## Abstract

Compound flooding from rainfall and storm tides is prone to occur in coastal cities. The identification of them is essential for controlling urban flooding. First, the dependence between rainfall and storm tides is quantified by Kendall's *τ*, Spearman's *ρ*, and tail dependence coefficient. Then, a bivariate copula-based probability distribution model is built to calculate the joint and conditional probability of rainfall and storm tides. Finally, MK and SQMK methods are employed to detect the trends of the dependence and joint probability. The results show that: (1) The dependence between strong rainfall and corresponding storm tides is much higher than that of small rainfall and storm tides, and the effect of tropical cyclones may be one of the reasons. (2) The dependence between rainfall and storm tides is the largest in October and the smallest in July. More attention should be paid to the compound flooding caused by rainfall and storm tides in October for Haikou. (3) The upper tail dependence coefficient of the rainfall and storm tides is significantly greater than the lower tail dependence coefficient and exhibits a significant positive trend. The results can provide additional insights into the effect of rainfall and storm tides for coastal flood management.

## HIGHLIGHTS

The compound effects of rainfall and storm tides were investigated from different perspectives.

Trends of the compound risk were analyzed by the copula and Mann–Kendall method.

Significant increasing trends of joint probability were detected after 1988–1992.

Compound event of rainfall and storm tides is more likely to occur when a high storm tide event has occurred.

## INTRODUCTION

With climate change and rapid urbanization, urban floods become more and more frequent, and flood risk presents an increasing trend (Xu *et al.* 2020). In China, coastal cities with high urbanization levels and rapid economic growth are vulnerable to floods caused by the compound effect of rainfall and storm tides, which are two main disaster-causing factors in coastal cities and are often driven by common meteorological conditions such as low atmospheric pressure and tropical cyclone (TC) (Zheng *et al.* 2014). In coastal cities, rainfall collected by drainage systems flows into the sea either directly, or it flows into tidal rivers first and is then pumped into the sea. High storm tide has an adverse influence on drainage capability and could directly cause coastal flooding. According to the report by the Intergovernmental Panel on Climate Change, extreme rainfall and sea level have rising trends in recent years. Thus, the compound effects of rainfall and storm tides will be more and more significant and it is necessary to investigate the trends of compounding rainfall and storm tide events in coastal cities.

There are numerous studies on the dependence between rainfall and storm tides. Svensson & Jones (2002) found that the dependence between precipitation and surge is strongest when precipitation preceded surge by 1 day in eastern Britain. However, in South and West Britain, they found the dependence between precipitation and daily maximum surge is strongest when they occurred on the same day, but is not particularly strong for any lag (Svensson & Jones 2004). Archetti *et al.* (2011) considered different rainfall and sea level conditions to estimate the threshold of flooding and proposed a simplified method to assess the urban flooding severity as a function of climate variables in Rimini (Italy). Zheng *et al.* (2013) employed a bivariate logistic threshold-excess model to quantify the dependence between extreme rainfall and storm surge. Statistically significant dependence is observed for the majority of locations along the Australian coastline and the strength of dependence varies with storm burst duration and the lag between extreme rainfall and storm surge events. More recently, Hurk *et al.* (2015) used an ensemble of regional climate model simulations to demonstrate that the combined occurrence of heavy precipitation and storm surge is physically related in a Dutch coastal polder area. They concluded that the role of the correlation between storm surge and heavy precipitation increases with inland storm tide up to a certain value, but its role decreases at the higher storm tides when tidal characteristics become increasingly important. Couasnon et al*.* (2018) proved that considering the compound effect through building a copula-based Bayesian network was crucial for flood risk assessment in coastal cities. Xu *et al.* (2022) investigated the amplification of flood Risks by the compound effects of precipitation and storm tides under the nonstationary scenario. The above analyses show that there is a certain dependence between rainfall and storm tides. Even though the dependence is often weak, it can have significant implications for flood risk estimation (Archetti *et al.* 2011; Zheng *et al.* 2014; Lian *et al.* 2017; Xu *et al*. 2023; Guan *et al*. 2023).

