## 3 Data and Methodology
### Time-Series Analysis
Naturally, we can use d (t\({}_{\text{h}}\), x, y) h = 1,..., N to represent the deformation time series we want to obtain and set the corresponding phase as \(\varphi\) (t\({}_{\text{h}}\), x, y), Then, there are  
\[\Phi(\text{t}_{\text{h}},\,\text{x, y})\ \approx\ \frac{4\pi}{\lambda}\text{d}( \text{t}_{\text{h}},\,\text{x, y}) \tag{3}\]  
After unwrapping the phase signal, we can select a corrected pixel as a reference and represent its corresponding N unknown phase values as vectors:  
\[\varphi\text{T}=[\varphi\,(\text{t}_{1}),\ \cdots,\ \varphi(\text{t}_{\text{N}})] \tag{4}\]  
The M values calculated from the differential interferogram are represented as vectors:  
\[\delta\varphi\text{T}=[\delta\varphi_{1},\ \cdots,\ \delta\varphi_{\text{M}}] \tag{5}\]  
Formula (4) can be defined using the following two vectors (HZ and HE correspond to the acquisition time series from the image and the main image in the image pair that generates the interferogram, respectively):  
\[\text{HZ}=[\text{HZ}_{1},\cdots,\text{HZ}_{\text{M}}]\qquad\text{HE}=[\text{ HE}_{1},\cdots,\text{HE}_{\text{M}}] \tag{6}\]The main image and the sub images are arranged in chronological order, i.e., HE\({}_{\rm k}\) \(>\) HZ\({}_{\rm k}\), \(\forall\)k = 1, \(\ldots\), M, that is, the following equation:  
\[\delta\varphi_{\rm k}=\varphi(t_{\rm Hek})-\varphi(t_{\rm HZk})\quad\forall_{\rm k }=1,\ldots,\rm M \tag{7}\]  
The system of equations consisting of M equations with N unknowns is represented in matrix form as follows:  
\[\rm A\varphi=\delta\varphi \tag{8}\]  
We can perform SVD decomposition on equation system (8) as follows:  
\[\rm A=UZV^{\rm T} \tag{9}\]  
U is an M \(\times\) M orthogonal matrix, with the first N rows being the eigenvectors of AAT, known as the left singular vector of A; V is an N \(\times\) M unitary matrix, and all its rows are the eigenvectors of ATA, called the right singular vector of A; A is an M \(\times\) N matrix, and its elements (singular values \(\sigma_{\rm h}\)) are the square roots of the corresponding eigenvalues of the M \(\times\) N matrix AAT. Usually, M \(>\) N, and there are M-N eigenvalues with 0. Moreover, due to the rank loss property of matrix A, there are L-1 additional zero eigenvalues, namely:  
\[Z=\rm diag(\sigma_{1},\ \cdots\cdots,\ \sigma_{N-L+1},\ 0,\ \cdots\cdots,\ 0) \tag{10}\]  
Under the constraint of least squares, we can calculate the value of \(\varphi\) as follows:  
\[\delta\Phi_{\text{k}}(\text{x, y})=\varphi(\text{t}_{\text{B}},\,\text{x, y})-\varphi(\text{t}_{\text{A}},\,\text{x, y})\ \approx\ \frac{4\pi}{\lambda}[\text{d}(\text{t}_{\text{B}},\,\text{x, y})-\text{d}(\text{t}_{\text{A}},\,\text{x, y})] \tag{2}\]0  
In the equation, \(Z^{+}=\rm diag(1/\sigma_{1},\ \cdots\cdots,\ 1/\sigma_{N-L+1},\ 0,\ \cdots\cdots,\ 0)\); therefore:  
\[\delta\Phi_{\text{k}}(\text{x, y})=\varphi(\text{t}_{\text{B}},\,\text{x, y})-\varphi(\text{t}_{\text{A}},\,\text{x, y})\ \approx\ \frac{4\pi}{\lambda}[\text{d}(\text{t}_{\text{B}},\,\text{x, y})-\text{d}(\text{t}_{\text{A}},\,\text{x, y})] \tag{2}\]1  
U\({}_{\rm h}\) and V\({}_{\rm h}\) are the row vectors of U and V, respectively, and the phase is converted to the average phase velocity:  
\[\delta\Phi_{\text{k}}(\text{x, y})=\varphi(\text{t}_{\text{B}},\,\text{x, y})-\varphi(\text{t}_{\text{A}},\,\text{x, y})\ \approx\ \frac{4\pi}{\lambda}[\text{d}(\text{t}_{\text{B}},\,\text{x, y})-\text{d}(\text{t}_{\text{A}},\,\text{x, y})] \tag{2}\]2  
Thus, a new matrix equation is obtained: D\({}_{\rm V}\) = \(\delta\varphi\); D is also an M \(\times\) N matrix. For the k-th row, the column located between the acquisition times of the main and auxiliary images, D(k, k) = t\({}_{\rm k+1}-\) t\({}_{\rm k}\), Other D (k, k) = 0. In this case, applying SVD decomposition to matrix D yields the minimum norm solution of velocity vector V. Furthermore, starting with the composition of differential phase, we know that in addition to the contribution of deformation phase, there is also a phase contribution of elevation error \(\Delta\)q.