## 2 Materials and Methods
### LIS Modelling, Data Processing and Analysis  
The characterization of grain sizes and phase variations can be achieved by analysis of the propagation of high-frequency ultrasonic waves in a LUS setup. Samples of 3.0 \(\times\) 3.0 \(\times\) 2.0 mm (length \(\times\) width \(\times\) height) have been used with a spatial resolution of 3.0 \(\upmu\)m. This sample size is restricted due to practical limitations regarding computer memory use and computation time. For LUS, a relatively large source has been defined, with a flat top and smoothened sides, as shown in Figure 5. The full width at half maximum (FWHM) is 1.5 mm, while the flat top is roughly 0.75 mm in diameter. Note that the size is limited by the sample size, which in turn is limited by computational capabilities. The full bottom of the sample has been recorded throughout the simulation. A receiver is defined with the same size as the source, but on the bottom side of the sample. Figure 4: (**a**) The elastic coefficients \(C_{11}\), \(C_{12}\), and \(C_{44}\) of ferrite plotted as function of temperature between room temperature and 900 \({}^{\circ}\)C as a solid line. An extrapolation down to the absolute minimum temperature is shown as a dotted line. Values as given by [PERSON] [41] are shown as crosses; (**b**) the isotropy \(\mathcal{Z}\) as derived from the temperature-dependent elasticity coefficients; and (**c**) the temperature-dependent densities of ferrite, martensite and of austenite [51; 52]. The excitation signal is a single pulse with a center frequency of 25 MHz and a bandwidth of 64 MHz (at \(-20\) dB), as shown in Figure 6, inducing a vertical compressional wave. The design of the spot size was optimized for grain size measurement, for texture measurement a much smaller spot size should be used. The signal was recorded at the bottom of the sample for a duration long enough to capture the primary wave and five or six passes through the thickness of the sample. Since the source was relatively large, the wave travelling up and down in the sample was regarded as a plane wave. Two cross-sections of a recording are shown in Figure 7, respectively, a recorder line at the bottom at \(y=0\) mm, and a single recorder point at \((x,y)=(0,0)\) mm. The receiver accumulates the signal from the bottom surface with radius \(r=2.1\) mm. This leads to spatial averaging of the grain scatter noise, while maintaining the shape of the plane wave. Figure 5: (**a**) The source profile in 1D; and (**b**) the source at the top the sample in 2D. Figure 6: (**a**) The excitation signal (**a**) in the time domain; and (**b**) in the frequency domain. The dashed line denotes the bandwidth at \(-20\) dB. The recorded signal shows a series of pulses. Those pulses are the primary compressional wave and internal reflections of the compressional wave. The smaller events in between the main pulses are mode conversions and noise due to grain scattering. As can be seen, the amplitude of the pulses decreases over time due to the mode conversion, energy spreading horizontally in the plate and grain scattering. [PERSON] mentions three main contributors to the attenuation of an ultrasound pulse in a medium with a microstructure [21]:  
1. Grain scattering, leading to wave energy dispersion due to mismatches in elasticity in neighboring grains (crystallographic orientation differences due to texture);
2. Diffraction of the ultrasonic wave in the medium, affecting both the frequency content as well as the amplitude;
3. Absorption of wave energy, which is assumed to be of minor importance to frequency-dependent attenuation. The first two aspects were incorporated in the finite-difference simulations; the third was not, but according to [PERSON], this would have had a negligible effect on the results. The wave velocity \(c\) can be determined by measuring the delay \(\Delta t\) between the arrival of pulses and the thickness of the sample \(H\), as  
\[c=2H/\Delta t. \tag{3}\]  
The factor 2 is due to the fact that the wave travels up and down through the sample in between two pulses.