# Uncertainty Analysis of Two Copula-Based Conditional Regional Design Flood Composition Methods: A Case Study of Huai River, China

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Copula Theory

^{2}. F

_{X}(x) and F

_{Y}(y) are the cumulative distribution functions (C.D.Fs) of X and Y, respectively, which are uniformly distributed random variables on [0,1] and are respectively denoted as u and v. According to Sklar’s Theorem [27], a two-dimensional copula can be represented as:

_{X,Y}(x,y) = C[F

_{X}(x),F

_{Y}(y);θ] = C(u,v;θ)

_{X,Y}(x,y) denotes the bivariate C.D.F of X and Y; and C(u,v;θ) represents the unique bivariate copula function with the parameter, θ.

#### 2.2. Regional Design Flood Composition

#### 2.2.1. Basic Concept of Regional Design Flood Composition

#### 2.2.2. Flood Regional Composition Methods Based on Copulas

_{p}, and the corresponding value (y) of the flood variable, Y, at the interval, B, is not fixed. A different y corresponds to a different probability of occurrence. The combination of x

_{p}and the conditional expected value of Y (E(y|x

_{p})) is denoted as the conditional expectation regional design flood composition (CEC), which denotes the average level of different composition schemes. E(y|x

_{p}) can be expressed by Equation (2):

_{p}) is the conditional probability density function (P.D.F), and f(Y ≤ y|X = x

_{p}) = c(u,v)f

_{Y}(y) according to the copula theory; c(u,v) is the density function of C(u,v), and $c\left(u,v\right)={\partial}^{2}C\left(u,v\right)/\partial u\partial v$; f

_{Y}(y) is the P.D.F of Y; and ${F}_{Y}^{-1}(\xb7)$ is the inverse C.D.F of Y.

_{p}, E(y|x

_{p})) will be obtained.

_{p}) (y

_{M}), the combination of x

_{p}and y

_{M}(i.e., (x

_{p}, y

_{M})) is regarded as the conditional most likely regional design flood composition (CMLC).

_{p}) can be derived as:

_{2}·f

_{Y}(y) + c·(f

_{Y}′(y)/f

_{Y}(y)) = 0

_{Y}’(y) are the corresponding derived functions of PDF f

_{Y}(y).

_{p},y

_{M}) can be obtained by solving the nonlinear Equation (4) with the Newton iteration method [33]. Similarly, when the interval flood volumes are taken as the condition, we also can get two corresponding regional design flood composition schemes.

_{p}) be equal to α/2 and 1 − α/2, respectively, where α is the significance level, then the corresponding value of the flood variable, Y, at the interval sub-basin, B, can be described as y

_{L}and y

_{U}, respectively. [y

_{L}, y

_{U}] is regarded as the corresponding confidence interval (CI) given X = x

_{p}, which can quantitatively evaluate the uncertainty associated with the composition schemes estimation and provide a basis to clarify the flood risk. The CEC and CMLC methods together are known as the conditional regional composition method.

#### 2.3. Conditional Copula-Based Parametric Bootstrapping (CC-PB) Procedure

- Fit the marginal distributions and parametric copula function for the original dataset (i.e., X and Y). The parameters of the chosen marginal distributions and copula function are estimated by the L-moment method and maximum pseudo-log-likelihood (ML) method, respectively.
- Predefine N
_{B}bivariate bootstrapping samplings of size n through the usage of the conditional simulation method [29], and then obtain Z* = (X*,Y*) = (x_{ij},y_{ij}) from the bivariate dependence structure via the probability integral transform (PIT) using the fitted parameters of the margins (i = 1,…,n; j = 1,…,N_{B}). - Estimate the parameters of marginal distributions and the parametric copula function of Z* utilizing the same estimation method used for the original dataset, then obtain N
_{B}pairs of F_{j}*(x_{ij},y_{ij}), (i = 1,…,n; j = 1,…,N_{B}). - Identify the CEC and CMLC realizations for different (x,y) pairs by Equations (2) and (4), respectively.
- Utilize these realizations to estimate the bivariate confidence intervals (BCIs) by employing the kernel density estimation (KDE) method [34].

