## 4 Results
### Differences Between Predicted and Observed Vertical Ground Displacements  
The maps of the differences between the expected best fit values at a given distance from the deformation center calculated with polynomial curves of different grades--i.e., third (Figure 11), fourth (Figure 12), fifth (Figure 13) and sixth (Figure 14)--and the observed (measured) values of vertical ground deformation allow us to visualize the spatial distribution of the residuals, i.e., the differences between the predicted and measured uplift values in SAR target points. The obtained dataset is made up of 15,900 points with values of differences between predicted and observed vertical ground deformation (i.e., residuals) ranging from about \(-100\) mm to about +70 mm in six years. Figure 11: Vertical ground deformation anomalies obtained by using a third-order polynomial curve compared with volcano-tectonic elements and seismic events. Figure 12: Vertical ground deformation anomalies obtained by using a fourth-order polynomial curve compared with volcano-tectonic elements and seismic events. Figure 13: Vertical ground deformation anomalies obtained by using a fifth-order polynomial curve compared with volcano-tectonic elements and seismic events. In these maps (Figures 11-14), areas where the observed uplift is significantly higher than the predicted values are characterized by negative values (lower than \(-15\) mm) and are mapped in yellow to red. Conversely, areas where the observed uplift is lower than what was predicted by the polynomial models have positive values (higher than \(+15\) mm) and are shaded in light to dark blue. The spatial representation of these differences provides relevant insights into the second-order patterns of deformation and allows for a better understanding of the areas experiencing significant differences in uplift or subsidence. It is important to observe how the residual deformation values vary in the maps obtained using the different polynomial curves (Figures 11-14). The second-degree curve was not used for the map because of the very high values of RMSE (Table 2 and Figure 6a). The map related to the third-degree polynomial curve (Figure 11), despite the low values of adjusted R\({}^{2}\) and RMSE (Table 2), shows anomalous concentric bands of residuals due to an oscillation in the tail of the retrieved polynomial function (Figure 6b) at distances of greater than 4-5 km from the center of deformation. The concentric bands are no longer visible in the map in Figure 12, relating to the fourth-order polynomial curve, and the distribution of residuals becomes stable and remains constant even in the maps relating to the fifth-order (Figure 13) and sixth-order (Figure 14) polynomial curves, with only minor differences in the peripheral areas of the caldera. The values of adjusted R\({}^{2}\) and RMSE (Table 2) are stable for these curves. Within the caldera area, two main anomalies are always well mapped (Figures 11-14); (i) a large area of negative values is located between Baia to the west and Pozzuoli city center to the east, and (ii) a sector with high positive values is located between Monte di Procida, Monte Olibano, la Pietra, and Bagnoli. These sectors with differences in uplift are separated by some lineaments, described as follows. Figure 14: Vertical ground deformation anomalies obtained by using a sixth-order polynomial curve compared with volcano-tectonic elements and seismic events. * A well-marked SW-NE-oriented line (A) goes from the caldera deformation center to Pisciarelli and Agnano and separates sectors characterized by about 60-75 mm of difference in vertical ground deformation. * An ill-defined SSW-NNE-oriented line (B) between Bagnoli and La Pietra separates sectors characterized by +15 mm to +45 mm of uplift anomaly (about 30 mm of difference in total). * A well-marked WNW-ESE-oriented line (C) goes from Baia to Fusaro Lake and separates sectors characterized by \(-\)15 mm to +15 mm of uplift anomaly (about 30 mm of difference in total).