# Uncertainty Assessment of Urban Hydrological Modelling from a Multiple Objective Perspective

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## Abstract

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^{3}/s in the calibration period, and by 7.56 m

^{3}/s in the validation period. The coverages increased by 20.3% in the calibration period, and by 3.2% in the validation period. The median estimates improved, with an increase of the Nash–Sutcliffe efficiency coefficients by 1.6% in the calibration period, and by 10.0% in the validation period. We conclude that the proposed GLUE-TOPSIS is a valid approach to assess the uncertainty of urban hydrological model from a multiple objective perspective, thereby improving the reliability of model results in urban catchment.

## 1. Introduction

## 2. Methods

- The performance of the parameter sets was comprehensively investigated by considering the threshold of each objective function. Only the parameter sets for which all objective function values exceeded their threshold were considered as behavioral.
- An integrated likelihood measure was carried out using TOPSIS, which was used as a weighting factor to derive the posterior probability density functions for both the parameters and the predictions. The prominent advantage of TOPSIS is that different types of objective functions can be easily integrated into a unified evaluation index by setting the benefit criteria and loss criteria.

- Instead of a unique likelihood measure, multiple thresholds from multiple criteria are used to determine the behavioral parameter sets. In urban stormwater management, not only the precision of the flood process but also the precision of the flood volume and flood peak is considered by the modeler [31]. A behavior parameter set should be able to achieve these objectives. The criteria typically used in flood prediction are as follows:$$\mathrm{NSE}=1-\frac{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{{(\mathrm{y}}_{\mathrm{obs},\mathrm{i}}{-\text{}\mathrm{y}}_{\mathrm{sim},\mathrm{i}})}^{2}}{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{{(\mathrm{y}}_{\mathrm{obs},\mathrm{i}}-\overline{{\mathrm{y}}_{\mathrm{obs},\mathrm{i}}})}^{2}}$$$$\mathrm{VB}=\frac{\left|{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{y}}_{\mathrm{obs},\mathrm{i}}-\text{}{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{y}}_{\mathrm{sim},\mathrm{i}}\right|}{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{y}}_{\mathrm{obs},\mathrm{i}}}$$$$\mathrm{PB}=\frac{\left|\underset{1\le \mathrm{i}\le \mathrm{n}}{\mathrm{max}}{\text{{}\mathrm{y}}_{\mathrm{obs},\mathrm{i}}\text{}}-\text{}\underset{1\le \mathrm{i}\le \mathrm{n}}{\mathrm{max}}{\text{{}\mathrm{y}}_{\mathrm{sim},\mathrm{i}}\text{}}\right|}{\underset{1\le \mathrm{i}\le \mathrm{n}}{\mathrm{max}}{\text{{}\mathrm{y}}_{\mathrm{obs},\mathrm{i}}\text{}}}$$$$\mathrm{R}=\frac{{{\displaystyle \sum}}^{\text{}}\left({\mathrm{y}}_{\mathrm{obs},\mathrm{i}}-\overline{{\mathrm{y}}_{\mathrm{obs}}}\right)\left({\mathrm{y}}_{\mathrm{sim},\mathrm{i}}-\overline{{\mathrm{y}}_{\mathrm{sim}}}\right)}{\sqrt{{{\displaystyle \sum}}^{\text{}}{\left({\mathrm{y}}_{\mathrm{obs},\mathrm{i}}-\overline{{\mathrm{y}}_{\mathrm{obs}}}\right)}^{2}{\left({\mathrm{y}}_{\mathrm{sim},\mathrm{i}}-\overline{{\mathrm{y}}_{\mathrm{sim}}}\right)}^{2}}}$$In the above formula, NSE is the widely used Nash–Sutcliffe efficiency index [32]. The flood volume bias (VB) and flood peak bias (PB) are the modified expressions for the flood volume and flood peak deviation [18,33], respectively. The parameter R represents the consistency between the observed flow and the simulated flow [34]. The threshold of these criteria can be defined with reference to practical demand [35]. The reasonable range of the four criteria is between 0 to 1. The optimum value of NSE and R is 1, and optimum value of VB and PB is 0, which means the simulation results of the model completely fit the measured results. In this study, the behavioral parameter sets whose likelihood values of the four criteria were greater than the corresponding thresholds were chosen for further analysis.
