Hydrological time series (HTS) are the key basis of water conservancy project planning and construction. However, under the influence of climate change, human activities and other factors, the consistency of HTS has been destroyed and cannot meet the requirements of mathematical statistics. Series division and wavelet transform are effective methods to reuse and analyse HTS. However, they are limited by the change-point detection and mother wavelet (MWT) selection and are difficult to apply and promote in practice. To address these issues, we constructed a potential change-point set based on a cumulative anomaly method, the Mann–Kendall test and wavelet change-point detection. Then, the degree of change before and after the potential change point was calculated with the Kolmogorov–Smirnov test, and the change-point detection criteria were proposed. Finally, the optimization framework was proposed according to the detection accuracy of MWT, and continuous wavelet transform was used to analyse HTS evolution. We used Pingshan station and Yichang station on the Yangtze River as study cases. The results show that (1) change-point detection criteria can quickly locate potential change points, determine the change trajectory and complete the division of HTS and that (2) MWT optimal framework can select the MWT that conforms to HTS characteristics and ensure the accuracy and uniqueness of the transformation. This study analyses the HTS evolution and provides a better basis for hydrological and hydraulic calculation, which will improve design flood estimation and operation scheme preparation.

Under multiple influences of human activities, atmospheric circulation and other factors, the original evolution of river runoff is featured by randomness, fuzziness, nonlinearity, non-stationarity and multi-timescale variation, which breaks the consistency in the “three properties” of hydrological time series (HTS; formed by the time arrangement of hydrological elements such as rainfall and runoff) (Chen et al., 2021; Fang and Shao, 2022). Independent and identically distributed (IID) is an assumption of mathematical statistics in hydrological and hydraulic calculation (Mat Jan et al., 2020). When the series cannot meet the IID, analysing its internal evolution and division will help to improve the accuracy and decision-making of the hydrological forecasting and operation scheme preparation by the mathematical model (Li et al., 2021).

In stochastic hydrology, HTS consist of deterministic components and
stochastic components. The analysis of their evolution involves the period, trend
and change point (Hobeichi et al., 2022). The period and trend mainly focus on
deterministic components, while change-point detection is used to explain the
stochastic components caused by various random and uncertain factors (Dang
et al., 2021). Change-point detection determines the starting and ending
points of period and trend division; thus it is the key to analysing HTS
evolution (Şen, 2021). However, affected by feature uncertainty,
change-point detection has become a complex problem because the extent,
number and occurrence time of change points must be determined at the same
time (Zhao et al., 2019). The

Commonly used change-point detection methods include graphical methods
(cumulative anomaly method, etc.), parametric methods (sliding

After the change-point detection, the period and trend of HTS can be further explored. These methods include a cumulative anomaly method, the Mann–Kendall (M-K) test, continuous wavelet transform (CWT) and mode decomposition (empirical or extreme point symmetric, etc.) (De Oliveira-Júnior et al., 2022; Qin et al., 2021). Among them, CWT has a relatively complete theoretical system, which can comprehensively analyse the evolution of HTS and reveal their localization characteristics in the time domain (time variation) and frequency domain (frequency and amplitude variation), so it has been widely used in hydrology (Zerouali et al., 2022). However, the analysis results of CWT highly depend on the selection of the mother wavelet (MWT). Moradi (2022) optimized MWT by comparing the similarity of cross-correlation function, signal-to-noise ratio and mean standard error between the denoised series and the original. Benhassine et al. (2021) determined the optimal MWT by comparing the minimum mean square error between the original image and the denoised. Strömbergsson et al. (2019) proposed and verified the validity of using the Shannon entropy of the wavelet coefficients as the index for selecting MWT. However, change-point detection has not been explored by scholars to optimize the MWT that conforms to the series characteristics.

To solve the above problems, we proposed the change-point detection criteria based on a cumulative anomaly method, the M-K test, wavelet change-point detection and the K-S test, which can detect the consistency of HTS and complete a reasonable division. Furthermore, based on the detection accuracy, a MWT optimal framework that conforms to series characteristics was proposed, and the evolution analysis was summarized by CWT. This work proposed, in a pioneering way, an efficient way to optimize the MWT based on variance and change-point detention. Using the optimal MWT in CWT is helpful in catching the HTS evolution accurately and fully mining its information, which provides a feasible way to use inconsistent measured data for hydrological and hydraulic calculations.

To solve the problems of incomplete change-point detection and non-unique MWT optimization, we followed the process of potential change-point set construction, change-point determination, MWT optimization and evolution analysis, and then we proposed the change-point detection criteria and the MWT optimization framework, as shown in Fig. 1.

Study framework and main modules of MWT optimization.

