## 3 Results
### Bedload Transport Rate on Dune Bedforms  
In the late 19 th century, [PERSON] in France initially proposed a theory of drag forces for bedload transport [47]. Subsequently, numerous researchers have expanded on this topic, developing a variety of formulas and theoretical frameworks [48]. Notable among these are the Meyer-Peter formula [26; 27] and the Engelund formula [29]. Later, Chinese scholars streamlined these bedload transport formulas and standardized their format [49]. Based on the bedload transport rate measured during the experiment and the flow conditions (Table 1), the bedload transport intensity and the flow intensity at the different slopes were computed. These calculations were performed using the representative bedload transport formulas derived from the experimental results, as illustrated in Figure 9.  
Figure 9 indicates that in the presence of dunes, the bedload transport rates estimated by the representative formulas for uniform sediment deviate from the measured sediment transport rates. These deviations are more pronounced at higher flow intensities and in more fully developed dune bedforms. Specifically, at a low flow intensity (\(\theta<0.1\)), the sediment transport rates calculated by the representative formulas approximate the average measured sediment transport rate; however, at a medium-to-high flow intensity (\(\theta\geq 0.1\)), the calculated rates generally exceed the average measured rate. At low flow intensities, significant variability was observed in the bedload transport rates measured in each experiment group, particularly in the experiments conducted on low-slope bedforms, where the maximum rate exceeded the minimum by more than fivefold. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{**Statistic**} & \multicolumn{3}{c}{**2D Bedforms**} & \multicolumn{3}{c}{**3D Bedforms**} \\ \cline{2-7}  & \multirow{2}{*}{**Distribution**} & \multicolumn{2}{c}{**Distribution Parameters**} & \multicolumn{2}{c}{**Distribution Parameters**} \\ \cline{3-6}  & & **Mean** & **Standard Deviation** & **Form** & **Shape Parameter** & **Scale Parameter** \\  & & **(\(\mu\))** & **(\(\sigma\))** & **(\(\sigma\))** & **(\(\beta\))** & **(\(\beta\))** \\ \hline \(\eta c\) & Normal & 1.0 & 0.334 & Gamma & 4.906 & 0.204 \\ \(\eta t\) & Normal & 1.0 & 0.416 & Gamma & 3.083 & 0.324 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of crest height and trough depth distributions for different bedforms. Figure 9: Comparison of measured and calculated bedload transport rates. This variability can be attributed to the group transport of bedload particles. When dunes are present on the bed surface, their movement during the low-flow-intensity tests (\(\theta<0.1\)) is relatively slow. Consequently, during the measurement of the bedload transport rates, the dune crests and troughs alternately enter the pit during different measurement periods, leading to substantial fluctuations in the measured transportation rates. Conversely, in the experiments conducted at medium-to-high flow intensities (\(\theta\geq 0.1\)), the dunes move more rapidly, and multiple dunes enter the pit during each measurement period, thereby reducing the variability in the measured bedload transport rates. This study employed a nonlinear least square fitting method to derive a bedload transport formula for dune bedforms based on the measured rates and flow intensities as follows. The meaning of each physical parameter is the same as above:  
\[\Phi=2.03(\theta-0.05)^{1.32} \tag{5}\]  
The comprehensive error of the bedload transport formula for dune bedforms was quantified using the Mean Square Error (MSE) between the experimentally measured and calculated sediment transport rates, as expressed in Equation (6). The results indicate a comprehensive error of 0.016, demonstrating the high accuracy of the proposed formula.