Figure
1.
Distribution of wind, rainfall and tide gauge stations in Hong Kong.
| Citation: | Shanshan Tao, Yunfei Hua, Sheng Dong. Hazard risk assessment of tropical cyclones based on joint probability theory[J]. Acta Oceanologica Sinica, 2023, 42(6): 89-99. doi: 10.1007/s13131-022-2143-9
|
A tropical cyclone is a cyclonic vortex that occurs in a tropical or subtropical ocean. Tropical cyclones are often accompanied by strong wind, rainstorm, and storm surge (Houze, 2014; Hoque et al., 2017; Kashem et al., 2019). They are destructive weather systems with high frequencies, wide ranges, and high intensities. Hong Kong is an important commercial port connecting East Asia, Southeast Asia, Australia and the Americas. However, because of its geographical location, it has been affected by tropical cyclone disasters year round, causing significant economic losses (Wang and Zhang, 2018). Therefore, a comprehensive assessment of the hazard-causing intensity of historical tropical cyclones is helpful for the relevant disaster prevention departments to take precautions in advance and reduce losses.
Most natural disasters involve the comprehensive action of multiple hazard-causing factors, and are difficult to fully be characterised the risk grades using a single hazard-causing factor. The main hazard-causing factors of tropical cyclones include strong wind, heavy rainfall, and storm surge (Hendricks, 2012; Yan et al., 2020; Ding et al., 2020). The intensity and frequency of these hazard-causing factors significantly affect the risk grades of tropical cyclones. Strong wind is one of the most important hazard-causing factors of tropical cyclones, as it often causes extensive damage to coastal areas and inland housing, infrastructure, and ecosystems (Kruk et al., 2010; Shao et al. 2018). In many studies, indicators related to wind speed have been used to measure the risk grades of tropical cyclones (Schroeder et al., 2009; Li and Duan, 2010). However, heavy rainfall, storm surge, and other hazard-causing factors cause large numbers of casualties and substantial economic losses (Cerveny and Newman, 2000; Phadke et al., 2003; Bloemendaal et al., 2019). Heavy rainfall often leads to severe urban inland inundation and floods, including flash floods (Czajkowski et al., 2013; Meyer et al., 2014; Yussouf et al., 2020). Storm surge can damage coastal protection facilities, submerge large coastal zones, cause casualties, and significantly affect the surrounding areas (Melton et al., 2010; Sun et al., 2015; Kim et al., 2019; Shi et al., 2020). In the present study, strong wind, heavy rainfall, and storm surge were selected as three representative factors to study the hazard-causing intensity of tropical cyclones.
At present, the comprehensive risk-assessment methods for tropical cyclones mainly include the construction of a comprehensive risk index and multivariate joint probability analysis. According to the intensity, the duration, and other data of tropical cyclones, Xiao et al. (2011) proposed the Tropical Cyclone Potential Impact Index (TCPI) and analysed the interannual variation and spatial distribution of the TCPI from 1949 to 2009. Saha and Wasimi (2014) developed a landfall potential index to reflect the possibility of tropical cyclone landing. Yu et al. (2017) proposed a comprehensive evaluation index for the hazard grades of strong winds and heavy rainfall of tropical cyclones and used it to evaluate the potential hazard grades of tropical cyclones in Southeast China. Chen et al. (2019) defined a joint index based on the wind and precipitation of a tropical cyclone to assess the combined effects of wind and precipitation.
The calculation of the multivariate joint probability is important for accurate assessment of the comprehensive risk of tropical cyclones. Copulas can connect the joint distribution function with their respective marginal distribution functions and are widely used in the comprehensive risk assessment of hazard-causing factors of natural disasters (Tsakiris et al., 2015; Lin-Ye et al., 2017; Radfar et al., 2021). Ye and Fang (2018) used copulas to study the combined wind and rain risk of historical tropical cyclones in coastal areas of China. Hu et al. (2010) used copulas to construct the joint distribution functions of typhoons and plum rain and calculated the probabilities of encountering typhoons and plum rain in the Taihu Lake Basin. Xu et al. (2019) used copulas to study the bivariate return periods of composite events of rainfall and storm surge. In addition, copulas are widely used for analysing sandstorms, droughts, and other natural disasters (Mirabbasi et al., 2012; Li et al., 2013; Liu et al., 2015). At present, most studies on the hazard-causing factors of tropical cyclones only involve bivariate situations, and there have been few studies involving three factors.
In this study, strong wind, rainfall, and storm surge are represented by the maximum hourly mean wind speed (V), total rainfall amount (R) and maximum tide level (L) of the tropical cyclone process, respectively. According to the data of V, R, and L for various observation stations in Hong Kong during the impact periods of 129 historical tropical cyclones from 1997 to 2020, three hazard-causing indices for tropical cyclones affecting Hong Kong are calculated: the strong-wind index (VI), total-rainfall index (RI), and tide-level index (LI). Then, the optimal marginal distribution and trivariate joint probability distribution of VI, RI, and LI are fitted and selected, and the joint return periods of VI, RI, and LI for tropical cyclones are calculated to evaluate the joint hazard-causing intensity. The interannual variation of the trivariate joint return period for 129 historical tropical cyclones is analysed. Finally, by constructing the limit state function of Hong Kong’s resistance to tropical cyclones, the regional risk of tropical cyclones affecting Hong Kong is evaluated.
The data of tropical cyclones were obtained from the Tropical Cyclone Annual Publications (Hong Kong Observatory, 1997−2020). The distribution of wind stations, rainfall stations, and tide gauge stations in Hong Kong is shown in Fig. 1. The names of these stations are presented in Table 1. The elevations of the anemometers can refer to the Tropical Cyclone Annual Publications (Hong Kong Observatory, 1997−2020).
| Wind station | Wind station | Rainfall station |
| Bluff Head | Ta Kwu Ling | Jordan Valley |
| Central Pier | Tai Mei Tuk | Kwai Chung |
| Cheung Chau | Tai Mo Shan | Mid Levels |
| Cheung Chau Beach | Tai Po Kau | Sha Tin |
| Green Island | Tap Mun East | Shau Kei Wan |
| HKIA | Tate’s Cairn | Shek Kong |
| Kai Tak | Tseung Kwan O | So Uk Estate |
| King’s Park | Tsing Yi Shell Oil Depot | Tai Mei Tuk |
| Lamma Island | TMGO | Tap Shek Kok |
| Lau Fau Shan | Waglan Island | Tung Chung |
| Ngong Ping | Wetland Park | TMR |
| North Point | Wong Chuk Hang | |
| Peng Chau | Tide gauge station | |
| Ping Chau | Rainfall station | Quarry Bay |
| Sai Kung | Hong Kong Observatory | Shek Pik |
| Sha Chau | HKIA | Tsim Bei Tsui |
| Sha Lo Wan | Cheung Chau | Tai Miu Wan |
| Sha Tin | Aberdeen | Tai Po Kau |
| Shek Kong | Fanling | Waglan Island |
| Star Ferry | High Island | − |
| Note: HKIA: Hong Kong International Airport; TMGO: Tuen Mun Government Offices; TMR: Tuen Mun Reservoir. − represents no station. | ||
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If a tropical cyclone enters within 800 km in Hong Kong, the Hong Kong Observatory issues corresponding warning signals (1, 3, 8, 9, and 10, in order of increasing risk) according to the impact of the cyclone on the area (Hong Kong Observatory, 1997−2020). Although tropical cyclones affecting Hong Kong can reach and surpass the typhoon grade during their lifecycle, they may be less intense when they influence Hong Kong, or their path may be far from the Hong Kong area, causing them to have little impact. Thus, it is reasonable to use the warning signals issued by the Hong Kong Observatory and the wind speed observations at various anemometers in Hong Kong (rather than the categories of tropical cyclones) to assess the wind hazard-causing intensity of a tropical cyclone affecting Hong Kong.
The annual tropical cyclone report records two types of wind speeds for each tropical cyclone at each station: the maximum gust peak speed and the maximum hourly mean wind speed. The maximum hourly mean wind speed corresponds to winds averaged over a 60-min interval ending on the hour. It can reflect the hazard-causing intensity of strong winds caused by tropical cyclones.
