# Case Study: On Objective Functions for the Peak Flow Calibration and for the Representative Parameter Estimation of the Basin

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}, bR

^{2}, NSE, MNS, RSR, SSQR, KGE, and PBIAS). They found that the different combinations of optimization algorithms and objective functions had the same performance, but different parameter ranges. Past research has optimized parameters to calibrate runoff models, and examined various calibration methods to accommodate the different model characteristics. Since the accuracy of flood runoff analysis varies with the choice of objective function, it is important to adopt adequate objective functions for parameter calibration. While various optimization techniques have been used to optimize parameters, most studies were limited to a single rain storm event. The parameters optimized for a single rain storm event may not be suitable for other rain storm events.

## 2. Rainfall–Runoff Model

#### 2.1 SSARR Model

_{s}, T

_{ss}, n

_{1}, n

_{2}and channel parameters: KTS, n, n

_{1}′) exist in the SSARR model. United States Army Corps of Engineers (USACE) [28] showed that three parameters, SMI, BII, and S-SS, in SSARR model can have a big influence in runoff estimation. These parameters can control peak flow runoff, total runoff in a specific period, and base flow runoff. However, USACE [28] also said that the other parameters can be used for the detailed calibration to derive more accurate runoff hydrograph. Therefore, this study used ten parameters in SSARR model for the reduction of uncertainty or error in the estimation of runoff hydrograph.

#### 2.1.1. Watershed Routing

_{s}

_{s}for routing of below-surface runoff, and T

_{s}for routing of above-surface runoff. Among the internal parameters of the SSARR model, the most sensitive parameters to runoff are SMI, BII, and S-SS. The remaining parameters have relatively smaller influence on runoff [26]. To simulate runoff using the SSARR model, this study calibrated seven parameters: SMI, BII, S-SS, T

_{ss}, T

_{s}, n

_{1}, and n

_{2}. T

_{s}and T

_{ss}are storage times, and n

_{1}and n

_{2}are the number of virtual reservoirs. The characteristics of key parameters are as follows:

- Soil Moisture Index (SMI)

_{1}and SMI

_{2}are SMI values before and after an event, MI is the moisture in the soil, RGP is the total runoff, PH is the calculation time, and ETI represents evapotranspiration.

- Baseflow Infiltration Index (BII)

_{1}and BII

_{2}are BII values before and after an event, RG is the total runoff ratio calculated as RGP/PH, and BIITS is the storage time.

- Surface-Subsurface Separation (S-SS)

#### 2.1.2. Channel Routing

_{s}is the storage time, KTS is the constant determined by the trial and error method, Q is the flow, and n is a coefficient relating the time of storage variation as a function of discharge. The range of the n parameter is between −1 and 1, n

_{1}′ is the number of phases, and T is travel time.

## 3. Calibration Methods

#### 3.1. Genetic Algorithm (GA)

#### 3.2. Pattern Search

_{1}and S2

_{1}are the starting points, and Δ is the increment. [a, b] is the direction vector, where a and b have values of −1, 0 and 1. S1

_{2}and S2

_{2}are starting points for different searches, and continue to change until the end condition is satisfied. That is, Δ and [a, b] change in a direction and size such that the objective functions of [S1

_{2}, S2

_{2}] become smaller. [S1

_{2}, S2

_{2}] is repeatedly calculated until a satisfactory objective function value is obtained or the end condition is satisfied. First, the starting point of the search, the range of possible solutions, and the end condition are determined. The objective function that is minimized is chosen among the four direction vectors ([−1, 0], [0, −1], [1, 0], [0, 1]), and the increment is increased until the objective function increases. When the objective function increases, the value obtained in the previous stage becomes the starting point, and the increment is decreased. The increment is further dropped until the objective function decreases. When the objective function decreases, the value obtained in the previous stage becomes the starting point, and the search resumes in a different direction. The pattern search method continues to vary the search direction and repeatedly increase/decrease the increment until the end condition is met or a satisfactory objective function is attained (Figure 3).

#### 3.3. Shuffled Complex Evolution Method Developed at the University of Arizona (SCE-UA)

## 4. Objective Functions

#### 4.1. Sum of Squared of Residual (SSR)

_{obs}is the observed runoff, and Q

_{sim}is the simulated runoff. SSR commonly adopted as an objective function when performing parameter optimization for hydrologic models. Given the cumulative nature of errors, the results are highly influenced by the number of data and abnormalities. SSR is widely used for the parameter optimization of the rainfall–runoff model in flood forecasting and warning [35]. Since SSR is accumulation of error, it is influenced by the number of data and abnormal data such as high flood level. In addition, SSR does not give good results all the time for peak flow value and so we may need another objective function, which can be used for the fit of peak flow in hydrograph.

