# Parametric Vine Copula Framework in the Trivariate Probability Analysis of Compound Flooding Events

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## Abstract

**:**

## 1. Introduction

## 2. Methodological Framework

#### 2.1. Trivariate Joint Probability Analysis via a Fully Nested Copula Framework

#### 2.2. A 3D Vine Copula Framework for the Trivariate Analysis

#### 2.3. Generating Random Observations from the Selected D-Vine Copula Structure

- Step 1: Estimating the first random variable, ${\mathrm{m}}_{1}={\mathrm{s}}_{1}$
- Step 2: Estimating the second random variable, ${\mathrm{m}}_{2}={\mathrm{K}}_{2|1}^{-1}({\mathrm{s}}_{2}|{\mathrm{m}}_{1})$, where ${\mathrm{K}}_{2|1}\left({\mathrm{m}}_{2}{|\mathrm{m}}_{1}\right)=\frac{\partial {\mathrm{C}}_{12}\left({\mathrm{m}}_{1},{\mathrm{m}}_{2}\right)}{\partial {\mathrm{m}}_{1}}$
- Step 3: Estimating the third random variable, ${\mathrm{m}}_{3}={\mathrm{K}}_{3|12}^{-1}({\mathrm{s}}_{3}|{\mathrm{m}}_{1},{\mathrm{m}}_{2})$, where ${\mathrm{K}}_{3|21}\left({\mathrm{m}}_{3}{|\mathrm{m}}_{1},{\mathrm{m}}_{2}\right)=\frac{\frac{{\partial}^{2}}{\partial {\mathrm{m}}_{1}\partial {\mathrm{m}}_{2}}{\mathrm{C}}_{123}\left({\mathrm{m}}_{1},{\mathrm{m}}_{2},{\mathrm{m}}_{3}\right)}{\frac{{\partial}^{2}}{\partial {\mathrm{m}}_{1}\partial {\mathrm{m}}_{2}}{\mathrm{C}}_{12}\left({\mathrm{m}}_{1},{\mathrm{m}}_{2}\right)}$
- Step 4: Finally, the corresponding value of the flood characteristics (rainfall (R), storm surge (SS), and river discharges (RD)) are estimated by taking the inverse of the univariate marginal cumulative distribution function, ${\mathrm{F}}^{-1}\left({\mathrm{m}}_{1}\right)=$ Simulated rainfall (R) observations; ${\mathrm{F}}^{-1}\left({\mathrm{m}}_{2}\right)=$ Simulated storm surge (SS) observation; ${\mathrm{F}}^{-1}\left({\mathrm{m}}_{3}\right)=$ Simulated river discharge (RD) observations.

#### 2.4. Probabilistic Analysis of Compound Flooding Events

#### 2.4.1. Joint and Conditional Return Periods

#### 2.4.2. Hydrologic Risk Evaluation of Flood Events

## 3. Application

#### 3.1. Study Area and Delineation of Compound Flooding Characteristics

^{3}/s, flowing for 1375 km and finally draining into the Strait of Georgia. This study is investigating the compound impact of storm surge, rainfall and river discharge events on flood risk in the coastal region. To achieve this task, the present work search for the annual maximum 24-h rainfall events and takes the highest storm surge and river discharge within a time range of ±1 days.

