## 3 Data and Methods
### Vertical Deformation Time-Series Combination  
The adopted combination scheme is now described. To this end, let us suppose that we have two sets of non-time-overlapped SAR data. The two datasets are composed of \(Q_{1}\) and \(Q_{2}\) SAR images, acquired at ordered times \(\mathbf{T}_{1}=\left[T_{1,1},T_{1,2},\ldots,T_{1,Q_{1}}\right]\) and \(\mathbf{T}_{2}=\left[T_{2,1},T_{2,2},\ldots,T_{2,Q_{2}}\right]\), respectively, and \(T_{2,1}>T_{1,Q_{1}}\).  
Vertical deformation time-series relevant to homologous highly coherent pixels of the two datasets, namely \(\mathbf{h}_{1}=\left[h_{1,1},h_{1,2},\ldots,h_{1,Q_{1}}\right]\) and \(\mathbf{h}_{2}=\left[h_{2,1},h_{2,2},\ldots,h_{2,Q_{1}}\right]\) (where, taking into account Equation (2), we have \(h_{1,1}\equiv 0\) and \(h_{2,1}\equiv 0\)) are, thus, combined to obtain a unique displacementtime-series, namely \(\mathbf{H}=\left[H_{1},H_{2},\ldots,H_{Q_{1}+Q_{2}}\right]\), which spans the overall set of \(Q_{1}+Q_{2}\) available SAR acquisition times. All the combined time-series also refer to one global reference time acquisition, which is selected as the time when the oldest SAR image was acquired, namely at time \(T_{1,1}\). Accordingly, taking into account that the two sets are not overlapped in time, the combined (unique) vertical deformation time-series vector, \(\mathbf{H}\), relevant to the highly coherent pixels, common to both SAR datasets, can be written as:  
\[\mathbf{H}\left(\hat{h}\right)=\left[h_{1,1},h_{1,2},\ldots,h_{1,Q_{1}},h_{2, 1}+\hat{h},h_{2,2}+\hat{h},\ldots,h_{2,Q_{2}}+\hat{h}\right] \tag{3}\]  
where \(\hat{h}\) is the (unknown) vertical deformation at the first time epoch of the second SAR dataset, estimated with respect to the global reference time. Note that the time gap between the SAR datasets makes the use of SVD, as is done in the original SBAS algorithm [28], completely unreliable. To face this problem, we have developed a methodology, which is detailed in what follows.  
The fundamental idea is to exploit the knowledge of an external model of the deformation, namely, \(\mathbf{m}=\mathbf{m}\left(T,\alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right)\), which analytically describes how the deformation is expected to evolve over the entire time, \(\mathbf{T}=\left[\mathbf{T}_{1},\mathbf{T}_{2}\right]\). The model also depends on the \(\alpha_{i},\,i=1,\ldots,K\) parameters. The best-fit model parameters, as well as the unknown \(\hat{h}\) term, are, thus, jointly estimated by solving the following nonlinear optimization problem:  
\[\alpha_{1},\alpha_{2},\ldots,\alpha_{k},\hat{h}=\operatorname{argmin}\left[ \left|\left|\mathbf{m}-\mathbf{H}\right|\right|_{2}\right] \tag{4}\]  
where the symbol \(\left|\left|\cdot\right|\right|_{2}\) represents the two-norm of a vector.  
We note that the developed algorithm is revealed to be particularly appropriate in the presence of non-linear deformation signals, when other external sources of information of ground displacements are not available, or when the spatial sampling of the measurement points (for instance, derived from Global Positioning System (GPS)/leveling campaigns) is not adequate for linking SAR data. Technical and implementation details of the proposed combination method are detailed in the following section.