#### 2.4. Metrics for Sampling Uncertainty

_{x}; vertical standard deviation, σ

_{y}; area of 25% BCI, S

_{25%}; area of 50% BCI, S

_{50%}; and area of 75% BCI, S

_{75%}, are utilized to quantify the sampling uncertainty. S

_{25%}, S

_{50%}, and S

_{75%}are highly dependent on the grids contained in the BCIs and are approximated by Riemann sums using the R-package ‘ks’ [34]. The other two metrics are calculated by the following equations:

## 3. Study Area and Data

^{4}km

^{2}, including the mainstream and five major tributaries, still termed as the Huai River Basin (Figure 2b). According to topographic relief and river characteristics, the middle of Huai River can be divided into two interval sections above and below the Zhengyangguan section. The drainage area above the Zhengyangguan section is 8.86 × 10

^{4}km

^{2}, and the length of the stream from Zhengyangguan to the Bengbu section is 119 km, which drains an area of 3.27 × 10

^{4}km

^{2}.

_{30d}) from Zhengyangguan Station and Bengbu Station were collected. The natural flood process of the Zhengyangguan section was routed to the Bengbu section with the Muskingum method [38]. By deducting this routed flood process from the natural flood process of the Bengbu section, we could obtain the flood process of the interval basin between Zhengyangguan and Bengbu (hereinafter referred to as the Zheng-Beng interval). Then, the Zheng-Beng interval, W

_{30d}, could be easily obtained.

_{30d}and Zheng-Beng interval W

_{30d}series. These two series exhibited similar changing variations throughout the 63-year data. The dispersion of these two series was measured by boxplot graphs (Figure 3b), and the symmetry of the boxplot implies that there is a certain level of right skew for both series. Figure 3c illustrates that Zhengyangguan W

_{30d}has a rough linear correlation with the Zheng-Beng interval W

_{30d}, and the Pearson correlation coefficient between the W

_{3}

_{0d}of the two sub-basins is 0.869, which connotes a comparatively marked linear relationship. Nevertheless, the dependence structure of the dataset is supposed to be more complicated than a straightforward linear correlation, which can be more flexibly analyzed based on copulas. Table 2 lists some basic statistical parameters of the two series, which describe the structure and overall situation of the data.

## 4. Results and Discussions

#### 4.1. Selection of Marginal Distribution

_{30d}at each sub-basin, the first step is to choose appropriate marginal distributions. Seven widely-used distributions in hydrology [37,39,40], namely Pearson type III (PE3), three parameters log-normal (LN3), generalized extreme value (GEV), generalized pareto (GP), gamma (GAM), gumbel (GUM), and generalized logistic (GLO), were picked as candidate distributions for flood volume. Figure 4 illustrates the distributions of the W

_{30d}series in each study sub-basin fitted by the seven candidate distributions. The quantile-quantile plot, C.D.F plot, and P.D.F plot are exhibited in this figure.

_{30d}series at each sub-basin. The Cramér–von Mises statistic (w

^{2}) is designed to quantify the distance between the empirical and fitted distributions [42]. Lower w

^{2}values and higher p-values mean a preferable imitative effect.

_{30d}data, yielding the minimum RMSE and AIC values, while MEV is the best fitting one for W

_{30d}at the Zheng-Beng interval. All of the p values listed in Table 3 are quite bigger than 0.05, indicating that seven theoretical distributions are capable of fitting the distributions of W

_{30d}at each sub-basin at the 5% significance level.

#### 4.2. Copula Function Construction

_{30d}series at Zhengyangguan and the Zheng-Beng interval. The contour density plots of three GH copulas (Figure 5) indicate that GH copulas have the advantage to simulate tail dependence, which may be weaker when displayed through the observed dataset. It is proven that improper selection of copula function types will bring significant uncertainty to the estimation of model simulation sequence design values [24,25]. Therefore, selecting and adjusting the best-fit copula function is a decisive step in the fitting process. Table 4 shows the result of the parameters of the three GH copulas using the maximum pseudo-likelihood method and corresponding AIC values. The given dataset cannot easily answer the question of tail dependence, so the upper tail-dependence coefficient, ${\hat{\lambda}}_{U}^{\mathit{CFG}}$, of the copulas estimated with the CFG estimator [43] supports the quantification of the tail dependence of extreme values. The ${\hat{\lambda}}_{U}^{\mathit{CFG}}$ value of the empirical copula is 0.582, slightly smaller than the ${\hat{\lambda}}_{U}^{\mathit{CFG}}$ values calculated straight from the three candidate GH copulas (Table 4). The two indicators’ values illustrated the good performance of the two-parameter GH copula.

