## 4 Results  
### PICMUS Simulation Dataset  
Table 3 shows the \(C_{beam}\) with the best image quality (minimal \(-6\)-dB LW and maximal CR) of different receiving aperture sizes represented by the corresponding receiving f-number (i.e., RxFn). Results show that when RxFn is equal to \(1.5\), the \(C_{beam}\) with the minimal \(-6\)-dB LW appears to deviate from the actual tissue SOS in the simulation (i.e., 1540 m/s) and the \(C_{beam}\) with the maximal CR matches the actual tissue SOS for all RxFn. Therefore, it is shown that the image quality metrics better correspond to the actual tissue SOS when the receiving aperture is large enough.  
Figure 3: B-mode images of Prodigy dataset for measurement of image quality metrics. (**a,b**) Two different FOVs of the same experimental phantom.  
Figure 2: B-mode images of PICMUS dataset for measurement of image quality metrics. (**a,b**) Two simulation phantoms for wires and anechoic cysts in speckle background, respectively. (**c,d**) Two different field of views (FOVs) of the same experimental phantom.  
Figure 4 shows the effect of different receiving aperture sizes on estimation of tissue SOS when the \(p\) value is set to 2 under four different channel SNR. For each channel SNR, there are 10 independent realizations of random noise to calculate the average and standard deviation of SOS estimation. As expected, results indicate that the estimated SOS of the three DMAS methods will become more accurate and robust when the noise level decreases. In fact, for any channel SNR larger than 0 dB, all DMAS methods can provide the estimated \(C_{opt}\) that accurately matches the actual tissue SOS of 1540 m/s in the PICMUS simulation data. Moreover, it is also shown that a larger receiving aperture (i.e., a smaller RxFn) will facilitate the estimation of tissue SOS. For example, when the channel SNR is \(-\)15 dB, the SOS estimation is 1544 \(\pm\) 4.9, 1543 \(\pm\) 4.6 and 1541 \(\pm\) 3 m/s, respectively, for 2D-DMAS, Rx-DMAS and Tx-DMAS beamforming with the RxFn of 1.25. The SOS estimation improves to 1540 \(\pm\) 0, 1540 \(\pm\) 0 and 1540 \(\pm\) 0, respectively, for 2D-DMAS, Rx-DMAS and Tx-DMAS beamforming when the RxFn decreases to 0.75. In other words, a larger receiving aperture and a lower noise level both help to ensure a reliable DCF value in the proposed method for accurate SOS estimation. It can be also found that Tx-DMAS generally has the lowest deviation and standard deviation compared to other DMAS beamforming methods. Based on the simulation results, we chose using Tx-DMAS method with a RxFn of 0.75 to estimate tissue SOS for subsequent PICMUS experimental data and Prodigy experimental data.  
The number of PW for estimating the \(C_{opt}\) in Figure 4 is 21. Figure 5 shows the effect of the number of PW on the DCF value when the beamforming method is Tx-DMAS and the \(p\) value is equal to 2. The results indicate that, when the number of PW is above 21, the DCF value tends to be stable. In order to reduce the computational complexity, we choose the number of PW equal to 21 for estimating the \(C_{opt}\).  
Figure 6 further shows that the effect of \(p\) values on SOS estimation in Tx-DMAS beamforming method. It was found that the three curves for \(p\) value of 2, 2.5, and 3 overlap and they provide lower deviation and standard deviation of SOS estimation compared to the case for \(p\) value of 1.25 and 1.5. In other words, when the \(p\) value has been larger than 2, it appears that further increasing the \(p\) value to emphasize the signal coherence for better SOS estimation is marginal in the DCF method. Therefore, in the following experiments, we use the setting of \(p\) = 2 to estimate the tissue SOS.  
Figure 5: Plot of DCF value in Tx-DMAS method as a function of number of PW.  
Figure 6: Plot of \(C_{opt}\) in Tx-DMAS method with different \(p\) values as a function of channel SNR.