# Extending the Applicability of the Meyer–Peter and Müller Bed Load Transport Formula

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data

^{2}and lies downstream of Platanovrysi Dam. The river basin outlet is located at Toxotes. The river basin terrain is covered by forest (48%), bush (20%), cultivated land (24%), urban area (2%) and no significant vegetation (6%). The altitude ranges between 80 m and 1600 m, whereas the length of Nestos River is 55 km. The mean slope of Nestos River in the basin is 0.35%. The stream flow rate and bed load transport rate measurements concerning Nestos River were conducted at a location between the outlet of Nestos River basin (Toxotes) and the river delta. The measurement procedures are described in [25].

_{Gm}(kg/(s m)) is the measured bed load transport rate per unit width, Q (m

^{3}/s) is the measured stream discharge, b (m) is the measured width of the assumed rectangular cross section, h (m) is the measured flow depth, u

_{m}(m/s) is the measured mean flow velocity, d

_{50}(m) is the median grain diameter of bed load, determined by the granulometric curves, and d

_{90}(m) is a characteristic grain size diameter (in case of taking a stream bed load sample, concerning the sample weight, 90% is composed of grains with size less than or equal to d

_{90}).

#### 2.2. Meyer–Peter and Müller (MPM) Bed Load Transport Formula

_{Gc}: computed bed load transport rate per unit width (kg/(s·m))

^{2})

_{F}: sediment density (kg/m

^{3})

_{w}: water density (kg/m

^{3})

_{o}: actual shear stress (N/m

^{2})

_{o}

_{,cr}: critical shear stress (N/m

^{2}), characterizing the incipient motion of bed grains

_{r}: energy line slope due to individual grains

_{s}: hydraulic radius of the specific part of the cross section under consideration which affects the bed load transport (m).

_{m}: mean diameter of bed load grains (m)

_{st}: Strickler coefficient, the value of which depends on the roughness due to individual grains, as well as to stream bed forms (m

^{1/3}/s).

_{r}: coefficient, with value depending on the roughness due to individual grain (m

^{1/3}/s)

_{90}: characteristic grain size diameter (m). It was defined for Table 1.

- Slope of energy line (I) from 0.04% to 2%
- Sediment particle size (d
_{50}) from 0.4 mm to 20 mm - Flow depth (h) from 0.01 m to 1.20 m
- Specific stream discharge (Q/b) from 0.002 m
^{2}/s to 2 m^{2}/s - Relative sediment density (ρ
_{F}/ρ_{w}) from 0.25 to 3.2 - Particle size > 1 mm, to avoid the effects of apparent cohesion
- Flow depth > 0.05 m, to assure Froude similitude.

^{2}) is the stream cross section, assuming a rectangular section, approximately, and where R (m) is the hydraulic radius and U (m) is the wetted perimeter. The indices s and w stand for bed and walls, respectively. The hydraulic radius R

_{w}is given by the familiar Manning formula:

_{m}(m/s) is the mean flow velocity through the cross-sectional area A and k

_{w}(m

^{1/3}/s), a coefficient depending on the roughness of the walls. It is assumed that k

_{w}= k

_{st}. Additionally, I is set equal to the longitudinal stream bed slope on the basis of the assumption of uniform flow.

_{s}, by combining Equations (3) and (4), turns out as

_{Gc}becomes dimensionless by means of Equation (7). The derivation of Equation (9) is given in Appendix A.

_{m}can be approximated by the median grain diameter d

_{50}. Therefore, Equation (6) acquires the simpler non-dimensional form:

_{p50}(Equation (8)), an explicit particle Reynolds number, Re* (Equation (9)), a shear Reynolds number, ρ′ (appearing in the third one of Equations (2)), the submerged specific gravity of the sediment.

#### 2.3. Calibration of an Enhanced Meyer–Peter and Müller (EMPM) Formula

_{m}, U

_{w}, U

_{s}and d

_{50}, as well as measured values of bed transport rates m

_{Gm}, denoted respectively as A

_{i}, u

_{mi}, U

_{wi}, U

_{si}, d

_{50,i}and m

_{Gmi}, for i = 1, 2,…, N, where N is the number of data points. These subscripted quantities are substituted into the corresponding Equations (7)–(10), giving the non-dimensional bed load transport rate of Equation (10) in terms of measured quantities:

_{st}emerges as an adjustment parameter. Therefore, the following expression can be used for the calibration of the MPM formula:

_{Gmi}, i = 1,…,N, denote measured values of bed load transport rate. Calibration with respect to one parameter only, namely k

_{st}, has already been tried (Sidiropoulos et al., 2018). In this paper, Equation (12) is further extended, so as to include more adjustment parameters:

**d**= (k

_{M}_{st}, α, β, γ) is the vector of parameters,

_{M}by Equation (17).

