Design tide hydrograph with a given risk threshold by a copula-based multivariate method

This paper describes the development of a T-year design tide hydrograph (DTH). A core innovation is that the proposed technique uses the design risk threshold and copula-based conditional risk probability to analyze the optimal combination of high waters and low waters of the DTH. A brief description of the method is presented. The in situ semi-diurnal tide data at the coast of Jiangsu Province in China are analyzed. Marginal distributions for high waters and low waters of tides are examined. Furthermore, the joint distributions, condition risk probabilities and risk thresholds of high waters and low waters are presented. Results of the DTH from the proposed method are compared with those from the traditional same-multiple enlarging design approach. It is demonstrated that the proposed method is preferable.


Introduction
Tides are the periodic rise and fall of the ocean surfaces. Tides are commonly diurnal tide (one high and one low water each day), semi-diurnal tide (two high waters and two low waters each day), or a mixed tide (two uneven tides a day, or sometimes one high and one low each day). The design tide hydrograph (DTH) is a large magnitude tide hydrograph used as ocean boundary conditions in many hydrodynamic models (Salisbury and Hagen, 2007;Lian et al., 2013). However, the probability (or return period) of occurrence of the DTH is unknown, as it is the joint probability of the high water, low water, tidal range and tidal period. In China, the tide hydrograph for a return period of T years is usually constructed by two design tide hydrograph enlarging methods. One is the same-frequency method; the other is the same-multiple method.
However, many hydrological variables (or events) are partially correlated. Examples are the wave height and water level (Hawkes et al., 2002), wind stress and wave height (Blake, 1991), rainfall and storm surge (Zheng et al., 2013), river flow and sea surge (Svensson and Jones, 2002), etc. Similarly, high water, low water and tidal range are partially dependent and this dependence means that T-year high water, T-year low water and T-year tidal range do not occur together. However, these empirical design methods of China mentioned above do not incorporate this dependence for the combination of high water, low water and tidal range in the design of tide hydrographs. This characteristic usually has the potential effect of a tide hydrograph being designed to the incorrect level. Fortunately, many studies, such as designs of hyetographs (Alfieri et al., 2008;Lee and Ho, 2008) and flood hydrographs (Mailhot et al., 2007;Xiao et al., 2009), show that this problem can be solved by analysis of the joint multivariable probability based on copula functions. The work of Veneziano and Villani (1999), Serinaldi and Kilsby (2013), De Michele (2004, 2010), Volpi andFiori (2012), andFavre et al. (2004) provide a nice theoretical framework for copula-based probability analysis.
DTH has not been studied as extensively as the design hyetograph and the design flood hydrograph. A tide hydrograph may be described by a multivariate function of the high water, low water, tidal range and tidal period. There is no standardized or specific definition of the return period of the DTH. Usually, the return period is used for designing structures to withstand an event that exceeds a given risk threshold. Depending on different purposes of the DTH, either the tidal level or the tidal range could be the more in-fluential parameter in determining this risk threshold. It is worth stressing that, when hydrological events are described by several correlated random variables, the choice of a return period of the hydrological events will lead to different critical combinations of the related random variables (Volpi and Fiori, 2012). Therefore, the risk threshold can be defined as a given high water, a given low water, a given tidal range or other design variables of the DTH. The problem is complex and this means that there is not a unique tide hydrograph for a given risk and return period. In other words, with different combinations of high water, low water, tidal range and other design variables, a set of design tide hydrographs can have the same design risk and return period. These combinations are generally not equivalent from a practical point of view. Mediero et al. (2010) defined the design risk of a flood hydrograph as the probability of exceeding the reservoir water elevation in any given year.
As previously stated, the probability of occurrence of the tide event is variable even if all the events share the same value of the design risk and return period. Therefore, the selection of design combinations of high water, low water and tidal range should take into account the specific characteristics of the design problem. In this paper, a methodology is presented to obtain the DTH for any given design risk. The primary aim is to provide a simple but effective rule to get optimal choices for the hydrological design variables of the DTH. To illustrate how the proposed methodology can be easily used in practice, only the risk threshold defined as a given high water is applied to the case studies of the DTH. The next section discusses the observed data set used in this work. The third section describes the method adopted to generate the design tide hydrographs. The fourth section discusses the results of a case study. The study's conclusions are drawn in the final section.