Overall, the above studies mainly focus on the dependence and joint probability between rainfall and storm tides. However, little attention has been paid to the trends and compound effects of rainfall and storm tides in different magnitudes, different years, and different months. The existence of trends is an indication of potential climate change and the identification of such trends in rainfall and storm tides is essential for planning and design of sustainable drainage measures in coastal cities. The aim of the study is to detect the trend of dependence between rainfall and storm tides in coastal cities. The main contributions include that (1) the compound effects of rainfall and storm tides were investigated from different perspectives (different magnitude, annual dependence, monthly dependence, tail dependence) and (2) trends of the compound risk were analyzed by a bivariate copula-based probability distribution (BCPD) model, Mann–Kendall (MK), and the sequential Mann–Kendall (SQMK) test method. The study area and used data are described in Section 2. Section 3 describes the method used in this study. In Section 4, the dependence between rainfall and storm tides is quantified by Kendall's *τ*, Spearman's *ρ*, and tail dependence coefficient first, and then the trends of the dependence are detected by the MK test method and the SQMK test method. Finally, a BCPD model is established to calculate the joint probability and conditional probability of rainfall and storm tides, and the trends of joint probability and conditional probability are also analyzed. The discussion and conclusions are summarized in Section 5.

## METHODS

In this study, the trend of compounding rainfall and storm tide events during the period from 1974 to 2012 are analyzed from the following three aspects. First, the annual dependence, monthly dependence and dependence between different magnitudes of rainfall and corresponding storm tides are investigated by Kendall's *τ*, Spearman's *ρ*, and their trends are detected by the MK and SQMK test methods. Secondly, tail dependence coefficients including upper and lower tail dependence coefficients are calculated by the BCPD model, and then the trends of upper and lower tail dependence coefficients are determined. Thirdly, the trends of the joint probability and conditional probability calculated by the BCPD model are also detected. The BCPD model and tail dependence coefficient are described as follows.

### Study area and data

This paper presents an assessment of Haikou city center with an area of 150 km^{2} as the study area. Daily rainfall data and storm tide data from 1974 to 2012 provided by Haikou Municipal Water Authority are collected from six rainfall stations and a storm tide station in the city center of Haikou respectively. The daily rainfall data used in the following analysis take the average of six rainfall stations by Thiessen Polygon Method. The maximum distance between the storm tide station and the rainfall station is about 9.5 km, and the nearest distance is about 3 km. The continuity of the data has been checked. The TC data are available from the best-track dataset by the Shanghai Typhoon Institute of China Meteorological Administration (http://tcdata.typhoon.org.cn/zjljsjj_zlhq.html).

### BCPD model of rainfall and storm tides

Copula functions have been widely applied to multivariate analysis of hydrologic events in recent years (Wang *et al.* 2017; Guo *et al.* 2018) because marginal properties and the dependence structure of random variables could be investigated separately. and are marginal distributions of *X* and *Y*, respectively. According to Sklar's theorem (Sklar 1959), if *u* and *v* are continuous, the copula function is unique. Commonly used copula functions are presented in Supplementary material, Table A1.

The marginal distribution is fitted using Lognorm, Gamma, Weibull and GEV distribution. Norm, Gamma, Weibull, and Generalized Extreme Value (GEV) distribution are employed to fit . The parameters of the above distributions are estimated by the maximum likelihood method. The best-fit marginal distribution and are selected by the Akaike information criterion (AIC) and ordinary least squares criteria (OLS).

*D*is defined as follows:where is the theoretical distribution of the measured samples;

*i*is the serial number of the measured samples in ascending order; and

*m*is the number of samples. When the statistic

*D*is less than the critical value , the test is accepted.

After determining the optimal marginal distribution and , the copulas in Supplementary material, Table A1 are used to build a joint distribution of rainfall and storm tides. The maximum likelihood method is employed to estimate the parameters in copulas. The best-fit copula is selected by AIC and OLS.

