## 2 Confocal Ellipsoidal Densification Model (CEDM)
### Mass Balance Equation for CEDM  
According to Assumption 3, the mass balance equation after DC may be expressed thus:  
\[\mathrm{W_{crater}}(\gamma_{0})+\mathrm{W_{WCZ}}(\gamma_{0})=\mathrm{W_{HCZ}}( \gamma_{\mathrm{max}})+\mathrm{W_{WCZ-HCZ}}[\gamma(\rho,\delta,\theta)] \tag{23}\]  
where \(\mathrm{W_{crater}}\) is the weight of soil related to the initial dry density of the crater cylinder before DC; \(\mathrm{W_{WCZ}}\) is the weight of soil related to the initial dry density of the WCZ ellipsoid before DC; \(\mathrm{W_{HCZ}}\) is the weight of soil related to the maximum dry density of the HCZ ellipsoid after the \(i\)th tamping; and \(\mathrm{W_{WCZ-HCZ}}[\gamma(\rho,\delta,\theta)]\) is the weight of soil related to the variational dry density of the weak compacted zone (\(\mathrm{V_{WCZ}}-V_{\mathrm{HCZ}}\)) after the \(i\)th tamping. The soil weight with the initial dry density corresponding to the volume of crater is  
\[\mathrm{W_{crater}}=\gamma_{0}\cdot\pi\mathrm{R}^{2}d_{i} \tag{24}\]  
The soil weight with initial dry density corresponding to the volume of the WCZ ellipsoid is calculated thus:  
\[\mathrm{W_{WCZ}}=\frac{2}{3}\pi\cdot\gamma_{0}\bigg{[}b_{i2}\cdot a_{i2}^{2} +\frac{3}{2}R^{2}(h_{i2}-d_{i})+\frac{b_{i2}}{a_{i2}}(a_{i2}^{2}-R^{2})^{ \frac{3}{2}}\bigg{]} \tag{25}\]  
The soil weight with maximum dry density corresponding to the volume of the HCZ ellipsoid is calculated thus:  
\[\mathrm{W_{HCZ}}=\frac{2}{3}\pi\cdot\gamma_{\mathrm{max}}\bigg{[}b_{i1}\cdot a _{i1}^{2}+\frac{3}{2}R^{2}(h_{i1}-d_{i})+\frac{b_{i1}}{a_{i1}}(a_{i1}^{2}-R^{2 })^{\frac{3}{2}}\bigg{]} \tag{26}\]  
The soil weight with variational dry density corresponding to the volume of the weak compacted zone (\(\mathrm{V_{WCZ}}-\mathrm{V_{HCZ}}\)) in the polar coordinates is calculated thus:  
\[\mathrm{W_{WCZ-HCZ}} =\underset{\mathrm{V_{WCZ}}-\mathrm{V_{HCZ}}}{\iint}\gamma(\rho, \delta,\theta)\mathrm{d}V=\underset{\mathrm{V_{WCZ}}-\mathrm{V_{HCZ}}}{\iint} \gamma(\rho,\delta,\theta)\rho^{2}\sin\theta\mathrm{d}\rho\mathrm{d}\delta \mathrm{d}\theta \tag{27}\] \[=4\int_{0}^{\pi/4}\mathrm{d}\delta\int_{0}^{\beta}\sin\theta \cdot\mathrm{d}\theta\frac{\rho}{\int\frac{e_{1}\bar{p}_{1}}{1-e_{ 1}\cos\delta}}\left[\gamma(\rho,\delta,\theta)\right]\cdot\rho^{2}\mathrm{d}\rho\]where \(\beta\) is the angle between the connecting line of the origin and point \((R,d_{i})\) and the vertical direction; this angle may be determined using Equation (28), as follows:  
\[\beta=\text{arccot}\Big{(}\frac{c_{i1}-h_{i1}+d_{i}}{R}\Big{)} \tag{28}\]  
According to Assumption 2, the soil dry density of any point in the range of WCZ is expressed as follows:  
\[\gamma(\rho,\delta,\theta)=\gamma_{0}+\frac{\rho-\frac{c_{i2}p_{i2}}{1-c_{i2} \cos\theta}}{\frac{c_{i1}p_{i1}}{1-c_{i1}\cos\theta}-\frac{c_{i2}p_{i2}}{1-c_{i 2}\cos\theta}}\cdot(\gamma_{\text{max}}-\gamma_{0}) \tag{29}\]  
where \(\gamma_{0}\) is the initial dry density of soil for unimproved area; \(\gamma_{\text{max}}\) is the maximum dry density of soil in HCZ; \(e_{i1}\) and \(e_{i2}\) are the eccentricity ratios of HCZ and WCZ ellipsoids, respectively; and \(p_{i1}\) and \(p_{i2}\) are the vertical distances from the focus to the datum lines for HCZ and WCZ ellipsoids, respectively. Substituting Equations (29) into Equation (27), we obtain  
\[