## 3 Method
### Online Phase  
The compressed FRFs and transmissibility functions can now be used to rapidly obtain the response at any plausible sensor location within a plate. For simplicity, in a first step, a scenario will be investigated where only a single wave mode is propagating, e.g., the \(S_{0}\)-mode, in an infinite plate with two features \(F_{1}\) and \(F_{2}\) and a single sensor (Figure 5). Therefore, three points of interest exist to which a wave packet, excited at an actuator, will travel: the two feature and the sensor locations. We first investigate the wave from the actuator to the sensor. The response of the ray, representing this propagating wave packet, can be evaluated through the FRF determined in the offline part. Further, the two rays pointing from the actuator to the features need to be investigated, as these initiate reflections at the features. When further tracking the ray, which points to feature \(F_{1}\), reflections to two points of interest occur (dashed rays in Figure 5), where the first ray points to the sensor. To compute the sensor response of this path, three steps are necessary: (1) compute the FRF response of the ray pointing from the actuator to the feature \(F_{1}\); (2) evaluate the transmissibility function for the ray pointing from the feature \(F_{1}\) to the sensor; and (3) multiply this evaluated transmissibility function with evaluate FRF from step (1). The further reflection from the feature \(F_{1}\) points to the feature \(F_{2}\), where reflections again occur at two points of interest: the sensor location and once again back in the feature \(F_{1}\) position (dotted in Figure 5). The response at the sensor location can be evaluated by multiplying the FRF and the respective transmissibility function belonging to the respective ray on this path. This ray tracing can be understood as a tree, which is displayed in Figure 5b. In theory, the branches of this tree can be tracked to infinity; however, the amplitude reduces with every reflection and the cumulative length of relevant rays. Moreover, propagation across longer distances requires more time for the wave to arrive at the sensor, which is eventually longer than the investigated time period. Therefore, from a practical viewpoint, it is sufficient to investigate a finite number of reflections.  
Next, this concept should be generalized to a plate, where several modes occur. Therefore, a plate of infinite size comprising one actuator, one feature, and one sensor will be investigated, where the frequency of the excitation is chosen so that only the two fundamental modes (\(S_{0}\) and \(A_{0}\)) propagate, as displayed in Figure 6. As two modes might be excited, for each point of interest, two rays need to be investigated, one for each mode. The two possible modes entail the possibility of mode conversion at the feature. This leads to a total of four wave packets propagating from the feature to the sensor. These wave packets pertain to the reflected \(S_{0}\) and mode-converted \(A_{0}\) modes, stemming from the arriving \(S_{0}\) mode, as well as the reflected \(A_{0}\) and mode-converted \(S_{0}\) modes, stemming from the arriving \(A_{0}\) mode. At the sensor location, this leads to a total of six incoming wave packets, which need to be summed. These wave packets can be determined through the evaluated FRFs for the direct waves and the multiplication of the FRFs for the waves propagating to the feature, with the corresponding transmissibility functions for the reflected and mode-converted waves.  
Figure 5: An infinite plate comprising two features (\(F_{1}\) and \(F_{2}\), e.g., defects or rivet holes) and one sensor. Possible travel paths up to third order (meaning paths with up to three rays). Solid: first-order rays. Dashed: second-order rays. Dotted: third-order rays. (**a**) Possible travel paths. (**b**) Tree corresponding to the rays in (**a**).