## 3 Method
### Offline Phase
This means that the response at a sensor location for a discrete frequency and a certain mode can be described as follows:  
\[u(\omega)=\frac{A}{\sqrt{d}}\cdot e^{i\left(\frac{d}{c(\omega)}\omega+\theta \right)} \tag{14}\]  
with the unknown parameters being the amplitude \(A\) and phase shift \(\theta\). The frequency of the periodic term with respect to the distance \(\omega_{dist}=\frac{\omega}{c(\omega)}\) can be evaluated, using the dispersion characteristics, as displayed in Figure 1. To avoid confusion, it should be emphasized at this point that \(\omega_{dist}\) is not a frequency with respect to time, but a spatial frequency with respect to the distance from the origin of the wave. In what follows, it is assumed that the sensor response along a radial line from a source can be adequately described by Equation (14). Next, the amplitude \(A\) and phase shift \(\theta\) need to be determined. With the amplitude \(A\) and the distance to the source \(d\) being strictly positive, the magnitude of the sensor response in Equation (14) becomes  
\[|u(\omega)|=\frac{A}{\sqrt{d}}\cdot|e^{i\left(\frac{d}{c(\omega)}\omega+ \theta\right)}|=\frac{A}{\sqrt{d}} \tag{15}\]  
And, therefore, as \(|e^{i\left(\frac{d}{c(\omega)}\omega+\theta\right)}|=1\),  
\[A=|u(\omega)|\cdot\sqrt{d} \tag{16}\]  
Figure 4: A line of sensors in increasing distance from the defect (feature). The first sensor needs to be placed at a certain distance to the defect, as otherwise evanescent waves will occur, which distort the sensor response. The former implies that the amplitude could be extracted from a single sensor along the investigated radial line. However, within a practical setting, it is preferable to evaluate this for several virtual sensors and evaluate the mean so as to account for the fact that the size of the feature is small, but not 0. The only parameter that remains unknown is the phase \(\theta\in[0,2\pi]\), which can be inversely evaluated through use of an optimization scheme needing only a few sensors. For Section 4 of this paper, a Particle Swarm Optimization (PSO) employing eight virtual sensors showed satisfying results. As Equation (14) is expected to approximate the response at a sensor location for a discrete frequency well, the PSO determines \(\theta\) with good approximation. At this point, it is possible to describe the response at various sensor locations along a line by only three parameters: the spatial frequency (\(\omega_{dist}\)), the amplitude (\(A\)), and the phase shift (\(\theta\)). This allows for, firstly, significantly reducing the amount of stored training data and, secondly, accurately extrapolating beyond the distance of the last sensor used for training. As the response along a radial line due to a specific excitation can be accurately described by Equation (14), it is of interest to obtain the response due to an arbitrary excitation. To achieve this, the concepts of Frequency Response Functions (FRFs) and transmissibility functions show promising results. These concepts, which are introduced in Section 2.3 and described by Equations (6) and (12), respectively, allow for evaluating the response in a sensor for inputs with different time-dependence. As stated above, the response for the sensors in Figure 4 due to the wave packets stemming from the feature is found through baseline subtraction. Following Equation (11), the transmissibility function for each sensor with respect to the baseline can be numerically evaluated as follows:  
\[\mathbf{T}^{bl}(\omega_{i})=\frac{\mathbf{u}_{u}^{bl}(\omega_{i})}{u_{k}^{bl}(\omega_ {i})} \tag{17}\]  
where the superscript \((\cdot)^{bl}\) denotes the baseline, \(\mathbf{u}_{u}^{bl}(\omega_{i})\) describes the frequency response at the sensors, and \(u_{k}^{bl}(\omega_{i})\) the frequency response in the feature location.