# Volume Loss Assessment with MT-InSAR during Tunnel Construction in the City of Naples (Italy)
## 3 Methodology
### Multi Temporal InSAR
Considering the geometrical scheme of Figure 8a, i.e., when the tunnel cross section is contained in the LOS plane and the satellite is acquiring in ascending geometry, the intensity p(x) and the direction \(\alpha(x)\) (to the vertical, positive if anticlockwise) of the actual (theoretical) displacement vector p(x) can be computed with the following:  
\[\mathrm{|}\,\mathrm{p}(\mathrm{x})\,\mathrm{|}=\mathrm{p}(\mathrm{x})=( \mathrm{w}(\mathrm{x})^{2}+\mathrm{u}_{\mathrm{x}}(\mathrm{x})^{2})^{0.5}, \tag{9}\]  
\[\alpha(\mathrm{x})=\mathrm{atan}(\mathrm{x}/\mathrm{z}0). \tag{10}\]  
Then, the intensity of the monitored displacements along the LOS can be computed as follows:  
\[\mathrm{d}_{\mathrm{LOS}}\,(\mathrm{x})=\mathrm{p}(\mathrm{x})\cdot\mathrm{ cos}(\beta(\mathrm{x}))=\mathrm{p}(\mathrm{x})\cdot\mathrm{cos}(\vartheta\,-\, \alpha(\mathrm{x})), \tag{11}\]  
with \(\beta(\mathrm{x})\) being the angle formed by the vectors p(x) and \(\mathrm{d}_{\mathrm{LOS}}(\mathrm{x})\). Finally, substituting Equation (11) into Equation (8), the projected displacement in the vertical direction is obtained as a function of the actual displacement intensity p(x):  
\[\mathrm{d}_{\mathrm{v}}(\mathrm{x})=\mathrm{p}(\mathrm{x})\cdot\mathrm{cos}( \vartheta\,-\,\alpha(\mathrm{x}))/\mathrm{cos}(\vartheta), \tag{12}\]  
Figure 8: (**a**) Influence of horizontal displacements on the assumption inherent in Equation (8), under the assumption that the cross section is contained in the LOS plane; (**b**) the effect of the angular deviation \(\gamma\) between the cross-sectional plane and the line-of-sight plane on the magnitude of horizontal monitored displacements. And, by taking the ratio with w(x), it is possible to assess the error \(\psi(x)\) implied in the assumption of pure vertical displacements:  
\[\psi(x)=d_{v}(x)/w(x)=\cos(\vartheta-\alpha(x))/[\cos(\vartheta)\cdot\cos( \alpha(x))]=1+x\cdot\tan(\vartheta)/z_{0}. \tag{13}\]  
From Figure 8a, it can be seen that, for negative values of x (i.e., left side of tunnel axis), the error is \(\psi(x<0)>1\), while for positive values of x, the error is \(0<\psi(x<0)<1\). Figure 9a shows the comparison between w(x) and d\({}_{v}(x)\). It can be noted that, in the case of tunneling monitoring, when one converts the LOS displacements in vertical direction, the shape of the settlement profile is preserved (i.e., Gaussian distribution), but there is an overall increase in magnitude and a shift toward the satellite of the settlement profile. The quantification of the maximum settlement's increment is explicated through the expression \(\psi\!\mid\!\max\!=\!\max\!\{d_{v}(x)\}/\!\max\!\{w(x)\}\), the determination of which depends exclusively on the local angle of incidence \(\vartheta\) and the geotechnical parameter K, as shown in Figure 9b. The shift of the settlement profile \(\Delta x_{v}\) depends upon the local incidence angle, the geotechnical parameter K, and the tunnel axis depth \(z_{0}\). The shift is determined using the following equation:  
Figure 6:0  
Thus, when converting the LOS monitored displacements into vertical settlements, the presence of horizontal displacements modifies the expected settlement profile shape according to the following equation:  
Figure 6:1  
Hence, \(\psi(x)\) can be seen as a modification function to be applied to the classic Gaussian curve, and Equation (15) will be used in Section 4 to predict the settlement profile. It is possible to demonstrate that the volume loss obtained from the function in Equation (15) is equal to the actual volume loss \(V_{L}\) in Equation (4) (see Appendix A). Consequently, for tunneling applications, the assumption of pure vertical displacements does not have any influence on the detection of volume loss via SAR remote sensing, but it has an influence on the location and magnitude of the settlement profile.