#### 3.2. Marginal Behaviour of the Targeted Flood Characteristics

#### 3.3. Incorporating Vine Copula in the Trivariate Flood Dependence Structure

#### 3.3.1. Approximating Bivariate Joint Dependence Structure via 2D Copulas

#### 3.3.2. Constructing the D-Vine Copula Structure in the Trivariate Analysis

#### 3.4. Assessing the Hydrologic Risk of Compound Flooding Events

#### 3.4.1. Primary OR and AND Joint Return Period

^{3}/s, the bivariate OR- and AND-joint RP is ${\mathrm{T}}^{\mathrm{OR}}=54.98{\mathrm{years}\mathrm{and}\mathrm{T}}^{\mathrm{AND}}=551.45\mathrm{years}$ (between flood pair rainfall–storm surges), ${\mathrm{T}}^{\mathrm{OR}}=51.80{\mathrm{years}\mathrm{and}\mathrm{T}}^{\mathrm{AND}}=1435.13\mathrm{years}$ (between flood pair rainfall–river discharge), and ${\mathrm{T}}^{\mathrm{OR}}=64.35{\mathrm{years}\mathrm{and}\mathrm{T}}^{\mathrm{AND}}=224.15\mathrm{years}$ (between flood pair storm surges–river discharge). Similarly, for a 100 year flood event with above mentioned flood characteristics (refer to Table 5), the trivariate OR- and AND-joint RPs is ${\mathrm{T}}^{\mathrm{OR}}=43.90{\mathrm{years}\mathrm{and}\mathrm{T}}^{\mathrm{AND}}=1280\mathrm{years}$. From the above explanation, it could be more practical to account for both OR- and AND cases of trivariate joint return periods. It must be observed from the estimated values, refer to Table 5, how much it could be effective when simultaneously integrating the joint distribution behaviour of the above flood variable instead of just considering pairwise dependency modelling. Accounting for either AND- or OR-joint cases would be problematic in the flood risk analysis; in other words, their importance will solely depend on the nature of the problem. Otherwise, it might underestimate or overestimate the hydrologic risk associated with compound flooding (CF) events.

#### 3.4.2. Conditional Joint Return Periods of CF Events

^{3}/s (25th percentile)), 41.86 years (when river discharge ≤ 1615 m

^{3}/s (50th percentile)), 35.25 years (when river discharge ≤ 2162.5 m

^{3}/s (75th percentile)), and 29.65 years (when river discharge ≤ 3100 m

^{3}/s (90th percentile)), and so on.

^{3}/s (25th percentile for both variables)), 43.25 years (when storm surge ≤ 0.23 m and river discharge ≤ 1615 m

^{3}/s (50th percentile for both variables)), and 29.97 years (when storm surge ≤ 0.45 m and river discharge ≤ 3100 m

^{3}/s (90th percentile for both variables)), and so on. Therefore, it is observed that the return period decreases with an increase in the percentile values of both conditional variables (storm surge and river discharge).

^{3}/s (99th percentile value), the return period decreases with an increase in the percentile value of the storm surges. For instance, in a flood event with flood characteristics, rainfall = 145.8 m and river discharge = 5377 m

^{3}/s (constant), the return period is 40.70 years (when storm surge ≤ 0.06 m), 32.57 years (when storm surge ≤ 0.23 m), and 30.71 years (when storm surge ≤ 0.35 m). In conclusion, the RPs are higher at the lower value of storm surge than the lower storm surge for the same specified values of rainfall events. Also, from Figure 7, the estimated RP is lower at a higher conditional river discharge with a constant storm surge (storm surge = 0.62 m, 99th percentile value) than the lower river discharge for the same specified values of rainfall events. For instance, in a flood event with flood characteristics, rainfall = 145.8 m and storm surge = 0.62 m (constant, 99th percentile value), the return period are 47.26 years (when river discharge ≤ 1085 m

^{3}/s), 41.94 years (when river discharge ≤ 1615 m

^{3}/s), and 35.32 years (when river discharge ≤ 2162.5 m

^{3}/s), and so on.