Rainfall data were obtained from a network of meteorological and rainfall stations operated by the Hong Kong Observatory, as well as rain gauges operated by the Geotechnical Engineering Office. Daily rainfall amounts are computed over a 24-h period ending at midnight Hong Kong Time. The total rainfall generated by a tropical cyclone is the sum of all the daily rainfall at a rainfall station during the cyclone. It can reflect the hazard-causing intensity of heavy rain caused by tropical cyclones.
The observations of each tide gauge station include the maximum sea tide level (above chart datum) and the maximum storm surge (above astronomical tide) during the tropical cyclone. The maximum tide level can reflect the hazard-causing intensity of storm surge caused by tropical cyclones.
There are many wind stations, rainfall stations, and tide gauge stations in Hong Kong. For a given tropical cyclone, the maximum hourly mean wind speed, total rainfall, and maximum sea level differ among the stations.
The joint hazard-causing intensity of the tropical cyclone mainly refers to the intensity, total amount and area of wind, rainfall, and storm surge caused by the tropical cyclone. Tropical cyclones cause strong wind, flood, or storm surge in the landing area and surrounding areas. For each tropical-cyclone hazard-causing factor, its intensity and scope should be considered. The damage caused by a tropical cyclone to different areas of Hong Kong varies significantly; thus, the data of wind speed, rainfall, and tidal level differ among different stations. Therefore, the V, R, or L of a single observation station during the impact period of a tropical cyclone is insufficient to reflect the hazard-causing intensity for the entire Hong Kong region. Hence, it is necessary to develop a comprehensive VI, RI, and LI covering different observation stations (Chen et al., 2009).
In this study, comprehensive hazard-causing intensity indices VI, RI, and LI are calculated using the data of V, R, and L for various observation stations in Hong Kong during the impact periods of 129 historical tropical cyclones from 1997 to 2020. These indices take into account the average of multiple tropical cyclones over many years, as well as the average and extreme values of the current tropical cyclone for different stations. They are defined as follows.
Before calculating the comprehensive hazard-causing intensity index of strong wind VI, the power law is used to unify the wind data. Because the anemometers of different weather stations are at different elevations, it is necessary to normalise the elevations to 10 m using Eq. (1).
| $$ V = {V_1} \times {\left( {\frac{z}{{{z_1}}}} \right)^m} ,$$ | (1) |
where V and V1 represent the wind speeds at heights z and z1 above the ground, respectively, and m is the power exponent, which is determined by the characteristics of underlying surface (it is generally between 1/2 and 1/8).
According to the geomorphological characteristics of Hong Kong and the surrounding environments of the stations, the woodland case is taken as the general performance of this area; thus, m is set as 0.28.
The hazard-causing intensity index of strong wind VI for a tropical cyclone is mainly measured by the maximum hourly mean wind speed for different wind stations. For the ith tropical cyclone, VIi is defined as follows:
| $$ V{I_i} = {\sum\limits_{j = 1}^{{n_i}} {{V_{ij}}} }/ {\left( {{n_i} \cdot AV} \right)} \times {\left[ {1 + {{\left( {{T_{8i}} + {T_{10i}}} \right)} / {{T_i}}}} \right]^2} ,$$ | (2) |
| $$ AV = \sum\limits_{i = 1}^m {{{\left( {{{\sum\limits_{j = 1}^{{n_i}} {{V_{ij}}} } / {{n_i}}}} \right)} / m}} , $$ | (3) |
where Vij represents the maximum hourly mean wind speed of the ith tropical cyclone at the jth wind station, ni represents the number of wind stations during the impact period of the ith tropical cyclone, m represents the total number of historical tropical cyclones affecting Hong Kong (for example, there were 129 tropical cyclones affecting Hong Kong between 1997 and 2020), T8i represents the duration of the ith tropical cyclone beyond warning signal 8, T10i represents the duration of the ith tropical cyclone beyond warning signal 10, Ti represents the total duration of the ith tropical cyclone affecting Hong Kong, and AV represents the average of the maximum hourly mean wind speeds at different wind stations during multiple historical tropical cyclones.
The strong-wind index VI is a dimensionless ratio obtained by dividing the average of the maximum hourly mean wind speeds at the wind stations during the impact period of a tropical cyclone by the average of all historical averages, taking into account extreme cases of strong or gale winds with warning signals of ≥8 and 10.
The hazard-causing intensity index of heavy rainfall RI for a tropical cyclone is mainly measured by the total rainfall amount for different rainfall stations. For the ith tropical cyclone, RIi is defined as follows:
| $$ R{I_i} = {{\sum\limits_{j = 1}^{{n_i}} {{R_{ij}}} } / {\left( {{n_i} \cdot AR} \right)}} \times {\left[ {1 + {{\left( {{T_{50i}} + {T_{100i}}} \right)} / {{T_i}}}} \right]^2} ,$$ | (4) |
| $$ AR = \sum\limits_{i = 1}^m {{{\left( {{{\sum\limits_{j = 1}^{{n_i}} {{R_{ij}}} } / {{n_i}}}} \right)} / m}} , $$ | (5) |
where Rij represents the total rainfall amount of the ith tropical cyclone at the jth rainfall station, ni represents the number of rainfall stations during the impact period of the ith tropical cyclone, m represents the total number of historical tropical cyclones affecting Hong Kong, T50i represents the number of days in which the daily rainfall exceeds 50 mm, T100i represents the number of days in which the daily rainfall exceeds 100 mm, Ti represents the total number of days with rainfall during the impact period of the tropical cyclone affecting Hong Kong, and AR represents the average of the total rainfall amount at different rainfall stations during multiple historical tropical cyclones.
The heavy-rainfall index RI is a dimensionless ratio obtained by dividing the average of the total rainfall amounts at the stations during the impact period of a tropical cyclone by the average of all historical averages, taking into account extreme cases of heavy rain with daily rainfall above 50 mm and 100 mm.
The hazard-causing intensity index of storm surge LI for a tropical cyclone is mainly measured by the maximum tide level for different tide gauge stations. For the ith tropical cyclone, LIi is defined as follows:
| $$ L{I_i} = {{\sum\limits_{j = 1}^{{n_i}} {{L_{ij}}} } / {\left( {{n_i} \cdot AL} \right)}} \times {\left[ {1 + {{\sum\limits_{k = 1}^{{{\tilde n}_i}} {\left( {{L_{i(k)}} - AL} \right)} } / {\left( {{n_i} \cdot AL} \right)}}} \right]^2} ,$$ | (6) |
| $$ AL = \sum\limits_{i = 1}^m {{{\left( {{{\sum\limits_{j = 1}^{{n_i}} {{L_{ij}}} } / {{n_i}}}} \right)} / m}} , $$ | (7) |
where Lij represents the maximum tide level of the ith tropical cyclone at the jth tide gauge station, ni represents the number of tide gauge stations during the impact period of the ith tropical cyclone, m represents the total number of historical tropical cyclones affecting Hong Kong, AL represents the average of the maximum tide levels at different tide gauge stations during multiple historical tropical cyclones,
The tide-level index LI is a dimensionless ratio obtained by dividing the average of the maximum tide levels at the stations during the impact period of a tropical cyclone by the average of all historical averages, taking into account extreme cases of a high tide level exceeding AL.
According to the joint probability distribution of hazard-causing indices VI, RI, and LI, the joint hazard-causing intensity of the ith tropical cyclone can be defined by using the joint return period of its indices VIi, RIi, and LIi. To obtain the joint probability distributions of VI, RI, and LI, Gumbel, Weibull, Pearson type 3 (P-III) and lognormal distributions are used to fit the best marginal distributions of VI, RI, and LI. Then, the Gaussian, Gumbel–Hougaard (G-H), Clayton, and Frank copulas are used to construct the joint distribution, and the Kolmogorov–Smirnov (K-S) test and Akaike information criterion (AIC) are used to select the best trivariate probability model.
