# Evaluation of Methods for Estimating Long-Term Flow Fluctuations Using Frequency Characteristics from Wavelet Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data

#### 2.2. FFI

^{3}/s) and median discharge (m

^{3}/s), respectively. ${Q}_{\mathsf{\sigma}}$ is the standard deviation (m

^{3}/s). ${CV}_{\mu}$ and ${CV}_{m}$ are the coefficients of variation based on ${Q}_{\mu}$ and ${Q}_{m}$, respectively. Table 2 summarizes the FFIs mentioned earlier and provides a brief description of each.

#### 2.3. Wavelet Transform

#### 2.4. Regionalization of Basins

## 3. Results

#### 3.1. Pre-Analysis and Estimation of FFI and L-FFI

^{3}/s for the minimum and 4922.2 m

^{3}/s for the maximum. Hence, using these values, an RRC of 1059.8 is derived. Lastly, for Method (3), using the entire dataset, the RRC value is determined from the minimum value of 0.1 m

^{3}/s (in 2010 or 2018) and the maximum value of 15,126.4 m

^{3}/s (in 2006), resulting in an RRC value of 151,264.

#### 3.2. Wavelet Analysis

#### 3.3. Evaluation of Methods for L-FFI Estimation

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Daily inflow (black solid line) and mass curve (blue dashed line) of Chungju Dam from 1986 to 2021.

**Figure 3.**Daily precipitation (black solid line) and mass curve (blue dashed line) of Chungju Dam from 1986 to 2021.

**Figure 8.**Wavelet analysis results using raw data. For each set of figures, (

**a**) denotes the data for wavelet analysis, (

**b**) represents the results of the wavelet analysis, and (

**c**) indicates the global power of the wavelet. The dark solid line in (

**b**) and dotted line in (

**c**) indicate the 95% confidence level.

**Figure 9.**Wavelet analysis results using seasonally differenced data. For each set of figures, (

**a**–

**c**) are the same as described in Figure 8.

**Figure 10.**Heatmaps of correlation coefficients among the LFFIs determined using different methods. Method (1): calculate the annual index and then average it; Method (2): average the annual flow characteristics and then calculate the index; and Method (3): calculate the index considering all available data.

**Figure 11.**Scatter plots of the M-GWP frequency vs. L-FFIs. In (

**a**), the red box zooms in on the red shadow region (RRC of 0–10,000).

**Figure 12.**Heatmap of (

**a**) correlation coefficient and (

**b**) determination coefficient among the M-GWP frequency vs. L-FFIs obtained using different methods.

**Figure 13.**Results of regionalization through cluster analysis. (

**a**–

**c**) utilized all L-FFIs considered in this study, while (

**d**) employed the global wavelet power for all scales.

Dam Name | Basin Area (km^{2}) | Channel Length (km) | Total Storage (10^{6} m^{3}) |
---|---|---|---|

Paldang | 23,800 | 377.6 | 244 |

Uiam | 7709 | 230.9 | 80 |

Chungju | 6648 | 270.6 | 2750 |

Hwacheon | 3901 | 178.1 | 1018 |

Daecheong | 3204 | 208.1 | 1490 |

Soyanggang | 2703 | 136.0 | 2900 |

Andong | 1584 | 142.4 | 1248 |

Seomjingang | 763 | 74.2 | 466 |

Symbol | Equation | Index | Description |
---|---|---|---|

$RRC$ | ${Q}_{\mathrm{max}}/{Q}_{\mathrm{min}}$ | River regime coefficient | The ratio of the maximum flow to the minimum flow |

${C}_{R}$ | ${Q}_{2.7}/{Q}_{97.3}$ | Flow regime coefficient | The ratio of the high flow to the low flow excepting for the extreme flows |

${C}_{F}$ | ${Q}_{\mathrm{m}\mathrm{a}\mathrm{x}}/{Q}_{50.7}$ | Flood coefficient | The ratio of the maximum flow to the approximate mode flow |

${C}_{A}$ | ${Q}_{26.0}/{Q}_{50.7}$ | Abundance coefficient | The ratio of the approximate third quartile flow to the approximate median flow |

${C}_{L}$ | ${Q}_{75.3}/{Q}_{50.7}$ | Low coefficient | The ratio of the approximate first quartile flow to the approximate median flow |

${C}_{D}$ | ${Q}_{97.3}/{Q}_{50.7}$ | Drought coefficient | The ratio of the high flow to the approximate median flow |

${C}_{VD}$ | ${Q}_{26.0}/{Q}_{97.3}$ | Variance of the drought coefficient | The ratio of the approximate third quartile flow to the low flow |

${C}_{VL}$ | ${Q}_{26.0}/{Q}_{75.3}$ | Variance of the low coefficient | The ratio of the approximate third quartile flow to the approximate first quartile flow |

${CV}_{\mu}$ | ${Q}_{\mathsf{\sigma}}/{Q}_{\mu}$ | Coefficient of variance based on mean | The variation of the flows to the mean flow |