Case studies
Jiangsu Province, located in the east of China, is adjacent to Shanghai. Jiangsu has a coastline (31°37′N-35°08′N) of approximately 1000 km along the Yellow Sea. The Yangtze River passes through the southern part of Jiangsu. The data of the spring tides recorded at three tide stations along the Jiangsu coastline, from 1972 to 2011, named Lianyungang, Sheyangzha and Tianshenggang (see Fig. 1), are used for case studies in this paper. These data show that the region's tides are commonly semi-diurnal tides with two high and two low waters during a lunar day. The two high waters are the higher high water (HHW) and the lower high water (LHW). Similarly, the two low waters are the higher low water (HLW) and the lower low water (LLW). The observed spring tides with the maximum higher high water from 1972 to 2011 at each station, respectively, are shown in Fig. 2 (vertical datum: Feihuanghe). Furthermore, the following data were extracted from these raw tide data recorded between 1972 and 2011 at each station, respectively: annual maximum higher high water (MHHW) and its corresponding lower high water (LHW), the first one (LW1) of two low waters, the second one (LW2) of two low waters.

Design risk probability and threshold
In China, the design tide hydrographs usually have the return periods of 20, 50 or 100 years based on the local economic and social conditions. In Jiangsu, a 50-year return period is usually adopted, which means that there is a 2% risk probability allowed by the design standard. The design risk probability (DRP) of the DTH can be given as: where T is the return period, and its value is determined by a specified design standard. Let F(mhhw) be the marginal cumulative distributions of the MHHW, where mhhw is the value of the MHHW. The corresponding risk threshold (mhhw) T defined as a given high water can be given as: where P is the non-exceedance probability and (mhhw) T is the design risk threshold of the DTH. The Pearson Type III (Griffis and Stedinger, 2009;Lian et al., 2013;Vogel and McMartin, 1991) distribution is adopted to fit the marginal cumulative distributions of the MHHW. Table 1 shows F(mhhw) and return periods of the MHHW at Lianyungang, Sheyangzha and Tianshenggang, respectively.

Copula-based multivariate method for DTH
To identify the best possible combination of HHW, LHW, LW1 and LW2 of the DTH, the objective of this section is to study a copula-based multivariate method for the DTH.

Marginal distributions of LHW, LW1 and LW2
The Pearson Type III distribution is adopted to fit the marginal cumulative distributions of LHW, LW1 and LW2, denoted by F(lhw), F(lw1) and F(lw2), where lhw, lw1 and lw2 are the values of variable LHW, LW1 and LW2, respectively. Fig. 3 shows the marginal cumulative distributions of LW1, LW2 and LHW, respectively.

Joint distributions of high and low waters
In the following analysis, joint distributions of (MHHW, LHW), (MHHW, LW1) and (MHHW, LW2) will be computed. With the Gumbel-Hougaard copula (Zhang and Singh, 2007), the theoretical joint distribution F(mhhw, lhw), F(mhhw, lw1) and F(mhhw, lw2) can be expressed as: where θ is the parameter of the copula. The parameter θ can be estimated by the correlation of the empirical and theoret-ical joint distribution. For representation of the empirical joint distributions, let two random variables be denoted as X and Y. The empirical joint distribution F' can be determined using the Gringorten plotting-position formula (Zhang and Singh, 2007) as: where (x i , y i ) can be obtained by assigning variables either values of X or Y; N' is the number of (x i , y i ) counted as X≤x i and Y≤y i ; N is the sample size. The probability-probability plots  show the excellent correlations (denoted by R 2 ) of the empirical and theoretical joint distributions with coefficients of 0.9782-0.9971 at each station. This means that the Gumbel-Hougaard copula is an ideal model. The optimal θ values at each station are presented in Figs. 4-6.

