Seasonality in a tidal reach: Existence, impact and a possible approach for design flood level estimation

Science of the Total Environment 714 (2020) 136478

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Seasonality in a tidal reach: Existence, impact and a possible approach for design flood level estimation Yi-han Tang, Xiao-hong Chen ⁎ Center for Water Resources and Environment, Sun Yat-sen University, Guangzhou 510275, China Guangdong Engineering Technology Research Center of Water Security Regulation and Control for Southern China, Sun Yat-sen University, Guangzhou 510275, China Key Laboratory of Water Cycle and Water Security in Southern China of Guangdong High Education Institute, Sun Yat-sen University, Guangzhou 510275, China

H I G H L I G H T S

G R A P H I C A L

A B S T R A C T

• Significant seasonality results in heterogeneous annual maximum flood level series in the tidal reach. • Statistical differences of the seasonal series are highly correlated to different procedures of flood generation. • In a single-distribution model, flood season mainly affects the upper tail and the non-flood season affects the lower tail. • The mixed-distribution model is superior in design flood level estimation for it keeps more non-flood season information.

a r t i c l e

i n f o

Article history: Received 6 June 2019 Received in revised form 31 December 2019 Accepted 31 December 2019 Available online 11 January 2020 Keywords: Seasonality Design flood level Mixed-distribution Tidal reach Pearl River Delta

a b s t r a c t Heterogeneity caused by seasonality could lead to the estimation error of the design flood level (DFL). This research intended to examine the existence of seasonality in the extreme water levels in a tidal reach and to quantify its impact on the DFL estimation. The mixed-distribution, a commonly used method for design value estimation with heterogeneous samples, was tested. A case study was carried out in the Pearl River Delta, South China. Results showed that a significant seasonality existed in the extreme water levels that were generated from the flood-tide interactions in the delta. If the DFL was estimated with a single distribution, the DFL with a return period smaller than 1.1 years would be underestimated and more information of the non-flood season would be lost. The mixed-distribution was superior in its consideration of seasonality, however, when the return period was over 10 years or smaller than 5 years, the DFL estimation results of this approach were only shifted by b1% from that of a single distribution. © 2020 Elsevier B.V. All rights reserved.

1. Introduction ⁎ Corresponding author at: Center for Water Resources and Environment, Sun Yat-sen University, Guangzhou 510275, China. E-mail address: [email protected] (X. Chen).

https://doi.org/10.1016/j.scitotenv.2019.136478 0048-9697/© 2020 Elsevier B.V. All rights reserved.

Over 200 million people are being threatened by the increasing frequencies of coastal floods all over the world (Nicholls, 2011; Arns et al., 2013). To protect deltas from flood disasters, reliable designs of flood

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Y. Tang, X. Chen / Science of the Total Environment 714 (2020) 136478

