# Multi-Fraction Bayesian Sediment Transport Model

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

_{35}or D

_{50}, or the grain sizes for which 35% and 50% of the particles are smaller) to represent the actual range of sizes in the sediment. Such simplifications can be useful, but cannot capture many sub-processes that drive grain sorting and can thereby influence overall transport.

#### 1.1. Background

#### 1.2. Objectives

## 2. Experimental Section

#### 2.1. Model Formulation

#### 2.1.1. Sediment Transport Governing Equations

_{rj}is the reference stress of the jth size fraction (ML

^{−1}T

^{−2}), and τʹ is the skin friction shear stress (ML

^{−1}T

^{−2}). ${W}_{j}^{*}$ is defined as:

^{−2}), q

_{s,j}is the unit bed load transport rate of the j

^{th}size fraction [L

^{2}/T], s is the specific gravity (2.65), F

_{j}is the proportion of size j on the bed surface, ${u}^{*}=\sqrt{\tau \u02b9/\rho}$(LT

^{−1}), and ρ is the density of the water (ML

^{−3}).

_{rj}is a surrogate for the critical stress at initial grain motion. Equation (1) is the basis of a similarity collapse using τ

_{rj}as the single adjustable parameter [3,15]. Equation (1) is similar in form to the equations proposed by Parker [16,17] and used in a previous Bayesian analysis by Schmelter et al. [4]. The relation can be applied to bulk transport by dropping the j subscript.

_{r,sm}is reference shear stress for the geometric mean particle size of the surface (ML

^{−1}T

^{−2}), D

_{sm}is the geometric surface mean diameter (L), and the exponent b (dimensionless) in (3) is:

_{r,sm}, which Wilcock and Crowe [6] found depended on the fraction of sand in the bed surface. It was determined that the Shields parameter, ${\tau}_{r,sm}^{*}$ (ML

^{−1}T

^{−2}), in the sediment experiments varied with sand content, F

_{S}, by:

#### 2.1.2. Multi-fraction Bayesian Transport Model

_{r,bk}~ TN(μ

_{bk},ψ

_{bk},a

_{bk},b

_{bk}) is the reference shear stress of the bulk transport rate (ML

^{−1}T

^{−2}) with hyperpriors μ

_{bk},ψ

_{bk},a

_{bk},b

_{bk}; ${\sigma}_{bk}^{2}$ ~ N(η

_{bk},γ

_{bk}

^{2}) is the variance of the bulk transport (ML

^{−1}T

^{−2})

^{2}with hyperpriors η

_{bk},γ

_{bk}

^{2}; Q

_{s;o;i;bk}is vector of n observations (for i = 1,...,n) of sediment transport summed across all size fractions for a given flow condition (shear stress) and Q

_{s,0}is the vector of sediment transport observations. The logarithm of Q

_{s,0}is used because transport rate is a power function of shear stress and is modeled as log(Q

_{s,o,i,bk}) ~ N(log(Q

_{s,i,bk}),σ

_{b}

^{2}), where the mean value of the distribution is the logarithm of the dimensionalized transport rate (i.e., Qs rather than W*) calculated by Equations (1) and (2). This first model evaluates the total bed load transport of all size fractions for a given flow condition in aggregate and results from the likelihood and priors described immediately above.

_{s,o,i,j}is an n × m matrix of sediment transport observations for each size fraction (i = 1,...,n observations and j = 1,...,m size fractions). The terms in bold are vector forms of those terms defined in Equation (7) due to the use of the multiple individual particle sizes.

_{s,o,i,j})|τ

_{c,j,}σ

_{j}

^{2}~ N(log(Q

_{s,i,j}),σ

_{j}

^{2})

_{s,i,j}is determined from Equation (1), using Equations (2)–(6). As demonstrated in previous work, the mean value of the likelihood can be any function. In the absence of a pre-defined model, the mean value of the likelihood could be a statistical function of forcing variables with unknown coefficients. In this case, the number of model parameters to estimate increases and priors would need to be specified for each of the new parameters.

