# Assessment of Uncertainty in Grid-Based Rainfall-Runoff Model Based on Formal and Informal Likelihood Measures

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Grid-Based Rainfall-Runoff Model (GRM) Definition

_{r}is return flow, $\mathsf{\Delta}$y is the width of control volume, A is cross-sectional area, Q is discharge, q

_{L}is lateral flow, q

_{ss}is subsurface flow, q

_{b}is baseflow, S

_{0}is surface slope, and S

_{f}is friction slope.

#### 2.3. Rainfall-Runoff Analysis

_{i}, and P

_{i}refer to the average rainfall in the watershed, the number of stations, the effect area for each station, and the measured precipitation for each point, respectively.

#### 2.4. Uncertainty Analysis Using Generalized Likelihood Uncertified Estimation (GLUE)

_{t}and P

_{t}refer to the actual and predicted flow at time t, and $\overline{O}$ refers to the average value of the actual flow.

^{2}refers to the variance of the simulated and observed values, as shown in Equation (11), and ε

_{i}

^{(θ)}refers to the vector value of the residual, which corresponds to the difference between the simulated and observed values at time t, as indicated in Equation (12).

#### 2.5. Automation Analysis and Parameter Setting of the GRM

## 3. Results

#### 3.1. Distribution of Parameters According to Likelihood

#### 3.1.1. Event 1 Dotty Plot

#### 3.1.2. Event 2 Dotty Plot

#### 3.2. Posterior Distribution of GRM Parameters

#### 3.2.1. Event 1 Parameter Sensitivity

#### 3.2.2. Event 2 Parameter Sensitivity

#### 3.3. Quantied Uncertainty in the GRM Model

#### 3.3.1. Event 1 95PPU

#### 3.3.2. Event 2 95PPU

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Spatial data of study site (

**a**) Digital Elevation Model; (

**b**) Land cover map; Attributes 1 to 7 represent city/dry area, agricultural area, forest area, meadow, wetland, bare land, and water; (

**c**) Land cover map; Attributes 1 to 8 represent silty clay loam, silt loam, sandy loam, sand, clay loam, clay, loamy sand, loam, respectively; (

**d**) Soil depth map; Attributes 1 to 4 represent very shallow, shallow, moderately deeper, deep, respectively.

**Figure 5.**Event 1 Dotty plot of behavioral parameters based on formal and informal likelihood. (

**a1-1**–

**a1-8**) LNSE results; (

**b1-1**–

**b1-8**) LPBIAS results; (

**c1-1**–

**c1-8**) LRSR results; (

**d1-1**–

**d1-8**) LLOG results.

**Figure 6.**Event 2 dotty plot of behavioral parameters based on formal and informal likelihood. (

**a2-1**–

**a2-8**) LNSE results; (

**b2-1**–

**b2-8**) LPBIAS results; (

**c2-1**–

**c2-8**) LRSR results; (

**d2-1**–

**d2-8**) LLOG results.

**Figure 7.**Event 1 parameter sensitivity (histogram/CDF). (

**a3-1**–

**a3-8**) LNSE results; (

**b3-1**–

**b3-8**) LPBIAS results; (

**c3-1**–

**c3-8**) LRSR results; (

**d3-1**–

**d3-8**) LLOG results.

**Figure 8.**Event 2 parameter sensitivity (histogram/CDF). (

**a4-1**–

**a4-8**) LNSE results; (

**b4-1**–

**b4-8**) LPBIAS results; (

**c4-1**–

**c4-8**) LRSR results; (

**d4-1**–

**d4-8**) LLOG results.

**Figure 9.**Event 1 95PPU. (

**e1-1**) LNSE results; (

**e1-2**) LPBIAS results; (

**e1-3**) LRSR results; (

**e1-4**) LLOG results.

**Figure 10.**Event 2 95PPU. (

**e2-1**) LNSE results; (

**e2-2**) LPBIAS results; (

**e2-3**) LRSR results; (

**e2-4**) LLOG results.

Rainfall Station Name | Rainfall Station Location | Station Weight (%) | |
---|---|---|---|

Latitude | Longitude | ||

Samjuk | 37°4′36.36″ N | 127°22′7.03″ E | 26.13 |

Geumdang Elementary School | 37°12′3.45″ N | 127°36′21.74″ E | 17.51 |

Beopcheon | 37°12′16.31″ N | 127°44′52.87″ E | 15.78 |

Yeoju Bridge | 37°17′43.27″ N | 127°38′31.92″ E | 14.03 |

Wonsam | 37°10′6.32″ N | 127°18′15.06″ E | 9.01 |

Seolseong | 37° 8′45.33″ N | 127°31′17.97″ E | 7.24 |

Saenggeu | 37° 2′4.37″ N | 127°36′3.93″ E | 4.56 |

Eumseong | 36°56′13.82″ N | 127°41′32.14″ E | 2.18 |

Namgok | 37°13′53.30″ N | 127°16′41.07″ E | 1.71 |

Angseong | 37° 5′11.35″ N | 127°44′43.87″ E | 1.50 |

Oryu | 36°58′21.40″ N | 127°28′40.99″ E | 0.35 |

Num | NAME | Start Date | End Date |
---|---|---|---|

1 | Event 1 | 29 June 2011 (10:00:00) | 1 July 2011 (17:30:00) |

2 | Event 2 | 31 July 2017 (3:00:00) | 1 August 2012 (07:30:00) |

Num | Parameters | Description | Range | Change Range | ||
---|---|---|---|---|---|---|

Lower | Upper | Lower | Upper | |||

1 | IniSaturation (ISSR) | Initial soil saturation ratio | 0 | 1 | 0 | 0.5 |

2 | MinSlopeOF (MSLS) | Minimum slope of land surface | 0.0001 | 0.01 | 0.0001 | 0.007 |

3 | ChRoughness (CRC) | Channel roughness coefficient | 0.008 | 0.2 | 0.008 | 0.14 |

4 | CalCoefLCRoughness (CLCRC) | Correction factor for land cover roughness coefficient | 0.6 | 1.3 | 0.6 | 1.3 |

5 | CalCoefSoilDepth (CSD) | Correction factor for soil depth | 0.8 | 1.2 | 0.9 | 1.1 |

6 | CalCoefPorosity (CSP) | Correction factor for soil porosity | 0.9 | 1.1 | 0.25 | 4 |

7 | CalCoefWFSuctionHead (CSWS) | Correction factor for soil wetting front suction head | 0.25 | 4 | 0.05 | 8 |

8 | CalCoefHydraulicK (CSHC) | Correction factor for soil hydraulic conductivity | 0.05 | 20 | 0.8 | 1.2 |

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**MDPI and ACS Style**

Seong, Y.; Choi, C.-K.; Jung, Y.
Assessment of Uncertainty in Grid-Based Rainfall-Runoff Model Based on Formal and Informal Likelihood Measures. *Water* **2022**, *14*, 2210.
https://doi.org/10.3390/w14142210

**AMA Style**

Seong Y, Choi C-K, Jung Y.
Assessment of Uncertainty in Grid-Based Rainfall-Runoff Model Based on Formal and Informal Likelihood Measures. *Water*. 2022; 14(14):2210.
https://doi.org/10.3390/w14142210

**Chicago/Turabian Style**

Seong, Yeonjeong, Cheon-Kyu Choi, and Younghun Jung.
2022. "Assessment of Uncertainty in Grid-Based Rainfall-Runoff Model Based on Formal and Informal Likelihood Measures" *Water* 14, no. 14: 2210.
https://doi.org/10.3390/w14142210