Wavelet transform can be divided into continuous wavelet transform (CWT) and
discrete wavelet transform (DWT). Its essence is to reveal the similarity
between the HTS to be analysed and the MWT. Therefore, the selection of MWT
is a key factor affecting the accuracy of wavelet transform. MWT (

Properties and application range of commonly used MWT systems.

Note that “

CWT can be used to determine whether there is periodicity in HTS and
identify the main timescales and their local trends. Let

The multi-timescale variation in wavelet transform refers to the
multi-level structure and localized features of

Since the measured HTS are usually discrete, by discretizing Eq. (1), we can
get

Both

According to the dyadic DWT, the theoretical maximum value

Variance is one of the important parameters to detect whether HTS has fundamentally changed. Wavelet change-point detection is based on the maximal overlap discrete wavelet transform (MODWT). By calculating the variance of wavelet coefficients to be analysed one by one (Strömbergsson et al., 2019), the number and location of change point at a confidence level of 95 % can be determined through the MATLAB software toolbox.

(1) MODWT multi-resolution analysis

Decompose

(2) MODWT variance decomposition

After a series of decompositions are performed on the variance of

Based on the above decomposition, the evolution of wavelet coefficient
variance of

Change point detection has always been a significant issue in hydrology. However, except for the deterministic runoff changes caused by human activities such as large-scale river regulation, reservoir construction or operation (seasonal and above regulation capacity), there exist many uncertain factors, such as whether there is a change point in HTS, how many change points exist and the specific occurrence time of each change point. Therefore, it is necessary to integrate multiple detection methods. The main methods used in this study are as follows.

The cumulative anomaly method is a graphic method. The cumulative anomaly value of

The cumulative anomaly curve can be obtained by drawing the cumulative anomaly value in chronological order. According to the curve fluctuation, the change trend and potential change point of HTS can be identified. If the cumulative anomaly value is greater than 0, it indicates that the HTS is in an up trend; otherwise, the HTS is in a downtrend. The point that changes the trend can be regarded as the potential change point.

The M-K test analyses the number, location, trend and significance of
change points in HTS by setting a confidence level

When

The K-S test can determine whether the distributions of the two series are
the same according to the maximum vertical distance between the two
empirical distributions. The empirical distribution of

The original hypothesis

Based on the change-point detection results of various methods, the potential
change-point set

By comparing

The Yangtze River originates from the southwest of the Tanggula Mountains on
the Qinghai–Tibet Plateau. Its main stream flows through central China from
west to east, with a total length of about 6300 km, and the total catchment
area is

Location of the study area.

Main hydrological parameters of Pingshan station and Yichang station.

The flood season of Pingshan station is from June to November, and the flood season of Yichang station is from May to October. The three months with the largest flow on the two stations are both from July to September (accounting for 49.96 % and 54.18 % of the year, respectively). In 2012, Pingshan station was moved down 24 km to Xiangjiaba hydrological station. In addition, the runoff of Pingshan station should consider the influence of the upstream Ertan Reservoir (seasonal regulation, water storage in May 1998), and Yichang station should consider the Three Gorges Reservoir (annual regulation, water storage in June 2003). Combining the above factors, the measured runoff data of Pingshan station (1950–2011) and Yichang station (1950–2016) were used to test the applicability of the change-point detection framework and the MWT optimization framework proposed in this study, and the runoff evolution of the two stations was analysed by CWT.

The statistical series of the two stations used in the study includes Pingshan annual mean runoff series (Pingshan annual series, PAS), Pingshan 6–11 mean runoff series (Pingshan flood season series, PFSS), Yichang annual mean runoff series (Yichang annual series, YAS) and Yichang 5–10 mean runoff series (Yichang flood season series, YFSS), collectively referred to as “4-Series”.

The cumulative anomaly method, M-K test and wavelet change-point detection were used to detect the potential change points in the 4-Series. At the same time, by comparing the annual series and the flood season series at the same station, we further analysed the sensitivity of the three methods to the variation of flow amplitude and the influence of flood season on the annual series.

The points causing the trend change can be regarded as potential
change points, and the detection results of the cumulative anomaly method are
shown in Fig. 3. At a confidence level of 95 % (the upper and lower critical
lines are

Potential change points of the cumulative anomaly method at Pingshan station and Yichang station.

Potential change points of the M-K test at Pingshan station and Yichang station.

The number of potential change points of 4-Series detected by the cumulative anomaly method is 15, 15, 16 and 18 (Fig. 3). However, the number detected by the M-K test is 2, 2, 0 and 0 (Fig. 4). In addition, there are differences in the potential change-point detection results between the annual series and the flood season series, indicating that the cumulative anomaly method has a certain response ability to flow changes. However, the consistent rate of potential change points in Pingshan station is 100 %, while Yichang station is 37.5 % and 33.33 %, respectively. This means that the response ability can only be reflected when the flow variation reaches a certain extent.