_{r,j}~ TN(μ

_{j},ψ

_{j},a

_{j},b

_{j})

_{j}, ψ

_{j}, a

_{j}, and b

_{j}are hyperpriors (mean, variance, lower support and upper support, respectively) of the truncated normal distribution on τ

_{r,j}. For the bulk transport model, the same specification applies by simply dropping the j subscripts. The prior standard deviation for each particle size j is distributed as:

_{j}) ~ N(η

_{j},γ

_{j}

^{2})

_{j}and γ

_{j}

^{2}are hyperpriors (mean and variance, respectively) of the normal distribution on log(σ

_{j}) (essentially, a lognormal distribution on σ

_{j}). As before, replacing the j subscript with bk specifies the prior variance for the bulk transport model.

#### 2.2. MCMC Algorithm

^{2}was conjugate with the likelihood) and a Metropolis update for the reference/critical stress. The Metropolis update [20] was manually tuned. The model presented here has many more dimensions that make manually tuning the MCMC algorithm prohibitive. In previous work [4] a single tuning parameter for the critical/reference stress was all that was required to make the Markov chain converge to the target distribution. The work in this paper uses five different sediment experiments, each consisting of 13 different size fractions. Further, the modification of the prior for σ precludes the use of a Gibbs sampler because Equation (11) is not conjugate with the likelihood, therefore two Metropolis updates are required. In short, a manual tuning in the present work would require the specification of 130 tuning parameters (assuming a single chain is utilized). To avoid this, an adaptive MCMC algorithm was employed.

^{2}, a seed tuning variance is specified that is generally appropriate and is selected by trial and error, and this initial tuning variance is adapted (modified) at each MCMC iteration such that the acceptance probability is optimal. The adapted tuning variance is defined as:

_{k}= σ

_{k}/k and effectively scales the adaptation such that with each iteration the tuning variance adapts less and less, $\alpha (x,y)=\mathrm{min}\left(1,\frac{\pi (y)}{\pi (x)}\right)$ π(x) is the likelihood of parameter value x, X

_{k}is the Markov chain for the parameter of interest, Y

_{k+1}is a new proposal Y

_{k}

_{+1}~ N(X

_{k},σ

_{k}

^{2}), and $\overline{\tau}\equiv 0.234$. A random variable U ~ Uniform(0,1) is sampled, and the newly proposed value is accepted when $U\le \alpha \left({X}_{k},{Y}_{k+1}\right)$ and rejected otherwise in which case X

_{k+1}= X

_{k}.

## 3. Results and Discussion

#### 3.1. Results

#### 3.1.1. Experimental Data

**Figure 1.**(

**a**) Grain-size distribution of sediments and (

**b**) bulk transport rates from the Wilcock et al. [18] experiments (W* is defined in Equations (1) and (2)).

#### 3.1.2. Bulk Transport

**Table 1.**Data summary for sediment mixtures. Inferred bulk transport posterior means for τ

_{r}, in Pa and σ in m

^{3}/m/s, number of observational runs for each sediment, and the proportion sand in both the surface and total mixture.

Parameter | Sediment Mixture | ||||
---|---|---|---|---|---|

J06 | J14 | J21 | J27 | BOMC | |

τ | 8.90 | 7.14 | 3.46 | 1.82 | 0.73 |

σ | 0.74 | 1.26 | 1.23 | 1.11 | 1.04 |

Observations | 10 | 9 | 8 | 10 | 10 |

Surface sand proportion | 0.0013 | 0.0130 | 0.0728 | 0.1990 | 0.4789 |

Total sand proportion | 0.0612 | 0.1484 | 0.2056 | 0.2730 | 0.3432 |

**Figure 2.**Trace plots for J06 sediment. Results are representative of all bulk sediments. Red line represents the prior distribution used in the model.

