# Volume Loss Assessment with MT-InSAR during Tunnel Construction in the City of Naples (Italy)
## 3 Methodology  
### Empirical Method for Tunnel-Induced Vertical Displacements  
The excavation of tunnels in greenfield conditions, or free-field (i.e., without buildings), induces a settlement trough at the surface. Far from the excavation face, plane strain  
Figure 4: Interferogram network (single Master PS analysis), where the timeline of data acquisition and normal baseline with reference to the Master Image can be noted (23 December 2019). Each circle marker represents a slave image, while each line represents the connection between the slave and the master image. Figure 3: (**a**) Reflectivity map (RM) and (**b**) subset of RM showing the analyzed area and, in red, the tunnel alignments. Both figures are viewed in the local satellite coordinate system (line and sample directions, i.e., azimuth and range directions). conditions are assumed, and the transverse section of the greenfield settlement trough is well described by a Gaussian curve [22; 23] as follows:  
\[\mathrm{w(x)=w_{max}\cdot exp[-x^{2}/(2\cdot i_{x}{}^{2})],} \tag{1}\]  
where \(\mathrm{w_{max}}\) is the maximum settlement and \(\mathrm{i_{x}}\) is the distance of the inflection point location from \(\mathrm{x}\) = 0, as displayed in Figure 5a for clarification. Integrating Equation (1), the volume of the settlement trough is equal to the following:  
\[\mathrm{V_{C}=\sqrt{(2\pi)\cdot i_{x}\cdot w_{max}}.} \tag{2}\]  
In general, it can be assumed that \(\mathrm{V_{C}}\) is a fraction of the excavated volume per unit length of advancement and this fraction is called 'volume loss', \(\mathrm{V_{L}}\). Therefore, the maximum settlement can be expressed as follows:  
\[\mathrm{w_{max}=V_{L}\cdot\sqrt{(2\pi)\cdot D^{2}/(8\cdot i_{x})}=0.31\cdot V_ {L}\cdot D^{2}/i_{x}.} \tag{3}\]  
Thus, it is evident that a correct prediction of the settlement trough depends on a reliable estimate of the parameter \(\mathrm{V_{L}}\) and \(\mathrm{i_{x}}\). In the case of mechanized excavation with a tunnel boring machine (TBM), in general, the volume loss \(\mathrm{V_{L}}\) rarely exceeds 1-1.5%, with usual values of 0.5%. For the inflection point distance \(\mathrm{i_{x}}\), it can be seen as a fraction K of the tunnel axis depth \(\mathrm{z_{0}}\) (\(\mathrm{i_{x}=K\cdot z_{0}}\)), with K ranging from 0.25 to 0.45 for sands and gravels and from 0.4 to 0.6 for clay [28; 29]. Therefore, substituting Equation (3) in Equation (1) leads to the following:  
\[\mathrm{w(x)=0.31\cdot V_{L}\cdot D^{2}/i_{x}\cdot exp[-x^{2}/(2\cdot i_{x}{}^ {2})].} \tag{4}\]  
In the case of twin tunnels, the superposition of the effects is often used to obtain the total ground surface settlement profile, as results of its speed and simplicity (Figure 5b,c). The superposition method involves a direct combination of two settlement troughs induced by each single tunnel excavated in greenfield conditions, without considering any interference between the two excavations. This can result in a misestimation of the total settlement  
Figure 5: (**a**) Gaussian profile of settlements from Equation (1) with the indication of the tunnel position (1), inflection point distance \(\mathrm{i_{x}}\), and maximum settlement \(\mathrm{w_{max}}\); (**b**) the black solid lines represent two Gaussian profiles of both tunnels (1 and 2) and the solid red line represents the superposition of effects using Equation (5) when \(\Delta\mathrm{x}<2\cdot i_{x}\); (**c**) superposition of effects using Equation (5) when \(\Delta\mathrm{x}<2\cdot i_{x}\). profile [30; 31]. When superimposing the effects of the twin tunnels, it is important to consider the distance \(\Delta\)x between them and the interaction between the two tunnels [32; 33]. The distance \(\Delta\)x between two tunnels determines the shape of the total settlement profile. Generally, when \(\Delta\)x \(<\) 2:i\({}_{x}\), the total settlement profile still preserves a Gaussian shape, as shown in Figure 5b.