Dynamically estimating deformations with wrapped InSAR based on sequential adjustment

Abstract

The Interferometric Synthetic Aperture Radar (InSAR) technique has been greatly improved in both scientific studies and engineering applications as more and more InSAR observations become available. However, the long-term and dynamic deformation analysis would be more challenging for ordinary users due to the so big volume of InSAR data. In this paper, we proposed a method, termed as SeaInSAR, which introduces the rationale of sequential adjustment (Sea) algorithm to dynamically estimate the surface deformations with wrapped interferometric phase. The proposed method focuses on the arcs of adjacent coherent points in the short-baseline interferograms to avoid the time-consuming and error-prone phase-unwrapping procedure. When dynamically monitoring the temporal deformations with respect to the area of interest, the Sea algorithm, based on the previous result together with newly observed data, can accelerate the procedure of estimating deformation parameters. Furthermore, by only involving the partial previous deformation time series result rather than all the result, the efficiency of the SeaInSAR method can be significantly improved without any compromise of the accuracy. The Sea algorithm is also used to realize the dynamic estimation of a fading signal in the multi-looked or filtered interferograms. This signal is induced by the temporally varying surface moisture and can bias the deformations when only short temporal-baseline interferograms are used. Its dynamic estimation benefits the high-accuracy and high-efficiency deformation monitoring in the era of big InSAR data. Simulated and real data experiments are conducted in this paper, demonstrating that the proposed SeaInSAR method can improve the computational efficiency by more than 20 times compared with the classical static method. With the availability of more and more SAR data as well as the increasing demand of InSAR engineering applications, the proposed SeaInSAR method has a great potential in the InSAR post-processing procedure.

Appendix

1.1 A. Dynamic estimation of deformation model parameters and topographic error

Given two functional models with respect to the archived and new observations, respectively,

$$ {\varvec{L}}^{\left( 1 \right)} = {\mathbf{\mathcal{A}}}^{\left( 1 \right)} \cdot {\mathbf{\mathcal{X}}} + {\mathbf{\rm E}}^{\left( 1 \right)} , {\varvec{P}}^{\left( 1 \right)} $$

(A1)

$$ {\varvec{L}}^{\left( 2 \right)} = {\mathbf{\mathcal{A}}}^{\left( 2 \right)} \cdot {\mathbf{\mathcal{X}}} + {\mathbf{\rm E}}^{\left( 2 \right)} , {\varvec{P}}^{\left( 2 \right)} $$

(A2)

the initial estimation of the unknown ${\hat{\mathbf{\mathcal{X}}}}^{\left( 1 \right)}$ and its variance and covariance matrix ${\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 1 \right)}$ can be derived based on Eq. (A1) by the WLS method

$$ {\hat{\mathbf{\mathcal{X}}}}^{\left( 1 \right)} = \left( {\left( {{\mathbf{\mathcal{A}}}^{\left( 1 \right)} } \right)^{T} {\varvec{P}}^{\left( 1 \right)} {\mathbf{\mathcal{A}}}^{\left( 1 \right)} } \right)^{ - 1} \left( {{\mathbf{\mathcal{A}}}^{\left( 1 \right)} } \right)^{T} {\varvec{P}}^{\left( 1 \right)} {\varvec{L}}^{\left( 1 \right)} $$

(A3)

$$ {\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 1 \right)} = \left( {\left( {{\mathbf{\mathcal{A}}}^{\left( 1 \right)} } \right)^{T} {\varvec{P}}^{\left( 1 \right)} {\mathbf{\mathcal{A}}}^{\left( 1 \right)} } \right)^{ - 1} $$

(A4)

