## 3 Results
### Hop Length  
After [PERSON]'s seminal work [51], sediment particle's hop distances have received a good amount of attention from several researchers such as [28; 29; 30; 31], as it could be used to quantify bedload transport rates, if the rate of transport was considered [17; 57]. The instrumented particle is transported with continuous hops throughout its advection downstream the test section and hop lengths were calculated from sensor data as the stream-wise distance travelled by one particle from the start of its motion to the end of its rebound. Numerous distribution fittings were assessed to obtain the best fit stochastic distributions for the different experimental runs, including Weibull, Beta, Gamma, normal, and exponential (Table 1). The Chi Square metric, describing how well each distribution fits the set of experimental observations by determining whether a significant difference between the observed values and measured values is expected under the model, was used to determine the best distribution fitting. The goodness of fit metric was used as follows:  
\[X^{2}=\sum\frac{(Oi-Ei)^{2}}{Ei} \tag{4}\]  
where \(X=\) goodness of the fit, \(O_{i}\) = observed value, and \(E_{i}\) = expected value. Table 1 shows the goodness of fit results for different distributions of particle hop lengths for the minimum and maximum flow rates used in the range of experiments. Considering the goodness of fit metrics, Gamma was found to be the optimal distribution for modeling particle hop distances. Hop length was fitted by the Gamma probability distribution function model, as follows:  
\[f_{x}(x)=\frac{1}{\Gamma(ax+1)\beta_{x}^{ax+1}}x^{ax}e^{\frac{-x}{Bx}} \tag{5}\]  
where \(x\) is the hop distance, \(ax\) and \(\beta x\) are the shape and rate (inverse scale) parameters, respectively, with \(ax>-1\) and \(\beta x>0\), and \(\Gamma\) is the gamma function. Figure 8 shows the probability density function (PDF) of measured hop lengths for the lowest flow rate of 49 (L/s), particle density of 1180 (kg/m\({}^{3}\)), bed surface A, where the x-axis is the hop lengths (m), the \(y\)-axis is the count of hop events, and the red curve represents the Gamma distribution fit to the experimentally observed data. As shown in Figure 8, the hop lengths were not hugely varied, and the Gamma function was the best fit for the measured hop lengths distributions, a similar finding \begin{table}
\begin{tabular}{c c c c} \hline
**Model Distribution** & **Goodness of Fit** & **Goodness of Fit** & **Rank** \\
**for Travel Distances** & **(for Q = 56 L/s)** & **(for Q = 42 L/s)** & **Rank** \\ \hline Weibull & 1.28 & 1.64 & 2 \\ \hline Beta & 3.54 & 1.92 & 3 \\ \hline Gamma & 0.34 & 0.58 & 1 \\ \hline Normal & 3.7 & 5.2 & 4 \\ \hline Exponential & 6.4 & 7.1 & 5 \\ \hline \end{tabular}
\end{table} 
Table 1: Goodness of fit results and corresponding ranking for a range of distributions modelling hop travel distances for the minimum and maximum flow rates used in the experiment. Figure 8: Empirical hop length data (m) and the best fit Gamma probability density function for the flow rate of 49 (L/s), instrumented particle density of 1180 (kg/m\({}^{3}\)), and bed surface A.  
to [58], which also suggests the same distribution. In addition, it is similar to past research that has found exponential or Gamma (hop distance) distributions [57; 58; 59; 60; 61; 62]. On the other hand, some researchers found different results, such as [29; 30], who reported best fit for the Weibull distribution, as well as [63], who proposed the exponential distribution. The difference in the fitting distributions could be attributed to different control parameter settings used in this experimental study in comparison to the previous studies, as the used particle in this study is lighter, bigger in size, has higher flow Reynolds number ranges, and bed surface particles smaller to the saltating.