## 2 Theory
### DCF of Multi-Angle PW Imaging for Tissue SOS Estimation  
In this study, the magnitude ratio of image pixel value of DMAS beamforming to that of 2D-DAS is defined as the DCF to indicate the degree of signal coherence. For low-coherent image targets like clutter, the DMAS image pixel value will decrease significantly relative to the 2D-DAS, so it will have a smaller DCF value. On the contrary, for high-coherent image targets like point reflector, its DMAS image pixel value will be similar to 2D-DAS and thus has a DCF value close to unity. Taking 2D-DMAS beamforming method as an example, the corresponding DCF value can be readily simplified as the \((p-1)\)-th power of the magnitude of phasor summation for every signal in the 2D echo matrix by assuming the echo signal of all RxCh at all transmit angle has the same amplitude (i.e., assuming \(a_{nk}\) is a constant):  
\[\text{DCF}_{\text{2D-DMAS}}\equiv\left|\frac{y_{\text{2D-DMAS}}}{y_{\text{2D- DAS}}}\right|=\left|\frac{\left|\frac{\frac{1}{M}\sum\limits_{k=1}^{M}\left(\frac{1}{N} \sum\limits_{n=1}^{N}\sqrt[\alpha_{nk}]e^{j\phi_{nk}}\right)\right|^{p}}}{ \frac{1}{M}\sum\limits_{k=1}^{M}\left(\frac{1}{N}\sum\limits_{n=1}^{N}a_{nk}e ^{j\phi_{nk}}\right)}\right|\approx\left|\frac{1}{MN}\sum\limits_{k=1}^{M} \sum\limits_{n=1}^{N}e^{j\phi_{nk}}\right|^{p-1}, \tag{6}\]  
Though the approximation in Equation (6) only holds for point-like reflector, the magnitude ratio of DMAS image to 2D-DAS image can still be used to measure the signal similarity in phase for diffused scatterers without loss of generality. With the same assumption, the DCF obtained from the other two DMAS beamforming methods can also be approximated as the following formula:  
\[\text{DCF}_{\text{Rx-DMAS}}\equiv\left|\frac{y_{\text{Rx-DMAS}}}{y_{\text{2D- DAS}}}\right|\approx\left|\frac{1}{N}\sum\limits_{n=1}^{N}e^{ja_{n}}\right|^{p-1}, \tag{7}\]  
\[\text{DCF}_{\text{Tx-DMAS}}\equiv\left|\frac{y_{\text{Tx-DMAS}}}{y_{\text{2D - DAS}}}\right|\approx\left|\frac{1}{M}\sum\limits_{k=1}^{M}e^{j\beta_{k}} \right|^{p-1}, \tag{8}\]  
where the definition of \(\alpha_{n}\) and \(\beta_{k}\) is provided in Equations (4) and (5), respectively. Thought the DCF proposed in this study can be understood as the magnitude of phasor summation in the echo matrix of PW imaging, it should not be confused with the phase coherent factor in [24] or the vector coherent factor in [25] because they are intrinsically different in both definition and calculation. In this study, we propose to use the aforementioned DCF to consider the signal coherence of echo matrix for tissue SOS estimation in multi-angle PW imaging. Figure 1a-c illustrate the DCF maps of different \(C_{beam}\). Figure 1a,c show that, when the \(C_{beam}\) mismatches the tissue SOS, the signal coherence and the corresponding DCF value of the 2D echo matrix will decrease. In contrast, Figure 1b shows that the DCF value will be significantly higher when the \(C_{beam}\) matches the tissue SOS. In our method, we will calculate the DCF map of different \(C_{beam}\) and select the speckle region-of-interest (ROI) at the depth of elevational focus of the array probe (as shown by the black square in Figure 1b) to observe the change of DCF value with the \(C_{beam}\). As shown in Figure 1d, the \(C_{beam}\) with the maximal DCF value is the \(C_{opt}\) that corresponds to the tissue SOS estimated in our method as expressed in Equation (9):  
\[C_{opt}=\underset{C_{beam}}{\text{argmax}}|\text{DCF}|, \tag{9}\]  
Note that, in the scenario of focused transmit imaging, the degree of signal coherence has also been used for estimation of tissue SOS so that the \(C_{beam}\) with the highest signal coherence corresponds to the tissue SOS [18,26]. Nonetheless, all these methods demand for additional channel-domain processing to estimate the degree of signal coherence.