The dynamic estimation of the unknowns ${\hat{\mathbf{\mathcal{X}}}}^{\left( 2 \right)}$ can be considered as the static estimation of combining Eqs. (A1) and (A2), i.e.,

$$ {\hat{\mathbf{\mathcal{X}}}}^{\left( 2 \right)} = {\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 2 \right)} \left( {\left( {{\mathbf{\mathcal{A}}}^{\left( 1 \right)} } \right)^{T} {\varvec{P}}^{\left( 1 \right)} {\varvec{L}}^{\left( 1 \right)} + \left( {{\mathbf{\mathcal{A}}}^{\left( 2 \right)} } \right)^{T} {\varvec{P}}^{\left( 2 \right)} {\varvec{L}}^{\left( 2 \right)} } \right) $$

(A5)

where ${\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 2 \right)} = \left( {\left( {{\mathbf{\mathcal{A}}}^{\left( 1 \right)} } \right)^{T} {\varvec{P}}^{\left( 1 \right)} {\mathbf{\mathcal{A}}}^{\left( 1 \right)} + \left( {{\mathbf{\mathcal{A}}}^{\left( 2 \right)} } \right)^{T} {\varvec{P}}^{\left( 2 \right)} {\mathbf{\mathcal{A}}}^{\left( 2 \right)} } \right)^{ - 1}$. Let

$$ \left( {{\mathbf{\mathcal{A}}}^{\left( 1 \right)} } \right)^{T} {\varvec{P}}^{\left( 1 \right)} {\varvec{L}}^{\left( 1 \right)} = \left( {{\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 1 \right)} } \right)^{ - 1} {\hat{\mathbf{\mathcal{X}}}}^{\left( 1 \right)} $$

(A6)

$$ \left( {{\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 1 \right)} } \right)^{ - 1} = {\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 2 \right)} - \left( {{\mathbf{\mathcal{A}}}^{\left( 2 \right)} } \right)^{T} {\varvec{P}}^{\left( 2 \right)} {\mathbf{\mathcal{A}}}^{\left( 2 \right)} $$

(A7)

Then, Eq. (A5) can be written as

$$ {\hat{\mathbf{\mathcal{X}}}}^{\left( 2 \right)} = {\hat{\mathbf{\mathcal{X}}}}^{\left( 1 \right)} + {\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 2 \right)} \left( {{\mathbf{\mathcal{A}}}^{\left( 2 \right)} } \right)^{T} {\varvec{P}}^{\left( 2 \right)} \left( {{\varvec{L}}^{\left( 2 \right)} - {\mathbf{\mathcal{A}}}^{\left( 2 \right)} {\hat{\mathbf{\mathcal{X}}}}^{\left( 1 \right)} } \right) $$

(A8)

Based on the matrix inversion formula, ${\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 2 \right)}$ can be represented as

$$ {\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 2 \right)} = \left( {\left( {{\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 1 \right)} } \right)^{ - 1} + \left( {{\mathbf{\mathcal{A}}}^{\left( 2 \right)} } \right)^{T} {\varvec{P}}^{\left( 2 \right)} {\mathbf{\mathcal{A}}}^{\left( 2 \right)} } \right)^{ - 1} = {\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 1 \right)} - {\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 1 \right)} \left( {{\mathbf{\mathcal{A}}}^{\left( 2 \right)} } \right)^{T} \left( {\left( {{\varvec{P}}^{\left( 2 \right)} } \right)^{ - 1} + {\mathbf{\mathcal{A}}}^{\left( 2 \right)} {\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 1 \right)} \left( {{\mathbf{\mathcal{A}}}^{\left( 2 \right)} } \right)^{T} } \right)^{ - 1} {\mathbf{\mathcal{A}}}^{\left( 2 \right)} {\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 1 \right)} $$

(A9)

Simultaneously, after premultiplication of Eq. (A7) by ${\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 2 \right)}$ and postmultiplication of Eq. (A7) by ${\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 1 \right)}$, we can obtain the following formula

$$ {\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 2 \right)} = {\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 1 \right)} - {\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 2 \right)} \left( {{\mathbf{\mathcal{A}}}^{\left( 2 \right)} } \right)^{T} {\varvec{P}}^{\left( 2 \right)} {\mathbf{\mathcal{A}}}^{\left( 2 \right)} {\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 1 \right)} $$