### Tail dependence coefficient

*τ*and Spearman's

*ρ*are not able to correctly describe the dependence of extremes. However, extreme events are of particular importance due to their severe impacts on the economy, environment and society (Aghakouchak

*et al.*2010). The degree of association between extreme values can be described by the tail dependence coefficient. Let

*X1*and

*X2*be two random variables and let and be their distribution functions, respectively. The upper tail dependence coefficient is defined as (Frahm

*et al.*2005):where

*w*is the extreme value threshold. Similarly, the lower tail dependence coefficient is defined as:

*et al.*2005; Domino

*et al.*2014). The Gumbel copula function is sensitive to the changes in the variables at the upper tail of the distribution, while the Clayton copula function is sensitive to the changes in the variables at the lower tail. Thus, the tail dependence coefficient can be calculated by following formulas (Domino

*et al.*2014).where is the parameter in the Gumbel copula function; is the parameter in the Clayton copula function.

## RESULTS AND DISCUSSION

### Analysis on dependence between rainfall and storm tides

#### Dependence between different magnitudes of rainfall and corresponding storm tides

According to the rainfall classification of the China Meteorological Administration, different magnitudes of rainfall include rainfall >1, >10, >25, and >50 mm. The number of occurrences exceeding the different thresholds is 3,690; 1,624; 739; and 259, respectively. The dependence between different magnitudes of rainfall and corresponding storm tides are quantified by Kendall's *τ* and Spearman's *ρ*. As shown in Table 1, the Spearman's *ρ* would more than double (increases from 0.124 to 0.276) when rainfall increases from higher than 1 mm to higher than 50 mm. The Kendall's *τ* has similar trends. It indicates that the dependence between strong rainfall and corresponding storm tides is greater than that between small rainfall and corresponding storm tides. The reason may be related to rainfall patterns of different magnitudes of rainfall. Small rainfall such as 1–10 mm rainfall is more susceptible to local small-scale climate (e.g., air humidity), but it has a small impact on storm tides, which results in weak dependence between rainfall and storm tides. Moderate rainfall and heavy rainfall are more susceptible to mesoscale weather systems, such as TCs. TCs are often accompanied by strong winds and rainstorms, which both have a significant impact on rainfall and storm tides. As shown in Table 2, in TC conditions, the Spearman's *ρ* of TC rainfall and storm tides is 0.274 with the significant level at 0.01 when rainfall is higher than 25 mm, which is much higher than that of non-TC rainfall and storm tides with the value of 0.04. It indicates that the dependence between rainfall and storm tides is enhanced by TCs for moderate rainfall and heavy rainfall in Haikou City.

Correlation Coefficients . | Dependence between different magnitudes of rainfall and corresponding storm tides . | |||
---|---|---|---|---|

Rainfall >1 mm . | Rainfall >10 mm . | Rainfall >25 mm . | Rainfall >50 mm . | |

Kendall's τ | 0.084*** | 0.103*** | 0.108*** | 0.183*** |

Spearman's ρ | 0.124*** | 0.153*** | 0.159*** | 0.276*** |

Correlation Coefficients . | Dependence between different magnitudes of rainfall and corresponding storm tides . | |||
---|---|---|---|---|

Rainfall >1 mm . | Rainfall >10 mm . | Rainfall >25 mm . | Rainfall >50 mm . | |

Kendall's τ | 0.084*** | 0.103*** | 0.108*** | 0.183*** |

Spearman's ρ | 0.124*** | 0.153*** | 0.159*** | 0.276*** |

***Significance at *α* = 0.01.

Correlation Coefficients . | Rainfall higher than 10 mm . | Rainfall higher than 25 mm . | ||
---|---|---|---|---|

TC rainfall and storm tides . | Non-TC rainfall and storm tides . | TC rainfall and storm tides . | Non-TC rainfall and storm tides . | |