**Figure 3.**Similarity collapse for bulk transport rates. The bold line is the calibrated Wilcock-Crowe Equation (1) for the bulk transport data. W* = 0.002 is the reference value that corresponds to the reference shear stress τ

_{r}.

**Figure 4.**Bulk rating curves. Red line shows the NLS prediction. Color coded darkest to lightest for the 95%, 90%, and 68% credible intervals.

#### 3.1.3. Fractional Transport

**Figure 5.**Fractional posterior distributions of τ

_{r;j}for J14. Red line represents the prior distribution used in the model.

**Figure 6.**Fractional posterior distributions of σ

_{j}for J14. Red line represents the prior distribution used in the model.

Size (mm) | Sediment Mixture | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

J06 | J14 | J21 | J27 | BOMC | ||||||

τ_{r,j} (Pa) | # Obs. | τ_{r,j} (Pa) | # Obs. | τ_{r,j} (Pa) | # Obs. | τ_{r,j} (Pa) | # Obs. | τ_{r,j} (Pa) | # Obs. | |

45.3 | 14.40 | 3 | 13.87 | 2 | 13.13 | 1 | 8.14 | 1 | 8.42 | 1 |

32.0 | 11.33 | 4 | 11.15 | 4 | 7.45 | 2 | 7.00 | 4 | 7.07 | 2 |

22.6 | 10.27 | 6 | 9.67 | 7 | 7.48 | 5 | 5.38 | 6 | 3.59 | 4 |

16.0 | 9.08 | 7 | 7.25 | 8 | 5.39 | 6 | 3.57 | 6 | 3.86 | 4 |

11.3 | 9.39 | 7 | 7.31 | 9 | 4.80 | 6 | 3.08 | 7 | 3.46 | 4 |

8.0 | 8.65 | 9 | 7.46 | 9 | 4.04 | 8 | 2.83 | 9 | 2.31 | 5 |

5.7 | 7.90 | 10 | 6.38 | 9 | 3.09 | 8 | 2.19 | 9 | 1.94 | 6 |

4.0 | 7.71 | 10 | 5.60 | 9 | 2.87 | 8 | 1.48 | 10 | 1.79 | 6 |

2.8 | 8.99 | 10 | 7.41 | 9 | 3.47 | 8 | 1.97 | 10 | 1.31 | 8 |

2.0 | 8.96 | 9 | 6.83 | 9 | 3.07 | 8 | 1.51 | 10 | 1.03 | 9 |

1.4 | 9.53 | 8 | 7.12 | 9 | 3.21 | 8 | 1.56 | 10 | 1.01 | 10 |

1.0 | na | 0 | 5.02 | 8 | 2.57 | 8 | 1.35 | 10 | 0.74 | 10 |

0.5 | 6.77 | 4 | 4.18 | 8 | 2.63 | 8 | 1.43 | 10 | 0.61 | 10 |

0.21 | na | 0 | na | 0 | na | 0 | na | 0 | 0.62 | 10 |

**Figure 9.**Individual sediment comparisons of original reference stress curves from Wilcock and Crowe [6] to inferred curves from Bayesian Model. Error bars are those from Wilcock and Crowe [6]. Credible intervals are color coded from darkest to lightest for 95%, 90%, and 68%. The 0.707 mm value was jittered so as to not overplot for J06. The solid black line shows the standard Shields Curve for incipient motion of uni-size sediment.

**Figure 10.**Posterior predictive distribution for J21, color coded darkest to lightest for the 95%, 90%, and 68% credible intervals.

**Figure 12.**Reference Shields stresses for the mean size of the bed surface ${\tau}_{r,sm}^{*}$ plotted against proportion sand on the bed surface, with credible intervals, color coded darkest to lightest for 95%, 90%, and 68%.

**Figure 13.**Fractional similarity collapse for multifraction observations and model prediction. The bold line though the data is the calibrated multifraction Wilcock-Crowe Equations (2)–(6) for the multifraction transport data). W* = 0.002 is the reference value that corresponds to the reference shear stress τ

_{r}.