_{M}of Equation (19) was executed by a genetic algorithm followed by a Nelder–Mead local search.

#### 2.4. Application of Machine Learning Schemes

#### 2.4.1. Random Forests

**p**

_{i}, serve as input variables in the RF learning scheme, while ${\mathrm{m}}_{\mathrm{Gmi}}$ will be the corresponding target for the output. The general form of Equations (18) and (19) can be used again for the formation of the objective function, as follows:

**d**

_{R}is the vector of the RF parameters (i.e., the set of decision trees that operate as an ensemble) that will be determined through training.

#### 2.4.2. Gaussian Processes Regression

**x**is derived by means of a Gaussian stochastic process with an assumed mean equal to 0 and with a variance σ

^{2}calculated in terms of covariances involving

**x**and the training data. A suitable covariance function is selected and parametrized, and finally, the hyperparameters involved are determined through optimization.

**d**

_{R}replaced by

**d**

_{G}for GPR. The vector

**d**

_{G}represents the internal parameters of the respective machine learning process, which will be optimally determined according to the above outline.

#### 2.5. Training and Testing Procedures

#### 2.6. Nearest Neighbor Smoothing

**p**

_{i}of input measured quantities (Equation (16)), the distances are computed to all other vectors

**p**

_{j}, j = 1, 2,…, N, where N is the number of available measurement points. The k nearest neighbors to

**p**

_{i}are then picked out and the average of these is taken, as well as the average of the corresponding bed load transport measurements. These averages will replace the original data. The process is formalized as follows:

**p**

_{i}and all

**p**

_{j}’s, including

**p**

_{i}itself.

_{i}from all other parameter vectors, as computed according to Equation (22).

_{i}of Equation (23) and let

_{M}, and in Equations (21) and (22) for the formation of objective function F

_{R}.

## 3. Results and Discussion

#### 3.1. Implementation of Algorithms

_{st}, α, β and γ, contained in the objective function of Equation (19). The last parameter γ is related to the so-called threshold referred to in the introduction, which turned out to be zero for our data and for all repetitions of the EMPM formula calibrations. The required minimization was executed by a genetic algorithm followed by a Nelder–Mead local search. As is well-known, the genetic algorithm points to the area within which the minimum needs to be sought and the sequent local search more accurately yields the location of the minimum [32].

#### 3.2. Error Metrics

^{2}).

_{i}= m

_{Gci}/m

_{Gmi}, i = 1, 2,…,N

^{2}.

^{2}is a well-established index in hydrologic modeling, being the same as the well-known Nash–Sutcliff Efficiency index (NSE). However, serious reservations have been reported in the pertinent literature as to its autonomous use for performance evaluation [34]. Various cases have been reported in which a good model could have low R

^{2}and a bad model high R

^{2}. Also, in the context of sediment transport modeling, due to inherent difficulties and due to the noise in field data, low R

^{2}values would not be a surprise.

^{2}, has been a primary realistic index, since it estimates errors in terms of ratios rather than differences between measured and computed values. Regarding discrepancy ratios in this study, a qualitative comparison of the present results to analogous results of the literature will be given in Section 3.4.

#### 3.3. Results of Error Metrics and Related Comparisons

- for RDEF, k
_{st}= 10.7379, α = 0.2000, β = 1.1199, γ = 0 and - for SDEF, k
_{st}= 30.00, α = 0.6519, β = 3.5374, γ = 0.

^{2}are not to be taken as a sole representative measure of performance. In comparative terms, the R

^{2}value for RDEF from Table 2 is 0.47962, while for RDOF it is 0.169077; i.e., R

^{2}is about three times greater for the enhanced versus the original MPM formula.

#### 3.4. Discrepancy Ratio Comparisons

_{1}, D

_{2}and D

_{3}, appear in relation to discrepancy ratios:

_{i}’s be the ratios defined in Equation (27). Then

- D
_{1}is defined as the percentage of dr_{i}’s such that 0.5 ≤ dr_{i}≤ 2. - D
_{2}is defined as the percentage of dr_{i}’s such that 0.25 ≤ dr_{i}≤ 4. - D
_{3}is defined as the percentage of dr_{i}’s such that 0.1 ≤ dr_{i}≤ 10.

