	0.76005
Aug.	Gaussian	−22.738	−21.337	0.056781	0.93601
	T	−25.139	−20.935	0.055456	0.93601
	Clayton	−20.37	−18.969	0.062795	0.93601
	Frank	−22.864	−21.463	0.056949	0.76005
	Gumbel	−23.461	−22.06	0.055815	0.93601
Sep.	Gaussian	−27.872	−26.471	0.04149	0.93601
	T	−24.378	−20.174	0.04191	0.93601
	Clayton	−29.112	−27.71	0.048493	0.99697
	Frank	−26.789	−25.388	0.039445	0.99697
	Gumbel	−21.555	−20.153	0.048105	0.93601
Oct.	Gaussian	−23.306	−21.905	0.044697	0.93601
	T	−19.894	−15.69	0.045214	0.93601
	Clayton	−23.443	−22.042	0.053884	0.93601
	Frank	−22.189	−20.788	0.043841	0.93601
	Gumbel	−18.182	−16.781	0.049622	0.99697
Nov.	Gaussian	−13.299	−11.898	0.05872	0.93601
	T	−9.3572	−5.1536	0.05921	0.93601
	Clayton	−16.91	−15.508	0.05981	0.93601
	Frank	−11.617	−10.215	0.056307	0.93601
	Gumbel	−7.6086	−6.2074	0.070685	0.76005
Dec.	Gaussian	−8.811	−7.4098	0.04506	0.99697
	T	−4.8183	−0.61473	0.045513	0.99697
	Clayton	−13.992	−12.591	0.045349	0.99697
	Frank	−6.8973	−5.4961	0.045843	0.93601
	Gumbel	−3.159	−1.7578	0.056435	0.76005

Note: Bold values indicate optimal Copula functions based on minimum AIC criterion. All K-S test p-values > 0.05 indicate adequate model fit at 95% confidence level.

). The corresponding optimal Copula functions are as follows: the Gaussian Copula is employed for the months of February, May, and July. For August, the t-Copula is utilized. The Clayton Copula is implemented for September, October, November, and December. Finally, the Gumbel Copula is applied to January, March, April, and June. The joint distributions and probability distributions of the Copula functions are illustrated in Figure 6 and Figure 7.

4.4. Improved Monthly Frequency Method for Ecological Flow Calculation

The conditional probability target for monthly average flow was determined using the 90% condition of the low-flow year type. The ecological flow guarantee rate was determined using a Copula function. The results for the guarantee rate of the 16th sub-basin river are shown in

Table 3. Monthly mean flow guarantee rate of basic ecological flow for the river in sub-basin No. 16.

Month	Jan.	Feb.	Mar.	Apr.	May	Jun.	Jul.	Aug.	Sep.	Oct.	Nov.	Dec.
Guarantee Rate (%)	89.69	90.08	89.55	89.22	87.49	88.74	84.11	83.42	81.83	82.15	82.78	83.42

. The established correlation between annual average flow and monthly average flow was taken into consideration when assessing the discrepancies in guarantee rates between the two. The conventional monthly frequency method frequently disregards the correlation between annual average flow and monthly average flow during the calculation of ecological flow, resulting in outcomes that may diverge from the actual hydrological conditions. Preliminary findings from the ecological flow calculation indicate that the ecological flow demand in the Hailang River basin is relatively high from July to October. The enhanced monthly frequency method has been demonstrated to offer a reasonable reflection of the annual variation patterns of flow processes. The calculation results have been shown to be consistent with the actual conditions of the river. The basic ecological flow process line calculated using the enhanced monthly frequency method is located above the ecological flow process line calculated using the traditional monthly frequency method (Figure 8). Furthermore, the ecological flow calculated using the improved monthly frequency method is significantly greater than the results obtained using the traditional monthly frequency method during August–October, thereby confirming the higher flow demand during this period. It can be observed that the improved monthly frequency method for calculating ecological flow is more effective in ensuring river ecological safety to a certain extent.

