# Robust Parameter Estimation Framework of a Rainfall-Runoff Model Using Pareto Optimum and Minimax Regret Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Framework

#### 2.2. Multi-Event Objective Function

_{n}(θ) is the objective function for the nth event to be simultaneously minimized with respect to the hydrological simulation model parameter set, θ. By randomly selecting different values for the weights allocated to the n event objectives, as many discrete optimal parameter sets as necessary can be generated to obtain an acceptable approximation of the continuous parameter set space. Alternatively, we can interactively guide the selection of weights until a satisfactory parameter set is discovered [25].

#### 2.3. Minimax Regret Approach (MRA)

#### 2.4. Standardization

_{ij}) in the matrix is replaced with the value (s

_{ij}) according to the following formula:

_{ij}is the impact of a scenario (j) with respect to a criterion (i); is the worst score of the criterion (i) with respect to all scenarios, i.e., the worst score in the row (i) of the payoff matrix; and is the “best” score of the criterion (i) with respect to all scenarios, i.e., the best score in the row (i) of the payoff matrix. This way, all scores in the payoff matrix are scaled between the values of 0.0 and 1.0 [28].

#### 2.5. SWMM Modeling

#### 2.5.1. Study Watershed and Rainfall Events

^{2}. The study area is located in a mountainous region with the elevation ranging from 130 to 1180 m. As shown in Figure 2, the Milyang Dam is located in the outlet of the study basin with water flowing from the northeast to southwest of the basin. The basin was divided into 24 sub-basins, and the river channels were divided into the 26 sub-channels for the SWMM. The rainfall data from two telemetry (TM) stations at the Milyang Dam and Seoli were used.

Events | Period (mm/dd/yyyy hh:mm) | Total Precipitation (mm) | Maximum Precipitation (mm/hr) | Peak Runoff (m^{3}/s) | Peak Time (mm/dd/yyyy hh:mm) | |||
---|---|---|---|---|---|---|---|---|

Milyang Dam | Seolli | Milyang Dam | Seolli | |||||

Pareto optimality | E1 20080726 | 07/26/2008 01:00 ~ 07/26/2008 23:00 | 73 | 105 | 24 | 21 | 233.3 | 07/26/2008 07:00 |

E2 20120828 | 08/28/2012 01:00 ~ 08/28/2012 23:00 | 64 | 106 | 6 | 25 | 119.1 | 08/28/2012 16:00 | |

Performance evaluation | E3 20090707 | 07/07/2009 01:00 ~ 07/08/2009 12:00 | 121 | 130 | 9 | 8 | 223.22 | 07/07/2009 17:00 |

E4 20100810 | 08/10/2010 1:00 ~ 08/12/2010 23:00 | 194 | 194 | 33 | 26 | 406.4 | 08/10/2010 11:00 | |

E5 20110625 | 06/25/2011 01:00 ~ 06/27/2011 12:00 | 169 | 249 | 8 | 12 | 422.4 | 06/26/2011 13:00 |

#### 2.5.2. Model Configuration

#### 2.5.3. Model Optimization

_{1}and f

_{2}are the NSEs for the two events, and w

_{1}and w

_{2}are the weighted values for the two events for the calibration (w

_{1}+ w

_{2}= 1). This study used 41 combinations of weights: (1.0 and 0.0, 0.975 and 0.025, 0.950 and 0.050, …, 0.0 and 1.0 for w

_{1}and w

_{2}.

Class | Parameters | Description | Unit | Lower Bound | Upper Bound |
---|---|---|---|---|---|

Basin | Pcnt. Imperv | Percent of impervious area | % | 0.8*Obs. value | 1.2*Obs. value |

Width | Characteristics width of the overland flow path | m | 0.8*Obs. value | 1.2*Obs. value | |

N-Imperv | Manning’s n of impervious area | - | 0.01 | 0.016 | |

N-Perv | Manning’s n of pervious area | - | 0.15 | 0.4 | |

S-Imperv | Depth of depression storage on impervious area | mm | 1.6 | 3.8 | |

S-Perv | Depth of depression storage on pervious area | mm | 3.8 | 6.4 | |

Pct Zero | Percent of the impervious area with no depression storage | % | 10 | 30 | |

CN | NRCS runoff curve coefficient | - | 0.8*Obs. value | 1.2*Obs. value | |

Channel | Manning N | Manning’s roughness coefficient | - | 0.02 | 0.05 |

Ground-water | Porosity | Porosity (volume of voids/total soil volume) | Fraction | 0.453 | 0.463 |

Wilt Point | Wilting point (soil moisture content at which plants cannot survive) | Fraction | 0.085 | 0.116 | |

Field Capac | Field capacity (soil moisture content after all free water has drained off) | Fraction | 0.19 | 0.232 | |

Hyd Cond | Soil’s saturated hydraulic conductivity | mm/h | 3.3 | 10.92 | |

Cond Slop | Average slope of log (conductivity) versus soil moisture deficit curve | - | 1 | 10 | |

Tens Slop | Average slope of soil tension versus soil moisture deficit curve | mm | 1,000 | 5000 | |