(A10)

By comparing Eqs. (A9) and (A10), the following equation can be established

$$ {\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 2 \right)} \left( {{\mathbf{\mathcal{A}}}^{\left( 2 \right)} } \right)^{T} {\varvec{P}}^{\left( 2 \right)} = {\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 1 \right)} \left( {{\mathbf{\mathcal{A}}}^{\left( 2 \right)} } \right)^{T} \left( {\left( {{\varvec{P}}^{\left( 2 \right)} } \right)^{ - 1} + {\mathbf{\mathcal{A}}}^{\left( 2 \right)} {\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 1 \right)} \left( {{\mathbf{\mathcal{A}}}^{\left( 2 \right)} } \right)^{T} } \right)^{ - 1} $$

(A11)

Combining (A8) and (A11), we can obtain the following equation,

$$ \begin{aligned} {\hat{\mathbf{\mathcal{X}}}}^{\left( 2 \right)} & = {\hat{\mathbf{\mathcal{X}}}}^{\left( 1 \right)} + {\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 1 \right)} \left( {{\mathbf{\mathcal{A}}}^{\left( 2 \right)} } \right)^{T} \\ &\quad \left( {\left( {{\varvec{P}}^{\left( 2 \right)} } \right)^{ - 1} + {\mathbf{\mathcal{A}}}^{\left( 2 \right)} {\varvec{Q}}_{{{\hat{\mathbf{\mathcal{X}}}}}}^{\left( 1 \right)} \left( {{\mathbf{\mathcal{A}}}^{\left( 2 \right)} } \right)^{T} } \right)^{ - 1} \left( {{\varvec{L}}^{\left( 2 \right)} - {\mathbf{\mathcal{A}}}^{\left( 2 \right)} {\hat{\mathbf{\mathcal{X}}}}^{\left( 1 \right)} } \right) \end{aligned}$$

(A12)

which is the same as the equations in Sect. 2.3.1.

1.2 B. Dynamic estimation of deformation time series

Given the functional model between the deformation time series and the InSAR observations corrected by topographic error,

$$ \tilde{\user2{L}}^{\left( 1 \right)} = {\varvec{A}}^{\left( 1 \right)} {\varvec{X}} + {\varvec{B}}^{\left( 1 \right)} {\varvec{Y}},{ }\tilde{\user2{P}}^{\left( 1 \right)} $$

(A13)

$$ \tilde{\user2{L}}^{\left( 2 \right)} = {\varvec{B}}^{\left( 2 \right)} {\varvec{Y}} + {\varvec{C}}^{\left( 2 \right)} {\varvec{Z}},{ }\tilde{\user2{P}}^{\left( 2 \right)} $$

(A14)

the initial estimation of deformation time series $\hat{\mathbb{X}}^{\left( 1 \right)} = \left[ {\begin{array}{*{20}c} {\left( {\hat{\user2{X}}^{\left( 1 \right)} } \right)^{T} } & {\left( {\hat{\user2{Y}}^{\left( 1 \right)} } \right)^{T} } \\ \end{array} } \right]^{T}$ can be derived by

$$ \hat{\mathbb{X}}^{\left( 1 \right)} = {\varvec{Q}}_{{\hat{\mathbb{X}}}}^{\left( 1 \right)} \left[ {\begin{array}{*{20}c} {{\varvec{A}}^{\left( 1 \right)} } & {{\varvec{B}}^{\left( 1 \right)} } \\ \end{array} } \right]^{T} \tilde{\user2{P}}^{\left( 1 \right)} \tilde{\user2{L}}^{\left( 1 \right)} $$

