## 2 Method  
### Wave-Field Extrapolation for Multi-Layered Structures  
In this subsection, the basic wave-field extrapolation is introduced, as well as the extension to multi-layered structures. Here, a linear array is assumed to detect a homogeneous medium with a sound velocity of \(c\). The \(x\)-axis is directed along the interface, and the \(z\)-axis is perpendicular to the interface, as shown in Figure 1. In FMC, the ultrasound waves excited by the \(u\)-th element will propagate to the scatterers and the reflected waves will be received by all array elements. Let \(p_{u}(x,z,t)\) be the ultrasound pressures reflected by the point scatterer. The propagation of the reflected pressures should satisfy the wave equation given in [18]:  
\[\
abla^{2}p_{u}(x,z,t)-\frac{1}{c^{2}}\frac{\partial^{2}p_{u}(x,z,t)}{\partial t ^{2}}=0 \tag{1}\]  
Meanwhile, \(p_{u}(x,z,t)\) can be expanded in the wavenumber domain as in [28]:  
\[p_{u}(x,z,t)=\iint P_{u}(k_{x},z,\omega)\cdot e^{ik_{x}x-i\omega t}dk_{x}d\omega \tag{2}\]  
where \(P_{u}(k_{x},z,\omega)\) is the Fourier expansion of \(p_{u}(x,z,t)\), \(k_{x}\) is the wavenumber in the \(x\) direction, and \(\omega\) is the angular frequency. By inserting Equation (2) into Equation (1), the Helmholtz equation can be obtained:  
\[\left(\frac{\partial^{2}}{\partial z^{2}}+k_{z}^{2}\right)P_{u}(k_{x},z, \omega)=0 \tag{3}\]  
where \(k_{z}\) is the wavenumber component in the \(z\) direction, defined as  
\[k_{z}^{2}=\omega^{2}/c^{2}-k_{x}^{2} \tag{4}\]  
The general solution of Equation (3) is given by [17]:  
\[P_{u}(k_{x},z,\omega)=A(k_{x},\omega)\cdot e^{ik_{z}z}+B(k_{x},\omega)\cdot e^ {-ik_{z}z} \tag{5}\]  
where \(A(k_{x},\omega)\cdot\exp(ik_{z}z)\) and \(B(k_{x},\omega)\cdot\exp(-ik_{z}z)\) represent the upward-traveling and downward-traveling waves, respectively. We assume that the array is placed at \(z=0\) and all reflectors are located in the half-space \(z>0\). Thus, only upward-traveling waves can be recorded by the ultrasonic array. Considering the boundary condition \(P_{u}(k_{x},z=0,\omega)\)  
Figure 1: Multi-layered object inspected by a linear array. which is the 2D Fourier transform of the received pressures \(p(x,z=0,t)\) at depth \(z=0\), the wave field at an arbitrary depth \(z\) can be extrapolated as in [17]:  
\[P_{u}(k_{x},z,\omega)=P_{u}(k_{x},z=0,\omega)\cdot e^{ik_{z}z} \tag{6}\]  
where  
\[k_{z}=-\text{sgn}(\omega)\cdot\sqrt{\omega^{2}/c^{2}-k_{x}^{2}},\text{ with }\omega^{2}/c^{2}\geq k_{x}^{2} \tag{7}\]  
The sign function in Equation (7) guarantees that the \(k_{z}\) value represents the waves traveling in the \(-z\) direction. Thus, the pressures \(p_{u}(x,z,t)\) in time-space coordinates can be reconstructed by an inverse Fourier transform  
\[p_{u}(x,z,t)=\iint_{\infty}P_{u}(k_{x},z=0,\omega)\cdot e^{ik_{z}z}e^{ik_{x}x -i\omega t}dk_{x}d\omega \tag{8}\]  
In the case of multi-layered structures where the sound velocity varies along the \(z\)-axis (Figure 1), Equation (6) cannot be applied directly. On the one hand, the phase factor \(\exp(ik_{z}z)\) in Equation (6) is determined by the sound velocity of the medium, and it should be modified in order to extend it to a multi-layer case. On the other hand, the wave transmissions through each interface must be considered. Theoretically, the transmission coefficient between different layers is a complex function of the incident angle and the acoustic impedances of the materials [29]. In practice, however, the transmission coefficient can be approximated as a constant based on the assumption of a narrow transducer beam [30]. A typical multi-layered structure is schematically illustrated in \[\
abla^{2}p_{u}(x,z,t)-\frac{1}{c^{2}}\frac{\partial^{2}p_{u}(x,z,t)}{\partial t ^{2}}=0 \tag{1}\]0 The sound velocity is different between layers, while it is homogeneous inside each layer. The layers are numbered \(m=1,2,\dots,L\), where \(c_{m}\) and \(d_{m}\) denote the sound velocity and thickness of layer \(m\), respectively.