- TOPSIS, which is a well-known MCDA method and can provide the ranking order of all alternatives [36,37], was employed in the calculation of the aggregate likelihood value L(${\mathsf{\theta}}_{\mathrm{i}}$) of the behavioral parameter set ${\mathsf{\theta}}_{\mathrm{i}}$.In the TOPSIS method, the four criteria of all parameter sets should be normalized by the classification of the benefit and cost criterion, where the benefit criterion means that a larger value is more valuable, and vice-versa for the cost criterion [37]. In this study, NSE and R are benefit criteria, while VB and PB are cost criteria; x
_{ij}is the ith criterion of the jth parameter set. For the benefit criteria, the normalized value (r_{ij}) is calculated as follows:$${\mathrm{r}}_{\mathrm{ij}}=\frac{{\mathrm{x}}_{\mathrm{ij}}}{\sqrt{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{x}}_{\mathrm{ij}}{}^{2}}}.$$For the cost criteria, the normalized value (r_{ij}) is calculated as follows:$${\mathrm{r}}_{\mathrm{ij}}=\frac{\frac{1}{{\mathrm{x}}_{\mathrm{ij}}}}{\sqrt{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{\frac{1}{{\mathrm{x}}_{\mathrm{ij}}}}^{2}}}.$$In many situations, the criterion should be weighted according its importance [36]. Because it is difficult to identify which criterion is more important, we assume that all criteria are equally important. The ideal solution ${\mathrm{R}}_{\mathrm{j}}{}^{+}$ and negative-ideal solution ${\mathrm{R}}_{\mathrm{j}}{}^{-}$ can be calculated as follows:$${\mathrm{R}}_{\mathrm{j}}{}^{+}{=\mathrm{max}\text{{}\mathrm{x}}_{1\mathrm{j}}{,\mathrm{x}}_{2\mathrm{j}}{,\text{\u2026},\mathrm{x}}_{\mathrm{nj}}\text{}}$$$${\mathrm{R}}_{\mathrm{j}}{}^{+}{=\mathrm{min}\text{{}\mathrm{x}}_{1\mathrm{j}}{,\mathrm{x}}_{2\mathrm{j}}{,\text{\u2026},\mathrm{x}}_{\mathrm{nj}}\text{}}.$$Then, the separation of each alternative from the ideal and negative-ideal solutions are expressed, respectively, as follows:$${\mathrm{D}}_{\mathrm{i}}^{+}=\sqrt{{\displaystyle \sum}_{\mathrm{j}=1}^{\mathrm{n}}{\left({\mathrm{r}}_{\mathrm{ij}}-{\mathrm{R}}_{\mathrm{j}}^{+}\right)}^{2}}$$$${\mathrm{D}}_{\mathrm{i}}^{-}=\sqrt{{\displaystyle \sum}_{\mathrm{j}=1}^{\mathrm{n}}{({\mathrm{r}}_{\mathrm{ij}}-{\mathrm{R}}_{\mathrm{j}}^{-})}^{2}}$$The aggregate likelihood value L(${\mathsf{\theta}}_{\mathrm{i}}$) is evaluated by comparing the distance from the ideal solution and the distance from the negative-ideal solution:$$\mathrm{L}\left({\mathsf{\theta}}_{\mathrm{i}}\right)=\frac{{\mathrm{D}}_{\mathrm{i}}{}^{+}}{{\mathrm{D}}_{\mathrm{i}}{}^{+}{+\mathrm{D}}_{\mathrm{i}}{}^{-}}$$ - Finally, the predictions from the behavioral parameter sets are ranked in the order of the likelihood weights W(i), which is defined as follows:$$\mathrm{W}\left(\mathrm{i}\right)=\frac{\mathrm{L}\left({\mathsf{\theta}}_{\mathrm{i}}\right)}{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{N}}\mathrm{L}\left({\mathsf{\theta}}_{\mathrm{i}}\right)}$$$${\mathrm{P}(\mathrm{Q}\mathrm{Q}}_{\mathrm{i}})=\frac{{{\displaystyle \sum}}_{\mathrm{j}=1}^{\mathrm{i}}\mathrm{W}\left(\mathrm{j}\right)}{{{\displaystyle \sum}}_{\mathrm{j}=1}^{\mathrm{n}}\mathrm{W}\left(\mathrm{j}\right)}$$

