## 4 Discussion
### Dependence of Hop Travel Time on Particle and Flow Features  
The mean hop travel times for particle motion can be obtained from exponential distribution fittings. Here, these are compared for different particle densities, flow Reynolds numbers, and bed surfaces. The mean particle hop travel times can be considered as an important characteristic for particle transport, as it gives an indication about the particle instantaneous velocities, as well as changes in the number of particles hop events. For high particle travel times, the same transport distances are transversed over a longer period and particles' instantaneous velocities are lower. Figure 13 shows comparisons of exponential distribution fittings for measured particle travel times, for different particle densities and the flow Reynolds numbers assessed herein. From Figure 13a,c, it is seen that with increasing flow Reynolds number, the probability of occurrence of higher particle travel times increases for both bed surfaces, and even more for the rougher bed surface (B). For example, in bed surface A (Figure 13a), when flow Reynolds number increased by about 15% (from 43,353 to 50,789), the probability of occurrence increased by 45% (from 0.426 to 0.624) for the same particle hop travel time of 0.09 s. In addition, the probability of occurrence of higher particle hop travel times increases for both bed surfaces as the particle density decreases (Figure 13b,d). For example, for bed surface B, when particle density decreased by about 15% (from 1224 to 1085 kg/m\({}^{3}\)), the probability of occurrence increased from 0.38 to 1.01 (165%) for the same particle hop travel time of 0.14 s. The effects of different particle densities and flow Reynolds number on the fitted exponential distribution parameters modelling hop travel times are also tabulated in Tables 6 and 7 in Appendix A, along with further discussion to allow for a clearer representation of the variations in the body and specifically the tail of the distribution (that may be hard to discern in Figure 13). Figure 14 shows the effect of different flow Reynolds  
Figure 13: Hop time modelled by the exponential distribution: (**a**) for different flow Reynolds numbers, bed surface A, and fixed instrumented particle density of 1132 kg/m\({}^{3}\), (**b**) for different instrumented particle densities, bed surface A, and fixed Reynolds number of 50,789, (**c**) for different flow Reynolds numbers, bed surface B, and fixed instrumented particle density of 1132 kg/m\({}^{3}\), and (**d**) for different instrumented particle densities, bed surface B, and fixed Reynolds number of 51,488.  
and Shields numbers and test particle densities on the average hop travel time modeled with the fit exponential distributions for bed surfaces A and B. Figure 14 shows that for bed surface A, when there is an increase in flow Reynolds number from 38,190 to 50,789, which is about 35%, or an increase in the Shields number by 50% from 0.091 to 0.135, with fixed particle density 1132 kg/m\({}^{3}\), the particle will interact with the bed less and remain for longer durations and there will be an increase in the mean time interval of particle hop, as it increases by 40% from 0.0264 to 0.0370 s.  
When the particle density decreases by about 10% from 1180 to 1085 kg/m\({}^{3}\), the mean particle hop travel time will increase by 40% from 0.0246 to 0.0348 s, with fixed flow Reynolds number of 43,353. In addition, particle velocities are low on average, so the net time travel of particle hops are short on average.