# Comparative Study of Regional Frequency Analysis and Traditional At-Site Hydrological Frequency Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Methods

#### 2.2.1. Regional Frequency Analysis

_{v}: $\tau ={\lambda}_{2}/{\lambda}_{1}$;

_{s}: ${\tau}_{3}={\lambda}_{3}/{\lambda}_{2}$;

_{k}: ${\tau}_{4}={\lambda}_{4}/{\lambda}_{2}$.

#### 2.2.2. Data Preparation

#### 2.2.3. Identification of Homogeneous Regions

- (1)
- The discordancy measure. The discordancy measure is mentioned in 2.2.2. In addition to being used to detect whether the original data is reasonable or not, the discordancy measure can also be used to test whether each site in the initially formed homogeneous region is a discordant point of the region. If the discordance statistic Di of a site exceeds the critical value, moving the site to another adjacent homogeneous region may be considered.
- (2)
- The heterogeneity measure. Hosking et al. [31] recommended the use of a heterogeneity measure to determine whether the region formed by preliminary division is a homogeneous region. The calculation formula for the heterogeneity measure is as follows:$${\mathrm{H}}_{1}=\frac{({\mathrm{V}}_{1}-{\mathsf{\mu}}_{\mathrm{v}})}{{\mathsf{\sigma}}_{\mathrm{v}}},$$$${\mathrm{V}}_{1}={\{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{n}}_{\mathrm{i}}}{({\mathrm{t}}^{\mathrm{i}}-{\mathrm{t}}^{\mathrm{R}})}^{2}/{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{n}}_{\mathrm{i}}}\}}^{\frac{1}{2}},$$$${\mathrm{t}}^{\mathrm{R}}={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{n}}_{\mathrm{i}}}{\mathrm{t}}^{\mathrm{i}}/{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{n}}_{\mathrm{i}}},$$
_{i}is the length of the rainfall data series of the i-th station in the homogeneous region, t_{i}is the L-C_{v}of the i-th station in the homogeneous region, N is the total number of stations in the homogeneous region, and μ_{v}and σ_{v}are the mean and mean square error of V_{1}obtained by Monte Carlo simulations, respectively. H_{1}< 1 indicated that the area was an acceptable homogeneous region, 1 ≤ H_{1}< 2 indicated that the area was a possible heterogeneous area, and H_{1}≥ 2 indicated that the area was a heterogeneous area. If the value of H_{1}did not meet the standard, the site within the region needed to be adjusted until it met the criteria to become an acceptable homogeneous region. According to experience, there were several useful ways to adjust the sites: (1) move a site or few sites from one region to the adjacent one; (2) subdivide the region; (3) merge two or more regions and redefine groups; and (4) break up the region by reassigning its sites to other regions.

#### 2.2.4. The Optimal Frequency Distribution

^{DIST}) to determine the optimal frequency distribution for each homogeneous region.

_{i}, and the sample L-moment ratios of the station are, respectively, t

^{(i)}, t

_{3}

^{(i)}, t

_{4}

^{(i)}, and the t

^{R}, t

_{3}

^{R}, and t

_{4}

^{R}are, respectively, the corresponding weighted average of the measured maximum annual rainfall sequence length of all stations in the homogeneous region, and t

_{4}

^{m}is the regional average linear kurtosis coefficient for the mth simulation result.

_{4}

^{m}is as follows:

_{k}of the corresponding simulation data is as follows:

^{DIST}of the goodness-of-fit test is:

^{DIST}was not more than 1.64, the fitting result of the frequency distribution was considered to be acceptable. And the closer to 0, the better the fitting effect of the frequency distribution was, and the corresponding frequency distribution was the optimal regional frequency distribution.

