Study on Flood Season Segmentation and Rationality Examination for Wuluwati Reservoir

Abstract

Scientific flood season segmentation serves as the foundation for determining the flood-limited operating water levels across different periods, providing crucial support for reservoir flood control safety operations and optimal water resource utilization. Under the background of climate change, the traditional static flood-limited water level management model based on fixed dates struggles to adapt to variations in flood season patterns. This study aims to establish a scientifically sound flood season segmentation scheme, providing a basis for dynamic control of flood-limited water levels across different periods, thereby improving water resource utilization efficiency while ensuring flood control safety. This study focuses on the Wuluwati Reservoir and employs the circular distribution method and the Fisher optimal partition method to conduct its flood season segmentation calculations. First, the circular distribution method is used to analyse the concentration and periodic characteristics of flood occurrences in the basin. Subsequently, the Fisher optimal partition method is applied to perform statistical segmentation of the historical hydrological series. Based on this analysis, the flood season of the Wuluwati Reservoir is comprehensively determined as: the pre-flood season from 1 June to 2 July, the main flood season from 3 July to 27 August, and the post-flood season from 28 August to 30 September. To objectively evaluate the rationality of the segmentation results, the improved Cunderlik method was employed to examine the rationality of 15 segmentation schemes based on relative superiority degree. The results show that the scheme with the main flood season from 3 July to 23 August achieves the highest relative superiority degree (0.930). The comprehensively determined segmentation of this study (3 July–27 August) encompasses this optimal interval, demonstrating that the flood season segmentation for the Wuluwati Reservoir is reasonable and effective.

1. Introduction

Reservoirs are key engineering structures for regulating basin water resources and balancing flood control and beneficial water use [1]. The precision of their operation directly impacts flood safety and water utilization efficiency. Currently, climate change has led to an increase in extreme hydrological events [2]. Traditional methods are based on the assumption of historical climatic stability. However, climate change has altered the timing, intensity, and frequency of extreme rainfall events, making the single flood-limited water level based on fixed dates either overly conservative (wasting water resources) or insufficiently safe. Therefore, more refined, statistically based segmentation is needed to support dynamic adjustments. scientifically segmenting the flood season and examining the rationality of such segmentation have become an important foundation for implementing dynamic reservoir operation [3], and coordinating safety with resource utilization.

Methods for flood season segmentation can be primarily categorized into qualitative analysis and quantitative analysis. Qualitative analysis methods mainly rely on historical flood investigations and extensive practical experience to subjectively determine the start and end dates of the flood season. A typical method is the causative analysis method. Ai et al. [4] studied the flood season segmentation of the Three Gorges Reservoir by analysing flood occurrence statistics at the Yichang Station. As qualitative analysis requires examining numerous causative factors, it possesses strong subjectivity and struggles to objectively reflect the inherent statistical law of hydrological processes [5].

With the application of mathematical statistical methods, research on flood season segmentation has entered a quantitative stage. Common quantitative analysis methods include the change-point analysis method [6], fuzzy set analysis method [7,8], fractal method [9], circular distribution method [10], Fisher optimal partition method [11], and set pair analysis method [12], among others. For example, Li et al. [13] analyzed the influence of different time-domain units on flood season segmentation using the Fisher optimal partition method; Zhou et al. [14] combined multiple methods, such as the fractal analysis method and MCP analysis method, to segment the flood season of the lower Yellow River reach; Wu et al. [15] employed an improved set pair analysis method for flood season segmentation of a tropical island river.

The quantitative methods mentioned above have all contributed valuable explorations to flood season segmentation. Taking the Wuluwati Reservoir in Hetian County, Xinjiang, China as a case study, this paper employs two mathematical statistical methods to conduct research on flood season segmentation. It also examines the rationality of the segmentation results and establishes a two-step framework that first segments the flood season and then assesses the reasonableness of the segmentation. This framework provides a scientific basis for rationally determining the flood-limited water levels for different periods at the Wuluwati Reservoir. The specific objectives of this study include: (1) applying the circular distribution method and the Fisher optimal partition method to calculate the flood season segmentation for the Wuluwati Reservoir; (2) integrating the results from both methods to derive the final segmentation; (3) employing the improved Cunderlik method to examine the rationality of the segmentation results.

2. Materials and Methods

2.1. Flood Season Segmentation

Flood season segmentation aims to identify sub-periods within the flood season that exhibit distinct hydrological characteristics. This study employs two complementary methods: the circular distribution method focuses on analyzing the temporal concentration of flood events, reflecting the seasonal aggregation patterns of floods through indicators such as the concentration period and concentration degree; the Fisher optimal partition method utilizes multiple hydrometeorological indicators (discharge, rainfall, temperature) to identify stages with similar variation patterns, enabling segmentation down to the daily scale. The comprehensively determined segmentation results will be examined for rationality using the improved Cunderlik method to enhance the robustness and physical consistency of the results. The principles of each method are briefly described as follows.

2.1.1. Circular Distribution Method

The circular distribution method treats flood occurrence times as vectors on a circle. By calculating the resultant direction (concentration period) and the resultant length (concentration degree) of these vectors, this method reveals whether floods tend to occur during a particular period of the year and the degree of their concentration, thereby determining the flood season segmentation.