The change-point detection results of M-K test at Pingshan station
(Fig. 4a and b) are concentrated around 1956 and 2005. During the same timescale, the
intersection of the flood season series is slightly later than the annual
series, but the amplitude of

Among the 16 commonly used MWT systems, 8 of them satisfy the biorthogonality (59 MWT systems in total). In this study, 59 MWT systems were used to detect the potential change points of 4-Series one by one, and the number of decomposition layers used is five. However, only five MWT systems can detect the change points of 4-Series, as shown in Table 3.

Wavelet change-point detection results of biorthogonal MWT at Pingshan station and Yichang station (number of decomposition layers is 5). Bold font represents the optimal MWT or change point. The number represents the HTS corresponding to the optimal MWT or change point.

The change point and the optimal MWT are marked with the same number (in the upper right corner) as the series.

From Table 3, the number of potential change points detected by a single MWT is between 1 and 3. The top two potential change points of the PAS are 1992 and 1999, of the PFSS 1999 and 2000, of the YAS 1961 and 1968, and of the YFSS 1975 and 2005. The number of 4-Series of change points detected is 19, 18, 19 and 17 respectively. Compared with the cumulative anomaly method and M-K test, the wavelet change-point detection has the highest contribution to the construction of the potential change-point set, followed by the cumulative anomaly method.

As the MWT changes, the detection results are quite different. For the same hydrological station and the same MWT, there is also a difference in the detection results between the annual series and the flood season series, indicating that the wavelet change-point detection is very sensitive to the flow variation of HTS. Furthermore, the detection results of Pingshan station are concentrated in 1959–2000, while those of Yichang station are concentrated in 1959–2004. Compared with the series length used in the study (Pingshan 1950–2011 and Yichang 1950–2016), the detection results are susceptible to marginal effects, and the potential change points at both ends of the series (before and after 10 years) may be ignored.

We deduplicated and sorted the above detection results as potential
change-point sets for each series, with capacities of 31, 30, 31 and 28,
respectively. The degree of change (

Change-point trajectory of Pingshan station and Yichang station (confidence level 99 %).

For PAS, the starting point of the change-point trajectory is 1950. We need
to find the grid point with

Based on the change-point detection criteria, the year in which the series
consistency has changed due to human factors (water storage of large
reservoirs, etc.) can be determined (Fig. 5b–e red line).
The change-point trajectory of PFSS is consistent with PAS, while YFSS lags
behind YAS by 1 year. The reason could be related to the interannual
variation of runoff. The flood season of Pingshan station is from June to
November, accounting for 81.34 % of the annual average runoff. The
upstream Ertan Reservoir (water storage in May 1998) has seasonal regulation
capacity, so it can have a direct impact on PFSS, which is divided into
1950–1997, 1998–2004 and 2005–2011. However, the flood season of Yichang station is from May to October, and the runoff in May accounts for 7.1 %
of the year. The annual mean runoff from 2001 to 2004 is 13154.73,
12454.25, 12991.84 and 13115.10

Based on the change-point trajectories, the detection accuracy of the three methods was calculated, and the MWT optimization can be completed according to the optimization framework in Sect. 2.4. The screening process is shown in Table 3, and the optimization results of MWT are shown in Table 4.

Change point and optimal MWT of Pingshan station and Yichang station (Confidence Level 99 %).

Combining the MWT optimization results in Tables 3 and 4, it is found that the change point is the key to series division, and optimization step (1) can quickly locate the MWT that conforms to the series characteristics. For Pingshan station, the annual series of MWT meeting optimization step (1) is db8, and the flood season series are db8 and fk8. The optimization step (2) is selected according to the runoff physical cause at the same station, which makes it easier to analyse the evolution of the two series from the time–frequency space of the same MWT. Therefore, the optimal MWT of PFSS is db8.

When the optimal MWT of the series is determined, the accuracy of wavelet change-point detection is generally higher than the cumulative anomaly method and the M-K test (Table 4). Except for YAS, the contribution rate of wavelet change-point detection to the overall potential change point is also higher than both of them. The results show that the MWT optimization framework proposed in this study can accurately screen the optimal MWT of each series. The wavelet transform based on the MWT conforming to the series characteristics is helpful to improve the rationality of the analysis.

Based on the optimization results of MWT in Table 4, the evolution of 4-Series was analysed by CWT. To further explore the influence of MWT, Haar, Morlet and Mexican hat (referred to as three common wavelets) were used in CWT of PAS, as shown in Fig. 6a. The analysis results of the optimal MWT are shown in Fig. 6b–e.

Results of CWT at Pingshan station and Yichang station (wavelet variance and real part of a contour map, with a confidence level of 99 %).