#### 4.2. Weighted Sum of Squared of Residual (WSSR)

_{obs}is the observed runoff, Q

_{sim}is the simulated runoff, Q

_{obs,peak}is the observed peak flow, Q

_{sim,peak}is the simulated peak flow, T

_{obs,peak}is the time of occurrence of observed peak flow, and T

_{sim,peak}is the time of occurrence of simulated peak flow. ${W}_{1}$ in WSSR is the relative error of peak flow runoff and it can be used as a weighting value for preventing overestimation and underestimation of peak flow runoff. ${W}_{2}$ is also the relative error of peak time and it can be used as a weighting value for diminishing lag time error. Thus, WSSR is an objective function that improves SSR for peak flow runoff and less influence of abnormal data.

## 5. Application to Mihocheon Stream Basin

#### 5.1. Study Area

^{2}, it accounts for 18.8% of the total area of the Geum River basin. Mihocheon stream has a length of 87.3 km. Sub-basins are indicated by numbers on the basin map (Figure 6). In this study, Mihocheon stream basin is divided into six sub-basins using 30 m × 30 m DEM (Digital Elevation Model) and then parameter calibration is performed after obtaining factors for basin and channel characteristics using soil map and land cover map.

#### 5.2. The Observed Rain Storm Events and Flood Runoff Hydrographs

_{s}, T

_{ss}, n

_{1}, n

_{2}) and three channel parameters (KTS, n, n

_{1}′) of each channel.

#### 5.3. Calibration of Flood Runoff Hydrograph

_{obs}is the observed runoff, Q

_{sim}is the simulated runoff, Q

_{obs,peak}is the observed peak flow, Q

_{sim,peak}is the simulated peak flow, Q

_{obs,ave}is the average observed runoff, and Q

_{sim,ave}is the average simulated runoff. The evaluation criteria values for each calibration method and watershed are presented in Table 2 and Table 3. And the excellent results are indicated in bold.

#### 5.4. Application of WSSR Objective Function

## 6. Objective Function for Representative Parameter Estimation in a Basin

#### 6.1. Suggestion of Objective Function for the Representative Parameter Estimation

#### 6.2. Estimation of the Representative Parameter

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 11.**The observed and simulated runoff hydrographs for verification in each water stage station (for rain event of 2004).

**Figure 12.**The observed and simulated runoff hydrographs for verification in each water stage station (for rain event of 2006).

Classification | Data Period |
---|---|

Calibration Event | 30 June 1997–10 July 1997 |

1 August 1999–6 August 1999 | |

4 August 2002–12 August 2002 | |

20 July 2003–27 July 2003 | |

Validation Event | 18 June 2004–24 June 2004 |

9 July 2006–25 July 2006 |

Objective Functions | Year of Rain Storm Events | Evaluation Criteria | Calibration Methods | ||
---|---|---|---|---|---|

GA | Pattern Search | SCE-UA | |||

SSR | 1997 | R^{2} | 0.9655 | 0.9633 | 0.9638 |

NRMSE | 0.0375 | 0.0367 | 0.0412 | ||

RE | 0.0008 | 0.0005 | 0.0614 | ||

1999 | R^{2} | 0.8989 | 0.9070 | 0.9131 | |

NRMSE | 0.0649 | 0.0667 | 0.0709 | ||

RE | 0.0180 | 0.1309 | 0.0990 | ||

2002 | R^{2} | 0.9288 | 0.9337 | 0.8854 | |

NRMSE | 0.0644 | 0.0610 | 0.0781 | ||

RE | 0.1744 | 0.1645 | 0.0073 | ||

2003 | R^{2} | 0.9606 | 0.9606 | 0.9303 | |

NRMSE | 0.0408 | 0.0385 | 0.0513 | ||

RE | 0.1071 | 0.1039 | 0.1492 |

Objective Functions | Year of Rain Storm Events | Evaluation Criteria | Calibration Methods | ||
---|---|---|---|---|---|

GA | Pattern Search | SCE-UA | |||

SSR | 1997 | R^{2} | 0.9611 | 0.9362 | 0.9393 |

NRMSE | 0.0344 | 0.0470 | 0.0586 | ||

RE | 0.0402 | 0.0092 | 0.0236 | ||

1999 | R^{2} | 0.9436 | 0.9689 | 0.9231 | |

NRMSE | 0.0639 | 0.0427 | 0.0823 | ||

RE | 0.2278 | 0.1693 | 0.2760 | ||

2002 | R^{2} | 0.8102 | 0.7996 | 0.7655 | |

NRMSE | 0.1487 | 0.1396 | 0.1629 | ||

RE | 0.4553 | 0.3879 | 0.4288 | ||

2003 | R^{2} | 0.9082 | 0.9568 | 0.9335 | |

NRMSE | 0.0772 | 0.0490 | 0.0783 | ||

RE | 0.2497 | 0.1336 | 0.2891 |

Calibration Methods | Model | Year of Rain Storm Events | Evaluation Criteria | Objective Functions | |
---|---|---|---|---|---|

