## 4 Minimum Acceleration Combination Technique: Rationale and Applications
Note that \(\vartheta\) and \(\varphi\) represent the RADAR side-looking and the satellite heading angles, respectively; the imaging geometries for ascending/descending data-tracks are shown in Figure 18. Extending to our case what initially proposed in [71], let us relate the available LOS deformations \(d^{(j)}(P),\,j=1,\ldots,K\) (for each high coherent pixel) to their unknown 3-D components. This leads to the writing of a system of Q-K independent linear equations with respect to the M = 3(Q - 1) unknowns representing the East-West (E-W), Up-Down (U-D) and North-South (N-S) deformation velocities components between adjacent time-acquisitions, namely \(\mathbf{V_{E}}=\left[V_{E_{1}},V_{E_{2}},\ldots,V_{E_{Q-1}}\right]\), \(\mathbf{V_{U}}=\left[V_{U_{1}},V_{U_{2}},\ldots,V_{U_{Q-1}}\right]\) and \(\mathbf{V_{N}}=\left[V_{N_{1}},V_{N_{2}},\ldots,V_{N_{Q-1}}\right]\). This system of linear equations can be expressed using matrix formalism as follows [71]:  
\[\mathbf{B}\cdot\left[ \begin{array}{c}\mathbf{V_{E}}\\ \mathbf{V_{U}}\\ \mathbf{V_{N}}\end{array} \right]=\left[ \begin{array}{c}\Gamma^{1}\Big{(}d^{ (1)}-d_{0}^{(1)}\Big{)}\\ \Gamma^{2}\Big{(}d^{(2)}-d_{0}^{(2)}\Big{)}\\ \ldots\\ \Gamma^{K}\Big{(}d^{(K)}-d_{0}^{(K)}\Big{)}\end{array} \right]=\widetilde{ \mathbf{d}} \tag{48}\]  
where \(\Gamma^{j}j=1,\ldots,K\) are the values of temporal coherence [110] associated with the \(K\) different datasets (representing a quality factor of the obtained LOS displacement time-series) and B is the incidence-like matrix of the linear transformation that converts LOS-projected measurements into their essential 3-D components (see [71], for further details). The system of Equations (47) and (48) has fewer linear independent equations (Q-K) than unknowns (M). Thus, it is an under-determined system that does not admit a unique solution. The matrix \(\mathbf{B}\) has singular values that gradually decay to zero, thus rendering any solution much sensitive to noise level corrupting the vector \(\widetilde{\mathbf{d}}\). It represents a canonical example of a linear discrete ill-posed problem whose meaningful solution can be obtained by replacing the \"original\" linear system (47) and (48) by a nearby system less sensitive to perturbations of the right-hand side of the system. This operation is known as regularization and can be performed using Tikhonov regularization [144], Truncated Singular Value Decomposition (TSVD) [145] and maximum entropy principle [146]. MinA algorithm accomplishes the regularization problem by adding to the original system (47) and (48) new equations that impose the condition the (unknown) 3-D (E-W, U-D, N-S) displacement time-series are with minimum curvature. That is to say, the velocity deformation differences (for all the 3-D components) between consecutive time intervals is minimal. Such conditions can formally be expressed by adding to (2) the following set of 3(Q - 2) additional equations:  
\[ \begin{array}{l}\kappa(V_{E_{i+1}}-V_{E_{i}})i=1,\ldots,Q-2\\ \kappa(V_{U_{i+1}}-V_{U_{i}})i=1,\ldots,Q-2\\ \kappa(V_{N_{i+1}}-V_{N_{i}})i=1,\ldots,Q-2\end{array}  \tag{49}\]  
Figure 18: SAR data acquisition geometries in the Up-East (**a**) and North-East (**b**) planes. where \(\kappa\) is a proper regularization factor. Accordingly, the regularized system of linear equations becomes:  
\[\left[ \begin{array}{c}\mathbf{B}\\ \mathbf{C}\end{array} \right]\cdot\left[ \begin{array}{c}\mathbf{V_{E}}\\ \mathbf{V_{U}}\\ \mathbf{V_{N}}\end{array} \right]=\left[ \begin{array}{c}\widetilde{\mathbf{d} }\\ \mathbf{0}\end{array} \right] \tag{50}\]  
where \(\mathbf{C}\) is the incidence-like matrix related to the minimum-acceleration-regularization linear transformation. The solution of (50) is finally obtained in the LS sense by applying Truncated SVD to the matrix \(\mathbf{\Omega}=\left( \begin{array}{c}\mathbf{B}\\ \mathbf{C}\end{array} \right)\). Finally, once the problem in (40) is solved, the 3-D velocity deformation components are independently time-integrated to recover the relevant 3-D displacement time-series.