Under the impact of climate change and human activities, the stationarity of hydrometeorological extreme value series has been losing in many regions, which makes occurrence rules of hydrometeorological extreme events more complicated. In this study, the efficiencies of trend test methods such as Spearman rank correlation test and Mann-Kendall test, as well as the efficiencies of change-point test methods such as moving T test, moving rank sum test, Pettitt test, and sequential Mann-Kendall test were analyzed quantitatively through Monte Carlo simulation. Five representative level stations in the Yangtze River estuary were selected, and the methods listed above were used in the trend and change-point detection of the annual maximum tidal level records in the period of 1950–2008. It was found that obvious rising tendency existed in the annual maximum tidal level series for all these 5 stations, and year 1980 (for 3 stations) and year 1979 (for 2 stations) were statistically significant change-points. Two subseries were divided with the change-point as the dividing point for all these actual series in the stations. Frequency analyses were carried out, respectively, for all of the subseries, and the impact of nonstationary changes in annual maximum tidal levels on probability distribution was evaluated quantitatively.
In a relatively stable environment, hydrometeorological extreme value series are often seen as pure random variables drawn independently and randomly from the identical population distribution. The statistical parameter of population distribution can be estimated from the instrument records by hydrological frequency analysis. However, under the impact of climate change and human activities, the stationarity of hydrometeorological extreme value series has been losing in many regions in the world. In view of the magnitude and ubiquity of the hydroclimatic change, Milly et al. [
By now, there are numerous studies focused on trend detection methods [
The area around the Yangtze River estuary is one of the most developed regions in China with a high density of population, high speed of urbanization, and high vitality of economic development. Considering its low and flat topography, together with the frequent occurrence of storm surges, flood prevention was mainly realized through embankment projects in this area. With the global climate change, rapid urbanization and hydraulic engineering constructions have taken place in this area. Therefore, it is necessary to detect the nonstationary change of annual maximum tidal level in the Yangtze River estuary and reevaluate the risk of annual maximum level exceeding certain extreme values, so as to provide scientific foundation for flood protection.
As the longest (6300 km) river in China, the Yangtze River extends from the Qinghai-Tibet Plateau and runs eastward into the East China Sea. Datong hydrological station, located at 642 km upstream of the river mouth, which is just free from tidal influences during low flow season, was selected as the control station of upstream discharge in this study. The study area is located in the Yangtze River estuary, which can be characterized as a system of tidal channels of three-order bifurcation with four outlets into the sea (Figure
Locations of typical tidal gauge stations in the Yangtze River estuary.
In this study, 5 national tide stations in the Yangtze River estuary were selected as the representative stations, namely, Jiangyin, Tianshenggang, Xuliujing, Wusong, and Gaoqiao, the locations of which were shown in Figure
Many methods have been developed to detect the tendency in hydrometeorological variable. In this study, Spearman rank correlation test and Mann-Kendall test were investigated.
The original sequence of time order
Statistic
The MK test is a nonparametric rank based test [
The statistic
In this study, we selected 0.05 as significance level in all statistical tests and considered only monotonically increasing or decreasing trend, even though there could be other patterns of trend.
Different methods have been developed to test the change-points in the hydrometeorological variables. In this study, moving T test [
In the moving T test, to find out the change-point, we successively set the change-point in different time point
Null hypothesis is assumed to be true, and statistic
Responding to the different time point
In the moving rank sum test, to find out the change-point, we successively set the change-point in different time point
Responding to the different time point
Because Pettitt test is one of the nonparametric tests, it is more robust against outliers and skewed distributions. The length of the time series
If
Mann-Kendall test was developed to detect some trends in the time series [
In hydrological frequency analysis, it is assumed that the samples are drawn independently and randomly from the identical population distribution, the function form of which has not been proved in theory. At present, there are more than 20 popular distribution function forms around the world, including P-III, LP-III, GEV, EV, LN, and K-M. P-III distribution is considered as the most suitable and widely used function for the frequency analysis of hydrologic variables in most regions of China. In this study, P-III distribution function was used as the fitting function for annual maximum tidal level, the probability density function of which is as follows:
Considering that the basis for hypothesis test is the small probability event principle, there may exist two kinds of typical errors: type-I error, where null hypothesis is rejected when the null hypothesis is true; type-II error, where null hypothesis is accepted when the alternative hypothesis is true. The probability of type-I error is equal to significance level. For selected significance level, low probability of type-II error indicates more powerful test. The efficiency of the test is defined as the probability of correctly detecting the trend when it is present. For selected
To analyze efficiencies of SRC test and MK test for detecting trend, taking the annual maximum tidal level series as example, Monte Carlo experiment scheme was designed as follows:
In this study, 9 combinations of 3 linear trends and 3 random fluctuations were investigated; for every combination, 7 different lengths were set, and with each length, 20000 series were simulated by Monte Carlo approach. The total number of the series is
In Figure With the other factors being kept the same, trend test efficiencies of SRC test and MK test were approximate. With the other factors being kept the same, with the increase of random fluctuation, the trend test efficiency decreased, and with the decrease of random fluctuation, the trend test efficiency increased, indicating that random fluctuation may generate impact on trend test. For short time series with small trend change magnitude and big random fluctuation, the trend test efficiency was low. No matter what the kind of the composition of random component and trend component was, trend test efficiency increased with the increase of sample size. If the sample size was above 50 years, the trend test efficiency was high in general.
