# Improved Mixed Distribution Model Considering Historical Extraordinary Floods under Changing Environment

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Variation Diagnosis System

#### 2.2. Mixed Distribution Model

#### 2.2.1. Mixed Distribution

_{i}, variation coefficient C

_{vi}and skewness coefficient C

_{si}. The formulas are given as follows: ${a}_{0i}=E{X}_{i}(1-\frac{2{C}_{vi}}{{C}_{si}})$, ${\alpha}_{i}=\frac{4}{{C}_{si}{}^{2}}$ and ${\beta}_{i}=\frac{2}{E{X}_{i}{C}_{vi}{C}_{si}}$. The initial value of the sample mean EX

_{1}and EX

_{2}estimated by the moment method can be considered as the unbiased estimate of the total. Thus, there are w, C

_{v}

_{1}, C

_{v}

_{2}, C

_{s}

_{1}and C

_{s}

_{2}up to five parameters to be estimated in mixed distribution f(x).

#### 2.2.2. Parameter Estimation

_{v}

_{1}, C

_{v}

_{2}, C

_{s}

_{1}and C

_{s}

_{2}, which can make the objective function attain the minimum and obtain the best fitted mixed distribution parameters. The main calculation procedures are as follows.

- (1)
- Use real number coding to generate an initialization population with a population size Np of 100. The initial parameter variation range of five parameters for mixed distribution should be constrained. For example, the weight coefficient w of each sub-series distribution should be between 0~1, the variation range of C
_{v}should be 0~2 according to the information of the Ankang hydrological station and the variation range of C_{s}/C_{v}should be between 2~2.5. This approach effectively avoids the large deviation between the estimated values of C_{v}and C_{s}/C_{v}as well as the recommended value. - (2)
- Calculate the fitness of the initial population. The fitness value of the initial population can be calculated through the objective function shown in Equation (4).
- (3)
- Set the population gap GGAP = 0.7, the crossover probability P
_{c}= 0.6 and the maximum number of iterations N_{G}= 150; the processes of multiple selection, crossover and mutation are carried out for the initial population. Each iteration is used to evaluate the fitness of the population to minimize the objective function value until the optimal parameter is obtained according to the maximum number of iterations.

#### 2.2.3. Model Evaluation Criterion and Goodness-of-Fit Test

_{v}and C

_{s}up to three parameters in the P3 distribution; thus, m = 3.

#### 2.3. Improved Mixed Distribution Model

#### 2.4. Monte Carlo Simulation

## 3. Study Area and Data Set

#### 3.1. Study Area

^{2}and a length of 1570 km. The basin is bounded by 30°10′ N to 34°20′ N latitude and 106°15′ E to 114°20′ E longitude. Originating in Hanzhong city of Shaanxi province, the main stream flows southeast through Shaanxi and Hubei provinces and returns to the Yangtze River in Wuhan city. The area controlled by the Ankang hydrological station, with a catchment area of 38,700 km

^{2}, is the study area (shown in Figure 1). The annual average discharge is 621 m

^{3}/s at the Ankang hydrological station. The study area has a subtropical continental monsoon climate, which is mild and has four distinctive seasons. The average annual temperature is 15~17 °C and the annual average rainfall is 800~1000 mm. Floods are mainly caused by rainfall, occurring over 3~10 months but mostly in summer and autumn. Summer floods mainly occur in July, mostly consisting of a heavy intensity and short duration rainstorms. Autumn floods often appear in September, generally consisting of stable and persistent rainfall. Thus, the floods in autumn often have a long duration and large volume. The hydrographs of the 1974 typical flood and the 1983 largest flood are shown in Figure 2.

