1. Introduction
Abundant wind energy is stored in the northern South China Sea, and ocean planners have recently focused on exploitation and utilization of marine wind energy (Zheng et al. 2016, 2018, 2019). An analysis of wind power in the northern South China Sea indicates that offshore wind power is a potential resource (Yang et al. 2017; Wang et al. 2018; Zheng et al. 2012, 2013). Wind plays an important role in the wind power field because of its power generation efficiency and its effect on structural safety. The South China Sea is located in the monsoon zone and is greatly affected by typhoons. Moreover, the waves and storm surges induced by typhoons occasionally destroy marine structures and cause disasters in coastal regions (Wu et al. 2014, 2018). Marine structures must be designed to defend against the extreme sea states induced by typhoons, and a number of experts have indicated that marine disasters occur because of the improper application of design standards (Mittal 2005; Carter 2005; Wisch and Ward 2007).
The wind speed design standard is expressed as the maximum speed with a return period. In general, the method of estimating extreme wind speed is to fit the probability distribution according to the sample and calculate the wind speed of a certain return period based on the distribution. Obtaining samples is the first step for extreme value estimation, and the samples will influence the estimated values. The type of wind speed data includes observation data, typhoon data, and reanalysis data. Observation data are accurate but expensive. Moreover, extreme value estimations need long-term data (several decades), and extensive and long-term observation data are usually not available. Typhoon data include certain key parameter information, and the wind speed needs to be calculated by parametric wind models. Yan et al. (2020) synthesized new typhoon data utilizing key typhoon factors based on the deductive method, which solved the problem of insufficient sample data. In addition, the track and intensity of a tropical cyclone (TC) can be synthesized stochastically (Emanuel et al. 2006; Vickery et al. 2000; Rumpf et al. 2009). Reanalysis data largely underpredict wind speed near the TC center. Shao et al. (2018) blended TC wind fields by combining reanalysis data with typhoon data using a formula, which improved the accuracy of wave simulation.
To estimate the extreme value of wind speed, the sample can be fitted by corresponding distributions. The most commonly recommended and adopted distribution for describing the distribution of wind speed is the Weibull distribution (Nedaei et al. 2014; Tizpar et al. 2014; Weisser 2003; Ohunakin and Akinnawonu 2012). In addition, the Gumbel and generalized extreme value (GEV) distributions are also widely used for extreme wind analysis (Wang et al. 2015; Lee et al. 2012; Lombardo et al. 2009; Yao et al. 2012). Fawad et al. (2019) selected 12 distributions that analyze annual maximum wind speed and used different criteria to determine the most suitable distribution for nine stations from Pakistan. There is no consensus on which distribution is best, and the performance varies in different areas. As a result, many extreme value estimation models are continuously discussed and improved, for example, the multivariate compound extreme value distribution (Liu et al. 2006; Pang et al. 2007), peak-over-threshold (Gao et al. 2018; Liang et al. 2019), and environment contour line (Haver and Winterstein 2009). Because of intense and rapid extreme events, extreme values may be underestimated (Zhang et al. 2018). The neural network model can predict the wind speed, and this method is more accurate than the traditional models (Sfetsos 2000; Wang et al. 2016). Wang et al. (2015) developed a generalized hyperbolic method based on artificial intelligence optimization algorithms to reanalyze future samples of extreme wind speeds. Lydia et al. (2016) built autoregressive models for short-term wind speed forecasting. The sampling and parameter estimation methods also affect the calculation results. The commonly used parameter estimation methods include the maximum likelihood estimate (MLE), least squares method (LSM), method of moments (MOM), and probability-weighted moment (PWM) (Smith 2002). Campos et al. (2019) calibrated hindcast data and improved return value estimates of extreme waves.
In this paper, extreme wind speeds are analyzed based on 40 years of ERA5 wind field and typhoon data in the northern South China Sea. The extreme value estimation model will be analyzed based on sampling, goodness of fit, uncertainty, and estimated results. The organization of the paper is as follows: In section 2, we provide an overview of the research region and data. In section 3, we introduce the distributions for estimating the extreme wind speed. In section 4, we analyze the sampling, goodness of fit and uncertainty of the extreme value estimation model. In section 5, the influencing factors of the estimated results are discussed. In section 6, a summary and a set of conclusions are provided.
