## 5 Empirical Model
### The Statistical Reliability of the Proposed Model  
A comparison of the results from the present work with those previously reported for cohesive soils was conducted to present the statistical reliability of the proposed model. The authors chose four well-known equations for damping ratio in wide shear strain range, suggested by the following:  
[PERSON] and [PERSON] [44]:  
\[D=\left(\frac{0.333\Big{(}1+e^{-0.0145I_{P}^{1.3}}\Big{)}}{2}\right)\ast\left[ 0.586\left(\frac{G}{G_{MAX}}\right)^{2}-1.547\left(\frac{G}{G_{MAX}}\right)+1 \right]; \tag{6}\]  
[PERSON] and [PERSON] [48]:  
\[D=17.83\left[0.56\left(\frac{G}{G_{MAX}}\right)^{2}-1.39\left(\frac{G}{G_{MAX} }\right)+1\right]; \tag{7}\]  
[PERSON] et al. [47]:  
\[D=2+\left[18-0.08(I_{P}-15)\right]\ast\left[1-\left(\frac{G}{G_{MAX}}\right) \right]; \tag{8}\]  
[PERSON] et al. [45]:  
\[D=10.6\left(\frac{G}{G_{MAX}}\right)^{2}-31.6\left(\frac{G}{G_{MAX}}\right)+2 1+\left[(0.008I_{P}+0.82)(\frac{\sigma^{\prime}0}{Pa})^{-0.5\text{K}}.\right. \tag{9}\]  
The equation proposed by [PERSON] and [PERSON] [44] was calibrated for cohesive soil with \(PI\) below 50% and effective stress below 1000 kPa. [PERSON] and [PERSON] [48] created an equation for the average damping ratio curve based on nine publications and considered different types of clay, for example, offshore clayy silt, Taganuma and Hommoku clay, Edger plastic kaolin, Mexico City clay, San Old Bay clay, et cetera. [PERSON] et al. [47] found an empirical model for cohesive soil with a \(PI\) from 0% to 100%. The equation created by [PERSON] et al. [45] was appropriated for Quaternary, Tertiary, and older soils with a \(PI\) from 0% to 132%, collected from a dept of 0 to 326 m. Thus, every considered literature models was suitable for the studied soils. In Figure 12a-e, the measured versus calculated damping ratio is shown for the five analyzed models. The red dashed line indicates equal points between measured and calculated values, while black dashed lines mean \(\pm\)20% error. In Figure 12a, the results for the model created by the authors of this article are presented. It can be seen that most of the calculated damping ratio fits in the \(\pm\)20% error range. Moreover, points are located more or less equally on both sides of the line M = C, which means that the equation neither understates nor overstates the results. The worst match of measured and calculated results can be seen in Figure 12b, for the equation suggested by [PERSON] and [PERSON] [44]. That model significantly underestimates the results, because almost all calculated values are above the +20% error line. In Figure 12c,d, a quite similar distribution of points is presented. However, the [PERSON] and [PERSON] [48] equation slightly underestimates, while the [PERSON] et al. [47] model rather overestimates the damping ratio. In Figure 12e, results for the equation constructed by [PERSON] et al. [45] are shown. This proposition has a problem with estimations of the small value of damping ratio because nearly every measured parameter from 2% to about 6% is above the +20% error line. In Figure 12a-e, it is clear that the best model for Quaternary cohesive soil from Warsaw is the equation proposed by the authors. In Table 6, more statistical information about the empirical model and other compered in this article is provided. Table 6 consists of values of the absolute and the relative error for studied equations, and average values of this error referred to the entire study and the coefficient of determination for all five models. Based on Table 6, it can be concluded that the smallest absolute and relative error for individual samples has Equation (5) proposed in this article. Moreover, the average relative error is over three times smaller than the second-best equation suggested by [PERSON] and [PERSON] [48]. The highest value of absolute (0.77) and relative (18.4%) error for Equation (5) was recognized in sample _K12_. However, it is still a very small value compared to the highest errors from other analyzed models.