_{30d}at each sub-basin. Furthermore, GH2 possesses its own unique features to characterize the high relevance between extraordinary flood volumes at the two study sub-basins.

#### 4.3. CEC and CMLC Point Identification

_{30d}that occurred at the Zhengyangguan section and Zheng-Beng interval, which have statistical basis and could satisfy the inherent disciplines of hydrologic events. For the convenience of subsequent analysis of the uncertainty of marginal distribution selection, an explored experiment combining different margins was carried out. Table 5 lists 13 analyzed combinations.

_{30d}, while PE3, LN3 GEV, GP, GAM, GUM, and GLO were applied for the Zheng-Beng interval W

_{30d}for the uncertainty analysis, which corresponds to the combinations, C1–C7. Analogously, when the design value occurred at the Zheng-Beng interval, the corresponding conditional expected value and conditional most likely value occurred at the Zhengyuangguan section, MEV was selected as a distribution for the Zheng-Beng interval W

_{30d}, PE3, LN3, GEV, GP, GAM, GUM, and GLO were selected as distributions for Zhengyuangguan W

_{30d}, which corresponded to the combinations, C3, C8–C13. Three candidate marginal distributions all passed the Cramér–von Mises test given α = 0.05. Then, GH2 was applied to estimate the CEC and CMLC points. Table 6 lists the CEC and CMLC points determined under different combinations given T = 20, 50, 100 years using Equations (2) and (4) in the study sub-basins. For example, when a 20-year design flood occurs at the Zhengyangguan section, the CEC and CMLC points under C3 are (199.22 × 10

^{8}m

^{3}, 47.10 × 10

^{8}m

^{3}) and (199.22 × 10

^{8}m

^{3}, 48.26 × 10

^{8}m

^{3}), respectively.

#### 4.4. Uncertainty Analysis

#### 4.4.1. Uncertainty Due to the Selection of Margins

_{30d}and Zheng-Beng W

_{30d}were selected, respectively, ranging from the moderate flood volume standard to the extreme one. The estimated CEC and CMLC points and their corresponding 90% CIs under different combinations are exhibited along with the observed data in Figure 6. By comparing these bivariate design realizations, it is expected that the impact of marginal distribution uncertainty will be discovered.

^{8}m

^{3}to 47.55 × 10

^{8}m

^{3}, with the variation ratio of 3.48% compared with C3, while given T = 100 years, the corresponding value ranges from 63.05 × 10

^{8}m

^{3}to 75.46 × 10

^{8}m

^{3}, with a reduction ratio of 13.54%. This phenomenon results from the CEC algorithm and the different fitting performance of selected marginal distributions. Equation (2) indicates that large conditional expected values correspond to large values of cumulative probability. When the cumulative probability of the Zhengyuangguan W

_{30d}event ranges from 0.95 to 0.99, the differences between the fitting performances of three margins increase with the increasing values of the cumulative probability. In Case 1, the corresponding cumulative probability of the conditional expected value of W

_{30d}at the Zheng-Beng interval given T = 100 years (0.9819–0.9854) is larger than that given T = 20 years (0.9235–0.9333), so the amplitude of variation of the former corresponding conditional expected value is larger than the latter. Similar analysis can be conducted for the CMLC scheme. In Case 2, the corresponding cumulative probability of the conditional most likely value of W

_{30d}at the Zheng-Beng interval given T = 100 years ranges from 0.9851 to 0.9876 compared with the ranges from 0.9332 to 0.9370 given T = 20 years. The finding is in line with that of Guo et al. [45] and Zhao et al. [15].

_{30d}at the two sub-basins. Similar results can be found in the rainfall frequency analysis, which implies the negligible impact of different marginal distributions. This is rational owing to the relative errors of the performances among the seven candidate univariate distributions being less than 20%, at least for the univariate return periods regarded in the conducted experiment.

