## 5 Discussion and Conclusions  
This paper combines multi-angle PW imaging and DMAS beamforming for tissue SOS estimation. In this study, the magnitude ratio of the image pixel value between DMAS beamforming and traditional CPWC is defined as the DCF of the pixel. When there is a mismatch between \(C_{beam}\) and tissue SOS, the signal coherence degrades and the corresponding DCF value also decreases. Therefore, by calculating the DCF at different \(C_{beam}\), the \(C_{beam}\) with the largest DCF is the optimal \(C_{beam}\) (i.e., \(C_{opt}\)) that matches the tissue SOS. Compared to other methods, the proposed DCF method not only considers the signal coherence among RxCh but also the signal coherence among TxEt. In the computational complexity, the DCF method only needs pixel-to-pixel division of the multi-angle PW DMAS image by the traditional CPWC image, and does not need to additionally retain the channel signal to calculate the signal coherence. Note that \(C_{beam}\) in this study is only used to adjust the time delay in the receive beamforming while the transmit delay for PW transmit steering is determined by the constant nominal SOS. In other words, the transmit delay (\(\tau_{TX}\)) for PW steering angle (\(\theta\)) is designed by using the nominal SOS as \(C_{TX}\) in \(\tau_{TX}=x\sin\theta/C_{TX}\) where \(x\) represents the lateral position of transmit element in the array. When the \(C_{TX}\) mismatches the actual tissue SOS, PW transmit angle will deviate from its designed value. For example, when \(C_{TX}\) is larger than the actual tissue SOS, it is expectable that the actual PW transmit angle will be smaller than the designed value such that the maximal PW transmit angle for SOS estimation is reduced. A supplementary experiment has been performed to study the robustness of the proposed DCF method in the presence of mismatch between the \(C_{TX}\) and the actual tissue SOS. Specifically, two sets of channel waveforms from Model 042 phantom are acquired using the Prodigy system with different \(C_{TX}\). Results in Figure 13 indicate that both DCF curves peak at the same value of \(C_{beam}\) regardless of the setting of \(C_{TX}\). In other words, the proposed DCF method will suggest a robust tissue SOS even with significant deviation of the \(C_{TX}\). Noted that the estimated tissue SOS in Figure 13 is different from that in Figure 11 possibly due to the change of ambient temperature in this two separate experiments. However, since PW imaging does not have transmit focusing, our method generally relies on the ROI at the elevational focal depth for DCF calculation. This is because the signal coherence is better maintained when the received echoes come from a smaller sample volume (e.g., the transmit focal zone). Therefore, it is expectable that the signal coherence among both receive channel and transmit event could improve by selecting the ROI at the elevational focal depth. At depth away from the elevational focus, the proposed method will introduce larger estimation error and thus may not be applicable to deep tissues. The possible solution is to use a 1.5D/2D array to provide elevational focusing at the region of depth. Figure 13: (a) DCF map of Tx-DMAS beamforming for the supplementary dataset to test the effect of mismatch between \(C_{TX}\) and tissue SOS. (b) Plot of averaged DCF value in (a) as a function of \(C_{beam}\) for different \(C_{TX}\). By using different Rx aperture sizes, channel SNR, beamforming methods, and \(p\) values in the PICMUS simulation data, the effect of the proposed DCF method for tissue SOS estimation are evaluated in this study. Results indicate that, when the Rx aperture size becomes larger and the channel SNR increases, the estimation results of tissue SOS will be more accurate. Compared to either Rx-DMAS or 2D-DMAS imaging methods, results also show that Tx-DMAS generally provides better performance in tissue SOS estimation when considering noise.