(A15)

where

$$ {\varvec{Q}}_{{\hat{\mathbb{X}}}}^{\left( 1 \right)} = \left( {\left[ {\begin{array}{*{20}c} {{\varvec{A}}^{\left( 1 \right)} } & {{\varvec{B}}^{\left( 1 \right)} } \\ \end{array} } \right]^{T} \tilde{\user2{P}}^{\left( 1 \right)} \left[ {\begin{array}{*{20}c} {{\varvec{A}}^{\left( 1 \right)} } & {{\varvec{B}}^{\left( 1 \right)} } \\ \end{array} } \right]} \right)^{ - 1} = \left[ {\begin{array}{*{20}c} {\left( {{\varvec{A}}^{\left( 1 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 1 \right)} {\varvec{A}}^{\left( 1 \right)} } & {\left( {{\varvec{A}}^{\left( 1 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 1 \right)} {\varvec{B}}^{\left( 1 \right)} } \\ {\left( {{\varvec{B}}^{\left( 1 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 1 \right)} {\varvec{A}}^{\left( 1 \right)} } & {\left( {{\varvec{B}}^{\left( 1 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 1 \right)} {\varvec{B}}^{\left( 1 \right)} } \\ \end{array} } \right]^{ - 1} $$

(A16)

and ${\varvec{Q}}_{{\hat{\mathbb{X}}}}^{\left( 1 \right)}$ is the variance and covariance matrix of $\hat{\mathbb{X}}^{\left( 1 \right)}$.

When the new observation $\tilde{\user2{L}}^{\left( 2 \right)}$ is available, the functional model will be established by taking the initial estimation $\hat{\mathbb{X}}^{\left( 1 \right)}$ into account, i.e.,where

$$ \left\{ {\begin{array}{*{20}c} {\Delta \hat{\mathbb{X}}^{\left( 1 \right)} = {\mathbf{\mathcal{I}}}\left[ {\begin{array}{*{20}c} {\Delta {\varvec{X}}^{{\varvec{T}}} } & {\Delta {\varvec{Y}}^{{\varvec{T}}} } & {{\varvec{Z}}^{{\varvec{T}}} } \\ \end{array} } \right]^{T} ,{\varvec{Q}}_{{\hat{\mathbb{X}}}}^{\left( 1 \right)} } \\ \\ {{{\varvec{\Gamma}}} = {\mathbb{A}}^{\left( 2 \right)} \left[ {\begin{array}{*{20}c} {\Delta {\varvec{X}}^{{\varvec{T}}} } & {\Delta {\varvec{Y}}^{{\varvec{T}}} } & {{\varvec{Z}}^{{\varvec{T}}} } \\ \end{array} } \right]^{T} ,\tilde{\user2{P}}^{\left( 2 \right)} } \\ \end{array} } \right. $$

$$ {{\varvec{\Gamma}}} = \tilde{\user2{L}}^{\left( 2 \right)} - {\varvec{B}}^{\left( 2 \right)} \cdot {\varvec{Y}}^{\left( 1 \right)} $$

(A17)

$$ {\mathbf{\mathcal{I}}} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{array} } \right] $$

(A18)

$$ {\mathbb{A}}^{\left( 2 \right)} = \left[ {\begin{array}{*{20}c} 0 & {{\varvec{B}}^{\left( 2 \right)} } & {{\varvec{C}}^{\left( 2 \right)} } \\ \end{array} } \right] $$

(A19)

Then, based on the weighted least square (WLS) method, the following equations will be satisfied

$$ \left( {{\mathbf{\mathcal{I}}}^{T} \left( {{\varvec{Q}}_{{\hat{\mathbb{X}}}}^{\left( 1 \right)} } \right)^{ - 1} {\mathbf{\mathcal{I}}} + \left( {{\mathbb{A}}^{\left( 2 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 2 \right)} {\mathbb{A}}^{\left( 2 \right)} } \right)\left[ {\begin{array}{*{20}c} {\Delta {\varvec{X}}^{{\varvec{T}}} } & {\Delta {\varvec{Y}}^{{\varvec{T}}} } & {{\varvec{Z}}^{{\varvec{T}}} } \\ \end{array} } \right]^{T} = \left( {{\mathbb{A}}^{\left( 2 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 2 \right)} {{\varvec{\Gamma}}} $$