defense structures are required. Since the 1960s, the design flood level (DFL) like the 20-year flood level or the 50-year flood level has been widely used as the design standard, and it is mainly calculated on the basis of extreme value theory (Gumbel, 1958). Since rising sea-level has lifted tidal levels and lead to the non-stationarity in records, many recent researches have studied the impact of non-stationarity on the DFL estimation (Dixon and Tawn, 1999; Haigh et al., 2013; Arns et al., 2013; Särkkä et al., 2017; Uranchimeg et al., 2018). However, except for non-stationarity, the heterogeneity of the recorded levels can also cause the estimation error of the DFL. The problem of heterogeneity widely exists in hydro-climate extreme series. It is raised by the fact that extreme events usually emerge from different physical processes (Iliopoulou et al., 2018). These events also have different statistical features and these features often belong to different populations. For example, the process features of the floods generated by rainstorms largely differed from that of the floods by snow-melt events; rainstorms caused by tropical cyclones have different statistical characteristics from those by frontal cloud systems. In extreme value theory, samples are assumed to come from the same population (Allamano et al., 2011). However, when selecting the extreme samples with the annual maxima sampling method, samples generated by different processes are extracted and they are assumed to be coming from the same population. Under this false assumption, a single probability density is often adopted to estimate the entirety and it can further lead to the bias of estimation results. Therefore, it is necessary to examine the heterogeneity of records before carrying out the frequency analysis of extreme events. Seasonality is a leading cause for heterogeneity, and to exclude the impact of seasonality, previous studies chose to either identify seasonality before sampling or replace the single distribution method with a mixed distribution model (Waylen and Woo, 1982; Burn, 1997). As for seasonality identification, methods like directional statistics (DS), relative frequency and variable measurement were widely applied (Cunderlik et al., 2004; Chen et al., 2013). In order to further differentiate events coming from different geneses, Fischer et al. (2012) shortened the time scale and cut seasonal series into sub-seasonal series which only included samples coming from the same flood type. As for the frequency analysis with a mixed distribution, there were mainly two types of mixed distribution models developed for either extreme rainfalls or floods. The first type was a product of two functions, and it required the physical process of each seasonal population to be completely independent (Waylen and Woo, 1982). Baratti et al. (2012) guaranteed the consistency between the annual and seasonal distribution and improved this method by introducing a subjective weight for each season. The second type contained a weight, and samples from different seasonal populations were not necessarily independent (Chen et al., 2010; Mascaro, 2018). Durrans et al. (2003) improved this method by considering the interrelationship between annual and seasonal maxima series. Allamano et al. (2011) examined the usage of this model by comparing the analyzing results of different sampling techniques like the Annual Maxima and the Peaks-over-threshold method. However, most previous works that considered heterogeneity caused by seasonality in the frequency analysis of hydro-climate events like extreme rainfalls and floods mainly focused on closed drainages rather than delta areas, and only a few works studied this issue from the aspect of DFL estimation (Bernier et al., 2007; Méndez Fernando et al., 2007). What's more, these previous works, which applied the series of seasonal maxima total sea level in the design level estimation, usually took the process of extracting seasonal maxima as an improvement in the sampling method by narrowing down the time-scale. They neither treated seasonality as the cause for heterogeneity nor tested the significance of seasonality or examined the impact of seasonality on DFL analysis (Devlin et al., 2017). In addition, previous studies often chose the spots along the coastline or at an estuary for their researches. In these areas, the extreme

water level was mainly affected by tidal fluctuation and storm surge (Wolf, 2008; Haigh et al., 2013; Bacopoulos, 2017). However, in the deltas that most international metropolis is located, floods are very likely to happen in tidal reaches where the extreme water level is mutually affected by the flood flow volume coming down from the upstream river basin and the extreme high tide going up from the downstream estuary (Moftakhari et al., 2017). The flood-tide compounding effect in tidal reaches increases the complexity of the statistical characteristics of extreme water levels, especially when the heterogeneity caused by seasonality is also taking effects. Under this circumstance, it is extremely important and necessary to study the seasonality as well as its impact on the DFL estimation in tidal reaches. Hence, this research aims to study the existence and the impact of the seasonality in tidal reaches by trying to answer the following questions: Is there any significant seasonality existing in the extreme water levels in a tidal reach? Will the heterogeneity caused by this seasonality lead to an estimation bias of DFL? A representative tidal reach in the Pearl River Delta in Southern China, where the Guangdong-Hong Kong-Macao-Greater-Bay locates, is used as the study area. The result of this research will show the pattern and the cause of the impact of seasonality on DFL calculation by exploring the procedure of DFL estimation as well as the statistical features of seasonal time series. It will also provide a possible solution to avoid the estimation bias caused by seasonality, which can be served as a reference for DFL estimation in other delta areas. 2. Study area and dataset 2.1. Study area The Pearl River in South China is one of the three largest rivers in China (Fang et al., 2018b; Han et al., 2018; Aa et al., 2019). It is comprised of the West, the East, and the North River. The Pearl River Delta (PRD) is located at the lower reach of the Pearl River. It is formed by the aforementioned three tributaries, covering an area of 42,657 km2 (Yang et al., 2015). The terrain in the PRD decreases from the northwest to the southeast. Flowing down the slope, runoff from three rivers converges and flows into the estuary through eight outlets in the PRD. The criss-crosses of river channels contribute to a high density of river network, which makes the PRD one of the most complex deltas all over the world (Yang et al., 2009). The West River Mainstream Channel is a major river channel in the whole PRD, and it goes straight from the confluence point of three rivers to one of the outlets (Fig. 1). All river channels in the PRD are tidal reaches, and the in-channel extreme water level is therefore highly affected by the discharge variation of the upstream river basin and the tidal propagation in the estuary (Zhang et al., 2017). In the upstream river basins, precipitation concentrates in a wet season which lasts from April to September and floods mainly happen between June and August. Meanwhile, high tides in the estuary, which are induced by either the astronomical impacts or the tropical cyclone, mainly arise between August and November (Tang et al., 2016). As a result, over 60% of floods in the PRD happen between June and August (Chen et al., 2009). 2.2. Dataset In this research, monthly maximum water levels and their occurring dates in three stations along the West River Mainstream Channel have been applied. The locations of these three stations are shown in Fig. 1. Makou, which is also the joint point of upstream rivers, is located in the upper reach of the river channel and is 129 km away from the outlet. Nanhua station is located in the middle reach and is 61 km away from the outlet. Meanwhile, Denglongshan station is 5.2 km away from the delta outlets and is close to the Pearl River Estuary. Data series in each station lasted from January 1958 to December 2011 and were all qualified for statistical analysis. The records between

