## 5 System Invariant Method
### Principle of System Invariant Method  
Weldment flaws signal \(V_{f}\) include the system, flaw scattering pattern, and specimen as variables. Among these, systems and specimens increase the complexity of the database, which can inhibit performance in classifying neural network flaws, so the database must be refined by extracting only the flaw scattering patterns. The schematic experiment setup is shown in Figure 5.  
Because the ultrasound system is linear time-invariant (LTI), \(V_{f}\) can be expressed in a time domain with convolution between system variables, as in Equation (5) [33]:  
\[V_{f}(t)=\beta(t)*C_{1}(t)*T_{12}(t)*C_{2}(t)*A(t)*C_{2}(t)*T_{21}(t)*C_{1}(t) \tag{5}\]  
where \(\beta\) is system factor, \(C_{1}\) is the diffraction correction in the wedge, \(T_{12}\) is the transmission coefficient of the wedge to the specimen, \(C_{2}\) is the diffraction correction in the specimen, \(A\) is the scattering pattern from the flaw, and \(T_{21}\) is the transmission coefficient of the specimen to the wedge. To eliminate system and specimen parameters, signals were acquired from a reference specimen using the same system used for the weldment signal acquisition. A reference signal was acquired using a circular reference block to obtain the same ultrasound path for every inspection angle for each transducer-system combination. Here, 21 reference signals  
Figure 5: Schematics of the experimental setup to obtain a flaw signal in the weldment. by 21 different types of transducers were obtained using the same ultrasonic pulser-receiver device. The schematic experimental setup for the reference signal is shown in Figure 6.  
Furthermore, the reference signal \(V_{R}\) can be described in a time domain as a convolution between system factors, as in Equation (6):  
\[V_{R}(t)=\beta(t)*C_{1}(t)*T_{12}(t)*C^{\prime}{}_{2}(t)*R(t)*C^{\prime}{}_{2}(t )*T_{21}(t)*C_{1}(t) \tag{6}\]  
where \(C^{\prime}{}_{2}\) is diffraction correction in the reference specimen and \(R\) is the reflection coefficient from the round surface. In the frequency domain, the signals \(V_{f}\) and \(V_{R}\) can be combined by simple multiplication, not convolution [34; 35]. Using fast Fourier transforms (FFT), \(V_{f}\) and \(V_{R}\) are described in the frequency domain as Equations (7) and (8):  
\[V_{f}(\omega)=\beta(\omega)C_{1}(\omega)T_{12}(\omega)C_{2}(\omega)A(\omega)C _{2}(\omega)T_{21}(\omega)C_{1}(\omega) \tag{7}\]  
\[V_{R}(\omega)=\beta(\omega)C_{1}(\omega)T_{12}(\omega)C^{\prime}{}_{2}(\omega )R(\omega)C^{\prime}{}_{2}(\omega)T_{21}(\omega)C_{1}(\omega) \tag{8}\]  
Divide these two terms, flaw scattering pattern \(A\) can be extracted as Equation (9):  
\[\frac{V_{f}(\omega)}{V_{R}(\omega)}=\bigg{(}\frac{C_{2}(\omega)}{C^{\prime}{}_ {2}(\omega)}\bigg{)}^{2}\frac{A(\omega)}{R(\omega)} \tag{9}\]  
The reflection \(R\) is affected only by the mechanical property of the two mediums and the incidence angle. Here, the round surface of the reference block fixes the incidence angle as normal when the transducer angle varies. Therefore, \(R_{i}\) is constant, while i represents each transducer with a different angle. Furthermore, the reflection coefficient for steel to air is significantly close to \(-1\) in this case. This means that Equation (9) is described as Equation (10):  
\[\frac{V_{f}(\omega)}{V_{R}(\omega)}\cong-\bigg{(}\frac{C_{2}(\omega)}{C^{ \prime}{}_{2}(\omega)}\bigg{)}^{2}A(\omega) \tag{10}\]  
Diffraction correction \(C(\omega)\) in far-field is represented as Equation (11):  
\[C(\omega)=\frac{-ika^{2}}{2z} \tag{11}\]  
\[V_{f}(t)=\beta(t)*C_{1}(t)*T_{12}(t)*C_{2}(t)*A(t)*C_{2}(t)*T_{21}(t)*C_{1}(t) \tag{5}\]0 Schematics of experiment setup to obtain reference signal. where \(k\) is the wavenumber, \(a\) is the radius of the transducer and \(z\) is the flight distance. \(C_{2}\) and \(C^{\prime}{}_{2}\) feature the same frequency and radius as the transducer, except for distance.