Relationship between trend test efficiency and sample size in different combinations.
Linear trend was 0.019 m/a and standard deviation was 0.350 m
Linear trend was 0.013 m/a, SRC test
Standard deviation was 0.350 m, MK test
To analyze the efficiencies of MT test, MRS test, Pettitt test, and SQMK test for detecting change-point, Monte Carlo experiment scheme was designed as follows taking the annual maximum tidal level series as an example:
For every one of the above mentioned 10 compositions of standard deviation and variation range of the mean, 20000 series with the length being 50 were generated by Monte Carlo approach. There were 10 × 20000 series in total. MT test, MRS test, Pettitt test, and SQMK test were carried out on all the series, and the ratio for correct detection of the change-point in every 20000 series was counted to evaluate the change-point test efficiency with different composition.
Relationship between change-point test efficiency and ratio of shift in mean to standard deviation of random component was presented in Table With the other factors being kept the same, the efficiencies of MT test, MRS test, and Pettitt test for detecting change-point were approximative with each other, while the efficiency of SQMK test for detecting change-point was obviously low. SQMK test is widely used in the change-point test in hydrological series currently, but based on the statistical experiment it was found that the efficiency of SQMK test for detecting change-point was quite low. With the fixed ratio of shift in mean to standard deviation of random component, the change-point test efficiencies of all these methods were approximate. With the increase of ratio of shift in mean to standard deviation of random component, test efficiency increased continuously, while the increase amplitude decreased gradually.
Efficiencies of different tests for detecting change-point under different compositions.
Standard deviation of random component | Shift in mean | Ratio of shift in mean to standard deviation of random component | Efficiency of test for detecting change-point | |||
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MT test | MRS test | Pettitt test | SQMK test | |||
0.350 | 0.350 | 1 | 25.6 | 29.2 | 32.9 | 11.0 |
0.175 | 0.175 | 1 | 26.1 | 29.7 | 33.4 | 11.1 |
0.525 | 0.525 | 1 | 26.2 | 29.6 | 33.3 | 11.2 |
0.350 | 0.700 | 2 | 65.1 | 65.8 | 67.1 | 18.9 |
0.175 | 0.350 | 2 | 65.5 | 66.0 | 67.3 | 19.1 |
0.525 | 1.050 | 2 | 65.7 | 66.0 | 67.3 | 19.1 |
0.350 | 1.050 | 3 | 87.5 | 86.3 | 86.8 | 21.2 |
0.175 | 0.525 | 3 | 88.5 | 86.8 | 87.3 | 22.0 |
0.525 | 1.575 | 3 | 88.4 | 86.9 | 87.3 | 21.3 |
0.175 | 0.700 | 4 | 94.9 | 95.8 | 96.0 | 22.0 |
0.175 | 1.050 | 6 | 98.9 | 99.7 | 99.7 | 22.2 |
Relationship between change-point test efficiency and ratio of shift in mean to standard deviation of random component.
SRC test and MK test were carried out for detecting trend in annual maximum tidal level series during the period from 1950 to 2008 in 5 representative tide level stations in Yangtze River estuary, namely, Jiangyin, Tianshenggang, Xuliujing, Wusong, and Gaoqiao, the results of which were presented in Table
Trend detection for annual maximum level series of 1950–2008 in 5 stations.
