As the most important parameter in the rigidity correlation method, the rockfill rigidity could be determined by additive mass method that simplify the vibrating rockfill into a single free linear spring system under the vibrating force by placing a series of additive masses on the rockfill. Based on the model, the vibration equations could be obtained as follows [1] :

where m = total vibrating mass; A = vibration acceleration of system; m₀ = rockfill mass; Δm = additive mass; K = system rigidity; Z = vibration displacement of system; ω = vibration circular frequency of system.

Therefore, it could be draw a line between ω⁻² and Δm that is measured by experiments, and the inverse slope of line is the rockfill rigidity.

According to the static rigidity equation and dynamic rigidity equation based on semi-infinite elastic foundation model, the analytic equation to density of circular foundation could be derived as follows [5] :

where r = density of medium; μ = Poisson’s ratio of medium; r = foundation radius; V_s = shear wave velocity of medium; K = foundation rigidity

From the Equation (4), we can see that the change of wave velocity would lead to the change in coefficient of rigidity, so as to a poor linear correlation between density and rigidity strictly. Due to the small rangeability of rockfill density, we could consider the correlation curve as a straight line in a small portion simply, where there is a linear correlation. In this case, dividing K by 4r, one-parameter correlation equation between density and normalized rigidity could be obtained as follows:

The rigidity correlation method for density inversion of rockfill is a direct correlation method whose model is established between rigidity and density, and studies have shown that there is a high correlation between rigidity and density of rockfill. Therefore, based on the model of rigidity correlation method, through taking the rigidity and density as samples, the correlation could be obtained accordingly under the condition of getting the accurate and representative data of samples.

There are three keys in inversion process of rigidity correlation method: the first one is whether the rigidity that is measured by additive mass method is correct; the second is whether the density that is measured by pit method is accurate for correlation curve; the last one is whether the density that is inverted by rigidity correlation method satisfies the requirements of high accuracy and the needing of practice engineering. There is a great deal of research to shows that the correlation coefficient between ω⁻² and Δm is more than 0.95 generally, using the additive mass method, so that the rigidity data are accurate [4] . Therefore, on the condition that the rigidity and density is accurate, it is necessary to do the research on applicability of rigidity correlation method for density inversion of rockfill.

The 13 sets of data, which are measured by additive mass method and pit method in Chinese Liyuan face rockfill dam whose rockfill is continuously graded limestone with maximum diameter of 300 mm, are shown in Table 1 [6] . From Table 1, it can be observed that the normalized rigidity values of various measuring points are distributed between 336.0 KN∙m⁻² and 643.6 KN∙m⁻², the wet density values are distributed between 2.350 g∙cm⁻³ and 2.496 g∙cm⁻³, and the dry density values are distributed between 2.323 g∙cm⁻³ and 2.391 g∙cm⁻³.

The results of linear regression between normalized rigidity and density which are in Table 1 are shown in Figure 1 and Figure 2.

Figure 1 and Figure 2 show that there is a linear correlation roughly between normalized rigidity and density of rockfill due to the zonal distribution of points. The determination coefficient between normalized rigidity and wet density is 0.85 while that is 0.87 between normalized rigidity and dry density, and accordingly there are lots of points which can explain the deviation of density with the change of rigidity. As such, it is proved that regression line has a good fitting degree so as to take a good result. The equation between normalized rigidity and wet density is shown in Equation (6) and between normalized rigidity and dry density is shown in Equation (7).

where r_W = wet density of rockfill; r_d = dry density of rockfill; K = rockfill rigidity; r = foundation radius.

In order to judge whether the regression equations above are meaningful or not, in other words, whether the density fluctuation is caused by the change of the parameters, the significance test of regression equations are carried out with F examination method based on the regression analysis theory [7] . According to F examination method, if the F value of equation is more than critical value, the equation is significant, but not vice versa. The F value could be solved by Equation (8).

where n = samples; Y_i = regression values; y_a = sample mean; y_i = sample values.

With calculation, the test results of regression equations are shown in Table 2.

The critical value of parameter samples is 4.60 when confidence level given is 0.05. The Table 1 shows that the F value of regression equations to calculate wet density and dry density is more than the critical values, and consequently the regression effect of the equations are highly significant when the significance level is 0.05. Therefore, the two regression equations are appropriate in density inversion of rockfill absolutely.

After pluging the normalized rigidity into the Equation (6) and Equation (7) respectively, we calculate to obtain the inversion values of wet density and dry density which are shown in Table 2.

From Table 3, compared with the density measured by pit method, the maximum relative error of wet density inverted by rigidity correlation method is 1.78% and the minimum one is −0.37%, while the maximum relative

error of dry density is 1.24% and minimum one is 0.00%. So the result shows that because the errors of wet density and dry density are within 2%, the inversion results of rigidity correlation method are in good agreement with that of pit method, and this method could be reliably used to in practical engineering.