4.5. Ecological Flow Validation

The findings of the enhanced monthly frequency method for calculating ecological flow were corroborated by means of the Tennant method. The Hailang River basin was divided into two distinct seasons, designated as the flood season (May–September) and the non-flood season (October–April of the following year). The grading standards for ecological flow calculations using the improved monthly frequency method based on the Copula function were validated based on this division. The results indicate that the ecological flow calculated using the improved monthly frequency method meets or exceeds the excellent standard throughout the year (

Table 4. Comparative validation of ecological flow for the river in sub-basin No. 16.

Month	Jan.	Feb.	Mar.	Apr.	May	Jun.	Jul.	Aug.	Sep.	Oct.	Nov.	Dec.
Percentage (%)	41.91	49.70	30.78	40.71	57.84	66.68	74.97	80.80	80.95	78.54	70.79	63.07
Grade Standard	Outstanding	Outstanding	Excellent	Outstanding	Excellent	Optimum range	Optimum range	Optimum range	Optimum range	Optimum range	Optimum range	Optimum range

). The ecological flow calculated using the improved monthly frequency method has been demonstrated to offer enhanced protection for the river’s ecological environment.

4.6. Ecological Flow Analysis in the Hailang River Basin

The calculation method employed for the 16th sub-basin was also utilized for the ecological flow calculations of all rivers within the SWAT model sub-basins of the Hailang River basin (Figure 9). This approach enabled the establishment of the spatiotemporal distribution of ecological flow in the basin. A comprehensive analysis revealed that the primary stem of the Hailang River basin exhibited the highest demand for water flow. This area demonstrated comparatively diminished ecological flows from January to April, with the majority of regions exhibiting ecological flows ranging from 0.00 to 7.03 m³/s. The lowest values occur from January to February, with most areas in the basin having ecological flows below 3.57 m³/s. From June to September, ecological flows reach their annual peak, with the northeastern and central regions reaching 22.58 to 96.93 m³/s. From November to December, ecological flow gradually decreases, with most areas dropping to the range of 7.03–22.58 m³/s. From a spatial distribution perspective, the ecological flow in the Hailang River basin exhibits significant regional differences and distinct spatiotemporal distribution patterns.

The spring ecological flow changes in the Hailang River basin exhibit unique glacial hydrological characteristics. The low ecological flow in the Hailang River basin from January to April is the result of multiple factors acting in concert. During this period, the average temperature remains below freezing, precipitation is scarce, and it predominantly occurs in solid form. Precipitation during this period accounts for only 10–15% of the annual total, and primarily exists as snow, which cannot be immediately converted into runoff. From March to May each year, as temperatures rise, snow begins to melt, forming snowmelt runoff that provides an important supplement to spring ecological flow. In the upstream areas situated at higher elevations, snowmelt occurs at a later date, while in the downstream areas, snowmelt occurs at an earlier date. The differential snowmelt process extends the duration of spring runoff, thereby facilitating a smooth transition of ecological flow. Soil freeze–thaw cycles also influence spring runoff processes. The presence of permafrost has been shown to impede the infiltration of precipitation and meltwater, thereby increasing surface runoff. However, as the soil undergoes thawing, its infiltration capacity undergoes a gradual recovery, resulting in a corresponding decrease in runoff coefficients. This finding aligns with the sensitivity of key SWAT model parameters calibrated for this cold-region watershed. Specifically, the shallow groundwater evaporation depth threshold (REVAPMN) was calibrated to 179.64 mm, and the shallow aquifer water level threshold when groundwater enters the main river channel (GWQMN) was set to 3961.22 mm. Other critical parameters include the temperature lapse rate (TLAPS = −0.058 °C/km), the plant absorption compensation coefficient (EPCO = 0.636), the groundwater re-evaporation coefficient (GW_REVAP = 0.127), and the groundwater runoff delay time (GW_DELAY = 162.31 days). These parameter values reflect the unique hydrological characteristics of cold-region watersheds, particularly the extended groundwater delay time and elevated aquifer threshold, which are consistent with slower groundwater movement in frozen soil conditions. The average ecological flow in the watershed from January to April was only 6.2 m³/s, approximately 40% of the annual average. This is a typical characteristic of cold-region rivers during the winter and spring seasons. Achieving this low-flow state necessitates judicious water resource scheduling to ensure that ecological necessities are met.