Station | SRC | MK | ||||
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Test statistic | Critical value | Test result | Test statistic | Critical value | Test result | |
Jiangyin | 3.149 | 2.003 | 1 | 2.956 | 1.960 | 1 |
Tianshenggang | 4.534 | 2.003 | 1 | 4.009 | 1.960 | 1 |
Xuliujing | 4.415 | 2.003 | 1 | 3.865 | 1.960 | 1 |
Wusong | 3.048 | 2.003 | 1 | 3.087 | 1.960 | 1 |
Gaoqiao | 3.684 | 2.003 | 1 | 3.531 | 1.960 | 1 |
“1” indicates upward trend.
Change-point detection for annual maximum levels in 5 stations.
Station | Method | Change-point of 1950–2008 | Change-point of 1955–2008 | Change-point of 1960–2008 | Change-point of 1950–1998 | Change-point of 1950–2003 |
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Jiangyin | MT | 1980 | 1980 | 1980 | 1980 | 1980 |
MRS | 1980 | 1980 | 1980 | 1980 | 1980 | |
Pettitt | 1980 | 1980 | 1980 | 1980 | 1980 | |
SQMK | 1987 | 1985 | 1981 | 1990 | 1990 | |
Synthetical result study |
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Tianshenggang | MT | 1989 | 1989 | 1989 | 1980 | 1989 |
MRS | 1980 | 1980 | 1980 | 1980 | 1980 | |
Pettitt | 1980 | 1980 | 1980 | 1980 | 1980 | |
SQMK | 1987 | 1985 | 1981 | 1990 | 1989 | |
Synthetical result |
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Xuliujing | MT | 1980 | 1980 | 1980 | 1980 | 1989 |
MRS | 1980 | 1980 | 1980 | 1980 | 1980 | |
Pettitt | 1980 | 1980 | 1980 | 1980 | 1980 | |
SQMK | 1982 | 1987 | 1980 | 1989 | 1988 | |
Synthetical result |
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Wusong | MT | 1989 | 1989 | 1989 | 1979 | 1989 |
MRS | 1979 | 1979 | 1979 | 1979 | 1979 | |
Pettitt | 1979 | 1979 | 1979 | 1979 | 1979 | |
SQMK | 1980 | 1979 | 1979 | 1988 | 1988 | |
Synthetical result |
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Gaoqiao | MT | 1989 | 1974 | 1989 | 1989 | 1989 |
MRS | 1979 | 1979 | 1979 | 1979 | 1979 | |
Pettitt | 1979 | 1979 | 1979 | 1979 | 1979 | |
SQMK | 1980 | 1979 | 1979 | 1988 | 1988 | |
Synthetical result |
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Time series of annual maximum level for 5 stations during 1950–2008.
Jiangyin
Tianshenggang
Xuliujing
Wusong
Gaoqiao
Sequential values of
The following can be seen from Table The results of MRS test and Pettitt test were almost the same. Even with the changes in samples and stations, results with high stability can be obtained. Change-points were concentrated in 1980 (Jiangyin, Tianshenggang, and Xuliujing stations) and 1979 (Wusong and Gaoqiao stations). As for the results of MT test, with the changes in samples and stations, stable results can also be obtained. Almost half of the first change-points were concentrated in 1989 instead of in 1980 or in 1979 (by contrast, under the same condition, for the series in Jiangyin, Tianshenggang, and Xuliujing stations, the 3rd and the 4th change-points appeared in 1980; for the series in Wusong and Gaoqiao stations, the 3rd, 4th, and 5th change-points appeared in 1979). The change-points detected by SQMK test changed with the samples and the stations, which was not consistent with the results of the other methods. Li et al. [ The final change-points in the annual maximum tidal level series in these stations were synthesized based on the change-points detected by MT test, MRS test, Pettitt test, and SQMK test, and the results were presented in Table
Tidal level in the Yangtze River estuary was mainly affected by such factors as upstream runoff, downstream tidal level, and the river channel storage capacity [
Datong hydrological station was selected as the control station for analysis of runoff in the Yangtze River estuary. Based on recorded data, annual maximum runoff in Datong station was concentrated mainly in July and August, and the annual maximum tidal level in the Yangtze River estuary was also concentrated in July and August. For example, in Jiangyin station and Tianshenggang station, probabilities of the annual maximum tidal level in July and August were 78.0% and 76.3%, respectively.