The cumulative probability distribution functions, i.e., F(x), of the Gumbel distribution, three-parameter Weibull distribution, P-III distribution, and three-parameter lognormal distribution are given by Eqs (8)–(11), respectively.
| $$ F(x) = \exp \left\{ { - \exp \left[ { - \alpha \left( {x - \mu } \right)} \right]} \right\} ,$$ | (8) |
| $$ F\left( x \right) = 1 - \exp \left[ { - {{\left( {\frac{{x - \mu }}{\sigma }} \right)}^\gamma }} \right],{\text{ }}x \geqslant \mu ,$$ | (9) |
| $$ F\left( x \right) = \int_a^x \frac{\beta_0 ^{\alpha }}{\Gamma (\alpha )} \cdot \left( {t - \mu } \right)^{\alpha - 1}\exp \left[ - \beta_0 \left( {t - \mu } \right) \right]{\rm{d}}t,t \geqslant \mu ,\alpha > 0 ,$$ | (10) |
| $$ F\left( x \right) = \int_a^x {\frac{1}{{(t - a)\sigma \sqrt {2\pi } }}} \cdot \exp \left\{ { - \frac{1}{{2{\sigma ^2}}}{{\left[ {\ln \left( {t - a} \right) - \mu } \right]}^2}} \right\}{\rm{d}}t,t > a ,$$ | (11) |
where μ and a are the location parameters; α (in Eq. (8)), β0 and σ are the scale parameters; α (in Eq. (10)) and γ are the shape parameters.
A copula is a function that describes multivariate distributions defined on the unit hypercube, and the univariate marginal distributions are uniform on the interval (0, 1). According to Sklar (1959) theorem, if the marginal probability functions FX (x) = P (X≤x), FY (y) = P (Y≤y), and FZ (z)=P (Z≤z) of random variables X, Y, and Z are known, the joint distribution function FXYZ (x, y, z) can be obtained as follows:
| $$\begin{split} {F_{XYZ}}(x,y,z)=& P\;(X \leqslant x,Y \leqslant y,Z \leqslant z) = \\ &C\left( {{F_X}(x),{F_Y}(y),{F_Z}(z)} \right) = C(u,v,w) ,\end{split} $$ | (12) |
where X, Y, and Z represent the attributes of the event (corresponding to VI, RI, and LI, respectively, in this study); x, y, and z represent the thresholds of the attributes; u, v, and w represent the marginal probability functions of the attribute.
In addition, if the probability density functions fX (x), fY (y), and fZ (z) exist, the joint probability density function fXYZ (x, y, z) of X, Y, and Z can be expressed as
| $$ \begin{split} {f_{XYZ}}(x,y,z) =& c\left( {{F_X}(x),{F_Y}(y),{F_Z}(z)} \right){f_X}(x){f_Y}(y){f_Z}(z) =\\ &c\left( {u,v,w} \right){f_X}(x){f_Y}(y){f_Z}(z) , \end{split} $$ | (13) |
where c(u, v, w) is defined as
| $$ c(u,v,w) = \frac{{{\partial ^3}C(u,v,w)}}{{\partial u\partial v\partial w}} , $$ | (14) |
where c(u, v, w) represents the joint probability density functions calculated using the copulas.
The commonly used trivariate copulas are the Gaussian, G-H, Clayton, and Frank copulas, and their joint probability distribution functions and joint density functions are presented below.
(1) Gaussian copula
| $$ \begin{split} C(u,v,w;{\boldsymbol{\text{Σ}}}) =& \int\limits_{ - \infty }^{{\Phi ^{ - 1}}(u)} \;\int\limits_{ - \infty }^{{\Phi ^{ - 1}}(v)} \;{\int\limits_{ - \infty }^{{\Phi ^{ - 1}}(w)} {\frac{1}{{{{(2{\text{π}})}^{\tfrac{3}{2}}}{{\left| {\boldsymbol{\text{Σ}}} \right|}^{\tfrac{1}{2}}}}}} }\times \\&\exp \left( { - \frac{1}{2}{{\boldsymbol{\text{ω} }}^{\rm{T}}}{{\boldsymbol{\text{Σ} }}^{ - 1}}{\boldsymbol{\text{ω} }}} \right){\text{d}}{{\omega }} ,\end{split}$$ | (15) |
| $$ c(u,v,w;{\boldsymbol{\text{Σ} }}) = \frac{1}{{{{\left| {\boldsymbol{\text{Σ} }} \right|}^{\tfrac{1}{2}}}}}\exp \left[ { - \frac{1}{2}\left( {{{{\text{ζ} }}^{\rm{T}}}{{\boldsymbol{\text{Σ}}}^{ - 1}}{{\text{ζ} }} - {{{\text{ζ} }}^{\rm{T}}}{{\text{ζ} }}} \right)} \right] .$$ | (16) |
Here, Φ−1(⋅) represents the inverse of the standard normal distribution, and
| $$ {\boldsymbol{\text{Σ} }}= \left( {\begin{array}{*{20}{c}} 1&{{\rho _{12}}}&{{\rho _{13}}} \\ {{\rho _{21}}}&1&{{\rho _{23}}} \\ {{\rho _{31}}}&{{\rho _{32}}}&1 \end{array}} \right);{\text{ }}{\rho _{ij}} = {\rho _{ji}},{\text{ }}i \ne j;{\text{ }} - 1 \leqslant {\rho _{ij}} \leqslant 1 . $$ | (17) |
Additionally, ω = [ω1, ω2, ω3]T represents the integrand variable matrix, and ζ = [Φ－1(u), Φ－1(v), Φ－1(w)] T.
(2) G-H copula
| $$ \begin{split}&C(u,v,w;\theta ) = \exp \left\{ { - {{\left[ {{{( - \ln u)}^\theta } + {{( - \ln v)}^\theta } + {{( - \ln w)}^\theta }} \right]}^{\tfrac{1}{\theta }}}} \right\},\\ &\theta \in \left[ {1, + \infty } \right),\\[-10pt]\end{split}$$ | (18) |
where θ is the correlated parameter.
| $$ \begin{split} c(u,v,w;\theta ) = &\frac{{{{( - \ln u\ln v\ln w)}^{\theta - 1}}}}{{uvw}}\exp \left( { - {\omega ^{\tfrac{1}{\theta }}}} \right)\times\\ &\left[ {{\omega ^{\tfrac{3}{\theta } - 3}} + \left( {3\theta - 3} \right){\omega ^{\tfrac{2}{\theta } - 3}} + \left( {\theta - 1} \right)\left( {2\theta - 1} \right){\omega ^{\tfrac{1}{\theta } - 3}}} \right] , \end{split}$$ | (19) |
where
(3) Clayton copula
| $$ C(u,v,w;\theta ) = {\left( {{u^{ - \theta }} + {v^{ - \theta }} + {w^{ - \theta }} - 2} \right)^{ - \tfrac{1}{\theta }}}{\text{, }}\theta \in ({\text{0,}}\;\infty ), $$ | (20) |
| $$\begin{split} c(u,v,w;\theta ) =& {u^{ - (\theta + 1)}}{v^{ - (\theta + 1)}}{w^{ - (\theta + 1)}}\left( {1 + \theta } \right)\left( {1 + 2\theta } \right)\times\\ &{\left( {{u^{ - \theta }} + {v^{ - \theta }} + {w^{ - \theta }} - 2} \right)^{ - \left( {\tfrac{1}{\theta } + 3} \right)}}. \end{split}$$ | (21) |
(4) Frank copula
| $$C(u,v,w;\theta ) = - \frac{1}{\theta }\ln \left[ {1 + \frac{{\left( {{{\text{e}}^{ - \theta u}} - 1} \right)\left( {{{\text{e}}^{ - \theta v}} - 1} \right)\left( {{{\text{e}}^{ - \theta w}} - 1} \right)}}{{{{\left( {{{\text{e}}^{ - \theta }} - 1} \right)}^2}}}} \right],\\ \theta \ne {\text{0}} $$ | (22) |
| $$ c(u,v,w;\theta ) = \frac{{{\theta ^2}{{\text{e}}^{ - \theta u}}{{\text{e}}^{ - \theta v}}{{\text{e}}^{ - \theta w}}{{\left( {{{\text{e}}^{ - \theta }} - 1} \right)}^2}\left[ {{{\left( {{{\text{e}}^{ - \theta }} - 1} \right)}^2} - \left( {{{\text{e}}^{ - \theta u}} - 1} \right)\left( {{{\text{e}}^{ - \theta v}} - 1} \right)\left( {{{\text{e}}^{ - \theta w}} - 1} \right)} \right]}}{{{{\left[ {{{\left( {{{\text{e}}^{ - \theta }} - 1} \right)}^2} + \left( {{{\text{e}}^{ - \theta u}} - 1} \right)\left( {{{\text{e}}^{ - \theta v}} - 1} \right)\left( {{{\text{e}}^{ - \theta w}} - 1} \right)} \right]}^3}}} .$$ | (23) |
In this study, the maximum likelihood method is used to estimate both marginal distributions and trivariate joint probability distributions, and the K-S test is used to judge whether the parameter estimation of univariate curves and copulas satisfies the requirements. The choice of the optimal models is mainly based on the average sum of deviation squares and the AIC.
