# Deep Learning Based Inversion of Locally Anisotropic Weld Properties from Ultrasonic Array Data
## 2 Methods  
### Weld Models  
In this paper, a 90\({}^{\circ}\) V-groove austenitic stainless steel welded constructed from 21 weld passes and joining two 15 mm thick steel plates, is considered (see Figure 1a). The parent steel plates are taken to be isotropic with longitudinal wave velocity \(V_{L}=5790\) m/s. Since austenitic stainless steel is anisotropic (that is the wave speed of probing ultrasonic waves is dependent on the incident wave direction), reconstruction of velocity fields through the structure (as is commonly the objective in seismic tomography [15; 16]) does not sufficiently capture the complexity of the underlying weld microstructure. Instead, a map of the orientations of the large grains which form during the welding process is required. To this end, knowledge of the longitudinal group velocity curve of the medium is required (note we are examining travel-time tomography and so consider only the longitudinal (fastest) wave modes). The group velocity curves for a transversely isotropic, cubic, austenitic weld are obtained by solving the Christoffel Equation [31] using a cubic stiffness tensor where \(c_{11}=203.6\) GPa, \(c_{12}=133.5\) GPa and \(c_{44}=129.8\) GPa, and the density is \(\rho=7874\) kg/m\({}^{3}\) (these values are taken from the weld properties used in [12; 32]). The resulting relationship between the incident wave direction and longitudinal group velocity is plotted in Figure 1b. Note that in this paper we assume a priori knowledge of the elastic stiffness constants, however in practice these would need to be approximated experimentally and the uncertainty associated with these approximations considered [33; 34].  
There are several models that describe how microstructural properties vary throughout a weld depending on various welding parameters [13; 29; 33; 35]. When estimating the spatially varying microstructural descriptors such as grain orientation using ultrasound data, such models can be used to constrain the inverse problem. Here the Ogilvy model [29] for a V-shaped weld is employed. With this approach, the anisotropic grain orientations \(\theta(x,y)\) at each point \((x,y)\) in the domain are calculated by:  
Figure 1: (**a**) Schematic of the considered weld geometry, a 90\({}^{\circ}\) V-groove austenitic stainless steel welded constructed from 21 weld passes; (**b**) the group velocity curve used to model austenitic stainless steel using a cubic stiffness tensor where \(c_{11}=203.6\) GPa, \(c_{12}=133.5\) GPa and \(c_{44}=129.8\) GPa and the density is \(\rho=7874\) kg/m\({}^{3}\) (from [12; 32]).  
\[\theta(x,y)=\tan^{-1}\left(\frac{W_{1}/2+(y-Z)\tan\alpha}{x}\right)+90^{\circ}, \tag{1}\]  
where \(Z\) is the weld depth, \(W_{1}\) is the width of the weld root and \(\alpha\) is the grain orientation at the fusion face. Grain orientations \(\theta\) for the weld geometry shown in Figure 1a, further discretised into 91 sub-regions of size 2 mm\({}^{2}\), where \(Z=15\) mm, \(W_{1}=1\) mm and \(\alpha=23^{\circ}\), are shown in Figure 2a. In this work, this model will be used as a background reference model for creating the suite of models generated for the training of the DNN. Ten thousand training models are generated, each consisting of random perturbations of up to \(\pm 10^{\circ}\) of each grain orientation in this background model. Some examples of these randomly perturbed weld maps are shown in Figure 2b. Of the full set of models, 8000 models are used for model training and 2000 models are used for validation purposes.