5. Discussion

5.1. Scientific Significance of Copula-Based Improved Monthly Frequency Method

The conventional monthly frequency method for calculating river ecological flow generally utilizes a fixed guarantee rate for calculations. This method involves the direct extraction of flow values corresponding to the guarantee rate from the monthly flow sequence, which are then designated as the ecological flow for that specific month. Uddin et al. [64] calculated the ecological base flow of the Guanzhong section of the Wei River using the monthly frequency method. Their approach involved setting different guarantee rates based on different wet and dry year types. They also compared multiple ecological flow calculation methods to select the optimal approach. Palau et al. [65] also employed the monthly frequency method and other methods to determine the minimum ecological flow. A comparison of multiple calculation methods revealed that the traditional monthly frequency method was not the optimal method for calculating ecological flow. The traditional monthly frequency method does not account for the complex relationship between annual average flow and monthly average flow. The relationship between the two guarantee rates is not merely a simple correspondence. When the same guarantee rate is applied, it may result in certain months during low-flow years having higher flow values than the same months during high-flow years. In contrast, the improved monthly frequency method used in this study yields monthly guarantee rates ranging from 82.15% to 90.08%, rather than the uniform guarantee rate of the traditional method. This discrepancy is indicative of the genuine statistical correlation between monthly and annual flows, thereby enhancing the congruence with the hydrological patterns exhibited by the watershed. As Liu et al. [66] have observed, in inland arid regions characterized by the rapid degradation of ecological environments, climate change and anthropogenic activities serve as catalysts for hydrological variability. A comparison was made of the Tennant method for ecological flow assessment, resulting in the confirmation that the traditional monthly frequency method exhibits significant uncertainty under extreme environmental conditions. The consistent performance of our improved monthly frequency method under various hydrological extreme conditions is primarily attributed to the Copula function’s ability to capture the tail dependence between interannual and intermonthly flows. A comparison of the Copula function-enhanced monthly frequency method with traditional methods reveals its superiority in calculating ecological flow values under extreme low-flow scenarios. This finding is significant for the field of ecological conservation.

5.2. Theoretical Advantages of Copula-Based Improvements

Copula functions serve as instrumental tools in establishing a link between marginal distributions and joint distributions, thereby providing a theoretical foundation for the relationship between annual average flow and monthly average flow. Copula functions have been extensively applied in hydrological modeling due to their ability to construct joint distributions with nonlinear, asymmetric, and complex dependency structures between hydrological variables [67]. These functions provide significant advantages in modeling multivariate hydrological relationships, as demonstrated in numerous water resources applications. For instance, Copula-based approaches have been successfully used for flood frequency analysis [68], drought analysis, and rainfall–runoff correlation studies. Among the various Copula functions, t-Copula provides enhanced flexibility for modeling high-dimensional dependencies in hydrological applications [69]. However, studies in hydroclimatic analysis have shown that t-Copula requires significantly larger sample sizes (n > 100 years) for reliable parameter estimation, which can be challenging for many hydrological datasets with limited record lengths [70]. In this study, we selected the most suitable Copula function for each month using an optimized approach, with four different types of Copula functions chosen for the 12 months. This differential selection is predicated on a robust hydrological foundation. The Gumbel Copula was identified as the optimal function for January, March, April, and June as its positive tail correlation characteristic indicates that monthly flow in these months exhibits stronger synchrony. The Clayton Copula is suitable from September to December, with its lower-tail correlation reflecting stronger synchrony during the wet season. The applicability of Gaussian and t-Copula is contingent upon the distinct transitional months, thereby reflecting the relatively stable correlation structure during these periods. This differential selection is not set at random; rather, it is objectively derived from rigorous statistical tests (AIC, BIC, etc.), which scientifically reflect the seasonal characteristics of the hydrological processes in the Hailang River basin. The improved method is predicated on establishing the conditional probability and joint probability relationship of monthly average flow under low-flow year conditions. The criterion for identifying low-flow years is set to P(X₀≥x₀, 90%), and the conditional probability target for monthly average flow is set to 90%. The corresponding joint probability and marginal probability are derived. This process ensures that the calculation of ecological flow for each month under low-flow year conditions accurately reflects the actual hydrological characteristics of that year type. The ecological flow calculated using the improved method is generally higher than that of the traditional method, particularly from August to October, providing a more adequate water supply during critical ecological periods. Furthermore, the alterations in ecological flow are in complete alignment with seasonal variations.

