## 4 Minimum Acceleration Combination Technique: Rationale and Applications  
The potential of an alternative technique, known as minimum-acceleration (MinA) combination technique, which does not require the simultaneous process of very large archives of DInSAR interferograms has been proposed in [32]. It consists of a straightforward post-processing stage that involves the analysis of sequences of independently processed (potentially, also with different DInSAR toolboxes) multi-platform LOS displacement time series. Before their combination, LOS displacement time series are also depurated from atmospheric artefacts and residual topographic errors, as provided within the used DInSAR toolboxes. Following the lines of [71; 142], the preliminary-computed LOS displacement time-series are connected to the unknown 3-D velocities among consecutive time acquisitions. This leads to the writing of an underdetermined system of independent linear equations, which might be solved in the LS sense by using the SVD. The obtained system of linear equations is regularized by imposing the additional constraint that the 3-D displacement components are with minimum acceleration, which means that the difference between consecutive velocities is minimal. Once estimated, the unknown 3-D velocities are eventually (and independently) time integrated to recover the E-W, N-S and U-D deformation time series. To introduce the right mathematical framework, let us assume the availability of K independent sets of multiple-platform SAR data collected at ordered times \(t^{(j)}=\left[t_{0}^{(j)},t_{1}^{(j)},\ldots,t_{Q_{j}-1}^{(j)}\right]^{T} \forall j=1,\ldots,K\) over the same area on the ground, consisting of \(Q_{j}\) distinctive time epochs, respectively. The MinA algorithm [32] requires the preliminary generation from each single SAR data-track of the original LOS-projected deformation time-series. This task can be achieved by independently applying either the SBAS technique [71] or other alternative multi-temporal DInSAR approaches to the available K SAR datasets. During this preliminary stage, the residual topography, as well as the APS artefacts corrupting differential SAR interferograms, might be estimated and successfully filtered out from the generated LOS-projected displacement time-series. The obtained LOS-projected time-series of deformation are geocoded to a common spatial grid of points where the subsequent combination stage is applied. During this preliminary step, the location of high coherent targets is also identified. Henceforth, let \(\mathbf{d}^{(j)}=\left[d_{0}^{(j)},d_{1}^{(j)},\ldots,d_{Q_{j}-1}^{(j)}\right] ^{T}\forall j=1,\ldots,K\) be the geocoded LOS-projected deformation time-series relevant to a pixel P that belongs to the group of high-coherent pixels common to all the available K SAR datasets. LOS-projected time-series of deformations are expressed with respect to the instants \(t_{0}^{(j)},\,\forall j=1,\ldots,K\), which are singularly taken as reference for each dataset, that is to say \(d_{0}^{(j)}=0,\,\forall j=1,\ldots,K\). Let us now describe how the algorithm works and let \(Q=\sum\limits_{j=1}^{K}Q_{j}\) be the total number of the available SAR images collected at the \"whole\" ordered times \(T=\cup_{j=1}^{K}t^{(j)}=\left[T_{0},T_{1},\ldots,T_{Q-1}\right]^{T}\). We observe that a generic LOS-projected deformation measurement, namely \(d_{\text{LOS}}\), can be related to its inherent 3-D components, namely \(\left[d_{E-W},d_{U-D},d_{N-S}\right]^{T}\), as [32]:  
\[d_{\text{LOS}}=\mathbf{d}\cdot\hat{\hat{i}}_{\text{LOS}}=\sin\vartheta\cos \varphi d_{East-West}-\cos\vartheta d_{Up-Down}+\sin\vartheta\sin\varphi d_{ North-South} \tag{47}\]  
where \(\hat{i}_{\text{LOS}}\) is the LOS-direction versor.