# Prediction of Typhoon-Induced Flood Flows at Ungauged Catchments Using Simple Regression and Generalized Estimating Equation Approaches

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## Abstract

**:**

## 1. Introduction

## 2. Case Study Data and Preliminary Data Analysis

#### 2.1. Geum River Region

- -
- Area is size of the catchment, which affects flood volumes;
- -
- ALTBAR is the mean altitude of catchment above sea level;
- -
- DPS is the mean of the catchment slope, which affects surface runoff response times;
- -
- DD is the Drainage Density, a measure of the total length of all the rivers in a catchment area divided by the total area of the catchment, which affects how efficiently a catchment is drained;
- -
- FF is the Form Factor, the ratio of the catchment area to the squared value of the total catchment drainage length. It varies from zero (in a highly elongated area) to 1 (in a perfectly circular area), affecting runoff response times. DD and FF are calculated based on equations provided by National Water Resources Management Information System (WAMIS) in Korea;
- -
- Curve Number (CN) is an empirical parameter for predicting direct runoff, developed by the US Soil Conservation System, and affects the volume of runoff during a storm [20];
- -
- FARL is Flood Attenuation Factor by Reservoir and Lakes, estimated based on the reservoir data and the catchment terrain database in WAMIS, affecting flood attenuation [21];
- -
- SAAR is Standard Annual Average Rainfall in the period 1981 to 2010, which represents effects of long-term catchment wetness.

#### 2.2. Typhoons in the Geum River Region

## 3. Method

#### 3.1. PDM Rainfall Runoff Model

_{k}is the rainfall, ae

_{k}is the actual evapotranspiration, S

_{k}is the soil moisture storage, Q is the runoff, c

_{k}is the soil moisture storage capacity, C

_{max}is the maximum soil moisture storage capacity of the catchment, and b is a parameter that defines the strength of spatial variation of c. c is distributed uniformly between values of 0 and C

_{max}if b is equal to 1; and tends towards a single value, C

_{max}, as b tends towards 0.

#### 3.2. Calibration

#### 3.3. Statistical Regression

#### 3.3.1. Conventional Regression

#### 3.3.2. Generalized Estimating Equations (GEEs)

#### 3.4. Validation

## 4. Results

#### 4.1. Calibration of the PDM Model

#### 4.2. Statistical Regression

^{2}of these equations varies between 0.1 (Cmax) to 0.79 (f).

^{2}values in Table 9 are low compared to those in Table 7. This is expected because of the much lower variance of the MPs when they are averaged over the typhoons so that only 23 data points are used, as was done to obtain the results in Table 7, compared to the 107 data points used to obtain the results in Table 9 (107 points rather than 113 because the 6 points from C4 are kept back for model validation). In particular, the non-identifiable MPs would increase variance of the errors more if they are not averaged over events. The results in Table 9 may also be influenced by the covariance of the MP values between flood events in each catchment, whereby if the matrix R Equation (6) could be more accurately specified it may improve the GEE result.

## 5. Conclusions

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- The application of a calibrated PDM model to modelling flood event flows in Geum River catchments shows, overall, an acceptable model performance. This supports the use of the PDM model or comparable rainfall-runoff models for simulating extreme flood events in Korea.
- -
- Using conventional regression equations to regionalize model parameters to ungauged catchments showed mixed success, for example when treating the C4 catchment as ungauged, there were good results for two flood events (typhoons Nari and Dianmu) and underestimated peaks for two events (typhoons Rusa and Kompasu).
- -
- The GEE model extends the conventional regression by including the inter-event variability in PDM model parameters as well as the inter-catchment variability. However only the model parameter b was found to be related to event properties; and validation results showed only slight improvement on the simpler regression approach. While for practical applications we would therefore recommend the simpler regression approach, refinements to the GEE approach may be explored, in particular its potential advantage for estimating an error model and confidence limits on predictions.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A. GEE for Regionalization of Rainfall Runoff Model

## References

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**Figure 2.**Box plot of the flood characteristics of the 113 flood events (Boxes are 25% and 75% quantiles, R = Rusa, M = Maemi, N = Nari, D = Dianmu, K = Kompasu, B = Bolaven, T = Total).

