# A Computational Platform for Automatic Signal Processing for Bender Element Sensors +
## 4 Validation Using the International Parallel Test Benchmarks
### Statistical Analysis of the Results
This test is conducted on pairs of distributions, resulting in multiple ANOVA values calculated on all possible combinations the outcomes, as depicted in Figure 21.  
The ANOVA tests yield consistent conclusions for both void ratios. The means of the cross-spectrum and first sharp methods are statistically dissimilar to all other interpretation techniques. GeoHyTE is statistically _not_ dissimilar to the first bump technique and, in the \(e=0.690\) case, to the cross-correlation method. The pairs (cross-correlation + first bump) and (zero crossing + peak-to-peak) are statistically very similar in the \(e=0.690\) case. In the \(e=0.770\) case, cross-correlation, peak-to-peak, and zero crossing methods are also very similar. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Void Ratio**} & \multirow{2}{*}{**Test No**} & \multicolumn{6}{c}{**\(G_{0}\) [MPa]**} \\ \cline{3-8}  & & **Cross-Corr.** & **Peak-to-Peak** & **GeoHyTE** & **First Sharp** & **First Bump** & **Zero Crossing** & **Cross-Spectrum** \\ \hline \multirow{8}{*}{0.77} & 1 & 36.17 & 35.50 & 40.10 & 48.68 & 40.53 & 35.80 & 34.79 \\ \cline{2-8}  & 2 & 35.74 & 35.20 & 39.20 & 44.12 & 38.23 & 36.79 & 25.44 \\ \cline{2-8}  & 3 & 36.17 & 35.50 & 42.00 & 47.82 & 39.88 & 35.97 & 32.42 \\ \cline{2-8}  & 4 & 36.11 & 35.50 & 41.00 & 47.82 & 39.88 & 35.97 & 31.75 \\ \cline{2-8}  & 5 & 34.83 & 34.14 & 38.90 & 43.87 & 36.78 & 33.64 & 24.24 \\ \cline{2-8}  & 6 & 35.93 & 35.32 & 39.30 & 45.39 & 38.43 & 35.98 & 26.99 \\ \cline{2-8}  & 7 & 36.54 & 35.62 & 40.20 & 44.62 & 38.42 & 37.05 & 27.35 \\ \cline{2-8}  & 8 & 36.23 & 35.93 & 39.80 & 43.95 & 39.04 & 37.11 & 28.61 \\ \hline Mean & 35.97 & 35.34 & 40.06 & 45.79 & 38.90 & 36.04 & 28.95 \\ \hline Std. deviation & 0.5131 & 0.5282 & 1.028 & 1.999 & 1.197 & 1.106 & 3.685 \\ \hline Std. error of the mean & 0.1814 & 0.1868 & 0.3635 & 0.7067 & 0.4232 & 0.3912 & 1.303 \\ \hline LL 95\% CI & 35.54 & 34.90 & 39.20 & 44.11 & 37.90 & 35.12 & 25.87 \\ \hline UL 95\% CI & 36.39 & 35.78 & 40.92 & 47.46 & 39.90 & 36.97 & 32.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Shear modulus values and statistical data for the tests performed on the sample with \(e=0.770\)[19]. The correlation test assesses whether the readings obtained from one interpretation method exhibit a linear relationship with those obtained from another interpretation method. Unlike the ANOVA test, which provides information on the _absolute_ magnitude of shear modulus values, the correlation test focuses on their _relative_ magnitudes. A higher correlation indicates that the shear modulus values from the two methods are more closely aligned. Figure 22 presents the correlation values between different interpretation methods. Figure 21: ANOVA tests of the shear modulus distributions. Figure 22: Correlations of the shear modulus distributions [19]. Overall, there is strong correlation between all pairs of interpretation techniques, except for the cross-spectrum method. The shear modulus values obtained with the cross-spectrum method are notably smaller than those obtained with other methods and show significant variance. This trend has been noted by previous researchers, for instance, in references [6; 12]. Finally, a comparison is made between the average values of the shear moduli obtained with each interpretation technique and those reported in the International Parallel Test. The results are presented in Figure 23.  
The shear moduli reported here generally align well with the bulk of readings documented in the International Parallel Test, except for those obtained with the cross-spectrum method. Predictions from GeoHyTE appear slightly more favourable compared to mainstream techniques like cross-correlation and peak-to-peak distance. However, this improvement is not statistically significant, as indicated by the ANOVA test, so this finding should not be generalised.