## 2 Methodology
The turbulent velocity fluctuations were estimated based on Equations (1) and (2) below:  
\[u^{\prime}_{i}=u_{i}-\overline{u} \tag{1}\]  
\[w^{\prime}_{i}=w_{i}-\overline{w} \tag{2}\]  
where \(u\) and \(w\) are the instantaneous flow velocity readings (stream-wise and normal to the bed surface components), \(u^{\prime}\) and \(w^{\prime}\) are the turbulent velocity fluctuations (stream-wise and normal to the bed surface components), \(\overline{u}\) and \(\overline{w}\) are the time-averaged velocity (stream-wise and normal components) and \(i\) is the ADV instantaneous reading. Using each instantaneous velocity reading, or basic ADV reading, the quadrant that corresponds to the reading was defined. An illustration of the analysis that was performed using the flow velocity measurements to estimate the flow impulses and determine the quadrants is shown in Figure 2 below. As discussed in [54], the frequency of entrainment of the instrumented particle was estimated using the normalised roll results by the inertial sensor fusion of the instrumented particle's logged readings during the experiment. As for the flow impulses, which were estimated based on the illustration shown in Figure 2, they were used to estimate the frequency of flow impulses by dividing the total number of flow impulses by the total ADV recording time in each position, including the position of investigation. Given that the contribution of the \(Q_{2}\) and \(Q_{4}\) quadrants is more significant near the bed surface, it is expected that for smaller particle-size-to-flow-depth ratios, these quadrants will play a greater role in particle entrainment. Thus, it is expected that smaller particle sizes will have an even more pronounced contribution of the \(Q_{2}\) and \(Q_{4}\) events than found herein. In the position of investigation, different theoretical critical impulse values (\(l_{cr}\)) were tested to determine the value that results in a good agreement between the frequency of entrainment of the instrumented particle and the frequency of flow impulses above that value, or above the critical impulse (\(f_{l}\)). Figure 3 below shows the selected critical value in addition to slightly higher and lower values for the critical impulse, for comparison purposes. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Run** & **Flow Vel.** & **Flow Depth** & **Fr** & **Re** & **Re\(\mathbf{\ast}\)\(=\)\(\frac{\overline{\mathbf{\mu}}\mathbf{b}\cdot\mathbf{D}}{\overline{\mathbf{\mu}}}\)** & \(\mathbf{\tau}\)\(=\)\(\frac{\overline{\mathbf{\tau}}_{\mathbf{a}}}{(\rho_{\mathbf{a}}-\rho_{\mathbf{a}}) \mathbf{b}\mathbf{D}}\)** & **[ entrainment/min]** \\ \hline Run1 & 0.345 & 11.4 & 0.32 & 39,416 & 4528.20 & 0.0047 & 0.42 \\ \hline Run2 & 0.346 & 11.5 & 0.32 & 39,768 & 4774.57 & 0.0052 & 0.97 \\ \hline Run3 & 0.357 & 11.7 & 0.33 & 41,880 & 5067.38 & 0.0059 & 1.21 \\ \hline Run4 & 0.358 & 11.8 & 0.33 & 42,231 & 5451.77 & 0.0068 & 4.91 \\ \hline Run5 & 0.359 & 11.9 & 0.33 & 42,760 & 5862.61 & 0.0079 & 6.12 \\ \hline \end{tabular}
\end{table} 
Table 1: The flow characteristics (mean flow velocity, mean flow depth, Froude number, flow and particle Reynolds number (\(Re\) and \(Re\ast\), respectively) and average bed surface Shields shear stresses) and resulting entrainment particle frequency for the 5 runs of the study. It is clear from Figure 3 that the critical impulse is very close to 0.0455 m\({}^{2}\)/s, i.e., the line of perfect agreement between the frequencies of flow impulses above the critical impulse and the frequency of entrainment of the instrumented particle. In a similar fashion, the theoretical equations presented in [21; 22] can be used to derive values of critical impulses, by offering different input about the flow, grain and bed surface properties. and/or utilising appropriate assumptions (e.g., flow uniformity, grain shape, uniformity and packing of the bed surface grain sizes, local bed surface and grain arrangement slope).