# River Bathymetry Model Based on Floodplain Topography

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Input Data for the Bathy-Supp Model

#### 2.2. Construction of the Bathy-Supp Model

- The user defines the number and location of the new cross-sections (location of cross-section endpoints) from which a new bathymetry model will be composed.
- Computation of the spatial terrain characteristics (predictor variables) derived from the floodplain DEM, and estimation of the model parameters m
_{1}and m_{2}. - Cross-section construction and transformation.
- River bed reconstruction.

#### 2.2.1. Location of the New Cross-Sections

#### 2.2.2. Explaining Model Parameters

_{1}, m

_{2}) are unknown. To establish them (without using an inverse problem-solving method), it is necessary to find the relationship between the search parameters and other explanatory variables. Spatial terrain characteristics can be used as possible explanatory variables [21,26]. The overall curvature, planar curvature, profile curvature, overall slope, slope in the x-direction, and slope in the y-direction were, in the present case, selected for predicting individual model parameters. The characteristics were determined as described by Zevenbergen and Thorne [29]. The source of terrain characteristics was the DEM input. Terrain characteristics were calculated for the all raster cells located behind the bank lines around the river. The terrain characteristics of the nearest raster cell were used to estimate the parameters of a given endpoint. As explanatory variables, the left bank terrain characteristics for model parameter m

_{1}and the right bank terrain characteristics for parameter m

_{2}were used.

_{i}are predictor variables, ε

_{i}is the error term, and M is the number of predictor variables. A mixed selection procedure, as described by Gareth et al. [30], was adopted for choosing the optimal number of variables.

_{i}are positive-definite matrices composed using variance and covariance matrices, β

_{x}are the linear parameters, x

_{i}are predictor variables, ε

_{i}is the error term, and M is the number of predictor variables. Again, the mixed selection procedure, as described by Gareth et al. [30], was adopted for choosing the optimal number of variables.

^{2}) was used to evaluate the reliability of model parameters. As a second quality assessment, vertical differences between models based on the best model parameters and a model based on estimated parameters were calculated. For this comparison, a similar approach is used in Section 2.1.

#### 2.2.3. Cross-Section Construction and Transformation

_{1}and m

_{2}are theoretical model parameters that are unique for each river cross-section. An example of the estimated shapes of the cross-section is shown in Figure 2.

_{1}and E

_{2}, for which the cross-section has been created. All internal points are placed on the line connecting the endpoints. Therefore, the coordinates of the internal points can be calculated based on the coordinates of the endpoints and its station value. The method of calculating the coordinates of internal points may vary depending on the coordinate system used. The lower of the altitudes of both endpoints is determined as the water level of this cross-section. The altitude (Z value) of each point is obtained by subtracting the water level and its depth. All cross-section points have coordinates X, Y, and Z and stationing after transformation.

#### 2.2.4. River Bed Reconstruction

#### 2.3. Model Suitability

#### Model Suitability Evaluation

_{MOD}represents the elevation (m) obtained from each model cross-section and Elev

_{REF}represents the equivalent reference point obtained from the measured cross-section (see Section 2.5.2). N represents the number of the cross-section points.

#### 2.4. Case Study Area

^{2}. The studied river reach is located into the lower part of the river (between 22.83 and 24.58 river km) and is 1.75 km long. The average depth of the river reach fluctuates around 1 m. The average bankfull width is between 22.8 and 52.7 m. The average annual flow rate is 23.4 m

^{3}/s. The average water level is 354.84 m above sea level. The flow rates for N–year floods in the Otava River reach are shown in Table 1. Q

_{a.a.}denotes the average annual discharge.

#### 2.5. Ground and Bathymetry Data

#### 2.5.1. Aerial Laser Scanning Data

#### 2.5.2. Acoustic Doppler Current Profiler Data

#### 2.5.3. Compared DEMs

#### 2.5.4. DEM evaluation

#### 2.6. Hydrodynamic Modelling

^{1/3}were determined for inundations and the value for the main channel was 0.031 s/m

^{1/3}. Selection of the Manning’s values was verified by a calibration-verification process. The validation of model parameters was based on a flood event from December 2012, where, for discharges of 143 m

^{3}/s, the recorded water level (hydrological station located in the model river reach) was equal to 356.43 m. The normal depth was used as the downstream boundary condition. The results of the basic hydrodynamic model (topography source was ADP DEM) for the given discharge provided a difference of 2 cm in the water level. That verified the accuracy of the model setup. All simulations began from a minimal start discharge, which then grew until eventually becoming steady [18,46,47]. Table 1 shows the selected N-year floods that were used as boundary conditions.

_{dif}represents the difference in inundation areas (%), IA

_{DEM}represents the inundation area (km

^{2}) of compared models (ALS, CRO, BAT), and IA

_{REF}represents the inundation area of the ADP model (km

^{2}).

## 3. Results and Discussion

#### 3.1. Bathymetric Model Suitability

_{1}ranged from 1.0202 to 1.8219, and for the m

_{2}parameter from 1.0929 to 2.0376. Overall, 375 measured cross-sections were evaluated. The mean RMSE and MAE values were 0.16 m and 0.11 m, respectively. The variability of the RMSE and MAE values is shown in Figure 4.

#### 3.2. Explaining Bathymetric Model Parameters

#### 3.3. DEM Comparison

#### 3.4. Thalweg Comparison

#### 3.5. Water Surface Elevations Comparison

_{a.a.}, the RMSE was more than 1.2 m, although the WSE variability was comparable to that for other models. For models CRO and BAT, the medians of the differences, as well as their variability, decreased with the increasing flow rate. The BAT model provided the smallest median values and the smallest variance for all rated flows. Table 8 presents RMSE and MAE values for WSE comparison.

