# Comparison of Local and Global Optimization Methods for Calibration of a 3D Morphodynamic Model of a Curved Channel

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Experimental Data

_{50}= 1 mm and a geometric standard deviation of ơ

_{g}= 2.5 was placed on the bed, having a final slope of 2‰. Further details regarding the sediment characteristics can be seen in Table 1, which also includes the fall velocity of particles calculated according to the empirical equation of Ferguson and Church [48], by assuming ρ

_{s}= 2650 kg/m

^{3}as the density of sediments. The base flow rate for all experiments was set to Q

_{0}= 0.02 m

^{3}/s, according to the incipient motion of sediment particles, with a flow depth of h

_{0}= 5.44 cm. During the experiments, the discharge was linearly increased up to a maximum value and then progressively lowered to the base flow discharge (triangular-shaped hydrograph). In total, five different experiments were conducted with different durations and peak discharge values.

#### 2.2. Numerical Model

#### 2.3. Calibration Procedure

#### 2.3.1. Local Optimization Method

#### 2.3.2. Global Optimization Methods

#### 2.3.3. Investigated Parameters

- Roughness height at the bed (k
_{s}): For fixed beds, this parameter is typically assumed to be proportional to the representative grain size d_{n}(the diameter where n% of sediment grains are finer) (i.e., Nikoradse’s equivalent grain roughness). For movable beds, however, the roughness caused by bedforms has to be added to the grain roughness, which may increase it with a higher factor than the grain roughness itself. A collection of different equations regarding the roughness height can be found in the literature (e.g., [87]). In this study, despite the dynamic nature of the roughness coefficient due to the formation of bedforms, the roughness height is considered to be uniform along the whole domain, because of the focus on the automatic calibration procedure. The range of this parameter is selected to be between d_{50}and 10d_{90}. - Active layer thickness (ALT): This parameter is described as a function of the representative grain diameter and the bedform height as a dynamic value that depends on the sediment properties and the flow conditions. ALT determines the maximum depth of erosion during one time-step in the numerical model. For this study, a constant value of ALT is also chosen, with a range between d
_{50}and 5d_{max}[88,89]. - The volume fraction of compacted sediments (VFS): This parameter describes the proportion of deposited sediments in the bed compared to the water content, which depends on the bulk density as a function of grain size distribution and packing of sediment depositions. This parameter’s range is adjusted between 40% and 60% in the calibration process.

#### 2.4. Work Structure

## 3. Results and Discussion

#### 3.1. Testing the Performance of Different Optimization Methods

_{s}= d

_{90}, ALT = d

_{max}, and VFS = 50%. However, for ensuring the survival of the gradient-based method by using PEST from being trapped in a local minimum point, three additional starting values are also proposed as:

- PEST#2 (k
_{s}= d_{90}, ALT = 2d_{max}, and VFS = 55%) - PEST#3 (k
_{s}= 2d_{90}, ALT = 4d_{max}, and VFS = 45%) - PEST#4 (k
_{s}= 5d_{90}, ALT = 2d_{max}, and VFS = 50%)

#### 3.2. Application of the Selected Calibration Method for Additional Numerical Setups

_{s}), gain higher values with increase of the flow discharge. It can be argued that the formation and evolution of bedforms along the channel bottom account for this effect.

^{2}) and the root mean squared error (RMSE) statistics, all three calibrated models using Wu’s formula have the best agreement with the observations, with R

^{2}values ranging between 0.89 and 0.95 and RMSE values ranging between 0.67 cm and 1.49 cm. The simulation results with van Rijn’s formula show similar statistical performance as those obtained with Wu’s formula. Nevertheless, the results for the models using Engelund-Hansen’s formula have the lowest correlation and the highest error compared to the measured data.

_{0}, where h

_{0}= 5.44 cm) are presented for all runs using Wu’s formula (right side), and are compared with the experimental data (left side).

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Initial and final bed levels of Run#1 along three longitudinal sections, (

**a**) 10 cm from the inner bank, (

**b**) the central line, and (

**c**) 10 cm from the outer bank.

**Figure 2.**Initial and final bed levels of Run#3 along three longitudinal sections, (

**a**) 10 cm from the inner bank, (

**b**) the central line, and (

**c**) 10 cm from the outer bank.

**Figure 3.**Initial and final bed levels of Run#5 along three longitudinal sections, (

**a**) 10 cm from the inner bank, (

**b**) the central line, and (

**c**) 10 cm from the outer bank.

**Figure 4.**Cross-sectional bed levels for all runs in the maximum deposition (

**a**,

**c**,

**e**) and erosion (

**b**,

**d**,

**f**) sections according to the experiments.

**Figure 5.**Plan view of the normalized bed deformations for (

**a**) Run#1, (

**b**) Run#3, and (

**c**) Run#5, comparing the simulations (

**right**) to the measurements (

**left**).

Sediment Size Classes | ||||||||
---|---|---|---|---|---|---|---|---|

Size (mm) | 0.25 | 0.42 | 0.84 | 1.19 | 2.00 | 3.36 | 4.76 | 8.52 |

Proportion (%) | 6.55 | 10.56 | 25.36 | 15.06 | 20.11 | 13.02 | 4.88 | 4.46 |

Cumulative proportion (%) | 6.55 | 17.11 | 42.47 | 57.53 | 77.64 | 90.66 | 95.54 | 100 |

