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.
National Natural Science Foundation of ChinaNo. 52179014No. 51641901National Key Research and Development Program of China2016YFC04022082016YFC04019032017YFC0405900Introduction
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 t test, the two-sample Kolmogorov–Smirnov (K-S) test
and the Shapiro–Wilk test are commonly used quantitative methods for series
variation. In particular, the K-S test can calculate the degree of change by
indicators such as asymptotic significance (two-tailed, p); therefore it
is widely used (Jia et al., 2022).
Commonly used change-point detection methods include graphical methods
(cumulative anomaly method, etc.), parametric methods (sliding t test and the
Lee–Heghinian test, etc.) and nonparametric methods (ordered clustering
method, Mann–Kendall test, and wavelet change-point detection, etc.). Graphical
methods have the advantages of simple calculation and intuitive results, but
the detection accuracy is low. Parametric methods assume that the series to
be analysed obey a known distribution, which have certain limitations (Liu
et al., 2022). Nonparametric methods have higher detection accuracy but are
easily affected by factors such as parameter settings and series marginal
effects (Stasolla and Neyt, 2019). Malki et al. (2022) used machine learning
to compare the gap between historical data and forecasts from real-time
monitoring data to determine whether the consistency of IoT energy
consumption data has changed. Shi et al. (2022) constructed a single
change-point test based on the covariance, cumulative sum and likelihood
ratio of forecast residuals to detect the potential change point in time
series. Corradin et al. (2022) constructed a Bayesian nonparametric
multivariate change-point detection method by combining prior distributions
with multivariate kernels and argued that the posterior probability of most
change points should be lower than the posterior estimate. Xie et al. (2022)
calculated the fitted local trend line based on the piecewise linear
representation algorithm and the Akaike information criterion to realize
change-point detection and series division and classified change points into
three categories with the help of the slope and intercept. Change-point detection
is of great significance to series division and is the basis for making full
use of HTS to carry out more research. It can be seen that there is no
unified standard to determine the change point of HTS. Therefore, this is a
field worthy of further study.
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.
Methodology
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 and change-point detection
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 (φ(t)) is a wave of finite length and zero mean, with
irregularity and asymmetry. The 16 commonly used MWT systems are shown in
Table 1 (Moradi, 2022; Nielsen, 2001).
Properties and application range of commonly used MWT systems.
Note that “√” means has this property. “√∗” means approximately having this property. “–” means does not have this property.
Continuous wavelet transform (CWT)
CWT can be used to determine whether there is periodicity in HTS and
identify the main timescales and their local trends. Let L2(R) denote the measurable square-integrable functions on the real axis.
If HTS X(t) (t=1,2,…,T) is a CWT
in L2(R), which can be expressed as
1WX(a,b)=∫-∞+∞X(t)φa,b∗(t)dt2φa,b(t)=1aφt-baa,b∈R,a≠0,
where WX(a,b) is the coefficient of CWT; φa,b∗(t) is the complex conjugate function of
φa,b(t); t is the time; a is the timescale
factor, which reflects the period length of MWT; and b is the time position
factor, which reflects the translation of MWT in time.
The multi-timescale variation in wavelet transform refers to the
multi-level structure and localized features of X(t) in the
time domain, which is usually analysed with the help of the real part or
modulus-square contour map of CWT coefficients. HTS evolution of a certain
year on different timescales can be observed by vertically intercepting the
contour map. At a certain period, the HTS evolution over time can be
observed by horizontally intercepting the contour map. In addition, the
positive wavelet coefficient corresponds to the wet season. The negative
wavelet coefficient corresponds to the dry season. The wavelet coefficient
is zero, which corresponds to the transition point of wet and dry. The
larger the absolute value of the wavelet coefficient, the more obvious its
change.
Discrete wavelet transform (DWT)
Since the measured HTS are usually discrete, by discretizing Eq. (1), we can
get
3WX(j,b)=∫-∞+∞X(t)φj,b∗(t)dt4φj,b(t)=a0-j2φ(a0-jt-kb0),
where WX(j,b) is the coefficient of DWT, a0 and
b0 are both constants, and j (j=1,2,…,J) is the
decomposition level.
