## 4 Results
### Identification and Estimation of Displacement Sources  
The observed postseismic displacement patterns indicate the presence of at least two deformation sources acting in the area, producing ground displacements over time but with different amplitudes and opposite signings. We postulate that the first deformation source is related to the same postseismic tectonic processes affecting the epicentral area to the northwest (i.e., afterslip and poroelastic stress changes), with the Sant'Agostino-Mirabello area also being affected by non-negligible coseismic ground displacements.  
Otherwise, widespread liquefaction evidence was observed after the main shock (red circles in Figure 4a-c). Such evidence, which often manifests at the surface with sand boils and water leaks from cracks, indicates the presence of liquefied layers at depth. These layers are also responsible for diffuse ground subsidence, not only during the earthquake shaking, but also in the following days to months because of the slow dissipation of excess pore pressure generated in the deeper layers by the earthquake [12].  
Liquefaction-induced ground displacements accumulate not only in places where liquefaction effects manifest on the ground, but also in places where the liquefaction phenomena are not recognized, as this depends on the area's alluvial stratigraphic architecture [41; 67]. Therefore, we assume that the second deformation source could be related to widespread post-liquefaction ground subsidence.  
According to the above assumptions, we hypothesize that almost all the observed LoS displacements result from vertical ground movements. Thus, we converted the LoS displacement time series in Figure 5d into equivalent vertical movements according to Equation (2). Then, we assumed that both the postseismic tectonic uplift and the liquefaction-induced consolidation trends follow an exponential decay, as shown in Equation (15):  
\[d=d_{0}\pm a\Big{(}1-e^{-\frac{t}{b}}\Big{)}, \tag{15}\]  
where \(d_{0}\) is a fixed displacement term, \(t\) is the time, and the \(a\) and \(b\) terms control the amplitude and shape of the exponential function, respectively. The \(\pm\) symbol at the second member of Equation (15) is intended to simulate the exponential postseismic uplift (plus symbol) and the postseismic subsidence (minus symbol), respectively.  
The algebraic sum of the tectonic and post-liquefaction ground movements should provide the observed median ground displacements retrieved by InSAR and represented in Figure 5c,d with a black line. Therefore, we calibrated the Equation (15) coefficients for tectonic uplift and post-liquefaction subsidence to best reproduce such displacements.  
The performed procedure provides the curves in Figure 6, where the green curve represents the theoretical coseismic and postseismic uplift of tectonic origin (U in Figure 6), the blue curve represents the ground subsidence (S in Figure 6), and the yellow curve is the sum of both the green and blue curves (U + S in Figure 6). The latter reasonably mimics the observed vertical displacements (dashed black line in Figure 6), with a mean absolute error (MAE) of approximately 0.38 cm. Figure 6 also reports the coefficients of Equation (15) for both tectonic uplift and post-liquefaction subsidence. For the post-seismic tectonic uplift, the coefficient \(b\) (=11 days) in Equation (15) was calibrated by fitting the post-seismic ground uplift observed in the epicentral area, as shown in Figure 5a.  
An asymptotic median consolidation subsidence of roughly \(-2.1\) cm is representative of the investigated area, with a consolidation time of approximately 100-150 days.