#### 4.4.2. Sampling Uncertainty Caused by the Limited Records

- Three values of T for the W
_{30d}at two sub-basins, respectively, are selected (viz., T = 20, 50, 100 years), ranging from the moderate flood volume standard to the extreme one. Similarly, three values of sample size (sz) are selected (viz., sz = 63, 200, 500). - Here, the selected model combination is C3, listed in Table 5. The CEC and CMLC events for C3 are estimated by fixing the value of sz (or T), and leaving the other variable vary in the corresponding subset. A triple of BCIs (viz., 25%, 50%, 75%) is exhibited under different schemes (sz, T). The larger the BCIs, the greater the uncertainty, and vice versa.
- To judge the plausibility and compare the performance of the two proposed composition methods, the contours of several selected joint probability levels [11] (viz., p-level = 0.99, 0.98, 0.95, 0.90, 0.80) together with the observed data are plotted on the same graphs as a reference.
- Five indexes mentioned above are also calculated (Table 7) to evaluate the sampling uncertainty of the 36 schemes.

_{30d}ranges from 150 to almost 1000 years. Evaluations of the sampling uncertainty also exists in previous studies, such as quantifying the uncertainty of hydrological droughts [15], estimating sampling uncertainty in bivariate flood quantiles estimation [34], and handling the overlap problem of the return periods of the bivariate design events [45]. As the findings of this study and previous research suggests, the large uncertainty was unable to be reduced with the observed dataset and poses a huge challenge for basin development, reservoir design, etc. [26], particularly for the Huai River basin, with high densities in both population and water projects.

_{1}) quantifies the different performance of the methods proposed, while the latter (Un

_{2}) describes the uncertainty related to the limited size of data, including the univariate estimation uncertainty. σ

_{x}and σ

_{y}are the uncertainty estimation indexes for one-dimensional space. When a given flood occurs at the Zhengyangguan section, the role played by σ

_{x}and σ

_{y}seem to measure Un

_{2}and Un

_{1}, respectively, and the contrary is the case that a given flood occurs at the Zheng-Beng interval. However, the CEC and CMLC realizations are exhibited in a two-dimensional plane, Un

_{1}and Un

_{2}together make up the overall sampling uncertainty, and the two-dimensional indexes (i.e., S

_{25%}, S

_{50%}, S

_{75%}) should be better utilized. The five uncertainty metrics for the CMLC method and their variation ratio contrasted with the results of the CEC method with a resampling size of 63 are also presented in Table 7. No matter which sub-basin a given flood occurs in, the CMLC realizations have a smaller uncertainty (the corresponding variation ratios are negative) than CEC realizations when the return period of the given flood is 50 or 100, but in the case that a 20-year flood occurs, the CEC realizations seem to perform better (the corresponding variation ratios are positive). The slight difference between the listed metrics indicates that the CMLC method performs more stably and satisfactorily for large floods, while in considered moderate and small floods, the CEC method is a better choice for uncertainty reduction.

## 5. Conclusions

_{30d}series at its two sub-basins (i.e., Zhengyangguan section, Zheng-Beng interval) were obtained. The copula function was adopted to capture the dependence structure of W

_{30d}at each sub-basin for its flexibility to select any complex marginal distributions. For the purpose of integrated river-basin development, the CEC and CMLC methods were utilized to obtain the design flood combinations. To figure out the sensibility of marginal distribution selection and the influence of sampling uncertainty caused by the limited records on the two composition methods, a corresponding comprehensive experiment was designed. Here, to clarify the former uncertainty source, seven candidate univariate distributions were combined according to certain rules to produce 13 combinations for fitting the W

_{30d}series at each sub-basin. For the quantification of the latter uncertainty source, the CC-PB procedure was developed and five evaluation indexes were calculated. The following conclusions have been drawn from this study:

_{30d}at the Zhengyangguan section, and MEV for the W

_{30d}at the Zheng-Beng interval. Taking the overall dependence structure and tail dependence into account, GH2 was awarded as the most appropriate model.

