# Nonstationary Analysis for Bivariate Distribution of Flood Variables in the Ganjiang River Using Time-Varying Copula

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study Region and Data

^{2}(Figure 1). This study area belongs to hilly region with the main land use type of woodland [33]. The Ganjiang River basin is located in the subtropical humid monsoon climate zone with distinct seasonal variations, where the annual mean precipitation is about 1680 mm. The heaviest rainfall occurs in the main flood season from April and June, and often lasts 15 to 20 days due to monsoon and typhoon rainstorms [34].

^{2}(Figure 1). The hydrological data used in this study include daily river discharge, daily water stage, daily suspended sediment rate and yearly section-cross elevation at the hydrological station from 1964 to 2013. These data are provided by the Hydrological Bureau of Jiangxi province (http://www.jxsw.cn/). To reflect the capacity of suspended sediment transport in a flood event, suspended sediment load that measures the absolution amount of sediment appears to be more reasonable than suspended sediment rate, since the latter is a relative value for sediment. The daily suspended sediment load (S) is calculated by multiplication between the daily discharge and the corresponding suspended sediment rate. Then, the hydrological series of Q, Z and S are extracted from the same flood event in each year with the maximum discharge value criterion. Besides, the annual forest cover rates of the region are obtained from the book of China Compendium of Statistics 1949–2008 [35] and Jiangxi provincial statistical yearbook from 1983 to 2014 (http://www.jxstj.gov.cn/).

## 3. Methodology

#### 3.1. Calculation of Explanatory Variables

_{L}stands for the total land area of the study region.

#### 3.2. Marginal Distribution with Time-Varying Parameters

#### 3.3. Time-Varying Bivariate Copula Model

#### 3.4. Joint and Conditional Probability under Nonstationary Framework

## 4. Results

#### 4.1. Temporal Trend Analysis

^{3}kg/day from 1964 to 2003, and then decrease to 20.96 m and 123.90 × 10

^{3}kg/day for the last decade. Through the aforementioned analysis, it is reasonable to conclude that both Z and S display significant nonstationarity during the period from 1964 to 2013, whereas Q is stationary. The trend identification of Q, Z and S are consistent with some previous research conclusions [52,53].

#### 4.2. Nonstationary Marginal Distributions

^{3}kg/day and 0.47 during 1964–1994. After 1995, the mean and Cv of S are getting smaller significantly, because massive afforestation activities have been implemented in the study basin with growth rate of forest cover at 1.17% year by year.

#### 4.3. Nonstationary Dependence of Bivariate Flood Variables

#### 4.4. Temporal Variation in Joint and Conditional Probabilities

^{3}kg/day to 151.58 × 10

^{3}kg/day for S. It can be seen that the suspended sediment load is more sensitive to flood event than the peak water stage from comparison of their value ranges. For the most likely combination, the peak discharge at Waizhou station in the Ganjiang River displays a slight increasing trend from 15.97 × 10

^{3}m

^{3}/s to 16.41 × 10

^{3}m

^{3}/s over time, while the values of peak water stage and suspended sediment load have obviously decreased, especially in the last decade. As shown in Figure 6b,d, it is indicated that Z and S in the most likely combination are obviously affected by riverbed down-cutting and the change of forest cover, respectively. Moreover, since the dependence structure of Z-Q is nonstationary as Equation (12), Q in the most likely combination is impacted by riverbed down-cutting as well.

## 5. Conclusions

- It is obvious that both the mean and variance of S have significantly decreased, while only the mean has reduced for Z, particularly in the recent decades. Furthermore, Gamma distribution with location parameter expressed as a function of MCE is best fitted distribution for Z, and Gamma with parameters of location and scale expressed as functions of FCA is for S, while the best fitted distribution of Q is the Gamma with constant parameters.
- It is found that the most fitted bivariate copulas for both Z-Q and Z-S are Frank copula, the parameters of which are expressed as the function of MCE. Therefore, riverbed down-cutting at Waizhou station plays the dominant role in strengthening dependences of both Z-Q and Z-S from 1964 to 2013.
- The results of joint probability and conditional probability show that the corner of contour lines enhanced more greatly due to the strengthening dependences over time, especially for the lower probability. In addition, it can be seen that values of Z and S fall rapidly in the last ten years due to the decreasing mean of these two variables.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Data analyses of four variable series in the Ganjiang River basin (GRB) during 1964–2013. (

**a**) is evolution of the main channel mean elevation (MCE) at Waizhou station, (

**b**) is evolution of forest coverage rate (FCR) of the GRB, and two correlation plots are between peak water stage (Z) and MCE (

**c**) and suspended sediment load (S) and FCR (

**d**), respectively. R

^{2}value is the square of the correlation coefficient.