Step 1: Data Transformation and Vectorization

Assume there are N flood samples, and the total number of days in the calculation period is T. For the i-th flood sample, its occurrence time is D_i, and the flood magnitude is q_i. The occurrence time (angle) of the i-th flood is then:

$${\alpha }_i=D_i{\displaystyle \frac{2\pi }{T}}, 0\le {\alpha }_i\le 2\pi$$

(1)

The coordinate values (x_i, y_i) representing the occurrence time of the i-th flood event are calculated as follows, both for scenarios disregarding and considering flood magnitude:

$$\left(\begin{array}{c}{x}_i,y_i\end{array}\right)=\left\{\begin{array}{c}\left(\begin{array}{c}\cos {\alpha }_i, \sin {\alpha }_i\end{array}\right)\hspace{1em} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{t} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{m}\mathrm{a}\mathrm{g}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{f}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d}\\ \left(\begin{array}{c}{q}_i\cos {\alpha }_i, q_i\sin {\alpha }_i\end{array}\right) \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{m}\mathrm{a}\mathrm{g}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{f}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d}\end{array}\right.$$

(2)

Step 2: The Calculation of the Resultant Vector

By synthesizing the coordinate values of all samples, the coordinate values of the resultant vector are obtained:

$$\left(\overline{x},\overline{y}\right)=\left({\displaystyle \frac{{\sum }_{i=1}^N{x}_i}{N}},{\displaystyle \frac{{\sum }_{i=1}^N{y}_i}{N}}\right)$$

(3)

Step 3: The Calculation of Concentration Period and Degree

Concentration Period ($\overline{\alpha }$): This represents the average direction of flood occurrence times, i.e., the directional angle of the resultant vector. Based on the coordinate values of the resultant vector, the concentration period ($\overline{\alpha }$) can be calculated using the following formula:

$$\overline{\alpha }=\left\{\begin{array}{cc}\arctan \overline{y}/\overline{x}& \overline{x}>0, \overline{y}>0\\ 2\pi +\arctan \overline{y}/\overline{x}& \overline{x}>0, \overline{y}<0\\ \pi +\arctan \overline{y}/\overline{x}& \overline{x}<0\\ \pi /2& \overline{x}=0, \overline{y}>0\\ 3\pi /2& \overline{x}=0, \overline{y}<0\end{array}\right.$$

(4)

The concentration day $\overline{D}$ corresponding to the concentration period $\overline{\alpha }$ is:

$$\overline{D}=\overline{\alpha }{\displaystyle \frac{\mathrm{T}}{2\pi }}$$

(5)

Concentration Degree (γ): This index describes the central tendency of flood occurrence times, with values ranging from 0 to 1. A value of γ closer to 1 indicates that flood occurrence times are more concentrated, while a value closer to 0 suggests greater dispersion. The formula for calculating the concentration degree γ is:

$$\mathsf{\gamma }=\left\{\begin{array}{c}\sqrt{{\overline{x}}^2+{\overline{y}}^2}\hspace{1em} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{t} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{m}\mathrm{a}\mathrm{g}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{f}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d}\\ \sqrt{{\overline{x}}^2+{\overline{y}}^2}/\overline{q} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{m}\mathrm{a}\mathrm{g}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{f}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d}\end{array}\right.$$

(6)

where $\overline{q}$ is the mean flood peak flow of the N flood samples.

Step 4: The Calculation of the Start and End Dates for the Peak Flood Period

Calculate the standard deviation S, which characterizes the degree of dispersion in the timing:

$$S =\sqrt{-2\mathrm{l}\mathrm{n}\mathsf{\gamma }}$$

(7)

Calculate the start and end dates (D₁, D₂) of the peak flood period, which defines the main flood season:

$$D_1={\displaystyle \frac{\overline{\alpha }-S}{2\pi }}T$$

(8)

$$D_2={\displaystyle \frac{\overline{\alpha }+S}{2\pi }}T$$

(9)

Thus, bounded by D₁ and D₂, the entire flood season can be segmented into the pre-flood season (from the flood season start to D₁), the main flood season (from D₁ to D₂), and the post-flood season (from D₂ to the end of the flood season).

2.1.2. Fisher Optimal Partition

The Fisher optimal partition method is a clustering technique for ordered samples. In this study, each ten-day period during the flood season is treated as an ordered sample, with each sample containing multiple hydrometeorological indicators (discharge, rainfall, temperature). By calculating the sum of squared deviations within classes, this method partitions the flood season into consecutive stages that are as internally homogeneous as possible, thereby identifying natural transition points in flood-related variables. It integrates information from multiple indicators, avoiding the one-sidedness of a single indicator and objectively dividing the flood season into stages.

Step 1: Data Preprocessing and Standardization

Assume there are n ordered samples in total, with each sample containing m indicators. A data matrix X can be constructed as follows:

$$\mathrm{X}=\left[\begin{array}{cccc}{x}_{11}& x_{12}& \cdots & x_{1m}\\ x_{21}& x_{22}& \cdots & x_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ x_{n1}& x_{n2}& \cdots & x_{nm}\end{array}\right]$$

(10)

To eliminate the influence of different measurement units, the range standardization method is applied:

$$x_{ij}^{\mathrm{*}}={\displaystyle \frac{x_{ij}-\underset{1\le i\le n}{min}{x}_{ij}}{\underset{1\le i\le n}{max}{x}_{ij}-\underset{1\le i\le n}{min}{x}_{ij}}}$$

(11)

where $x_{ij}^{*}$ represents the normalized value, ranging within [0,1]. $\underset{1\le i\le n}{max}{x}_{ij}$, $\underset{1\le i\le n}{min}{x}_{ij}$ are the maximum and minimum values of the j-th indicator column, respectively.