Y. Tang, X. Chen / Science of the Total Environment 714 (2020) 136478

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Fig. 1. Map of the Pearl River Delta and the locations of three gauging stations. The West River Mainstream Channel is highlighted in orange. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

1959 and 1989 and between 2006 and 2011 were extracted from the Hydrological Year Book (Information Center of Water Resources, 1959-1989, 2006-2011). The data between 1990 and 2005 were provided by the Hydrology Bureau of Guangdong Province. There was no missing data in Nanhua. In Denglongshan, the data in 1958 was missing. In Makou, the data in 1959 (May–August), 1966 (January–December), 1968 (June–August), 1969 (June–August), and 1978 (June) were missing. Before putting all the data into use, all the missing data in Makou and Denglongshan were interpolated from the data in their neighboring stations using the BP neural network method (Fang et al., 2018a; Wang et al., 2018; Wang et al., 2019).

3. Methodology 3.1. Seasonality identification Three approaches have been adopted to identify the seasonality of extreme water levels in this research. Rao's spacing test is a simple and feasible omnibus test to assess the uniformity of a circular dataset, and it is considered as the basic procedure to deal with circular data. This method uses the space between observations to examine whether the samples have directionality, and the result only contains one single value U, which is in a positive correlation with the significance of seasonality (Yan et al., 2017). Another index P is built up to show the tendency of the occurrence of annual maximum water level (AMWL) in each month. The value of P

can be achieved based on the following equation,

P¼

ni  100%ði ¼ 1; 2; …; 12Þ N

ð1Þ

where ni is the number of AMWL occurrences in the ith month; N is the number of years. Last but not least, DS is applied (Cunderlik et al., 2004; Chen et al., 2010). This method can depict the statistical characteristics of the seasonality in hydrological extremes by generating indexes like the concentration date (D), the concentration period (from Df to Dl), as well as the degree (r) and the standard deviation (s) of sample concentration. Assuming that there are T days in a computation period, then the occurrence date of the jth AMWL sample (Dj) can be transformed by trigonometric function and then represented by a set of coordinates (xj,yj), which can be calculated as follows:

aj ¼ Dj

2π 0≤a j ≤2π T



 x j ; y j ¼ q j cosα j ; q j sinα j

ð2Þ

ð3Þ

where aj is an angle, and qj is the magnitude of the jth AMWL sample. In this way, all the dates of AMWL occurrences can make up a matrix

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et al., 2017). Therefore, it has been used in this research, and it can be calculated as follows,

(X, Y): 0

1 N N ∑ j¼1 x j ∑ j¼1 x j @ A ; N ðX; Y Þ ¼ N ∑ j¼1 q j ∑ j¼1 q j

ð4Þ

where N is the number of all samples. Then D, Df and Dl can be estimated by the following equation, 8 T > > D¼σ > > 2π < σ −s Df ¼ > 2π > > σ þs > : Dl ¼ 2π

ð5Þ

AIC ¼ −2 L þ 2p

ð9Þ

where L is the log-likelihood maximum of the mixed-distribution and p is the number of parameters used in the mixed-distribution. The PPCC test can be carried out regardless of the model type. It tests the goodness of fitting by examining the correlation coefficient between the records xk (k = 1,2,3, ……) and the expectation series yk (Bezak et al., 2014). And yk is the estimation value calculated by the probability distribution function with an empirical frequency Pk. The greater the correlation coefficient is, the better the fitting efficiency is.

