## 2 Fundamental Theory
### Features for Fatigue Crack Characterization
#### 2.2.2 Energy-Based DIs  
The discrepancy in fatigue cracks (different distances to an actuator-sensor path, severities, and orientations [25]) can result in a distinct damage-scattered magnitude of ultrasonic guided waves. Therefore, the deviations of wave energy, which depend on the ultrasonic guided waves signals captured from the target structure (called damage or current signal) and its healthy counterpart acquired from a pristine benchmark (denoted baseline signal), can be employed to develop a DI. In consideration of the correlations between the current and baseline signals, as follows:  
\[D(i,j)_{energy\_k\_t}=\frac{\int_{I1}^{I2}\left[u(i,j)_{B\_k}(t)-u(i,j)_{D\_k}( t)\right]^{2}dt}{\int_{I1}^{I2}u(i,j)_{B\  
unwanted signal features such as ambient noise and measurement uncertainties, multiple guided waves modes conversion, and reflections from structural boundaries. These unwanted signal features possibly dilute crack-associated features and weaken the proportion from a single actuator-sensor path [25]. Therefore, based on the DI weights of different paths, the signal representations of the fused DIs are embodied by Equations (6)-(9). The ultimate fitting result is a straight line, which can be used to indicate the fatigue crack propagation effectively.  
To summarize the above, each actuator-sensor path (N in total) offers an original DI via a pitch-catch configuration. Then, a hybrid fusion scheme is proposed to coalesce all the available DIs of each path in the sensor network, as follows:  
\[\text{DI(i,j)}_{\text{magnitude-fusion-t}}=\sum_{\text{k=1}}^{\text{N}}\bigl{(} \alpha_{\text{magnitude-k-t}}\text{DI(i,j)}_{\text{magnitude-k-t}}\bigr{)} \tag{6}\]  
\[\text{DI(i,j)}_{\text{amplitude-fusion-f}}=\sum_{\text{k=1}}^{\text{N}}\bigl{(} \alpha_{\text{amplitude-k-f}}\text{DI(i,j)}_{\text{amplitude-k-f}}\bigr{)} \tag{7}\]  
\[\text{DI(i,j)}_{\text{energy-fusion-t}}=\sum_{\text{k=1}}^{\text{N}}\bigl{(} \alpha_{\text{energy-k-t}}\text{DI(i,j)}_{\text{energy-k-t}}\bigr{)} \tag{8}\]  
\[\text{DI(i,j)}_{\text{energy-fusion-f}}=\sum_{\text{k=1}}^{\text{N}}\bigl{(} \alpha_{\text{energy-k-t}}\text{DI(i,j)}_{\text{energy-k-f}}\bigr{)} \tag{9}\]  
where \(\text{DI(i,j)}_{\text{magnitude-fusion-t}}\) and \(\text{DI(i,j)}_{\text{energy-fusion-t}}\) are the ultimate fusion DIs in the time domain, meanwhile, \(\text{DI(i,j)}_{\text{amplitude-fusion-f}}\) and \(\text{DI(i,j)}_{\text{energy-fusion-f}}\) are the synthetic DIs in the frequency domain. The k stands for the kth actuator-sensor path, while \(\alpha\) refers to the weights of each path. The incentive to build such a hybrid fusion scheme is twofold:  
(i) An arithmetic fusion equally takes into account prior perceptions from each actuator-sensor path and well decentralizes individual contributions. Although the arithmetic fusion fully encompasses previous perceptions from all paths, information including ambient noise and measurement uncertainties are also engaged, which might overstate the presence of fatigue cracks and lead to false alarms, and  
(ii) The synthesized fusion DI method is obtained by implementing a probability weighting operation on the original DIs of each actuator-sensor path. A hybrid DI scheme is constructed to enhance the approach tolerance for environmental noise, measurement uncertainties, and erroneous perceptions from individual paths.  
The hybrid fusion scheme can be used to represent the fatigue crack propagation quantitatively, which effectively strengthens the individual original DIs of crack-related information and enhances the availability, stability, and signal-to-noise ratio of the fitting signals.