## 2 Methodology
### Temporarily-Coherent Point Identification  
Temporarily coherent points (TCPs) are the points in the interferograms that can remain coherent during one or several intervals of SAR acquisitions, but do not need to be visible over the whole time span. We identify these points primarily based on the fact that the offset estimation at the points with high coherence is less affected by window size and oversampling factors compared with low coherent points. By statistical analysis of offset vectors estimated with changing window sizes at each pixel, the TCPs can be isolated. The theoretical analysis and real data test of the method can be found in [30; 36]. Here, only the key steps of the method are listed briefly. First, the scene is divided into a set of large patches and the range and azimuth offsets for each patch are estimated to determine the offset polynomial functions between two images. Second, the estimated offset is taken as an initial input and cross-correlation is implemented at every single pixel using smaller patches. We can obtain an offset matrix:  
\[\mathbf{O}_{l\times m}=\left[ \begin{array}{cccc}o_{1,1}&o_{1,2}&\cdots&o_{1,m }\\ o_{2,1}&o_{2,2}&\cdots&o_{2,m}\\ \vdots&\vdots&&\vdots\\ o_{l,1}&o_{l,2}&\cdots&o_{l,m}\\ \end{array} \right], \tag{1}\]  
where \(o_{ij}(i=1,2,\cdots,l;\ j=1,2,\cdots,m)\) is the offset of pixel \((i,j)\). TCP candidates can be selected according to a 2D histogram:  
\[ \begin{array}{l}o_{c}=peak\{hist_{2D}(O_{l\times m})\}\\ |o_{i,k}-o_{c}|<A\\ \end{array} , \tag{2}\]  
where \(A\) is the tolerated interval used in the 2D histogram. Third, the TCP candidates are further evaluated by changing the size of the patches and the oversampling factor for the cross-correlation estimation to find sub-pixel offsets. For the sake of simplicity, a fixed oversampling factor can be used. We can obtain a set of offsets for a given TCP candidate (\(j\)). The TCP candidates whose standard errors of offsets are less than 0.1 pixels, which is usually taken as a required co-registration accuracy, are selected:  
\[ \begin{array}{l}OT_{j}=\left[\begin{array}{cccc}ot_{j1}&ot_{j2}&\cdots&ot_{ jN}\\ std(OT_{j})<0.1\\ \end{array}\right],\\ \end{array}  \tag{3}\]  
Figure 3a shows an example of the 2D histogram of the offsets estimated for an image pair where the color shows the number of points with the corresponding azimuth and range offsets. Figure 2: The amplitude image of Donghai Bridge from (**a**) Phased Array type L-band Synthetic Aperture Radar (PALSAR) HH polarization data, and (**b**) PALSAR HV polarization data. The figure is plotted in the synthetic aperture radar (SAR) coordinate system. The histograms of the standard errors of the range and azimuth offsets by using different window sizes (from 256 to 32 with a step of 16) are exhibited in Figure 3b,c, respectively. Fourth, a smooth polynomial is used to fit the offsets of TCP candidates, and the TCPs are finally identified by discarding pixels whose offsets do not fit the polynomial well. An interpolator kernel is employed to resample the TCPs in the slave image based on the polynomial estimated especially from the TCPs. The TCPs identified from two interferograms are usually under different range/Doppler image coordinates. To perform time series analysis, these points should be under the same reference frame. By coincidence in this study the selected interferograms are with the same master image, therefore, no more work needs to be performed to combine the selected TCPs. The TCPs that remain coherent in all image pairs are shown as the dots in Figure 4a,b. With the proposed strategies, the coherence at the TCPs has an improvement of 0.12 on average.