Step 2: Definition of Class Diameter

Let a class contain the samples $\{x_i,x_{i+1},\dots ,x_j\}$ (where j ≥ i), denoted as G_ij. The mean vector for this class is:

$${\overline{x}}_{ij}={\displaystyle \frac{1}{j-i+1}}\sum _{t=i}^j{x}_t$$

(12)

The class diameter D(i,j) is defined as the sum of squared deviations of all samples within the class from the mean:

$$D\left(i,j\right)=\sum _{t=i}^j(x_t-{\overline{x}}_{ij}{)}^T(x_t-{\overline{x}}_{ij})$$

(13)

A smaller diameter indicates greater similarity among the samples within that class.

Step 3: Objective Function and Recurrence Formula

Dividing n samples into k classes is denoted as partition P(n,k). Its objective function is defined as the sum of the diameters of all classes:

$$e\left[P\left(n,k\right)\right]=\sum _{r=1}^{k}D(i_r,i_{r+1}-1)$$

(14)

where $1=i_1<i_2<\cdots <i_k<i_{k+1}=n+1$ represents the partition points.

The optimal partition $P^{*}(n,k)$ is the one that minimizes this objective function:

$$e\left[P^{\mathrm{*}}\left(n,k\right)\right]=\mathrm{m}\mathrm{i}\mathrm{n}\left\{e\left[P\left(n,k\right)\right]\right\}$$

(15)

This can be solved using the following recurrence formulas:

$$e\left[P^{\mathrm{*}}\left(n,2\right)\right]=\underset{2\le j\le n}{min}[D(1,j-1)+D(j,n)]$$

(16)

$$e\left[P^{\mathrm{*}}\left(n,k\right)\right]=\underset{k\le j\le n}{min}\{e[P^{\mathrm{*}}(j-1,k-1)]+D(j,n)\}$$

(17)

Step 4: Solution of the Optimal Partition

Calculate all possible D(i,j) (1 ≤ i ≤ j ≤ n) and store them in a diameter matrix; Use the recurrence formulas to compute $e\left[P^{*}\right(n,k\left)\right]$ for all values of k; Backtrack to identify the partition points: First, find j_k that minimizes $e\left[P^{\mathrm{*}}\right(n,k\left)\right]$. The last class is then $\{j_k,j_k+1,\dots ,n\}$. Proceed forward iteratively to successively determine j_k-1, j_k-2, …, j₂. Obtain the optimal k-class partition result.

Step 5: Determination the Optimal Number of Classes (k)

Calculate the rate of change in the objective function value with respect to k:

$$f\left(k\right)={\displaystyle \frac{e\left[P^{\mathrm{*}}\right(n,k-1\left)\right]-e\left[P^{\mathrm{*}}\right(n,k\left)\right]}{k-(k-1)}}=e[P^{\mathrm{*}}(n,k-1)]-e[P^{\mathrm{*}}(n,k)]$$

(18)

Plot the f(k)∼k curve. The value of k corresponding to a distinct inflection point in this curve is selected as the optimal number of classes. To quantify this, the change in slope can be calculated as:

$$\beta \left(k\right)=\left|f\right(k)-f(k+1\left)\right|$$

(19)

The maximum value of β(k) corresponds to the inflection point on the f(k)∼k curve, indicating that this particular k is the optimal value.

2.2. Rationality Examination

To examine the rationality of the flood season segmentation results. The improved Cunderlik method can be used to test whether the frequency of flood occurrences within each segmentation stage deviates significantly from a uniform random distribution. This method is based on the flood seasonality testing framework proposed by Cunderlik et al. [16], incorporating improvements by Chen et al. [17] that integrate fuzzy optimum selection with nonparametric bootstrap sampling techniques. By constructing confidence intervals under the assumption of a uniform distribution and accounting for sampling uncertainty through bootstrap sampling, this method quantitatively evaluates how well different segmentation schemes reflect the seasonality of floods. Finally, a fuzzy optimum selection model is used to compare multiple candidate schemes and identify the most reasonable segmentation.

Step 1: Calculation of Relative Frequency

To quantify the frequency characteristics of flood occurrences within each segmentation stage, the relative frequency for each stage is first calculated. Annual Maximum (AM) sampling is adopted. From the daily inflow data spanning M years, M flood events are extracted. The number of these M flood events falling within each segmentation stage j is counted as b_j. The relative frequency is then calculated as:

$$R{F}_j={\displaystyle \frac{b_j}{M}},j=\mathrm{1,2},\dots ,n$$

(20)

As the number of days in each segmentation stage varies, the relative frequency RF_j is adjusted with a weighting factor based on the number of days to eliminate the influence of differing stage lengths on frequency comparison:

$$R{F}_j^{\prime }={\displaystyle \frac{R{F}_j\cdot a}{n\cdot a_j}}$$

(21)

In the formula, $R{F}_j$ is the original relative frequency, $a$ is the total number of days in the flood season, $a_j$ is the number of days in the j-th segmentation stage, and n is the number of stages. By multiplying by ${\displaystyle \frac{a}{a_j}}$, the frequencies of stages with different lengths can be normalized to a comparable basis.

Step 2: Construction of Confidence Intervals

Assume that N flood events occur uniformly and randomly within the flood season spanning $a$ days; Set the number of flood events N to increment from 30 to 500. For each N, randomly generate the occurrence dates of N floods, count the number falling into each segmentation stage, and calculate the adjusted relative frequency RF_j′. This simulation process is repeated N_sim times, yielding N_sim × n values of RF_j′. All RF_j′ values are then sorted in ascending order. The values at the α and 1−α quantiles are taken as the lower limit y₁ and upper limit y₂ of the confidence interval (typically with α = 0.05). Curve fitting is performed for the corresponding y₁(N) and y₂(N) for each N.

