## 6 Discussions of Measurement Errors  
The velocity profile distribution in silt carrying flow can be measured using ultrasound imaging and the iterative multi-grid deformation technique. The key to the visual measurement of the displacement or feature vector in the image is based on the correlation analysis between two interrogation windows. There are some errors in the flow velocity gradient across the window, because not all the particle images consistently appear in the first interrogation window and then in the next interrogation window. When the interrogation windows have not deformed, the displacement will be close to the actual displacement because the template matching method assumes that the particle image displacement is uniform over the interrogation. When the velocity gradient increases, the measurement error arises because the image deforming algorithm allows some particle images to be detected in an interrogation field. The image deforming algorithm can tolerate a much higher velocity gradient, which is why the measured velocity profile with the deforming algorithm is in great agreement with the standard profile. The image deforming algorithm has better performance than the template matching approach.  
Furthermore, we take the values of the ADCP method as the reference values, and then recalculate all the RMS errors of vectors in the interrogation window of the single ultrasonic image. The cumulative RMS error is the sum of all vectors' RMS errors in the image at a time in order to facilitate comparison. The cumulative RMS errors of the WDI64-32, WT64, WT32, and WT16 methods are shown in Figure 11.  
Figure 11 shows the cumulative RMS errors computed with the different algorithms. The mean RMS error relative to standard velocity is limited to 0.148 of the WDI64-32 algorithm, 0.4214 of the WT64 algorithm, 0.8796 of the WT32 algorithm, and it reaches 1.4148 of the WT16 algorithm. (The velocity profile value obtained by the ADCP method at eleven measurement points is taken as the standard value.)  
The comparisons of relative RMS errors along with the depth are shown in Figure 12. From Figure 12, the curve of the WDI64-32 algorithm presents the RMS errors relative to the standard value and is relatively flat. As for the curves of the WDI64-32 and WT64 algorithms, the image deformation algorithm is more suitable for the matching than the WT64 algorithm when considering the gradient. As is shown in the curves of the WT64, WT32, and WT16 methods in Figure 12, the relative RMS error increases with the window size decrease for the measurement of the average velocity in certain conditions. When the  
Figure 11: Cumulative RMS error of the WDI64-32, WT64, WT32 and WT16 methods.  
interrogation window size decreases from 64 \(\times\) 64 to 32 \(\times\) 32, then up to 16 \(\times\) 16 pixels, the relative RMS error is strongly influenced because the efficient particle image number decreases rapidly. To 16 \(\times\) 16 pixels, there is no sufficient particle image to estimate the centroid of the correlation peak, so the RMS reaches the maximal RMS error. Therefore, the WDI64-32 algorithm gives minimal RMS error and is in excellent agreement with the reference case.  
All in all, the spatial resolution increases when the window size becomes smaller from 64 to 32. However, the RMS error will increase dramatically when the window size becomes smaller from 32 to 16. The reason is that there are not enough particle images to determine the flow velocity in some positions. These results give an idea of the performance of the ultrasound PIV method compared with the conventional PIV method. The iterative multi-grid deformation algorithm has some advantages in coping with velocity gradients. The WDI64-32 algorithm shows optimal performance.