(A20)

$$ \left[ {\begin{array}{*{20}c} {\left( {{\varvec{A}}^{\left( 1 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 1 \right)} {\varvec{A}}^{\left( 1 \right)} } & {\left( {{\varvec{A}}^{\left( 1 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 1 \right)} {\varvec{B}}^{\left( 1 \right)} } & 0 \\ {\left( {{\varvec{B}}^{\left( 1 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 1 \right)} {\varvec{A}}^{\left( 1 \right)} } & {\left( {{\varvec{B}}^{\left( 1 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 1 \right)} {\varvec{B}}^{\left( 1 \right)} + \left( {{\varvec{B}}^{\left( 2 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 2 \right)} {\varvec{B}}^{\left( 2 \right)} } & {\left( {{\varvec{B}}^{\left( 2 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 2 \right)} {\varvec{C}}^{\left( 2 \right)} } \\ 0 & {\left( {{\varvec{C}}^{\left( 2 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 2 \right)} {\varvec{B}}^{\left( 2 \right)} } & {\left( {{\varvec{C}}^{\left( 2 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 2 \right)} {\varvec{C}}^{\left( 2 \right)} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\Delta {\varvec{X}}} \\ {\Delta {\varvec{Y}}} \\ {\varvec{Z}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 \\ {\left( {{\varvec{B}}^{\left( 2 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 2 \right)} {{\varvec{\Gamma}}}} \\ {\left( {{\varvec{C}}^{\left( 2 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 2 \right)} {{\varvec{\Gamma}}}} \\ \end{array} } \right] $$

(A21)

The relationship between $\Delta {\varvec{X}}$ and $\Delta {\varvec{Y}}$ can be derived from Eq. (A21) as

$$ \Delta {\varvec{X}} = - \left( {\left( {{\varvec{A}}^{\left( 1 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 1 \right)} {\varvec{A}}^{\left( 1 \right)} } \right)^{ - 1} \cdot \left( {{\varvec{A}}^{\left( 1 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 1 \right)} {\varvec{B}}^{\left( 1 \right)} \Delta {\varvec{Y}} $$

(A22)

Combining Eqs. (A21) and (A22), we can get the following equation

$$ \left[ {\begin{array}{*{20}c} {{\mathbf{\mathcal{P}}} + \left( {{\varvec{B}}^{\left( 2 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 2 \right)} {\varvec{B}}^{\left( 2 \right)} } & {\left( {{\varvec{B}}^{\left( 2 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 2 \right)} {\varvec{C}}^{\left( 2 \right)} } \\ {\left( {{\varvec{C}}^{\left( 2 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 2 \right)} {\varvec{B}}^{\left( 2 \right)} } & {\left( {{\varvec{C}}^{\left( 2 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 2 \right)} {\varvec{C}}^{\left( 2 \right)} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\Delta {\varvec{Y}}} \\ {\varvec{Z}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\left( {{\varvec{B}}^{\left( 2 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 2 \right)} {{\varvec{\Gamma}}}} \\ {\left( {{\varvec{C}}^{\left( 2 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 2 \right)} {{\varvec{\Gamma}}}} \\ \end{array} } \right] $$

(A23)

where

$$ \begin{aligned} {\mathbf{\mathcal{P}}} & = \left( {{\varvec{B}}^{\left( 1 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 1 \right)} {\varvec{B}}^{\left( 1 \right)} - \left( {{\varvec{B}}^{\left( 1 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 1 \right)} {\varvec{A}}^{\left( 1 \right)} \\ &\quad\quad \left( {\left( {{\varvec{A}}^{\left( 1 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 1 \right)} {\varvec{A}}^{\left( 1 \right)} } \right)^{ - 1} \left( {{\varvec{A}}^{\left( 1 \right)} } \right)^{T} \tilde{\user2{P}}^{\left( 1 \right)} {\varvec{B}}^{\left( 1 \right)} \end{aligned} $$

(A24)

So far, we have verified those equivalence.