## 3 Materials and Methods  
We use Field II simulations and experimentally acquired data from a commercial phantom (Model 040 GSE, CIRS, Norfolk, VA, USA) provided on the PICMUS platform (IEEE IUS 2016) [27] to validate the proposed DCF method for SOS estimation in multi-angle PW imaging. In addition to the PICMUS dataset, channel waveform from another experimental phantom (Model 042, CIRS) was also collected using Prodigy ultrasonic imaging system (S-sharp, New Taipei City, Taiwan). In this study, the effects of Rx aperture size, channel SNR and DMAS beamformer on tissue SOS estimation were firstly evaluated using PICMUS simulation phantoms, and the result of tissue SOS estimation was justified with image quality metrics such as -6-dB lateral width (LW) of wire targets and the CR of anechoic cyst. After choosing the proper Rx aperture size and DMAS beamformer in the simulations, the same setting is used to estimate the tissue SOS of the PICMUS experimental phantom. In order to determine the exact SOS of our own experimental phantom at the time of measurement, a 0.5 MHz single-element probe (Panametrics NDT A301S, Waltham, MA, USA) is used with a pulse transmitter/receiver (Panametrics 5072 PR) and an oscilloscope to measure the pulse-echo time from the bottom of the phantom. Since the dimension of the experimental phantom is known, the actual tissue SOS of the phantom is calculated from the measured pulse-echo time. Note that the collection of channel data from 21 PW transmissions evenly distributed between \(-16^{\circ}\) and \(+16^{\circ}\) was accomplished within one hour after the pulse-echo measurement in order to avoid possible SOS variation of the phantom due to the change of ambient temperature [28]. The proposed DCF method is also tested using in-vivo carotid dataset on the PICMUS platform. The imaging parameters of PICMUS dataset are listed in Table 1. The \(-6\)-dB LW of the simulation and experimental phantoms is calculated from the point targets in blue rectangular ROIs of Figure 2a,c. The CR is calculated based on the anechoic cysts in green circular ROIs and background area in blue  
Figure 1: **(a–c)** Typical DCF maps of the simulation phantom for \(C_{beam}\) greater than, equal to and less than the actual SOS, respectively. **(d)** Plot of average DCF value as a function of \(C_{beam}\) in different DMAS beamforming methods. rectangular ROIs of Figure 2b,d using Equation (10) where \(\mu_{\text{background}}\) and \(\mu_{\text{cyst}}\) represents the mean pixel values before log compression in the background area and in the cyst area, respectively. Note that the \(C_{beam}\) with the minimal \(-6\)-dB LW and the maximal CR will be determined for comparison to the estimated \(C_{opt}\) in the proposed method. The imaging parameters of Prodigy dataset are also provided in Table 2. The average \(-6\)-dB LW of the experimental phantom in the Prodigy dataset is calculated from the point target in blue rectangular ROI of Figure 3a, and the CR is calculated from the cyst area in green circular ROI and background area in blue rectangular ROI of Figure 3b. The \(C_{beam}\) used for calculating image quality and DCF value ranges from 1480 m/s to 1600 m/s for the PICMUS data and from 1370 m/s to 1490 m/s for Prodigy data. The increment of \(C_{beam}\) is 10 m/s. It should be noted that the \(C_{beam}\) is used to calculate the receive delay in both the DAS and DMAS beamforming. On the other hand, since the actual tissue SOS should be unknown during PW transmission, the calculation of transmit delay for PW steering is based on the nominal SOS in Tables 1 and 2, respectively, for the PICMUS and the Prodigy datasets:  
\[\text{CR}=20\log_{10}(\mu_{background}/\mu_{cyst}), \tag{10}\]