Kendall's τ | 0.171*** | 0.051*** | 0.189*** | 0.027 |

Spearman's ρ | 0.249*** | 0.076*** | 0.274*** | 0.04 |

Correlation Coefficients . | Rainfall higher than 10 mm . | Rainfall higher than 25 mm . | ||
---|---|---|---|---|

TC rainfall and storm tides . | Non-TC rainfall and storm tides . | TC rainfall and storm tides . | Non-TC rainfall and storm tides . | |

Kendall's τ | 0.171*** | 0.051*** | 0.189*** | 0.027 |

Spearman's ρ | 0.249*** | 0.076*** | 0.274*** | 0.04 |

***Significance at *α* = 0.01.

#### Annual dependence analysis

*τ*and Spearman's

*ρ*in each year increase with the increase of rainfall. For higher than 1 mm rainfall, the average value of Spearman's

*ρ*is 0.11. When rainfall is higher than 25 mm, the maximum value of Spearman's

*ρ*is 0.633 (occurred in the 1977 year) and the average value is 0.20 significance at

*α*= 0.01.

*τ*and Spearman's

*ρ*. When rainfall is higher than 1 mm, UF and UB statistics of the SQMK test experience more than one non-significant change point and a decreasing trend is detected after 1989 (Figure 3(a) and 3(b)). When rainfall is higher than 10 mm, the UF statistics depict that, after a change in 1992, Kendall's

*τ*and Spearman's

*ρ*lean toward an increasing trend (Figure 3(c) and 3(d)). When the rainfall is higher than 25 mm, an increasing trend is detected after 1995 (Figure 3(e) and 3(f)). Results from the SQMK test reveal the approximate year of the start of a significant trend between 1989 and 1995, which is consistent with the urbanization time of Haikou.

#### Monthly dependence analysis

Month . | Jun. . | Jul. . | Aug. . | Sep. . | Oct. . |
---|---|---|---|---|---|

The number of days of heavy rains | 3 | 3 | 8 | 5 | 12 |

Average wind speed of TCs (m/s) | 19.20 | 21.25 | 20.29 | 21.49 | 25.49 |

Month . | Jun. . | Jul. . | Aug. . | Sep. . | Oct. . |
---|---|---|---|---|---|

The number of days of heavy rains | 3 | 3 | 8 | 5 | 12 |

Average wind speed of TCs (m/s) | 19.20 | 21.25 | 20.29 | 21.49 | 25.49 |

### Analysis on tail dependence coefficient between rainfall and storm tides

In order to verify this conclusion, events that exceeded both the rainfall and storm tides marginal thresholds (90th percentile) in the different decades are shown in Supplementary material, Figure A1. The red points in the figure represent extreme rainfall observations co-occurring with extreme storm tides on the same day. As can be seen from the figure, joint extremes in the period of 2003–2012 occur more frequently than in other decades. It indicates that there is an increasing trend of dependence between extreme rainfall and extreme storm tides in recent years.

### Analysis on joint probability distribution of rainfall and storm tides

#### Joint probability analysis

*H*and

*Z*is estimated through Equation (5). The MK test statistics of the joint probability for different return periods (RPs) of

*H*and

*Z*are shown in Supplementary material, Table A2. Supplementary material, Table A2 illustrates that there is a significantly increasing trend in the joint probability of different combinations of

*H*and

*Z*. The results of the SQMK test statistics in different combinations of

*H*and

*Z*are shown in Figure 7. A positive UF value indicates an increasing trend of

*H*and

*Z*, and a negative curve indicates a decreasing trend. When the UF value exceeds the critical value, it indicates a significant trend of change. The intersection of the curves UF and UB represents the change point of rainfall and storm tide events. From the figures a common characteristic can be seen: the joint probability shows a decreasing trend before 1988 and the changes are not significant, but a significantly increasing trend is observed after 1988–1992. This period is consistent with the analysis of Section 3.1.2.