#### 3.4.3. Analysing the Hydrologic Risk of Flooding Events

## 4. Research Summary and Conclusions

- No significant trend and serial correlation are identified within the time series of annual maximum 24-h rainfall and maximum river discharge (time interval = ±1 day). Moreover, both series exhibited homogenous behaviour. The maximum storm surge (time interval = ±1 day) showed a significant time trend and non-homogenous behaviour.
- The association among these mutually correlated flood variables is examined and used for dependency structure modelling of 3D copulas. The graphical and analytical investigation found that the dependence structure is statistically significant with positive dependency. Finally, the copula-based methodology is adopted for risk analysis of compound events.
- Firstly, BB7, Gumbel–Hougaard (G–H), and Survival BB7 copula are the most appropriate for describing dependence structures for flood pair rainfall–storm surge, storm surge–river discharge, and rainfall–river discharge, respectively. Besides this, upper tail dependence coefficient assessments confirm that the selected 2D copulas capture the extreme of observed data well. The copula dependence parameters are estimated using the maximum pseudo-likelihood (MPL) estimation procedure.
- Three different forms of the D-vine copula are constructed by the permutation of a conditioning variable in the first tree (Tree 1, refer to Figure 3). It is observed that developing a vine structure by changing the location of the conditional variable facilitates high flexibility and is much more practical. The best-fitted D-vine structure is selected by comparing the estimated AIC, BIC, and model LL values. The D-vine structure 1 with river discharge as a conditioning variable is the best. The performance of the selected D-vine structure is compared further with frequently used asymmetric copulas analytically and graphically. The selected D-vine copula structure 1 outperforms asymmetric copulas and is thus employed in estimating trivariate JCDFs and their associated joint and conditional return periods. The D-vine framework can approximate heterogeneous dependency of compound events much better than asymmetric FNA structure due to the conditional mixing approach. In reality, assigning a fixed trivariate structure to the given observation is not a comprehensive way of constructing joint density; the given flood characteristics can exhibit a different strength of dependency between them. Table 4 points out the reliability and suitability of the incorporated vine framework. The selected D-vine structure regenerates the dependence structure of historical flood characteristics more effectively than the FNA framework because of the minimum gap observed between theoretical and empirical Kendal’s $\tau $ correlation measures.
- The return periods for trivariate and bivariate OR- and AND-joint cases and univariate cases are estimated, and comparative analysis is performed. It is concluded that the AND-joint case produces a higher return period than the OR-joint case for the same flood variables. Estimating trivariate return periods of compound events is vital to understand the risk of flood extreme and their magnitude of influence if they occur simultaneously. The return period’s importance depends solely on the nature of the undertaken problem. The importance of different return periods cannot be interchanged, and it is not easy to select them consistently. The appropriate choice of return period can depend on the impact of design variable quantiles. Besides the importance of joint RP, the significance of the conditional joint return periods is often crucial in water infrastructure design. It is found that, at the lower value of both conditional variables, storm surge and river discharge, return periods are higher than those obtained at a lower value of the above conditional variables for the same specified value of the rainfall events. Also, trivariate return periods of rainfall and storm surges, conditional to the river discharge series, increase with a decrease in the value of the conditional variable (river discharge). In addition, the return periods of one variable conditioning to the second variable with the constant value of the third variable are also estimated. In summary, the trivariate return period of rainfall events decreases with an increase in the conditional variable storm surge at the fixed value of river discharge events. Similarly, the trivariate return period of rainfall events is lower at a higher value of river discharge events with a constant value of storm surge events.
- The conditional return periods for bivariate joint cases are also estimated. It is observed that higher return periods can result in higher rainfall events when conditioning to storm surge events and vice-versa. It is also inferred that higher return periods are obtained when conditioned to rainfall events than when considering storm surge events as a conditioning variable. Similarly, when observing the conditional relationship between river discharge and rainfall events, return periods of river discharge (or rainfall) events are inversely proportional to the percentile value of rainfall (or river discharge) events. It is also observed that higher return periods have resulted when conditioning to rainfall events than river discharge series.
- The estimated trivariate and bivariate joint CDFs are used further to assess the risk of failure associated with trivariate (and bivariate) return periods. It is concluded that the failure probability would be an underestimation if the trivariate joint probability analysis is ignored in compounding the collective impact of the selected flood variables. The trivariate flood events produced higher FP than the bivariate (or univariate) events. The investigation also revealed that trivariate (also bivariate) hydrologic risk decreases with increased return periods. At the same time, FP increases with the increase in the service lifetime of the water infrastructure. Changes in the bivariate hydrologic risk following rainfall events in differently designed storm surges and river discharge are also examined, derived from CDF of best-fitted 2D copulas. Both designed events are considered for different RPs (refer to Figures S17a–c and S18a–c), and three different project design lifetimes are considered (e.g., 100 years, 50 years, and 30 years).