The three common wavelets have great differences in the analysis results of the main periods of PAS, namely 10a and 35a, 10a and 29a, and 3a and 10a (Fig. 6a). Furthermore, they frequently alternate between wet and dry in the short time period and exhibit a distinct “wet–dry–wet” evolution over the long time period. Compared with Fig. 6b, the CWT of three common wavelets is relatively scattered in the timescale of 0 to 60a, and the Morlet and Mexican hat wavelets show a wet period after 1998, which does not reflect the regulation effect of the Ertan Reservoir on Pingshan station, and the accuracy of the analysis results is questionable. According to historical records, during the flood season in June 1998, a basin-wide flood occurred in the middle and lower Yangtze River due to continuous heavy rain in Dongting Lake and Panyang Lake below Yichang station (Zhang et al., 2021). From the timescale (Fig. 6b and c), Pingshan station and Yichang station suffer continuous dry years, which is consistent with the actual situation. Based on the analysis of integrated moisture transport, land-falling atmospheric rivers geometric metrics and large-scale climatic circulations, Ayantobo et al. (2022) believed that the extreme rainfall in the Yangtze River basin had a declining period after 1999, which was consistent with the analysis results of this study. We believe that optimizing the MWT that conform to series characteristics based on the change-point detection is a suitable approach.

According to the analysis, the main periods of PAS are 10a and 30a, and the flood season series are 10a and 29a. The long-period scale of flood season is slightly earlier than the annual series, indicating that the annual adjustment of Pingshan station has a certain buffer capacity. On the short-period scale 10a, the two series show the phenomenon of frequent alternation of wet and dry seasons, but the consecutive dry seasons from 1926 to 1968 and 1998 to 2004 have a serious impact on the series. Especially after 1998, due to the operation of Ertan Reservoir, the runoff reduction in the annual series is larger than that in flood season, so attention should be paid to the annual water demand of river channels and cities along the route. From 2005 to 2011, Pingshan station had the wet season, and attention should be paid to flood control and flood resource utilization. The main periods of YAS are 9a and 27a, and the main periods of flood season series are 9a and 31a. Similarly, Yichang station frequently alternates between wet and dry on the short-period scale. The annual series shows the evolution of “wet–dry–wet–dry–wet” on the long-period scale, while the flood season series shows “wet–dry–wet–dry”. After 2002–2003, YFSS did not enter the wet season as the annual series, indicating that the operation of the Three Gorges Reservoir has a large reduction in the flood season. On the premise of ensuring the storage of the downstream reservoir at the end of the flood season, it is helpful to adjust the annual and interannual distribution of the runoff in the Yangtze River and improve the utilization efficiency of water resources.

Hydrological time series (HTS) is the basis of water conservancy project planning and construction. However, under the multiple effects of human activities and other factors, the consistency of HTS is destroyed. It is necessary to analyse its evolution to ensure the rationality of hydrological and hydraulic calculation. Wavelet transform is one of the widely used analysis tools of evolution in hydrology, but the its analysis accuracy is closely related to mother wavelet (MWT). To solve these two problems, with the help of the cumulative anomaly method, the Mann–Kendall (M-K) test and wavelet change-point detection, we proposed the change-point detection criteria and a MWT optimization framework in this study and took Pingshan station and Yichang station on the Yangtze River as study cases to test their effectiveness. The main conclusions are as follows:

It is found that the change points of the Pingshan annual series and the Pingshan flood season series both are 1998 and 2005, the Yichang annual series are 2002 and 2011, and the Yichang flood season series are 2003 and 2012. In addition, the optimal MWT of 4-Series is db8, db8, db6 and fk8 respectively. The Ertan Reservoir has a greater impact on the annual runoff of Pingshan station, while the Three Gorges Reservoir only reduces the runoff of the Yichang station to a large extent during the flood season. Limited by the data, we did not explore the evolution of the two stations after 2017. It is also found that the wavelet change-point detection is not sufficient enough to detect the potential change point of 10 years before and after the series.

Data for this study can be downloaded from the Yangtze River Hydrological
Network (

JL: conceptualization, validation, writing (review and editing), supervision, project administration and funding acquisition. JH: conceptualization, methodology, software, formal analysis, resources, writing (original draft) and visualization. LZ: methodology, software, formal analysis and data curation. WZ: software, validation, investigation and visualization.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors would like to give special thanks to the anonymous reviewers.

This research has been supported by the National Natural Science Foundation of China (grant nos. 52179014 and 51641901) and the National Key Research and Development Program of China (grant nos. 2016YFC0402208, 2016YFC0401903 and 2017YFC0405900).

This paper was edited by Carlo De Michele and reviewed by Mohammad Nazeri Tahroudi and Geoff Pegram.