**Figure 3.**Quantiles–Quantiles (QQ) plot of the selected marginal distributions (

**a**) and bivariate copula (

**b**).

**Figure 4.**Scatter plots of Z-Q (

**a**) and Z-S (

**b**) shown comparison of observed data with sets of 1500 generated random samples based on the selected bivariate copulas. Solid circles in blue color are observed data and gray dots are simulated samples.

**Figure 5.**Joint probability contours with the given ${P}^{\wedge}$ (AND) of Z-Q (

**a**) and Z-S (

**b**) using the best fitted bivariate copulas at the Waizhou station in the years of 1970, 1990 and 2010.

**Figure 6.**Time variation of joint probability contours ${P}^{\wedge}=0.1$ for Z-Q (

**a**) and Z-S (

**c**) and corresponding the most likely combinations for Z-Q (

**b**) and Z-S (

**d**) derived from the best fitted bivariate copula at the Waizhou station from 1964 to 2013.

**Figure 7.**Time variation of conditional probability ${P}^{Q|Z}$ (

**a**) and ${P}^{S|Z}$ (

**b**) with the given warning water stage (Z ≥ 23.5 m) at the Waizhou station from 1964 to 2013.

**Figure 8.**Conditional probability ${P}^{Q|Z}$ (

**a**) and ${P}^{S|Z}$ (

**b**) with various given water stages when main channel elevation and forest coverage rate are assumed to be 6 m and 70%, respectively.

Copula | Cumulative Distribution Function with Time-Varying Parameters | Parameters |
---|---|---|

Clayton | $C(u,\nu |{\theta}_{}^{t})={\left({(u)}_{}^{-{\theta}_{}^{t}}+{(\nu )}_{}^{-{\theta}_{}^{t}}-1\right)}^{-1/{\theta}_{}^{t}}$ | ${\theta}_{}^{t}>0$ |

Gumbel–Hougaard | $C(u,\nu |{\theta}_{}^{t})=\mathrm{exp}\left(-{\left({\left(-\mathrm{ln}u\right)}^{{\theta}_{}^{t}}+{\left(-\mathrm{ln}\nu \right)}^{{\theta}_{}^{t}}\right)}^{1/{\theta}_{}^{t}}\right)$ | ${\theta}_{}^{t}>1$ |

Frank | $C(u,\nu |{\theta}_{}^{t})=-\frac{1}{{\theta}_{}^{t}}\mathrm{ln}\left(1+\left(\mathrm{exp}\left(-u{\theta}_{}^{t}\right)-1\right)\times \left(\mathrm{exp}\left(-\nu {\theta}_{}^{t}\right)-1\right)/\left(\mathrm{exp}\left(-{\theta}_{}^{t}\right)-1\right)\right)$ | ${\theta}_{}^{t}\ne 0$ |

Series | Annual Mean | MK | Spearman | Kendall | |
---|---|---|---|---|---|

1964–2003 | 2004–2013 | ||||

$Q$ | 12.01 | 10.47 | −1.33 | −0.16 | −0.12 |

$Z$ | 23.25 | 20.96 | −2.92 ** | −0.41 ** | −0.29 ** |

$S$ | 418.23 | 123.90 | −5.33 ** | −0.70 ** | −0.52 ** |

^{3}m

^{3}/s); Z: peak water stage (m); S: suspended sediment load (10

^{3}kg/d). –: delineates negative trends; * and ** delineate significant trend at 0.05, 0.01 significance level, respectively.

**Table 3.**Parameters and goodness-of-fit of the candidate marginal distributions fitted to Q, Z, and S at the Waizhou station during 1964–2013, respectively.

Variable | Distribution | Estimated Parameters | AICc | KS-Test | |
---|---|---|---|---|---|