In this study, trends and change-points for 3 series such as annual maximum discharge, average discharge in July, and average discharge in August during the period of 1950–2008 in Datong station were detected using SRC test, MK test, MT test, MRS test, and Pettitt test. It was found that there was no significant increasing tendency and change-point for these 3 discharge series, which was in coincidence with the conclusions of Qin et al. [
Climate change made the extreme weather events more frequently. In Figure
Relationship between annual maximum level in Jiangyin station and corresponding discharge in Datong station. (Series A: 1950–1979, without impact of typhoon; Series B: 1980–2005, without impact of typhoon; Series C: 1950–1979, with impact of typhoon; Series D: 1980–2005, with impact of typhoon.)
It is found from Figure
For all these 5 stations, the annual maximum tidal level series during 1950–2008 were split into two subseries (before and after the change-point) and trend tests were performed for each of the two subseries separately. Considering the precondition for frequency analysis, samples should be drawn from the identical population, which should also be independent of each other. Sample autocorrelation test was carried out for evaluation of independence of the two subseries for all these stations. Since sample autocorrelation test was widely used [
Trend and autocorrelation test results for subseries before and after the change-point in 5 stations.
Station | Subseries before change-point | Subseries after change-point | ||||||||||
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SRC test | MK test | Autocorrelation test | SRC test | MK test | Autocorrelation test | |||||||
Test statistic | Test result | Test statistic | Test result | Autocorrelation coefficients | Test result | Test statistic | Test result | Test statistic | Test result | Autocorrelation coefficients | Test result | |
Jiangyin | –1.267 | 0 | –1.124 | 0 | –0.093 | √ | –0.018 | 0 | 0.094 | 0 | 0.165 | √ |
Tianshenggang | –0.709 | 0 | –0.678 | 0 | –0.192 | √ | 0.899 | 0 | 0.882 | 0 | 0.176 | √ |
Xuliujing | –0.570 | 0 | –0.571 | 0 | –0.199 | √ | 0.159 | 0 | 0.169 | 0 | 0.128 | √ |
Wusong | 0.438 | 0 | 0.482 | 0 | –0.244 | √ | 0.079 | 0 | 0.169 | 0 | 0.036 | √ |
Waigaoqiao | 0.218 | 0 | 0.375 | 0 | –0.229 | √ | 0.243 | 0 | 0.300 | 0 | 0.017 | √ |
“0” indicates “no trend”, “1” indicates “upward trend”, “−1” indicates “downward trend”, and “√” indicates “independence”.
To carry out quantitative analysis on the probability distribution changes in these stations, frequency analysis was carried out on the annual maximum tidal level subseries before and after the change-point in each station. P-III distribution function was selected as the fitting function, the 3 parameters of which were determined with curve-fitting method, and the initial parameters of the curve-fitting method were estimated by L-moment method. Through frequency analysis, annual maximum tidal level probability distributions before and after the change-point in Jiangyin, Tianshenggang, Xuliujing, Wusong, and Gaoqiao stations were obtained, which were presented in Figure
Distribution parameters and design values of the annual maximum tidal level subseries before and after the change-point in each station.
Station | Series | EX |
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Frequency (%) | |||||
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0.1 | 0.2 | 1 | 2 | 5 | 10 | |||||
Jiangyin | B | 3.99 | 0.090 | 1.40 | 5.82 | 5.63 | 5.17 | 4.96 | 4.69 | 4.47 |
A | 4.35 | 0.100 | 1.50 | 6.63 | 6.38 | 5.80 | 5.54 | 5.20 | 4.93 | |
A-B | 0.36 | 0.81 | 0.76 | 0.63 | 0.58 | 0.51 | 0.46 | |||
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Tianshenggang | B | 3.65 | 0.088 | 1.40 | 5.29 | 5.11 | 4.70 | 4.52 | 4.27 | 4.08 |
A | 4.04 | 0.106 | 1.80 | 6.46 | 6.18 | 5.54 | 5.26 | 4.89 | 4.60 | |
A-B | 0.39 | 1.17 | 1.07 | 0.84 | 0.74 | 0.62 | 0.53 | |||
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Xuliujing | B | 3.34 | 0.087 | 1.40 | 4.82 | 4.66 | 4.29 | 4.13 | 3.90 | 3.73 |
A | 3.72 | 0.109 | 1.90 | 6.06 | 5.79 | 5.16 | 4.89 | 4.53 | 4.25 | |
A-B | 0.38 | 1.24 | 1.13 | 0.87 | 0.76 | 0.62 | 0.52 | |||
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Wusong | B | 3.08 | 0.088 | 1.40 | 4.46 | 4.31 | 3.97 | 3.81 | 3.61 | 3.44 |
A | 3.41 | 0.128 | 1.80 | 5.87 | 5.59 | 4.94 | 4.65 | 4.28 | 3.99 | |
A-B | 0.33 | 1.41 | 1.28 | 0.97 | 0.84 | 0.67 | 0.54 | |||
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Gaoqiao | B | 3.15 | 0.085 | 1.40 | 4.59 | 4.43 | 4.06 | 3.89 | 3.68 | 3.51 |
A | 3.42 | 0.125 | 1.90 | 5.89 | 5.60 | 4.94 | 4.65 | 4.27 | 3.98 | |
A-B | 0.27 | 1.30 | 1.17 | 0.88 | 0.76 | 0.60 | 0.47 |
“B” indicates subseries before change-point, “A” indicates subseries after change-point, and “A-B” indicated the difference between the subseries before and after change-point.