Furthermore, r, which is the index depicting the concentration tendency of αi, and the standard deviation s can be calculated as, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X2 þ Y 2 r¼ ; 0≤r≤1 N ∑ j¼1 q j s¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −2 ln r

4. Results and discussions ð6Þ

ð7Þ

The greater this r is, the more concentrated the samples are. And when the value of r is N0.55, occurrences of events concentrate in a period with an obvious significance; when the value of r is smaller than 0.35, events are assumed to happen evenly within a year. 3.2. The mixed-distribution function The mixed-distribution has been proposed for frequency analysis of sample series that contains heterogeneity, and there are four major types based on the structure of the mixed-distribution (Alila and Mtiraoui, 2002). This research has developed the mixed-distribution model on the basis of the third type, which assumed the annual maximum samples to be a mix of different populations while each population had a homogeneous distribution. This approach does not require a prior separation of flood processes, and the equation of the probability density function (PDF) is displayed in Eq. (8) (Fiorentino et al., 1985), f ðxÞ ¼ αp1 ðxÞ þ ð1−α Þp2 ðxÞ

ð8Þ

where α is the weight showing the relative contribution of each population. Parameters were estimated by the Genetic Algorithm carried out in Matlab (Sharman and McClurkin, 1989). The difference between the model adopted in this research and this third type mixed-distribution lies in the selection process of each element PDF. Since no prior separation is required in the third type, each element PDF is not selected based on the statistical characteristics of each seasonal ensemble. It is determined when the whole model reaches the best fitting efficiency. However, this ignores the different physical processes of different seasonal ensembles. Therefore, each element PDF has been selected based on the best-fit model of each seasonal series in this research. 3.3. Methods for the best-fit model selection To evaluate the efficiency of distribution fitting, two methods for model selection have been used in this research. One is the Akaike Information Criterion (AIC), and another is the PPCC method (Akaike, 1974; Bezak et al., 2014). AIC is a commonly used criterion method for selecting the best-fit model in flood frequency analysis. It has been applied in model selection for both singular distribution and mix-distribution models. It can avoid the over-fitting problem caused by the mixture of two distributions by adding a penalty term for the increased number of parameters (Yan

4.1. Existence of seasonality The results of Rao's spacing test were 226.08, 243.3 and 205.28 in Makou, Nanhua, and Denglongshan. The values of Us all passed 163.60, which is the critic value of level 0.001. In other words, monthly extreme levels in three stations all had significant seasonality. In addition, according to the different values in three stations, Makou and Nanhua had stronger seasonality than Denglongshan, which demonstrated that the seasonality was stronger in the upper and middle reach of the river channel. Then the contribution of each month to annual maxima in three representative stations was detected by index P (Fig. 2). In both Makou and Nanhua, annual maxima happened the most in June and July, and the P of either month was over 30%. In Denglongshan, however, the highest probability of annual extreme occurrence was only in one month, i.e., July. As for the results of DS (Fig. 2), the annual extreme level in Makou mainly occurred between June 29th and September 2nd, and the most on July 31st. In Nanhua, the concentration period lasted from June 4th to August 5th, and the concentration date was July 5th. The concentration period in Denglongshan, however, lasted 20 days longer than the other two stations, which covered the period from June 20th to September 17th. What's more, the values of r were higher than 0.5 in all three stations, and values in Makou (0.86) and Nanhua (0.87) were both higher than the value in Denglongshan (0.73). On the contrary, the deviation s in Denglongshan (0.8) was much higher than that in Makou (0.55) and Nanhua (0.53). Overall, the seasonality features detected by three approaches indicated the same phenomenon: (1) Two stations located in the upper and middle reach of river channel, i.e., Makou and Nanhua, shared similar features of seasonality in both the significance and the distribution of months; (2) Three stations all had significant seasonality in their extreme water levels, yet the station located closer to the estuary, i.e., Denglongshan, was prone to have longer concentration period, greater dispersion and comparatively lower degree of seasonality; (3) Although the concentration period lasted longer in Denglongshan, the month that contributed to annual maximum sample was more concentrated. The greater impact of tidal fluctuation from the estuary possibly caused the abnormity in Denglongshan. Compared to the stations in the upper reach, extreme level in this station was more affected by tropical cyclones, storm surges, rising sea-level as well as the spring rather than the upstream flood flow (Tang et al., 2016; Zhang et al., 2017). Compared to the upstream flood, the impact of high tides could last for one-month longer (Fischer et al., 2012). As a result, the occurrences of annual maxima distributed in longer periods, and the month, when the upstream flood and the downstream high tides were the most likely to join, had contributed the most to the annual maxima series.

Y. Tang, X. Chen / Science of the Total Environment 714 (2020) 136478

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Fig. 2. Seasonality of extreme water level in three stations in the Pearl River Delta. (a) Makou; (b) Nanhua; (c) Denglongshan. The values of the index P are displayed in the left panel (Unit: %); Results of DS method, i.e., concentration date, concentration period, r, and s are demonstrated in the right panel.

