## 2 LiCSAR and LiCSBAS
### Time Series Analysis
#### 2.4.3 Step 1-3: Small Baseline Network Inversion  
In order to perform an estimate of the velocity of a surface pixel through time based upon a series of displacement data, we perform an SB inversion on the network of interferograms. Suppose that we have a stack of \(M\)-unwrapped interferograms \(\mathbf{d}=[d_{1},\dots,\ d_{M}]^{\text{T}}\) produced from \(N\) images acquired at \((t_{0},\ \dots,t_{N-1})\), \(N\)-1 incremental displacement vector \(\mathbf{m}=[m_{1},\dots,m_{N-1}]^{\text{T}}\) (i.e., \(m_{i}\) is the incremental displacement between time \(t_{i\text{-}1}\) and \(t_{i}\)) can be derived by solving  
\[\mathbf{d}=\mathbf{G}\mathbf{m} \tag{2}\]  
where \(\mathbf{G}\) is a \(M\times\)(\(N\)-1) design matrix of zeros and ones describing the relationship between the network of the interferograms and incremental displacements, considering that the unwrapped interferogram (i.e., displacement between two acquisitions) is the sum of corresponding successive incremental displacements [52]. Cumulative displacements for each acquisition (i.e., the displacement time series) are simply calculated by summing the incremental displacements. The mean displacement velocity is then derived from the cumulative displacements by least-squares. Even though Sentinel-1 data are constantly being acquired with short spatial and temporal baselines in most areas (unlike with predecessors such as Envisat and ERS), there could still be gaps in the SB network (i.e., separated subsets of the network with no interferograms spanning them) due to decorrelation associated with factors such as vegetation, cultivation, or snow cover, and especially after the network refinement Step 1-2. Additionally, some frames may have large acquisition gaps (several months or longer). It is noteworthy that even if the refined network of interferograms is fully connected on an image-by-image basis, network gaps may still exist on a pixel-by-pixel basis. This is because each interferogram has a different coverage of valid unwrapped data due to the coherence-based masking applied to the LiCSAR products (see Section 2.1). In such cases, the design matrix \(\mathbf{G}\) is rank deficient; Equation (2) can nevertheless be solved using singular value decomposition (SVD). However, the minimum norm solution provided by SVD gives incremental displacements of zero in any network gaps, which is not realistic even for very small gaps (e.g., 6 days) where the true displacement would be almost zero, because noise in the interferograms between the gaps would not be close to zero. Figure 2: Example of the loop closure for the region of the Echigo plain, Japan (Section 4). Top and bottom row images are interferograms consisting of five images and loop phases calculated from the interferograms by Equation (1), respectively. Black and red lines between them denote which three interferograms are used to calculate each loop phase. Three loop phases with a red frame (\(\Phi_{124},\ \Phi_{234},\Phi_{245}\)) have areas with \(\pm 2\pi\) phase and a large (\(>\)2 rad) root mean square (RMS). Then, the interferogram \(\phi_{24}\) with a red frame can be identified as a bad interferogram including significant unwrapping errors, because all loops associated with it are problematic, whilst the other interferograms generate at least one nonproblematic loop. To obtain the more realistic time series of the displacement even with a disconnected network, we follow the NSBAS method [19; 50; 53] that imposes a temporal constraint,  
\[\left[ \begin{array}{c}\mathbf{d}\\ \mathbf{0}\end{array} \right]=\left[\begin{array}{cccccc}\left[ \begin{array}{ ccccc}&&\mathbf{G}&&&\mathbf{0}&\mathbf{0}\\ \end{array} \right]\\ \gamma\left[ \begin{array}{cccccc}1&0&\ldots&\ldots&0&-t_{1}&-1\\ \vdots&\ddots&\ddots&&\vdots&-t_{2}&\vdots\\ 1&\ldots&1&\ddots&\vdots&\vdots&\vdots\\ \vdots&&\vdots&\ddots&0&\vdots&\vdots\\ 1&\ldots&1&\ldots&1&-t_{N-1}&-1\end{array} \right]\left[ \begin{array}{c} \mathbf{m}\\ v\\ c\end{array}