# Modeling the Influence of River Cross-Section Data on a River Stage Using a Two-Dimensional/Three-Dimensional Hydrodynamic Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of Study Area

^{2}, which includes part of the southwestern rugged foothills and fertile coastal plains. The tide is the primary tidal constituent at the river mouth and has a mean tidal range below 1 m. Based on the tidal classification [21], the Tsengwen River mouth can be classified as a microtidal estuary [22,23]. The average annual rainfall for the drainage basin is 2643 mm, with a contrasting rainfall pattern between dry and wet seasons. The dry and wet seasons are October–April and May–September, respectively. Thus, the river discharge varies seasonally with a high discharge of 411 × 10

^{3}m

^{3}/day in the wet season and a low discharge of 14 × 10

^{3}m

^{3}/day in the dry season. In the wet season, episodic flooding from heavy monsoon rains and typhoons is not unusual and critically affects the river discharge and the suspended load.

## 3. Materials and Methods

#### 3.1. Methods for Resampling River Cross-Section Data

#### 3.2. Interpolation Methods

#### 3.2.1. Linear Interpolation

#### 3.2.2. Inverse Distance Weighting (IDW)

#### 3.2.3. Natural Neighbor (NN)

#### 3.3. Three-Dimensional (3D) Hydrodynamic Model

#### 3.4. Two-Dimensional (2D) Hydrodynamic Model

#### 3.5. Model Implementation

#### 3.6. Assessment of the Model Performance

## 4. Results

#### 4.1. Simulation of the River Stage Using Different Cross-Section Data

^{3}/s. Although the simulated river stages using the original cross-section data show an acceptable result at the Mashan Bridge (Figure 10a), the simulated river stage cannot be lowered, even if there has been a decrease in the river flows (Figure 11a). This result means that the riverbed elevations interpolated by the original cross-section data are higher than the actual ones. The simulated river stages using the resampled cross-section data based on the IDW and NN methods extremely overestimate the measured river stages at both the Mashan Bridge and Xinzhong stations shown in Figure 10c,d and Figure 11c,d.

#### 4.2. Comparison of the Simulated River Stage Using the 2D and 3D Models

^{−4}m

^{2}/s [22].

#### 4.3. Model Sensitivity

## 5. Discussion

^{3}/s. It can be explained using the simplified momentum equation. If the Coriolis force, tidal effect, atmospheric pressure at the free surface, and wind shear stress are neglected, and a steady state is assumed, the momentum equation can be expressed as:

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 1.**Maps of the study area (white, blue, and cyan represent land, ocean, and river, respectively).

**Figure 2.**(

**a**) Conceptual diagram of the resampling method (L and R represent the main channel at left part and right part); and (

**b**) comparison of the original and resampled cross-sectional data points generated by the linear interpolation method.

**Figure 5.**River bed elevation used for hydrodynamic modeling: (

**a**) original cross-sectional data; and resampled cross-section data using: (

**b**) the interpolation method; (

**c**) the inverse distance weighting (IDW) method; and (

**d**) the natural neighbor (NN) method.

**Figure 6.**A 3D view of the river bed elevation of the hydrodynamic model using: (

**a**) the original data; and resampled data according to: (

**b**) the linear interpolation method; (

**c**) the IDW method; and (

**d**) the NN method. The red dashed box in the sub-figure represents the location of the 3D view river segment in the Tsengwen River.

**Figure 7.**A comparison of the simulated and observed river stages at the Mashan Bridge under the low flow condition using the 2D model with the: (

**a**) original cross-section data; and resampled cross-section data according to: (

**b**) the linear interpolation method; (

**c**) the IDW method; and (

**d**) the NN method.

**Figure 8.**A comparison of the simulated and observed river stages at the Xinzhong station under low flow conditions using the 2D model with the: (

**a**) original cross-section data; and resampled cross-section data according to: (

**b**) the linear interpolation method; (

**c**) the IDW method, and (

**d**) the NN method.

**Figure 9.**Simulated river stage distribution at 618 1:00 on 5 June 2012, under the low flow condition using the 2D model with the: (

**a**) original cross-section data; and resampled cross-section data according to: (

**b**) the linear interpolation method; (

**c**) the IDW method; and (

**d**) the NN method.

**Figure 10.**A comparison of the simulated and observed river stages at the Mashan Bridge under the high flow condition using the 2D model with the: (

**a**) original cross-section data; and resampled cross-section data according to: (

**b**) the linear interpolation method; (

**c**) the IDW method; and (

**d**) the NN method.

**Figure 11.**A comparison of the simulated and observed river stages at the Xinzhong station under high flow conditions using the 2D model with the: (

**a**) original cross-section data; and resampled cross-section data according to: (

**b**) the linear interpolation method; (

**c**) the IDW method; and (

**d**) the NN method.

**Figure 12.**Simulated river stage distributions at 12:00 on 12 June 2012, under the high flow condition using the 2D model with the: (

**a**) original cross-section data; and resampled cross-section data according to: (

**b**) the linear interpolation method; (

**c**) the IDW method; and (

**d**) the NN method.

**Figure 13.**Comparison of the simulated river stages using the 2D and 3D models with the resampled cross-section data under the low flow condition at: (