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Y. Tang, X. Chen / Science of the Total Environment 714 (2020) 136478

4.2. Statistical differentiation in seasonal population Although seasonality existed in three stations, it was not necessary that heterogeneity existed. Therefore, this section examined whether seasonal series in these stations belonged to different populations, and the statistical characteristics of seasonal series were calculated and compared. Also, the statistical characteristics of seasonal series were compared to that of the annual one. Monthly extremes were first generated and divided into the flood and non-flood season series. To keep in accordance with the results in Section 4.1, and to unify the number of samples in different stations, flood seasons in all three stations were set to last for three months, i.e., June to August in Makou and Nanhua, and July to September in Denglongshan. Except for the months in the flood season, other months all belonged to the non-flood season. 4.2.1. General features Box plots of annual and seasonal series in three stations were displayed in Fig. 3. In general, obvious differences existed between seasonal series. In all stations, flood season had a greater maximum, median, and mean than a non-flood season. The upper interquartile of the non-flood season was close to the lower one of the flood season. The difference between seasonal means was the greatest in Nanhua (0.99 m), then in Makou (0.42 m), and the least in Denglongshan (0.25 m). Based on the seasonal differences of general statistics, samples from flood and non-flood season could be assumed to belong to different populations. Annual series had similar median, mean, interquartile, outliers, maximum value as well as the difference between the median and the mean with flood season series. To conclude, the annual series and flood season series had similar patterns in most statistics. It was partly due to the over 65% contribution of flood season to annual maximum samples, which was shown in Fig. 2 in Section 4.1. Meanwhile, the deviation of the annual series stayed between two seasonal series, and the minimum value of the annual series was close to that of the non-flood season. The following four statistics in Denglongshan were different from the other stations: (1) The difference between the maximum and minimum value of the series: in Denglongshan, this gap of the flood season was 2.2 times that of the non-flood season. However, this gap of flood season was similar to that of non-flood season in both Makou and Nanhua; (2) The difference between the median and the mean: it was smaller in flood season than non-flood season in Makou and Nanhua,

yet in Denglongshan, it was greater in flood season; (3) The seasonal minimum value: it was greater in flood season in Makou and Nanhua, but it was 0.21 m higher in non-flood season in Denglongshan. (4) The annual minimum value: it was similar to flood season in Makou and Nanhua, but it was closer to the non-flood season in Denglongshan.

4.2.2. Specific features Since extreme values played an important role in DFL estimation, and the deviation of the sample was not explicitly demonstrated in Fig. 3, in this part, the standard deviations and 95th quantiles of annual and seasonal time series were further compared and displayed in Table 1. In all three stations, the 95th quantiles of flood season were higher than that of the non-flood season. On the contrary, however, standard deviations in Makou and Nanhua were all higher in non-flood season. Overall, flood season series had greater values of samples while the non-flood season series had a greater deviation in samples. The gap between seasonal 95th quantiles was the greatest in Nanhua and the slightest in Makou. Annual quantiles were always higher than the seasonal ones except for the flood season in Denglongshan. In addition, in Makou and Nanhua, the annual series had the smallest deviation, while in Denglongshan, the non-flood season had the least deviation. Just like the general features, the specific features in Denglongshan were also different from the other two stations. Also, it could be deducted from the pattern of standard deviations that the fluctuation of extreme water levels decreased along the direction of the river channel from Makou to Denglongshan. All the statistical differences in annual and seasonal series contributed to the different best-fit distribution selection results, which also reflected different features of these populations (Table 2). As was shown in Table 2, the best-fit models for annual series and flood season series were the same in all stations. Seasonal best-fit models were different in Makou and Nanhua. Although flood season and non-flood season had the same distribution in Denglongshan, the relative difference between seasonal Xs, Cvs and Css could reach 33.3%. The patterns of these specific features proved that seasonal series belonged to different populations, which was concluded from the results in Section 4.2.1. What's more, the annual series was more affected by the flood season one.

Fig. 3. Box plots of three series in three stations. box: interquartile range (excluding the median value); line in the middle of the box: median; whiskers: minimum and maximum value; dots: outliers; crosses: mean value.

