## 5 Empirical Model
### The Statistical Reliability of the Results
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c} \hline \hline
**No.** & _K1_ & _K2_ & _K3_ & _K4_ & _K5_ & _K6_ & _K7_ & _K8_ & _K9_ & _K10_ & _K11_ & _K12_ & _K13_ & _K14_ & _K15_ & **Avx.** \\ \hline
**S** & 0.21 & 0.16 & 0.19 & 0.21 & 0.08 & 0.14 & 0.06 & 0.09 & 0.11 & 0.09 & 0.13 & 0.10 & 0.10 & 0.08 & 0.10 & 0.12 \\
**I** & 0.64 & 0.48 & 0.59 & 0.64 & 0.24 & 0.42 & 0.19 & 0.23 & 0.34 & 0.26 & 0.37 & 0.29 & 0.29 & 0.24 & 0.28 & 0.37 \\
**W** & 0.07 & 0.03 & 0.05 & 0.08 & 0.01 & 0.03 & 0.01 & 0.01 & 0.01 & 0.01 & 0.04 & 0.01 & 0.01 & 0.01 & 0.01 & 0.03 \\
**M** & 4.90 & 2.70 & 5.30 & 5.94 & 3.39 & 4.03 & 2.33 & 2.72 & 5.14 & 2.85 & 6.48 & 4.11 & 4.3 & 4.43 & 5.24 & - \\
**SE** & 0.07 & 0.05 & 0.06 & 0.07 & 0.03 & 0.04 & 0.02 & 0.03 & 0.04 & 0.03 & 0.04 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 & 0.04 \\
**AE** & 0.17 & 0.12 & 0.15 & 0.11 & 0.07 & 0.11 & 0.05 & 0.07 & 0.09 & 0.07 & 0.1 & 0.09 & 0.08 & 0.07 & 0.08 & 0.09 \\
**RE** & 3.55 & 4.41 & 2.61 & 2.64 & 1.53 & 2.66 & 1.42 & 3.67 & 1.94 & 2.00 & 1.31 & 2.23 & 1.46 & 1.46 & 1.44 & 2.29 \\ \hline \hline \end{tabular} Notes: S—standard deviation, I—interval, W—variation, M—median, SE—standard error of the mean value, AE—absolute measurement error, RE—relative measurement error [%], and Avxr.—average value of each statistical parameter referred to all specimens.  
\end{table} 
Table 4: Averaged statistical parameters calculated for measurements of the damping ratio (\(D\)) of all tested samples. The analysis of the impact of the plasticity index on the damping ratio allowed to determine the threshold value of \(PI=20\%\), which indicates the change of the \(D_{MIN}\) trend from decreasing to increasing value (Section 4.3. and Figure 9). Therefore, the tested soils were divided into two groups, namely, low and medium cohesive soil with a \(PI\) below 20% and very cohesive soil with a \(PI\) above 20%. In Table 5, the correlation matrix for tested specimens from both groups and the values of the Pearson correlation coefficient are shown. It can be seen that the most significant influence on the damping ratio in both groups of soil has normalized the shear modulus (\(G/G_{MAX}\)) and shear strain (\(\gamma\)). As was recognized in Section 2.2, \(G/G_{MAX}\) is strictly related to shear strain and can replace \(\gamma\) in an empirical damping model, which is rather hard to determine in situ. In the group of low and medium cohesive soil, another parameter with a high value of \(R\) is the plasticity index. It is worth noting that the sign of the correlation coefficient for the \(PI\) is different for both groups. That means, as mentioned above, a change in the dependence of the plasticity index on the damping ratio. Additionally, significant influence on \(D\) has effective stress (\(p^{\prime}\)). However, \(R\) for \(p^{\prime}\) is higher for very cohesive soils than for low and medium cohesive soils. Moreover, \(p^{\prime}\) in soils with \(PI>20\%\) has an absolute value of \(R\) higher by \(0.11\) than \(PI\). Besides the mentioned parameters, the void ratio in soils with \(PI<20\%\) has enough of a high correlation coefficient to be used in an empirical model. Nevertheless, the authors specially selected the three same parameters to build a universal model core for both groups. To create a correct and reliable empirical model, the authors analyzed every important factor affecting damping properties. Due to the nonlinear characteristic of the damping ratio, the authors eliminated from the analysis linear regression. Moreover, the investigation of Figures 6 and 8 provided the basic function regarding the dependence of damping on deformation (or \(G/G_{MAX}\)), i.e., quadratic function. Subsequently, after investigating Table 5, \(PI\) was selected as a parameter which describes physical soil properties in the created model. The examination of Figure 9 provided two opposite trends, increasing and decreasing \(D_{MIN}\) with the increase of \(PI\); thus, two linear functions had to describe this phenomenon.