_{75%}of two regional design flood composition events covered more than one p-level curve. Consequently, it leads to an undervaluation or overvaluation of the risk related to hydrological designs [23].

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**(

**a**) Sketch map of a typical regional flood composition issue. (

**b**) Location of Bengbu and Zhengyangguan hydrological stations in the Huai River basin.

**Figure 3.**(

**a**) Zhengyangguan W

_{30d}and Zheng-Beng interval W

_{30d}series. (

**b**) Boxplot graph of Zhengyangguan W

_{30d}and Zheng-Beng interval W

_{30d}series. (

**c**) Scatter plot of Zhengyangguan W

_{30d}and Zheng-Beng interval W

_{30d}together with linear regression line and histogram of marginal distributions on the side panels.

**Figure 4.**Fitting distributions to Zhengyangguan W

_{30d}and Zheng-Beng interval W

_{30d}: The quantile-quantile plot (

**left**), C.D.F plot (

**middle**), and P.D.F plot (

**right**).

**Figure 5.**Contours of copula densities scaled to the standard normal distribution (N(0,1)) margins and plots of empirical Pickand’s dependence function compared with three fitted ones.

**Figure 6.**CEC and CMLC points under combinations of specified marginal distributions. Case 1 and Case 3 denote the CEC and CMLC points in the condition that the design flood (T = 20, 50, 100) occurs at the Zhengyuanguan section, respectively, while Case 2 and Case 4 denote the CEC and CMLC points in the condition that the design flood occurs at the Zheng-Beng interval, respectively. For better visualization, the horizontal and vertical coordinates of Case 2 and Case 4 have been exchanged respectively.

**Figure 7.**Comparison of CEC realizations under different schemes when the designed values occur in the Zhengyangguan section.

**Figure 8.**Comparison of CEC realizations under different schemes when the designed values occur in the Zheng-Beng interval.

**Figure 9.**Comparison of CMLC realizations under different schemes when the designed values occur in the Zhengyangguan section.

**Figure 10.**Comparison of CMLC realizations under different schemes when the designed values occur in the Zheng-Beng interval.

Name | Descriptions |
---|---|

Symmetric GH copula | $C\left(u,v;\theta \right)=\mathrm{exp}\left\{-{\left[{\left(-\mathrm{log}u\right)}^{\theta}+{\left(-\mathrm{log}v\right)}^{\theta}\right]}^{1/\theta}\right\},\theta \in [1,+\infty )$ |

Two-para GH copula | $C\left({u,v;\beta}_{1},{\beta}_{2}\right)={\left\{{\left[{\left({u}^{{-\beta}_{2}}-1\right)}^{{\beta}_{1}}+{\left({v}^{{-\beta}_{2}}-1\right)}^{{\beta}_{1}}\right]}^{1/{\beta}_{2}}+1\right\}}^{-1/{\beta}_{2}},$ ${\beta}_{1}\in [1,+\infty ){,\text{}\beta}_{2}\in [0,+\infty )$ |

Asymmetric GH copula | $C\left({u,v;\theta ,\pi}_{2}{,\pi}_{3}\right)=\mathrm{exp}\left[-A\left({-\mathrm{log}u,\text{}-\mathrm{log}v;\theta ,\pi}_{2}{,\pi}_{3}\right)\right],$ $A\left({x,y;\theta ,\pi}_{2},{\pi}_{3}\right)={\left[{\left({\pi}_{2}x\right)}^{\theta}+{\left({\pi}_{3}y\right)}^{\theta}\right]}^{1/\theta}+(1-{\pi}_{2})x+(1-{\pi}_{3})y,$ $\theta \in [1,+\infty ),{\pi}_{2}\in \left(0,1\right),{\pi}_{3}\in \left(0,1\right)$ |

u∈[0, 1], v∈[0, 1] ${\theta ,\text{}\beta}_{1}{,\text{}\beta}_{2}{,\text{}\pi}_{2}{,\text{}\pi}_{3}$: Copula parameter C: Copula function A: Marshall-Olkin copulas |

Zhengyangguan W_{30d} (10^{8} m^{3}) | Zheng-Beng Interval W_{30d} (10^{8} m^{3}) | |
---|---|---|