Both WX(a,b) and WX(j,b) are the
values output by X(t) through the unit impulse response
filter, which can reflect the evolution of X(t) in the time
domain and frequency domain at the same time. In practical applications, it
is often decomposed with the help of dyadic DWT, i.e. a0=2 and b0=1, and Eq. (4) can be expressed as
φj,b(t)=2-j2φ(2-jt-k).
According to the dyadic DWT, the theoretical maximum value J of
decomposition level j is
J=[log2(TX(t))],
where [⋅] represents the rounding operation, and TX(t)
represents the length of the X(t).
Wavelet change-point detection
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 X(t) into T-dimensional column vectors W1,W2,…,WJ and VJ, where WJ is
calculated from the MODWT wavelet coefficient of X(t) within
τjΔt, and VJ consists of τj+1Δt and higher
dimensional MODWT scaling coefficients. X(t) can be expressed
as
X=∑j=1JDj+Sj,
where Dj=WjFkFhj∗ (k=0,1,…,T-1) is the jth maximal-overlap detail. Sj=VjFkFgj∗ is the jth maximal-overlap smooth. hj and
gj are the high-frequency filter and the low-frequency filter,
respectively. F is a T×T dimensional matrix that cyclically shifts
hj by one unit.
(2) MODWT variance decomposition
After a series of decompositions are performed on the variance of X(t) part by part, on the premise that the wavelet coefficient is
stable, it can be expressed as
‖X‖2=∑j=1J‖Wj‖2+‖Vj‖2.
Based on the above decomposition, the evolution of wavelet coefficient
variance of X(t) with time in different timescales can be
obtained, and the point where the variance changes can be recorded as the
change point. It is worth noting that the MWT used for change-point detection
needs to be biorthogonal (see Table 1).
Traditional change-point detection method
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.
Cumulative anomaly method
The cumulative anomaly method is a graphic method. The cumulative anomaly value of
X(t) at a certain time can be expressed as
JP[X(t)]=∑t=1N[X(t)-X‾],
where JP[⋅] is the cumulative anomaly value of X(t), and T and X‾ are the length and mean of X(t), respectively.
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.
Mann–Kendall (M-K) test
The M-K test analyses the number, location, trend and significance of
change points in HTS by setting a confidence level α and calculating
statistics (UFk and UBk). The UFk statistics of
X(t) is calculated as follows:
UFk[X(t)]=SkX(t)-E[SkX(t)]Var[SkX(t)],
where UFk[X(t)] is the statistical
series of X(t) calculated in order, and SkX(t) is the rank sum of time k in X(t), which is the
cumulative value of the numbers at time k greater than time i (1≤k≤i). E[SkX(t)] and
Var[SkX(t)] are the mean and
variance of SkX(t), respectively.
When UFk[X(t)]>0, X(t)
shows an upward trend; on the contrary, it shows a downward trend. The statistic UBk[X(t)] is obtained by repeating Eq. (10) in the
reverse order. Draw UFk[X(t)] and
UBk[X(t)] in the same figure. If the
two statistics intersect within the confidence interval U0.05=±1.96
(confidence level 95 %), the time corresponding to the intersection is the
change point of X(t).
Kolmogorov–Smirnov (K-S) test
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 X(t)
is
Fn[X(t)]=1T∑t=1TI[-∞,T]n[X(t)],
where I[-∞,T]n[X(t)] is the indicator function of X(t).
The original hypothesis H0 is as follows: F1[X(t)]=F2[X(t)]; that is, the empirical
distribution of the two series is consistent. The alternative hypothesis H1 is as follows: F1[X(t)]≠F2[X(t)]; that is, the empirical distribution is inconsistent. To
quantify the difference between the empirical distributions, a maximum
difference D is proposed, calculated as
D=sup-∞<X(t)<∞|F1[X(t)]-F2[X(t)]|.DT,α is used to represent the rejection domain when the series capacity is T at
significant level α. When D≥DT,α, reject H0;
otherwise, accept H0. To further quantify the significance of the
difference, p is introduced to concretize α. The value of α is usually 95 % or 99 %, and the corresponding p is 0.05 and 0.01.
If p≤0.01, it indicates that the determination result is strong and
H0 should be rejected; that is, the two series obey different
distributions and are not consistent. If 0.01≤p≤0.05, the
determination result is weak. In this case, p is considered to be
marginal, and H0 is usually rejected. If p>0.05, H0 is
acceptable.