Step 3: Bootstrap Sampling

The nonparametric bootstrap method is employed. The M-year daily inflow series serves as the initial sample. Using a random number generator, data for M years are drawn with replacement from the initial sample to form a bootstrap sample. This process is repeated m = 1000 times, resulting in 1000 bootstrap samples, each with a size of M. For each bootstrap sample, the Annual Maximum method is applied to extract M flood events. The number of floods in each segmentation stage is counted, and the adjusted relative frequency RF′_k,j is calculated (where k denotes the k-th bootstrap sample).

Step 4: Fuzzy Optimum Selection and Calculation of Relative Membership Degree

To comprehensively compare the rationality of different segmentation schemes, a generalized distance measure and a fuzzy optimum selection model are employed to calculate the relative membership degree for each scheme.

Define the generalized distance d_k,j to quantify the deviation of the bootstrap sample frequency RF′_k,j from the confidence interval bounds:

$$d_{k,j}=\left\{\begin{array}{c}{\left[{\displaystyle \sum _{i=1}^m}{\left(R{F}_{k,j}^{\prime }-y_2\right)}^p\right]}^{1/p}(\mathrm{M}\mathrm{a}\mathrm{i}\mathrm{n} \mathrm{F}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d} \mathrm{S}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{o}\mathrm{n})\\ {\left[{\displaystyle \sum _{i=1}^m}{\left(y_1-R{F}_{k,j}^{\prime }\right)}^p\right]}^{1/p}(\mathrm{N}\mathrm{o}\mathrm{n}\text{-}\mathrm{M}\mathrm{a}\mathrm{i}\mathrm{n} \mathrm{F}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d} \mathrm{S}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{o}\mathrm{n})\end{array}\right.$$

(22)

where i indexes the sampling iterations (1000 times in total), and p defines the distance metric, this study adopts p = 1 (Hamming distance).

Based on the sign (positive/negative) of d_k,j, the corresponding formula is selected to compute the relative membership degree:

(1)

If d_k,j > 0 (indicating a larger value is better):

$$r_{k,j}={\displaystyle \frac{d_{k,j}}{max\left(d_{k,j}\right)}}$$

(23)

(2)

If d_k,j < 0 (indicating a smaller value is better):

$$r_{k,j}={\displaystyle \frac{min\left(\right|d_{k,j}\left|\right)}{\left|d_{k,j}\right|}}$$

(24)

(3)

If the signs of d_k,j are mixed (both positive and negative values exist):

$$r_{k,j}={\displaystyle \frac{d_{k,j}-min\left(d_{·,j}\right)}{max\left(d_{k,j}\right)-min\left(d_{k,j}\right)}}$$

(25)

where max(d_k,j) and min(d_k,j) represent the maximum and minimum values of the generalized distance across all schemes for the j-th segmentation stage, respectively.

Step 5: Calculation of Fuzzy Relative Superiority

The two-stage fuzzy membership degree formula is employed to calculate the fuzzy relative superiority u_k for each segmentation scheme k:

$$u_k={\left[1+{\displaystyle \frac{{\sum }_{j=1}^n{\left[{\omega }_j\left(r_{k,j}-1\right)\right]}^p}{{\sum }_{j=1}^n{\left({\omega }_j{r}_{k,j}\right)}^p}}\right]}^{-1}$$

(26)

In this formula, ω_j represents the weight of the j-th segmentation stage. Equal weights are typically assigned ω_j = 1/n; The parameter p is set to 2. u_k ∈ [0,1]. A larger value of u_k indicates a more optimal segmentation scheme. The value u_k = 1 corresponds to the ideal optimal scheme.

3. Results

3.1. Overview of the Reservoir

The Wuluwati Reservoir is located in Hetian County, Xinjiang Uygur Autonomous Region, with a total storage capacity of 333.6 million m³. It is a large-scale (Type II, according to the Chinese reservoir classification standard for reservoirs with storage capacity between 100 million and 1 billion m³) reservoir serving comprehensive purposes, including irrigation, flood control, and power generation, belonging to the Kalakashi River Basin. The average annual precipitation in the basin is 81.1 mm, classifying it as an arid region. The downstream irrigation areas have high agricultural water demands, requiring a water supply reliability rate for agricultural irrigation of over 75% (i.e., in at least 75 out of 100 years, the irrigation water demand is fully met). Concurrently, it is essential to ensure ecological water demand downstream while integrating high comprehensive utilization needs, such as power generation and flood control. However, water resources are concentrated during the flood season, leading to a mismatch between the timing of water availability and water demand.

Floods in the basin include rainstorm floods, snowmelt floods, and mixed floods. Precipitation is concentrated during the four months of the flood season (June to September), which easily combines with snowmelt to form significant mixed floods. The basin faces the dual pressures of water scarcity and flood disasters, making research on floodwater resource utilization urgently necessary. As the only large-scale water control project in the basin, scientifically and rationally determining the flood-limited operating water level control based on flood season segmentation for the Wuluwati Reservoir holds significant practical importance for alleviating the contradiction between water supply and demand in this region. The Wuluwati Reservoir and its basin are shown in Figure 1. The data used in this study were obtained from the Wuluwati Hydrological Station (with data from 1997 to 2001 sourced from the Tuoman Hydrological Station).

3.2. Flood Season Segmentation Result

3.2.1. Flood Season Segmentation Based on the Circular Distribution Method

Taking the daily average inflow during the flood season from 1957 to 2012 at the Wuluwati Reservoir as the fundamental dataset, the circular distribution method was applied to conduct flood season segmentation research.