[Min, Max] | [12.95, 313.47] | [2.72, 74.29] |

Median | 75.27 | 19.22 |

Mean | 85.76 | 22.51 |

Standard deviation | 57.75 | 15.39 |

Skewness | 1.32 | 1.45 |

Kurtosis | 2.22 | 2.11 |

Interquartile range | 65.04 | 15.66 |

Region | Series | Functions | Parameters | CvM Test | RMSE | AIC | ||
---|---|---|---|---|---|---|---|---|

Name | Estimated Value | w^{2} | p | |||||

Zhengyangguan | W_{30d} | PE3 | [α, β, γ] | [1.921, 0.024, 4.870] | 0.027 | 0.985 | 2.881 | 668.32 |

LN3 | [μ_{log}, σ_{log}, ζ] | [4.599, 0.497, −26.673] | 0.030 | 0.978 | 3.003 | 672.85 | ||

MEV | [ξ, α, κ] | [58.042, 40.019, −0.105] | 0.034 | 0.964 | 3.013 | 674.06 | ||

GP | [ξ, α, κ] | [16.994, 84.403, 0.227] | 0.042 | 0.924 | 3.214 | 673.34 | ||

GAM | [β, α] | [39.100, 2.193] | 0.028 | 0.983 | 3.085 | 669.58 | ||

GUM | [ξ, α] | [60.053, 44.544] | 0.051 | 0.869 | 3.324 | 674.90 | ||

GLO | [ξ, α, κ] | [73.944, 28.046, −0.239] | 0.049 | 0.887 | 3.119 | 676.88 | ||

Zheng-Beng Interval | W_{30d} | PE3 | [α, β, γ] | [1.434, 0.077, 3.899] | 0.037 | 0.949 | 2.138 | 500.80 |

LN3 | [μ_{log}, σ_{log}, ζ] | [3.064, 0.579, −2.801] | 0.024 | 0.993 | 2.084 | 489.11 | ||

MEV | [ξ, α, κ] | [15.037, 9.770, −0.161] | 0.023 | 0.994 | 2.025 | 488.53 | ||

GP | [ξ, α, κ] | [5.366, 19.379, 0.130] | 0.060 | 0.815 | 2.355 | 501.80 | ||

GAM | [β, α] | [10.109, 2.227] | 0.040 | 0.936 | 2.651 | 499.99 | ||

GUM | [ξ, α] | [15.811, 11.614] | 0.069 | 0.762 | 3.170 | 504.70 | ||

GLO | [ξ, α, κ] | [18.972, 7.066, −0.278] | 0.028 | 0.982 | 2.369 | 503.51 |

**Table 4.**Estimated parameters; two indicators: AIC and ${\hat{\lambda}}_{U}^{\mathit{CFG}}$ of the copulas.

Copula Model | Parameter Name | Estimated Parameter | AIC | ${\hat{\mathbf{\lambda}}}_{\mathit{U}}^{\mathit{CFG}}$ |
---|---|---|---|---|

Symmetric GH Copula | $\theta $ | 2.7576 | −519.544 | 0.714 |

Two-parameter GH Copula | ${[\beta}_{1}$, ${\beta}_{2}]$ | [2.1241, 0.7176] | −520.997 | 0.614 |

Asymmetric GH Copula | $[\theta $, ${\pi}_{2}$, ${\pi}_{3}$] | [2.7569, 1.0000, 1.0000] | −515.480 | 0.714 |

**Table 5.**Experimental design for CEC and CMLC point identification considering the uncertainty of marginal distribution selection.

Combination | Copula | Zhengyuangguan W_{30d} Distribution | Zheng-Beng Interval W_{30d} Distribution |
---|---|---|---|

C1 | GH2 | PE3 | PE3 |

C2 | GH2 | PE3 | LN3 |

C3 | GH2 | PE3 | GEV |

C4 | GH2 | PE3 | GP |

C5 | GH2 | PE3 | GAM |

C6 | GH2 | PE3 | GUM |

C7 | GH2 | PE3 | GLO |

C8 | GH2 | LN3 | GEV |

C9 | GH2 | GEV | GEV |

C10 | GH2 | GP | GEV |

C11 | GH2 | GAM | GEV |

C12 | GH2 | GUM | GEV |

C13 | GH2 | GLO | GEV |

Combination | Conditional Design Regional Flood Composition Points | ||||||
---|---|---|---|---|---|---|---|

