## 1 Introduction  
Ultrasound single plane-wave (PW) imaging only needs a single shot to reconstruct one image and thus has a very high frame rate. Because of the lack of transmit focusing, however, the image resolution, contrast (CR) and signal-to-noise ratio (SNR) degrade in single PW imaging. Therefore, previous studies have proposed multi-angle PW coherent compounding imaging (CPWC) [1] that can simultaneously maintain a high frame rate and achieve the quality of traditional transmit focused images. This method uses multiple transmit events (TxEt) of different PW angles to synthesize high-resolution images. The receiving channel (RxCh) echoes are firstly processed in parallel to obtain low-resolution sub-images of each PW angle, and then these sub-images are coherently summed to form a high-resolution compounded image. In other words, the final CPWC image pixel value is obtained by two-dimensional (2D) summation of the echo matrix in both the RxCh and the TxEt directions. Compared with CPWC imaging, the signal coherence of the 2D echo matrix can be further used to attenuate low coherent clutters in the image for better image resolution and CR. For example,delay-multiply-and-sum (DMAS) beamforming is based on signal coherence to improve the image quality of ultrasound array system. It remedies the low image resolution and high side lobe level of conventional delay-and-sum (DAS) beamforming by using mutual multiplication of possible channel pairs to suppress any spatially uncorrelated clutter artifacts and thermal noises [2; 3; 4; 5; 6]. Note that DMAS beamforming has been adopted in not only medical ultrasound but also photoacoustic imaging [7; 8; 9; 10]. Several works have been developed to extend DMAS beamforming for PW imaging using either the signal coherence of the echo matrix in the TxEt direction [11] or in the RxCh direction [12]. Recently, a novel DMAS operation has also been proposed to extract the 2D spatial coherence of echo matrix for further improvement of image quality [13]. Similar to ultrasound imaging with focused transmit, the imaging quality of ultrasonic multi-angle PW imaging also depends on the accuracy of beamforming. In the process of beamforming, it is usually assumed that the tissue speed of sound (SOS) of propagating medium is a known fixed value for designing the delay profile among channels to achieve focusing. However, the actual tissue SOS varies widely according to the composition of tissues. Therefore, when the beamforming velocity (\(C_{beam}\)) does not match the actual tissue SOS, a beamforming error occurs and results in a decrease in image resolution and CR. In the literatures tissue SOS estimation for finding the optimal \(C_{beam}\) (\(C_{opt}\)) can be roughly separated into two categories: pre-beamforming array signal analysis and post-beamforming image quality analysis. The former includes the RxCh waveform curvature fitting to the geometric position of every channel in an array using different \(C_{beam}\). The \(C_{beam}\) that best fits the echo curvature is the \(C_{opt}\)[14]. The minimum average phase variance method is to calculate the minimal phase variances of channel signal using different \(C_{beam}\) to determine the \(C_{opt}\)[15]. The phase variance can be also represented in the form of channel differential phase gradient between left and right sub-apertures [16; 17]. On the other hand, the channel autocorrelation function of different \(C_{beam}\) with the largest area under curve also correspond to the \(C_{opt}\)[18]. Post-beamforming methods includes image spatial frequency spectrum analysis in the lateral direction to calculate the largest spatial spectral area within a specified frequency range as the \(C_{opt}\)[19]. Similarly, the edge sharpness of image can be estimated using anisotropic diffusion filtering to locate the \(C_{opt}\) the highest edge sharpness [20]. Autocorrelation of background speckle pattern also serve as the focus quality factor to quantify the image resolution.