Y. Tang, X. Chen / Science of the Total Environment 714 (2020) 136478 Table 1 Standard deviations and 95th quantiles of three series in three stations. Station

Makou Nanhua Denglongshan

Index

Time series

Standard deviation 95th quantile Standard deviation 95th quantile Standard deviation 95th quantile

Annual

Flood season

Non-flood season

1.55 9.20 0.974 5.72 0.329 1.59

2.13 9.09 1.02 5.52 0.334 1.92

2.33 8.98 1.19 4.85 0.16 1.43

4.3. Impact of seasonality on DFL estimation with a singular distribution The present technique of DFL calculation often adopted a single distribution to fit annual samples and ignored the diverse populations of seasonal series. This section took a further look into the fitting efficiency of a singular distribution on the annual series composed of heterogeneous samples. As was shown in Table 2, the generalized extreme value distribution was the best model in Makou and Nanhua, and the Pearson-III type was the best model in Denglongshan. It could be seen that, regardless of the tail features, the best probability distribution functions were all 3-parameter ones, which accorded with the results of Yang et al.'s (2009) research. Therefore, only three 3-parameter distributions were used in this section. The flood frequency curves of three 3-parameter distributions in three stations were demonstrated and compared in Fig. 4. The general pattern of frequency curves was looked into through three portions: (1) the upper tail, i.e., the frequency curve with an exceeding probability smaller than 10%; (2) the body, i.e., the exceeding probability staying between 10% and 90%; (3) the lower tail, i.e., the exceeding probability being over 90%. Slight gaps existed in the body between different frequency curves. In the upper tail, gaps between curves were in a negative correlation with the exceeding probability. Gaps in the lower tail were all greater than the body but smaller than the upper tail. Meanwhile, these gaps amplified along with the increase of exceeding probability. It was not hard to see that fitting efficiency of three distributions had almost no difference in the body, and the fitting effects of the upper and lower tails made the decisive impact on the selection result of a best-fit singular model. Fitting effect for each realm of exceeding probability was further detected by the root-mean-square error (RMSE) respectively (Hosking,

7

1986), and the results were shown in Table 3. As was shown, the bestfit model had the highest efficiency in fitting both the body and the upper tail. In the lower tail in all three stations, however, generalized logistics distribution rather than other distributions had the highest fitting effect. Somehow, it was not the best-fit singular function for any of these stations. In other words, when estimate DFLs with a bestfit singular distribution, the fitting efficiency of the lower tail was sacrificed. What's more, since the design value estimated by a generalized logistics distribution was higher than the others, when estimating DFLs with a singular model, the ones with a return period smaller than 1.1 years were all underestimated (Fig. 4). According to the seasonal contribution to the annual maxima in different scopes of exceeding probability (Table 4), almost all three annual maximum series in all scopes were composed of both seasonal samples. In general, flood season made a greater contribution to the annual series than the nonflood season. The contribution of the non-flood season in the lower tail, however, was similar to the body and the upper tail in Makou, and greater than the body in both Nanhua and Denglongshan. In other words, the features of the non-flood season were more reflected in the lower tail than the other parts of the frequency curve. Therefore, when fitting frequency curves ignoring the efficiency in the lower tail, more features of the non-flood season would be lost at the same time. 4.4. DFL estimated by mixed-distribution To explore a possible solution for the DFL estimation, this research tested the mixed-distribution model, which was commonly used in previous researches for frequency analysis of heterogeneous hydrological samples in closed river basins. DFLs of different return periods calculated by singular and mixed-distribution were compared. Previous studies usually chose the same distribution for each population in the mixed-distribution model, neglecting the possibility that different populations might have different best-fit distributions just like Makou and Nanhua in this research (Alila and Mtiraoui, 2002). To increase the reliability of the results and to make full use of recorded data, this research intended to apply the best-fit model of each population to build up the mixed-distribution. However, in Makou and Nanhua, the mixeddistribution failed in reaching constrictions in parameter estimation. It was assumed to be the result of the improved construction of the mixed-distribution applied in this research, which was composed of different probability function distributions and each element function was selected according to the best-fit model of each seasonal series. This

Table 2 Results of best-fit distribution selection. (1) Akaike Information Criterion (AIC) test results of three series in three stations.

Makou

Nanhua

Denglongshan

Annual Flood season Non-flood season Annual Flood season Non-flood season Annual Flood season Non-flood season

P3

GEV

GLO

WBL

LN2

209.48 250.84 252.62 155.67 181.70 147.61 −8.40 −3.49 −61.30

202.66 234.79 251.11 154.65 174.56 149.96 4.59 8.59 −59.10

205.98 239.13 247.19 158.21 177.90 152.59 5.85 10.08 −58.37

230.49 260.95 261.65 163.21 194.04 148.65 18.08 16.29 −58.17

221.22 271.00 274.87 159.16 193.47 148.02 26.50 24.47 −48.39

(2) Comparison of statistical characteristics of three series in Denglongshan. X Annual Flood season Non-flood season

