# Uncertainty Estimation Using the Glue and Bayesian Approaches in Flood Estimation: A case Study—Ba River, Vietnam

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## Abstract

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## 1. Introduction

**Input error**: The input uncertainty estimation was reported in only a few studies in which the inputs or the key variables of input were simulated as random processes. For example, Kuczera [1] applied the Bayesian inference to estimate rainfall uncertainty characterized by two parameters k and rMult for Abercrombie River’s daily flow. Rainfall flood events were simulated as random processes (e.g., [1,4,5,6,7,8,9]). The model parameter error and output error, in this case, were combined into a single random process.

**Model error:**The model error is estimated mainly by analysis of model parameter uncertainty. A well-known approach in uncertainty analysis of model parameters is the behavioral approach [10,11]. Examples of this approach are as follows: Generalized Likelihood Uncertainty Estimation (GLUE) method [12,13,14]; Bayesian method using Metropolis-Hasting algorithm and Adaptive Metropolis (AM) algorithm [1,15,16]; and Markov chain Monte Carlo [17,18,19]. In this approach, the input and output errors are ignored and the threshold of objective functions for selecting parameter values were determined.

^{2}distribution at three degrees of freedom and a probability level of 0.9 [4].

**Output error:**In design flood estimation, output error in flood quantiles is estimated from observed flow by using Flood Frequency Analysis method (FFA) reported in Kuczera [33]. As a result, the uncertainty of flood quantiles is estimated.

## 2. Methodology

#### 2.1. Bayesian Inference Background

#### 2.2. Bayesian Approach in Parameter Estimation

#### 2.3.GLUE Approach in Parameter Estimation

## 3. Test Catchment

#### 3.1. Catchment Location and Climate Description

^{2}area of the catchment upstream of An Khe gauge. This portion of the catchment comprises high to moderate mountainous areas, and is located mostly in the eastern part of the central highlands of Vietnam [35].

#### 3.2. Rainfall and Flow Data

## 4. Model Implementation

#### 4.1. Software

#### 4.2. Rainfall Model

#### 4.3. Model Parameters—Vector θ

_{1}) and variation coefficient (K

_{2}), were introduced for each category. The mean coefficient represents the average value of the 16 parameters over the catchment, while the variation coefficient represents the variation of these values across the catchment.

_{2}) for all the parameter categories results in noise or accumulation of parameter adjusted values. Therefore, selection of one representative variation coefficient for each rainfall-runoff process was preferred. As a result, the variation coefficient was applied to only three parameter categories: Curve Number, Subcatchment Roughness, and Channel Manning. The system, therefore, decreased to eight parameters, namely vector θ. This included K

_{1}for Curve Number (K

_{1}—CN); K

_{2}for Curve Number (K

_{2}—CN); K1 for Subcatchment Slope (K

_{1}—Slope); K

_{1}for Subcatchment Length (K

_{1}—length); K

_{1}for Subcatchment Roughness (K

_{1}—Catchment Roughness); K

_{2}for Subcatchment Roughness (K

_{2}—Catchment Roughness); K

_{1}for Channel Manning (K

_{1}—Channel Manning); and K

_{2}for Channel Manning (K

_{2}—Channel Manning).

#### 4.4. Uncertainty in Design Flood Flows—FFA Method

_{e}[standard deviation (log

_{e}flow)]) is −0.74320 and the standard deviation of scale parameter is 0.15064. Finally, the most probable value of the shape parameter (skewness) is −0.56875 and the standard deviation is 0.45883 (Table 2).

#### 4.5. Likelihood Function

#### 4.6. Calibration Process

_{sim}. The model parameter sets that resulted in vector β

_{sim}met the objective function were selected.

## 5. Results

#### 5.1. Parameter Uncertainty

_{1}—CN; K

_{2}—CN; and K

_{2}—Catchment roughness. The differences were indicated by the p value being less than 0.05. The most probable values of K

_{1}—CN estimated by the GLUE approach were 1.277, while those values estimated by Bayesian approach were 1.3881. This explained the 8% difference. The difference in K

_{2}—CN and K

_{2}—catchment roughness was 24% and 35%, respectively.

#### 5.2. Model Goodness of Fit

^{−8}. The design flow quantiles of the selected ARI were estimated and shown in Figure 6.

#### 5.3. Uncertainty in Design Flow Quantiles Due to Parameter Uncertainty

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Flood frequency curve at An Khe gauge [34].

**Figure 5.**Histogram of mean and variation coefficients estimated by Bayesian and Generalized Likelihood Uncertainty Estimation (GLUE) method. (

**a**) K

_{1}-catchment roughness, (

**b**) K

_{1}—CN, (

**c**) K

_{2}—Catchment Roughness and (

**d**) K

_{2}—CN.