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## List of Important Symbols and Acronyms

CF | Compound Flooding |

CDF | Cumulative Distribution Function |

JCDF | Joint Cumulative Distribution Function |

PDF | Probability Density Function |

FNA | Fully Nested Archimedean |

FP | Failure Probability |

PCC | Pair-copula Construction |

RP | Return Period |

MLE | Maximum Likelihood Estimation |

MPL | Maximum Pseudo-likelihood |

${\mathsf{\phi}}_{1}$${\mathsf{\phi}}_{2}$ | Laplace Transforms |

$\circ $ | Composite Function |

$\varnothing \left(\xb7\right)$ | Generator Function of Archimedean Copulas |

${\varnothing}^{-1}$ | Inverse of Archimedean’s Copula Generator |

$\mathsf{\theta}$ | Copula Dependence Parameter |

C-vine | Canonical Vine |

D-vine | Drawable Vine |

CCDF | Conditional Cumulative Distribution Function |

R | Rainfall or Annual Maximum 24-h Rainfall |

SS | Storm Surge or Maximum Storm Surge (Time interval = ±1 days) |

RD | River Discharge or Maximum River Discharge (Time interval = ±1 days) |

KDE | Kernel Density Estimation |

UTDC | Upper Tail Dependence Coefficient |

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**Figure 2.**Schematic diagram of the parametric D-vine copula construction in modelling trivariate flood dependence structure. Note: the above diagram only illustrates river discharge (RD) series as a conditioning variable (centred variable in the above D-vine structure). The other D-vine structures can be obtained by placing rainfall or storm surge as a conditioning variable.

**Figure 3.**Schematic diagram of different ways of constructing the D-vine copula structure in this study (

**a**) Case 1: river discharge (variable 3) as a conditioning variable (

**b**) Case 2: storm surge (variable 2) as a conditioning variable (

**c**) Case 3: rainfall series (variable 1) as a conditioning variable.

**Figure 4.**Vine tree plot of selected D-vine copula structure (D-vine structure 1 (Case 1), refer to Table 2).

**Figure 5.**Trivariate conditional joint return periods (in years) of flood characteristics (for case ${\mathrm{T}}_{\mathrm{RAIN}|\mathrm{STORM}\mathrm{SURGES}\le \mathrm{storm}\mathrm{surge}\left(\mathrm{threshold}\right),\mathrm{RIVER}\mathrm{DISCHARGE}\le \mathrm{river}\mathrm{discharge}\left(\mathrm{threshold}\right)}$) of rainfall at a given storm surge and rainfall events.

**Figure 6.**Trivariate conditional joint return periods (in years) of rainfall given various percentile value of storm surge with constant river discharge, ${\mathrm{T}}_{\mathrm{RAIN}|\mathrm{STORM}\mathrm{SURGE}\le \mathrm{storm}\mathrm{surge}\left(\mathrm{threshold}\right),}$${}_{\mathrm{RIVER}\mathrm{DISCHARGE}=\mathrm{constant}}$.

**Figure 7.**Trivariate conditional joint return periods (RPs) of rainfall given various percentile value of river discharge with constant storm surge value, ${\mathrm{T}}_{\mathrm{RAIN}|\mathrm{RIVER}\mathrm{DISCHARGE}\le \mathrm{river}\mathrm{discharge}\left(\mathrm{threshold}\right),}$${}_{\mathrm{STORM}\mathrm{SURGE}=\mathrm{constant}}$.

