Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives Gap-filling of InSAR displacement time series Alexandre Hippert-Ferrer 1 , Yajing Yan 1 , Philippe Bolon 1 1 Laboratoire d’Informatique, Systèmes, Traitement de l’Information et la Connaissance (LISTIC), Annecy, France Wednesday, October 16 Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 1 / 20
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Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives
Gap-filling of InSAR displacement time series
Alexandre Hippert-Ferrer1, Yajing Yan1, Philippe Bolon1
1Laboratoire d’Informatique, Systèmes, Traitement de l’Information et la Connaissance (LISTIC), Annecy, France
Wednesday, October 16
Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 1 / 20
Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives
Introduction
Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 2 / 20
Missing data is a frequent issue in SAR-derived products in both space and time.
It can prevent the full understanding of the physical phenomena under observation.
Signal learned as empirical orthogonal functions (EOFs).
Low rank structure of the sample temporal covariance matrix.
Reconstruction with appropriate initialization of missing values 1.
Expectation-Maximization (EM)-type algorithm for resolution.
1. [1] Beckers and Rixen, “EOF calculations and data filling from incompleteoceanographics datasets,” J. Atmos. Oceanic Technol., vol.20(12), pp.1836-1856, 2003
Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 4 / 20
Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives
EM-EOF : data representation and initialization
Let X(s, t) be a spatio-temporal field containing the values of X at position s andtime t :
(xij )1≤i≤p,1≤j≤n is the value at position si and time tj and may be missing.
Missing values are then initialized by an appropriate value (first guess).
Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 5 / 20
Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives
EM-EOF : covariance estimation and decomposition
The sample temporal covariance is first estimated :
Σ̂ =1
p − 1(X − 1nX̄ )T (X − 1nX̄)
EOFs (ui )0≤i≤n are the solution of the eigenvalue problem :
Σ̂U = UΛ
EOFs can be used to express Σ̂ in terms of EOF modes :
Σ̂ = λ1u1uT1 + λ2u2uT
2 + · · ·+ λnunuTn
Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 6 / 20
Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives
EM-EOF : covariance estimation and decomposition
The sample temporal covariance is first estimated :
Σ̂ =1
p − 1(X − 1nX̄ )T (X − 1nX̄)
EOFs (ui )0≤i≤n are the solution of the eigenvalue problem :
Σ̂U = UΛ
EOFs can be used to express Σ̂ in terms of EOF modes :
Σ̂ = λ1u1uT1 + λ2u2uT
2 + · · ·+ λnunuTn
Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 6 / 20
Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives
EM-EOF : covariance estimation and decomposition
The sample temporal covariance is first estimated :
Σ̂ =1
p − 1(X − 1nX̄ )T (X − 1nX̄)
EOFs (ui )0≤i≤n are the solution of the eigenvalue problem :
Σ̂U = UΛ
EOFs can be used to express Σ̂ in terms of EOF modes :
Σ̂ = λ1u1uT1 + λ2u2uT
2 + · · ·+ λnunuTn
Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 6 / 20
Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives
EM-EOF : reconstruction of the field
X ′ is reconstructed with M number of EOFs :
X ′ =n∑
i=1
ai uTi → X̂
′=
M�n∑i=1
ai uTi
with ai = X ′ui are the Principal Components (PCs) of the anomaly field (X ′).
The first EOF modes capture the main temporal dynamical behavior of the signalwhereas other modes represent various perturbations 2.
Goal : find the optimal M
2. [3] R. Prébet, Y. Yan, M. Jauvin and E. Trouvé, “A data-adaptative EOF based method for displacement signalretrieval from InSAR displacement measurement time series for decorrelating targets”, IEEE Trans. Geosci. RemoteSens., vol. 57(8), pp. 5829-5852, 2019
Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 7 / 20
Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives
Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives
Case 1 : Gorner Glacier
Number of EOF modes : 3
Consistent pattern in missing data areas
Missing interferogram is reconstructed by adding the temporal mean to theanomaly.
Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 14 / 20
Initial Reconstructed Residual
A. Hippert-Ferrer, Y. Yan and P. Bolon, Em-EOF : gap-filling inincomplete SAR displacement time series, 2019, in revision.
Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives
Case 2 : Miage Glacier
Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 15 / 20
Initial Reconstructed Residual
Number of EOF modes : 2
Discontinuities in theresiduals due to phasejumps in the originalinterferogram.
Detection and correction ofinconsistencies.
Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives
Case 3 : Argentière Glacier
Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 16 / 20
Initial Reconstructed Residual
Very low SNR and strongcorrelated gaps in space and time
Strong mixing betweendisplacement signal and noise
Global agreement betweenreconstructed and initial fields
Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives
Conclusion
ConclusionEM-EOF : a new method to handle complex cases
Missing interferogramsDiscontinuities due to phase jumps (coherence loss)
Allows to increase the effective size of a time series.Limitations
More sensitive to SNR than to % of gaps.Argentière case : some breakdown points→ potential for improvement
PerspectivesEstimation of a covariance matrix with missing dataTime series of complex interferograms before unwrappingApplications : slow slip event, glacier velocities from optical data...
Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 17 / 20
Références
Questions
Thank you for your attention.
[1] J. M. Beckers and M. Rixen. EOF calculations and data filling from incompleteoceanographics datasets. J. Atmos. Oceanic Technol., 20(12) :1836–1856, 2003.
[2] R. Fallourd, O. Harant, E. Trouvé, and P. Bolon. Monitoring temperate glacierdisplacement by multi-temporal TerraSAR-X images and continuous GPSmeasurements. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens., 4(2) :372–386, 2011.
[3] R. Prébet, Y. Yan, M. Jauvin, and E. Trouvé. A data-adaptative eof based methodfor displacement signal retrieval from insar displacement measurement time seriesfor decorrelating targets. IEEE Trans. Geosci. Remote Sens., 57(8) :5829–5852,2019.
This work has been supported by the Programme National de Télédétection Spatiale (PNTS,http ://www.insu.cnrs.fr/pnts), grant PNTS-2019-11, and by the SIRGA project.
Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 18 / 20
Références
Worst case scenarii
Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 19 / 20
Correlated gaps :
Random gaps :
Références
Computation of a correlated noise
From an autocorrelation function c(r) = r−β and a white noise image b :
1. Compute power spectral density of c : Γ(c) = |F{c}|2. Compute FT of b : F{b}3. Do some filtering : F{b}Γ(c)
4. Compute F−1{F{b}Γ(c)}
Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 20 / 20