The maximum t-day flow values were sequentially extracted from the 53-year dataset and denoted as q_Lt, where the year index L = 1, 2, …, 53, and the sampling period length for maximum flow t = 1, 3, 5, 7. Based on Equations (1)–(9), key indicators such as the concentration period, concentration day, and concentration degree were calculated for different sampling periods under scenarios both considering and not considering flood magnitude. The calculation results are presented in

Table 1. Results of Flood Season Segmentation Using the Circular Distribution Method.

Sampling Period t/d	Whether to Consider the Flood Magnitude	Concentration Degree γ	$\mathbf{Concentration} \mathbf{Period} \overline{\mathit{\alpha }}$	$\mathbf{Concentration} \mathbf{Day} \overline{\mathit{D}}$	Main Flood Season
1	No	0.822	2.996	7/28	7/16–8/9
	Yes	0.493	3.004	7/28	7/5–8/20
3	No	0.805	3.082	7/29	7/17–8/11
	Yes	0.495	3.065	7/29	7/6–8/21
5	No	0.797	3.118	7/30	7/17–8/12
	Yes	0.477	3.096	7/30	7/6–8/22
7	No	0.806	3.189	7/30	7/19–8/13
	Yes	0.480	3.153	7/30	7/7–8/23

.

When flood magnitude is not considered, all flood events have equal weight, reflecting the concentration of flood occurrence times. When flood magnitude is considered, major floods are assigned higher weights, reflecting the temporal concentration pattern of flood magnitudes. Comparing the two can reveal the important physical characteristic of whether the timing of major floods differs from that of ordinary floods. When flood magnitude is not considered, the concentration degree γ is consistently above 0.80, indicating that flood occurrence times are highly concentrated. However, when flood magnitude is taken into account, the γ value decreases to a range of 0.45–0.50, suggesting that major flood events are more dispersed over time. This phenomenon reveals the inherent characteristics of flood composition in the basin where the Wuluwati Reservoir is located: floods include rainstorm floods, snowmelt floods, and mixed floods resulting from their superposition. Major floods are often formed by the combination of extreme rainfall or high-temperature snowmelt, and such meteorological events can exhibit multiple peaks within the June–September period, leading to the occurrence times of major floods not being entirely concentrated within a narrow timeframe. Therefore, the results of the circular distribution method considering flood magnitude better align with the actual occurrence patterns of major floods in the basin. Under the scenario considering flood magnitude, the concentration degree γ across different sampling periods ranges from 0.45 to 0.50, the concentration period $\overline{\alpha }$ ranges from 3.00 to 3.20, and the corresponding concentration day $\overline{D}$ falls between 28 July and 30 July. According to the scenario that considers flood magnitude, the maximum coverage range of the main flood season for different sampling lengths is 5 July to 23 August. Adhering to the principle of prioritizing operational safety, the main flood season is determined as 5 July to 23 August. The flood season for the basin where the Wuluwati Reservoir is located spans June to September. Correspondingly, the pre-flood season is defined as 1 June to 4 July, and the post-flood season as 24 August to 30 September.

3.2.2. Flood Season Segmentation Based on the Fisher Optimal Partition Method

The daily average inflow, daily rainfall, and daily temperature data during the flood season from 1957 to 2012 (June–September) at the Wuluwati Reservoir were used as the fundamental dataset. The following ten-day period (decadal) average indicators were calculated as sample indices: multi-year average decadal discharge (Q), multi-year average maximum 1-day discharge (Q₁), multi-year average maximum 3-day discharge (Q₃), multi-year average decadal precipitation (P), multi-year average maximum 1-day precipitation (P₁), multi-year average decadal temperature (T), and multi-year average maximum 1-day temperature (T₁).

The hydro-meteorological factors, which were normalized using Equation (11), and the results are presented in

Table 2. Normalized Results of Annual Average Values for Hydro-meteorological Factors at Wuluwati.

Serial Number	Ten-Day Period	P (mm)	P1 (mm)	Q (m3/s)	Q1 (m3/s)	Q3 (m3/s)	T (°C)	T1 (°C)
1	Early June	1.000	1.000	0.218	0.275	0.26	0.512	0.512
2	Mid-June	0.716	0.844	0.292	0.377	0.348	0.545	0.545
3	Late June	0.568	0.773	0.5	0.586	0.567	0.779	0.779
4	Early July	0.58	0.767	0.634	0.695	0.694	0.791	0.791
5	Mid-July	0.457	0.667	0.796	0.862	0.86	0.922	0.922
6	Late July	0.346	0.501	0.915	0.976	1.000	0.961	0.961
7	Early August	0.358	0.516	1.000	1.000	1.000	1.000	1.000
8	Mid-August	0.457	0.643	0.761	0.815	0.807	0.823	0.823
9	Late August	0.198	0.236	0.476	0.514	0.517	0.65	0.65
10	Early September	0.296	0.33	0.262	0.274	0.28	0.419	0.419
11	Mid-September	0.012	0.000	0.105	0.111	0.11	0.272	0.272
12	Late September	0.000	0.012	0.000	0.000	0.000	0.000	0.000

. From Table 2, it can be observed that the normalized indicator for mean annual rainfall shows a decreasing trend from June to September over time. The normalized indicator for mean flow is highest in early August. The normalized indicator for mean temperature increases from June to August, reaching its maximum in early August, after which the indicator gradually decreases.