**Figure 8.**Hydrologic risk assessments of CF events for different return periods (

**a**) 100 years, (

**b**) 50 years, (

**c**) 20 years, (

**d**) 10 years, and (

**e**) 5 years.

Dependency Measure Statistics | Compound Flood Variables | ||
---|---|---|---|

Annual Maximum 24-h Rainfall-Maximum Storm Surge (Time Interval = ±1 Days) | Maximum Storm Surge (Time Interval = ±1 Days)—Maximum River Discharge (Time Interval = ±1 Days) | Annual Maximum 24-h Rainfall-Maximum River Discharge (Time Interval = ±1 Days) | |

Pearson | 0.301 | 0.469 | 0.118 |

Kendall | 0.208 | 0.341 | 0.094 |

Spearman | 0.297 | 0.504 | 0.122 |

**Table 2.**Overall summary table of fitted 3D vine copula frameworks constructed via permutation of conditioning variables (flood variable centred at the selected D-vine structure).

Vine Structure (Conditioning Variable) | Tree Level | Flood Attribute Pairs | Most Parsimonious or Best-Fitted Copula | $\mathbf{Copula}\mathbf{Dependence}\mathbf{Parameters}\left(\theta \right)$ | Log-Likelihood (LL) | Akaike Information Criterion (AIC) | Bayesian Information Criterion (BIC) | |
---|---|---|---|---|---|---|---|---|

Case 1 (1-3-2) * | $\mathrm{D}-\mathrm{vine}\mathrm{structure}1(\mathrm{Variable}3\mathrm{or}\mathrm{Maximum}\mathrm{river}\mathrm{discharge}(\mathrm{Time}\mathrm{interval}=\pm 1days)$ is placed in the center) | Tree 1 | 1-3 (Rain-River discharge) | Survival BB7 (Rotated BB7 180 degrees) | $\theta \left(theta\right)=par=1.0957$$;\delta \left(delta\right)=par2=0.1504$ | 9.53788 | −10.87909 | −3.564521 |

3-2 (Storm surge-River discharge) | Gumbel | $\theta \left(theta\right)$= par= 1.554 | ||||||

Tree 2 | 12|3 | Clayton | $\theta \left(theta\right)=par=$0.3688 | |||||

Case 2 (1-2-3) | $\mathrm{D}-\mathrm{vine}\mathrm{structure}2(\mathrm{Variable}2\mathrm{or}\mathrm{Maximum}\mathrm{storm}\mathrm{surge}(\mathrm{Time}\mathrm{interval}=\pm 1\mathrm{days})$ placed in the center) | Tree 1 | 1-2 (Rain—Storm surge) | BB7 (Joe-Clayton) | $\mathsf{\theta}\left(\mathrm{theta}\right)=\mathrm{par}=1.153$$;\mathsf{\delta}\left(\mathrm{delta}\right)=\mathrm{par}2=0.433$ | 8.194824 | −6.389647 | 2.75356 |

2-3 (Storm surge—River discharge) | Gumbel | $\theta \left(\mathrm{theta}\right)$= par = 1.554 | ||||||

Tree 2 | 13|2 | Rotated BB8 270 degrees | $\mathsf{\theta}\left(\mathrm{theta}\right)$= par = −1.083$\mathsf{\delta}$(delta) = par2 = −1.000 | |||||

Case 3 (2-1-3) | D-vine structure 1 (Variable 1 or Annual maximum 24-h rainfall placed in the center) | Tree 1 | 2-1 (Storm surge-Rain) | BB7 (Joe-Clayton) | $\mathsf{\theta}\left(\mathrm{theta}\right)=\mathrm{par}=1.153$$;\mathsf{\delta}\left(\mathrm{delta}\right)=\mathrm{par}2=0.433$ | 9.34687 | −8.693741 | 0.449466 |

3-1 (River discharge-rainfall) | Survival BB7 (Rotated BB7 180 degrees) | $\mathsf{\theta}\left(\mathrm{theta}\right)=\mathrm{par}=1.0957$$;\mathsf{\delta}\left(\mathrm{delta}\right)=\mathrm{par}2=0.1504$ | ||||||

Tree 2 | 23|1 | Frank | $\mathsf{\theta}\left(\mathrm{theta}\right)=\mathrm{par}=$3.689 |

**Table 3.**Estimation of dependence parameters of the fitted 3D FNA copulas and comparing their performance with selected best-fitted D-vine copula (D-vine structure 1).