_{1}, D

_{2}, D

_{3}) is equal to (3%, 7%, 9%), especially for the MPM formula resulting from 6319 values of a field dataset regarding sand and gravel bed streams in USA. In [16], D

_{1}= 3%, especially for the MPM formula applied to Ebro River (Spain) with the gravel bed.

- Raw data, original formula: (D
_{1}, D_{2}, D_{3}) = (32%, 45%, 60%). - Raw data, enhanced formula: (D
_{1}, D_{2}, D_{3}) = (55%, 81%, 97%).

#### 3.5. Examination of Possible Overfitting

#### 3.6. Statistical Comparisons

^{−18}) indicated that the null hypothesis of all the algorithms performing the same could safely be rejected. Then, post-hoc tests followed for all possible pairs of algorithms using the Wilcoxon signed rank test [38]. Because of the multiple pairwise tests, the p-values that resulted were adjusted using the Benjamini and Hochberg method [39], which controls the false discovery rate (Table 5 and Table 6). Table 5 shows the adjusted p-values below the main diagonal and at the upper diagonal positions, showing the estimated differences between methods. Table 6 shows the corresponding quantities for smoothed data. Indeed, the p-values indicate statistical differences, and it is noted that the enhanced formula not only shows better performance compared to the original formula, but also compared to the machine learning methods.

#### 3.7. Indicative Scatter Plots

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Derivation of Equation (6)

_{o}and τ

_{o}

_{,cr}are replaced by their values from the two first Equations (2):

_{50}: median grain size):

#### Appendix A.2. Derivation of Equation (9)

_{50}: median grain diameter (m)

^{2}/s).

_{m}: mean flow velocity (m/s)

_{o}: bed shear stress (N/m

^{2})

_{w}: water density (kg/m

^{3})

^{2})

_{s}regarding the specific part of the cross section which affects the bed load transport:

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**Figure 1.**The Nestos River basin. The red filled circle symbolizes the location of the bed load measurements.

**Figure 2.**Boxplots for: (

**a**) out-of-the-bag RMSE for all methods and bootstrapping repetitions using raw data; (

**b**) out-of-the-bag RMSE for all methods and bootstrapping repetitions using smoothed data.

**Figure 3.**Predicted values for a random out-of-the-bag dataset using the optimal tuned parameters of each model versus observed values of bed load.

**Figure 4.**Predicted values for a random out-of-the-bag dataset using the optimal tuned parameters of each model versus smoothed values of bed load.

**Table 1.**The average statistical properties of bed load related values. SD is an abbreviation for standard deviation.

Variable | Min | Mean | Median | Max | SD | Skew | Kurtosis |
---|---|---|---|---|---|---|---|

${\mathrm{m}}_{\mathrm{Gm}}$ (kg/(s·m)) | 0 | 0.0225 | 0.0175 | 0.0883 | 0.0201 | 1.0790 | 0.7741 |

Q (m^{3}/s) | 0.100 | 2.457 | 1.760 | 11.020 | 2.242 | 1.872 | 3.393 |

b (m) | 6.00 | 16.20 | 15.70 | 32.00 | 7.35 | 0.40 | −0.82 |

h (m) | 0.10 | 0.32 | 0.31 | 0.61 | 0.11 | 0.33 | −0.21 |

${\mathrm{u}}_{\mathrm{m}}$ (m/s) | 0.20 | 0.47 | 0.44 | 1.48 | 0.21 | 1.92 | 5.02 |

d_{90} (m) | 0.0014 | 0.0028 | 0.0030 | 0.0037 | 0.0006 | −0.8150 | −0.2193 |

d_{50} (m) | 0.0008 | 0.0045 | 0.0014 | 0.0235 | 0.0064 | 1.6859 | 1.0979 |

**Table 2.**Mean and standard deviation of error metrics for raw data and for the out-of-the-bag dataset from bootstrap resampling. RMSE units: kg/(s·m).