Fall velocity (m/s) | 0.03 | 0.06 | 0.11 | 0.14 | 0.20 | 0.26 | 0.32 | 0.43 |

Run# | Peak Flow Discharge (m^{3}/s) | Peak Flow Depth (cm) | Duration (min) |
---|---|---|---|

1 | 0.0750 | 12.9 | 180 |

3 | 0.0613 | 11.3 | 240 |

5 | 0.0436 | 9.10 | 420 |

**Table 3.**Calibration results for the numerical model Run#5 by local and global optimization methods, using Wu’s sediment transport formula.

Calibration Results | Calibration Method | ||||||
---|---|---|---|---|---|---|---|

PEST#1 | PEST#2 | PEST#3 | PEST#4 | SCE-UA | CMA-ES | PSO | |

k_{s} (cm) | 0.904 | 0.894 | 0.899 | 0.903 | 0.898 | 0.892 | 0.904 |

ALT (cm) | 1.639 | 1.644 | 1.629 | 1.638 | 1.635 | 1.631 | 1.647 |

VFS (%) | 48.1 | 47.9 | 48.4 | 48 | 48.1 | 48.1 | 47.9 |

Minimum of SSR (cm) | 0.28249 | 0.28251 | 0.28240 | 0.28249 | 0.28247 | 0.28247 | 0.28241 |

Number of model runs | 26 | 30 | 28 | 27 | 176 | 440 | 448 |

_{s}, roughness height at the bed; ALT, active layer thickness; VFS, the volume fraction of compacted sediments; SSR, sum of the squared residuals.

Calibration Parameters | Sediment Transport Formula | ||||||||
---|---|---|---|---|---|---|---|---|---|

Wu | Van Rijn | Engelund-Hansen | |||||||

Run#1 | Run#3 | Run#5 | Run#1 | Run#3 | Run#5 | Run#1 | Run#3 | Run#5 | |

k_{s} (cm) | 1.52 | 1.34 | 0.90 | 0.63 | 0.61 | 0.37 | 0.48 | 0.31 | 0.25 |

ALT (cm) | 2.04 | 1.85 | 1.64 | 2.20 | 1.94 | 1.24 | 1.31 | 1.12 | 0.74 |

VFS (%) | 49 | 51 | 48 | 60 | 60 | 60 | 52 | 51 | 53 |

Goodness of Fit | Sediment Transport Formula | ||||||||
---|---|---|---|---|---|---|---|---|---|

Wu | Van Rijn | Engelund-Hansen | |||||||

Run#1 | Run#3 | Run#5 | Run#1 | Run#3 | Run#5 | Run#1 | Run#3 | Run#5 | |

R^{2} (-) | 0.90 | 0.89 | 0.95 | 0.90 | 0.89 | 0.94 | 0.88 | 0.83 | 0.90 |

RMSE (cm) | 1.49 | 1.04 | 0.67 | 1.57 | 1.08 | 0.69 | 1.63 | 1.33 | 0.81 |

^{2}, the coefficient of determination; RMSE, the root mean squared error.

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## Share and Cite

**MDPI and ACS Style**

Shoarinezhad, V.; Wieprecht, S.; Haun, S.
Comparison of Local and Global Optimization Methods for Calibration of a 3D Morphodynamic Model of a Curved Channel. *Water* **2020**, *12*, 1333.
https://doi.org/10.3390/w12051333

**AMA Style**

Shoarinezhad V, Wieprecht S, Haun S.
Comparison of Local and Global Optimization Methods for Calibration of a 3D Morphodynamic Model of a Curved Channel. *Water*. 2020; 12(5):1333.
https://doi.org/10.3390/w12051333

**Chicago/Turabian Style**

Shoarinezhad, Vahid, Silke Wieprecht, and Stefan Haun.
2020. "Comparison of Local and Global Optimization Methods for Calibration of a 3D Morphodynamic Model of a Curved Channel" *Water* 12, no. 5: 1333.
https://doi.org/10.3390/w12051333