#### 3.2. Data Set

^{3}/s is the largest flood that has been encountered since the establishment (1935) of the Ankang hydrological station. In addition, combined with the historical extraordinary floods investigation results by Yang [51,52], the historical extraordinary flood data of 36,000 m

^{3}/s in 1583, 30,000 m

^{3}/s in 1867 and 26,000 m

^{3}/s in 1921 were selected. Due to the lack of historical extraordinary flood volume data, the correlation relationship (shown in Figure 3) between the flood peak and volume of the Ankang hydrological station was used to calculate the historical maximum 24-h and 72-h flood volume data corresponding to the historical maximum peak discharge. The three series of flood samples that consider the discontinuity of historical extraordinary floods are formed.

## 4. Results and Discussion

#### 4.1. Results of the Variation Diagnosis System

#### 4.1.1. Primary Diagnosis

#### 4.1.2. Detailed Diagnosis

#### 4.1.3. Comprehensive Diagnosis

#### 4.2. Results of Monte Carlo Simulation and Uncertainty Analysis

#### 4.3. Analysis of the Design Flood Results under Changing Environments

- Case 1: original flood characteristic series (1968–2013) that do not consider the variation but add the historical extraordinary flood data, denoted as series ${A}_{i1}$.
- Case 2: observed flood characteristic series before the change point (1968–1987) with the addition of the historical extraordinary flood data, denoted as series ${A}_{i2}$.
- Case 3: observed flood characteristic series after the change point (1987–2013) with the addition of the historical extraordinary flood data, denoted as series ${A}_{i3}$.
- Case 4: only the observed flood characteristic series after the change point (1987–2013), denoted as series ${A}_{i4}$.

#### 4.4. Analysis of the Design Flood Results Based on IMD in Consideration of Historical Extraordinary Floods

## 5. Conclusions

- (1)
- Hydrological series diagnosis was performed by using a variant diagnostic system. The trends of AMPDS, 24-h AMFVS and 72-h AMFVS at the Ankang hydrological station all decreased significantly at the 5% significance level. However, the final variant form was the change point, which illustrated that the change points of all flood characteristic series were in the year of 1987. This result was mainly related to the construction of the Ankang reservoir.
- (2)
- Based on the principle of MD, we proposed the methods of IMD, for which the genetic algorithm was applied, to estimate the parameters and the information of historical extraordinary floods was supplemented in the series after the change point. Meanwhile, the superiority of IMD was demonstrated by the consideration of both environment changes and historical extraordinary floods. Although the design flood of IMD was slightly larger than MD at the Ankang hydrological station, adding historical extraordinary flood data into both sub-series divided by the change point improved the theoretical mechanism of the mixed distribution. The new design flood based on IMD provides the basis for the regulation of reservoir floods in the upper reaches of the Hanjiang River.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The hydrograph of the 1974 typical flood and the 1983 largest flood. (

**a**) The hydrograph of the typical flood in 1974; and (

**b**) The hydrograph of the largest flood in 1983.

**Figure 3.**Correlation relationship between the flood peak and volume at the Ankang hydrological station.

**Figure 4.**Variation between the design flood value of annual maximum peak discharge series (AMPDS) simulated by Monte Carlo and estimated by genetic algorithm (GA). (

**a**) Design flood values of AMPDS simulated by Monte Carlo and estimated by GA at different return periods; and (

**b**) Quantile-Quantile plots between design flood values of AMPDS simulated by Monte Carlo and estimated by GA.

**Figure 5.**95% confidence intervals computed using the bootstrap method for flood characteristic series. (

**a**) 95% confidence intervals of the annual maximum peak discharge; (

**b**) 95% confidence intervals of the annual maximum 24-h flood volume; and (

**c**) 95% confidence intervals of the annual maximum 72-h flood volume.

**Figure 6.**Fitting results for each flood characteristic series under four conditions. (

**a**) Fitting results of the annual maximum peak discharge; (

**b**) Fitting results of the annual maximum 24-h flood volume; and (

**c**) Fitting results of the annual maximum 72-h flood volume.

**Figure 7.**Fitting results of the MD and IMD methods. (

**a1**) MD fitting results of AMPDS; (

**a2**) IMD fitting results of AMPDS; (

**b1**) MD fitting results of annual maximum 24-h flood volume series (24-h AMFVS); (

**b2**) IMD fitting results of 24-h AMFVS; (

**c1**) MD fitting results of 72-h AMFVS; and (

**c2**) IMD fitting results of 72-h AMFVS.

Correlation Function C(t) | Hurst Exponent h | Variation Degree |
---|---|---|

$0\le C(t)<{r}_{\alpha}$ | $0.5\le h<{h}_{\alpha}$ | No variation or Weak variation |

${r}_{\alpha}\le C(t)<0.6$ | ${h}_{\alpha}\le h<0.839$ | Medium variation |

$0.6\le C(t)<0.8$ | $0.839\le h<0.924$ | Strong variation |

$0.8\le C(t)\le 1.0$ | $0.924\le h\le 1.0$ | Vast variation |

^{2h−1}− 1; ${h}_{\alpha}=\frac{1}{2}\left[1+\mathrm{ln}(1+{r}_{\alpha})/\mathrm{ln}2\right]$.

Name of Reservoir | Drainage Area (km ^{2}) | Annual Average Flow (m ^{3}/s) | Design Flood Flow (m ^{3}/s) | Storage Capacity (10 ^{8} m^{3}) | Completion Year |
---|---|---|---|---|---|

Huangjinxia | 17,950 | 259 | 18,000 | 0.92 | Not built yet |

Shiquan | 23,400 | 343 | 21,500 | 1.80 | 1974 |

Xihe | 25,207 | 361 | 21,800 | 0.20 | 2006 |

Ankang | 35,700 | 621 | 36,700 | 14.72 | 1992 |

Flood Characteristics Series | Hurst Exponent | $\mathit{C}(\mathit{t})\text{}$ | Variation Degree |
---|---|---|---|

AMPDS | 0.8047 | 0.5256 | Medium variation |

24-h AMFVS | 0.7352 | 0.3855 | Medium variation |

72-h AMFVS | 0.8107 | 0.5384 | Medium variation |

Variation Component | Methods | AMPDS | 24-h AMFVS | 72-h AMFVS |
---|---|---|---|---|

Trends | Spearman rank correlation statistics | −2.318 | −2.225 | −2.384 |

Kendall rank correlation statistics | −2.320 | −2.452 | −2.566 | |

Change point | Lee-Heghinian method | 1987 | 1987 | 1985 |

Sequential clustering method | 1987 | 1987 | 1985 | |

Pettitt test | 1993 | 1994 | 1994 | |

Mann-Kendall test | 1990 | 1989 | 1988 | |

R/S analysis method | 1987 | 1987 | 1987 | |

Possible change points | 1987 | 1987 | 1985 and 1987 |

**Table 5.**Kendall rank correlation statistic values for the flood series before and after the change point.

Flood Characteristics Series | Kendall Rank Correlation Statistic U | |
---|---|---|

1968–1986 | 1987–2013 | |

AMPDS | 1.224 | −0.063 |

24-h AMFVS | 0.735 | −0.271 |

72-h AMFVS | 0.734 | −0.229 |

Flood Characteristic Series | Trend Variation Efficiency Coefficients | Change Point Efficiency Coefficients | Final Variation Forms |
---|---|---|---|

AMPDS | 0.0857 | 0.2272 | Change point |

24-h AMFVS | 0.1032 | 0.2346 | Change point |

72-h AMFVS | 0.1051 | 0.2202 | Change point |

Statistical Parameters | AMPDS | 24-h AMFVS | 72-h AMFVS |
---|---|---|---|

NMB | 0.0046 | 0.0157 | 0.0208 |

RRMSE | 0.0091 | 0.0217 | 0.0281 |

Estimated Parameters | AMPDS (m^{3}/s) | 24-h AMFVS (10^{8} m^{3}) | 72-h AMFVS (10^{8} m^{3}) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{A}}_{11}\text{}$ | ${\mathit{A}}_{12}\text{}$ | ${\mathit{A}}_{13}\text{}$ | ${\mathit{A}}_{14}\text{}$ | ${\mathit{A}}_{21}\text{}$ | ${\mathit{A}}_{22}$ | ${\mathit{A}}_{23}$ | ${\mathit{A}}_{24}$ | ${\mathit{A}}_{31}$ | ${\mathit{A}}_{32}$ | ${\mathit{A}}_{33}$ | ${\mathit{A}}_{34}$ | |