The K-S test is a robust goodness-of-fit test method for determining whether samples obey the selected model. The statistic of the K-S test is
| $$ {D_n} = \mathop {\max }\limits_{1 \leqslant i \leqslant n} \left| {{{\hat F}_i} - {F_i}} \right| , $$ | (24) |
where
The AIC value is a commonly used standard for evaluating model fit. It is defined as follows:
| $$ {\rm{AIC}} = n\ln \left( {{\rm{MSE}}} \right) + 2k , $$ | (25) |
where n represents the length of the sample, k represents the number of probability model parameters, and MSE represents the mean squared error, which is defined as
| $$ {\rm{MSE}} = \frac{1}{n}\sum\limits_{i = 1}^n {{{\left( {{{\hat F}_i} - {F_i}} \right)}^2}} . $$ | (26) |
A smaller AIC value corresponds to a better fitting effect of the joint probability distribution model.
The joint return period of a trivariate characterised event can be described in RP∩, which is defined as the return period of an event with three variables all above a specific threshold (X≥x, Y≥y, and Z≥z). The joint return period is expressed as follows:
| $$ \begin{split} R{P_ \cap } =& \frac{{E(t)}}{{P\;(X \geqslant x,Y \geqslant y,Z \geqslant z)}} =\\ & \frac{{E(t)}}{{\displaystyle\int\limits_x^{ + \infty } {\int\limits_y^{ + \infty } {\int\limits_z^{ + \infty } {{f_{XYZ}}(x,y,z){\text{d}}x{\text{d}}y{\text{d}}z} } } }}=\\ &\frac{{E(t)}}{{\displaystyle\int\limits_x^{ + \infty } {\displaystyle\int\limits_y^{ + \infty } {\displaystyle\int\limits_z^{ + \infty } {c(u,v,w){f_X}(x){f_Y}(y){f_Z}{\text{(}}z{\text{)d}}x{\text{d}}y{\text{d}}z} } } }} = \\ &\frac{{E(t)}}{{\displaystyle\int\limits_u^1 {\displaystyle\int\limits_v^1 {\int\limits_w^1 {c(u,v,w){\text{d}}u{\text{d}}v{\text{d}}w} } } }} , \end{split} $$ | (27) |
where, for the ith tropical cyclone, x, y, and z represent VIi, RIi, and LIi, respectively; E(t) represents the mean interval between two successive tropical cyclones (in this study, a total of 129 tropical cyclones occurred between 1997 and 2020; thus, E(t) = 24/129).
First, the VI, RI, and LI of each tropical cyclone affecting Hong Kong from 1997 to 2020 were selected and fitted by the Gumbel, Weibull, P-III, and lognormal distributions. The data sequences are shown in Fig. 2, and the fitting results are presented in Table 2 and Fig. 3.
| Indices | Type | A1 | A2 | A3 | Dn(0.05) | $\hat{D}_n $ | Q |
| Strong-wind index | Gumbel | 0.830 | 3.202 | – | 1.197×10−1 | 8.096×10−2 | 7.077×10−4 |
| Weibull | 0.346 | 0.750 | 1.619 | 1.197×10−1 | 7.185×10−2 | 9.110×10−4 | |
| P-III | 2.233 | 3.469 | 0.373 | 1.197×10−1 | 5.553×10−2 | 5.035×10−4 | |
| Lognormal | 0.157 | −0.266 | 0.486 | 1.197×10−1 | 5.607×10−2 | 4.017×10−4 | |
| Total-rainfall index | Gumbel | 0.904 | 1.607 | – | 1.347×10−1 | 1.255×10−1 | 3.926×10−3 |
| Weibull | 0.118 | 1.293 | 1.285 | 1.347×10−1 | 1.268×10−1 | 1.854×10−3 | |
| P-III | 0.956 | 0.976 | 0.330 | 1.347×10−1 | 7.837×10−2 | 7.015×10−4 | |
| Lognormal | 0.109 | −0.134 | 0.868 | 1.347×10−1 | 9.940×10−2 | 8.670×10−4 | |
| Tide-level index | Gumbel | 0.932 | 6.892 | – | 1.197×10−1 | 9.046×10−2 | 2.268×10−3 |
| Weibull | 0.598 | 0.458 | 2.602 | 1.197×10−1 | 8.894×10−2 | 2.296×10−3 | |
| P-III | 4.183 | 12.718 | 0.678 | 1.197×10−1 | 8.644×10−2 | 1.715×10−3 | |
| Lognormal | −0.187 | 0.168 | 0.131 | 1.197×10−1 | 6.826×10−2 | 8.945×10−4 | |
| Note: $\hat{D}_n $ is the K-S stastic, Dn(0.05) is the K-S tests value, Q is the average sum of deviation squares; the corresponding relationship of the parameters is as follows: (1) Gumbel distribution, A1 and A2 represent location parameter u and scale parameter $\alpha $ respectively; (2) Weibull distribution, A1, A2 and A3 represent the position parameter μ, shape parameter $\sigma $ and scale parameter γ, respectively; (3) P-III distribution, A1, A2 and A3 represent $\alpha $, scale parameter $\ \beta_0 $ and μ, respectively; (4) lognormal distribution, A1, A2 and A3 represent the parameters a, μ and $\sigma $, respectively. − represents no data. | |||||||
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Because the index RI has zero-value data, the zero-value sequences are ignored first, and the frequency of nonzero-value sequences
| $$ P\left( {Y \leqslant y} \right) = 1 - \frac{n}{N} + \frac{n}{N}P\left( {Y \leqslant y\left| {Y > 0} \right.} \right) , $$ | (28) |
where n represents the number of Y > 0 values in the sequence, and N represents the total number of values in the sequence.
The results in Table 2 indicate that at a confidence level of 0.05, the four distributions of VI all pass the K-S tests. The Q (sum of squares of deviation) values of the distributions increase in the following order: lognormal distribution, P-III distribution, Gumbel distribution, and Weibull distribution. As shown in Fig. 3, the fitting effect of the lognormal distribution on the upper tail and lower tail is slightly better than those of the other distributions. Therefore, the lognormal distribution function is the optimal fitting function for the marginal distribution of VI.
Similarly, the P-III and lognormal distribution functions are selected as the optimal fitting functions of the indices RI and LI, respectively.
According to the foregoing optimisation results, the marginal probability distributions of VI and LI are both lognormal distributions, the marginal probability distribution of RI is a P-III distribution. The parameter estimation results are presented in Table 2.
On the basis of the univariate marginal distributions u = FX(x), v = FY(y), and w = FZ(z), the trivariate Gaussian, G-H, Clayton, and Frank copulas C(u, v, w) are used to construct the joint probability distributions of (VI, RI, LI). Then, the optimal trivariate distribution is selected, and the joint return period of (VI, RI, LI) is calculated.