Trivariate Distribution Framework | Copula Function | Log-Likelihood (LL) | Akaike Information Criterion (AIC) | Bayesian Information Criterion (BIC) | |
---|---|---|---|---|---|

Parametric 3D Vine or Pair-Copula Construction (PCC) | D-Vine Copula * | 9.53788 | −10.87909 | −3.564521 | |

Estimated Parameters | Log-Likelihood (LL) | ||||

Asymmetric or fully nested Archimedean (FNA) framework (parametric marginals) | Gumbel copula | ${\mathsf{\theta}}_{1}=1.2;{\mathsf{\theta}}_{2}=1.54$ | 9.063 | −0.408 | 3.248 |

Clayton copula | ${\mathsf{\theta}}_{1}=0.17;{\mathsf{\theta}}_{2}=0.59$ | 5.627 | 0.544 | 4.201 | |

Frank copula | ${\mathsf{\theta}}_{1}=1.51;{\mathsf{\theta}}_{2}=3.46$ | 8.594 | −0.302 | 3.355 |

**Table 4.**Examining the reliability of the developed 3D parametric vine copula structure vs. 3D FNA Gumbel–Hougaard copula by comparing Kendall’s $\mathsf{\tau}$ correlation coefficient estimated using the generated random samples (size N = 1000) obtained from the above-selected model with Empirical Kendall’s $\tau $ values estimated from historical observations.

Flood Attribute Pairs | $\mathbf{Kendall}\u2019\mathbf{s}\mathit{\tau}\mathbf{Estimated}\mathbf{from}\mathbf{Historical}\mathbf{Observations}(\mathbf{Empirical}\mathbf{Estimates})$ | $\mathbf{Kendall}\u2019\mathbf{s}\mathit{\tau}\mathbf{Estimated}\mathbf{from}\mathbf{Best}-\mathbf{Fitted}\mathbf{Fully}\mathbf{Nested}\mathbf{Gumbel}\mathbf{Copula}\mathbf{with}\mathbf{Parametric}\mathbf{Marginals}(\mathbf{Theoretical}\mathbf{Estimates})$ | $\mathbf{Kendall}\u2019\mathbf{s}\mathit{\tau}\mathbf{Correlation}\mathbf{Coefficient}\mathbf{Estimated}\mathbf{from}\mathbf{the}\mathbf{Best}-\mathbf{Fitted}\mathbf{D}-\mathbf{Vine}\mathbf{Copula}\mathbf{Structure}(\mathbf{Sample}\mathbf{Size}=\mathbf{N}=1000)$ |
---|---|---|---|

Annual maximum 24-h rainfall (mm)-Maximum storm surge (m) (Time interval = $\pm 1\mathrm{days})$ | 0.207 | 0.192 | 0.196 |

Maximum storm surge (m) (Time interval = $\pm 1\mathrm{days})-{\mathrm{Maximum}\mathrm{river}\mathrm{discharge}(\mathrm{m}}^{3}/\mathrm{s})(\mathrm{Time}\mathrm{interval}=\pm 1\mathrm{days})$ | 0.341 | 0.358 | 0.342 |

Annual maximum 24-h rainfall (mm)—Maximum river discharge (m^{3}/s) (Time interval = $\pm 1\mathrm{days})$ | 0.093 | 0.156 | 0.120 |

**Table 5.**Univariate, bivariate and trivariate joint return periods (JRPs) estimated for various combinations of selected flood characteristics.