SSR | WSSR | ||||

Pattern Search | SSARR | 1997 | R^{2} | 0.9633 | 0.9552 |

NRMSE | 0.0367 | 0.0382 | |||

RE | 0.0005 | 0.0000 | |||

1999 | R^{2} | 0.9070 | 0.9157 | ||

NRMSE | 0.0667 | 0.0770 | |||

RE | 0.1309 | 0.0000 | |||

2002 | R^{2} | 0.9337 | 0.9337 | ||

NRMSE | 0.0610 | 0.0598 | |||

RE | 0.1645 | 0.1441 | |||

2003 | R^{2} | 0.9606 | 0.9579 | ||

NRMSE | 0.0385 | 0.0372 | |||

RE | 0.1039 | 0.0653 |

Calibration Methods | Model | Year of Rain Storm Events | Evaluation Criteria | Objective Functions | |
---|---|---|---|---|---|

SSR | WSSR | ||||

Pattern Search | SSARR | 1997 | R^{2} | 0.9362 | 0.9303 |

NRMSE | 0.0470 | 0.0544 | |||

RE | 0.0092 | 0.1070 | |||

1999 | R^{2} | 0.9689 | 0.9688 | ||

NRMSE | 0.0427 | 0.0421 | |||

RE | 0.1693 | 0.1574 | |||

2002 | R^{2} | 0.7996 | 0.7991 | ||

NRMSE | 0.1396 | 0.1367 | |||

RE | 0.3879 | 0.3727 | |||

2003 | R^{2} | 0.9568 | 0.9558 | ||

NRMSE | 0.0490 | 0.0478 | |||

RE | 0.1336 | 0.1054 |

Parameter | SMI | BII | S-SS | T_{s} | T_{ss} | n_{1} | n_{2} | |
---|---|---|---|---|---|---|---|---|

Subbasin | ||||||||

19 | 1.00 | 1.00 | 0.50 | 3 | 17 | 2 | 1 | |

20 | 2.00 | 1.00 | 0.50 | 0 | 25 | 2 | 1 | |

21 | 2.00 | 1.00 | 0.50 | 7 | 28 | 2 | 1 | |

22 | 1.63 | 0.69 | 1.03 | 2 | 14 | 2 | 1 | |

23 | 2.78 | 0.00 | 0.50 | 4 | 23 | 2 | 1 | |

24 | 5.00 | 0.13 | 0.50 | 5 | 17 | 2 | 1 |

Parameter | KTS | n | n_{1}′ | |
---|---|---|---|---|

Channel | ||||

G1 | 7.00 | 0.20 | 2 | |

G2 | 7.00 | 0.20 | 2 | |

G3 | 29.38 | 0.83 | 1 | |

H | 3.00 | 0.20 | 2 | |

J1 | 5.69 | 0.20 | 2 | |

I | 2.25 | 0.20 | 2 |

Sub-Basin | Evaluation Criteria | Representative Parameter | Parameter in 2004 |
---|---|---|---|

Values | Values | ||

22 | R^{2} | 0.8472 | 0.8134 |

NRMSE | 0.0953 | 0.1121 | |

RE | 0.3272 | 0.3575 | |

23 | R^{2} | 0.9112 | 0.9646 |

NRMSE | 0.1389 | 0.1144 | |

RE | 0.2647 | 0.5034 |

Sub-Basin | Evaluation Criteria | Representative Parameter | Parameter in 2006 |
---|---|---|---|

Values | Values | ||

22 | R^{2} | 0.9743 | 0.8589 |

NRMSE | 0.0236 | 0.0814 | |

RE | 0.0859 | 0.0435 | |

23 | R^{2} | 0.9110 | 0.9290 |

NRMSE | 0.0609 | 0.0752 | |

RE | 0.0606 | 0.2896 |

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## Share and Cite

**MDPI and ACS Style**

Kim, J.; Kim, D.; Joo, H.; Noh, H.; Lee, J.; Kim, H.S.
Case Study: On Objective Functions for the Peak Flow Calibration and for the Representative Parameter Estimation of the Basin. *Water* **2018**, *10*, 614.
https://doi.org/10.3390/w10050614

**AMA Style**

Kim J, Kim D, Joo H, Noh H, Lee J, Kim HS.
Case Study: On Objective Functions for the Peak Flow Calibration and for the Representative Parameter Estimation of the Basin. *Water*. 2018; 10(5):614.
https://doi.org/10.3390/w10050614

**Chicago/Turabian Style**

Kim, Jungwook, Deokhwan Kim, Hongjun Joo, Huiseong Noh, Jongso Lee, and Hung Soo Kim.
2018. "Case Study: On Objective Functions for the Peak Flow Calibration and for the Representative Parameter Estimation of the Basin" *Water* 10, no. 5: 614.
https://doi.org/10.3390/w10050614