Change-point detection criteria
Based on the change-point detection results of various methods, the potential
change-point set PCP(n) (n=1,2,…,N) of
HTS is constructed with deduplication and sorting. To determine the
change point, it is necessary to further calculate the degree of change (p)
before and after potential change points with the help of the K-S test. At
a confidence level of 99 %, first, record the starting point and ending point
of X(t) as PCP(0) and PCP(N+1) respectively, and arrange the potential change-point set in
chronological order. Secondly, take PCP(0) as the
starting point and PCP(1) as the change point, and use
K-S test to successively calculate the p of the end point from PCP(2) to PCP(N+1). Finally, the change point
and its trajectory (connection of change points) of X(t) are
determined according to the change-point detection criteria:
Criterion 1. Before and after the change point of X(t), p<0.01.
Criterion 2. The change point can realize the continuous division of X(t) from PCP(0) to PCP(N+1).
Criterion 3. The trajectory contains the largest number (m=1,2,…,M) of change points.
Criterion 4. The p of M-1 in the trajectory is the minimum value.
MWT optimization framework
By comparing RCP(n) and the results of wavelet
change-point detection, a MWT that conforms to HTS characteristics can be
selected. The MWT optimization framework includes the construction of
potential change-point set, change-point detection and optimal MWT
determination. Among them, the potential change-point set is built to improve
the efficiency of change-point detection, and the specific optimization steps
are as follows:
Optimization step (1). Select candidate wavelet with the highest change-point
detection accuracy.
Optimization step (2). When two or more candidate wavelets have the same
detection accuracy, the MWT or the MWT system with the highest frequency in
different statistic series (length, flow, etc.) of the same hydrological
station is selected as the optimal one.
After optimization, we can perform CWT according to the MWT conforming to
HTS characteristics and analyse its evolution. For DWT, HTS can be more
accurately decomposed and reconstructed, providing a good basis for
hydrological forecasting and reservoir operation scheme formulation.
Data and study area
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 1.8×106km2, accounting for about 18.8 % of the total
area of China. The main stream from Yibin to Yichang is called the upstream,
with a length of about 4504 km and an area of about 1×106km2.
With the superposition and collection of upstream floods to the Yichang
hydrological station (Yichang station), it tends to form a process of high
peaks and large volumes (Wang et al., 2021). The Pingshan hydrological
station (Pingshan station) on the Jinsha River controls about half of
catchment area and one-third of the flood season average flow of Yichang station and is the basic source of upstream flooding. Therefore, exploring
the runoff evolution at Pingshan station and Yichang station will help to
scientifically arrange the watershed storage space to alleviate the frequent
floods in flood seasons and water shortages in dry seasons in the middle and
lower Yangtze River. The overview of the upper Yangtze River is shown in
Fig. 2, and the hydrological parameters of the tow stations are shown in
Table 2.
Location of the study area.
Main hydrological parameters of Pingshan station and Yichang station.
River JinshaYangtzeHydrological station PingshanYichangCatchment areaArea (km2)485 0991 005 501Proportion (%)48.2100Annual average water volumeVolume (108m3)11473410Proportion (%)33.6100Annual distribution of runoffFlood season (month)6–115–10Flow (m3s-1)44 850127 700Proportion (%)81.3478.67
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.
Results and discussion
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”.
Construction of potential change-point set
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.
Results of cumulative anomaly method and M-K test
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 ±1.96), the intersection of UFk and UBk
is a potential change point, and the M-K test results are shown in Fig. 4.
Potential change points in the two figures were marked in red.
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 UFk and UBk is lower, which
indirectly reflects the flood season in Pingshan station being relatively
gentle, but the difference between the wet and dry seasons of the year is
obvious. The YFSS is the opposite. In addition, the detection results of M-K
test for 4-Series are basically consistent, insensitive to flow variation.
The detected number of potential change points is small. It can be included
that the cumulative anomaly method is more suitable for constructing the
potential change-point set of HTS. A more accurate locating of the
change point needs other methods.
Results of wavelet change-point detection
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.
Results of change-point detection
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 (p) before and after each potential
change point was calculated by the K-S test. Traditional change-point
detection often adopts the method of traversal series. Take PAS as an
example (62 years in total); because the starting point, change point and end
point are changing, its p value is calculated
∑n=160∑i=1ni=35990 times.