#### 2.2.5. Rainfall Estimation

- (1)
- The annual maximum daily rainfall data of each station in the homogeneous region is averaged to generate a new rainfall data sequence. The calculation expression is as follows:$${q}_{\mathrm{ij}}={Q}_{\mathrm{ij}}/\overline{{Q}_{i}},$$
_{ij}represents the measured rainfall data sequence of the i-th site; and $\overline{{Q}_{i}}$ is the average rainfall value of the measured rainfall data sequence of the i-th site. - (2)
- The L-moment parameters L-C
_{v}, L-C_{s}, and L-C_{k}, and the weighted average of the L-moment parameters of each station in each homogeneous region are obtained, that is, the regional average L-moment parameters of the homogeneous region is obtained. Its calculation expression is:$${t}^{R}=\sum _{i-1}^{N}{n}_{i}{t}^{\left(i\right)}/\sum _{i-1}^{N}{n}_{i}$$$${t}_{r}^{R}=\sum _{i-1}^{N}{n}_{i}{t}_{r}^{\left(i\right)}/\sum _{i-1}^{N}{n}_{i}\xb7\mathrm{r}=3,4,$$^{R}and t_{r}^{R}are the weighted average of the L-moment parameters of each station in each homogeneous region. N is the number of sites in the region, n_{i}is the measured rain data sequence length for the i-th site. - (3)
- Estimate the distribution function parameters of the optimal frequency distribution of the region determined by the goodness-of-fit test, and then the quantiles under different return periods are obtained by the quantile functions of the chosen frequency distribution.
- (4)
- The principle of the index flood method is to treat the rainfall at each site as two components, one representing the rainfall unique to the site and the other representing the rainfall that reflects the rainfall characteristics common to the homogeneous region. The quantiles under different return periods represent the rainfall characteristics shared by the reaction area and the unique rainfall characteristics of the site, that is, the multi-year average rainfall of the site. The expression of the rainfall estimation is as follows:$${Q}_{T,i,j}={q}_{T,i}\times \overline{{x}_{i,j}},$$
_{T,i,j}is the estimation of the rainstorm in which the return period of the j-th station is in the i-th region. q_{T,i}is the quantile corresponding to the return period of the i-th region. $\overline{{x}_{i,j}}$ is the average annual maximum daily rainfall for site j.

#### 2.2.6. Traditional At-Site Hydrological Frequency Analysis

## 3. Results

#### 3.1. The Differences of Optimal Frequency Distributions

#### 3.2. Comparison of Fitting Results

#### 3.3. The Differences between the Estimations

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Schematic diagram of the study area and the selected representative meteorological site map.

**Figure 2.**(

**a**–

**r**) are the results of the design frequency curves obtained by the two methods fitted to observed data for 18 representative stations. RFA: regional frequency analysis; At-site: traditional at-site hydrological frequency analysis; Observations: observed annual maximum daily precipitation data series.

**Figure 3.**(

**a**–

**r**) are the results of the differences of rainfall design values under different return periods obtained by using regional frequency analysis and traditional at-site hydrological frequency analysis of 18 representative stations. (RFA: regional frequency analysis; At-site: traditional at-site hydrological frequency analysis; Absolute difference: the rainfall design value obtained by At-site minus the corresponding result obtained by RFA; Relative difference: the Absolute difference relative to the corresponding result obtained using RFA).

Region | Name | Number | Discordancy | Length (Years) | Average (mm) | C_{v} | C_{s} | C_{k} |
---|---|---|---|---|---|---|---|---|

