# A Data Assimilation Approach to the Modeling of 3D Hydrodynamic Flow Velocity in River Reaches

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. TELEMAC System

#### 2.1.1. The TELEMAC-3D Model

_{s}(m) is the free surface elevation, and Zf (m) denotes the bottom depth. In the above, p

_{atm}(pa) is the atmospheric pressure, g (m/s

^{2}) is the gravitational acceleration, ϑ (m

^{2}/s) is the kinematic viscosity and tracer diffusion coefficients, Δρ (kg/m

^{3}) is the variation in density around the reference density, t (s) is the simulation time, x and y (m) are the horizontal space components, and z (m) is the vertical space component. U, V, W, and h are the unknown quantities in the above formulations and act as computational variables.

#### 2.1.2. The TELAPY Module

#### 2.2. Particle Filter

- (1)
- The Simulation Step

- (2)
- The Updating Step

#### 2.3. Practical Experiment

#### 2.3.1. Study Area

^{2}, and its mean annual river discharge is 544 m

^{3}/s. The water flows from the Qu River and Jinhua River into the Lanxi River. Several three-stage gauging stations are set up for the upstream and downstream of the considered experiment to monitor the water level upstream and downstream.

#### 2.3.2. Observations

- 1.
- The water level data

- 2.
- The flow velocity data

#### 2.3.3. TELEMAC-3D Setup

#### 2.3.4. Particle Filter Setup

- (1)
- Set the number of particles N, the model error Err
_{m}, and the observed error Err_{o}(see Section 3.1); - (2)
- Call the specific function for TELEMAC-3D from FORTRAN API, then load and initialize the TELEMAC-3D configuration with the computed conditions and start the velocity simulation;
- (3)
- Obtain the velocity state variable at each grid node and each time step;
- (4)
- Determine whether there are observations. If no observations exist, continue the simulation process. Otherwise, determine the locations of the HADCP velocity data according to the HADCP installation elevation (the middle layer of the 3D mesh), the grid resolution (the HADCP cell length of 4.4 m), and the coordinates of HADCP;
- (5)
- Generate the N particles of velocity states in each grid node of the HADCP locations by adding the noises generated by the normal distribution N ~ (0, Err
_{m}) at time t; - (6)
- Add the HADCP velocities in U and V directions;
- (7)
- Compute the weight of each particle and perform normalization according to Equations (6) and (7);
- (8)
- To inhibit particle degeneration, use residual resampling to copy high-weight particles and eliminate low-weight particles based on the weights from the previous step. The new particles contain similar weights;
- (9)
- Obtain the assimilated velocity state variables for each grid node of the HADCP locations and set them as the initial velocities at time t + 1;
- (10)
- Repeat Steps (4)–(9) until the end of the simulation period;
- (11)
- Output the assimilated velocities.

#### 2.4. Model Evaluation

#### 2.5. Model Calibration and Validation

## 3. Results and Discussion

#### 3.1. Preliminary Sensitivity Analysis

_{m}), observation error (Err

_{o}), and the number of particles (N) are essential PF parameters directly affecting the assimilation effectiveness of the PF. To determine the values of these parameters, we performed a sensitivity analysis using the coupled model, and the results are shown in Table 3.

_{m}, Err

_{o}, and N, were 0.01, 0.001, and 100, respectively. Figure 7 also illustrates the impact of Err

_{m}on the PF. When observation error and number of particles were set to the initial values of 0.001 and 100, respectively, the RMSE of the coupled model was significantly dropped. The values of SKILL and DASS were also significantly increased, with a model error between 0.001 and 0.1. The RMSE value also reached the minimum value (0.045 m/s), and SKILL and DASS had the maximum values of 0.9536 and 0.9077, respectively, with a model error of 0.2. After that, RMSE slightly increased, while SKILL and DASS values were slightly reduced.

_{o}) from 0.001 to 0.01 for the model error, and the number of particles was 0.2 and 100, respectively. For the observation errors lower than 0.005, the RMSE showed a steady decline, while SKILL and DASS increased. By increasing the observation error to 0.005, the changes in the three curves followed the opposite trend. Similar to the case of the model error, the three parameters reached their extreme values (0.0438, 0.961, and 0.9174) for an observation error of 0.005.

#### 3.2. Simulation with Updated Data Assimilation Period

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Overview of the TELAPY module [29].

**Figure 3.**(

**a**) Map of Lanxi River; (

**b**) a 2D computation mesh of the considered experiment reach; (

**c**) the 10 cell locations of HADCP and mesh size of the considered experiment reach; (

**d**) details of the 10 cell locations of HADCP.

**Figure 4.**The flowchart of the proposed DA scheme for soft velocity measurement coupled with the TELEMAC-3D model and PF.

**Figure 5.**The calibration exercise compares the simulated (orange line) and observed (blue line) flow velocity from 06:00 to 23:59 on 5 January 2022.

**Figure 6.**The validation exercise compares the simulated (orange line) and observed (blue line) flow velocity on 6 January 2022.

**Figure 10.**Comparison of the simulated (orange solid line), observed (blue solid line), and assimilated (green dotted line) flow velocity during assimilating period.