Condition risk probability (CRP) of different variable combinations
When the design risk probability DRP(T) and its corresponding risk threshold (mhhw) T are determined, the conditional risk probabilities of different variable combinations for a design tide hydrograph are of greater interest.   Eng., 2017, Vol. 31, No. 4, P. 504-509 (1) The conditional risk probability (CRP) of MHHW and LW1, given MHHW≤(mhhw) T , is (2) The conditional risk probability of MHHW and LW2, (3) The conditional risk probability of MHHW and LHW, given MHHW≤(mhhw) T , is 4.4 Optimal combinations of high waters and low waters of DTH As illustrated in Fig. 7, when the DRP(T)=2% at Tianshenggang, the CRP may be larger than 2% and this will lead to under-design. Conversely, the CRP may be smaller than 2% and this will lead to over-design. By measuring the coordinates of the critical point shown in Fig. 7, we can ob-tain the proper critical values of high water and low water, which yields where (lw1) T , (lw2) T and (lhw) T are the critical values at the location of the critical points. For DTH with a given risk threshold (mhhw) T : HHW of DTH is equal to (mhhw) T ; LHW of DTH is equal to (lhw) T ; two low waters of DTH are equal to (lw1) T and (lw2) T , respectively. Detailed data on the CRP at each station are shown in Tables 2-4. Using these tabular data, we can find the proper critical values and obtain the best combinations of high waters and low waters of the DTH, e.g., at Sheyangzha, when DRP(50) =2%, LLW, HHW, HLW and LHW of the DTH are -1.37, 3.77, -1.06 and 2.93, respectively.

Results
In traditional methods of constructing the design tide hydrographs, two design tide hydrograph enlarging methods are usually considered. One is the same-frequency method; the other is the same-multiple method. According to the same-multiple method, the typical spring tide hyetographs (see Fig. 2) are scaled to fit the HHWs imposed by the design return period. Fig. 8 shows the 50-year design  YANG Xing, WANG Wei China Ocean Eng., 2017, Vol. 31, No. 4, P. 504-509 507 tide hydrographs obtained by the proposed method and the traditional method at Lianyungang, Sheyangzha and Tianshenggang, respectively. The results of the study show that there are significant differences between the two methods. The tidal range obtained by the proposed method is consid- Fig. 7. Illustration of the CRP with DRP(T)=2% at Tianshenggang.  erably larger, and the low waters obtained by the proposed method are considerably smaller.

Conclusions
A design tide hydrograph is characterized by several correlated variables (e.g., high water, low water, and tidal period). Therefore, without considering the probability of the joint occurrences of these correlated variables usually has the potential effect of DTH being designed to the incorrect level. A methodology for generating tide hydrographs that takes the advantage of the correlation properties of the high waters and low waters of tides was developed. The primary aim is to provide a procedure to estimate the optimal combination of these variables in the design tide hyetograph analysis with a specified risk level.
The work is divided into three main parts: (a) the establishment of the marginal and joint cumulative distributions of high waters and low waters; (b) the establishment of a copula-based multivariate method for the DTH; (c) case studies at three in situ stations from Jiangsu of China. The facts that are deduced from this study can be summarized as follows: (1) The joint cumulative distributions and conditional risk probabilities of high waters and low waters are estimated using the bivariate copula function. The probabilityprobability plots in this paper show that the Gumbel-Hougaard copula is an ideal bivariate copula function.
(2) From a view point of economy, a specified risk level is usually allowed in the DTH. Therefore, the conditional risk probability is much more valuable than the joint probability from the aspect of determining the optimal combination of design variables of the DTH.
(3) A comparison between the traditional method and the proposed method shows that there are significant differences between the two methods. The tidal range obtained by the proposed method is considerably larger and the low waters obtained by the proposed method are considerably smaller.