## 7 Conclusions  
This paper describes a method for the visual measurement of velocity profile distribution in silt carrying flow using the ultrasound PIV and iterative multi-grid deformation techniques with a kind of physical model experiment. A portable medical ultrasound imaging device with a convex array probe is used to acquire the ultrasonic image in the sediment-laden flow. Conventional techniques, such as the optical PIV technique, require a clear flow condition or optical access to capture particle images. However, our ultrasound PIV method does not have this limitation. It has the advantage of simplicity and accessibility with the help of an interface software in actual applications. An ultrasound imaging measurement system can realize the visual measurement of the flow field and velocity profile distribution in silt carrying flow. The biggest advantages of this system are the portable device and the good penetrability for silt carrying flow during kinds of physical model tests.  
This paper also provides an iterative multi-grid deformation method to improve the spatial resolution of ultrasonic images and the processing speed of the flow velocity measurement. A hierarchical approach is used to improve the spatial resolution through which the sampling grid is continually decreased, and the interrogation window size is also simultaneously decreased. The interrogation process is from coarse to fine. However, the decrease in interrogation window size can decrease the peak's signal-to-noise ratio. Physical model experiments assess the performance of the PIV technique based on the  
Figure 12: Comparison of RMS error point by point.  
ultrasonic imaging system. The velocity vectors obtained by using the WDI64-32 algorithm are the most in agreement with standard values (ADCP algorithm) compared with the WT64 algorithm, the WT32 algorithm, and the WT16 algorithm. Their RMS errors are 0.148, 0.4214, 0.8796, and 1.4148 at the certain condition, respectively. Additionally, profiles of streamwise velocity are often irregular and have a great velocity gradient. Therefore, the window deformation technique is necessary to compensate for the velocity gradient effect to optimize the accuracy of the real-time and visual measurement.  
Conceptualization, X.Z. and W.H.; methodology, W.H.; software, H.S.; validation, X.Z., W.H. and H.S.; formal analysis, X.Z.; investigation, B.C.; resources, W.H.; data curation, W.H.; writing--original draft preparation, X.Z.; writing--review and editing, X.Z. and H.S.; visualization, X.Z.; supervision, B.C.; project administration, X.Z.; funding acquisition, X.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (42007266) and the Natural Science Foundation of Hubei Province (2019 CFB345).  
Not applicable.  
All data, models, and code generated or used during the study appear in the submitted article.  
All the images and data are from our actual tests and permitted by the owners. We are compliant with Ethical Standards, and all authors declare that this paper has no conflict of interest. Finally, we are grateful for the many helpful and constructive comments from many anonymous reviewers.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.  
The descriptions of abbreviations in the text.