Frequency curves of the annual maximum tidal level subseries before and after the change-point in each station.
Jiangyin
Tianshenggang
Xuliujing
Wusong
Gaoqiao
As for Jiangyin, Tianshenggang, Xuliujing, Wusong, and Gaoqiao stations, before the change-point, probability of annual maximum tidal levels being above 5.82 m, 5.29 m, 4.82 m, 4.46 m, and 4.59 m was 0.1% (1000-year return period); after the change-point, probabilities of annual maximum tidal levels being above 5.82 m, 5.29 m, 4.82 m, 4.46 m, and 4.59 m were 1.0%, 1.9%, 2.4%, 3.2%, and 2.4%, respectively (corresponding to 104-, 53-, 41-, 31-, and 42-year return periods); before the change-point, probability of annual maximum tidal levels being above 5.17 m, 4.70 m, 4.29 m, 3.97 m, and 4.06 m was 1% (100-year return period); after the change-point, probabilities of annual maximum tidal levels being above 5.17 m, 4.70 m, 4.29 m, 3.97 m, and 4.06 m were 5.5%, 8.3%, 9.3%, 10.5%, and 8.6%, respectively (corresponding to 18-, 12-, 11-, 10-, and 12-year return periods). Accordingly, after the change-point, probability of annual maximum level exceeding the same value increased significantly.
In this paper, the efficiencies of different trend and change-point detection methods were investigated through Monte Carlo simulation, and the nonstationarity of annual maximum level records in the Yangtze River estuary was analyzed. The main conclusions were as follows: Based on statistical experiments, it was found that the efficiency of MK test was almost the same as that of the SRC test. Test efficiency depended on the amplitude and duration of trend changes, and the amplitude of random fluctuation. For short time series with small trend change magnitude and big random fluctuation, the trend test efficiency was low, while for long time series with big trend changes magnitude and small random fluctuation test efficiency was above 90%. Based on statistical experiments, it was found that the efficiencies of MT test, MRS test, and Pettitt test were almost the same. Test efficiency depended on the amplitude of shift in mean and the amplitude of random fluctuation. For the sequence with big shift in mean and small random fluctuation, test efficiency was high. For SQMK test widely used in change-point detection currently, the detection efficiency was very low. It was suggested that other methods with higher efficiency than SQMK be used in related studies in the future. Significant tendency changes existed in annual maximum tidal level series at all selected stations in the Yangtze River estuary. 1980 and 1979 were statistically significant change-points. Frequency increase of typhoons and storage capacity decrease of the local river channel were main causes for nonstationary change of annual maximum tidal level. Results of trend and independence test for the subseries before and after the change-points showed that the subseries can be accepted as stationary series without serial correlation. Through frequency analysis on the subseries before and after the change-point in all these stations, it was found that obvious changes took place in probability distribution of annual maximum tidal level in the Yangtze River estuary. For every station, compared with the conditions before the change-point, design high tidal level for the same frequency after the change-point increased significantly. Meanwhile, risks of exceeding the same extreme after the change-point also increased significantly. The changes in probability distribution should be considered in determining design high tidal levels for flood protection measures.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was supported by the National Natural Science Foundation of China (no. 51479061) and the Main Program of National Natural Science Foundation of China (no. 51190091). The authors would like to thank the anonymous reviewers for their comments permitting improvement of the paper.