According to the K-S tests and AIC values, the optimal trivariate copula model is selected. The results are presented in Table 3. All the copula models pass the K-S tests, and the G-H copula is the best, as it has the smallest AIC value.
| Copula function | $\hat{\theta} $ | $\hat{D}_n $ | Dn(0.5) | AIC |
| Frank | 2.0497 | 0.1010 | 0.1197 | −872.3280 |
| Clayton | 0.3392 | 0.0845 | 0.1197 | −876.6350 |
| G-H | 1.2018 | 0.0795 | 0.1197 | −927.4228 |
| Gaussian | (0.4065, 0.4913, −0.0173) | 0.1108 | 0.1197 | −856.0544 |
| Note: AIC: Akaike information criterion. | ||||
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According to Eq. (27), the joint return periods RP∩ are calculated. The joint return periods for the historical tropical cyclones from 1997 to 2020 are shown in Fig. 4.
As shown in Fig. 4, the joint return periods of most tropical cyclones’ indices (VIi, RIi, LIi) are <30 a. Typhoon Sam (1999), Typhoon York (1999), Typhoon Hagupit (2008), Severe Typhoon Vicente (2012), Super Typhoon Hato (2017), and Super Typhoon Mangkhut (2018) have joint return periods of >30 a; the corresponding joint return periods are 193 a, 35 a, 131 a, 30 a, 115 a, and 2 274 a, respectively.
Among these tropical cyclones, Mangkhut (2018) has the largest VIi and LIi values. Mangkhut does not have the largest RIi, but it has the longest joint return period. Typhoon Sam (1999) has a VIi of 1.609, ranking 12th in the dataset, an RIi of 6.650 (ranking 1st), and an LIi of 1 (ranking 11th). Its joint return period is ranked 2nd. Typhoon Hagupit (2008) has a wind index VIi of 1.846 (ranking 7th), a RIi of 1.207 (ranking 41st), and a LIi of 1.520 (ranking 2nd). Its joint return period is ranked 3rd.
The joint return periods of York, Vicente, and Hato are also relatively long. Their hazard-causing intensity indices VIi, RIi, and LIi are not the largest but are relatively large. The combination of VI, RI, and LI influences the joint return periods of tropical cyclones, which reflect their comprehensive hazard-causing intensity.
To study the interannual variations of the comprehensive hazard-causing intensity for tropical cyclones, the joint return periods of historical tropical cyclones are analysed. Typhoon Mangkhut (2018) is excluded, because as shown in Fig. 4a, it has a significantly longer joint return period than the other tropical cyclones and can be seen as an abnormal tropical cyclone. The mean value of the joint return periods of other tropical cyclones in 2018 is substituted for the joint return period of Mangkhut.
The hazard-causing intensity of tropical cyclones each year is defined by the sum of the joint return periods of all the tropical cyclones in the year. Figure 5a presents the annual change in the hazard-causing intensity (redline) and the total number (blue line) of tropical cyclones. As shown, there is no significant change in the number of tropical cyclones per year; the annual hazard-causing intensity of tropical cyclones exhibits obvious periodic change, and the period is approximately 9 a. If Typhoon Mangkhut is not taken into account, the peak value of the annual hazard-causing intensity of tropical cyclones in each cycle exhibits a decreasing trend.
Additionally, the average hazard-causing intensities of tropical cyclones in 3 a or 6 a are calculated (defined as the sum of annual hazard-causing intensities of all tropical cyclones in 3 a or 6 a). The trend change of the hazard-causing intensity of tropical cyclones can be analysed by this moving average, as shown in Figs 5b and c. In Fig. 5c, the return period in the first 12 a exhibits a decreasing trend, and the return period in the last 6 a exhibits an increasing trend. If Typhoon Mangkhut is taken into account, the increasing trend becomes more significant.
In addition, the maximum joint return period of the all tropical cyclones in each year is extracted to represent the highest hazard-causing intensity of tropical cyclones in the year, as shown in Fig. 6. Additionally, the average joint return period of all tropical cyclones in each year (i.e., the sum of the joint return periods divided by the number of tropical cyclones in the year) is extracted to represent the average hazard-causing intensity of tropical cyclones in the current year, as shown in Fig. 7.
Similar to Fig. 5a, Figs 6 and 7 show that the maximum and average annual hazard-causing intensities of tropical cyclones also have a 9-a cycle. Without consideration of Typhoon Mangkhut, the maximum value of each cycle exhibits a decreasing trend.
Hong Kong is frequently affected by tropical cyclones; thus, it is necessary to assess the overall risk of tropical cyclones in this region. In this section, it is assumed that in the structural design of Hong Kong, the design values satisfy the corresponding return period standard of the strong-wind index, total-rainfall index, or tide-level index, and the limit state function of the regional risk to tropical cyclones can be constructed accordingly. According to the joint probability distribution of the hazard-causing intensity indices VI, RI, and LI of tropical cyclones, a regional risk assessment of tropical cyclones affecting Hong Kong is performed.
According to the foregoing calculation results, the optimal marginal curves of VI, RI, and LI follow lognormal, P-III, and lognormal distributions, respectively, and corresponding parameter estimations are presented in Table 2. The return values xT of VI, RI, or LI can be calculated as follows:
| $$ T = \frac{{E(t)}}{{P(X \geqslant {x_T})}} = \frac{{E(t)}}{{1 - F({x_T})}}{\text{ }} \Rightarrow {\text{ }}{x_T} = {F^{ - 1}}\left( {1 - \frac{{E(t)}}{T}} \right) , $$ | (29) |
where X is a random variable (VI, RI, or LI), T represents the return period, x represents the T-year return value, F(x) represents the probability distribution function of X, E(t) represents the mean time interval between two successive tropical cyclones (in this study, a total of 129 tropical cyclones occurred between 1997 and 2020; thus, E(t) can be estimated as 24/129).
The estimated return values of the indices VI, RI, and LI under different return periods are presented in Table 4.
| Return period/a | 10 | 25 | 50 | 100 | 120 | 200 |
| Strong-wind index (VI) | 2.2667 | 2.6600 | 2.9713 | 3.2958 | 3.3834 | 3.6343 |
| Total-rainfall index (RI) | 4.3139 | 5.2448 | 5.9498 | 6.6554 | 6.8411 | 7.3614 |
| Tide-level index (LI) | 1.3672 | 1.4404 | 1.4927 | 1.5428 | 1.5557 | 1.5912 |
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According to the hazard-causing intensity indices VI, RI, and LI of tropical cyclones and the assumed return values V, R, and L for structural design in Hong Kong, the limit state function of the hazard-causing factors (strong wind, heavy rainfall, and storm surge) is constructed as follows:
| $$ Z = 1 - \left( {{a_1}\frac{{VI}}{{{V_{{T_1}}}}} + {a_2}\frac{{RI}}{{{R_{{T_2}}}}} + {a_3}\frac{{LI}}{{{L_{{T_3}}}}}} \right) , $$ | (30) |
where a1, a2, and a3 are linear coefficients of VI, RI, and LI, respectively, and a1 + a2 + a3 = 1;
Under the condition that the return values V, R, and L are known, according to the joint probability distribution of VI, RI, and LI for tropical cyclones, the regional reliability under tropical cyclones affecting Hong Kong can be calculated as
| $$ {P_s} = P\left( {Z > 0} \right) = \iiint\limits_{Z > 0} {{f_{XYZ}}(x,y,z)}{\rm{d}}x{\rm{d}}y{\rm{d}}z , $$ | (31) |
where Z represents the limit state function about VI, RI, and LI in Eq. (30); f(x, y, z) represents the joint probability density function of VI, RI, and LI; and Ps represents the regional risk reliability under tropical cyclones affecting Hong Kong.