After constructing the potential change-point set, the number of calculation
is reduced to ∑n=129∑i=1ni=4060, and the efficiency is improved by 88.72 %, and the calculation
results are shown in Fig. 5a. The change-point trajectories (marked with
red lines and blue dots) and alternative trajectories of 4-Series were
determined according to the detection criteria in Sect. 2.3, as shown in
Fig. 5b and c.
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 p<0.01 in Fig. 5a-1. Then, with the
change point as the starting point and the ending point as the change point,
find the grid point with p<0.01 until 2011. At a confidence level of 99 %,
there are three points in Fig. 5a-1 that meet the requirements of Criterion 1, namely 1950–1998–2005 (Trajectory 1), 1950–1998–2007 (Trajectory 2) and
1950–1999–2005 (Trajectory 3), and p is shown in Fig. 5b. It can be seen
that Criterion 1 can effectively narrow the selection range of
change points from many potential points. Criterion 2 requires further
search extending to 2011, which can fully explore the change point and ensure
the continuity of the trajectory. When there are multiple alternative
trajectories with an inconsistent number of change points, Criterion 3 requires
to select the one with the most points, which helps to divide the series in
detail. Figure 5b–e show all alternative trajectories that
meet the requirements of the above three detection criteria. According to
Criterion 4, select the year with small p of the first M-1 change points
one by one, which can make the series before and after the change point have
a large degree of change.
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 m3s-1 respectively.
The monthly mean runoff in flood season from 2001 to 2004 is 20010.98, 18895.22, 20690.22 and 19841.30 m3s-1
respectively. For the hydrological regime, 2002 is a year with less water
inflow, while 2003 is the opposite. However, affected by the Three Gorges
Reservoir, the water inflow in 2002 is closer to 2003–2010 in the flood
season series, while the annual series is closer to 1950–2001. It indirectly
shows that the change-point detection framework proposed in this study
considers the influence of both human factors and hydrological regime on the
series. The HTS division results of Pingshan station and Yichang station are
shown in Fig. 5b–e. Dividing series helps ensure
consistency of HTS and provides a basis for better information mining
through statistical analysis methods.
Results of MWT optimization
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 %).
∗ Contribution refers to the percentage of change points provided by the detection method for the potential change-point set.
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.
Analysis of HTS evolution based on CWT
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.
Conclusion
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:
Change-point detection criteria. Based on the three change-point detection
methods, a potential change point set of HTS is constructed, which can make
up for the limitations of a single method affected by factors such as
parameter settings and marginal effects and improve the calculation
efficiency. In addition, with the help of the Kolmogorov–Smirnov (K-S) test, we
proposed the detection criteria to quickly confirm the change-point
trajectory from the beginning to the end of HTS. While ensuring the
uniqueness of the result, the change point formed by the combined action of
multiple factors can be accurately identified to complete the series
division.
MWT optimization framework. Based on the change-point detection accuracy
of wavelet change-point detection, the MWT consistent with the series
characteristics can be selected to ensure the accuracy of wavelet transform
to analyse the HTS evolution and provide a good basis for hydrological and
hydraulic calculation.
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.
Acronym list.
OrderAcronymFull name1HTSHydrological time series2MWTMother wavelet3IIDIndependent and identically distributed4K-SKolmogorov–Smirnov5M-KMann–Kendall6CWTContinuous wavelet transform7DWTDiscrete wavelet transform8MODWTMaximal overlap discrete wavelet transform9PASPingshan annual series10PFSSPingshan flood season series11YASYichang annual series12YFSSYichang flood season seriesData availability
Data for this study can be downloaded from the Yangtze River Hydrological
Network (http://www.cjh.com.cn/, Hydrological Bureau of the Yangtze River Commission, 1950). In this study, the wavelet change-point
detection is based on the MATLAB (R2020b) toolbox, and the rest of the codes
(PyCharm 2021.2.2) are available from the corresponding author upon
reasonable request.
Author contributions
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.
Competing interests
The contact author has declared that none of the authors has any competing interests.
Disclaimer
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Acknowledgements
The authors would like to give special thanks to the anonymous reviewers.
Financial support
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).
Review statement
This paper was edited by Carlo De Michele and reviewed by Mohammad Nazeri Tahroudi and Geoff Pegram.
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