**Figure 11.**(

**a**) Comparison of RMSE for the simulation and DA; (

**b**) comparison of SKILL of the simulation and DA; (

**c**) DASS of the experiment during the assimilation period.

**Table 1.**Velocity estimation methods and their characteristics [2].

Method | Operational Complexity | Cost-Effectiveness | Accuracy | Time-Effectiveness | Ecological Impact |
---|---|---|---|---|---|

Float method | Easy | Inexpensive | Low | Efficient | Non-polluting |

Dilution gauging method | Difficult | Inexpensive | Low | Efficient | Affects the stream ecosystem |

Trajectory method | Difficult | Inexpensive | High | Inefficient | Non-polluting |

Current meter method | Difficult | Expensive | High | Efficient | Non-polluting |

Acoustic Doppler’s current profiler method | Difficult | Expensive | High | Efficient | Non-polluting |

Electromagnetic method | Difficult | Expensive | High | Efficient | Non-polluting |

Remote sensing method | Difficult | Expensive | Low | Efficient | Non-polluting |

Particle image velocimetry | Difficult | Expensive | High | Efficient | Non-polluting |

**Table 2.**The RMSE and SKILL values of the calibration and validation exercises of the TELEMAC-3D model.

Trial | Verification Metrics | Number of Cell | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Mean | ||

CalibrationExercise | RMSE | 0.114 | 0.147 | 0.088 | 0.134 | 0.161 | 0.237 | 0.176 | 0.183 | 0.183 | 0.178 | 0.16 |

SKILL | 0.915 | 0.884 | 0.934 | 0.894 | 0.865 | 0.779 | 0.84 | 0.831 | 0.824 | 0.85 | 0.86 | |

ValidationExercise | RMSE | 0.142 | 0.171 | 0.103 | 0.156 | 0.179 | 0.248 | 0.192 | 0.19 | 0.177 | 0.199 | 0.175 |

SKILL | 0.879 | 0.849 | 0.913 | 0.863 | 0.839 | 0.768 | 0.819 | 0.824 | 0.83 | 0.821 | 0.84 |

Trial | Parameters | Verification Metrics | ||||
---|---|---|---|---|---|---|

Model Error (m/s) | Observation Error (m/s) | Number of Particles (-) | RMSE (m/s) | SKILL (-) | DASS (-) | |

1 | 0.01 | 0.001 | 100 | 0.1862 | 0.9055 | −0.0505 |

2 | 0.05 | - | - | 0.1542 | 0.9141 | 0.3023 |

3 | 0.08 | - | - | 0.091 | 0.9326 | 0.7523 |

4 | 0.1 | - | - | 0.0533 | 0.9471 | 0.8974 |

5 | 0.2 | - | - | 0.045 | 0.9536 | 0.9077 |

6 | 0.2 | 0.002 | 0.0449 | 0.9545 | 0.9098 | |

7 | 0.2 | 0.003 | 0.0446 | 0.9567 | 0.9126 | |

8 | 0.2 | 0.004 | 0.0443 | 0.9591 | 0.9144 | |

9 | 0.2 | 0.005 | - | 0.0438 | 0.961 | 0.9174 |

10 | 0.2 | 0.005 | 200 | 0.0416 | 0.9615 | 0.9225 |

11 | 0.2 | 0.005 | 500 | 0.041 | 0.9619 | 0.9234 |

12 | 0.2 | 0.005 | 1000 | 0.0401 | 0.9622 | 0.9251 |

Trial | Verification Metrics | Number of Cell | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Mean | ||

SIM | RMSE | 0.142 | 0.172 | 0.108 | 0.16 | 0.186 | 0.259 | 0.197 | 0.204 | 0.19 | 0.201 | 0.182 |

SKILL | 0.882 | 0.849 | 0.91 | 0.861 | 0.836 | 0.756 | 0.815 | 0.808 | 0.819 | 0.819 | 0.836 | |

DA | RMSE | 0.07 | 0.053 | 0.049 | 0.052 | 0.046 | 0.028 | 0.029 | 0.021 | 0.022 | 0.051 | 0.042 |

SKILL | 0.94 | 0.954 | 0.959 | 0.956 | 0.958 | 0.958 | 0.971 | 0.968 | 0.985 | 0.957 | 0.96 | |

DASS | 0.759 | 0.904 | 0.791 | 0.894 | 0.939 | 0.989 | 0.978 | 0.99 | 0.986 | 0.934 | 0.92 |

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**MDPI and ACS Style**

Sun, Y.; Zhang, L.; Liu, J.; Lin, J.; Cui, Q. A Data Assimilation Approach to the Modeling of 3D Hydrodynamic Flow Velocity in River Reaches. *Water* **2022**, *14*, 3598.
https://doi.org/10.3390/w14223598

**AMA Style**

Sun Y, Zhang L, Liu J, Lin J, Cui Q. A Data Assimilation Approach to the Modeling of 3D Hydrodynamic Flow Velocity in River Reaches. *Water*. 2022; 14(22):3598.
https://doi.org/10.3390/w14223598

**Chicago/Turabian Style**

Sun, Yixiang, Lu Zhang, Jiufu Liu, Jin Lin, and Qingfeng Cui. 2022. "A Data Assimilation Approach to the Modeling of 3D Hydrodynamic Flow Velocity in River Reaches" *Water* 14, no. 22: 3598.
https://doi.org/10.3390/w14223598