## 2 Materials and Methods  
### Plane Wave [PERSON]'s f-k Method  
[PERSON]'s f-k beamforming method was first proposed based on the exploding reflector model (ERM), and was widely used in seismic imaging and other fields. [PERSON] et al. [21] applied [PERSON]'s f-k method to ultrasonic plane wave imaging. In seismology, the ERM model assumes that all sound sources in the underground imaging area generate sound waves at the same time, and the sound waves only propagate upward. In order to apply the [PERSON]'s f-k method to ultrasonic plane wave imaging, [PERSON] et al. [21] first demonstrated that the ERM model can be transformed into plane waves at different angles. Under the premise of the same echo signal, if the real scatterer position in the plane wave is obtained, the corresponding virtual scatterer position in the ERM model can be calculated according to the conversion method. On this basis, [PERSON] et al. [21] used [PERSON]'s f-k method to process plane wave signals at various angles, and obtained the virtual scatterer position under the ERM model. Then the real scatterer position can be calculated by the migration principle. Coherent compound imaging can be implemented by correcting delay and aligning spectrum's coordinates of different angles. Shown as Figure 1, for the plane wave emitted with an angle of \(\theta\), the longitudinal position of the inclined echo signal due to the emission angle will be distorted. Before using the [PERSON]'s f-k method, the position of received RF data should be corrected at first. We assume that \((x_{s},z_{s})\) is the position of a scatterer, then the travel time of echo signal is:  
\[\tau_{s}(x)=\frac{1}{c}(\sin(\theta)(x_{s}-x)+\cos(\theta)z_{s}+\sqrt{\left(x _{s}-x\right)^{2}+z_{s}{}^{2}}) \tag{1}\]  
The signal travels from emitter to scatterer and back to the receiver, depending on the transducer position \((x,0)\). To be compatible with the ERM model, the virtual exploding source is positioned at \((\hat{x}_{s},\hat{z}_{s})\) with a propagation speed of \(\hat{c}\). \(\hat{c}\) is a one-way speed from scatterer to the receiver. Then the following ERM travel time is:  
\[\hat{\tau}_{s}(x)=\frac{1}{\hat{c}}\sqrt{\left(\hat{x}_{s}-x\right)^{2}+\hat{ z}_{s}^{2}} \tag{2}\]  
By equalizing \(\tau_{s}(x)\) and \(\hat{\tau}_{s}(x)\), the relationships of the sound speed and coordinates between real situation and that in ERM model are obtained by the coefficients as below:  
\[\hat{\tau}_{s}(x)=\frac{1}{\alpha c}\sqrt{\left(x_{s}+\gamma z_{s}-x\right)^{ 2}+\beta^{2}{z_{s}}^{2}} \tag{3}\]  
where \((\alpha,\beta,\gamma)\) are defined as follows:  
\[\left\{ \begin{array}{l}\alpha=1/\sqrt{1+\cos(\theta)+\sin^{2}(\theta)}\\ \beta=\frac{\left(1+\cos(\theta)\right)^{3/2}}{1+\cos(\theta)+\sin^{2}(\theta) }\\ \gamma=\frac{\sin(\theta)}{2-\cos(\theta)}\end{array} \right. \tag{4}\]  
In the equation, \(\alpha\) relates the virtual propagation speed \(\hat{c}\) and real speed \(c\). \(\beta\) represents the scaling ratio between the virtual axial coordinate \(\hat{z}_{s}\) and the real axial depth \(z_{s}\). \(\gamma\) refers to the shift operation between virtual lateral coordinate \(\hat{x}_{s}\) and real lateral position \(x_{s}\). The relationship between virtual arguments and actual arguments are shown as follow:  
\[\left\{ \begin{array}{l}\hat{c}=\alpha c\\ \hat{z}_{s}=\hat{\beta}z_{s}\\ \hat{x}_{s}=x_{s}+\gamma z_{s}\end{array} \right. \tag{5}\]  
Assume that a plane wave is emitted at the angle of \(\theta\), and the RF data of the echo signal is \(\psi_{\theta}\). If the plane wave is backscattered in receiving, the signal received at the probe  
Figure 1: Scheme of route for the echo signal travelling from transmitter to receiver. whose coordinate is \((x,z=0)\) can be present as \(\psi_{\theta}(x,z=0,t)\). Fourier transform is applied to \(\psi_{\theta}(x,z=0,t)\) over time \(t\) to obtain the spectrum \(G_{\theta}(x,z=0,f)\), where \(f\) is the temporal frequency.