**Figure 4.**Schematic description of the PDM model [28].

**Figure 8.**Comparison between the NSE values obtained using calibration, conventional regression and GEEs.

Catchments | Area (km^{2}) | ALTBAR (m) | FF () | DD () | SAAR (mm) | FARL () | CN () | DPS (°) | PG () | |
---|---|---|---|---|---|---|---|---|---|---|

Bukil | C1 | 907.1 | 151 | 0.3 | 2.1 | 1222 | 0.92 | 67.4 | 9.7 | 4 |

Boksu | C2 | 162.5 | 229 | 0.1 | 4 | 1274 | 1 | 70.8 | 15.2 | 2 |

Cheoncheon | C3 | 291.2 | 555 | 0.3 | 3.3 | 1093 | 0.96 | 65.4 | 14.6 | 1 |

Cheongju | C4 | 166.3 | 159 | 0.2 | 2.4 | 1192 | 0.98 | 71.7 | 9.7 | 2 |

Cheongseong | C5 | 491 | 270 | 0.1 | 3.1 | 1198 | 0.95 | 65.6 | 12.9 | 2 |

Donghyang | C6 | 165.4 | 647 | 0.2 | 1.7 | 1231 | 0.96 | 60 | 16.2 | 3 |

Gideagyo | C7 | 353.1 | 284 | 0.3 | 3.1 | 1181 | 0.97 | 65.4 | 12.9 | 1 |

Guryong | C8 | 208.2 | 173 | 0.1 | 3 | 1258 | 0.99 | 72.3 | 12.3 | 2 |

Gasangyo | C9 | 183.9 | 141 | 0.3 | 1.7 | 1199 | 0.95 | 68.8 | 6.3 | 2 |

Habgang | C10 | 1853 | 136 | 0.2 | 2.4 | 1225 | 0.95 | 69.1 | 8.4 | 6 |

Indong | C11 | 58.7 | 217 | 0.1 | 3.4 | 1244 | 1 | 64.6 | 14.7 | 2 |

Juengpyung | C12 | 124.1 | 150 | 0.4 | 2.4 | 1206 | 0.96 | 69.3 | 8.6 | 2 |

Mihogyo | C13 | 1596 | 137 | 0.3 | 2.4 | 1192 | 0.98 | 63 | 9.7 | 2 |

Muju | C14 | 390.2 | 615 | 0.1 | 1.5 | 1154 | 0.99 | 59 | 18.9 | 4 |

Nonsan | C15 | 467.7 | 148 | 0.2 | 2.3 | 1318 | 0.91 | 62.5 | 11.4 | 4 |

Ochang | C16 | 564 | 154 | 0.2 | 1.9 | 1221 | 0.91 | 66 | 9.8 | 2 |

Simcheon | C17 | 664.3 | 372 | 0.2 | 2.2 | 968 | 0.98 | 61 | 14.7 | 2 |

Seokdong | C18 | 162.4 | 78 | 0.2 | 2.7 | 1271 | 0.96 | 66.2 | 8.1 | 3 |

Sangyegyo | C19 | 464.9 | 272 | 0.2 | 3.1 | 1195 | 0.96 | 65.6 | 12.9 | 2 |

Songcheon | C20 | 623.4 | 385 | 0.2 | 2.2 | 969 | 0.97 | 60.6 | 14.8 | 2 |

Seokhwa | C21 | 1594.1 | 143 | 0.3 | 2.3 | 1223 | 0.94 | 68.6 | 9.7 | 5 |

Tanbugyo | C22 | 81.7 | 363 | 0.2 | 2.5 | 1090 | 0.93 | 54.2 | 17 | 1 |

Woogon | C23 | 131.8 | 45 | 0.3 | 2.8 | 1235 | 0.98 | 68.9 | 5 | 1 |

Yuseong | C24 | 251.1 | 189 | 0.1 | 3.5 | 1315 | 0.99 | 62.8 | 12.1 | 2 |

Typhoon | Period |
---|---|

Rusa | 23 August 2002~6 September 2002 |

Maemi | 6 September 2003~19 September 2003 |

Nari | 13 September 2007~22 September 2007 |

Dianmu | 8 August 2010~17 August 2010 |

Kompasu | 29 August 2010~8 September 2010 |

Bolaven | 20 August 2012~3 September 2012 |

Typhoon | Rusa | Maemi | Nari | Dianmu | Kompasu | Bolaven |
---|---|---|---|---|---|---|

