## 5 Discussion and Conclusions
The superiority of Tx-DMAS beamforming in tissue SOS estimation should come from the fact that its DCF calculation is based on signals with higher SNR. Assuming that the imaging depth for SOS estimation is about 20 mm and the channel SNR is \(-20\) dB, Tx-DMAS first performs coherent summation along the RxCh direction and thus improves the signal SNR for DMAS calculation. Specifically, since the number of RxCh will be about 67 when the RxFn is 1 at imaging depth of 20 mm, the equivalent SNR for DMAS beamforming in the direction of the TxEt is \(-20+20\text{log}_{10}(67)=16.52\) dB. For Rx-DMAS beamforming, on the other hand, it first performs coherent summation along the TxEt direction (i.e., 21 PWs in this study), and the equivalent SNR for DMAS beamforming in the RxCh direction is \(-20+20\text{log}_{10}(21)=6.44\) dB. Note that the 2D-DMAS doesn't perform any coherence summation and thus its SNR for DMAS beamforming in both directions of RxCh and TxEt remains at \(-20\) dB. Consequently, since the signal of Tx-DMAS has the highest SNR for DMAS calculation, Tx-DMAS can use a smaller receiving aperture to provide the best tissue SOS estimation with smallest deviation and standard deviation, especially in the scenario of low channel SNR. On the contrary, 2D-DMAS usually has the worst tissue SOS estimation due to its lowest SNR for DMAS calculation. In addition, performance of the proposed SOS estimation will noticeably degrade when the \(p\) value in DMAS beamforming is smaller than 2. Note that a smaller \(p\) value reduces the degree of signal coherence in DMAS beamforming. Therefore, it is expectable that the calculation of DCF value may fail when the degree of signal coherence is insufficient. Using Tx-DMAS beamforming in the experiments, our results show that the \(C_{opt}\) estimated by the proposed DCF method does have the best image quality for the PICMUS experimental dataset. For the Prodigy dataset, the estimated tissue SOS is \(1426\pm 6\) m/s which is very close to the actual tissue SOS of 1427 m/s. The estimated SOS also corresponds to the \(C_{opt}\) with the minimal \(-6\)-dB LW and the maximal CR within an error of 10 m/s. For comparison, both 2D-DMAS and Rx-DMAS are also applied to the Prodigy dataset and the corresponding tissue SOS is estimated to be \(1428\pm 4\) m/s for both methods. In the experiments, the channel SNR can be estimated from the magnitude difference in power spectrum between the received channel signal and noise at PW angle of \(0^{\circ}\). Since the channel SNR in the Prodigy dataset is estimated to be about 20 dB, it is expectable that all three DMAS beamforming methods can provide comparable estimation of the tissue SOS in this high SNR condition. Nonetheless, it should be noted that Tx-DMAS has the lowest computational complexity since it does not need to retain the complete channel signal of each TxEt as in Rx-DMAS or to do \(p\)-root magnitude scaling for every entry in the 2D echo matrix as in 2D-DMAS. In fact, Tx-DMAS beamforming can be easily implemented in current PW imaging system by adopting DMAS processing instead of the conventional coherent summation of low-resolution sub-images. Therefore, the DCF method with Tx-DMAS beamforming should also outperform the Rx-DMAS and 2D-DMAS counterparts in terms of computational complexity. Note that 2D-DMAS would degenerate to Rx-DMAS when the number of PW involved in the DCF calculation reduces to 1. Since there is no synthetic transmit focusing in single PW imaging, the received echoes actually come from the unfocused PW transmission. In this case, the estimated \(C_{opt}\) of Prodigy imaging system dataset for Rx-DMAS method is \(1474\pm 31\) m/s which evidently over-estimates the actual SOS of the imaged phantom. This is because, when the received channel data comes from a large sample volume due to unfocused transmission, the corresponding phase curve would have a reduced curvature due to the superimposition of several diffuse scatterers in the lateral direction [29].