## 2 Method
### Oblique Incidence Compensation  
The multi-layer algorithm presented above is only applicable for horizontal multi-layered structures as shown in Figure 1. However, in practical phased array inspection, such as weld inspection, a wedge between the array and the object is used to create an oblique beam and obtain excellent coverage. Consequently, a tilt angle is generated between the normal direction of the transducer surface and the \(z\)-axis. The wave-field extrapolation of the multi-layered structures represented by Equation (11) is no longer applicable. Therefore,we propose an oblique incidence compensation method to extrapolate the measured wave field to a virtual measurement line that is parallel to the object's surface. Considering the case of oblique incidence, as shown in Figure 2, a wedge with angle \(\theta\) is added between the linear array and the multi-layered structure. The sound velocity of the wedge is \(c\). A classical Cartesian coordinate system is established using the center of the first array element as the origin. Another coordinate system \(Oxr^{2}\) is obtained by rotating the coordinate system \(Oxz\) by an angle \(\theta\), so that the \(x^{\prime}\)-axis is parallel to the surface of the multi-layered structure. The tilt of the wedge is compensated for by extrapolating the measured wave field from \(z=0\) to the virtual measurement line \(z^{\prime}=0\). The original element position \((x,0)\) is extrapolated to the new position \((x_{new},z_{new})\), which can be expressed in the original system as in [31]:  
\[\left\{ \begin{array}{l}x_{new}=bx\\ z_{new}=ax\end{array} \right.,\text{ where }\left\{ \begin{array}{l}b=\cos \theta\\ a=\sin\theta\end{array} \right.. \tag{12}\]  
Thus, for the oblique incidence situation, the wave field along the virtual measurement line \(z^{\prime}=0\) can be obtained by inserting Equation (12) into Equation (8):  
\[p_{u}(x^{\prime},z^{\prime}=0,t)=\iint\limits_{\infty}P_{u}(k_{x},z=0,\omega) \cdot e^{ik_{z}ax}e^{ik_{z}bx}e^{-i\omega t}dk_{x}d\omega \tag{13}\]  
After combining the \(\exp(ik_{z}ax)\) and \(\exp(ik_{x}bx)\) terms, Equation (13) is rewritten as  
\[p_{u}(x^{\prime},z^{\prime}=0,t)=\iint\limits_{\infty}P_{u}(k_{x},z=0,\omega) \cdot e^{ik_{x}x}e^{-i\omega t}dk_{x}d\omega \tag{14}\]  
where  
\[k_{x}^{\prime}=bk_{x}+ak_{z}=bk_{x}-a\cdot\text{sgn}(\omega)\cdot\sqrt{\omega^ {2}/c^{2}-k_{x}^{2}} \tag{15}\]  
Rearranging Equation (15), we obtain  
\[k_{x}(k_{x}^{\prime},\omega,a)=\frac{1}{a^{2}+b^{2}}\left(bk_{x}^{\prime}+a \sqrt{(a^{2}+b^{2})\cdot\omega^{2}/c^{2}-k_{x}^{\prime}{}^{2}}\right) \tag{16}\]  
With the integrals transformation, replacing \(k_{x}\) with \(k_{x}^{\prime}\), Equation (14) is expressed as  
\[p_{u}(x^{\prime},z^{\prime}=0,t)=\iint\limits_{\infty}P_{u}(k_{x}^{\prime},z^ {\prime}=0,\omega)\cdot e^{ik_{x}^{\prime}x^{\prime}}e^{-i\omega t}dk_{x}^{ \prime}d\omega \tag{17}\]  
Figure 2: Illustration of oblique incidence for a multi-layered structure. where  
\[P_{u}(k^{\prime}_{x},z^{\prime}=0,\omega)=\Phi(k^{\prime}_{x},\omega,a)\cdot P_{u }(k_{x}(k^{\prime}_{x},\omega,a),z=0,\omega) \tag{18}\]  
and  
\[\Phi(k^{\prime}_{x},\omega,a)=\frac{\partial k_{x}}{\partial k^{\prime}_{x}}= \frac{1}{a^{2}+b^{2}}\left(b-\frac{ak^{\prime}_{x}}{\sqrt{(a^{2}+b^{2})\cdot \omega^{2}/c^{2}-{k^{\prime}_{x}}^{2}}}\right) \tag{19}\]  
After the rotation transformation, the wavenumber-domain wave field \(P_{u}(k^{\prime}_{x},z^{\prime}=0,\omega)\) given by Equation (17) can be used directly as the start point for wave-field extrapolation, giving  
\[p_{u}(x^{\prime},z^{\prime},t)=\iint\limits_{\infty}\Phi(k^{\prime}_{x}, \omega,a)\cdot P_{u}(k_{x}(k^{\prime}_{x},\omega,a),z=0,\omega)\cdot e^{ik^{ \prime}_{x}z^{\prime}}e^{ik^{\prime}_{x}x^{\prime}-i\omega t}dk^{\prime}_{x}d\omega \tag{20}\]  
where \(k^{\prime}_{x}\) is the wavenumber along \(z^{\prime}\)-axis. Then, for the multi-layered structures, the wave field at depth \(z^{\prime}\) within layer \(L\) in an oblique incidence system can be extrapolated as:  
\[