The reliability Ps can be determined via the direct integration method using the following formula:
| $$ \begin{split} {P_s} =& \iiint\limits_{Z > 0} {{f_{XYZ}}(x,y,z)}{\rm{d}}x{\rm{d}}y{\rm{d}}z = \iiint\limits_{{R^3}} I\left[ {Z(x,y,z)} \right] \times\\ &{f_{XYZ}}(x,y,z){\rm{d}}x{\rm{d}}y{\rm{d}}z \approx \int_{{a_1}}^{{b_1}} \int_{{a_2}}^{{b_2}} \int_{{a_3}}^{{b_3}} I\left[ {Z(x,y,z)} \right] \times\\ & {f_{XYZ}}(x,y,z) {\rm{d}}x{\rm{d}}y{\rm{d}}z \approx \sum\limits_{{i_1} = 1}^{{k_1}} \sum\limits_{{i_2} = 1}^{{k_2}} \sum\limits_{{i_3} = 1}^{{k_3}} I\left[ {Z({a_{1{i_1}}},{a_{2{i_2}}},{a_{3{i_3}}})} \right] \times\\ & {f_{XYZ}}({a_{1{i_1}}},{a_{2{i_2}}},{a_{3{i_3}}}) {\Delta} {a_{1{i_1}}}{{{\Delta}}} {a_{2{i_2}}}{\Delta} {a_{3{i_3}}} , \end{split} $$ | (32) |
where k1 k2 and k3 represent the number of partitions dividing the intervals [a1, b1], [a2, b2], and [a3, b3] into intercells, respectively;
| $$ I\left[ {Z(x,y,z)} \right] = \left\{ {\begin{array}{*{20}{c}} {1,}&{Z(x,y,z) < 0} \\ {0,}&{Z(x,y,z) > 0} \end{array}} \right. . $$ | (33) |
The upper and lower limits of the integral a and b can be determined using the following formula:
| $$ \int_a^b {f(x)} {\rm{d}}x \geqslant 1 - {\varepsilon _0} , $$ | (34) |
where ε represents the calculation accuracy.
The β can be calculated using the regional reliability Ps.
| $$ \beta = - {\Phi ^{ - 1}}(1 - {P_s}) , $$ | (35) |
where Φ−1(·) represents the inverse function of the standard normal distribution.
Assume that the return periods T1, T2, and T3 in Eq. (30) are identical, e.g., T = T1 = T2 = T3 = 100 a. As the linear coefficient combination (a1, a2, a3) in Eq. (30) changes, the β can be calculated according to Eqs (30), (32), and (35), as shown in Table 5 and Fig. 8. The regional risk reliability index can reflect the change in regional reliability with the change in regional attention (a1, a2, a3) to the strong wind, heavy rainfall, or storm surge caused by tropical cyclones under the condition that the overall structural design of Hong Kong considers the return periods of the hazard-causing indices VI, RI, and LI.
| a1 | a2 | ||||||||||
| 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | |
| 0 | 2.9582 | 3.1114 | 3.1685 | 3.1654 | 3.1478 | 3.1167 | 3.0692 | 3.0319 | 2.9920 | 2.9448 | 2.9029 |
| 0.1 | 3.0894 | 3.1910 | 3.2053 | 3.1986 | 3.1768 | 3.1472 | 3.1082 | 3.0680 | 3.0265 | 2.9850 | – |
| 0.2 | 3.1621 | 3.2068 | 3.2158 | 3.2074 | 3.1921 | 3.1658 | 3.1327 | 3.0957 | 3.0609 | – | – |
| 0.3 | 3.1748 | 3.1962 | 3.2070 | 3.2026 | 3.1908 | 3.1722 | 3.1450 | 3.1138 | – | – | – |
| 0.4 | 3.1436 | 3.1747 | 3.1898 | 3.1908 | 3.1856 | 3.1708 | 3.1506 | – | – | – | – |
| 0.5 | 3.1038 | 3.1459 | 3.1660 | 3.1732 | 3.1709 | 3.1638 | – | – | – | – | – |
| 0.6 | 3.0631 | 3.1108 | 3.1374 | 3.1499 | 3.1553 | – | – | – | – | – | – |
| 0.7 | 3.0353 | 3.0734 | 3.1058 | 3.1234 | – | – | – | – | – | – | – |
| 0.8 | 2.9836 | 3.0339 | – | – | – | – | – | – | – | – | – |
| 0.9 | 2.9486 | – | – | – | – | – | – | – | – | – | – |
| 1.0 | 2.8709 | – | – | – | – | – | – | – | – | – | – |
| Note: – represents no data. | |||||||||||
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Assume that the return periods T1, T2, and T3 in Eq. (30) have the same values, e.g., T = T1 = T2 = T 3= 50 a. As the linear coefficient combination (a1, a2, a3) in Eq. (30) changes, the β can be calculated, as shown in Table 6 and Fig. 9.
| a1 | a2 | ||||||||||
| 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | |
| 0 | 2.4992 | 2.8912 | 2.9532 | 2.9468 | 2.9284 | 2.8894 | 2.8349 | 2.7955 | 2.7540 | 2.7104 | 2.6743 |
| 0.1 | 2.8705 | 2.9672 | 2.9923 | 2.9795 | 2.9484 | 2.9148 | 2.8728 | 2.8333 | 2.7900 | 2.7488 | – |
| 0.2 | 2.9492 | 2.9890 | 3.0008 | 2.9857 | 2.9585 | 2.9358 | 2.8985 | 2.8619 | 2.8177 | – | – |
| 0.3 | 2.9470 | 2.9808 | 2.9860 | 2.9834 | 2.9672 | 2.9459 | 2.9140 | 2.8784 | – | – | – |
| 0.4 | 2.9197 | 2.9566 | 2.9701 | 2.9734 | 2.9588 | 2.9440 | 2.9214 | – | – | – | – |
| 0.5 | 2.8811 | 2.9270 | 2.9467 | 2.9545 | 2.9513 | 2.9319 | – | – | – | – | – |
| 0.6 | 2.8353 | 2.8909 | 2.9175 | 2.9297 | 2.9310 | – | – | – | – | – | – |
| 0.7 | 2.7996 | 2.8488 | 2.8844 | 2.9023 | – | – | – | – | – | – | – |
| 0.8 | 2.7610 | 2.8091 | – | – | – | – | – | – | – | – | – |
| 0.9 | 2.7116 | – | – | – | – | – | – | – | – | – | – |
| 1.0 | 2.6584 | – | – | – | – | – | – | – | – | – | – |
| Note: – represents no data. | |||||||||||
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As indicated by Tables 5 and 6, if the return periods of the overall design of the regional structures decrease, the regional ability to defend against tropical cyclone disasters (i.e., regional risk reliability) decreases.
According to four types of univariate probability distributions and four types of trivariate copulas, the optimal marginal and joint distribution functions of VI, RI, and LI were selected through K-S tests and AIC values. The joint return periods of VI, RI, and LI during a tropical cyclone were determined to evaluate the joint hazard-causing intensity. The interannual variation of the trivariate joint return period for 129 historical tropical cyclones was analysed. Finally, by constructing the limit state function of Hong Kong’s resistance to tropical cyclones, the regional reliability of Hong Kong under tropical cyclones was evaluated.
The results indicated that the joint return period of VI, RI, and LI can reflect the joint hazard-causing intensity of strong wind, heavy rain, and storm surge caused by tropical cyclones. In the analysis of the interannual variation of the comprehensive hazard-causing intensity of tropical cyclones (excluding Super Typhoon Mangkhut), the annual, maximum, and average annual hazard-causing intensities of tropical cyclones exhibited obvious periodic change, and the period was approximately 9 a. Without considering the Super Typhoon Mangkhut, the maximum value of each cycle exhibited a decreasing trend.
According to the calculation of the regional reliability index of tropical cyclones affecting Hong Kong, if the return periods of the overall design of the regional structures decrease, the regional ability to defend against tropical cyclone disasters will be degraded.