**a**) Mashan Bridge; and (

**b**) Xinzhong station.

**Figure 14.**Comparison of the simulated river stages using the 2D and 3D models with the resampled cross-section data under the high flow condition at: (

**a**) Mashan Bridge; and (

**b**) Xinzhong station.

**Figure 15.**Simulated river stage distribution using the 3D model with the resampled cross-section data: (

**a**) at 1:00 on 5 June 2012 (low flow condition); and (

**b**) at 12:00 on 12 June 2012 (high flow condition).

**Table 1.**Statistical error between the simulated and measured river stages under the low flow condition.

Condition | Station | MAE (m) | RMSE (m) | PBIAS (%) |
---|---|---|---|---|

2D modeling with original cross-section data | Mashan Bridge | 0.304 | 0.346 | −17.13 |

Xinzhong | 3.627 | 3.750 | 87.090 | |

2D modeling with the linear interpolation method | Mashan Bridge | 0.203 | 0.232 | 1.229 |

Xinzhong | 0.133 | 0.151 | −3.285 | |

2D modeling with the IDW method | Mashan Bridge | 1.190 | 1.250 | 214.385 |

Xinzhong | 4.220 | 4.367 | 104.055 | |

2D modeling with the NN method | Mashan Bridge | 3.328 | 3.353 | 599.405 |

Xinzhong | 3.760 | 3.774 | 92.715 | |

3D modeling with the linear interpolation method | Mashan Bridge | 0.216 | 0.253 | 3.390 |

Xinzhong | 0.096 | 0.105 | −1.404 |

**Table 2.**Statistical error between the simulated and measured river stages under the high flow condition.

Condition | Station | MAE (m) | RMSE (m) | PBIAS (%) |
---|---|---|---|---|

2D modeling with original cross-section data | Mashan Bridge | 1.018 | 1.238 | 22.981 |

Xinzhong | 2.794 | 2.956 | 34.254 | |

2D modeling with the linear interpolation method | Mashan Bridge | 0.911 | 1.135 | 19.670 |

Xinzhong | 0.934 | 1.093 | 10.499 | |

2D modeling with the IDW method | Mashan Bridge | 3.038 | 3.063 | 77.010 |

Xinzhong | 3.192 | 3.293 | 39.133 | |

2D modeling with the NN method | Mashan Bridge | 3.613 | 3.623 | 91.595 |

Xinzhong | 2.690 | 2.711 | 32.978 | |

3D modeling with the linear interpolation method | Mashan Bridge | 0.858 | 0.950 | 19.428 |

Xinzhong | 0.546 | 0.645 | 6.267 |

Condition | Station | Maximum Rate of River Stage (%) |
---|---|---|

2D modeling with increasing 50% BDC | Mashan Bridge | 0.615 |

Xinzhong | 0.081 | |

2D modeling with decreasing 50% BDC | Mashan Bridge | −1.037 |

Xinzhong | −0.069 | |

3D modeling with increasing 50% BDC | Mashan Bridge | 0.208 |

Xinzhong | 0.002 | |

3D modeling with decreasing 50% BDC | Mashan Bridge | −0.353 |

Xinzhong | −0.005 | |

3D modeling with increasing 50% VEV | Mashan Bridge | 0.001 |

Xinzhong | 0.009 | |

3D modeling with decreasing 50% VEV | Mashan Bridge | −0.006 |

Xinzhong | −0.007 |

Condition | Station | Maximum Rate of River Stage (%) |
---|---|---|

2D modeling with increasing 50% BDC | Mashan Bridge | 2.612 |

Xinzhong | 5.960 | |

2D modeling with decreasing 50% BDC | Mashan Bridge | −0.267 |

Xinzhong | −4.217 | |

3D modeling with increasing 50% BDC | Mashan Bridge | 1.309 |

Xinzhong | 0.329 | |

3D modeling with decreasing 50% BDC | Mashan Bridge | −0.0003 |

Xinzhong | −0.0001 | |

3D modeling with increasing 50% VEV | Mashan Bridge | 0.943 |

Xinzhong | 0.506 | |

3D modeling with decreasing 50% VEV | Mashan Bridge | −0.945 |

Xinzhong | −0.508 |

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**MDPI and ACS Style**

Chen, W.-B.; Liu, W.-C.
Modeling the Influence of River Cross-Section Data on a River Stage Using a Two-Dimensional/Three-Dimensional Hydrodynamic Model. *Water* **2017**, *9*, 203.
https://doi.org/10.3390/w9030203

**AMA Style**

Chen W-B, Liu W-C.
Modeling the Influence of River Cross-Section Data on a River Stage Using a Two-Dimensional/Three-Dimensional Hydrodynamic Model. *Water*. 2017; 9(3):203.
https://doi.org/10.3390/w9030203

**Chicago/Turabian Style**

Chen, Wei-Bo, and Wen-Cheng Liu.
2017. "Modeling the Influence of River Cross-Section Data on a River Stage Using a Two-Dimensional/Three-Dimensional Hydrodynamic Model" *Water* 9, no. 3: 203.
https://doi.org/10.3390/w9030203