## 3 Results  
We performed numerical simulations using the k-WAVE toolbox (version 1.4) to demonstrate the reconstruction results using the DAS algorithm and multi-DASE algorithm, respectively [29]. The simulation was based on a homogeneous propagation medium with the sound speed set to 1500 m/s. The computing area is a square area with borders (21.6 \(\times\) 21.6 mm\({}^{2}\)) as shown in Figure 5(a). In total, 108 transducers are uniformly placed at the top of the area, and the circular sound source is placed in the middle of the area with a radius of 1.5 mm. Firstly, the DAS algorithm is used for reconstruction, and there are two obvious cambered artifacts in the reconstruction results, as shown in Figure 5(b). In order to better illustrate the superiority of the multi-DASE method, we also used the DMAS method to reconstruct the source, and the reconstruction result is shown in Figure 5(c). Although the artifacts have been effectively suppressed, it is evident that the reconstructed images exhibit discontinuities, and the reconstruction results still demonstrate a certain degree of shape distortion. When applying Equation (11) for reconstruction, the value of \(k\) needs to be determined first. The suppression effect corresponding to the smaller \(k\) is not obvious, while the larger \(k\) may lead to excessive suppression, which results in unsatisfactory reconstruction results. To ensure the scientific validity of selecting the \(k\) value, a quantitative criterion was established: when \(inv(x,y;t)=0.5\), introducing \(k\) should enable the suppression factor to reach at least \(inv^{k}(x,y;t)=0.1\). Based on this criterion, and after conducting numerical calculations and experimental validations, the optimal parameter value of \(k=3\) was ultimately determined. The reconstruction result is shown in Figure 5(d). From this result, it can be seen that, compared with the first two methods, artifacts are effectively suppressed, and the shape of the reconstructed result is well preserved. However, the internal structure of the source is also suppressed, leading to a darker appearance in the reconstructed region and causing a distinct \"gap\" to appear. Since different k values correspond to different degrees of suppression, we attempt to use different values of \(k\) for reconstruction. This was carried out to investigate whether the selection of k values affected the internal reconstruction results. To make the results more clear, we chose a circular sound source with a radius of 1.5 mm for verification. Figure 7 shows reconstructed images using Equation (11) with different \(k\) values, respectively. When k = 0, the reconstruction result of multi-DASE, compared to the reconstruction result of DAS, eliminates the negative components, resulting in more complete boundaries. However, in terms of image quality, there is no significant improvement. Since it does not suppress artifacts, the reconstruction results of multi-DASE still exhibit strong artifacts, as shown in Figure 6(a). As the value of \(k\) increases, artifacts are increasingly suppressed, with larger \(k\) values leading to the stronger suppression of artifacts. We observed that when the value of \(k\) is relatively small, the problem of missing the internal structure of the reconstructed source also exists, as shown in Figure 6(b). As \(k\) increases, the problem of the missing internal structure of the reconstructed source becomes more and more obvious, as shown in Figure 6(c),d. Through this experiment, it can be determined that the problem of the missing internal structure of the reconstructed source is not directly determined by the value of \(k\). We restored the complete signal at the pixel point at the center of the circular sound source shown in Figure 6, and calculated its envelope. The result is shown in Figure 7(b). The envelope signal observed is not a single peak but instead exhibits a bimodal distribution, which is what we refer to as signal overshoot.