## 3 Materials and Methods
### Sistem--DInSAR and GPS Integrated Data  
To estimate the total deformation occurring at the volcano during the entire crisis, we produced a high-resolution map by integrating the DInSAR data with the GPS displacements, calculated through PPP (Precise Point Positioning) processing by the GIPSY-OASIS package on a daily basis over the same InSAR time window. This processing is independent and widely different from the high-rate (HR) one, not only due to the different observation times but also being performed on daily standard RINEX (Receiver INdependent EXchange) data, decimated from the original 1 to 30 s rate, processed in static mode to achieve a unique, precise and not-moving solution for each day. 3D GPS and 1D/LOS DInSAR displacements were integrated by the SISTEM algorithm [26], a software that allows the complete integration of different ground deformation datasets in order to calculate, point-by-point with the DInSAR ground resolution, the 3D components of the ground motion over the 12-day period covered by DInSAR data. In this way, an integrated map can be produced, similar to an interferogram for spatial resolution but containing, for each pixel, the 3D motion of the ground (instead of the LOS component of the DInSAR image). The main source of error of the DInSAR measurements is due to atmosphere noise, especially related to the strong topography at Etna (often more than a fringe from base to summit, see [27, 28] and references therein). For this reason a preliminary step, before their use, is the validation-correction of the DInSAR data with GPS and/or other independent measures. The SISTEM algorithm not only integrates but also simultaneously validates the DInSAR data with GPS by taking advantage of the positive features of both these techniques, i.e., the high spatial resolution of the DInSAR and the 3-D measurements and  
Figure 7: DInSAR interferogram of the ascending orbit pair from 2 to 14 December 2015. A fringe (28 mm) of deflation is visible, affecting the whole upper part of the volcano, above 1500–2000 m a.s.l., and across the upper Pernicana-Provenzana fault system on the middle NE flank. The topography is represented with white contour lines (every 500 m). subcentimeter accuracy level of the GPS. This integrated and cross-validated information is able to give a more reliable interpretation of the geophysical phenomena producing ground deformations. The general pattern evidenced by DInSAR data (Figure 7) is mostly coherent with the deformation highlighted by the 12-day GPS displacement pattern (thick arrows in Figure 8), both evidencing, even with different resolution and completeness of information (1D but high-resolution for InSAR and 3D but low-resolution for GPS), a contraction of the volcano. The integration of both datasets (thin arrows and colors in Figure 8) images a general deflation of the edifice well, especially evident on its western and southern slopes. The match between the integrated and original observed GPS and DInSAR datasets has been checked and validated for looking at possible artifacts. Results of the validation process evidence a mean difference between observed GPS displacements and SISTEM displacements at GPS points of about 1-1.5 mm (abundantly within the GPS errors) and differences within 5 mm for most of the DInSAR interferogram (figures GPS vs. SISTEM and SAR vs. SISTEM provided in the Supplementary Materials). The main deflation seems to affect the upper part of the volcano, above 1700-1800 m a.s.l. (in agreement with the interferogram, see Figure 7), which shows an evident radial contraction and a subsidence of the ground. Figure 8: 3D displacement map obtained by SISTEM algorithm over the December 2 to 14 period. Integrated horizontal displacements are shown by (sub-sampled) blue vectors, while the vectors showing the GPS-only measured displacements are evidenced by thicker arrows with white borders. The color map shows the integrated vertical displacements, according to the color scale on the bottom left.