1 | Litang | 56,257 | 0.16 | 57 | 35.68 | 0.1338 | 0.1562 | 0.1660 |

Xinlong | 56,251 | 1.96 | 50 | 28.38 | 0.1167 | 0.0577 | 0.2052 | |

2 | Jiuzhaigou | 56,097 | 0.2 | 50 | 30.83 | 0.1440 | 0.1723 | 0.1747 |

A’ba | 56,171 | 1.93 | 55 | 33.22 | 0.1631 | 0.3773 | 0.3467 | |

3 | Liangzhong | 57,306 | 0.02 | 51 | 100.65 | 0.2101 | 0.1757 | 0.1748 |

Xichong | 57,309 | 1.75 | 49 | 92.22 | 0.2042 | 0.1660 | 0.2385 | |

4 | Beichuan | 56,194 | 0.19 | 49 | 153.43 | 0.2218 | 0.1998 | 0.1557 |

Shifang | 56,197 | 2.37 | 49 | 116.17 | 0.2531 | 0.2122 | 0.0773 | |

5 | Shehong | 57,401 | 0.05 | 50 | 98.34 | 0.2074 | 0.1931 | 0.1840 |

Pengxi | 57,402 | 2.86 | 52 | 97.71 | 0.2757 | 0.2967 | 0.1890 | |

6 | Minshan | 56,280 | 0.06 | 50 | 125.29 | 0.2081 | 0.2135 | 0.1950 |

Tianquan | 56,278 | 2.8 | 51 | 100.42 | 0.1591 | 0.0583 | 0.1662 | |

7 | Yuexi | 56,475 | 0.38 | 56 | 61.97 | 0.1732 | 0.2695 | 0.2384 |

Xide | 56,478 | 1.58 | 50 | 68.88 | 0.2017 | 0.2868 | 0.2440 | |

8 | Xuyong | 57,608 | 0.11 | 52 | 78.60 | 0.2175 | 0.2795 | 0.2333 |

Hejiang | 57,603 | 2.02 | 52 | 88.77 | 0.1752 | 0.2042 | 0.1831 | |

9 | Miyi | 56,670 | 0.07 | 53 | 80.44 | 0.1573 | 0.1627 | 0.1650 |

Yanbian | 56,665 | 2.65 | 51 | 83.73 | 0.1794 | 0.3564 | 0.3705 |

Number of Sites in Region | Critical Value |
---|---|

5 | 1.333 |

6 | 1.648 |

7 | 1.917 |

8 | 2.14 |

9 | 2.329 |

10 | 2.491 |

11 | 2.632 |

12 | 2.757 |

13 | 2.869 |

14 | 2.971 |

≥15 | 3 |

**Table 3.**Monte Carlo simulation test results for each representative site and the optimal frequency distribution for each homogeneous region.

Region | Site | Z^{DIST} | At-Site Optimal Frequency Distribution | Regional Optimal Frequency Distribution | ||||
---|---|---|---|---|---|---|---|---|

GLO | GEV | GNO | PE3 | GPD | ||||

1 | 56257 | −0.83 | −1.41 | −1.39 | −1.47 | −2.57 | GLO | GLO |

56251 | −1.38 | −2.37 | −2.2 | −2.23 | −4.25 | GLO | ||

2 | 56097 | 0.21 | −0.46 | −0.55 | −0.78 | −1.95 | GLO | GLO |

56171 | −1.36 | −1.54 | −1.83 | −2.32 | −2.14 | GLO | ||

3 | 57306 | 0.25 | −0.43 | −0.52 | −0.76 | −1.91 | GLO | GLO |

57309 | −1.59 | −2.24 | −2.3 | −2.51 | −3.65 | GLO | ||

4 | 56194 | 0.8 | 0.11 | −0.04 | −0.38 | −1.46 | GNO | GEV |

56197 | 2.99 | 2.15 | 1.92 | 1.45 | 0.21 | GPD | ||

5 | 57401 | 0.14 | −0.46 | −0.58 | −0.85 | −1.81 | GLO | GLO |

57402 | 0.67 | 0.28 | −0.02 | −0.53 | −0.77 | GNO | ||

6 | 56280 | 0.05 | −0.47 | −0.63 | −0.92 | −1.7 | GLO | GLO |

56278 | 0.02 | −0.98 | −0.81 | −0.85 | −2.88 | GLO | ||

7 | 56475 | −0.41 | −0.79 | −1.01 | −1.39 | −1.75 | GLO | GLO |

56478 | −0.35 | −0.67 | −0.89 | −1.27 | −1.51 | GLO | ||

8 | 57608 | −0.16 | −0.5 | −0.72 | −1.09 | −1.38 | GLO | GLO |

57603 | 0.23 | −0.35 | −0.49 | −0.78 | −1.68 | GLO | ||

9 | 56670 | 0.36 | −0.37 | −0.43 | −0.65 | −1.94 | GLO | GLO |

56665 | −2.08 | −2.29 | −2.56 | −3.03 | −2.93 | Wakeby |

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**MDPI and ACS Style**

Li, M.; Li, X.; Ao, T.
Comparative Study of Regional Frequency Analysis and Traditional At-Site Hydrological Frequency Analysis. *Water* **2019**, *11*, 486.
https://doi.org/10.3390/w11030486

**AMA Style**

Li M, Li X, Ao T.
Comparative Study of Regional Frequency Analysis and Traditional At-Site Hydrological Frequency Analysis. *Water*. 2019; 11(3):486.
https://doi.org/10.3390/w11030486

**Chicago/Turabian Style**

Li, Mengrui, Xiaodong Li, and Tianqi Ao.
2019. "Comparative Study of Regional Frequency Analysis and Traditional At-Site Hydrological Frequency Analysis" *Water* 11, no. 3: 486.
https://doi.org/10.3390/w11030486