# Accelerating Predictions of Morphological Bed Evolution by Combining Numerical Modelling and Artificial Neural Networks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Theoretical Aspects

^{2}] is the mean bed shear stress over a wave cycle, ${\mathsf{\tau}}_{\mathrm{w}}$ [kg∙m/s

^{2}] is the bed shear stress due to the effect of the waves and ${\mathsf{\tau}}_{\mathrm{c}}$ [kg∙m/s

^{2}] is the current-only bed shear stress contribution.

^{3}] is the water density and f

_{w}[-] is the wave friction factor.

_{s}[-] is the Nikuradse equivalent sand grain roughness usually related to the mean sediment diameter d

_{50}.

^{1/3}/s] is the Manning friction coefficient and g [m/s

^{2}] is the acceleration of gravity.

^{3}] is the density of the sediment grains and ${\mathrm{d}}_{50}$ [m] is the median sediment diameter.

^{2}] as the acceleration of gravity, $\mathrm{s}=\frac{{\mathsf{\rho}}_{\mathrm{s}}}{{\mathsf{\rho}}_{\mathrm{w}}}$ [-] is the ratio of the sediment (${\mathsf{\rho}}_{\mathrm{s}})$ to the water (${\mathsf{\rho}}_{\mathrm{w}})$ density, ν [m

^{2}/s] is the kinematic viscosity of the water and d

_{50}[m] is the median sediment diameter.

_{e}[rad] denotes the yearly equivalent incidence wave angle, with the subscript “b” denoting the value at the breaker line. In the right-hand side of Equation (11) the subscript “i” denotes individual wave events, with ${\mathrm{f}}_{\mathrm{i}}$ being the frequency of occurrence of each sea-state.

#### 2.2. Methodology Outline

_{50}by specifying a Nikuradse roughness ${\mathrm{k}}_{\mathrm{s}}=2{.5\mathrm{d}}_{50}$. The correspondence between the values within the surf zone and offshore sea-state characteristics ${\mathrm{H}}_{\mathrm{s}}{,\mathrm{T}}_{\mathrm{p}}{,\mathrm{a}}_{\mathrm{o}}$ can be obtained by implementing a non-linear parabolic mild slope wave model [24] for wave propagation and a flow model to estimate wave-induced current velocities.

- Three idealized alongshore uniform bathymetries were set up, each characterized by a different value of the bed slope tanβ, namely 1:5, 1:25, and 1:50.
- Determination of a number of combinations of the Input Parameters (IP) of $\left\{{\mathrm{H}}_{\mathrm{s}}{,\mathrm{T}}_{\mathrm{p}}{,\mathrm{a}}_{\mathrm{o}}{,\mathrm{d}}_{50},\mathrm{tan}\mathsf{\beta}\right\}$ to be inserted in the numerical modelling simulations.

- 3.
- For each set of IP, nearshore wave propagation simulations were carried out utilizing a spectral version of a nonlinear parabolic mild slope wave model and the resulting radiation stresses were then inserted as forcing input to a flow model based on the depth-averaged Shallow Water Equations, providing the current speed and current direction.
- 4.
- Subsequently calculations of the value of ${\mathsf{\tau}}_{\mathrm{max}}$ over a wave cycle were carried out for a cross-section in the middle of the domain in the alongshore direction through a post-processing algorithm. The value of ${\mathsf{\tau}}_{\mathrm{max}}$ is taken as the average value computed shoreward the maximum depth where initiation of breaking occurs, according to the criterion of [27], out of all the 3200 distinct wave records. The set of IP $\left\{{\mathrm{H}}_{\mathrm{s}}{,\mathrm{T}}_{\mathrm{p}}{,\mathrm{a}}_{\mathrm{o}}{,\mathrm{d}}_{50},\mathrm{tan}\mathsf{\beta}\right\}$ are consequently linked to a unique value of the Output Parameter (OP) $\left\{{\mathsf{\tau}}_{\mathrm{max}}\right\}$. The corresponding values of IP and OP are provided for the training and validation of an ANN, which will predict ${\mathsf{\tau}}_{\mathrm{max}}$ for any values within the ranges defined in Table 1.
- 5.
- Through Equations (8) and (9) estimation of ${\mathsf{\tau}}_{\mathrm{cr}}$ follows. Each wave record that satisfies the condition ${\mathsf{\tau}}_{\mathrm{max}}{\mathsf{\tau}}_{\mathrm{cr}}$ is eliminated, since it is considered unable to initiate sediment motion and thus produce significant morphological changes. Each eliminated wave record is considered to contribute to the “calm conditions” (i.e., waves from directions exiting the numerical domain) hence it retains its individual frequency of occurrence ${\mathrm{f}}_{\mathrm{i}}$. After elimination, a reduced dataset containing only the sea-states satisfying the Shield’s criterion of incipient sediment motion is obtained.
- 6.
- Utilizing the reduced dataset as input and implementing Equations (12) and (13) a set of annual “equivalent” wave characteristics are defined. The equivalent wave characteristics are used to force the coastal area model and obtain predictions of the coastal bed evolution. In our implementations, we used a combination of a parabolic mild slope wave model (PMS) a hydrodynamic model (HYD), and an initial sedimentation/erosion and morphological model (SDT).