#### Conditional probability analysis

*Z*exceeds (i.e., 50-year RP storm tides) when

*H*exceeds (i.e., 50-year RP rainfall). is the probability that

*H*exceeds when exceeds . Figure 8 illustrates that: (a) an increasing trend is observed in , but the change of is not obvious. (b) with the average value of 0.13 is significantly larger than with the average value of 0.05, that is, the occurrence probability of strong rainfall in high storm tide conditions is higher than that of high storm tide in strong rainfall conditions. Thus, compound flooding from rainfall and storm tides is more likely to occur when a high storm tide event has occurred.

### Limitation and future work

Similar to most research, some limitations also exist in this study. One is the limited record length and the quality of the observed data. The available record length of the observed data in this study lasted for 39 years. This record length is consistent with that of many studies (Yilmaz & Perera 2015; Zhang *et al.* 2018). Trend analysis on different temporal scales has attracted great concern over the past century. Long-term data covering more than 80 years (Haylock & Nicholls 2015; Bisht *et al.* 2017) and short-term data covering less than 30 years (Li *et al.* 2015; Naidu *et al.* 2015) have both been used in trend analysis on rainfall. It is noted that the trend analysis results are more reliable with a long time span and high-quality data. Another limitation of this investigation is the weak correlation between rainfall and storm tides when we establish the joint probability distribution between the two variables. The weak correlation between variables may have a potential influence on the joint probability analysis results. Even though the correlation between rainfall and storm tides is often weak, there are still many studies about their joint probability analysis (Zheng *et al.* 2014). For instance, Lian *et al.* (2013) used Gumbel copula and Zheng *et al.* (2013) proposed a bivariate logistic threshold-excess model to quantify the dependence between rainfall and storm surge. It is noted that a reasonable model which can reveal the weak correlation between the two variables should be selected to build the joint probability distribution.

## CONCLUSION

In this paper, trends of compounding rainfall and storm tide events in a coastal city are investigated integrating Kendall's *τ*, Spearman's *ρ*, tail dependence coefficient, BCPD model, and the MK and SQMK test methods. From an annual variability point of view, statistically decreasing trends are observed when rainfall is higher than 1 mm. However, it is worth pointing out that increasing trends are detected when rainfall is higher than 10 and 25 mm. Furthermore, the upper tail dependence coefficient has a significantly positive trend. This indicates that the dependence between extreme rainfall and high storm tides has shown an increasing trend in recent years. For urban flood management, it is necessary to pay attention to the compound events from extreme rainfall and high storm tides. When any extreme event occurs, the compound effect of rainfall and storm tides on urban floods should be considered. From a monthly variability point of view, the dependence between rainfall and storm tides is the largest in October and the smallest in July. So, the urban disaster prevention strategy should consider the compound effect of rainfall and storm tides in October.

For joint probability, significant increasing trends are detected in different combinations of rainfall and storm tides after 1988–1992. This period is consistent with the result of annual dependence variability and the urbanization time of Haikou. Furthermore, flooding would occur when either rainfall or storm tides exceeds the design standard. Thus, it is necessary to consider the compound effect of rainfall and storm tides when urban flooding standards are developed in coastal cities. For coastal cities with two kinds of hazard factors (rainfall and storm tides), the traditional univariate analysis (e.g., rainfall analysis) cannot describe the dependence between the two hazard factors, and cannot fully reflect the factual flood mechanism of coastal flood disaster. Therefore, it is necessary to study the encounter probability of rainfall and storm tides based on the two-variable frequency analysis, so as to determine the flooding standard in coastal cities. For conditional probability, an increasing trend is observed in, and with the average value of 0.13 is significantly larger than with the average value of 0.05. Thus, compound flooding from rainfall and storm tides is more likely to occur when a high storm tide event has occurred.

The results contained herein will provide additional insights into the effect of climate change in Haikou City. The existence of the trends is essential for the planning and design of sustainable drainage measures in coastal cities. The impacts of the trends on urban drainage engineering need to be further explored.

## ACKNOWLEDGEMENTS

This work was supported by the National Natural Science Foundation of China (Grant No. U20A20316), the Foundation for Innovative Research Groups of the Natural Science Foundation of Heibei Province (Grant No. E2020402074).

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

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