**Figure 6.**Uncertainty of design flood quantiles estimated using the GLUE (

**a**) and Bayesian (

**b**) methods.

**Table 1.**Model parameter categories and their available ranges [40].

Models | Parameter Categories | Range |
---|---|---|

Loss models | Curve Number | 20–90 |

Kinematic wave (Overland flow planes) | Typical length | |

Representative slope | 0.0001–1 | |

Overland-flow roughness coefficient | 0.35–0.8 | |

Area represented by plane | ||

Muskingum-Cunge routing (The main channel) | Main channel length | |

Description of main channel shape | Rectangular | |

Channel slope | 0.0001–1 | |

Channel width | ||

Representative Manning’s Roughness coefficient | 0.035–0.08 |

**Table 2.**Parameter values of LP-III distribution (vector β) and ranges of acceptable LP-III parameters at An Khe gauge.

N | Parameters | Most Probable Value (α) | Standard Deviation (σ) | Maximum | Minimum |
---|---|---|---|---|---|

1 | Mean (loge flow) | 7.042 | 0.090 | 7.132 | 6.952 |

2 | Loge (Std dev (loge flow)) | −0.743 | 0.151 | −0.593 | −0.894 |

3 | Skew (loge flow) | −0.569 | 0.459 | −0.110 | −1.028 |

Parameters | GLUE Approach | Bayesian Approach | t-Test | ||||
---|---|---|---|---|---|---|---|

Mean | STD | p-Value in Normality Test | Mean | STD | p-Value in Normality Test | p-Value in Similarity Test | |

K_{1}—CN | 1.278 | 0.314 | 0.111 | 1.388 | 0.308 | 0.691 | 0.0051 |

K_{2}—CN | 1.038 | 0.474 | 0.811 | 1.369 | 0.475 | 0.409 | 1.422 × 10^{−7} |

K_{1}—Slope | 0.664 | 0.125 | 0.376 | 0.647 | 0.189 | 0.911 | 0.462 |

K_{1}—Length | 1.812 | 0.379 | 0.994 | 1.785 | 0.377 | 0.440 | 0.567 |

K_{1}—Catchment Roughness | 1.495 | 0.461 | 0.045 | 1.427 | 0.474 | 0.181 | 0.252 |

K_{2}—Catchment Roughness | 1.514 | 0.506 | 0.241 | 1.115 | 0.155 | 0.649 | 2.2 × 10^{−16} |

K_{1}—Channel Manning | 1.345 | 0.404 | 0.550 | 1.431 | 0.381 | 0.226 | 0.075 |

K_{2}—Channel Manning | 1.435 | 0.552 | 0.224 | 1.539 | 0.506 | 0.086 | 0.109 |

Parameters | Most Probable Values | Standard Deviation | p-Value in Normality Test | Observed Most Probable Values | p-Value in Similarity Test |
---|---|---|---|---|---|

Location parameter LP-III distribution | 7.042 | 0.086 | 0.916 | 7.051 | 0.358 |

Scale parameter LP-III distribution | −0.762 | 0.031 | 0.174 | −0.743 | 1.114 × 10^{−6} |

Shape parameter LP-III distribution | −0.4588 (mean) | −0.409/0.550 (range) | 0.0005 | −0.569 | |

Q-10 (m^{3}/s) | 2036 | 206.8 | 0.670 | 2029 | 0.774 |

Q-20 (m^{3}/s) | 2323 | 241.2 | 0.769 | 2299 | 0.379 |

Q-50 (m^{3}/s) | 2668 | 282.0 | 0.858 | 2614 | 0.092 |

**Table 5.**Most probable values and confidence limits of flood flows (observed flows versus estimates).

ARI | Observed Quantiles (m^{3}/s) | Quantiles Estimated by the Bayesian Approach (m^{3}/s) | Quantiles Estimated by the GLUE Approach (m^{3}/s) | ||||||
---|---|---|---|---|---|---|---|---|---|

Most Probable Values | Lower Confidence Limit | Upper Confidence Limit | Most Probable Value | Lower Confidence Limit (95%) | Upper Confidence Limit (95%) | Mean | Lower Threshold | Upper Threshold | |

10 | 2029 | 1768 | 2437 | 2035 | 1623 | 2447 | 2043 | 1827 | 2273 |

20 | 2299 | 1978 | 2920 | 2322 | 1840 | 2804 | 2333 | 2075 | 2607 |

50 | 2614 | 2182 | 3732 | 2667 | 2105 | 3229 | 2682 | 2370 | 3007 |

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**MDPI and ACS Style**

Cu Thi, P.; Ball, J.E.; Dao, N.H.
Uncertainty Estimation Using the Glue and Bayesian Approaches in Flood Estimation: A case Study—Ba River, Vietnam. *Water* **2018**, *10*, 1641.
https://doi.org/10.3390/w10111641

**AMA Style**

Cu Thi P, Ball JE, Dao NH.
Uncertainty Estimation Using the Glue and Bayesian Approaches in Flood Estimation: A case Study—Ba River, Vietnam. *Water*. 2018; 10(11):1641.
https://doi.org/10.3390/w10111641

**Chicago/Turabian Style**

Cu Thi, Phuong, James E Ball, and Ngoc Hung Dao.
2018. "Uncertainty Estimation Using the Glue and Bayesian Approaches in Flood Estimation: A case Study—Ba River, Vietnam" *Water* 10, no. 11: 1641.
https://doi.org/10.3390/w10111641