Acknowledgements: Thanks to Shuntao Fan, Jialing Song and Haoshuang Song for their help in data collation.|
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Figures(9) / Tables(6)
Supported by:
Beijing Renhe Information Technology Co. Ltd
| Wind station | Wind station | Rainfall station |
| Bluff Head | Ta Kwu Ling | Jordan Valley |
| Central Pier | Tai Mei Tuk | Kwai Chung |
| Cheung Chau | Tai Mo Shan | Mid Levels |
| Cheung Chau Beach | Tai Po Kau | Sha Tin |
| Green Island | Tap Mun East | Shau Kei Wan |
| HKIA | Tate’s Cairn | Shek Kong |
| Kai Tak | Tseung Kwan O | So Uk Estate |
| King’s Park | Tsing Yi Shell Oil Depot | Tai Mei Tuk |
| Lamma Island | TMGO | Tap Shek Kok |
| Lau Fau Shan | Waglan Island | Tung Chung |
| Ngong Ping | Wetland Park | TMR |
| North Point | Wong Chuk Hang | |
| Peng Chau | Tide gauge station | |
| Ping Chau | Rainfall station | Quarry Bay |
| Sai Kung | Hong Kong Observatory | Shek Pik |
| Sha Chau | HKIA | Tsim Bei Tsui |
| Sha Lo Wan | Cheung Chau | Tai Miu Wan |
| Sha Tin | Aberdeen | Tai Po Kau |
| Shek Kong | Fanling | Waglan Island |
| Star Ferry | High Island | − |
| Note: HKIA: Hong Kong International Airport; TMGO: Tuen Mun Government Offices; TMR: Tuen Mun Reservoir. − represents no station. | ||
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| Indices | Type | A1 | A2 | A3 | Dn(0.05) | $\hat{D}_n $ | Q |
| Strong-wind index | Gumbel | 0.830 | 3.202 | – | 1.197×10−1 | 8.096×10−2 | 7.077×10−4 |
| Weibull | 0.346 | 0.750 | 1.619 | 1.197×10−1 | 7.185×10−2 | 9.110×10−4 | |
| P-III | 2.233 | 3.469 | 0.373 | 1.197×10−1 | 5.553×10−2 | 5.035×10−4 | |
| Lognormal | 0.157 | −0.266 | 0.486 | 1.197×10−1 | 5.607×10−2 | 4.017×10−4 | |
| Total-rainfall index | Gumbel | 0.904 | 1.607 | – | 1.347×10−1 | 1.255×10−1 | 3.926×10−3 |
| Weibull | 0.118 | 1.293 | 1.285 | 1.347×10−1 | 1.268×10−1 | 1.854×10−3 | |
| P-III | 0.956 | 0.976 | 0.330 | 1.347×10−1 | 7.837×10−2 | 7.015×10−4 | |
| Lognormal | 0.109 | −0.134 | 0.868 | 1.347×10−1 | 9.940×10−2 | 8.670×10−4 | |
| Tide-level index | Gumbel | 0.932 | 6.892 | – | 1.197×10−1 | 9.046×10−2 | 2.268×10−3 |
| Weibull | 0.598 | 0.458 | 2.602 | 1.197×10−1 | 8.894×10−2 | 2.296×10−3 | |
| P-III | 4.183 | 12.718 | 0.678 | 1.197×10−1 | 8.644×10−2 | 1.715×10−3 | |
| Lognormal | −0.187 | 0.168 | 0.131 | 1.197×10−1 | 6.826×10−2 | 8.945×10−4 | |
| Note: $\hat{D}_n $ is the K-S stastic, Dn(0.05) is the K-S tests value, Q is the average sum of deviation squares; the corresponding relationship of the parameters is as follows: (1) Gumbel distribution, A1 and A2 represent location parameter u and scale parameter $\alpha $ respectively; (2) Weibull distribution, A1, A2 and A3 represent the position parameter μ, shape parameter $\sigma $ and scale parameter γ, respectively; (3) P-III distribution, A1, A2 and A3 represent $\alpha $, scale parameter $\ \beta_0 $ and μ, respectively; (4) lognormal distribution, A1, A2 and A3 represent the parameters a, μ and $\sigma $, respectively. − represents no data. | |||||||
DownLoad:
CSV
| Copula function | $\hat{\theta} $ | $\hat{D}_n $ | Dn(0.5) | AIC |
| Frank | 2.0497 | 0.1010 | 0.1197 | −872.3280 |
| Clayton | 0.3392 | 0.0845 | 0.1197 | −876.6350 |
| G-H | 1.2018 | 0.0795 | 0.1197 | −927.4228 |
| Gaussian | (0.4065, 0.4913, −0.0173) | 0.1108 | 0.1197 | −856.0544 |
| Note: AIC: Akaike information criterion. | ||||
DownLoad:
CSV
| Return period/a | 10 | 25 | 50 | 100 | 120 | 200 |
| Strong-wind index (VI) | 2.2667 | 2.6600 | 2.9713 | 3.2958 | 3.3834 | 3.6343 |
| Total-rainfall index (RI) | 4.3139 | 5.2448 | 5.9498 | 6.6554 | 6.8411 | 7.3614 |
| Tide-level index (LI) | 1.3672 | 1.4404 | 1.4927 | 1.5428 | 1.5557 | 1.5912 |
DownLoad:
CSV
| a1 | a2 | ||||||||||
| 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | |
| 0 | 2.9582 | 3.1114 | 3.1685 | 3.1654 | 3.1478 | 3.1167 | 3.0692 | 3.0319 | 2.9920 | 2.9448 | 2.9029 |
| 0.1 | 3.0894 | 3.1910 | 3.2053 | 3.1986 | 3.1768 | 3.1472 | 3.1082 | 3.0680 | 3.0265 | 2.9850 | – |
| 0.2 | 3.1621 | 3.2068 | 3.2158 | 3.2074 | 3.1921 | 3.1658 | 3.1327 | 3.0957 | 3.0609 | – | – |
| 0.3 | 3.1748 | 3.1962 | 3.2070 | 3.2026 | 3.1908 | 3.1722 | 3.1450 | 3.1138 | – | – | – |
| 0.4 | 3.1436 | 3.1747 | 3.1898 | 3.1908 | 3.1856 | 3.1708 | 3.1506 | – | – | – | – |
| 0.5 | 3.1038 | 3.1459 | 3.1660 | 3.1732 | 3.1709 | 3.1638 | – | – | – | – | – |
| 0.6 | 3.0631 | 3.1108 | 3.1374 | 3.1499 | 3.1553 | – | – | – | – | – | – |
| 0.7 | 3.0353 | 3.0734 | 3.1058 | 3.1234 | – | – | – | – | – | – | – |
| 0.8 | 2.9836 | 3.0339 | – | – | – | – | – | – | – | – | – |
| 0.9 | 2.9486 | – | – | – | – | – | – | – | – | – | – |
| 1.0 | 2.8709 | – | – | – | – | – | – | – | – | – | – |
| Note: – represents no data. | |||||||||||
DownLoad:
CSV
| a1 | a2 | ||||||||||
| 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | |
| 0 | 2.4992 | 2.8912 | 2.9532 | 2.9468 | 2.9284 | 2.8894 | 2.8349 | 2.7955 | 2.7540 | 2.7104 | 2.6743 |
| 0.1 | 2.8705 | 2.9672 | 2.9923 | 2.9795 | 2.9484 | 2.9148 | 2.8728 | 2.8333 | 2.7900 | 2.7488 | – |
| 0.2 | 2.9492 | 2.9890 | 3.0008 | 2.9857 | 2.9585 | 2.9358 | 2.8985 | 2.8619 | 2.8177 | – | – |
| 0.3 | 2.9470 | 2.9808 | 2.9860 | 2.9834 | 2.9672 | 2.9459 | 2.9140 | 2.8784 | – | – | – |
| 0.4 | 2.9197 | 2.9566 | 2.9701 | 2.9734 | 2.9588 | 2.9440 | 2.9214 | – | – | – | – |
| 0.5 | 2.8811 | 2.9270 | 2.9467 | 2.9545 | 2.9513 | 2.9319 | – | – | – | – | – |
| 0.6 | 2.8353 | 2.8909 | 2.9175 | 2.9297 | 2.9310 | – | – | – | – | – | – |
| 0.7 | 2.7996 | 2.8488 | 2.8844 | 2.9023 | – | – | – | – | – | – | – |
| 0.8 | 2.7610 | 2.8091 | – | – | – | – | – | – | – | – | – |