Based on the normalized annual average values for different elements in Table 2, the class diameters for various segmentations were calculated using Equations (12) and (13). The results are shown in

Table 3. Class Diameters for Various Segmentations.

i	j = 2	j = 3	j = 4	j = 5	j = 6	j = 7	j = 8	j = 9	j = 10	j = 11	j = 12
1	0.068	0.401	0.740	1.405	2.319	3.090	3.173	3.631	4.527	6.431	8.983
2		0.161	0.327	0.732	1.334	1.830	1.848	2.338	3.370	5.363	7.994
3			0.030	0.206	0.539	0.803	0.812	1.408	2.624	4.748	7.481
4				0.084	0.272	0.410	0.445	1.123	2.435	4.575	7.262
5					0.057	0.096	0.195	0.944	2.308	4.386	6.941
6						0.006	0.159	0.895	2.162	4.014	6.258
7							0.135	0.745	1.733	3.193	4.961
8								0.311	0.826	1.721	2.861
9									0.169	0.558	1.162
10										0.170	0.457
11											0.097

. From Table 3, it is evident that as the value of j increases, the class diameter gradually increases, indicating growing differences among the samples. Conversely, as the value of i increases, the class diameter gradually decreases, suggesting that the samples become more similar.

The objective function values e[P*(n,k)] were calculated using Equations (14) and (15), as shown in

Table 4. Calculation Results of the Objective Function.

n	k = 2	k = 3	k = 4	k = 5	k = 6	k = 7	k = 8	…	k = 12
3	0.068 (3)	0.0 (3)						…
4	0.098 (3)	0.03 (3)	0.0 (4)
5	0.274 (3)	0.098 (5)	0.03 (5)	0.0 (5)
6	0.607 (3)	0.155 (5)	0.087 (5)	0.03 (6)	0.0 (6)
7	0.811 (4)	0.195 (5)	0.105 (6)	0.036 (6)	0.006 (6)	0.0 (7)
8	0.846 (4)	0.293 (5)	0.195 (8)	0.105 (8)	0.036 (8)	0.006 (8)	0.0 (8)
9	1.476 (4)	0.846 (9)	0.293 (9)	0.195 (9)	0.105 (9)	0.036 (9)	0.006 (9)
10	2.692 (4)	1.015 (9)	0.463 (9)	0.293 (10)	0.195 (10)	0.105 (10)	0.036 (10)
11	3.731 (9)	1.404 (9)	0.852 (9)	0.463 (11)	0.293 (11)	0.195 (11)	0.105 (11)	…
12	4.088 (10)	1.933 (10)	1.112 (11)	0.56 (11)	0.391 (11)	0.292 (11)	0.195 (12)	…	0.0 (12)

(the number in parentheses indicates the sample serial number corresponding to the minimum objective function value; for example, when dividing 5 samples into 2 classes, the number 3 indicates that segmenting the samples {1~5} into {1~2} and {3~5} is optimal). As shown in Table 4, the value of the objective function continuously decreases as k increases, while it increases as n increases. Furthermore, the sample sequence number corresponding to the objective function value also increases progressively.

Based on Table 4, the optimal k-partition of the samples was performed. Subsequently, the β(k) values were calculated according to Equations (18) and (19), with the results presented in

Table 5. Optimal Partition Results.

k	e[P*(n,k)]	β(k)	Segmentation Status
2	4.088		{1~9} {10~12}
3	1.933	1.335	{1~3} {4~9} {10~12}
4	1.112	0.268	{1~3} {4~8} {9~10} {11~12}
5	0.56	0.383	{1~2} {3~4} {5~8} {9~10} {11~12}
6	0.391	0.07	{1~2} {3~4} {5~8} {9} {10} {11~12}
7	0.292	0.002	{1~2} {3~4} {5~7} {8} {9} {10} {11~12}
8	0.195	0.007	{1~2} {3~4} {5~7} {8} {9} {10} {11} {12}
9	0.105	0.022	{1~2} {3~4} {5} {6~7} {8} {9} {10} {11} {12}
10	0.036	0.039	{1} {2} {3~4} {5} {6~7} {8} {9} {10} {11} {12}
11	0.006		{1} {2} {3} {4} {5} {6~7} {8} {9} {10} {11} {12}

. Based on the data in Table 5, curves depicting the variation in the objective function value e[P*(n,k)] and the slope change β(k) with the number of classes k were plotted, as shown in Figure 2.

As shown in Figure 2, when k = 3, β(k) reaches its maximum value. Therefore, k = 3 is determined as the optimal number of segments. According to Table 5, when k = 3, the segmentation corresponds to the 1st–3rd ten-day periods, 4th–9th ten-day periods, and 10th–12th ten-day periods, which align with the pre-flood season, main flood season, and post-flood season, respectively. Consequently, applying the Fisher optimal partition method, the flood season of the Wuluwati Reservoir can be divided as follows: the pre-flood season from 1 June to 30 June, the main flood season from 1 July to 31 August, and the post-flood season from 1 September to 30 September.

3.2.3. Comprehensive Analysis of Flood Season Segmentation

Based on the above analysis, the main flood season determined by the circular distribution method ranges from 5 July to 23 August, while the Fisher optimal partition method defines it as 1 July to 31 August. To further analyze the characteristics and relationship between the two methods, a comparison chart of the segmentation results is plotted, as shown in Figure 3.