#### 2.3. Overview of the Numerical Models

#### 2.3.1. The Parabolic Mild Slope Wave Model (PMS)

#### 2.3.2. The Hydrodynamic Model (HYD)

^{3}] is the seawater density, $\mathrm{h}$ [m] is the total water depth, $\mathrm{f}$ is the Coriolis coefficient, $\mathrm{g}$ is the acceleration of gravity, ${\mathrm{v}}_{\mathrm{h}}$ is the horizontal turbulent eddy viscosity coefficient, ${\mathsf{\tau}}_{\mathrm{sx}}$, ${\mathsf{\tau}}_{\mathrm{sy}}$ [kg·m/s

^{2}] are the components of the wind shear stress in the x and y axis respectively, $\mathrm{S}$ are external discharges added or subtracted in case of a point source or sink respectively, ${\mathrm{S}}_{\mathrm{x}},{\mathrm{S}}_{\mathrm{y}}$ are external discharges added in case of an external point source. Wave-generated currents can also be simulated considering the effect of the radiation stress components ${\mathrm{S}}_{\mathrm{xx}},{\mathrm{S}}_{\mathrm{xy}}$ and ${\mathrm{S}}_{\mathrm{yy}}$.

#### 2.3.3. The Sediment Transport and Morphological Model (SDT)

#### 2.4. ANN Parametrization and Training

- An input layer sending input data to the network
- A hidden layer composed of 32 units and a rectified linear unit (relu) activation function
- A hidden layer composed of 64 units using also a reLU activation function
- An output layer with an identity function

^{−5}was obtained along with a correlation factor R equal to 0.998. Figure 1a showcases the evolution of the MSE at each epoch during the ANN training, while Figure 1b presents the ANN predicted values plotted against the respective targets of the generalization dataset.

## 3. Methodology Implementation

#### 3.1. Idealized Test Case Validation

- A “brute force” simulation set (BF) containing a detailed representation of the wave climate composed of 36 sea-states propagating from the dominant wave directions.
- A simulation set with the elimination of sea-states considered unable to initiate sediment motion by implementing the developed ANN to reduce the input dataset. Thereafter seven “annual equivalent” wave representatives are obtained from the reduced dataset, one for each dominant wave direction and are used to force the integrated models. This simulation will be thereafter denoted as “EW w/ANN”
- A simulation set with the forcing conditions being the seven “annual equivalent” wave representatives considering the full wave climate. This simulation will be thereafter denoted as “EW”

#### 3.2. Real-Field Test Case

- A “brute force” simulation set (BF) containing a robust representation of the wave climate composed of 78 sea-states.
- A simulation set with the elimination of sea-states considered unable to initiate sediment motion by implementing the developed ANN to reduce the input dataset. This simulation set will be thereafter denoted as “EW w/ANN” and contains a representative sea-state for each dominant wave direction.
- A simulation set with the forcing conditions being the seven “annual equivalent” wave representatives after the elimination of wave records by employing an arbitrary threshold of ${\mathrm{H}}_{\mathrm{s}}$ < 0.5. This simulation will be thereafter denoted as “EW w/threshold”.

## 4. Results and Discussion

#### 4.1. Idealized Test Case

#### 4.1.1. Obtained Representative Wave Conditions

#### 4.1.2. Morphological Modelling Results and Evaluation

#### 4.2. Real-Field Case Study

#### 4.2.1. Obtained Representative Wave Conditions

#### 4.2.2. Morphological Modelling Results and Evaluation

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**Error and performance metrics of the training procedure: (

**a**) Evolution of the MSE during the training and validation of the generalization dataset (

**b**) ANN predicted values against the respective targets of the generalization dataset.

**Figure 3.**Selected overview of the bathymetry of the idealized case, evaluation area of the BSS (enclosed in the red rectangle). The shadow area of the breakwater is denoted by the black lines.

**Figure 6.**Retained (blue markers) and eliminated (light blue markers) sea-states by implementing the proposed methodology for the idealized case of bed slope 1:50 for (

**a**) d

_{50}= 1 mm and (

**b**) d

_{50}= 2 mm. The red data points denote the representatives for the simulation EW w/ANN and the orange those by implementing EW in the full dataset.

