## 3 Multi-Track/Multi-Satellite InSAR Methodologies  
Nowadays there is an increasing availability of large archives of SAR data collected by several RADAR instruments mounted on board space-borne platforms, which operate at different wavelengths and with distinctive looking angle geometries (and also possibly through different acquisition modes). This poses the issue of how to effectively combine the complementary pieces of information embedded by the different SAR datasets. In particular, the combination of multi-platform (multi-angle) LOS displacement time series can improve our ability to retrieve the 3-D, i.e., East-West (E-W), North-South (N-S), Up-Down (U-D) components of the measured surface displacement; thus, overcoming the main limitation of DInSAR to measure only the satellite LOS projection of the displacement. Such a problem has already been faced in the literature and a few technical solutions capable of combining multiple-orbit/multiple-angle DInSAR-based measurements, as well as merging DInSAR data products with other external information (such as the one derived from GPS stations), have been  
Figure 15: Deformation velocity map for the city of Assisi, Italy, generated by processing COSMO-SkyMed data acquired from 2009 to 2012, superimposed on the Landslide Inventory map. Shades of blue show ancient and relict landslides; shades of red show recent landslides (Adapted with permission from [24], Elsevier, 2014). proposed [124; 125; 126; 127; 128; 129; 130; 131; 132; 133; 134; 135]. Let us first consider the combination of InSAR data acquired from two satellites flying along ascending and descending orbits with \(\vartheta_{1}\) and \(\vartheta_{2}\) side-looking angles, respectively (see Figure 16). Note that, for the sake of convenience, the N-S deformation components have been considered negligible. This is due to the consideration that almost all modern space-borne satellites move along near-polar orbits (i.e., satellite azimuthal direction is approximately parallel to N-S) and, thus, there is no diversity in the viewing geometries to accurately measure N-S displacements. The LOS components of the deformation \(\mathbf{d}\) can be related to the E-W, namely \(d_{H}\) and U-D, namely \(d_{V}\), components through the following equations [136]:  
\[\left\{ \begin{array}{c}d_{LOS_{1}}=\mathbf{d}\cdot\mathbf{\hat{i}}_{LOS_{1}} =d_{H}\sin\vartheta_{1}+d_{V}\cos\vartheta_{1}\\ d_{LOS_{2}}=\mathbf{d}\cdot\mathbf{\hat{i}}_{LOS_{2}}=-d_{H}\sin\vartheta_{2 }+d_{V}\cos\vartheta_{2}\end{array} \right. \tag{43}\]  
The system of Equation (43) can be expressed using matrix formalism as:  
\[\left( \begin{array}{cc}\sin\vartheta_{1}&\cos\vartheta_{1}\\ -\sin\vartheta_{2}&\cos\vartheta_{2}\end{array} \right)\cdot\left( \begin{array} []{c}d_{H}\\ d_{V}\end{array} \right)=\left( \begin{array}{c}d_{LOS_{1}}\\ d_{LOS_{2}}\end{array} \right) \tag{44}\]  
In this simplified case, a direct solution can be formally achieved as follows:  
\[\left( \begin{array}{c}d_{H}\\ d_{V}\end{array} \right)=\left( \begin{array}{cc}\sin\vartheta_{1}&\cos \vartheta_{1}\\ -\sin\vartheta_{2}&\cos\vartheta_{2}\end{array} \right)^{-1}\cdot\left( \begin{array} []{c}d_{LOS_{1}}\\ d_{LOS_{2}}\end{array} \right)=\frac{\left( \begin{array}{c}d_{LOS_{1}}\cos \vartheta_{2}-d_{LOS_{2}}\cos\vartheta_{1}\\ d_{LOS_{1}}\sin\vartheta_{2}+d_{LOS_{2}}\sin\vartheta_{1}\end{array} \right)}{ \sin(\vartheta_{1}+\vartheta_{2})} \tag{45}\]  
In the more simplified case that ascending and descending SAR data are acquired from the satellite with the same side-looking angle \(\vartheta\), the results of the combination particularize as follows:  
\[\left( \begin{array}{c}d_{H}\\ d_{V}\end{array} \right)=\frac{\left( \begin{array}{c}d_{LOS_{1}}\cos \vartheta-d_{LOS_{2}}\cos\vartheta\\ d_{LOS_{1}}\sin\vartheta+d_{LOS_{2}}\sin\vartheta\\ \frac{d_{LOS_{1}}+d_{LOS_{2}}}{2\cos\vartheta}\end{array} \right)}{\sin(2 \vartheta)}=\left( \begin{array}{c}\frac{d_{LOS_{1}}-d_{LOS_{2}}}{2\sin \vartheta}\\ \frac{d_{LOS_{1}}+d_{LOS_{2}}}{2\cos\vartheta}\end{array}