| 0.9 | 2.7116 | – | – | – | – | – | – | – | – | – | – |
| 1.0 | 2.6584 | – | – | – | – | – | – | – | – | – | – |
| Note: – represents no data. | |||||||||||
DownLoad:
CSV
| Wind station | Wind station | Rainfall station |
| Bluff Head | Ta Kwu Ling | Jordan Valley |
| Central Pier | Tai Mei Tuk | Kwai Chung |
| Cheung Chau | Tai Mo Shan | Mid Levels |
| Cheung Chau Beach | Tai Po Kau | Sha Tin |
| Green Island | Tap Mun East | Shau Kei Wan |
| HKIA | Tate’s Cairn | Shek Kong |
| Kai Tak | Tseung Kwan O | So Uk Estate |
| King’s Park | Tsing Yi Shell Oil Depot | Tai Mei Tuk |
| Lamma Island | TMGO | Tap Shek Kok |
| Lau Fau Shan | Waglan Island | Tung Chung |
| Ngong Ping | Wetland Park | TMR |
| North Point | Wong Chuk Hang | |
| Peng Chau | Tide gauge station | |
| Ping Chau | Rainfall station | Quarry Bay |
| Sai Kung | Hong Kong Observatory | Shek Pik |
| Sha Chau | HKIA | Tsim Bei Tsui |
| Sha Lo Wan | Cheung Chau | Tai Miu Wan |
| Sha Tin | Aberdeen | Tai Po Kau |
| Shek Kong | Fanling | Waglan Island |
| Star Ferry | High Island | − |
| Note: HKIA: Hong Kong International Airport; TMGO: Tuen Mun Government Offices; TMR: Tuen Mun Reservoir. − represents no station. | ||
| Indices | Type | A1 | A2 | A3 | Dn(0.05) | $\hat{D}_n $ | Q |
| Strong-wind index | Gumbel | 0.830 | 3.202 | – | 1.197×10−1 | 8.096×10−2 | 7.077×10−4 |
| Weibull | 0.346 | 0.750 | 1.619 | 1.197×10−1 | 7.185×10−2 | 9.110×10−4 | |
| P-III | 2.233 | 3.469 | 0.373 | 1.197×10−1 | 5.553×10−2 | 5.035×10−4 | |
| Lognormal | 0.157 | −0.266 | 0.486 | 1.197×10−1 | 5.607×10−2 | 4.017×10−4 | |
| Total-rainfall index | Gumbel | 0.904 | 1.607 | – | 1.347×10−1 | 1.255×10−1 | 3.926×10−3 |
| Weibull | 0.118 | 1.293 | 1.285 | 1.347×10−1 | 1.268×10−1 | 1.854×10−3 | |
| P-III | 0.956 | 0.976 | 0.330 | 1.347×10−1 | 7.837×10−2 | 7.015×10−4 | |
| Lognormal | 0.109 | −0.134 | 0.868 | 1.347×10−1 | 9.940×10−2 | 8.670×10−4 | |
| Tide-level index | Gumbel | 0.932 | 6.892 | – | 1.197×10−1 | 9.046×10−2 | 2.268×10−3 |
| Weibull | 0.598 | 0.458 | 2.602 | 1.197×10−1 | 8.894×10−2 | 2.296×10−3 | |
| P-III | 4.183 | 12.718 | 0.678 | 1.197×10−1 | 8.644×10−2 | 1.715×10−3 | |
| Lognormal | −0.187 | 0.168 | 0.131 | 1.197×10−1 | 6.826×10−2 | 8.945×10−4 | |
| Note: $\hat{D}_n $ is the K-S stastic, Dn(0.05) is the K-S tests value, Q is the average sum of deviation squares; the corresponding relationship of the parameters is as follows: (1) Gumbel distribution, A1 and A2 represent location parameter u and scale parameter $\alpha $ respectively; (2) Weibull distribution, A1, A2 and A3 represent the position parameter μ, shape parameter $\sigma $ and scale parameter γ, respectively; (3) P-III distribution, A1, A2 and A3 represent $\alpha $, scale parameter $\ \beta_0 $ and μ, respectively; (4) lognormal distribution, A1, A2 and A3 represent the parameters a, μ and $\sigma $, respectively. − represents no data. | |||||||
| Copula function | $\hat{\theta} $ | $\hat{D}_n $ | Dn(0.5) | AIC |
| Frank | 2.0497 | 0.1010 | 0.1197 | −872.3280 |
| Clayton | 0.3392 | 0.0845 | 0.1197 | −876.6350 |
| G-H | 1.2018 | 0.0795 | 0.1197 | −927.4228 |
| Gaussian | (0.4065, 0.4913, −0.0173) | 0.1108 | 0.1197 | −856.0544 |
| Note: AIC: Akaike information criterion. | ||||
| Return period/a | 10 | 25 | 50 | 100 | 120 | 200 |
| Strong-wind index (VI) | 2.2667 | 2.6600 | 2.9713 | 3.2958 | 3.3834 | 3.6343 |
| Total-rainfall index (RI) | 4.3139 | 5.2448 | 5.9498 | 6.6554 | 6.8411 | 7.3614 |
| Tide-level index (LI) | 1.3672 | 1.4404 | 1.4927 | 1.5428 | 1.5557 | 1.5912 |
| a1 | a2 | ||||||||||
| 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | |
| 0 | 2.9582 | 3.1114 | 3.1685 | 3.1654 | 3.1478 | 3.1167 | 3.0692 | 3.0319 | 2.9920 | 2.9448 | 2.9029 |
| 0.1 | 3.0894 | 3.1910 | 3.2053 | 3.1986 | 3.1768 | 3.1472 | 3.1082 | 3.0680 | 3.0265 | 2.9850 | – |
| 0.2 | 3.1621 | 3.2068 | 3.2158 | 3.2074 | 3.1921 | 3.1658 | 3.1327 | 3.0957 | 3.0609 | – | – |
| 0.3 | 3.1748 | 3.1962 | 3.2070 | 3.2026 | 3.1908 | 3.1722 | 3.1450 | 3.1138 | – | – | – |
| 0.4 | 3.1436 | 3.1747 | 3.1898 | 3.1908 | 3.1856 | 3.1708 | 3.1506 | – | – | – | – |
| 0.5 | 3.1038 | 3.1459 | 3.1660 | 3.1732 | 3.1709 | 3.1638 | – | – | – | – | – |
| 0.6 | 3.0631 | 3.1108 | 3.1374 | 3.1499 | 3.1553 | – | – | – | – | – | – |
| 0.7 | 3.0353 | 3.0734 | 3.1058 | 3.1234 | – | – | – | – | – | – | – |
| 0.8 | 2.9836 | 3.0339 | – | – | – | – | – | – | – | – | – |
| 0.9 | 2.9486 | – | – | – | – | – | – | – | – | – | – |
| 1.0 | 2.8709 | – | – | – | – | – | – | – | – | – | – |
| Note: – represents no data. | |||||||||||
| a1 | a2 | ||||||||||
| 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | |
| 0 | 2.4992 | 2.8912 | 2.9532 | 2.9468 | 2.9284 | 2.8894 | 2.8349 | 2.7955 | 2.7540 | 2.7104 | 2.6743 |
| 0.1 | 2.8705 | 2.9672 | 2.9923 | 2.9795 | 2.9484 | 2.9148 | 2.8728 | 2.8333 | 2.7900 | 2.7488 | – |
| 0.2 | 2.9492 | 2.9890 | 3.0008 | 2.9857 | 2.9585 | 2.9358 | 2.8985 | 2.8619 | 2.8177 | – | – |
| 0.3 | 2.9470 | 2.9808 | 2.9860 | 2.9834 | 2.9672 | 2.9459 | 2.9140 | 2.8784 | – | – | – |
| 0.4 | 2.9197 | 2.9566 | 2.9701 | 2.9734 | 2.9588 | 2.9440 | 2.9214 | – | – | – | – |
| 0.5 | 2.8811 | 2.9270 | 2.9467 | 2.9545 | 2.9513 | 2.9319 | – | – | – | – | – |
| 0.6 | 2.8353 | 2.8909 | 2.9175 | 2.9297 | 2.9310 | – | – | – | – | – | – |
| 0.7 | 2.7996 | 2.8488 | 2.8844 | 2.9023 | – | – | – | – | – | – | – |
| 0.8 | 2.7610 | 2.8091 | – | – | – | – | – | – | – | – | – |
| 0.9 | 2.7116 | – | – | – | – | – | – | – | – | – | – |
| 1.0 | 2.6584 | – | – | – | – | – | – | – | – | – | – |
| Note: – represents no data. | |||||||||||
DownLoad:
