# Revision of Frequency Estimates of Extreme Precipitation Based on the Annual Maximum Series in the Jiangsu Province in China

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}and is located downstream of Yangtze River and Huaihe River basins. The terrain is dominated by plains, accounting for more than 70% of the area. Hills are concentrated in the southwest, accounting for 14.3% of the total area. The terrain slopes from west to east. The river network is intricate and includes the three major river systems of the Yishusi River drainage: Downstream of the Huaihe River, the Yangtze River, and Taihu Lake stream. Jiangsu is located in a transitional subtropical to warm temperate climate zone. The area is characterized by four distinct seasons, which are cold and dry in winter, and warm and humid with plum rains in the late spring and early summer, and typhoons in summer and autumn. The annual average rainfall is 996 mm. Precipitation gradually increases from south to north and is greater on the coast than inland. Rainstorm zones are mainly located in the south of Yimeng Mountain. The elevation, stream network, and meteorological stations of the study area are shown in Figure 1.

#### 2.2. Data

_{l}, x

_{2}, x

_{3},……, x

_{N}is a collection of the maximum data for each year, where N is the number of years in the observed time series. The partial duration series y

_{l}, y

_{2}, y

_{3},……, y

_{M}is a collection of exceedance over a certain truncation level, the M

^{th}largest in the whole time series of N

_{yr}. In this study, the threshold value was equal to three. That is, the three largest daily rainfalls were selected from each year and form the PDS. The PDS was sorted and intercepted the largest N events in descending order, which includes the AES. Therefore, the AES may be regarded as a special case of the PDS. The frequency estimations for extreme precipitation based on AMS and AES were assessed and compared.

#### 2.3. Methodology

#### 2.3.1. Regional L-Moments Method

_{T,j,i}can be computed by a regional component that reflects the common precipitation character and a local component that reflects the site-specific scaling factor. The formula can be written as:

_{j}. It can be determined by a set of regional parameters that are weighted average values over N sites for a selected distribution. For example, the regional Linear coefficient of deviation (L-C

_{v}) can be written as follows:

_{v}and the single station L-C

_{v}at site i.

#### 2.3.2. Identification of Homogeneous Regions

_{i}is the site’s record lengths. Hosking and Wallis [13] suggested that a region may be considered “acceptable homogeneous” if H < 1, “possibly heterogeneous” if 1 ≤ H < 2, “definitely heterogeneous” if H ≥ 2, and “possibly correlated” if H < 0.

_{i}) is used to identify data that are grossly discordant with the region as a whole [13]. The critical values for discordancy experiments are dependent on the number of sites in the region [13]. More detailed information on these procedures can be found in [12,13].

#### 2.3.3. The Goodness-of-Fit

_{k}scale. For each distribution, the goodness-of-fit measure is defined as follows:

^{th}simulated region, after the regional average L-kurtosis ${t}_{4}^{[m]}$ obtained, the bias (B

_{4}) and standard deviation (σ

_{4}) of ${t}_{4}^{R}$ can be calculated as follows:

_{k}of the real data at N sites to accurately evaluate the distribution pattern. The RMSE is calculated for each of the plausible distributions as follows:

_{i,L-Ck}is the sample L-C

_{k}at site i and D

_{i,L-Ck}is the distribution’s L-C

_{k}at sample L-Cs of site i. The distribution with the smallest RMSE is selected as the most appropriate distribution based on this experiment. More details of the MC and RMSE methods can be found in the literature [12,13,20].

#### 2.3.4. Conversion of AES-AMS

_{AMS}and T

_{AES}corresponding to the same event, as follows:

_{AMS}and T

_{AES}are, respectively, the return period of AMS and AES.

_{NON}). However, the computer program cannot be computed if P

_{NON}equals zero. From Equation (10), it is clear that it is not computable for a 1-year event under AMS data. If Chow’s equation is applicable to this study area, we can not only correct the frequency estimation at low return periods based on AMS data, but can also compute quantiles for a 1-year recurrence interval based on Chow’s equation.

## 3. Results

#### 3.1. Results and Analysis of the Goodness-of-Fit

#### 3.2. Comparison between Exceedance Frequency and Exceedance Probability

#### 3.3. Verification of the Applicability of Chow’s Equation in the Study

#### 3.4. Reliable Frequency Estimation and Spatiotemporal Analysis

#### 3.5. Validation of Frequency Estimations of Extreme Precipitation

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Lin, B.Z.; Shao, Y.H.; Yan, G.X.; Zhang, Y.H. The core research on the development of engineering hydrology calculation promoted by hydrometeorology. New development of hydrological science and technology. In China Hydrology Symposium Proceedings; Hohai University Press: Nanjing, China, 2012; pp. 50–63. [Google Scholar]
- Sun, R.C.; Yuan, H.L.; Liu, X.L.; Jiang, X.M. Evaluation of the latest satellite–gauge precipitation products and their hydrologic applications over the Huaihe River basin. J. Hydrol.
**2016**, 536, 302–319. [Google Scholar] [CrossRef] - Ba, H.T.; Cholette, M.E.; Borghesani, P.; Zhou, Y.; Ma, L. Opportunistic maintenance considering non-homogenous opportunity arrivals and stochastic opportunity durations. Reliab. Eng. Syst. Safe
**2017**, 160, 151–161. [Google Scholar] - Hailegeorgis, T.T.; Alfredsen, K. Analyses of extreme precipitation and runoff events including uncertainties and reliability in design and management of urban water infrastructure. J. Hydrol.
**2017**, 544, 290–305. [Google Scholar] [CrossRef] - Nguyen, T.H.; Outayek, S.E.; Lim, S.H.; Nguyen, V.T.V. A systematic approach to selecting the best probability models for annual maximum rainfalls—A case study using data in Ontario (Canada). J. Hydrol.
**2017**, 553, 49–58. [Google Scholar] [CrossRef] - Haddad, K.; Rahman, A. Selection of the best fit flood frequency distribution and parameter estimation procedure—A case study for Tasmania in Australia. Stoch. Environ. Res. Risk Assess.
**2011**, 25, 415–428. [Google Scholar] [CrossRef] - Merz, B.; Thieken, A.H. Flood risk curves and uncertainty bounds. Nat. Hazards
**2009**, 51, 437–458. [Google Scholar] [CrossRef] - Cunnane, C. Statistical Distributions for Flood Frequency Analysis; Operational Hydrological Report No. 5/33; World Meteorological Organization (WMO): Geneva, Switzerland, 1989. [Google Scholar]
- England, J.F. Flood frequency and design flood estimation procedures in the United States: Progress and challenges. Austral. J. Water Resour.
**2011**, 15, 33–46. [Google Scholar] [CrossRef] - Nagy, B.K.; Mohssen, M.; Hughey, K.F.D. Flood frequency analysis for a braided river catchment in New Zealand: Comparing annual maximum and partial duration series with varying record lengths. J. Hydrol.
**2017**, 547, 365–374. [Google Scholar] [CrossRef] - Mohssen, M. Partial duration series in the annual domain. In Proceedings of the 18th World IMACS and MODSIM International Congress, Cairns, Australia, 13–17 July 2009; pp. 2694–2700. [Google Scholar]
- Shao, Y.H.; Wu, J.M.; Li, M. Study on quantile estimates of extreme precipitation and their spatiotemporal consistency adjustment over the Huaihe River basin. Theor. Appl. Climatol.
**2017**, 127, 495–511. [Google Scholar] [CrossRef] - Hosking, J.R.M.; Wallis, J.R. Regional Frequency Analysis: An Approach Based on L-Moments; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]
- Neves, M.; Gomes, D.P. Geo-statistics for spatial extremes: A case study of maximum annual rainfall in Portugal. Procedia Environ. Sci.
**2011**, 7, 246–251. [Google Scholar] [CrossRef][Green Version] - Kuo, Y.M.; Chu, H.J.; Pan, T.Y.; Yu, H.L. Investigating common trends of annual maximum rainfalls during heavy rainfall events in southern Taiwan. J. Hydrol.
**2011**, 409, 749–758. [Google Scholar] [CrossRef] - Shao, Y.H.; Wu, J.M.; Ye, J.Y.; Liu, Y.H. Frequency analysis and its spatiotemporal characteristics of precipitation extreme events in China during 1951-2010. Theor. Appl. Climatol.
**2015**, 121, 775–787. [Google Scholar] [CrossRef] - Tiwari, H.; Rai, S.P.; Sharma, N.; Kumar, D. Computational approaches for annual maximum river flow series. Ain Shams Eng. J.
**2017**, 8, 51–58. [Google Scholar] [CrossRef] - Chow, V.T.; Maidment, D.R.; Mays, L.W. Applied Hydrology; McGraw-Hill: New York, NY, USA, 1988; pp. 380–385. [Google Scholar]
- Madsen, H.; Pearson, C.P.; Rosbjerg, D. Comparison of annual maximum series and partial duration series methods for modeling extreme hydrologic events 1. At-site modeling. Water Resour. Res.
**1997**, 33, 747–757. [Google Scholar] [CrossRef] - Lin, B.Z.; Bonnin, G.M.; Martin, D.; Parzybok, T.M.; Riley, D. Regional frequency studies of annual extreme precipitation in the United States based on regional L-moments analysis. In Proceedings of the World Environmental and Water Resource Congress, Omaha, NE, USA, 21–25 May 2006; pp. 1–11. [Google Scholar]
- Chow, V.T. Handbook of Applied Hydrology; McGraw-Hill: New York, NY, USA, 1964. [Google Scholar]
- Wu, J.M.; Lin, B.Z.; Pu, J. Underestimation of precipitation quantile estimates based on AMS data. J. Nanjing Univ. Inf. Sci. Technol. Nat. Sci. Ed.
**2016**, 8, 374–379. [Google Scholar] - Bhunya, P.K.; Berndtsson, R.; Jain, S.K.; Kumar, R. Flood analysis using negative binomial and Generalized Pareto models in partial duration series (PDS). J. Hydrol.
**2013**, 497, 121–132. [Google Scholar] [CrossRef] - Karim, F.; Hasan, M.; Marvanek, S. Evaluating Annual Maximum and Partial Duration Series for Estimating frequency of Small Magnitude Floods. Water
**2017**, 9, 481. [Google Scholar] [CrossRef][Green Version] - Agilan1, V.; Umamahesh, N.V. Non-Stationary Rainfall Intensity-Duration-Frequency Relationship: A comparison between Annual Maximum and Partial Duration Series. Water Resour. Manag.
**2017**, 31, 1825–1841. [Google Scholar] [CrossRef] - Begueria, S. Uncertainties in partial duration series modeling of extremes related to the choice of threshold value. J. Hydrol.
**2005**, 303, 215–230. [Google Scholar] [CrossRef][Green Version] - Nibedita, G.; Ramakar, J. Flood Frequency Analysis of Tel Basin of Mahanadi River System, India using Annual Maximum and POT Flood Data. Aquat. Procedia
**2015**, 4, 427–434. [Google Scholar] - Ahmadi, F.; Radmaneh, F.; Parham, G.A.; Mirabbasi, R. Comparison of the performance of power law and probability distributions in the frequency analysis of flood in Dez Basin, Iran. Nat. Hazards
**2017**, 87, 1313–1331. [Google Scholar] [CrossRef] - United States Water Resources Council. Guidelines for Determining Flood Flow Frequency; Bull. 17B; Hydrology Community, Water Resources Council: Washington, DC, USA, 1982.
- Franchini, M.; Galeati, G.; Lolli, M. Analytical derivation of the flood frequency curve through partial duration series analysis and a probabilistic representation of the runoff coefficient. J. Hydrol.
**2005**, 303, 1–15. [Google Scholar] [CrossRef] - Claps, P.; Laio, F. Can continuous stream flow data support flood frequency analysis? An alternative to the partial duration series approach. Water Resour. Res.
**2003**, 39, 12–16. [Google Scholar] [CrossRef][Green Version] - Takeuchi, K. Annual maximum series and partial duration series-evaluation of Langbein’s formula and Chow’s discussion. J. Hydrol.
**1984**, 68, 275–284. [Google Scholar] [CrossRef] - Ghahraman, B.; Khalili, D. A revisit to partial duration series of short duration rainfalls. Iran. J. Sci. Technol.
**2004**, 28, 547–558. [Google Scholar] - SAS. SAS/STAT User’s Guide, Release 6.03; SAS Institute: Cary, NC, USA, 1988. [Google Scholar]
- Pilon, P.J.; Condie, R.; Harvey, K.D. Consolidated Frequency Analysis Package (CFA), User Manual for Version 1-DEC Pro Series; Water Resources Branch, Inland Waters Directorate, Environment Canada: Ottawa, ON, USA, 1985. [Google Scholar]
- Adamowski, K.; Liang, G.C.; Patry, G.G. Annual maxima and partial duration flood series analysis by parametric and non-parametric methods. Hydrol. Process
**1998**, 12, 1685–1699. [Google Scholar] [CrossRef] - Adamowski, K. Regional analysis of annual maximum and partial duration flood data by nonparametric and L-moment methods. J. Hydrol.
**2000**, 229, 219–231. [Google Scholar] [CrossRef] - Salinas, J.L.; Castellarin, A.; Kohnová, S.; Kjeldsen, T.R. Regional parent flood frequency distributions in Europe-Part 2: Climate and scale controls. Hydrol. Earth Syst. Sci.
**2014**, 18, 4391–4401. [Google Scholar] [CrossRef][Green Version] - Ball, J.; Babister, M.; Nathan, R.; Weeks, W.; Weinmann, E.; Retallick, M.; Testoni, I. Australian Rainfall and Runoff: A Guide to Flood Estimation; Geoscience Australia: Canberra, Australia, 2016.
- SL44. Regulation for Calculating Design Flood of Water Resources and Hydropower Projects in China; China WaterPower Press: Beijing, China, 2006. [Google Scholar]
- Vogel, R.M.; Fennessey, N.M. L-Moment diagrams should replace product moment diagrams. Water Resour. Res.
**1993**, 29, 1745–1752. [Google Scholar] [CrossRef] - Delicadoa, P.; Goriab, M.N. A small sample comparison of maximum likelihood, moments and L-moments methods for the asymmetric exponential power distribution. Comput. Stat. Data Anal.
**2008**, 52, 1661–1673. [Google Scholar] [CrossRef] - Gubareva, T.S.; Gartsman, B.I. Estimating distribution parameters of extreme hydrometeorological characteristics by L-moments method. Water Resour.
**2010**, 37, 437–445. [Google Scholar] [CrossRef] - Norbiato, D.; Borga, M.; Sangati, M.; Zanon, F. Regional Frequency Analysis of Extreme Precipitation in the eastern Italian Alps and the August 29, 2003 Flash Flood. J. Hydrol.
**2007**, 345, 149–166. [Google Scholar] [CrossRef] - Zakaria, Z.A.; Shabri, A. Regional frequency analysis of extreme rainfalls using partial L-moments method. Theor. Appl. Climatol.
**2013**, 113, 83–94. [Google Scholar] [CrossRef] - Du, H.; Xia, J.; Zeng, S.D. Regional frequency analysis of extreme precipitation and its spatio-temporal characteristics in the Huai River Basin, China. Nat. Hazards
**2014**, 70, 195–215. [Google Scholar] [CrossRef] - Neykov, N.M.; Neytchev, P.N.; Gelder, P.H.A.J.M.V.; Todorov, V.K. Robust detection of discordant sites in regional frequency analysis. Water Resour. Res.
**2007**, 43, 1–10. [Google Scholar] [CrossRef][Green Version] - Yang, T.; Shao, Q.X.; Hao, Z.C.; Chen, X.; Zhang, Z.X.; Xu, C.Y.; Sun, L.M. Regional frequency analysis and spatio-temporal pattern characterization of rainfall extremes in the Pearl River Basin, China. J. Hydrol.
**2010**, 380, 386–405. [Google Scholar] [CrossRef] - Hu, C.; Xia, J.; She, D.X.; Xu, C.Y.; Zhang, L.P.; Song, Z.H.; Zhao, L. A modified regional L-moment method for regional extreme precipitation frequency analysis in the Songliao River Basin of China. Atmos. Res.
**2019**, 230, 104629. [Google Scholar] [CrossRef] - Deng, P.D. Review on probability and application of two sampling methods for urban storm. Water Wastewater Eng.
**2006**, 32, 39–42. [Google Scholar]

**Figure 4.**Map of quantile estimates for (

**a**) 1-year, (

**b**) 10-year, (

**c**) 25-year, and (

**d**) 50-year return periods.

**Figure 5.**Scatterplot of estimations and observations at the same frequency for each homogeneous region.

${\mathbf{T}}_{\mathbf{AES}}\left(-\mathbf{year}\right)$ | ${\mathbf{T}}_{\mathbf{AMS}}\left(-\mathbf{year}\right)$ | $\mathbf{P}=1/{\mathbf{T}}_{\mathbf{AMS}}$ | ${\mathbf{P}}_{\mathbf{NON}}=1-1/{\mathbf{T}}_{\mathbf{AMS}}$ |
---|---|---|---|

N/A | 1 | 1.0 | 0.0 * |

1.44 | 2 | 0.50 | 0.50 |

4.48 | 5 | 0.20 | 0.80 |

9.49 | 10 | 0.10 | 0.90 |

24.50 | 25 | 0.04 | 0.96 |

49.50 | 50 | 0.02 | 0.98 |

Test Index | Distribution | Homogeneous Region | ||||
---|---|---|---|---|---|---|

I | II | III | IV | V | ||

Z | GLO | −0.17 | 2.63 | −0.28 | 2.00 | 2.47 |

GEV | −1.39 | 0.26 | −1.66 | 0.05 | 0.37 | |

GNO | −2.05 | −0.25 | −2.05 | −0.78 | −0.56 | |

GPA | −4.46 | −5.11 | −4.87 | −4.71 | −4.79 | |

PE3 | −3.21 | −1.35 | −2.83 | −2.27 | −2.24 | |

Z_{min} | GLO | GNO | GLO | GEV | GEV | |

RMSE | GLO | 0.0404 | 0.0650 | 0.0401 | 0.0591 | 0.0655 |

GEV | 0.0424 | 0.0445 | 0.0509 | 0.0395 | 0.0447 | |

GNO | 0.0834 | 0.0466 | 0.0604 | 0.0452 | 0.0470 | |

GPA | 0.0555 | 0.0885 | 0.1239 | 0.0811 | 0.0773 | |

PE3 | 0.1057 | 0.0546 | 0.0797 | 0.0632 | 0.0628 | |

RMSE_{min} | GLO | GEV | GLO | GEV | GEV |

Region | Return Period (R.P.)/Exceedance Probability (E.P.) | |||||
---|---|---|---|---|---|---|

2-yr | 5-yr | 10-yr | 25-yr | 50-yr | 100-yr | |

0.50 | 0.20 | 0.10 | 0.04 | 0.02 | 0.01 | |

I | 0.518 | 0.210 | 0.124 | 0.051 | 0.022 | 0.012 |

II | 0.505 | 0.201 | 0.106 | 0.045 | 0.021 | 0.011 |

III | 0.500 | 0.201 | 0.120 | 0.044 | 0.020 | 0.010 |

IV | 0.508 | 0.217 | 0.101 | 0.041 | 0.022 | 0.013 |

V | 0.502 | 0.202 | 0.106 | 0.044 | 0.021 | 0.009 |

Average E.P. | 0.507 | 0.206 | 0.111 | 0.045 | 0.021 | 0.011 |

Real R.P. | 1.97-yr | 4.85-yr | 8.99-yr | 22.25-yr | 47.49-yr | 91.87-yr |

**Table 4.**Quantile estimates of extreme precipitation for a 1-day duration with different return periods in homogeneous region I.