1.70 1.65 1.41

E1 (%)

Cv

2.9 17.1

0.09 0.10 0.06

E1 (%)

Cs

E1 (%)

11.1 33.3

0.34 0.35 0.28

2.9 17.6

Note: (1) “P3” is short for Pearson-III type; “GEV” is short for Generalized Extreme Value distribution; “GLO” is short for Generalized Logistics distribution; “WBL” is short for Weibull distribution; “LN2” is short for Two-parameter Log-normal distribution; (2) E1 was the absolute value with the value of annual maximum as the denominator; X was the expectation; Cv represented the coefficient of variation, and Cs represented the coefficient of skewness.

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Y. Tang, X. Chen / Science of the Total Environment 714 (2020) 136478

Fig. 4. Comparing fitting effects of three 3-parameter distribution in frequency curves. (a) Makou; (b) Nanhua; (c) Denglongshan.

structure raised a higher requirement for the basis of mathematical theory, which was still beyond the applications of temporal researches (Alila and Mtiraoui, 2002; Baratti et al., 2012; Arns et al., 2013; Yan et al., 2017). Since the intention of this research was to examine the mixed-distribution as a possible solution, and the realization in statistical theory was not the major focus, only design levels in Denglongshan, the seasonal series of which both obeyed the Pearson-III type function, had been estimated and compared in both singular and mixeddistribution model. The mixed-distribution in Denglongshan was composed of two Pearson-III type functions, and the value of weight α was 0.6296. The results were shown in Table 5. As was shown in Table 5, the singular model had higher efficiency in fitting samples in the test results of AIC, but mixed-distribution had a better fitting efficiency in the PPCC test. This difference resulted from the punishment term added to the AIC test (Section 3.3). However, in the results of either AIC or PPCC, the differences between the fitting effects of two DFL estimation models were smaller than 0.5. As for the results of DFLs, the estimation of mixed-distribution was higher than the singular distribution when the return period was N10 years, and this gap increased along with the return period. The 100-year DFL estimated by a mixed-distribution was 0.03 m higher than that by a singular model; the 2-year DFL estimated by a mixeddistribution was 0.02 m higher than that by a singular model. This result was not only in accordance with the results in Section 4.3 but also proved that ignoring seasonality would lead to a downward bias in design values and a loss in the physical meaning of model (Allamano et al., 2011). By using the singular model rather than the mixed-distribution, the commonly used return levels, i.e., 100-year, 50-year, 20-year, 10-year, 5-year, and 2-year DFL, were underestimated by b1%, especially when the return period was N10 years or smaller than 5 years (Table 5). Table 3 Fitting efficiency in different scopes of exceeding probability.

PDF

Makou

Nanhua

Denglongshan

P3 GEV GLO P3 GEV GLO P3 GEV GLO

Exceeding probability Upper tail Body (<10%) (10–90%) 0.46 0.18 0.12 0.14 0.18 0.20 0.07 0.07 0.05 0.06 0.13 0.10 0.10 0.03 0.14 0.05 0.16 0.05

Lower tail (>90%) 0.80 0.54 0.43 0.13 0.13 0.12 0.04 0.022 0.021

Note: PDF is short for Probability Distribution Function. The results for the best-fit singular PDF are highlighted in BLUE, and the distribution with the highest efficiency in each realm was marked with a Bold format.

In closed river basins, seasonality would result in a great estimation error in the design peak flood volume (Alila and Mtiraoui, 2002). But in the coastal area, where the extreme sea-level was generated by the tidesurge interaction, ignoring the surge seasonality would only lead to little bias of estimation results (Dixon and Tawn, 1999). In this research, although the extreme water level in a tidal reach was affected by the flood-tide interaction rather than the tide-surge process, the impact of seasonality on the DFL estimation was slight and close to the one in the coastal area. Overall, the mixed-distribution took advantage of taking the heterogeneity caused by seasonality into consideration in DFL estimation. But both the fitting effect and the estimation results of this method were similar to that of a single distribution. Regardless of the seasonal information and the physical process, the application of the single distribution model was still feasible. 5. Conclusions The existence and impact of heterogeneity caused by seasonality in extreme water level series in a tidal reach were examined and discussed in this study, with the whole research carried out in the Pearl River Delta, South China. Significant seasonality existed in all three representative stations, i.e., Makou, Nanhua, and Denglongshan. Yet, the flood season in two stations that were located in the upper and middle reach of the tidal river channel (Makou and Nanhua) was earlier than the one close to the estuary (Denglongshan). Also, seasonal series in Makou and Nanhua shared similar statistical features, which were different from Denglongshan. In Denglongshan, the extreme water level was largely affected by high tides and surges, and lead to greater deviation in seasonal samples and the later time of flood season. In all three stations, extreme water level series in flood season and non-flood season had different tail types and belonged to two different populations. The difference between the two seasonal 95th quantiles could reach up to 0.67 m. In statistical features like mean, upper and lower bound as well as the 95th quantile, annual series in all three stations shared similar

