# Deep Learning Based Inversion of Locally Anisotropic Weld Properties from Ultrasonic Array Data
## 3 Results
#### TFM Reconstruction Results  
As shown in [3; 4], estimations of the grain maps can be used to improve the ability to detect, locate and characterise any defects embedded in the weld. The total focusing method (TFM) is a reference imaging algorithm for detecting and characterising defects in the NDE community [1]. It exploits the full matrix capture (FMC) dataset, which is the set of all signals (or A-scans) generated by transmitting on each ultrasonic transmitting transducer element \(s_{i}\) in turn, where \(i=1,..,L\) and \(L\) is the number of transmitters, and recording at each receiving transducer element \(r_{j}\) simultaneously, where \(j=1,...,M\) and \(M\) is the number of receivers (note \(M=L\) in this work), over time \(t\). The amplitude of a recorded A-scan at time \(t\) is therefore denoted as \(A_{ij}(t)\). In the TFM algorithm, the A-scans are summed in order to synthesise a focus at every point \((x,y)\) in the imaging domain [1; 30]. The travel times from each transmitting element \(s_{i}\) and each receiving element \(r_{j}\), to every imaging point \((x,y)\) are estimated, representing the wave's journey between \(s_{i}\) and \(r_{j}\) via a point scatterer at \((x,y)\). The time this journey takes can then be related to an amplitude on the relevant A-scan \(A_{ij}\). The intensity of each pixel \(I(x,y)\) in the TFM image is calculated by summing these amplitudes over all transmitter-receiver pairs:  
Figure 8: Comparison of absolute error (in degrees\({}^{\circ}\)) between predicted and true grain orientations for Test Model 1 (as in Figure 6) using the pulse-echo transducer array configuration. The error distributions are shown for case of the inverse crime, with AMSFMM generated data (**left**), and for FEA generated data (**right**). Figure 7: Mean absolute error (MAE) for each weld sub-region of the predicted grain orientations (tomographic reconstructions) averaged over 100 test models for (**a**) the through-transmission, (**b**) offset through-transmission, and (**c**) pulse-echo transducer array configurations. \[I(x,y)=\left|\sum_{i=1}^{L}\sum_{j=1}^{M}\mathcal{H}(A_{ij}(\tau^{s_{i}}(x,y)+ \tau^{r_{j}}(x,y)))\right| \tag{5}\]  
where \(\mathcal{H}(\cdot)\) denotes the Hilbert transform, \(\tau^{s_{i}}(x,y)\) is the travel-time between the transmitter \(s_{i}\) and pixel \((x,y)\) and \(\tau^{r_{j}}(x,y)\) is the travel time between the pixel \((x,y)\) and the receiver \(r_{j}\). In the case that a constant wave speed is assumed throughout the domain, \(\tau^{s_{i}}(x,y)\) and \(\tau^{r_{j}}(x,y)\) are the calculated travel times along straight rays between the transmitters/receivers and the point \((x,y)\). Alternatively, to account for spatial variations of the locally anisotropic material microstructure, and improve focussing capabilities in post processing, \(\tau^{s_{i}}(x,y)\) and \(\tau^{r_{j}}(x,y)\) can be extracted from travel time maps generated by the AMSFMM (which accounts for the varying wave speeds, anisotropy and the subsequent refraction). This approach has previously been referred to as TFM+ [4; 5]. To test the success of our DNN reconstructed maps in the TFM+ framework, the Onscale FE package is used to simulate an FMC dataset arising from the 2 MHz (corresponding to a wavelength approximately of 2.5 mm) pulse-echo inspection of a weld containing three embedded circular voids, each with a 2 mm diameter (labelled A, B and C in Figure 9a, left image), and where the model parameters are as described in Section 2.3. The microstructure within the weld is parameterised using the sub-region grain orientations of Test Model 1 (shown in Figure 6). The TFM algorithm (Equation (5)), in which a constant isotropic wave speed is assumed throughout the domain (that of the parent material, where \(V_{L}=5790\) m/s), was then applied to this generated FMC dataset. The resulting TFM image is shown in Figure 9b (left image) and exhibits peaks of image intensity \(I\) associated with the locations of defects A and B.