*2.1. InSAR Phase Noise Principle*

The interferogram Γ is defined as the conjugate product of a pair of single-look complex SAR images.

$$\Gamma = \mathbf{z}\_1 \cdot \mathbf{z}\_2^\* = |\mathbf{z}\_1 \cdot \mathbf{z}\_2^\*| e^{j(\boldsymbol{\Phi})} \tag{1}$$

where **z**1 and **z**2 are the two complex SAR images; \* indicates the complex conjugate; and ϕ denotes the measured interferometric phase with noise. The phase noising model with additive noise is as follows.

$$
\mathfrak{q} = \mathfrak{q}\_{\mathfrak{c}} + \mathfrak{n}\_{\mathfrak{Y}} \tag{2}
$$

where ϕ**c** denotes the clear interferometric phase and **<sup>n</sup>**γ is the zero-mean additive Gaussian noise associated with the coherence coefficient *γ*, which is expressed as:

$$\gamma = \frac{E(\mathbf{z}\_1 \cdot \mathbf{z}\_2^\*)}{\sqrt{E\left(|\mathbf{z}\_1|^2\right) \cdot E\left(|\mathbf{z}\_2|^2\right)}}\tag{3}$$

The interferometric phase noise level is correlated with the coherence coefficient *γ*. The higher the coherence coefficient, the lower the noise level.

It is worth noting that the value range of the interferometric phase is distributed in (− π, π]. Hence, phase denoising is implemented in the complex field in for the subsequent phase unwrapping steps. Employing a mathematical manipulation on Equation (2), we obtain the following expression.

$$\mathbf{p} = \mathbf{e}^{j\mathbf{q}} = \mathbf{e}^{j(\mathbf{q}\_c + \mathbf{n}\_\gamma)} = \mathbf{q}\_{\bar{r}} + j\mathbf{q}\_{\bar{i}} \tag{4}$$

where **p** indicates the interferogram and ϕ*r* and ϕ*i* denote the real and imaginary parts of **p**, respectively. According to the analysis of [45], ϕ*r* and ϕ*i* can be given by.

$$\begin{aligned} \mathfrak{sp}\_{\mathcal{I}} &= \cos(\mathfrak{sp}\_{\mathcal{L}} + \mathfrak{n}\_{\mathcal{I}}) = \aleph\_{\mathcal{L}}\cos(\mathfrak{sp}\_{\mathcal{L}}) + \mathfrak{n}\_{\mathcal{I}\mathfrak{c}} \\ \mathfrak{sp}\_{i} &= \sin(\mathfrak{sp}\_{\mathcal{L}} + \mathfrak{n}\_{\mathcal{I}}) = \aleph\_{\mathcal{L}}\sin(\mathfrak{sp}\_{\mathcal{L}}) + \mathfrak{n}\_{\mathcal{I}\mathfrak{s}} \end{aligned} \tag{5}$$

where **<sup>n</sup>***γc* and **<sup>n</sup>***γs* are defined as the zero-mean additive noise, and *Nc* is a phase quality evaluation metric related to the coherent coefficient *γ*. Finally, combining Equations (4) and (5), the formulation of the interferometric phase is derived as follows.

$$\mathbf{p} = \mathbf{N}\_{\mathcal{E}} \cdot \mathbf{p}\_{\mathcal{E}} + \mathbf{n} \tag{6}$$

where **n** = **<sup>n</sup>***γc* + *j***<sup>n</sup>***γs* denotes the noise of the interferogram and **p***c* = *ej*<sup>ϕ</sup>*<sup>c</sup>* is the ideal interferogram.

### *2.2. InSAR Phase SR Filtering Model*

The conventional SR model for recovering a single from a measurement is as follows.

$$y = \Phi x + n\tag{7}$$

where *y* ∈ R*<sup>M</sup>* is the measurement; **Φ** ∈ R*M*×*<sup>N</sup>* is dubbed the measurement matrix; *x* ∈ R*<sup>N</sup>* is the recovered signal; and *n* ∈ R*<sup>M</sup>* is the noise.