TR (Total Rainfall, (mm)) | 129 | 72 | 92 | 90 | 41 | 107 |

TD (Total Discharge, (mm)) | 114 | 70 | 79 | 46 | 16 | 66 |

MRI (Maximum Rainfall Intensity, (mm/h)) | 14 | 10 | 10 | 42 | 28 | 21 |

PF (Peak Flood, (mm/h)) | 6 | 3 | 5 | 4 | 3 | 5 |

RD (Rainfall Duration, (h)) | 17 | 9 | 11 | 7 | 2 | 15 |

LT (Lag Time, (h)) | 4 | 3 | 6 | 4 | 1 | 3 |

PE (Potential Evapotranspiration, (mm)) | 21 | 12 | 5 | 3 | 3 | 15 |

RR (Runoff Ratio to rainfall, ()) | 0.88 | 0.98 | 0.85 | 0.52 | 0.38 | 0.62 |

P5 (mm/5 days) | 31 | 43 | 91 | 153 | 44 | 10 |

MPs | Description | Range (Units) |
---|---|---|

C_{max} | Maximum storage capacity of the catchment | 1~500 (mm) |

b | Degree of spatial variability of storage capacity in the catchment | 0~2 () |

rtq | Residence time of the quick flow reservoir | 0~15 (h) |

rts | Residence time of the slow flow reservoir | 50~500 (h) |

f | Fraction of effective rainfall that enters the quick flow reservoir and 1-f of effective rainfall that enters the slow flow reservoir | 0~1 () |

Typhoon | C_{max} (mm) | b () | rtq (h) | rts (h) | f () | NSE |
---|---|---|---|---|---|---|

Rusa | 367.6 | 0.25 | 9.9 | 51.0 | 0.77 | 0.96 |

Maemi | 211.4 | 0.47 | 14.2 | 69.9 | 0.96 | 0.86 |

Nari | 201.0 | 0.32 | 7.8 | 273.1 | 0.99 | 0.97 |

Dianmu | 497.8 | 0.10 | 8.2 | 427.3 | 0.84 | 0.90 |

Kompasu | 453.0 | 0.05 | 3.5 | 276.0 | 0.81 | 0.90 |

Bolaven | 206.5 | 0.22 | 7.2 | 59.3 | 0.87 | 0.95 |

Averaged | 322.9 | 0.23 | 8.5 | 192.8 | 0.87 | 0.80 |

Catchment | Rusa | Maemi | Nari | Dianmu | Kompasu | Bolaven | No. > 0.5 NSE | No. Events | % |
---|---|---|---|---|---|---|---|---|---|

