# Anisotropy Corrected FMC/TFM Based Phased Array Ultrasonic Imaging in an Austenitic Buttering Layer
## 2 FMC/TFM with Anisotropic Velocity Correction  
In the FMC-TFM mode, every element in the transducer is fired sequentially, and the individual signals received by all elements (limited by the maximum number of active channels in the instrument) are processed using the TFM algorithm to obtain a focused image in the ROI. The ROI in the inspection medium is divided into grids (pixels), as shown in Figure 1. During TFM processing, the time delays between every transmission-reception pair are calculated for each pixel in the ROI. The time delay corresponding to a pixel is the sum of the time taken for the ultrasonic beam to reach the pixel location (\(x_{p}\), \(y_{p}\)) from the transmitter location (\(x_{t}\), \(y_{t}\)) and from the pixel location to the receiver location (\(x_{r}\), \(y_{r}\)). Once the time delays are calculated, amplitude values corresponding to each transmission-reception pair are obtained by accessing the corresponding individual A-scan signals. In an isotropic material, ultrasonic velocity is constant in all directions. Thus, it is a straightforward method to implement the delay and sum rule for different transmission and reception pairs for the reconstruction of the FMC data to generate a TFM image. However, in the case of an anisotropic material, ultrasonic velocity is direction dependent, and each transmission and reception pair requires individual velocity values to estimate the time delay. Time delay in an anisotropic medium is given by:  
\[Time\ delay=\frac{\sqrt{\left(x_{t}-x_{p}\right)^{2}+\left(y_{t}-y_{p}\right)^ {2}}}{V(\theta_{1})}+\frac{\sqrt{\left(x_{p}-x_{r}\right)^{2}+\left(y_{p}-y_{ r}\right)^{2}}}{V(\theta_{2})} \tag{1}\]  
where \(t\), \(p\), and \(r\) represent the transmitter, pixel, and receiver, respectively. \(V(\theta_{1})\) and \(V(\theta_{2})\) are the ultrasonic velocities corresponding to beam directions \(\theta_{1}\) and \(\theta_{2}\) with respect to the surface normal (Figure 1). In the case of an isotropic material, \(V(\theta_{1})=V(\theta_{2})=V\). The angle (\(\theta\)) varies in a discrete manner with a very small increment. Increment in \(\theta\) is governed by the size of the pixel in the ROI.  
Figure 1: Schematic of TFM methodology.  
Various approaches have been proposed in the literature to obtain ultrasonic velocities as a function of \(\theta\). These include performing measurements in specimens extracted at different angles [27] and performing velocity measurements at different angles by orienting the anisotropic specimen at different angles with respect to the beam directions [28]. Angle-dependent ultrasonic velocities can also be measured by acquiring FMC data in through transmission (TT) mode by using two identical PA probes placed on opposite sides of the specimen, as shown in Figure 2. Different transmitter-receiver combinations correspond to different angles, and by knowing the pitch of the transducers and the thickness of the specimen, distance \(y\) and angle \(\theta\) can be obtained. The velocity values at different angles can then be fitted to a polynomial equation, and the equation can be used to obtain \(V(\theta_{1})\) and \(V(\theta_{2})\), as required in Equation (1).