EX | 10,226 | 13,278 | 7995 | 7782 | 7.35 | 9.59 | 5.71 | 5.57 | 16.02 | 20.70 | 12.60 | 12.32 |

Cv | 0.59 | 0.42 | 0.76 | 0.77 | 0.56 | 0.38 | 0.72 | 0.8 | 0.51 | 0.37 | 0.69 | 0.77 |

Cs/Cv | 2 | 2 | 2.15 | 2.21 | 2 | 2 | 2 | 2.17 | 2 | 2 | 2.03 | 2.35 |

Evaluation Indicators | AMPDS | 24-h AMFVS | 72-h AMFVS | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{A}}_{11}\text{}$ | ${\mathit{A}}_{12}\text{}$ | ${\mathit{A}}_{13}\text{}$ | ${\mathit{A}}_{14}\text{}$ | ${\mathit{A}}_{21}\text{}$ | ${\mathit{A}}_{22}\text{}$ | ${\mathit{A}}_{23}\text{}$ | ${\mathit{A}}_{24}\text{}$ | ${\mathit{A}}_{31}\text{}$ | ${\mathit{A}}_{32}\text{}$ | ${\mathit{A}}_{33}\text{}$ | ${\mathit{A}}_{34}\text{}$ | |

${D}_{n}(\alpha )$ | 0.1940 | 0.2796 | 0.2400 | 0.2640 | 0.1940 | 0.2796 | 0.2400 | 0.2640 | 0.1940 | 0.2796 | 0.2400 | 0.2640 |

D | 0.0831 | 0.1571 | 0.0912 | 0.1051 | 0.0881 | 0.1375 | 0.0741 | 0.0928 | 0.1035 | 0.1411 | 0.0615 | 0.0708 |

AIC | −313.5 | −119.4 | −195.2 | −156.0 | −291.4 | −118.2 | −200.3 | −165.0 | −279.4 | −126.9 | −204.9 | −179.6 |

Flood Characteristic Series | Return Periods in Years | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

10,000 | 5000 | 1000 | 500 | 300 | 200 | 100 | 20 | 5 | ||

AMPDS (m^{3}/s) | 1999 design report | 48,100 | 45,500 | 39,300 | 36,700 | 34,600 | 32,800 | 30,000 | 23,000 | 16,100 |

${A}_{11}$ | 48,640 | 45,802 | 39,106 | 36,165 | 33,971 | 32,210 | 29,152 | 21,730 | 14,662 | |

Difference (1) (%) | 1.1 | 0.7 | −0.5 | −1.5 | −1.8 | −1.8 | −2.8 | −5.5 | −8.9 | |

${A}_{12}$ | 44,466 | 42,340 | 37,270 | 35,016 | 33,320 | 31,950 | 29,550 | 23,579 | 17,604 | |

${A}_{13}$ | 52,927 | 49,325 | 40,912 | 37,262 | 34,559 | 32,405 | 28,699 | 19,939 | 12,066 | |

${A}_{14}$ | 53,001 | 49,339 | 40,794 | 37,093 | 34,354 | 32,173 | 28,427 | 19,601 | 11,733 | |

Difference (2) (%) | 19.0 | 16.5 | 9.8 | 6.4 | 3.7 | 1.4 | −2.9 | −15.4 | −31.5 | |

Difference (3) (%) | 0.1 | 0.0 | −0.3 | −0.5 | −0.6 | −0.7 | −0.9 | −1.7 | −2.8 | |

24-h AMFVS (10^{8} m^{3}) | ${A}_{21}$ | 32.98 | 31.11 | 26.69 | 24.75 | 23.29 | 22.13 | 20.10 | 15.15 | 10.40 |