Table 4 Seasonal contribution to annual series in different exceeding probability scopes. Contribution

Makou Nanhua Denglongshan

Flood season Non-flood season Flood season Non-flood season Flood season Non-flood season

Exceeding probability b10%

10–90%

N90%

80 20 100 0 20 80

75 25 81.82 18.18 68.18 31.82

80 20 60 40 60 40

Y. Tang, X. Chen / Science of the Total Environment 714 (2020) 136478 Table 5 Comparison of fitting efficiency and the DFL results of different return periods in Denglongshan.

Singular model Mixed-distribution

Fitting efficiency

Return period (unit: m)

AIC

PPCC

100a

50a

20a

10a

5a

2a

11.4156 11.8097

0.9935 0.9954

2.90 2.93

2.66 2.68

2.35 2.36

2.12 2.12

1.89 1.89

1.58 1.60

Note: t represented time, β1 was a parameter, and β0 was a constant.

values with the flood season ones. Annual samples with an exceeding probability of b90% were mainly affected by samples from flood season. Only when the exceeding probability was over 90%, samples from nonflood season took a major impact. The DFL estimation with singular distribution models like the Pearson-III type or the Generalized Extreme Value distribution sacrificed the information from the non-flood season, and DFL with a return period smaller than 1.1 years was underestimated. The mixed-distribution model, which took seasonality into concern, could make full use of information in recorded series from different seasons. However, it had little advantage in fitting effect than the traditionally used single distribution, and the alteration in the DFL estimation caused by mixed-distribution was smaller than 1% when the return period was over 10 years or smaller than 5 years. This research adopted the commonly used approach in closed river basins, i.e., the mixed-distribution with a weight as a constant, to deal with seasonality. Due to the limitation of data and statistical restrictions, the mixed-distribution model was tested and compared with the singular model only in Denglongshan station. The feasibility, as well as the fitting effect of the mixed-distribution, could be further verified in other study areas. Meanwhile, other approaches to solve the problem of seasonality were waited to be tested in future studies. Funding The research is financially supported by Acknowledgement: The research is financially supported by National Key Research and Development Program of China (2017YFC0405900), National Natural Science Foundation of China (Grant No. 91547202, 51861125203, 51479216), the Chinese Academy of Engineering Consulting Project (2015-ZD-0704-03), the Project for Creative Research from Water Resources Department of Guangdong Province (Grant No. 2016-07, 2016-01), Research program of Guangzhou Water Authority (2017). Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References Aa, Y., Wang, G., Liu, T., Xue, B., Kuczera, G., 2019. Spatial variation of correlations between vertical soil water and evapotranspiration and their controlling factors in a semi-arid region. J. Hydrol. 574, 53–63. Akaike, H., 1974. A new look at the statistical model identification. IEEE Trans. Autom. Control 19 (6), 716–723. https://doi.org/10.1109/TAC.1974.1100705. Alila, Y., Mtiraoui, A., 2002. Implications of heterogeneous flood-frequency distributions on traditional stream-discharge prediction techniques. Hydrol. Process. 16 (5), 1065–1084. https://doi.org/10.1002/hyp.346. Allamano, P., Laio, F., Claps, P., 2011. Effects of seasonality on the distribution of hydrological extremes. Hydrol. Earth Syst. Sci. Discuss. 8, 4789–4811. Arns, A., Wahl, T., Haigh, I.D., Jensen, J., Pattiaratchi, C., 2013. Estimating extreme water level probabilities: a comparison of the direct methods and recommendations for best practise. Coast. Eng. 81 (16), 51–66. Bacopoulos, P., 2017. Tide-surge historical assessment of extreme water levels for the St. Johns River: 1928–2017. J. Hydrol. 553, 624–636. Baratti, E., Montanari, A., Castellarin, A., Salinas, J.L., 2012. Estimating the flood frequency distribution at seasonal and annual time scale. Hydrol. Earth Syst. Sci. 16 (12), 4651–4660.

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