## 5 Satellite Remote Sensing Techniques for Infrastructure Monitoring
### Persistent Scatterer Interferometry
Under the assumption of the same clutter pertaining to the pixel and its surroundings, the SCR can be evaluated for each pixel by calculating the phase standard deviation (\(\sigma_{\phi}\)). The relation amongst pixels can be written as:  
\[\sigma_{\Phi}=\frac{1}{\sqrt{2r_{sc}}} \tag{6}\]  
In this method, a pixel is recognised as a PS when \(\sigma_{\Phi}\) is less than a defined threshold. A main limitation of this method is related to the condition that if adjacent pixels contain several point scatterers, the clutter may be overestimated, resulting in the rejection of suitable PSCs [74]. In addition, considerable computation time is needed for the implementation of this method. The last methods are based on indicators of correlation by calculating the values of the interferometric coherence \(\gamma\). This is an important indicator of suitability of the data obtained by radar remote sensing systems. Since the backscatter of PSs is almost consistent in the images collected at different times, a higher correlation for the PSs should be maintained in interferometric pairs. If the correlation in a pixel is higher than a given threshold, it can be considered as a PSC. The correlation coefficient \(\gamma\) is estimated from the pixel and its surroundings in a given window size, reducing the quantity of data in the selection of PSs, and it is expressed as follows:  
\[\gamma=\frac{\left|\sum_{i=1}^{m}\sum_{j=1}^{n}M(i,j)S^{*}(i,j)\right|}{\sqrt {\sum_{i=1}^{m}\sum_{j=1}^{n}\left|M(i,j)\right|^{2}\,\sum_{i=1}^{m}\sum_{j=1 }^{n}\left|S(i,j)\right|^{2}}} \tag{7}\]  
where \(M\) and \(S\) are the local information sets of pixels on two SAR images of an interferometric pair (Master and Slave) and \({}^{*}\) represents the complex conjugation operator. \(\gamma\) takes values in the interval \([0,1]\); values higher than 0.5 relate to good data correlation, whereas values close to zero correspond to areas of high decorrelation, which are not suitable for interferometric data processing. The calculation of the correlation coefficient depends on the selection of the window size. A larger size of the windows may result in reduced reliability of the number of PSs detected, whereas choosing windows of smaller size can reduce the value of the correlation coefficient. In this context, PS methods based on \(D_{a}\), such as the PS-InSAR technique [14; 15], appear to be highly suitable for the monitoring of transport infrastructure in relation to the points density, the possibility of identifying individual stable scatterers for the target structure and the relatively low computational load. Furthermore, significant results have been obtained using other methods, amongst which the SBAS and the TomoSAR methods have been successfully reported in the literature [76; 77; 78]. After the selection process of the pixels as PSCs, these are used in a combined spatial and temporal analysis to estimate the atmospheric phase screen (APS). This is associated with the atmospheric delay affecting the phase term for each interferogram. After removing the contribution of APS from the interferograms, an estimate of the deformations and errors in the DEM can be performed by examining both the temporal evolution and the geometric baseline variation in each pixel. An overview of the PS-InSAR algorithm is given in [14; 15]. This was the first algorithm implementing the persistent scatterer approach. The method is suitable for analysing small areas at the same time, up to a maximum of 5 km \(\times\) 5 km dimensions. This is linked to the main assumption that the atmosphere can be modelled effectively as a linear plane, meaning that several sub-areas can be processed to cover wider areas.