Mean | SD | ||
---|---|---|---|

RDOFRaw Data Original Formula | RMSE | 0.05646 | 0.00564 |

MDR | 6.20336 | 1.44002 | |

R^{2} | 0.16077 | 0.12808 | |

RDEFRaw Data Enhanced Formula | RMSE | 0.01928 | 0.00217 |

MDR | 3.06172 | 0.92239 | |

R^{2} | 0.47922 | 0.08730 | |

RDRFRaw Data Random Forests | RMSE | 0.01832 | 0.00247 |

MDR | 3.49139 | 1.49874 | |

R^{2} | 0.54513 | 0.10054 | |

RDGPRaw Data Gaussian Processes | RMSE | 0.01999 | 0.00280 |

MDR | 3.36799 | 1.82963 | |

R^{2} | 0.34135 | 0.10937 |

**Table 3.**Mean and standard deviation of error metrics for smoothed data and for the out-of-the-bag dataset from bootstrap resampling. RMSE units: kg/(s·m).

Mean | SD | ||
---|---|---|---|

SDOFSmoothed Data Original Formula | RMSE | 0.04159 | 0.006065 |

MDR | 2.37244 | 0.719034 | |

R^{2} | −0.94236 | 0.235836 | |

SDEFSmoothed Data Enhanced Formula | RMSE | 0.00926 | 0.001925 |

MDR | 1.20260 | 0.27857 | |

R^{2} | 0.48299 | 0.20773 | |

SDRFSmoothed Data Random Forests | RMSE | 0.01030 | 0.002019 |

MDR | 1.31873 | 0.344024 | |

R^{2} | 0.40220 | 0.123037 | |

SDGPSmoothed Data Gaussian Processes | RMSE | 0.01180 | 0.002842 |

MDR | 1.27084 | 0.356768 | |

R^{2} | 0.22199 | 0.172782 |

Training Set | Test Set | |
---|---|---|

RDOF | 0.05427 | 0.05646 |

RDEF | 0.01785 | 0.01928 |

RDRF | 0.01367 | 0.01833 |

RDGP | 0.00690 | 0.01999 |

SDOF | 0.04476 | 0.04159 |

SDEF | 0.01037 | 0.00926 |

SDRF | 0.00696 | 0.01030 |

SDGP | 0.00191 | 0.01180 |

**Table 5.**To the right of the diagonal stand the estimated differences of RMSE between models for raw data. To the left of the diagonal stand the adjusted p-values for the H

_{0}(null hypothesis): difference = 0.

RDOF | RDGP | RDEF | RDRF | |
---|---|---|---|---|

RDOF | - | 0.0363 | 0.0377 | 0.0381 |

RDGP | 7.9 × 10^{−18} | - | 0.0014 | 0.0017 |

RDEF | 7.9 × 10^{−18} | 0.02 | - | 0.0003 |

RDRF | 7.9 × 10^{−18} | 4.3 × 10^{−5} | 0.01 | - |

**Table 6.**To the right of the diagonal stand the estimated differences of RMSE between models for smoothed data. To the left of the diagonal stand the adjusted p-values for the H

_{0}(null hypothesis): difference = 0.

SDOF | SDGP | SDRF | SDEF | |
---|---|---|---|---|

SDOF | - | 0.0303 | 0.0313 | 0.0323 |

SDGP | 7.9 × 10^{−18} | - | 0.0010 | 0.0020 |

SDRF | 7.9 × 10^{−18} | 8.2 × 10^{−4} | - | 0.0010 |

SDEF | 7.9 × 10^{−18} | 7.6 × 10^{−9} | 5.3·10^{−4} | - |

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**MDPI and ACS Style**

Sidiropoulos, E.; Vantas, K.; Hrissanthou, V.; Papalaskaris, T.
Extending the Applicability of the Meyer–Peter and Müller Bed Load Transport Formula. *Water* **2021**, *13*, 2817.
https://doi.org/10.3390/w13202817

**AMA Style**

Sidiropoulos E, Vantas K, Hrissanthou V, Papalaskaris T.
Extending the Applicability of the Meyer–Peter and Müller Bed Load Transport Formula. *Water*. 2021; 13(20):2817.
https://doi.org/10.3390/w13202817

**Chicago/Turabian Style**

Sidiropoulos, Epaminondas, Konstantinos Vantas, Vlassios Hrissanthou, and Thomas Papalaskaris.
2021. "Extending the Applicability of the Meyer–Peter and Müller Bed Load Transport Formula" *Water* 13, no. 20: 2817.
https://doi.org/10.3390/w13202817