**Figure 7.**Spatial distribution of (

**a**) significant wave height simulated with PMS and (

**b**) hydrodynamic circulation simulated with HYD, for the idealized case with slope 1:50.

**Figure 8.**Integrated annual rate of bed level change for the: (

**a**) “BF” (

**b**) “EW w/ANN”, and (

**c**) “EW” simulation sets.

**Figure 9.**Retained (blue markers) and eliminated (light blue markers) sea-states by implementing the proposed methodology and by employing an arbitrary threshold (green markers). The red data points denote the representatives for the simulation “EW w/ANN” and the orange those by following the “EW w/threshold” approach.

**Figure 10.**Integrated annual rate of bed level change for the (

**a**) “BF” (

**b**) “EW w/ANN”, and (

**c**) “EW w/threshold” simulation sets.

${H}_{s}\left[m\right]$ | ${T}_{p}\left[s\right]$ | ${a}_{o}[\xb0]$ | ${d}_{50}\left[\mathrm{mm}\right]$ | $\mathrm{tan}\mathsf{\beta}[-]$ | |
---|---|---|---|---|---|

minimum | 0 | 0 | −90 | 0.06 | 1:50 |

maximum | 7 | 15 | 90 | 2.0 | 1:5 |

**Table 2.**Examined scenarios and combinations of the considered values of ${\mathrm{d}}_{50}$ and $\mathrm{tan}\mathsf{\beta}$.

Test ID | ${d}_{50}\left[\mathrm{mm}\right]$ | $\mathbf{tan}\mathsf{\beta}$ |
---|---|---|

IC1 | 0.1 | 1:50 |

IC2 | 1.0 | 1:50 |

IC3 | 2.0 | 1:50 |

IC4 | 0.1 | 1:20 |

IC5 | 1.0 | 1:20 |

IC6 | 2.0 | 1:20 |

**Table 3.**Classification for the BSS (Adapted with permission from Ref [45], 2004, Elsevier).

BSS | |
---|---|

Excellent (E) | 1.0–0.5 |

Good (G) | 0.5–0.2 |

Reasonable/Fair (R/F) | 0.2–0.1 |

Poor (P) | 0.1–0.0 |

Bad (B) | <0.0 |

Sector | EW w/ANN | EW | ||||||
---|---|---|---|---|---|---|---|---|

H_{e} (m) | T_{e} (s) | MWD (°) | f (%) | H_{e} (m) | T_{e} (s) | MWD (°) | f (%) | |

SSW | 0.96 | 7.94 | 202.5 | 0.9765 | 0.62 | 6.04 | 202.5 | 13.30 |

SW | 1.04 | 8.35 | 225.0 | 1.6568 | 0.60 | 6.07 | 225.0 | 20.46 |

WSW | 0.98 | 7.87 | 247.5 | 0.2747 | 0.52 | 5.55 | 247.5 | 5.78 |

W | 1.02 | 7.08 | 270.0 | 0.2447 | 0.53 | 5.18 | 270.0 | 5.93 |

WNW | 1.15 | 6.72 | 292.5 | 2.5289 | 0.72 | 5.10 | 292.5 | 27.86 |

NW | 0.99 | 6.75 | 315.0 | 0.7542 | 0.51 | 4.58 | 315.0 | 24.93 |

NNW | 0.81 | 7.36 | 337.5 | 0.0034 | 0.46 | 5.89 | 337.5 | 0.50 |

Test ID | EW w/ANN | EW |
---|---|---|

IC1 | 0.92 (E) | 0.65 (E) |

IC2 | 0.77 (E) | 0.24 (G) |

IC3 | 0.65 (E) | 0.22 (G) |

IC4 | 0.86 (E) | 0.45 (G) |

IC5 | 0.86 (E) | 0.19 (R/F) |

IC6 | 0.72 (E) | 0.04 (P) |

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**MDPI and ACS Style**

Papadimitriou, A.; Chondros, M.; Metallinos, A.; Tsoukala, V.
Accelerating Predictions of Morphological Bed Evolution by Combining Numerical Modelling and Artificial Neural Networks. *J. Mar. Sci. Eng.* **2022**, *10*, 1621.
https://doi.org/10.3390/jmse10111621

**AMA Style**

Papadimitriou A, Chondros M, Metallinos A, Tsoukala V.
Accelerating Predictions of Morphological Bed Evolution by Combining Numerical Modelling and Artificial Neural Networks. *Journal of Marine Science and Engineering*. 2022; 10(11):1621.
https://doi.org/10.3390/jmse10111621

**Chicago/Turabian Style**

Papadimitriou, Andreas, Michalis Chondros, Anastasios Metallinos, and Vasiliki Tsoukala.
2022. "Accelerating Predictions of Morphological Bed Evolution by Combining Numerical Modelling and Artificial Neural Networks" *Journal of Marine Science and Engineering* 10, no. 11: 1621.
https://doi.org/10.3390/jmse10111621