## 3. Study Area and SWMM Model

#### 3.1. Study Area

^{2}and is located upstream of the Liangshui River basin in Beijing, between 39°48’–39°55’ N and 116°90’–116°24’ E. In the catchment, the terrain exhibits a downward trend from the western mountains to the eastern plains. The annual average precipitation is 522.4 mm, and 80% of the precipitation occurs during the period from June to September. The river systems and hydro-meteorological stations of the Dahongmen catchment are shown in Figure 1.

#### 3.2. SWMM Model

#### 3.3. Interval Evaluation Index

## 4. Results

#### 4.1. Comparison of Different Acceptability Thresholds

#### 4.2. Comparison of Posterior Distribution

#### 4.3. Uncertainty Estimation of Discharge Simulation

## 5. Discussion

## 6. Conclusions

- 10,000 parameter sets generated by the Monte Carlo sampling in GLUE framework revealed that none of the four commonly used objective criteria could fully represent the urban flow process. Notably, the NSE, which is widely used in assessing the performance of hydrological models also cannot describe the flow characteristics alone, which highlights the need for adopting multi-criteria methods.
- The GLUE-TOPSIS method provided more precise uncertainty bounds and median estimates than traditional GLUE method which used NSE as single criterion. The GLUE-TOPSIS method reduced the bandwidth and deviation of the uncertainty bounds with a higher coverage than these from single criterion. The median estimates of GLUE-TOPSIS are also superior to these from single criterion according to the four objective criteria.
- The SWMM model performed well in the flood simulation of Dahongmen catchment in Beijing, PRC. Most observed flows fell within the 90% uncertainty interval, which suggests that the parameter uncertainty analysis has a relatively high contribution toward improving the simulation accuracy of flood prediction. The comparison results for the posterior distribution revealed that the parameters characterizing the impervious area had obvious high frequency intervals, owing to the large impervious area of Dahongmen catchment.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**MCDA analysis results for rainstorm events 20110623, 20110726, and 20110814 in calibration period.

**Figure 5.**Multiple-criteria decision analysis (MCDA) analysis results for rainstorm event 20120624, 20120721 in validation period.

Category | Parameter | Description | Units | Distribution |
---|---|---|---|---|

Basic characteristic | %Imperv | Percent of impervious area | % | 60–80 |

Manning roughness | N-Imperv | Manning coefficient for impervious area | 0.025–0.045 | |

N-perv | Manning coefficient for pervious area | 0.1–0.5 | ||

N-river | Manning coefficient for riverway | 0.03–0.06 | ||

N-conduit | Manning coefficient for conduit | 0.01–0.03 | ||

Reservoir in depressions | D-imperv | Depth of depression storage on impervious area | mm | 15–40 |

D-perv | Depth of depression storage on pervious area | mm | 20–50 | |

Infiltration parameters | MaxRate | Maximum rate on Horton infiltration curve | mm/h | 100–150 |

MinRate | Minimum rate on Horton infiltration curve | mm/h | 10–90 | |

Decay | Decay constant for the Horton infiltration curve | h^{−1} | 0–50 |

Criterion | Number | Criterion | Number | Criterion | Number | Criterion | Number |
---|---|---|---|---|---|---|---|

NSE ≥ 0.7 | 2506 | NSE ≥ 0.7 and VB ≤ 0.3 | 2226 | NSE ≥ 0.7 and VB ≤ 0.3 and PB ≤ 0.2 | 1598 | NSE ≥ 0.7 and VB ≤ 0.3 and PB ≤ 0.2 and R ≥ 0.8 | 1598 |

VB ≤ 0.3 | 3958 | NSE ≥ 0.7 and PB ≤ 0.2 | 1772 | NSE ≥ 0.7 and VB ≤ 0.3 and R ≥ 0.8 | 2226 | ||

PB ≤ 0.2 | 4215 | NSE ≥ 0.7 and R ≥ 0.8 | 2506 | NSE ≥ 0.7 and PB ≤ 0.2 and R ≥ 0.8 | 1772 | ||