${CV}_{m}$ | ${Q}_{\mathsf{\sigma}}/{Q}_{m}$ | Coefficient of variance based on median | The variation of the flows to the median flow |

Year | Min | Max | RRC | Year | Min | Max | RRC |
---|---|---|---|---|---|---|---|

1986 | 10.3 | 3683.9 | 357.7 | 2004 | 4.1 | 4154.6 | 1013.3 |

1987 | 14.4 | 6731.8 | 467.5 | 2005 | 2.1 | 1586.5 | 755.5 |

1988 | 8.1 | 6121.9 | 755.8 | 2006 | 4.4 | 12,599.5 | 2863.5 |

1989 | 10.0 | 7043.6 | 704.4 | 2007 | 1.3 | 5575.2 | 4288.6 |

1990 | 17.0 | 13,142.1 | 773.1 | 2008 | 1.0 | 7808.1 | 7808.1 |

1991 | 12.9 | 2427.0 | 188.1 | 2009 | 0.4 | 3084.2 | 7710.5 |

1992 | 6.0 | 1692.0 | 282.0 | 2010 | 0.2 | 15,126.4 | 75,632.0 |

1993 | 16.0 | 4250.0 | 265.6 | 2011 | 0.2 | 2735.2 | 13,676.0 |

1994 | 9.6 | 4003.4 | 417.0 | 2012 | 0.2 | 6662.3 | 33,311.5 |

1995 | 7.1 | 8776.2 | 1236.1 | 2013 | 0.8 | 5012.7 | 6265.9 |

1996 | 6.1 | 1496.3 | 245.3 | 2014 | 0.1 | 4586.1 | 45,861.0 |

1997 | 8.0 | 3428.5 | 428.6 | 2015 | 0.3 | 5059.9 | 16,866.3 |

1998 | 14.8 | 5897.2 | 398.5 | 2016 | 1.4 | 3373.7 | 2409.8 |

1999 | 5.5 | 8156.7 | 1483.0 | 2017 | 0.9 | 4379.9 | 4866.6 |

2000 | 4.1 | 4154.6 | 1013.3 | 2018 | 0.4 | 989.1 | 2472.8 |

2001 | 2.1 | 1586.5 | 755.5 | 2019 | 0.3 | 700.4 | 2334.7 |

2002 | 4.4 | 12,599.5 | 2863.5 | 2020 | 0.2 | 3663.7 | 18,318.5 |

2003 | 1.3 | 5575.2 | 4288.6 | 2021 | 0.2 | 2467.1 | 12,335.5 |

Index | Method (1) | Method (2) | Method (3) |

$RRC$ | 8723.5 | 1059.8 | 151,264.0 |

${C}_{R}$ | 200.4 | 108.1 | 167.7 |

${C}_{F}$ | 100.9 | 96.2 | 305.6 |

${C}_{A}$ | 2.2 | 2.2 | 2.2 |

${C}_{L}$ | 0.5 | 0.5 | 0.5 |

${C}_{D}$ | 0.2 | 0.2 | 0.1 |

${C}_{VD}$ | 20.7 | 12.0 | 17.3 |

${C}_{VL}$ | 4.3 | 4.3 | 4.3 |

${CV}_{\mu}$ | 2.6 | 2.7 | 3.1 |

${CV}_{m}$ | 8.4 | 8.1 | 9.7 |

Dam | M-GWP [(m^{3}/s)^{2}] | Period (year) | Frequency (1/y) |
---|---|---|---|

Paldang | 107,505,864 | 0.076 | 13.1 |

Uiam | 13,904,177 | 0.045 | 22.1 |

Chungju | 25,609,269 | 0.032 | 31.2 |

Hwacheon | 5,821,673 | 0.045 | 22.1 |

Daecheong | 6,286,009 | 0.038 | 26.3 |

Soyanggang | 5,658,015 | 0.027 | 37.1 |

Andong | 979,627 | 0.023 | 44.2 |

Seomjingang | 488,983 | 0.023 | 44.2 |

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**MDPI and ACS Style**

Lee, J.; Moon, G.; Lee, J.; Jun, C.; Choi, J.
Evaluation of Methods for Estimating Long-Term Flow Fluctuations Using Frequency Characteristics from Wavelet Analysis. *Water* **2023**, *15*, 2968.
https://doi.org/10.3390/w15162968

**AMA Style**

Lee J, Moon G, Lee J, Jun C, Choi J.
Evaluation of Methods for Estimating Long-Term Flow Fluctuations Using Frequency Characteristics from Wavelet Analysis. *Water*. 2023; 15(16):2968.
https://doi.org/10.3390/w15162968

**Chicago/Turabian Style**

Lee, Jinwook, Geonsoo Moon, Jiho Lee, Changhyun Jun, and Jaeyong Choi.
2023. "Evaluation of Methods for Estimating Long-Term Flow Fluctuations Using Frequency Characteristics from Wavelet Analysis" *Water* 15, no. 16: 2968.
https://doi.org/10.3390/w15162968