The circular distribution method and the Fisher optimal partition method reflect the hydrological characteristics of the flood season from different perspectives, and the differences in their results have clear statistical and hydrological physical backgrounds. The circular distribution method is based on directional statistics of flood occurrence times, focusing on describing the temporal concentration of flood events. Since reservoir flood control operation focuses more on the occurrence patterns of major floods, the results without considering magnitude show a highly concentrated flood occurrence time (γ > 0.8), but this may be masked by numerous small floods. When magnitude is considered, γ decreases to 0.45–0.50, revealing the true characteristic that major floods are more dispersed over time. This reflects that major floods in the basin may be triggered by different weather systems or snowmelt processes at different times, rather than being uniformly concentrated within a brief period—a finding consistent with the mixed rainstorm and snowmelt flood-generating mechanisms in the basin. Therefore, the results considering flood magnitude are more practically significant for engineering, and are adopted as the segmentation outcome of the circular distribution method. In contrast, the Fisher optimal partition method is based on the clustering of multiple indicators (flow, rainfall, temperature), identifying segments of temporal similarity in meteorological and hydrological elements. Its results more comprehensively reflect the phased characteristics of the basin’s hydrological processes.

Therefore, a single method cannot fully capture the complex seasonal patterns of the flood season at Wuluwati Reservoir. To integrate information from both methods and enhance the robustness and physical consistency of the segmentation results, this study adopts a result fusion strategy—namely, the confidence interval overlap method—to synthesize the outcomes: taking the union of the main flood season time ranges under different sampling periods from the circular distribution method (5 July–23 August) and overlapping it with the main flood season from the Fisher optimal partition method (1 July–31 August). This yields 5 July–23 August as the candidate core period for the main flood season. Furthermore, to mitigate the influence of extreme samples, the median time of the results from both methods is considered, ultimately determining the main flood season as 3 July–27 August.

Consequently, by synthesizing both methods, the flood season segmentation for the Wuluwati Reservoir is determined as follows: the pre-flood season from 1 June to 2 July, the main flood season from 3 July to 27 August, and the post-flood season from 28 August to 30 September.

3.3. Rationality Examination Result

Following Steps 1 and 2 in Section 2.2, the process begins by assuming that N = 30 flood events occur uniformly and randomly within the flood season. The number of flood occurrences within each segmentation period is counted, and the relative frequency is calculated. This process is repeated 100 times, yielding 300 relative frequency values. Next, all relative frequency values are sorted in ascending order. The frequency values at the 5th percentile (the 15th value) and the 95th percentile (the 285th value) are identified as the lower and upper limits of the confidence interval, respectively. Subsequently, the number of flood events N is gradually increased from 30 to 500 in steps of 1, and the above procedure is repeated to obtain the upper and lower limit frequencies under the assumption of a uniform distribution. Based on these results, a theoretical frequency curve plot with a 90% confidence interval for the Wuluwati Reservoir is constructed, as shown in Figure 4.

Following Steps 3 to 5 in Section 2.2, using the 53-year daily flow data as the initial sample, the nonparametric bootstrap method is applied. Based on the comprehensive segmentation results derived from the circular distribution method and the Fisher optimal partition method, the partition points of the segmentation scheme are adjusted by advancing or delaying them by 1 to 3 days. This generates 15 candidate reservoir flood season segmentation schemes (main flood season start dates: 1 July, 3 July, 5 July; main flood season end dates: 23 August, 25 August, 27 August, 29 August, 31 August). For each scheme, 1000 bootstrap samples are generated. The relative frequency, generalized distance, relative membership degree, and relative superiority degree are calculated according to Equations (21)–(25), respectively. A comparison chart of the relative superiority degree for each scheme is then plotted, as shown in Figure 5.

As shown in Figure 5, the scheme with segmentation points on 3 July and 23 August achieves the highest relative superiority degree of 0.930. The scheme with points at 3 July and 25 August ranks second with a relative superiority degree of 0.896, while the scheme with points at 3 July and 27 August ranks fourth with a value of 0.835. When the first segmentation point is set at 3 July, the relative superiority degrees for all second-point options reach their maximum values, forming the upper envelope compared to other first-point choices. Therefore, selecting 3 July as the first segmentation point is more reasonable. When the second segmentation point is set at 23 August, the relative superiority degree attains its maximum value across all three schemes with 3 July as the first point. Consequently, 23 August is determined as the second segmentation point.

In summary, the comprehensively determined flood season segmentation for the Wuluwati Reservoir (3 July–27 August) encompasses the range defined by these optimal segmentation points. Thus, it can be concluded that the comprehensive segmentation results align well with the actual conditions.

4. Discussion

Flood season segmentation is not merely a statistical division of time series but an objective reflection of the seasonal patterns of floods within a basin. This study, through the combination of the circular distribution method and the Fisher optimal partition method, and utilizing the inflow hydrological data during the flood season from 1957 to 2012 at the Wuluwati Reservoir, achieved a multi-method fusion for flood season segmentation of the reservoir and conducted a rationality examination using an improved Cunderlik method.

The circular distribution method [18], based on the concentration trend of flood occurrence times, focuses on describing the temporal characteristics of flood events. The Fisher optimal partition method, on the other hand, comprehensively considers multiple factors such as flow, rainfall, and temperature to identify similar stages in hydrometeorological processes. Differences exist between the two methods regarding the start and end dates of the main flood season (a 4-day difference in start date and an 8-day difference in end date). These differences arise because the circular distribution method focuses on the temporal concentration of flood occurrence, while the Fisher optimal partition method emphasizes the phased nature of the coordinated variation in multiple factors. Nevertheless, the main flood season ranges derived from the two methods exhibit substantial overlap (5 July–23 August vs. 1 July–31 August), indicating that they reveal fundamentally consistent seasonal patterns from different perspectives and confirming the robustness of the segmentation results.

