## 2 Method
### Full-Matrix Imaging in Wavenumber Domain  
In the case of FMC data, we consider the rotated coordinate system \(O\)\(x^{\prime}z^{\prime}\), where the linear array is rotated to \(z^{\prime}=0\). For simplicity, we assume that the \(u\)-th element emits ultrasound waves at \(t=0\), and the scatterer is located at \((x^{\prime},z^{\prime})\) in the \(L\)-th layer of the oblique multi-layer structure. The ultrasound waves reach the scatterer at \(t=\tau_{u}(x^{\prime},z^{\prime})\), which is determined by the relative position of the scatterer and the element. At this moment, the reflected wave field from the scatterer is tightly focused and is not affected by the other scatterers. If there is an array element just above it to receive the reflected signals, an optimally focused image can be reconstructed [17]. Then, the imaging condition becomes \(t=\tau_{u}(x^{\prime},z^{\prime})\) instead of \(t=0\). Thus, the focused image line at depth \(z^{\prime}\) can be reconstructed according to [28]:  
\[I_{u}(x^{\prime},z^{\prime})=p_{u}(x^{\prime},z^{\prime},\tau_{u}(x^{\prime}, z^{\prime}))=\iint\limits_{\infty}P_{u}(k^{\prime}_{x},z^{\prime},\omega) \cdot e^{ik^{\prime}_{x}x^{\prime}-i\omega\tau_{u}(x^{\prime},z^{\prime})}dk^ {\prime}_{x}d\omega \tag{22}\]  
However, the computational burden of calculating \(\tau_{u}(x^{\prime},z^{\prime})\) is very heavy for FMC data, especially in multi-layered structures. To solve this problem, we extend the received wave-field extrapolation to the transmitted wave field, to compensate for the propagation time. For the \(u\)-th active element, the transmitted pressures can be written as \(s_{u}(x,z=0,t)\). Note that the transmitted pressures travel along the \(+z\) direction. According to Equation (5), \(A(k_{x},\omega)\) should be set to zero, and the extrapolated wave field from the active array element \(u\) can be expressed as  
\[S_{u}(k_{x},z,\omega)=S_{u}(k_{x},z=0,\omega)\cdot e^{-ik_{x}z} \tag{23}\]  
where \(S_{u}(k_{x},z=0,\omega)\) is the spectrum of \(s_{u}(x,z=0,t)\). Combining the wave-field extrapolation and the tilt compensation operations in oblique multi-layered structures, we can obtain the extrapolated transmitted wave field at position \((x^{\prime},z^{\prime})\) in the \(L\)-th layer, which is expressed as  
\[S_{u}(k^{\prime}_{x},z^{\prime},\omega)\propto S_{u}(k^{\prime}_{x},z^{\prime }=0,\omega)\cdot e^{-ik^{\prime}_{x}z}(z^{\prime}-z^{\prime}_{L-1})\prod_{m=1} ^{L-1}\exp(-ik_{z,m}d_{m}) \tag{24}\]  
where  
\[S_{u}(k^{\prime}_{x},z^{\prime}=0,\omega)=\Psi(k^{\prime}_{x},\omega,a)\cdot S _{u}(k^{*}_{x}(k^{\prime}_{x},\omega,a),z=0,\omega) \tag{25}\]and  
\[k_{x}^{*}(k_{x}^{\prime},\omega,a)=\frac{1}{1+a^{2}}\bigg{(}k_{x}^{\prime}-a \sqrt{(1+a^{2})\!\cdot\!\omega^{2}/c^{2}-k_{x}^{\prime\,2}}\bigg{)} \tag{26}\]  
\[\Psi(k_{x}^{\prime},\omega,a)=\frac{\partial k_{x}^{*}}{\partial k_{x}^{\prime }}=\frac{1}{1+a^{2}}\left(1+\frac{ak_{x}^{\prime}}{\sqrt{(1+a^{2})\!\cdot\! \omega^{2}/c^{2}-k_{x}^{\prime\,2}}}\right) \tag{27}\]  
It should be noted that Equations (26) and (27) differ from Equations (16) and (19) due to the physical differences between forward and backward propagations. Thus, the sign function in Equations (7) and (15) should be different when it is used in the transmitted wave-field extrapolation. After extrapolating the transmitted wave field at the point of interest, the image can be reconstructed by taking a correlation between the forward-extrapolated wave field of \(s_{u}(x,z=0,t)\) and the backward-extrapolated wave field of \(p_{u}(x,z=0,t)\), depth by depth. The correlation in the frequency domain can be written as  
\[C_{u}(x^{\prime},z^{\prime},\omega)=P_{u}(x^{\prime},z^{\prime},\omega)\! \cdot\!S_{u}(x^{\prime},z^{\prime},\omega) \tag{28}\]  
where \(S_{u}(x^{\prime},z^{\prime},\omega)\) and \(P_{u}(x^{\prime},z^{\prime},\omega)\) are the inverse Fourier transforms of \(S_{u}(k_{x}^{\prime},z^{\prime},\omega)\) and \(P_{u}(k_{x}^{\prime},z^{\prime},\omega)\) over \(k_{x}^{\prime}\), respectively.