T = 20 | T = 50 | T = 100 | |||||

CEC | CMLC | CEC | CMLC | CEC | CMLC | ||

Given flood occurs at the Zhengyuanguan section | C1 | (199.22, 47.55) | (199.22, 48.98) | (244.62, 59.23) | (244.62, 62.01) | (278.18, 68.39) | (278.18, 71.76) |

C2 | (199.22, 47.34) | (199.22, 48.37) | (244.62, 60.34) | (244.62, 61.66) | (278.18, 71.27) | (278.18, 73.34) | |

C3 | (199.22, 47.10) | (199.22, 48.26) | (244.62, 60.83) | (244.62, 61.00) | (278.18, 72.92) | (278.18, 73.72) | |

C4 | (199.22, 47.70) | (199.22, 50.37) | (244.62, 58.26) | (244.62, 62.14) | (278.18, 65.94) | (278.18, 70.15) | |

C5 | (199.22, 46.50) | (199.22, 48.18) | (244.62, 56.85) | (244.62, 59.57) | (278.18, 64.83) | (278.18, 67.97) | |

C6 | (199.22, 45.56) | (199.22, 46.91) | (244.62, 55.38) | (244.62, 57.68) | (278.18, 63.05) | (278.18, 65.81) | |

C7 | (199.22, 46.47) | (199.22, 46.67) | (244.62, 61.34) | (244.62, 59.15) | (278.18, 75.46) | (278.18, 73.28) | |

Given flood occurs at the Zheng-Beng interval section | C3 | (178.98, 52.27) | (181.18, 52.27) | (220.24, 68.13) | (230.76, 68.13) | (252.20, 81.69) | (264.54, 81.69) |

C8 | (178.56, 52.27) | (180.98, 52.27) | (223.53, 68.13) | (229.34, 68.13) | (260.37 81.69) | (269.52, 81.69) | |

C9 | (178.19, 52.27) | (179.39, 52.27) | (225.20, 68.13) | (228.18, 68.13) | (264.89, 81.69) | (271.74, 81.69) | |

C10 | (179.48, 52.27) | (186.67, 52.27) | (213.83, 68.13) | (229.82, 68.13) | (237.23 81.69) | (253.09, 81.69) | |

C11 | (177.86, 52.27) | (184.30, 52.27) | (217.72, 68.13) | (228.17, 68.13) | (248.45, 81.69) | (260.56, 81.69) | |

C12 | (174.16, 52.27) | (179.31, 52.27) | (211.80, 68.13) | (220.63, 68.13) | (241.24, 81.69) | (251.81, 81.69) | |

C13 | (176.10, 52.27) | (175.22, 52.27) | (228.12, 68.13) | (223.65, 68.13) | (276.11, 81.69) | (272.51, 81.69) |

T | Method | sz | ${\mathsf{\sigma}}_{\mathbf{x}}\left({10}^{8}\text{}{\mathbf{m}}^{3}\right)$ | ${\mathsf{\sigma}}_{\mathbf{y}}\left({10}^{8}\text{}{\mathbf{m}}^{3}\right)$ | S_{25%} (10^{16} m^{3}·m^{3}) | S_{50%} (10^{16} m^{3}·m^{3}) | S_{75%} (10^{16} m^{3}·m^{3}) | |
---|---|---|---|---|---|---|---|---|

Given flood occurs in the Zhengyangguan section | 20-year | CEC | 63 | 25.2 | 8.05 | 165 | 417 | 852 |

