# Statistical Roughness Properties of the Bed Surface in Braided Rivers

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Arrangements

_{10}= 0.293 mm, D

_{50}= 0.904 mm, and D

_{90}= 1.895 mm. In this work, four constant discharge experiments were carried out with discharge rates of 0.8 L/s, 1.3 L/s, 1.9 L/s, and 2.5 L/s. The experiments were divided into two phases: free evolution and experimental measurements. The experimental conditions are shown in Table 1.

## 3. Calculation Methods

#### 3.1. Data Collection and Processing

_{50}of the model at 0.904 mm.

#### 3.2. Morphological Parameters

#### 3.3. Statistical Parameters

#### 3.4. Variogram

- The nugget variance ${C}_{0}$ reflects the magnitude of the randomness of the regionalized variables. Theoretically, when the sampling scale $h=0$, the value of $\gamma \left(h\right)$ should be equal to 0. However, due to the presence of microstructural variations (i.e., internal variability that occurs at scales smaller than the sampling scale $h$) and errors associated with sampling, measurement, and analysis, $\gamma \left(h\right)\ne 0$ when two sampling points are very close, leading to the existence of a nugget variance.
- The sill ${C}_{0}+C$ reflects the magnitude of the variability of the regionalized variables, indicating the intensity of the variation within the study area. The ratio of the nugget variance to the sill is defined as the nugget coefficient, denoted as ${R}_{C}={C}_{0}/\left({C}_{0}+C\right)$, which represents the proportion of spatial variability caused by the random component to the total variability, indicating the strength of spatial variability. If ${R}_{C}<25\%$, it indicates strong spatial correlation of the variable; if $25\%\le {R}_{C}\le 75\%$, it indicates moderate spatial correlation; if ${R}_{C}>75\%$, it indicates weak spatial correlation. A larger ${R}_{C}$ suggests that the variability is mainly due to random factors.
- The correlation length $a$ reflects the extent of spatial autocorrelation of the regionalized variables and represents the point where the spatial correlation transitions from existence to non-existence. When the sampling scale $h<a$, there is spatial correlation and mutual influence between two points in space, with the influence decreasing as the distance between the points increases. When the sampling scale $h\ge a$, there is no spatial correlation between the two points.

## 4. Results and Discussion

#### 4.1. Morphological Parameters of Braided Rivers

#### 4.2. Relationship between Morphological Active Width, Stream Power, and Bedload Transport Rate

#### 4.3. Elevation Probability Distribution and Statistical Parameters of the Bed Surface

#### 4.4. Two-Dimensional Variograms of the Bed Surface Elevations

#### 4.5. One-Dimensional Variograms of the Bed Surface Elevations

## 5. Conclusions

- The morphological change area of the reach becomes more continuous and extensive with increasing discharge. The average morphological active width increases, while the average morphological active depth remains almost unchanged.
- There is a significant positive correlation between the bedload transport rate and the reach-averaged morphological active width and the stream power. This indicates that the easily measured parameters such as discharge, bed surface particle size, wetted width, and active width, which characterize morphological change, can be used to predict the mass of bedload transport that is difficult to measure directly.
- The bed elevation probability distribution in braided rivers is negatively skewed and leptokurtic. There is a relatively significant correlation between skewness and the dimensionless bedload transport rate, providing a simple index for the preliminary prediction of bedload transport rate in braided rivers. The two-dimensional variogram values of the bed surface elevation differ in each direction, indicating anisotropy in bed surface roughness within the research reach.
- As flow intensity increases, bed surface roughness, the corresponding sill, and correlation length increase. The bed elevation variable in the $x$-axis direction has a strong spatial correlation, and elevation variation caused by random factors can be ignored. The sill and correlation length of the bed surface elevation variogram can be used to estimate the bedload transport rate in the constant discharge experiments of braided rivers.
- This study primarily focused on different flow intensities. In the future, more extensive and systematic research should be conducted to analyze the influence of bed surface particle sizes, slopes, and different flow and sediment conditions on the statistical roughness properties of the bed surface in braided rivers to enrich the relevant research achievements.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic of the river modeling flume. (

**a**) plan view; (

**b**) side view. (Note: the unit in the figure is cm.)

**Figure 4.**DEMs of difference generated using the final two DEMs from the four constant discharge experiments. Flow was from left to right. (

**a**) Run 1: Q = 0.8 L/s. (

**b**) Run 2: Q = 1.3 L/s. (

**c**) Run 3: Q = 1.9 L/s. (

**d**) Run 4: Q = 2.5 L/s.

**Figure 5.**Reach-averaged morphological active width as a function of (

**a**) total stream power (Ω) and (

**b**) experimental run time for the constant discharge experiments.

**Figure 6.**Bedload transport rates for each total stream power of the constant discharge experiments.

**Figure 7.**(

**a**) Bedload transport rate plotted against reach-averaged morphological active width; (

**b**) dimensionless bedload transport rate plotted against dimensionless stream power for constant discharge experiments. The dashed line represents the linear regression through all observations.