C1 | 0.87 | 0.79 | 0.79 | 0.48 | 0.77 | 0.83 | 5 | 6 | 83 |

C2 | 0.92 | - | - | 0.53 | 0.76 | 0.72 | 3 | 4 | 75 |

C3 | - | - | 0.76 | 0.89 | 0.90 | 0.96 | 4 | 4 | 100 |

C4 | 0.96 | 0.86 | 0.97 | 0.90 | 0.90 | 0.95 | 6 | 6 | 100 |

C5 | - | 0.24 | 0.27 | −0.43 | −0.44 | 0.87 | 1 | 5 | 20 |

C6 | - | - | 0.75 | 0.69 | 0.8 | 0.89 | 4 | 4 | 100 |

C7 | - | - | 0.8 | 0.55 | 0.55 | 0.83 | 4 | 4 | 100 |

C8 | 0.73 | 0.42 | 0.74 | 0.35 | 0.28 | −0.02 | 2 | 6 | 33 |

C9 | - | - | - | 0.73 | 0.74 | 0.86 | 3 | 3 | 100 |

C10 | - | - | 0.26 | 0.04 | 0.66 | 0.7 | 1 | 3 | 33 |

C11 | 0.85 | 0.79 | 0.96 | 0.4 | 0.62 | 0.67 | 5 | 6 | 83 |

C12 | 0.83 | 0.83 | - | 0.67 | 0.55 | 0.84 | 5 | 5 | 100 |

C13 | - | - | - | 0.12 | −13.5 | 0.5 | 1 | 3 | 33 |

C14 | - | - | 0.43 | 0.68 | 0.34 | 0.3 | 1 | 4 | 25 |

C15 | 0.56 | −0.64 | 0.28 | −0.92 | 0.28 | 0.71 | 2 | 6 | 33 |

C16 | 0.76 | 0.25 | 0.64 | 0.68 | −0.32 | - | 3 | 5 | 60 |

C17 | - | - | 0.49 | 0.77 | 0.24 | 0.79 | 2 | 4 | 50 |

C18 | - | - | −1.5 | −0.56 | 0.49 | 0.39 | 0 | 4 | 0 |

C19 | - | - | 0.57 | −0.14 | −0.60 | 0.71 | 2 | 4 | 50 |

C20 | - | 0.82 | 0.73 | 0.83 | 0.47 | 0.47 | 3 | 4 | 75 |

C21 | 0.79 | 0.72 | 0.43 | 0.67 | −1.12 | 0.68 | 3 | 6 | 50 |

C22 | - | - | 0.9 | −0.37 | −0.14 | 0.03 | 1 | 4 | 25 |

C23 | 0.12 | −0.32 | 0.34 | −0.23 | 0.44 | 0.47 | 0 | 6 | 0 |

C24 | 0.72 | - | 0.82 | 0.52 | 0.73 | 0.87 | 5 | 5 | 100 |

Max | 0.92 | 0.86 | 0.97 | 0.89 | 0.9 | 0.96 | Missing values in observed data | ||

Min | 0.12 | −0.64 | −1.5 | −0.92 | −13.5 | −0.02 | |||

No. > 0.5 NSE | 10 | 6 | 12 | 13 | 12 | 17 | |||

No. Cat. | 11 | 11 | 20 | 24 | 24 | 23 | |||

% | 91 | 55 | 60 | 54 | 50 | 74 |

**Table 7.**The conventional regression equations for PDM model parameters when using C4 as the validation catchment.

Model Parameter | Regression Equation | R^{2} | p-Value |
---|---|---|---|

Cmax | 248.3 − 0.04 Area | 0.10 | 0.16 |

b | 2.35 − 0.0238 CN | 0.10 | 0.17 |

rtq | 32.05 + 0.003 Area − 0.01 ALTBAR − 0.3 CN | 0.52 | 0.00 |

rts | −728.56 − 0.06 Area − 77.16 DD + 1211.31 FARL | 0.55 | 0.00 |

f | −0.6 − 0.0002 Area − 0.12 DD + 1.84 FARL | 0.79 | 0.00 |

Model Parameter | C_{max} | b | rtq | rts | f |
---|---|---|---|---|---|

Value | 242 | 0.64 | 9.86 | 264 | 0.88 |

Model Parameter | GEE | R^{2} |
---|---|---|

Cmax | 232.33 − 0.03 Area | 0.01 |

b | 1.29 − 0.004 TR | 0.11 |

rtq | 22.75 + 0.003 Area − 0.007 ALTBAR − 0.15 CN | 0.24 |

rts | 678.12 − 0.09 Area − 50.46DD − 0.26 SAAR | 0.13 |

f | 1.70 − 0.0002 Area − 0.06 DD − 0.0006 SAAR | 0.08 |

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## Share and Cite

**MDPI and ACS Style**

Lee, H.; McIntyre, N.; Kim, J.; Kim, S.; Lee, H.
Prediction of Typhoon-Induced Flood Flows at Ungauged Catchments Using Simple Regression and Generalized Estimating Equation Approaches. *Water* **2018**, *10*, 647.
https://doi.org/10.3390/w10050647

**AMA Style**

Lee H, McIntyre N, Kim J, Kim S, Lee H.
Prediction of Typhoon-Induced Flood Flows at Ungauged Catchments Using Simple Regression and Generalized Estimating Equation Approaches. *Water*. 2018; 10(5):647.
https://doi.org/10.3390/w10050647

**Chicago/Turabian Style**

Lee, Hyosang, Neil McIntyre, Joungyoun Kim, Sunggu Kim, and Hojin Lee.
2018. "Prediction of Typhoon-Induced Flood Flows at Ungauged Catchments Using Simple Regression and Generalized Estimating Equation Approaches" *Water* 10, no. 5: 647.
https://doi.org/10.3390/w10050647