Flood Quantiles Estimated Using the Inverse of the Best-Fitted Marginal Cumulative Distribution Functions (CDFs) | Bivariate Joint Return Periods (JRPs) | Trivariate Joint Return Periods (JRPs) Estimated Using the Best-Fitted D-Vine Copula Structure (Case 1, Refer to Table 2) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Return Period (RPs) (Years), T | Annual Maximum 24-h Rainfall (R) (mm) | Maximum Storm Surge (m) (SS) (Time Interval = ±1 Days) | Maximum River Discharge (RD) (m^{3}/s) (Time Interval = ±1 Days)) | $\mathbf{OR}-\mathbf{JRP},{\mathbf{T}}_{\mathbf{R}\mathbf{S}}^{\mathbf{O}\mathbf{R}}$ | $\mathbf{AND}-\mathbf{JRP}{\mathbf{T}}_{\mathbf{R}\mathbf{S}}^{\mathbf{A}\mathbf{N}\mathbf{D}}$ | $\mathbf{OR}-\mathbf{JRP},{\mathbf{T}}_{\mathbf{R}\mathbf{R}\mathbf{D}}^{\mathbf{O}\mathbf{R}}$ | $\mathbf{AND}-\mathbf{JRP}{\mathbf{T}}_{\mathbf{R}\mathbf{R}\mathbf{D}}^{\mathbf{A}\mathbf{N}\mathbf{D}}$ | $\mathbf{OR}-\mathbf{JRP},{\mathbf{T}}_{\mathbf{S}\mathbf{S}\mathbf{R}\mathbf{D}}^{\mathbf{O}\mathbf{R}}$ | $\mathbf{AND}-\mathbf{JRP},{\mathbf{T}}_{\mathbf{S}\mathbf{S}\mathbf{R}\mathbf{D}}^{\mathbf{A}\mathbf{N}\mathbf{D}}$ | $\mathbf{OR}-\mathbf{JRP},{\mathbf{T}}_{\mathbf{R}\mathbf{S}\mathbf{S}\mathbf{R}\mathbf{D}}^{\mathbf{O}\mathbf{R}}$ | $\mathbf{AND}-\mathbf{JRP},{\mathbf{T}}_{\mathbf{R}\mathbf{S}\mathbf{S}\mathbf{R}\mathbf{D}}^{\mathbf{A}\mathbf{N}\mathbf{D}}$ |

5 | 102.90 | 0.145 | 2412.582 | 3.01 | 14.58 | 2.91 | 17.43 | 3.40 | 9.40 | 2.380 | 19.31 |

10 | 120.68 | 0.221 | 3374.429 | 5.72 | 39.47 | 5.52 | 52.75 | 6.60 | 20.57 | 4.40 | 50.98 |

20 | 138.75 | 0.284 | 4685.815 | 11.18 | 94.71 | 10.71 | 150.38 | 13.01 | 43.13 | 8.52 | 133.76 |

50 | 163.73 | 0.354 | 7215.067 | 27.59 | 265.73 | 26.17 | 557.63 | 32.26 | 110.99 | 21.48 | 964.97 |

100 | 183.68 | 0.401 | 10,005.067 | 54.98 | 551.45 | 51.80 | 1435.13 | 64.35 | 224.15 | 43.90 | 1280 |

200 | 204.69 | 0.444 | 13,886.119 | 109.78 | 1121.95 | 102.87 | 3575.25 | 128.52 | 450.51 | 79.19 | 1002.40 |

500 | 234.24 | 0.497 | 21,446.821 | 274.22 | 2830.45 | 255.57 | 11,454.75 | 321.05 | 1129.56 | 188.38 | 1596.42 |

1000 | 258.04 | 0.533 | 29,822.745 | 548.27 | 5678.59 | 509.45 | 26,954.17 | 641.93 | 2261.42 | 364.00 | 2508.15 |

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**MDPI and ACS Style**

Latif, S.; Simonovic, S.P.
Parametric Vine Copula Framework in the Trivariate Probability Analysis of Compound Flooding Events. *Water* **2022**, *14*, 2214.
https://doi.org/10.3390/w14142214

**AMA Style**

Latif S, Simonovic SP.
Parametric Vine Copula Framework in the Trivariate Probability Analysis of Compound Flooding Events. *Water*. 2022; 14(14):2214.
https://doi.org/10.3390/w14142214

**Chicago/Turabian Style**

Latif, Shahid, and Slobodan P. Simonovic.
2022. "Parametric Vine Copula Framework in the Trivariate Probability Analysis of Compound Flooding Events" *Water* 14, no. 14: 2214.
https://doi.org/10.3390/w14142214