2. Research area and data
a. Overview of the research area
The study area is from longitude 105° to 125°E and from latitude 12° to 26°N. The South China Sea is located to the south of the Chinese mainland in the western part of the Pacific Ocean. The natural sea area is approximately 3.5 million square kilometers, of which China’s territorial sea area is approximately 2.1 million square kilometers. The mean water depth is approximately 1212 m, and the maximum depth is 5559 m.
Two kinds of typhoons influence the South China Sea: typhoons generated in the western Pacific typhoon and those generated in the South China Sea. From every September to April or May, approximately 50 periods of cold waves occur, and the wind speed can research over 30 m s−1. Cold air is moved from the north to the south, and continental winds are dominant. Regarding the location of the tropical depression, the southernmost range can reach 5.5°N, the westernmost range can reach 107.5°E, and the northernmost range can reach 24°N. The location of the origin of the TC moves northward starting in April. From July to September, the origin is concentrated at 15°–20°E, 110°–120°N, and its position begins to move to the northeast from June to September, reaching the northernmost and easternmost position in September. After September, the origin moves southward, but the concentration is not obvious. The northern South China Sea is frequently hit by typhoons, and considerable ocean engineering has been performed.
b. Data introduction
Data from both the China Meteorological Administration Shanghai Typhoon Institute (CMA-STI) and the European Centre for Medium-Range Weather Forecasts (ECMWF) are utilized in this paper. The best-track dataset of the northwestern Pacific TC from CMA-STI records includes the year of occurrence, typhoon number, typhoon name, maximum wind speed, longitude and latitude of the typhoon center, and central pressure (Ying et al. 2014). The data include the whole process of tropical depression growth to a typhoon and then weakening to a tropical depression, with 4 observations collected per day and recordings performed every 6 h. Wind data collected over 40 years from 1979 to 2018 are analyzed in this paper. During these four decades, 1208 TCs were recorded with approximately 30 per year on average. Figure 1 shows the track and track density of TCs in the northern South China Sea. Cyclone activity is concentrated in the region of 111°–120°E, 15°–21°N. The cyclone tracks indicate that typhoons landing in the South China Sea affect Hainan, the south of Yunnan, Guangxi, Guangdong, Fujian, Taiwan, etc.
(left) Track and (right) track density throughout the research area from 1979 to 2018.
Citation: Journal of Applied Meteorology and Climatology 59, 10; 10.1175/JAMC-D-20-0046.1
The ECMWF provides the latest global atmospheric numerical forecast reanalysis data to global users. ERA5, the dataset collected by the ECMWF, is also utilized in this study. This dataset covers the period from 1950 to the present. The dataset period utilized in this study ranges from 1979 to 2018, with intervals of 1 h. The wind field is from the 10-m level and includes two components, that is, the u wind component and υ wind component. The spatial resolution of ERA5 wind data is 0.25°.
In addition, three calculated points (C1, C2, and C3) are selected for analyzing extreme wind speeds. The positions of the points are shown using a cyan asterisk in Fig. 2. These three points are located in areas where typhoon activity is most frequent, which can be seen in Fig. 1. The longitude and latitude of the calculated points are shown in Table 1.
Profile of the research area: E1 and E2 are observation points verifying two typhoons; R1 and R2 are observation points verifying the ERA5 wind data; and C1, C2, and C3 are the calculated points.
Citation: Journal of Applied Meteorology and Climatology 59, 10; 10.1175/JAMC-D-20-0046.1
Longitude and latitude of the sites.
c. Verification of ERA5 dataset
To verify the ERA5 data, the wind speed data at two sites (R1 and R2) in 2010 are selected for comparison with observation data. The observation stations are located in Xisha, and their position are shown as in Fig. 2 and Table 1. The hourly u component and υ component of wind speed in the whole year are shown in Fig. 3. The figure shows that the ERA5 data are consistent with the observation data at two sites. The correlation coefficients of the data shown in the four panels of Fig. 3 are more than 0.9, which indicates that the ERA5 data are highly correlated with the observed data. However, certain research studies have indicated that the reanalysis data largely underpredict wind speeds near the TC center (Jiang et al. 2003; Pan et al. 2016; Shao et al. 2018). To obtain the accurate extreme wind speeds, the wind speeds near the TC center can be calculated by the parametric wind model.
Verification of ERA5 wind speed data.
Citation: Journal of Applied Meteorology and Climatology 59, 10; 10.1175/JAMC-D-20-0046.1
d. Blend wind
The parametric wind model can calculate the wind speed induced by TCs based on key TC parameters. This study uses the model of Holland et al. (2010). The wind speed at a nominal height of 10 m is calculated as follows:
where bs is estimated by the central pressure deficit Δp. The r is the distance from the TC center to the calculated point,
where Δps is in hectopascals, ∂pcs/∂t is the intensity change in hectopascals per hour, φ is the absolute value of latitude in degrees, and υt is the cyclone translation speed in meters per second. The radius of the maximum wind speed Rmax is calculated by the following formula (Graham 1959):
where the TC parameters (Δps and υt) can be obtained through the CMA-STI dataset.
For improving the data accuracy near the center of the TC, the ERA5 data and parametric wind calculated data are blended according to the method proposed by Shao et al. (2018):
where VH is the wind speed calculated by the Holland parametric wind model, VE is the wind speed recorded in the ERA5, and VB is the blended wind speed. Two typhoons (Mangkut in 2018 and Conson in 2010) are selected to verify the blended wind speed. The track of the typhoons and observation sites (E1 and E2) are shown in Fig. 2. The longitude and latitude of the observation sites are shown in Table 1. A comparison of the wind speed is shown in Fig. 4, which indicates that the blended wind is better fit with the observations while the value of ERA5 is much smaller than the observations. Therefore, the blended wind improves the accuracy of the TC wind speed, which can make extreme value estimations more accurate.
Verification of the blended wind speed.
Citation: Journal of Applied Meteorology and Climatology 59, 10; 10.1175/JAMC-D-20-0046.1
3. Method
a. Extreme value distribution
Extreme sea states caused by typhoons can be described by extreme value distributions. The probability of extreme wind speeds can be calculated by extreme value distributions. As a representation of typhoon strength, wind speed is one of the environmental factors considered in the design of marine structures. In general, the design code for ocean engineering takes the extreme wind speed with a particular return period, which can be calculated by the extreme value distribution. According to the Fisher–Tippett extremal type theorem, there are three types of extreme value distributions: type I, type II, and type III. Among them, type III, which is also known as the Weibull distribution, is one of the most commonly used distribution for calculating extreme wind speed. The distribution function of the two-parameter Weibull (Weibull-2par) is shown as follows:
where α is the shape parameter. If both the scale parameter σ and the location parameter μ are drawn into the distribution, the distribution is called the three-parameter Weibull distribution (Weibull-3par), and its distribution function is as follows:
The GEV can unify the three types of extreme value distributions into one form. Its distribution function is as follows:
where ξ is the shape parameter and σ and μ are scale parameter and location parameter, respectively.
b. Lognormal distribution
In addition to the extreme value distribution mentioned above, the lognormal distribution is often used to estimate the extreme wind speed. A lognormal distribution refers to the normal distribution of random variables after taking the logarithm. The probability density function of the lognormal distribution is as follows:
c. GP distribution
The generalized Pareto (GP) distribution is a stable peak-over-threshold (POT) distribution. Its distribution function is as follows:
where μ is the location parameter, σ is the scale parameter, and ξ is the shape parameter.
The sample used in the extreme value distribution is generally the annual maximum value, while the POT method selects samples as long as the samples are over a certain threshold. The POT method is used by the GP distribution. Before choosing the threshold, the minimum time span Δt should be determined. Both the threshold and minimum time span are significant concerns with the POT approach. The time span should be long enough to guarantee the independence of the sample.
The threshold μ plays a significant role in the GP distribution. The threshold cannot be too high or too low. The sample size is too small when the threshold is too high, and the calculations are inaccurate when the threshold is too low. There are two methods commonly used to select thresholds: parameter stability plots and mean residual life plots. For sample X1, …, Xn, the mean excess function is defined as follows:
where Nμ is the number of excess points. Point set
defines the mean residual lift plot. The selection of the threshold should make en(μ) linear at μ ≥ μ0, which validates the GP model. The other method of selecting the threshold is plotting the parameter estimates against a range of values of μ. If the parameter estimates are stable above the threshold, this model is applicable.
d. Goodness-of-fit test
The Kolmogorov–Smirnov (KS) test is a commonly used method for determining the goodness of fit for extreme value distributions. The principle is described as follows.
The distribution function of total X is F(X) but unknown. The X1, X2, …, Xn is a sample from X, and its sample value is x1, x2, …, xn. Let original hypothesis H0 and alternative hypothesis H1 be
Considering the difference between the experimental frequency and theoretical frequency according to each sample point, the following statistics are constructed:
where Fn(x) is the experimental frequency distribution; F0(x) is the distribution form for the waiting test; n is the sample size; and Dn is the KS statistic, which represents the maximum absolute difference value between the experimental distribution and theoretical distribution. The smaller the value of Dn, the better the fit.
We let
Thus, Dn can be expressed as follows:
The critical value Dn(α) of the corresponding KS test can be checked from tables for different sample capacities. If Dn < Dn(α), then we accept the original hypothesis H0; otherwise, we reject it. The p value of the KS test is the probability that the null hypothesis is correct.
An alternative method to test the goodness of fit is the chi-square test. The degree of deviation between the actual observed value and the theoretical inferred value determines the size of the chi-square value. If the chi-square value is larger, the deviation between them is larger. The hypothesis to be tested is as follows:
H0: X obeys the F(x) distribution;
H1: X does not obey the F(x) distribution.The steps for testing are as follows:
Divide the population X into k mutually disjoint cells A1, A2, A3, …, Ak.
The number of samples in Ai is denoted as fi, which is called the group frequency, and the sum of the group frequencies is equal to the sample size n.
Calculate the probability pi of X in the interval Ai, and npi is the theoretical frequency of the interval Ai sample value.
Calculate the test statistics:
Given the significance level α, the reject field is
When the calculated chi-square value fails in the rejection field, the null hypothesis H0 will be rejected.
4. Model analysis
a. Sample selection
Different sampling methods are used for different distribution models. The threshold sampling method is taken for the GP distribution, that is, the data will be selected as the sample when it exceeds the threshold. The interval of the original data is 1 h. To ensure the independence of the sample, the minimum time span is taken 5 days (120 h) in this study. First, the maximum value for every five days is taken as the primary sample. Because a TC process lasts less than five days at a site, the primary sample is independent. Then, the threshold is select according to the parameter stability plot and mean residual life plot. Take point C1 as an example, Fig. 5 shows the parameter values and mean residual life en with different thresholds. When the threshold is from 19 to 40, the slopes of en and parameter values are stable. In addition, the average annual sample size λ should be 2–5 generally. In this study, 22 is selected as threshold and the sample size is 105, with 2.625 per year on average. The extreme value estimation with return period T is as follows:
The selection of the threshold is subjective to some extent, and its influence on extreme value estimates is discussed in section 5c.
Parameter stability plots and mean residual life plots used for selecting thresholds.
Citation: Journal of Applied Meteorology and Climatology 59, 10; 10.1175/JAMC-D-20-0046.1
The sample for the extreme value distribution is the maximum value in a certain period. To prevent the influence of seasonal variation of TCs on the sample, the maximum value of each year is selected as the sample. The blended wind improves the extreme wind speed relative to the ERA5 dataset. Figure 6 shows the annual extreme value sample from the blended wind and ERA5 dataset at three sites (C1, C2, and C3). For the mean value of samples, the blended wind values are 17.4, 17.6, and 13.3 m s−1 stronger than that of ERA5 at the three sites. The extreme value estimate with return period T is as follows:
The events selected as the sample are from different TCs, and they are distributed identically. Therefore, the samples are independent and identically distributed for both threshold sampling and annual extreme value sampling.
Comparison of the annual extreme value sample between the blended wind and ERA5 at three sites (C1, C2, and C3).
Citation: Journal of Applied Meteorology and Climatology 59, 10; 10.1175/JAMC-D-20-0046.1
b. Goodness of fit
Accurate estimates need a suitable fitting function. The goodness of fit reflects the fitting effect of the distribution function on the data. In general, the results estimated by distributions with a higher goodness of fit are more reliable. In this study, five probability distribution functions (GP, GEV, Weibull-3par, Weibull-2par, and lognormal) are selected to analyze the extreme wind speed at three sites (C1, C2, and C3). The P value of the KS test and chi-square test can reflect the goodness of fit as shown in Table 2. The results show that the GEV, Weibull-2par and Weibull-3par fit the sample well and all the mean p values of the KS test for the samples at three points exceed 0.7. The GP fitting was good at C2 but poor at C1 and C3. The lognormal value has a poor fit relative to the other four distributions. Therefore, the extreme value distribution fit the annual extreme value sample better than lognormal distribution.
The p values of the KS/chi-square tests for different distributions and sites.
In addition, the Q–Q plots of all five distributions at the three sites are shown in Fig. 7, which can reflect the goodness of fit intuitively. In the figure, the closer the blue point is to the red line, the better the fitting effect, and it shows the fitting is good except for the lognormal distribution.
The Q–Q plot fitted by different distributions at the three sites.
Citation: Journal of Applied Meteorology and Climatology 59, 10; 10.1175/JAMC-D-20-0046.1
c. Uncertainty analysis
A good distribution function has lower uncertainty, and the calculated results are stable. The bootstrapping method (Naess and Hungnes 2002) is based on resampling from a given sample, which is used to analyze uncertainty of model in this study. Supposing there are n elements in the sample that are each stochastically sampled and then returned. Repeating n times, a bootstrap sample with n elements was obtained. Different extreme values can be estimated by different bootstrap samples, which reflect the model uncertainty. A similar estimation by different bootstrap sample indicates a stable model. Figure 8 shows the error bar of five distributions at three sites based on 100 bootstrap samples. The length of the error bar of the lognormal distribution is longest, while the Weibull-2par is shortest. Therefore, the Weibull-2par distribution has the lowest uncertainty and the lognormal distribution has the largest uncertainty. Moreover, the results estimated by the lognormal distribution are much larger than those of the other four distributions, which may be related to the poor fitting effect.
Error bars for the estimates at the three sites.
Citation: Journal of Applied Meteorology and Climatology 59, 10; 10.1175/JAMC-D-20-0046.1
5. Estimation results and discussion
a. Results estimated by five distributions
The tail shape of each distribution is different; as a result, the estimated extreme wind speed is also different. To analyze the difference estimated by different distributions, the extreme wind speed is calculated, which is shown in Fig. 9. The results indicate that the difference in estimation increases as the return period increases. Moreover, the results calculated by different distributions are very close except for those of the lognormal distribution, and their difference of return level with an annual exceedance probability of 10−3 is no more than 5 m s−1. However, the lognormal distribution largely overestimates the extreme wind speed compared with the other four distributions. In addition, considering the poor goodness of fit, we do not recommend the lognormal distribution for calculating the extreme wind speed, especially for long return periods.
Comparison of return levels fitted by different distributions at the three sites.
Citation: Journal of Applied Meteorology and Climatology 59, 10; 10.1175/JAMC-D-20-0046.1
b. The estimation difference between blend wind and ERA5 dataset
In comparison with observation data, the blended wind improved the accuracy. The annual maximum samples of blended wind are much larger than the ERA5 samples. Thus, the extreme value wind speeds estimated by these two samples must be different. Table 3 shows the return levels at 100 years estimated by the blended wind and ERA5 dataset at three sites. For the GEV and Weibull distribution, the results estimated by blended wind are almost twice as large as the results estimated by ERA5. For the lognormal distribution, these results show a difference greater than twofold. The difference will have a significant influence on the designed wind speed. Therefore, the accuracy of the original data is very important.
Return levels (m s−1) of 100 years estimated by the blended wind and ERA5 dataset at the three sites.
c. Estimation by different threshold
The selection of thresholds is subjective to some extent, and a different threshold may have a different estimate. Table 4 shows the return levels and goodness of fit for different thresholds. The thresholds range from 20 to 28, which is suitable for parameter stability plots and mean residual life plots. Table 4 indicates that the estimation is very close for different thresholds. The difference in the estimates at C1 and C2 is no more than 0.05 m s−1 for once-in-a-century wind, which has little effect on the design. Therefore, the effect of the threshold on the estimate is negligible. In addition, the p value of the KS test is not same for different threshold, although obvious regularities are not observed.
Comparison of estimates based on the GP distribution with different threshold.
d. Estimates by different parameter estimation methods
The parameter estimation method also plays a significant role in obtaining the proper design wind speed. The parameter estimation method used above is the MLE method. The principle of the MLE is that the parameter estimates maximize the probability of sample occurrence. In addition to the MLE method, LSM and PWM method are used to estimate the parameters of the GEV distribution. The LSM is used to estimate the parameters according to the criterion of minimum square deviation, and the PWM method sets the sample moment equal to the corresponding population moment to obtain the estimated value for the parameter. The PWM takes the probability distribution as the weight and applies the basic principle of the moment estimation method to estimate the parameters of totality. The calculated return levels using the MLE, LSM, and PWM parameter estimation methods are shown in Table 5. The results indicate that the LSM results are less than the MLE and PWM results, and the calculated MLE results are largest among them. However, the greatest difference is less than 5 m s−1 for extreme wind speeds with 100-yr return periods.
Return levels (m s−1) of 100 years estimated by the MLE, LSM, and PWM at the three sites.
6. Conclusions
The best-track datasets for TCs over the northwestern Pacific Ocean and ERA5 wind field datasets are applied to analyze the extreme wind speed in the northern South China Sea. The wind speed representing a blend between ERA5 wind data and parametric wind represents the sample. The parametric wind is calculated from the best-track datasets. Compared with the ERA5 wind field, the blended wind significantly improved the accuracy of wind speed data. Moreover, the wind speed estimated by the blended wind is approximately twice as large as that estimated by the ERA5 dataset. Three typical points are selected to calculate the extreme wind speeds with the GEV, GP, Weibull-3par, Weibull-2par and lognormal distributions. An analysis using the KS test and chi-square test shows that the extreme value distribution has the best goodness of fit, followed by the GP distribution, with the lognormal presenting the worst fit. Moreover, the lognormal distribution presents greater uncertainty and a much larger estimation than the other four distributions. Therefore, we do not recommend using the lognormal distribution for extreme value estimation.
In addition, some key factors that influenced the calculation model are analyzed. Extreme wind speeds were calculated by different distributions at three sites. The results suggest that the return levels of the wind speed are similar except for that of the lognormal distribution. When the extreme wind speed is estimated by the GP distribution using different thresholds, the results are very close. The MLE, LSM, and PWM parameter estimation methods are applied to estimate the parameters of the GEV distribution. The biggest difference in their estimation is less than 5 m s−1 at the three sites. A comparison of the influence of the distribution function, data, threshold, and parameter estimation method on the extreme value estimation shows that the data have the greatest influence on the calculated result, followed by the distribution function and parameter estimation method, with the threshold presenting the least influence.
Acknowledgments
This work was supported by the NSFC-Joints Funds of Shandong (U1706226).
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