Site Name | Quantile Estimates Based on AES Data | |||||
---|---|---|---|---|---|---|

2-yr | 5-yr | 10-yr | 25-yr | 50-yr | 100-yr | |

Fengxian | 92.2 | 117.3 | 139.7 | 175.1 | 206.8 | 243.3 |

Peixian | 98.4 | 125.2 | 149.1 | 186.9 | 220.7 | 259.7 |

Pizhou | 106.6 | 135.6 | 161.5 | 202.4 | 239.0 | 281.2 |

Xuzhou | 103.2 | 131.3 | 156.4 | 196.0 | 231.5 | 272.4 |

Xinyi | 98.2 | 124.9 | 148.8 | 186.4 | 220.2 | 259.1 |

Donghai | 98.9 | 125.9 | 149.9 | 187.9 | 221.9 | 261.1 |

Suining | 115.4 | 146.8 | 174.9 | 219.2 | 258.9 | 304.6 |

Suyu | 116.5 | 148.2 | 176.5 | 221.2 | 261.3 | 307.5 |

Siyang | 104.9 | 133.4 | 158.9 | 199.1 | 235.2 | 276.7 |

Sihong | 104.5 | 132.7 | 154.9 | 185.5 | 209.6 | 234.5 |

Quantile estimates based on AMS data and Chow’s equation | ||||||

Fengxian | 91.0 | 116.6 | 139.1 | 174.9 | 207.9 | 246.9 |

Peixian | 99.4 | 127.4 | 152.0 | 191.1 | 227.2 | 269.8 |

Pizhou | 106.0 | 135.9 | 162.1 | 203.8 | 242.2 | 287.6 |

Xuzhou | 103.9 | 133.2 | 158.8 | 199.7 | 237.3 | 281.9 |

Xinyi | 96.8 | 124.1 | 148.0 | 186.1 | 221.2 | 262.7 |

Donghai | 99.0 | 126.9 | 151.4 | 190.3 | 226.2 | 268.6 |

Suining | 111.7 | 143.2 | 170.8 | 214.7 | 255.2 | 303.1 |

Suyu | 111.9 | 143.4 | 171.0 | 215.0 | 255.5 | 303.5 |

Siyang | 105.6 | 135.3 | 161.4 | 202.9 | 241.1 | 286.4 |

Sihong | 100.0 | 128.2 | 152.9 | 192.2 | 228.5 | 271.4 |

Mean RE (%) | 1.72 | 1.60 | 1.41 | 1.68 | 2.49 | 3.64 |

Homogeneous Regions | RE (%) | RMSE (mm) | r |
---|---|---|---|

Region I | 5.47 | 0.101 | 0.975 |

Region II | 5.31 | 0.093 | 0.975 |

Region III | 4.77 | 0.09 | 0.971 |

Region IV | 5.88 | 0.118 | 0.965 |

Region V | 5.96 | 0.119 | 0.961 |

All | 5.56 | 0.107 | 0.969 |

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**MDPI and ACS Style**

Shao, Y.; Zhao, J.; Xu, J.; Fu, A.; Wu, J. Revision of Frequency Estimates of Extreme Precipitation Based on the Annual Maximum Series in the Jiangsu Province in China. *Water* **2021**, *13*, 1832.
https://doi.org/10.3390/w13131832

**AMA Style**

Shao Y, Zhao J, Xu J, Fu A, Wu J. Revision of Frequency Estimates of Extreme Precipitation Based on the Annual Maximum Series in the Jiangsu Province in China. *Water*. 2021; 13(13):1832.
https://doi.org/10.3390/w13131832

**Chicago/Turabian Style**

Shao, Yuehong, Jun Zhao, Jinchao Xu, Aolin Fu, and Junmei Wu. 2021. "Revision of Frequency Estimates of Extreme Precipitation Based on the Annual Maximum Series in the Jiangsu Province in China" *Water* 13, no. 13: 1832.
https://doi.org/10.3390/w13131832