Statistic | p-Value | ||||

Q | LNO | m = 9.309, σ = 0.346 | 970.57 | 0.083 | 0.881 |

WEI | μ = 13090, σ = 3.154 | 973.81 | 0.101 | 0.683 | |

LOG | μ = 11451, σ = 2305 | 976.57 | 0.108 | 0.602 | |

GAM | μ = 11703, σ = 0.339 | 970.48 | 0.097 | 0.737 | |

PⅢ | μ = 11694, σ = 0.350, γ = 0.474 | 971.89 | 0.080 | 0.910 | |

Z | LNO | μ = exp (1.081 + 0.004MCE^{t}) | 177.31 | 0.084 | 0.843 |

σ = 0.059 | |||||

WEI | μ = exp (3.005 + 0.011MCE^{t}) | 181.46 | 0.101 | 0.646 | |

σ = exp (2.123 + 0.061MCE^{t}) | |||||

LOG | μ = 18.461 + 0.326MCE^{t} | 179.09 | 0.081 | 0.873 | |

σ = 0.782 | |||||

GAM | μ = exp (2.946 + 0.014MCE^{t}) | 177.24 | 0.089 | 0.793 | |

σ = 0.059 | |||||

PⅢ | μ = exp(2.936+0.014MCE^{t}) | 178.67 | 0.093 | 0.745 | |

Σ = 0.060, γ = 0.251 | |||||

S | LNO | μ = exp (2.150 − 0.886FCR^{t}) | 646.40 | 0.097 | 0.701 |

σ = exp (-1.332 + 1.378FCR^{t}) | |||||

WEI | μ = exp (8.072 − 4.544FCR^{t}) | 648.35 | 0.136 | 0.285 | |

σ = exp (1.476 − 1.448FCR^{t}) | |||||

LOG | μ=950.510 − 12.693FCR^{t} | 657.23 | 0.131 | 0.332 | |

σ = exp (6.137 − 3.412FCR^{t}) | |||||

GAM | μ = exp (7.943 − 4.530FCR^{t}) | 645.95 | 0.113 | 0.515 | |

σ = exp (−1.336 + 1.291FCR^{t}) | |||||

PⅢ | μ = exp (8.098 − 4.854FCR^{t}) | 646.05 | 0.084 | 0.845 | |

σ = 0.537, γ = 0.702 |

**Table 4.**Parameters and goodness-of-fit of the candidate bivariate copulas fitted to Z-Q and Z-S at the Waizhou station during 1964–2013, respectively.

Copula | Parameter (θ) | AICc RMSE | CM-Test | |||
---|---|---|---|---|---|---|

Statistic | p-Value | |||||

Z-Q | Clayton | 5.702 | −103.91 | 0.031 | 0.053 | 0.457 |

exp (1.811 − 0.005MCE^{t}) | −101.93 | 0.031 | 0.053 | 0.458 | ||

GH | 4.008 | −94.76 | 0.039 | 0.056 | 0.426 | |

exp (1.743 − 0.026MCE^{t}) | −93.33 | 0.039 | 0.055 | 0.433 | ||

Frank | 17.683 | −102.79 | 0.035 | 0.055 | 0.434 | |

41.713 − 1.747MCE^{t} | −104.25 | 0.035 | 0.054 | 0.448 | ||

Z-S | Clayton | 1.250 | −22.70 | 0.040 | 0.048 | 0.538 |

exp (−1.760 + 0.063FCR^{t}) | −25.11 | 0.039 | 0.038 | 0.716 | ||

exp (1.739 − 0.114MCE^{t}) | −24.01 | 0.040 | 0.044 | 0.596 | ||

exp (−5.025 + 0.118FCR^{t} − 0.113MCE^{t}) | −23.48 | 0.040 | 0.052 | 0.479 | ||

GH | 1.796 | −28.04 | 0.038 | 0.050 | 0.506 | |

exp (0.350 + 0.007FCR^{t}) | −26.25 | 0.039 | 0.047 | 0.541 | ||

exp (1.195 − 0.045MCE^{t}) | −27.40 | 0.038 | 0.047 | 0.556 | ||

exp (3.563 − 0.039FCR^{t} − 0.128MCE^{t}) | −26.76 | 0.039 | 0.057 | 0.410 | ||

Frank | 5.319 | −28.77 | 0.032 | 0.035 | 0.759 | |

−1.528 + 0.226FCR^{t} | −28.79 | 0.032 | 0.029 | 0.855 | ||

16.169 − 0.782MCE^{t} | −30.29 | 0.031 | 0.028 | 0.870 | ||

24.186 − 0.132FCR^{t} − 1.072MCE^{t} | −28.45 | 0.032 | 0.030 | 0.840 |

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## Share and Cite

**MDPI and ACS Style**

Wen, T.; Jiang, C.; Xu, X.
Nonstationary Analysis for Bivariate Distribution of Flood Variables in the Ganjiang River Using Time-Varying Copula. *Water* **2019**, *11*, 746.
https://doi.org/10.3390/w11040746

**AMA Style**

Wen T, Jiang C, Xu X.
Nonstationary Analysis for Bivariate Distribution of Flood Variables in the Ganjiang River Using Time-Varying Copula. *Water*. 2019; 11(4):746.
https://doi.org/10.3390/w11040746

**Chicago/Turabian Style**

Wen, Tianfu, Cong Jiang, and Xinfa Xu.
2019. "Nonstationary Analysis for Bivariate Distribution of Flood Variables in the Ganjiang River Using Time-Varying Copula" *Water* 11, no. 4: 746.
https://doi.org/10.3390/w11040746