#### 3.6. Inundation Areas Comparison

_{a.a.}These differences were diminishing with increasing flow, but nevertheless were >20% for the Q

_{100}flow rate. Similar results had been presented in the works of Bures et al. [18] and Roub et al. [42]. In both those cases, software-modified DEMs (CRO, BAT) provided better results than the ALS model. The CRO model underestimated the ADP model values by as much as 4.5%. The BAT model underestimated them by as much as 3.8%. The fact that CRO and BAT consistently underestimate the results may reflect their similar model settings. The flow area for model cross-section transformation was set the same for the two models.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Defining the position of the new cross-sections. (width (W), endpoint (E), lowest point search radius (R)).

**Figure 4.**Comparing ability of the bathymetric model (with ideal parameters) to represent a measured cross-section. Shown are variances of root mean square error (RMSE) and mean absolute error (MAE) values.

**Figure 7.**Error maps comparing evaluated digital elevation models (ALS, CRO, BAT) and the reference model (ADP).

**Figure 9.**Variance of raster cell differences in water surface elevation between the compared digital elevation models (ALS, CRO, BAT) and the reference model (ADP).

**Table 1.**The flow rates for N-year floods in the Otava River reach [39].

N-year Floods | Q_{a.a.} | Q_{1} | Q_{5} | Q_{10} | Q_{50} | Q_{100} |
---|---|---|---|---|---|---|

Flow rate (m^{3}/s) | 23.4 | 146 | 300 | 394 | 680 | 837 |

Parameter | Value |
---|---|

Calculation method | Longitudinal gradient |

Lowest point search radius (m) | 5 |

Manning’s roughness (s/m^{1/3}) | 0.031 |

Bank slope 1:m | 2 |

Flow rate (m^{3}/s) | 36.8 |

Water surface slope calculation (m) | 100 |

Parameter | Value |
---|---|

Lowest point search radius (m) | 5 |

Manning’s roughness (s/m^{1/3}) | 0.031 |

Flow rate (m^{3}/s) | 36.8 |

Digital Elevation Model | Floodplain Data | Bathymetry Data | Resolution (m) |
---|---|---|---|

ALS | LiDAR | -- | 0.5 |

CRO | LiDAR | CroSolver | 0.5 |

BAT | LiDAR | Bathy-supp | 0.5 |

ADP | LiDAR | ADCP | 0.5 |

**Table 5.**Coefficients of determination for the best-fit model parameters in comparing the estimation techniques linear model (LM), the extended linear model with no random effects (GLS), and the random forest (RF).

LM | GLS | RF | |
---|---|---|---|

R^{2} (m_{1}) | 0.145 | 0.161 | 0.918 |

R^{2} (m_{2}) | 0.085 | 0.118 | 0.914 |

ALS | CRO | BAT | |
---|---|---|---|

RMSE (m) | 1.19 | 0.46 | 0.30 |

MAE (m) | 1.06 | 0.36 | 0.23 |

ALS | CRO | BAT | |
---|---|---|---|

RMSE (m) | 1.52 | 0.37 | 0.30 |

MAE (m) | 1.49 | 0.31 | 0.21 |

Model | Q_{a.a.} | Q_{1} | Q_{10} | Q_{100} | |
---|---|---|---|---|---|

RMSE (m) | ALS | 1.24 | 0.87 | 0.80 | 0.85 |

CRO | 0.15 | 0.14 | 0.10 | 0.07 | |

BAT | 0.13 | 0.06 | 0.04 | 0.03 | |

MAE (m) | ALS | 1.24 | 0.87 | 0.79 | 0.84 |

CRO | 0.13 | 0.12 | 0.09 | 0.05 | |

BAT | 0.10 | 0.04 | 0.03 | 0.02 |

Model | Q_{a.a.} | Q_{1} | Q_{10} | Q_{100} | |
---|---|---|---|---|---|

Inundation | ADP | 0.0540 | 0.0911 | 0.1288 | 0.1608 |

Area | ALS | 0.0830 | 0.1182 | 0.1415 | 0.1953 |

(km^{2}) | CRO | 0.0536 | 0.0870 | 0.1277 | 0.1587 |

BAT | 0.0519 | 0.0898 | 0.1283 | 0.1598 | |

Difference | ADP | - | - | - | - |

in Area | ALS | 53.83 | 29.73 | 9.84 | 21.49 |

(%) | CRO | 0.6 | 4.5 | 0.87 | 1.27 |

BAT | 3.76 | 1.44 | 0.35 | 0.62 |

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**MDPI and ACS Style**

Bures, L.; Sychova, P.; Maca, P.; Roub, R.; Marval, S.
River Bathymetry Model Based on Floodplain Topography. *Water* **2019**, *11*, 1287.
https://doi.org/10.3390/w11061287

**AMA Style**

Bures L, Sychova P, Maca P, Roub R, Marval S.
River Bathymetry Model Based on Floodplain Topography. *Water*. 2019; 11(6):1287.
https://doi.org/10.3390/w11061287

**Chicago/Turabian Style**

Bures, Ludek, Petra Sychova, Petr Maca, Radek Roub, and Stepan Marval.
2019. "River Bathymetry Model Based on Floodplain Topography" *Water* 11, no. 6: 1287.
https://doi.org/10.3390/w11061287