#### 3.2. Discussion

#### 3.2.1. General

#### 3.2.2. Hiding Function and Similarity Collapse

_{r,sm}. The hiding function fitted to the inferred reference stresses does not suggest any deficiencies of the original function. By comparing Figure 3 of [6] to Figure 11 in the present analysis, the only discerning feature is that the Bayesian fit seems to have more uncertainty than the original fit as judged by the spread of the data points in the figure.

_{r,sm}controls the overall mobility of the mixture. Wilcock and Crowe [6] plotted τ

_{r,sm}versus sand content and demonstrated that the overall mixture mobility depends on the sand content of the bed (Equation (14), Figure 12). The notion that τ

_{r,sm}would decrease with increasing sand content was a departure from previous thought. The fitted relation used the best estimate for the reference stresses and was sensitive to uncertainty in the data because only five points (one for each sediment mixture) were available to fit the trend. As established above, the reference stresses are distributions and are not fixed. The analysis leading to Equation(14) had no basis for accounting for uncertainty in the five values of τ

_{r,sm}. Using the posterior mean values for the mean surface stresses, the present analysis resulted in a somewhat different relationship Equation (15), with the reference stress for BOMC being larger than what was originally specified in [6] for the same sediment. Because τ

_{r,sm}for BOMC is a leverage point, small shifts in its value result in larger changes to the trend. The reference stress itself is a random variable, and if we account for this, credible intervals can be developed (Figure 12). The regions defined in Figure 12 generally support the validity of the original equation, especially when one considers that for each of the 9–10 flume experiments for a given sediment, the mean surface diameter, while well-defined, also had some variability.

_{r,sm}directly. Although the original relation for τ

_{r,sm}proposed by Wilcock and Crowe [6] (14) and the modified relation developed here Equation (15) are largely contained within the credible regions in Figure 12, the new relation Equation (15) exceeds the credible interval for a surface sand content of 20%. Thus, while prediction of τ

_{r,sm}using the proposed equations is not needed for a Bayesian implementation of this sediment transport model, the original equation is supported by the Bayesian model results.

_{r,j}. The results indicate that the model parameters estimated by the Bayesian algorithm provide a well-defined similarity collapse quite similar to that originally produced in Figure 6 of [6]. Any of the sensitivities that were observed in the hiding and Shields parameter variation plots are not an issue in the similarity collapse indicating that the general similarity phenomenon is a robust characteristic of sediment transport rather than an experiment-specific result. In the linear portion of the curve (τ/τ

_{r,j}< 2) the data points are tightly and symmetrically clustered about the deterministic prediction, and the over-prediction that was observed in the bulk similarity collapse in Figure 3 does not appear to be an issue in the multi-fraction collapse. The original similarity collapse of Wilcock and Crowe [6] does not have this same symmetry in the linear region of the plot. However, the collapse shown in Figure 13 does suggest a small bias (under-prediction) relative to the observations, something that the original collapse in Wilcock and Crowe [6] does not have.

#### 3.2.3. Motivation and Implementation on New Datasets

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Schmelter, M.L.; Wilcock, P.; Hooten, M.; Stevens, D.K.
Multi-Fraction Bayesian Sediment Transport Model. *J. Mar. Sci. Eng.* **2015**, *3*, 1066-1092.
https://doi.org/10.3390/jmse3031066

**AMA Style**

Schmelter ML, Wilcock P, Hooten M, Stevens DK.
Multi-Fraction Bayesian Sediment Transport Model. *Journal of Marine Science and Engineering*. 2015; 3(3):1066-1092.
https://doi.org/10.3390/jmse3031066

**Chicago/Turabian Style**

Schmelter, Mark L., Peter Wilcock, Mevin Hooten, and David K. Stevens.
2015. "Multi-Fraction Bayesian Sediment Transport Model" *Journal of Marine Science and Engineering* 3, no. 3: 1066-1092.
https://doi.org/10.3390/jmse3031066