The Tennant method has been demonstrated to facilitate a comprehensive consideration of the seasonal needs of aquatic organisms, thereby ensuring a more precise alignment with the temporal patterns of river ecosystems. For instance, in a study by Guo et al. [71], it was proposed that the Tennant method can reasonably analyze ecological flow thresholds for fish spawning and fattening periods, as well as general water use periods. Khatar and Shokoohi [72] demonstrated the direct relationship between Tennant flow categories and habitat simulation methods. The grading criteria established by the Tennant method can assist in determining the grading criteria for improving the monthly frequency of ecological flow thresholds. The ecological flow calculated using the improved methodology achieved an “excellent” rating or higher throughout the entire year, thereby significantly enhancing the overall level of ecological conservation. The fundamental reason for this superiority lies in the improved method’s more accurate identification of the actual low-flow conditions in each month during the low-flow year, thereby providing a more scientifically sound and reasonable ecological flow guarantee. This study provides robust hydrological validation through the Tennant method. The calculated ecological flows meet “excellent” or higher standards throughout the year (Table 4). However, this approach does not account for species-specific habitat requirements or the actual response of organisms to flow mechanisms. While widely accepted in hydrological practice, the Tennant method represents a simplified approach, relying primarily on flow-size relationships rather than detailed assessments of ecosystem responses. This verification gap is particularly significant for cold-region rivers such as the Hailang River basin. In these environments, ice dynamics, freeze–thaw cycles, and seasonal temperature variations impose unique ecological constraints. These constraints cannot be fully captured by methods developed for temperate climates. Future applications of this refined monthly frequency approach should prioritize comprehensive ecological validation through fish habitat suitability modeling, benthic invertebrate community assessments, and integration with local fisheries management data. This will enhance the ecological credibility and practical applicability of the proposed methodology.

5.3. Limitations and Future Research Directions

This study better preserves the statistical correlations between interannual and intermonthly flows compared to the internationally prevalent RVA method. The ecological flow process line also more closely aligns with natural hydrological patterns [73]. However, certain limitations exist. Under climate change conditions, the stability of Copula function parameters faces challenges. Rising temperatures and shifting precipitation patterns may alter the correlation structure between annual and monthly average flows, causing drift in established Copula model parameters. It is recommended that future research should incorporate time-varying Copula models, dynamically adjusting parameters to accommodate non-stationary conditions.

Additionally, the spatial data resolution inconsistency issues present in this study may introduce certain uncertainties in the analysis results. The research utilized multi-source spatial data with varying resolutions, including 30 m DEM data, point-based meteorological station data, and land use and soil data with different spatial accuracies [74]. These resolution disparities among multi-source datasets may introduce uncertainties during data fusion and spatial interpolation processes, particularly in meteorological data spatialization and sub-basin parameter determination. Although the SWAT model has the capability to handle multi-resolution data, resolution mismatches may still affect the spatial representation accuracy of the model and the local precision of ecological flow calculations [75]. Future research should consider adopting more consistent spatial resolution datasets or developing more advanced multi-scale data fusion methods to reduce uncertainties caused by resolution inconsistencies.

Furthermore, a fundamental challenge encountered in this study pertains to the representativeness of historical data within the paradigm of climate change [76,77]. The intensification of global climate change has led to a significant increase in the frequency and intensity of drought events, coupled with a persistent decline in river runoff. This poses a severe challenge to the applicability of ecological flow calculation methods based on historical data in current and future scenarios. The year 2016 was selected as the endpoint for this analysis primarily due to increasing limitations in accessing detailed meteorological observation data in recent years, which prevented extending the analysis time series further. Notwithstanding the fact that this constraint has a certain effect on the timeliness of the findings, the long-term data sequence from 1956 to 2016 still provides a sufficiently large sample size for robust statistical analysis of extreme hydrological events and long-term evolution patterns. Moreover, the present study has accurately characterized the statistical correlation between annual and monthly average flows, a correlation structure that exhibits relative stability in the short term. This stability lends the current findings’ strong practical value. Future research may focus on three directions: firstly, incorporating climate change scenario analysis to integrate future climate projections into ecological flow estimation frameworks; secondly, developing non-stationary statistical models to accommodate hydrological sequence characteristics under changing conditions; and thirdly, establishing dynamic ecological flow regulation mechanisms based on real-time monitoring data to effectively address uncertainties arising from climate change.

