## 2 Materials and Methods
### LIS Modelling, Data Processing and Analysis
The wavelength \(\lambda\) corresponding to frequency \(f\) is given by  
\[\lambda=c/f. \tag{4}\]  
Due to grain scattering the spectral content of each pulse changes. Higher frequencies are more attenuated due to grain scattering than lower frequencies. This can be seen in Figure 8, where the weighted average frequency decreases for each subsequent echo. Note that this attenuation also heavily depends on the average grain sizes of the material. Figure 7: A cross-section of a recording (**a**) along a line (red colors denote a positive signal and blue a negative); and (**b**) from a single receiver in the center. In general, three grain scattering regimes are considered, depending on the ratio of wavelength and grain size [24; 53]:  
1. The Rayleigh regime, where \(\lambda\gg d\), leading to an attenuation coefficient \(\alpha~{}\sim~{}d^{3}~{}f^{4}\);
2. The stochastic regime, where \(\lambda\approx d\), leading to an attenuation coefficient \(\alpha~{}\sim~{}d~{}f^{2}\); and
3. The diffusion or geometric regime, where \(\lambda\ll d\), leading to an attenuation coefficient \(\alpha~{}\sim~{}d^{-1}\). The attenuation coefficient over all these regions can be generalized to \(\alpha~{}\sim~{}d^{\gamma-1}~{}f^{\gamma}\), covering all three regions and bridging the gaps in between. An illustration of the ratio between grain diameter and frequency is shown in Figure 9.  
Figure 8: (**a**) The windowed echoes in the time domain; and (**b**) their corresponding spectra in the frequency domain. The four colors correspond in both subplots. Figure 9: The Rayleigh and stochastic scattering regimes and the transition regime in between. The box on the upper left is the area of interest, bound by the used frequencies and the modelled grain sizes. For typical velocities of 5750 m/s and frequencies between 5 and 50 MHz, the wave lengths are between 1.2 and 0.12 mm respectively. Compared to the considered grain sizes, 5 to 20 \(\upmu\)m, the wave lengths are (much) larger, indicating that the Rayleigh scattering regime is mostly applicable here, but close to the stochastic regime, with an expected attenuation coefficient \(\alpha~{}\sim f^{\gamma}\) with \(3\leq\gamma\leq 4\). The frequency-dependent attenuation can be obtained in two ways, as described by [PERSON] [21]:  
1. Compare two echoes of a single measurement, where one echo has travelled a longer distance through the medium than the other and hence has undergone more attenuation due to the medium's microstructure;
2. Compare two echoes from two different measurements (both the same echo number), of which one originates from a well-known medium and the other from the medium to be characterized. Assumed is that the media have the same thickness and hence the waves have travelled the same distance. Following the second approach of [PERSON], i.e., comparing echoes from two different measurements, the frequency-dependent attenuation \(\alpha_{n,m}(f)\) of sample \(n\) for echo \(m\) is obtained as  
Figure 4:0  
where \(H\) is the sample thickness and hence \((2m-1)H\) corresponds with the total travelled distance of the signal providing echo \(m\). \(S_{\text{REF},m}\) and \(S_{n,m}\) are the spectra of the reference sample and the sample to be analysed, respectively. Echo 1 is the primary wave, which has travelled only once through the sample. As a result, the influence of scattering and attenuation on this echo is not very pronounced. The second echo has travelled three times through the sample and each next echo adds two more passes compared with the previous one. Due to the limited path length, the first echo is left out in further analysis, and only the second, third and fourth echoes are used, hence \(m\in[2,3,4]\). Echoes 5 and beyond are not considered, since the signal (echo) to noise (scattering) ratio for those echoes decreases too drastically, especially for samples with larger grain sizes.