A1 | Groundwater flow coefficient | - | 0.0001 | 0.1 | |

B1 | Groundwater flow exponent | - | 1 | 5 |

#### 2.5.4. Performance Measures

## 3. Results and Discussions

#### 3.1. Calibration of ROPS Candidates

**Figure 4.**NSEs for E1 and E2 with 41 parameter sets: (

**a**) scatter plot between two NSEs (i.e., Pareto optimal sets) and (

**b**) standardized NSEs along the 41 parameter sets.

# of Alt. | Weights | Calibration | Evaluation | Final Rank | |||
---|---|---|---|---|---|---|---|

E1 | E2 | Averaged Regret to Ideal NSEs | Rank | Averaged Regret to Ideal NSEs | Rank | ||

9 | 0.8 | 0.2 | 0.173 | 2 | 0.015 | 2 | 1 |

11 | 0.75 | 0.25 | 0.334 | 13 | 0.090 | 9 | 7 |

14 | 0.675 | 0.325 | 0.191 | 3 | 0.036 | 5 | 3 |

15 | 0.65 | 0.35 | 0.384 | 16 | 0.143 | 10 | 9 |

19 | 0.55 | 0.45 | 0.327 | 10 | 0.015 | 2 | 6 |

20 | 0.525 | 0.475 | 0.277 | 6 | 0.315 | 23 | 10 |

21 | 0.5 | 0.5 | 0.139 | 1 | 0.077 | 8 | 2 |

22 | 0.475 | 0.525 | 0.230 | 4 | 0.060 | 7 | 5 |

24 | 0.425 | 0.575 | 0.240 | 5 | 0.024 | 4 | 4 |

33 | 0.2 | 0.8 | 0.426 | 22 | 0.018 | 3 | 8 |

#### 3.2. Selection of ROPS

#### 3.2.1. NSE-Only with Both Calibration and Evaluation Sets

**Figure 5.**Simulated hydrographs of E3, E4, and E5 with 41 calibrated parameter sets. (

**a**) E3 (20090707); (

**b**) E4 (20100810); (

**c**) E5 (20110625).

**Figure 6.**NSEs for E3, E4, and E5 with 41 parameter sets: (

**a**) scatter plot between two NSEs (red circles for E3 vs. E4 and blue triangles for E3 vs. E5) and (

**b**) standardized NSEs along the 41 parameter sets.

#### 3.2.2. Multiple Performance Measures with Evaluation Set Only

Performance Measures | NSE | PFE | RMSE | PBIAS |
---|---|---|---|---|

NSE | 1 | 0.2539 | 0.7243 | 0.2719 |

PFE | -- | 1 | 0.2537 | 0.2940 |

RMSE | -- | -- | 1 | 0.1522 |

PBIAS | -- | -- | -- | 1 |

Number of Parameter Set | Weights | Performance Measures | Average Ranking | Final Ranking | ||||
---|---|---|---|---|---|---|---|---|

E1 | E2 | NSE | PFE | RMSE | PBIAS | |||

5 | 0.900 | 0.100 | 4 | 2 | 2 | 12 | 5 | 1 |

35 | 0.150 | 0.875 | 6 | 11 | 4 | 14 | 8.75 | 2 |

12 | 0.725 | 0.275 | 1 | 27 | 1 | 8 | 9.25 | 3 |

16 | 0.625 | 0.375 | 13 | 1 | 11 | 22 | 11.75 | 4 |

21 | 0.500 | 0.500 | 18 | 6 | 15 | 10 | 12.25 | 5 |

8 | 0.825 | 0.175 | 10 | 34 | 8 | 1 | 13.25 | 6 |

13 | 0.700 | 0.300 | 12 | 5 | 9 | 27 | 13.25 | 6 |

6 | 0.875 | 0.125 | 8 | 23 | 7 | 18 | 14 | 8 |

4 | 0.925 | 0.075 | 2 | 13 | 38 | 5 | 14.5 | 9 |

18 | 0.575 | 0.425 | 16 | 8 | 12 | 25 | 15.25 | 10 |

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Kim, Y.; Chung, E.-S.; Won, K.; Gil, K. Robust Parameter Estimation Framework of a Rainfall-Runoff Model Using Pareto Optimum and Minimax Regret Approach. *Water* **2015**, *7*, 1246-1263.
https://doi.org/10.3390/w7031246

**AMA Style**

Kim Y, Chung E-S, Won K, Gil K. Robust Parameter Estimation Framework of a Rainfall-Runoff Model Using Pareto Optimum and Minimax Regret Approach. *Water*. 2015; 7(3):1246-1263.
https://doi.org/10.3390/w7031246

**Chicago/Turabian Style**

Kim, Yeonjoo, Eun-Sung Chung, Kwangjae Won, and Kyungik Gil. 2015. "Robust Parameter Estimation Framework of a Rainfall-Runoff Model Using Pareto Optimum and Minimax Regret Approach" *Water* 7, no. 3: 1246-1263.
https://doi.org/10.3390/w7031246