## 5 Analysis of Velocity Profile Distribution
Figure 9d is the vector map obtained by cross-correlation with the size of window templates 16 \(\times\) 16, referred to as WT16. They are the measured results of velocity profile vectors in the same silt carrying flow and were calculated for effect comparison nearly simultaneously. To estimate the accuracy of the measured profile velocities, firstly, a statistical value is obtained at each sampling position and taken as a true value. Secondly, the last obtained profile velocities are compared with the standard velocities, which are obtained by the ADCP method. Lastly, the velocity profile distribution of the silt carrying flow is obtained by statistical averages of each layer, referred to as averaged depth among 50 ultrasonic images and listed in Table 2 and Figure 10.  
The statistical averages that are the depth-averaged velocities of profile flow field may be affected by the roughness of the coarse bed, and the depth-averaged profiles are valid over the entire flow depth [38]. The velocity profile obtained by the ADCP is taken to be the standard velocities at the same position, which has been verified. From Table 2 and  
Figure 10: Profiles obtained by five different methods. Figure 10, we concluded that the velocity profile obtained by the WT64 is smaller than the actual velocity (ADCP). The velocity profile obtained by the WDI64-32 is similar to actual velocity. The stability of the velocity profile obtained by the WT16 is poor. In contrast, the velocity profile obtained by the WT16 has the worst among them. The reason for the above results can be analyzed and explained as follows. The test flow is laminar, and the viscosity of the fluid will give a resistance to flow, so the distribution of profile velocities has the logarithmic shape of the velocity profile. Due to the velocity gradient, the measurement error of the velocity profile can be expressed as follows:  
\[\overline{V}_{m}(y)\ =\ \frac{1}{\Delta h}\int_{y-\Delta h/2}^{y+\Delta h/2}V_{A}( y)dy \tag{7}\]  
\[\Delta\overline{V}\ =\ \overline{V}_{m}(y)-V_{A}(y) \tag{8}\]  
where \(V_{A}(y)\) is the actual velocity at depth \(y\), \(\overline{V}_{m}(y)\) is the averaged velocity at depth \(y\), \(\Delta\overline{V}\) is the measured error between \(V_{A}(y)\) and \(\overline{V}_{m}(y)\), and \(\Delta h\) is the width of the interrogation. The Formulas (9) and (10) are obtained according to Formulas (7) and (8), which can explain the effect of flow field. From the Formula (10), \(\frac{\Delta\overline{V}}{\mu_{*}}<0\) means the average velocity \(\overline{V}_{m}(y)\) at depth \(y\) is close to a value, which is lower than the actual velocity \(\overline{V}_{A}(y)\). If the interrogation window size is smaller, the measurement error will also become less:  
\[\frac{\overline{V}_{A}(y)}{\mu_{*}}\ =\ [\frac{1}{\kappa}\ln\frac{\sqrt{y^{2}- \left(\Delta h/2\right)^{2}}\mu_{*}}{v}+B]+\frac{1}{\kappa}(\frac{y}{\Delta h }\ln\frac{y+\Delta h/2}{y-\Delta h/2}-1) \tag{9}\]  
Figure 9:0  
where \(\mu_{*}\) is the friction velocity, \(\kappa\) is the von Karman constant value, which has a given typical value 0.41, and \(B\) is around 5.0 for a smooth wall. From Figure 10 and the above analysis, the measurement precision can be increased from profile WT64 and profile WT32. The reason is that the spread of the correlation peak becomes wider if the window is larger. The dynamic range of the displacement scales aligns with the dimension of the interrogation window at the same displacement gradient. This will result in a rising tendency for the correlation peak width. As a result, a smaller interrogation window is more advantageous and can produce a higher particle image density. Consider an interrogation region of a given size: the interrogation region contains many more particle-image pairs with a small displacement than with a large displacement for the real-time measurement in a region with a strong velocity gradient. Therefore, smaller displacements are more favorable for the local mean displacement than large displacements.