${A}_{22}$ | 30.00 | 28.63 | 25.36 | 23.90 | 22.80 | 21.91 | 20.35 | 16.45 | 12.52 | |

${A}_{23}$ | 35.47 | 33.14 | 27.69 | 25.32 | 23.55 | 22.15 | 19.72 | 13.93 | 8.63 | |

${A}_{24}$ | 39.80 | 37.01 | 30.50 | 27.69 | 25.61 | 23.95 | 21.10 | 14.42 | 8.49 | |

Difference (2) (%) | 18.2 | 15.8 | 9.2 | 5.9 | 3.3 | 1.1 | −3.1 | −15.3 | −31.0 | |

Difference (3) (%) | 12.2 | 11.7 | 10.2 | 9.4 | 8.7 | 8.1 | 7.0 | 3.5 | −1.7 | |

72-h AMFVS (10^{8} m^{3}) | ${A}_{31}$ | 65.07 | 61.57 | 53.28 | 49.62 | 46.88 | 44.67 | 40.82 | 31.39 | 22.19 |

${A}_{32}$ | 61.79 | 59.07 | 52.55 | 49.64 | 47.44 | 45.66 | 42.54 | 34.70 | 26.72 | |

${A}_{33}$ | 72.32 | 67.71 | 56.91 | 52.20 | 48.69 | 45.89 | 41.05 | 29.46 | 18.73 | |

${A}_{34}$ | 86.21 | 80.13 | 65.97 | 59.86 | 55.33 | 51.74 | 45.57 | 31.13 | 18.41 | |

Difference (2) (%) | 17.0 | 14.6 | 8.3 | 5.2 | 2.6 | 0.5 | −3.5 | −15.1 | −29.9 | |

Difference (3) (%) | 19.2 | 18.3 | 15.9 | 14.7 | 13.6 | 12.7 | 11.0 | 5.7 | −1.7 |

**Table 11.**Parameter estimation results for the mixed distribution (MD) and improved mixed distribution (IMD) methods.

Flood Characteristic Series | Method | α | EX_{1} | Cv_{1} | Cs_{1} | EX_{2} | Cv_{2} | Cs_{2} |
---|---|---|---|---|---|---|---|---|

AMPDS (m^{3}/s) | MD | 0.346 | 13278 | 0.643 | 1.287 | 7782 | 0.649 | 1.299 |

IMD | 0.319 | 13278 | 0.643 | 1.287 | 7995 | 0.650 | 1.300 | |

24-h AMFVS (10^{8} m^{3}) | MD | 0.262 | 9.59 | 0.647 | 1.295 | 5.57 | 0.648 | 1.296 |

IMD | 0.236 | 9.59 | 0.652 | 1.303 | 5.71 | 0.646 | 1.291 | |

72-h AMFVS (10^{8} m^{3}) | MD | 0.254 | 20.70 | 0.588 | 1.175 | 12.32 | 0.603 | 1.206 |

IMD | 0.229 | 20.70 | 0.595 | 1.190 | 12.60 | 0.599 | 1.198 |

_{1}and EX

_{2}represent the mean values of the two sub-series divided by the change point; Cv

_{1}and Cv

_{2}represent the variation coefficient of the two sub-series; and Cs

_{1}and Cs

_{2}represent the skewness coefficient of the two sub-series.