According to Section 2.1, the InSAR phase filtering model can be modeled as.

$$\mathbf{p} = \Phi \mathbf{p}\_c + \mathbf{n} \tag{8}$$

Aiming to solve Equation (8), it is transformed into the following convex optimization problem.

$$\hat{\mathbf{p}}\_{\varepsilon} = \underset{\mathbf{p}\_{\varepsilon}}{\operatorname{argmin}} \frac{1}{2} \|\boldsymbol{\Phi}\mathbf{p}\_{\varepsilon} - \mathbf{p}\|\_{2}^{2} + \lambda \|\boldsymbol{\Psi}\mathbf{p}\_{\varepsilon}\|\_{1} \tag{9}$$

where ψ**p***c* is a sparse representation of **p***c*; ψ denotes a certain transform such as Wavelet, Fourier and so on; and *λ* is a tunable soft threshold parameter.

The main iterative steps of solving Equation (9) by the ISTA algorithm are as follows.

$$\mathbf{h}^{(k)} = \mathbf{p}\_{\varepsilon}^{\ (k-1)} - \kappa \boldsymbol{\Phi}^{\mathbf{H}} \left( \boldsymbol{\Phi} \mathbf{p}\_{\varepsilon}^{\ (k-1)} - \mathbf{p} \right) \tag{10}$$

$$\mathbf{p}\_c^{(k)} = \boldsymbol{\Psi}^{\mathbf{H}} \text{soft}\left(\boldsymbol{\Psi} \mathbf{h}^{(k)}, \boldsymbol{\lambda}\right) = \boldsymbol{\Psi}^{\mathbf{H}} \text{sign}(\boldsymbol{\Psi} \mathbf{h}^{(k)}) \text{max}\left\{ \left| \boldsymbol{\Psi} \mathbf{h}^{(k)} \right| - \boldsymbol{\lambda}, 0 \right\} \tag{11}$$

where *α* indicates the step size; sign(·) denotes a sign function; **Φ<sup>H</sup>** is the conjugate transpose of **Φ**; and **h**(*k*) is the residual error in iteration *k*. However, the sparse transform ψ and parameters such as *α* and *λ* are hand-crafted, which results in the algorithm being nonadaptive. Moreover, ISTA usually suffers from a huge calculative burden due to its large number of iterative steps.

We chose the identity matrix as the measurement matrix **Φ** by the formulation of the interferometric phase. It can be seen from the above analysis that the phase filtering is operated in the complex domain to achieve high-precision phase unwrapping. Therefore, taking advantage of the idea of separating real and imaginary parts, the noisy phase **p**, the ideal phase **p***c*, and the measurement matrix **Φ** are transformed as

$$\mathbf{p}\_{\mathbb{R}} = \left( \begin{array}{c} \mathfrak{q}\_{\mathbb{T}} \\ \mathfrak{q}\_{i} \end{array} \right) \mathbf{p}\_{\mathbb{c}\mathbb{R}} = \left( \begin{array}{c} \mathfrak{q}\_{\mathbb{c}\mathbb{T}} \\ \mathfrak{q}\_{\mathbb{c}i} \end{array} \right) \tag{12}$$

$$
\Phi\_{\mathbb{R}} = \begin{pmatrix} \Phi\_r & -\Phi\_i \\ \Phi\_i & \Phi\_r \end{pmatrix} \tag{13}
$$

where ϕ*cr* and ϕ*ci* are the real and imaginary parts of the ideal interferogram **p***c*, and **Φ***r* and **Φ***i* denote the real and imaginary parts of the measurement matrix **Φ**, respectively.

In the end, the filtered interferometric phase is obtained by the real part ϕ *r* and imaginary part ϕ *i*of the recovered interferometric phase as follows.

$$\mathbf{q}' = \angle(\mathbf{q}'\_r + j\mathbf{q}'\_i) \tag{14}$$