Specifically, for the circular distribution method, after “considering flood magnitude,” the concentration degree γ decreased from above 0.80 to between 0.45 and 0.50 (Table 1), indicating that major flood events are more dispersed over time. This is closely related to the flood composition of the basin: floods in the Wuluwati Reservoir basin include rainstorm floods, snowmelt floods, and mixed floods resulting from their superposition. Major floods are often triggered by different weather systems or high-temperature snowmelt at different times, leading to their occurrence times not being concentrated within a single narrow period. In contrast, the Fisher optimal partition method, from a multi-factor comprehensive perspective, identifies July–August as the period of coordinated high values in hydrometeorological elements, making it more systematically representative [7].

To reconcile the two segmentation results, this study took the union of the main flood season ranges under various sampling scenarios from the circular distribution method (5 July–23 August) and overlapped it with the main flood season from the Fisher optimal partition method (1 July–31 August), yielding 5 July–23 August as the core main flood season. Subsequently, based on the median of key time points from both methods, the main flood season was determined as 3 July–27 August. This process respects the statistical characteristics of each method while enhancing the robustness of the results through overlap interval analysis and median smoothing, thereby strengthening the scientific basis of the segmentation conclusion.

The comprehensive segmentation results align closely with the climatic and hydrological processes of the basin. The pre-flood season (1 June–2 July) corresponds to rising temperatures and the initial stage of snowmelt, where floods are primarily snowmelt-type and of smaller magnitude. The main flood season (3 July–27 August) coincides with the period of concentrated rainfall overlapping with high-temperature snowmelt, making it prone to the formation of major mixed floods and thus the critical period for flood control. The post-flood season 28 August–30 September ) sees a significant reduction in flood frequency and intensity as precipitation decreases and temperatures drop.

The examination of 15 segmentation schemes using the improved Cunderlik method shows that the scheme with segmentation points on 3 July and 23 August achieves the highest relative superiority degree (0.930). The comprehensive result of this study (3 July–27 August) encompasses this optimal segmentation range, and its own corresponding relative superiority degree remains at a relatively high level (0.835). This indicates that the segmentation results not only adhere to the statistical optimal principle but also possess practical engineering operability.

Based on the derived flood season segmentation (pre-flood season: 1 June–2 July; main flood season: 3 July–27 August; post-flood season: 28 August–30 September), the following dynamic flood control strategies can be implemented for the reservoir: ➀ Pre-flood season (1 June–2 July): Temperatures rise and snowmelt begins; floods are primarily snowmelt-driven with relatively small magnitudes. The flood-limited water level can be appropriately raised (e.g., by 1.5 m compared to the main flood season level) to increase beneficial storage, while incorporating short-term weather forecasts to ensure timely pre-release in the event of significant rainfall. ➁ Main flood season (3 July–27 August): Rainstorms coincide with snowmelt, easily forming major floods, making this the critical period for flood control. The designed flood-limited water level should be strictly maintained to reserve sufficient flood control capacity, with pre-release operations conducted based on real-time forecasts. ➂ Post-flood season (28 August–30 September): Precipitation decreases and temperatures drop; flood frequency and magnitude significantly reduce. The flood-limited water level can be raised again (e.g., by 2.0 m compared to the main flood season level) to store water for the dry season, ensuring that the water level can be drawn down to a safe level before any potential subsequent major flood events. This approach can alleviate the contradiction between the uneven spatiotemporal distribution of water resources in the basin while ensuring flood control safety. In the context of increasing extreme hydrological events due to climate change, such dynamic operation methods based on segmentation hold significant value for adaptation and broader application.

The limitations of this study are summarized as follows. First, the hydrological series ends in 2012, and recent climate and underlying surface changes may have influenced flood patterns. Future work could extend the series by reconstructing the reservoir outflow process. Second, the segmentation methods are based on historical statistics and do not account for frequency changes under non-stationary climatic conditions. Future research could incorporate climate model outputs for dynamic segmentation. Furthermore, while this study primarily fuses two methods, subsequent efforts could attempt to introduce models such as machine learning or copula functions to better capture the non-linear relationships in flood genesis.

5. Conclusions

To scientifically and reasonably segment the flood season of the Wuluwati Reservoir, this study establishes a two-step methodological framework that combines flood season segmentation with rationality examination. Two segmentation methods and one rationality examination method are employed to ultimately determine a reasonable flood season partition. The main conclusions of this study are summarized as follows:

(1)

The circular distribution method and the Fisher optimal partition method are applied to calculate the flood season segmentation for the Wuluwati Reservoir. The two methods yield highly consistent segmentation results with substantial overlap in the main flood season, and the average of the start and end dates derived from both methods is therefore adopted as the final segmentation result. Accordingly, the flood season of the Wuluwati Reservoir is determined as follows: the pre-flood season from June 1 to July 2, the main flood season from July 3 to August 27, and the post-flood season from August 28 to September 30.

(2)

To examine the rationality of the flood season segmentation, the improved Cunderlik method is employed to evaluate segmentation performance based on relative superiority degree. The results indicate that the segmentation points with the highest superiority degree occur on July 3 and August 23. These optimal points fall within the range identified by the comprehensive segmentation results, thereby confirming the rationality of the final flood season partition.

(3)

Based on the segmentation results, specific flood-limiting operation measures can be proposed: dynamically raising the flood-limited water level during the pre-flood and post-flood seasons to increase beneficial storage, while strictly controlling the water level during the main flood season to ensure flood control capacity; establishing pre-release rules near segmentation boundaries in combination with short-term inflow forecasts to achieve smooth transitions between different periods. These measures provide operable technical solutions for the dynamic operation of the Wuluwati Reservoir and offer methodological references for flood season segmentation and flood control operation of reservoirs with similar hydrological characteristics.