## 6 Conclusions  
In this paper, we have presented a simple method for detecting second-order deformation anomalies by filtering the primary radial deformation signal observed during phases of unrest at Campi Flegrei caldera. The procedure is based on the polynomial fit of the vertical displacement data, assuming they only depend on the distance from the deformation center and on the computation of the differences between theoretical radial vertical displacements and observed ones. This approach emphasizes the study's main objective: highlighting second-order deformation anomalies without introducing excessive model complexity. The use of polynomial fitting, particularly the fourth-order curve, ensures that the main axial-symmetric pattern is preserved and successfully subtracted from the global field, so that the residuals indicate smaller-scale, geologically meaningful deviations. In this way, we can generate an anomaly map that highlights sectors with peculiar deformation patterns during unrest. We identified, using a polynomial function of the sole distance from the maximum deformation center, the best coefficients to fit the ground displacement pattern derived from the MT-InSAR Sentinel-1 vertical dataset referring to the 2016-2021 period. In addition to polynomial fitting, we fitted our data using a Gaussian curve and the Mogi model. As can be seen (Figures 4 and 5), however, the best fit obtained with such models yielded lower R\({}^{2}\) values and were less effective in describing the observed deformation pattern, as both of them are in fact unable to simulate the observed shape over the whole range of distance. The polynomial fitting approach resulted in the most suitable method for our study, as it was able to provide, with few coefficients, an almost perfect fit to the axial-symmetrical part of the ground uplift, which likely represents the effect of the main volcanological source, whatever it is. We confirmed that the first-order geometry of vertical displacement is radially symmetric; we also tested the best fitting function obtained from MT-InSAR data against the one obtained from coeval GPS data. We obtained very similar functions with the two different data sets, thus confirming the robustness of the procedure. Finally, we explored the obtained second-order anomalies. There are many models in the literature for the main volcanic source of deformation [7; 21; 22; 23; 24; 113], and all of these models predict an axial-symmetric ground deformation. We used the most general axial-symmetrical function, built as a polynomial, for obtaining the best fit to the observed data. By subtracting such a best fitting axial-symmetric field, we succeeded in showing the very clear second-order features of the ground deformation field, hidden by the main field. Such features appear to be very well correlated with the most seismically active lineaments, thus corroborating their interpretation in terms of relevant volcano-tectonic structures. The resulting anomalies are non-random and show well-defined differential behavior between two contiguous areas: one with significant positive residuals, another with negative ones. The separation between the two areas is remarkably linear, with most of the largest earthquakes occurring in a cluster around this line. We then interpreted such discontinuity as a main structural feature of the area. The obtained results show that seismic structures that have not yet been fully activated can be recognized in the early stages of ground deformation from differential ground displacement anomalies, if appropriately filtered and analyzed. The spatial distribution of earthquakes occurring since 2022 confirms the clustering of most of the largest-magnitude events around this structural feature, as detected from the analysis of deformations measured in 2015-2021, thus proving the robustness of the results obtained by such analyses.