200 | 12.5 | 4.06 | 50.4 | 122 | 251 | |||

500 | 7.91 | 2.64 | 21.5 | 49.9 | 102 | |||

CMLC | 63 | 25.4 (0.79%) ^{1} | 7.77 (−3.48%) | 162 (−1.82%) | 422 (1.20%) | 858 (0.70%) | ||

200 | 12.5 | 3.85 | 49.7 | 124 | 251 | |||

500 | 7.93 | 2.48 | 20.7 | 50.6 | 99.9 | |||

50-year | CEC | 63 | 32.2 | 12.4 | 397 | 957 | 1946 | |

200 | 17.7 | 7.05 | 130 | 313 | 631 | |||

500 | 11.1 | 4.45 | 51.8 | 124 | 254 | |||

CMLC | 63 | 31.9 (−0.93%) | 11.8 (−4.83%) | 373 (−6.05%) | 891 (−6.90%) | 1866 (−4.11%) | ||

200 | 17.8 | 6.37 | 116 | 281 | 564 | |||

500 | 11.1 | 4.01 | 48.9 | 112 | 226 | |||

100-year | CEC | 63 | 40.1 | 18.2 | 679 | 1656 | 3369 | |

200 | 21.9 | 10.1 | 232 | 563 | 1146 | |||

500 | 13.9 | 6.41 | 94.2 | 230 | 462 | |||

CMLC | 63 | 40 (−0.25%) | 16.7 (−8.24%) | 626 (−7.81%) | 1537 (−7.19%) | 3301 (−2.02%) | ||

200 | 21.6 | 8.93 | 199 | 489 | 967 | |||

500 | 13.9 | 5.74 | 85.2 | 201 | 409 | |||

Given flood occurs in the Zheng-Beng interval | 20-year | CEC | 63 | 23.8 | 6.86 | 166 | 405 | 812 |

200 | 13.4 | 3.91 | 56.1 | 133 | 262 | |||

500 | 8.55 | 2.51 | 22.8 | 53.9 | 108 | |||

CMLC | 63 | 24.6 (3.36%) | 6.95 (1.30%) | 174 (4.82%) | 410 (1.23%) | 846 (4.19%) | ||

200 | 13.8 | 3.94 | 55 | 137 | 275 | |||

500 | 8.6 | 2.51 | 21.7 | 53.1 | 109 | |||

50-year | CEC | 63 | 34.1 | 12.2 | 420 | 1018 | 2106 | |

200 | 19.3 | 7.03 | 144 | 341 | 684 | |||

500 | 12.1 | 4.41 | 56.9 | 137 | 280 | |||

CMLC | 63 | 34 (−0.29%) | 12.3 (0.82%) | 405 (−3.57%) | 997 (−2.06%) | 2073 (−1.56%) | ||

200 | 18.4 | 6.87 | 134 | 317 | 631 | |||

500 | 11.6 | 4.43 | 56.6 | 129 | ||||

100-year | CEC | 63 | 42.2 | 18.2 | 739 | 1793 | ||

200 | 23.9 | 10.2 | 260 | 638 | ||||

500 | 14.9 | 6.41 | 102 | 253 | ||||

CMLC | 63 | 41.9 (−0.72%) | 18.2 (0.00%) | 712 (−3.65%) | 1780 (−0.73%) | |||

200 | 22.6 | 10.3 | 250 | 599 | ||||

500 | 14.1 | 6.58 | 101 | 235 |

^{1}Values shown in parenthesis are variation ratios compared to the CEC method.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mou, S.; Shi, P.; Qu, S.; Ji, X.; Zhao, L.; Feng, Y.; Chen, C.; Dong, F.
Uncertainty Analysis of Two Copula-Based Conditional Regional Design Flood Composition Methods: A Case Study of Huai River, China. *Water* **2018**, *10*, 1872.
https://doi.org/10.3390/w10121872

**AMA Style**

Mou S, Shi P, Qu S, Ji X, Zhao L, Feng Y, Chen C, Dong F.
Uncertainty Analysis of Two Copula-Based Conditional Regional Design Flood Composition Methods: A Case Study of Huai River, China. *Water*. 2018; 10(12):1872.
https://doi.org/10.3390/w10121872

**Chicago/Turabian Style**

Mou, Shiyu, Peng Shi, Simin Qu, Xiaomin Ji, Lanlan Zhao, Ying Feng, Chen Chen, and Fengcheng Dong.
2018. "Uncertainty Analysis of Two Copula-Based Conditional Regional Design Flood Composition Methods: A Case Study of Huai River, China" *Water* 10, no. 12: 1872.
https://doi.org/10.3390/w10121872