R ≥ 0.8 | 6918 | VB ≤ 0.3 and PB ≤ 0.2 | 2557 | VB ≤ 0.3 and PB ≤ 0.2 and R ≥ 0.8 | 2396 | ||

VB ≤ 0.3 and R ≥ 0.8 | 3413 | ||||||

PB ≤ 0.2 and R ≥ 0.8 | 3927 |

Method | Criteria | B(m³/s) | CR (%) | RD (m³/s) | |
---|---|---|---|---|---|

Calibration period | 20110623 | Single criteria | 48.308 | 54.5 | 13.606 |

GLUE-TOPSIS | 35.17 | 54.5 | 12.804 | ||

20110726 | Single criteria | 50.563 | 20.0 | 41.454 | |

GLUE-TOPSIS | 43.723 | 40.0 | 40.551 | ||

20110814 | Single criteria | 39.829 | 50.0 | 18.995 | |

GLUE-TOPSIS | 37.908 | 62.5 | 16.276 | ||

Average | Single criteria | 46.233 | 41.5 | 24.685 | |

GLUE-TOPSIS | 38.934 | 52.3 | 23.211 | ||

Validation period | 20120624 | Single criteria | 17.100 | 10.0 | 17.100 |

GLUE-TOPSIS | 14.100 | 10.0 | 14.150 | ||

20120721 | Single criteria | 53.953 | 81.8 | 17.140 | |

GLUE-TOPSIS | 41.839 | 84.8 | 13.358 | ||

Average | Single criteria | 35.526 | 45.9 | 17.120 | |

GLUE-TOPSIS | 27.970 | 47.4 | 13.754 |

**Table 4.**Simulation results of the median Generalized Likelihood Uncertainty Estimation (GLUE) estimates.

Method | Criteria | NSE | VB | PB | R | |
---|---|---|---|---|---|---|

Calibration period | 20110623 | Single criteria | 0.863 | 0.003 | 0.118 | 0.994 |

GLUE-TOPSIS | 0.876 | 0.02 | 0.112 | 0.995 | ||

20110726 | Single criteria | 0.936 | 0.086 | 0.096 | 0.955 | |

GLUE-TOPSIS | 0.949 | 0.123 | 0.077 | 0.941 | ||

20110814 | Single criteria | 0.966 | 0.184 | 0.085 | 0.962 | |

GLUE-TOPSIS | 0.982 | 0.103 | 0.138 | 0.991 | ||

Average | Single criteria | 0.921 | 0.091 | 0.099 | 0.970 | |

GLUE-TOPSIS | 0.936 | 0.082 | 0.109 | 0.975 | ||

Validation period | 20120624 | Single criteria | 0.429 | 0.402 | 0.375 | 0.790 |

GLUE-TOPSIS | 0.530 | 0.300 | 0.428 | 0.663 | ||

20120721 | Single criteria | 0.936 | 0.041 | 0.077 | 0.989 | |

GLUE-TOPSIS | 0.974 | 0.005 | 0.059 | 0.962 | ||

Average | Single criteria | 0.683 | 0.222 | 0.226 | 0.890 | |

GLUE-TOPSIS | 0.752 | 0.153 | 0.244 | 0.812 |

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## Share and Cite

**MDPI and ACS Style**

Pang, B.; Shi, S.; Zhao, G.; Shi, R.; Peng, D.; Zhu, Z.
Uncertainty Assessment of Urban Hydrological Modelling from a Multiple Objective Perspective. *Water* **2020**, *12*, 1393.
https://doi.org/10.3390/w12051393

**AMA Style**

Pang B, Shi S, Zhao G, Shi R, Peng D, Zhu Z.
Uncertainty Assessment of Urban Hydrological Modelling from a Multiple Objective Perspective. *Water*. 2020; 12(5):1393.
https://doi.org/10.3390/w12051393

**Chicago/Turabian Style**

Pang, Bo, Shulan Shi, Gang Zhao, Rong Shi, Dingzhi Peng, and Zhongfan Zhu.
2020. "Uncertainty Assessment of Urban Hydrological Modelling from a Multiple Objective Perspective" *Water* 12, no. 5: 1393.
https://doi.org/10.3390/w12051393