6. Conclusions

This study addresses the critical issue that traditional monthly frequency methods neglect interannual and intermonthly statistical correlations in ecological flow calculations. It constructs an improved monthly frequency method framework based on Copula functions. Through the Mann–Whitney–Pettitt method and ordered cluster analysis, a significant breakpoint (α = 0.05) was identified in the 1956–2016 flow series of the Hailang River basin at 1966, confirming that the 1956–1965 data represent the basin’s natural hydrological conditions. The SWAT natural runoff model constructed using this period’s data demonstrated excellent performance, with an R² = 0.84 and NSE = 0.83 during the calibration period and R² = 0.82 and NSE = 0.81 during the validation period, providing a reliable data foundation for ecological flow calculations.

The study systematically evaluated the fitting performance of five Copula functions (Gaussian, t-Copula, Clayton, Frank, and Gumbel) across 12 months using multiple statistical metrics including AIC, BIC, RMSE, and K-S tests, thereby establishing a monthly differentiated Copula model selection mechanism. The results indicate that Gaussian Copula is suitable for February, May, and July; t-Copula for August; Clayton Copula for September to December; and Gumbel Copula for January, March, April, and June. This monthly optimized Copula function selection strategy scientifically reflects the seasonal characteristics of hydrological processes in the Hai Lang River basin, providing a replicable methodological framework for diverse climatic regions.

The core methodological innovation of this research lies in establishing a quantitative relationship between the conditional probability and joint probability of monthly mean flow under the identification criterion for low-flow years defined by P(X₀≥x₀, 90%) = 0.9. This addresses the theoretical shortcoming in conventional approaches where annual mean flow and monthly mean flow guarantee rates are treated independently. The guaranteed flow rate range determined by the improved method for Sub-basin 16 spans 81.83% to 90.08% across months. Compared to the uniform 90% guarantee rate of traditional methods, this better reflects the actual hydrological characteristics of different months and effectively avoids the unreasonable phenomenon where the flow rate in a given month during a low-flow year exceeds that of the same month in a high-flow year. Validation using the Tennant method indicates that the ecological flow calculated by the improved method meets the “excellent” standard or higher throughout the year. Specifically, from January to May, it accounts for 30.78% to 57.84% of the multi-year average natural flow, while from June to December, it accounts for 63.07% to 80.95%. Compared to the traditional method, this provides more adequate water supply during the critical ecological period from August to October.

Based on research findings, a seasonal ecological flow regulation strategy for the Hai Lang River basin is proposed: (1) a spring storage period (March–May): appropriately increase water storage whilst meeting basic ecological flow requirements, preparing for the summer peak water demand; (2) a critical Summer–Autumn period (June–September): strictly implement flow in accordance with the ecological flow standards determined by this study to ensure the integrity of the river’s ecological functions; (3) a winter protection period (December–February): maintain a base flow of no less than 6.2 m³/s to safeguard fish overwintering habitat requirements. This scheduling framework provides scientific rationale and technical support for ecological flow management in cold-region rivers.

This study has certain limitations: (1) the representativeness of historical data faces challenges in the context of climate change; future research should incorporate climate change scenario analysis; (2) inconsistent spatial resolution across multi-source datasets may introduce uncertainty, necessitating higher-precision, consistent data; (3) whilst hydrological plausibility was validated using the Tennant method, ecological validation specific to species habitat requirements is lacking. It is recommended to conduct comprehensive assessments integrating biogeographical approaches such as fish habitat suitability modeling.

The improved monthly frequency method based on Copula functions established in this study provides a novel theoretical framework and technical pathway for calculating ecological flow in cold-region rivers, demonstrating promising prospects for broader application. This approach is not only applicable to the Haiwang River basin but also offers valuable insights for similar basins facing dual impacts from climate change and human activities. Future research should focus on (1) dynamic adjustment mechanisms for Copula parameters under non-stationary conditions; (2) integrated assessment of ecological flow requirements across multiple timescales; (3) synergistic optimization strategies for ecological flows and water resource allocation.