## 6 Result and Discussion
### Target Localization Results  
All pseudo-targets are shown in Figure 9 (in our case, the number of pseudo-targets was 2145). The black dots represent different pseudo-targets. Some pseudo-targets were close to the target, while others were far from it. The IMWLE method considers the distribution of pseudo-targets to improve the accuracy of localization. The localization results after multiple iterations and the real target location are shown in Figure 10. The yellow dot represents the real target location, \((64.33,93.93,43.33)\), where the ultrasonic waves interacted with the target. The blue dot represents the estimated target location. The grey dots and their grayscale represent different pseudo-targets and their weight factor. The higher the greyscale, the greater the possibility of detecting the target. Figure 8: The scattered signals received by the sensor at receiving position 1. The blue line represents the signals and the red line represents the envelope obtained by Hilbert transform. Figure 9: Distribution of pseudo-targets. The number of pseudo-targets is 2145. Figure 10: Target localization results using IMWLE after each iteration. The distribution of pseudo-targets after (**a**) the first iteration, (**b**) the second iteration, (**c**) the third iteration, (**d**) the fourth iteration, (**e**) the fifth iteration, and (**f**) the sixth iteration. Figure 10a shows the distribution of pseudo-targets after the first iteration. All grey dots had the same grayscale representing same weight factors. Therefore, the estimated target location \(\mu_{1}\) was the mean of coordinates of all pseudo-targets. Figure 10b-f show the distribution of pseudo-targets after the second, third, fourth, fifth, and sixth iterations, respectively. The weight factors of the pseudo-targets that were far away from the real target were gradually reduced after iteration so that corresponding greyscale became too low to observe. Therefore, the grayscale region became more converged with each iteration and the estimated target location came closer to the real target location. The error of localization can be defined as the distance between real target location and estimated target location,  
\[error=\sqrt{\left(x^{real}-x^{estimated}\right)^{2}+\left(y^{real}-y^{ estimated}\right)^{2}+\left(z^{real}-z^{estimated}\right)^{2}}, \tag{9}\]  
where \(\left(x^{real},y^{real},z^{real}\right)\) and \(\left(x^{estimated},y^{estimated},z^{estimated}\right)\) are the coordinates of the real target and the estimated target, respectively. The errors of localization after each iteration are listed in Table 2 and shown in Figure 11. In early iterations (in our case \(j=1\)-\(6\)), the influence of pseudo-targets far from the real target were reduced. The error gradually decreased from the first to the sixth iteration. After the sixth iteration, the influence of pseudo-targets that were close to the real target had also reduced. The number of pseudo-targets that took effect in localization algorithm were reduced, which means that the statistical method gradually became inappropriate. Therefore, the fluctuation of error appeared. The stopping condition could stop the iteration at \(j=7\) and \(\mu_{6}\), which was the final estimated target location. The probability-based ToF method and the IMWLE method both provided the estimated target area. By comparing the two results, it can be observed that the probability-based ToF method gave