Flood Characteristic Series | D | AIC | ||||
---|---|---|---|---|---|---|

P3 | MD | IMD | P3 | MD | IMD | |

AMPDS | 0.0831 | 0.1082 | 0.1052 | −313.5 | −272.6 | −274.5 |

24-h AMFVS | 0.0881 | 0.1511 | 0.1497 | −291.4 | −240.7 | −242.6 |

72-h AMFVS | 0.1035 | 0.1174 | 0.1177 | −279.4 | −254.3 | −255.0 |

Flood Characteristic Series | Return Periods in Years | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

10,000 | 5000 | 1000 | 500 | 300 | 200 | 100 | 20 | 5 | ||

AMPDS (m^{3}/s) | 1999 design report | 48,100 | 45,500 | 39,300 | 36,700 | 34,600 | 32,800 | 30,000 | 23,000 | 16,100 |

P3 | 48,640 | 45,802 | 39,106 | 36,165 | 33,971 | 32,210 | 29,152 | 21,730 | 14,662 | |

MD | 51,161 | 48,026 | 40,652 | 37,425 | 35,022 | 33,097 | 29,764 | 21,730 | 14,193 | |

IMD * | 51,174 | 48,037 | 40,659 | 37,430 | 35,025 | 33,099 | 29,764 | 21,726 | 14,187 | |

Difference (1) (%) | 6.39 | 5.58 | 3.46 | 1.99 | 1.23 | 0.91 | −0.79 | −5.54 | −11.88 | |

Difference (2) (%) | 5.21 | 4.88 | 3.97 | 3.50 | 3.10 | 2.76 | 2.10 | −0.02 | −3.24 | |

Difference (3) (%) | 0.026 | 0.023 | 0.016 | 0.012 | 0.009 | 0.006 | 0.000 | −0.018 | −0.047 | |

24-h AMFVS (10^{8} m^{3}) | P3 | 32.978 | 31.108 | 26.689 | 24.745 | 23.293 | 22.125 | 20.096 | 15.151 | 10.402 |

MD | 35.005 | 32.858 | 27.809 | 25.599 | 23.954 | 22.636 | 20.353 | 14.854 | 9.696 | |

IMD * | 35.051 | 32.902 | 27.845 | 25.633 | 23.985 | 22.665 | 20.380 | 14.873 | 9.709 | |

Difference (2) (%) | 6.29 | 5.77 | 4.33 | 3.59 | 2.97 | 2.44 | 1.41 | −1.84 | −6.67 | |

Difference (3) (%) | 0.1310 | 0.1310 | 0.1309 | 0.1309 | 0.1308 | 0.1308 | 0.1308 | 0.1306 | 0.1306 | |

72-h AMFVS (10^{8} m^{3}) | P3 | 65.067 | 61.568 | 53.278 | 49.617 | 46.876 | 44.669 | 40.824 | 31.385 | 22.191 |

MD | 69.726 | 65.629 | 55.969 | 51.729 | 48.565 | 46.027 | 41.622 | 30.939 | 20.786 | |

IMD * | 69.743 | 65.645 | 55.981 | 51.739 | 48.574 | 46.035 | 41.629 | 30.942 | 20.785 | |

Difference (2) (%) | 7.19 | 6.62 | 5.07 | 4.28 | 3.62 | 3.06 | 1.97 | −1.41 | −6.34 | |

Difference (3) (%) | 0.024 | 0.023 | 0.021 | 0.020 | 0.019 | 0.018 | 0.016 | 0.009 | −0.001 |

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## Share and Cite

**MDPI and ACS Style**

Li, J.; Zheng, Y.; Wang, Y.; Zhang, T.; Feng, P.; Engel, B.A.
Improved Mixed Distribution Model Considering Historical Extraordinary Floods under Changing Environment. *Water* **2018**, *10*, 1016.
https://doi.org/10.3390/w10081016

**AMA Style**

Li J, Zheng Y, Wang Y, Zhang T, Feng P, Engel BA.
Improved Mixed Distribution Model Considering Historical Extraordinary Floods under Changing Environment. *Water*. 2018; 10(8):1016.
https://doi.org/10.3390/w10081016

**Chicago/Turabian Style**

Li, Jianzhu, Yanchen Zheng, Yimin Wang, Ting Zhang, Ping Feng, and Bernard A. Engel.
2018. "Improved Mixed Distribution Model Considering Historical Extraordinary Floods under Changing Environment" *Water* 10, no. 8: 1016.
https://doi.org/10.3390/w10081016