## 3 Validation by the Simulated Case
### Results  
Both the classic two-step and the new three-step methods were used to process the simulated images and to quantify the designed bed-load motion pattern. Image processing parameters were kept the same in the two methods. If a particle moves out of the image, its information is not included in the final statistics. For each case, the 500-frame image series were processed, and thereafter statistical samples were established for calculating various instant parameters including the number of active particles between two images \(n\), particle velocity in the streamwise direction \(u\), particle velocity in spanwise direction \(v\), flight length \(l\), and flight time \(t\). The corresponding capital letters represent the mean value of parameters. The true values of the number of active particles, streamwise velocity, flight length and flight time are nonzero. Hence, the relative error (mean and standard deviation) of the average value was used as an indicator to compare the performance of the two methods. For instance, the mean relative error in the number of active particles is defined to be  
\[\varepsilon_{N}\ =\ \frac{N-N_{s}}{N_{s}} \tag{8}\]  
in which, \(N\) and \(N_{s}\) represent the mean measured and the real number of active particles in one pair of images, respectively. The relative error in streamwise velocity, flight length and flight time is similarly defined, denoted to be \(\varepsilon_{U}\), \(\varepsilon_{L}\) and \(\varepsilon_{T}\), respectively. The mean and standard deviation of the relative error provides a direct measure of the performance of the two methods. Positive and negative values of the mean value \(\varepsilon_{N}\) indicate overestimation and underestimation of the real situation, respectively. The standard deviation of relative errors represents the dispersion degree of measured parameters. The results are shown in Figure 9. The blue point and red triangle represent relative error by the present method and classic method, respectively. As shown in Figure 9, the measured values of all parameters by the present method are close to real value with the mean value of relative errors from \(-12\%\) to \(16\%\). The mean value of \(\varepsilon_{N}\) and \(\varepsilon_{U}\) by the classic method is from \(-26\%\) to \(6\%\) within acceptable range, but slightly more than those by the present method. However, the absolute value of \(\varepsilon_{L}\) and \(\varepsilon_{T}\) are very large, even greater than \(50\%\), which means the flight length and time are significantly underestimated by the classic method. The standard deviation (error bars are shown in Figure 9) of \(\varepsilon_{N}\), \(\varepsilon_{U}\), \(\varepsilon_{L}\) and \(\varepsilon_{T}\) by the present method are smaller than those by classic method, which indicates that the present method is more stable and reliable. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & \(\mathbf{U_{s}}\) **(Pixel/s)** & \(\mathbf{V_{s}}\) **(Pixel/s)** & \(\mathbf{L_{s}}\) **(Pixel)** \\ \hline C1 & 8 & \(\sim\)N(0, 1.6) & 40 \\ C2 & 15 & \(\sim\)N(0, 3) & 75 \\ C3 & 23 & \(\sim\)N(0, 4.6) & 115 \\ C4 & 30 & \(\sim\)N(0, 6) & 150 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summarizes the major parameters of the simulated images. Meanwhile, the absolute mean values of \(\varepsilon_{L}\), \(\varepsilon_{T}\) and \(\varepsilon_{U}\) increase with increased \(U_{s}\) by classic method, as shown in Figure 9b-d. However, there is no obvious tendency of \(\varepsilon_{N}\) with \(U_{s}\) by classic method, as shown in Figure 9a. These phenomena show that the classic method is more sensitive to article distance between two images, i.e., particle velocity and sampling frequency. The present method performs well in all cases. For the number of active particles and spanwise velocity, the probability density distributions are set as negative binomial distribution and Gaussian distribution in separate. Therefore, the PDFs of the true and measured value can be used for testing errors.