**Figure 8.**Elevation probability distribution of the bed surface. (

**a**) Run 1: Q = 0.8 L/s. (

**b**) Run 2: Q = 1.3 L/s. (

**c**) Run 3: Q = 1.9 L/s. (

**d**) Run 4: Q = 2.5 L/s.

**Figure 9.**Average statistical roughness parameters as a function of discharge for the constant discharge experiments. (

**a**) Variance ${\sigma}^{2}$; (

**b**) skewness ${S}_{k}$; (

**c**) kurtosis ${K}_{u}$. The dashed line represents the linear regression through all observations.

**Figure 10.**Dimensionless stream power and dimensionless bedload transport rate plotted against skewness for the constant discharge experiments. (

**a**) The variation in ${\omega}^{\ast}$ with ${S}_{k}$; (

**b**) the variation in ${q}_{b}^{\ast}$ with ${S}_{k}$; (

**c**) the variation in average ${q}_{b}^{\ast}$ with average ${S}_{k}$. The dashed line represents the linear regression through all observations.

**Figure 11.**Two-dimensional variograms of the bed surface elevations. (

**a**) Run 1: Q = 0.8 L/s. (

**b**) Run 2: Q = 1.3 L/s. (

**c**) Run 3: Q = 1.9 L/s. (

**d**) Run 4: Q = 2.5 L/s.

**Figure 13.**Spherical model parameters of the variograms plotted against dimensionless stream power. (

**a**) The variation in ${C}_{0}/{\sigma}^{2}$ with ${\omega}^{\ast}$; (

**b**) the variation in $\left({C}_{0}+C\right)/{\sigma}^{2}$ with ${\omega}^{\ast}$; (

**c**) the variation in $a/{\sigma}^{2}$ with ${\omega}^{\ast}$. The dashed line represents the linear regression through all observations.

**Figure 14.**Average spherical model parameters of the variograms plotted against average dimensionless stream power. (

**a**) The variation in average ${C}_{0}/{\sigma}^{2}$ with average ${\omega}^{\ast}$; (

**b**) the variation in average $\left({C}_{0}+C\right)/{\sigma}^{2}$ with average ${\omega}^{\ast}$; (

**c**) the variation in average $a/{\sigma}^{2}$ with average ${\omega}^{\ast}$. The dashed line represents the linear regression through all observations.

**Figure 15.**Average spherical model parameters of the variograms plotted against average skewness. (

**a**) The variation in average ${C}_{0}/{\sigma}^{2}$ with average ${S}_{k}$; (

**b**) the variation in average $\left({C}_{0}+C\right)/{\sigma}^{2}$ with average ${S}_{k}$; (

**c**) the variation in average $a/{\sigma}^{2}$ with average ${S}_{k}$. The dashed line represents the linear regression through all observations.

**Figure 16.**Average spherical model parameters of the variograms plotted against average dimensionless bedload transport rate. (

**a**) The variation in average ${C}_{0}/{\sigma}^{2}$ with average ${q}_{b}^{\ast}$; (

**b**) the variation in average $\left({C}_{0}+C\right)/{\sigma}^{2}$ with average ${q}_{b}^{\ast}$; (

**c**) the variation in average $a/{\sigma}^{2}$ with average ${q}_{b}^{\ast}$. The dashed line represents the linear regression through all observations.

Experiment | Slope | Discharge | Stream Power | Evolution Time | Experimental Run Time | Threshold |
---|---|---|---|---|---|---|

S | Q | Ω | ||||

(%) | (L/s) | (W/m) | (h) | (h) | (mm) | |

Run 1 | 1.5 | 0.8 | 0.12 | 15 | 16 | 2.42 |

Run 2 | 1.5 | 1.3 | 0.19 | 15 | 16 | 2.18 |

Run 3 | 1.5 | 1.9 | 0.28 | 12 | 16 | 1.98 |

Run 4 | 1.5 | 2.5 | 0.37 | 10 | 16 | 1.70 |

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**MDPI and ACS Style**

Ren, B.; Pan, Y.; Lin, X.; Yang, K.
Statistical Roughness Properties of the Bed Surface in Braided Rivers. *Water* **2023**, *15*, 2612.
https://doi.org/10.3390/w15142612

**AMA Style**

Ren B, Pan Y, Lin X, Yang K.
Statistical Roughness Properties of the Bed Surface in Braided Rivers. *Water*. 2023; 15(14):2612.
https://doi.org/10.3390/w15142612

**Chicago/Turabian Style**

Ren, Baoliang, Yunwen Pan, Xingyu Lin, and Kejun Yang.
2023. "Statistical Roughness Properties of the Bed Surface in Braided Rivers" *Water* 15, no. 14: 2612.
https://doi.org/10.3390/w15142612