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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-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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Page 1: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

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

Page 2: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

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.

Causes : rapid surface changes, missing image, technical limitations.

Argentiere glacier, offset tracking of TerraSAR-X in Summer 2010 [2]

Shen et al. 2015

Interferogram over land area,Mexico (Isterre)

Page 3: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives

Motivation of the study

Handling missing data in InSAR displacement time seriesClassical approach : spatial interpolationNot exploited (yet) : temporal information

→ Manage spatio-temporal missing data in time series←

Proposed : a statistical gap-filling method addressing1. Randomness and possible space time correlation of

NoiseMissing data

2. Mixed frequencies displacement patterns (complex signals)

Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 3 / 20

Page 4: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives

Expectation Maximization-Empirical Orthogonal Functions

Key components of the proposed method :

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

Page 5: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

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 :

X =(x1, x2, . . . xn

)=

x11 x12 x13 · · · x1nx21 x22 x23 · · · x2nx31 x32 x33 · · · x3n

......

.... . .

...xp1 xp2 xp3 · · · xpn

(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

Page 6: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

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

Page 7: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

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

Page 8: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

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

Page 9: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

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

Page 10: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives

Cross-validation

Cross-RMSE : cross-validation root-mean-square error :

E(k) =

[1N

N∑k=1

|x̂k − x|2]1/2

k : number of EOF modes used in the reconstruction

Key parameter : the optimal number of EOF modes M, estimated by :

arg minM∈[1,n]

E(k)

Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 8 / 20

Page 11: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives

A 2-stage method

Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 9 / 20

EM-typeiterations

Page 12: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives

Numerical simulations : setup

Displacement fields with different complexity are computed :

g(r , t) Order

g1 (1− 0.5r)t + sin(2πf1t) cos(2πf1r) + 0.5 cos(2πf2t) cos(2πf3r) 3

g2 g1 + 0.1 sin(2πf4t) cos(2πf5r) 4

g3 A + b exp(−t/τe) + ct -

TABLE – f1 = 0.25, f2 = 0.75, f3 = 2.5, f4 = 1.25, f5 = 5.

Type of noise : random ∼ N (0, 1), spatially and spatio-temporally correlated

Type of gaps : random, correlated

SNR = [0.5,4.5]

Gaps : [0,80]%

Initialization value : spatial mean, spatial mean + random noise, spatial mean +correlated noise

Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 10 / 20

Page 13: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives

Numerical simulations : setup

Displacement fields with different complexity are computed :

g(r , t) Order

g1 (1− 0.5r)t + sin(2πf1t) cos(2πf1r) + 0.5 cos(2πf2t) cos(2πf3r) 3

g2 g1 + 0.1 sin(2πf4t) cos(2πf5r) 4

g3 A + b exp(−t/τe) + ct -

TABLE – f1 = 0.25, f2 = 0.75, f3 = 2.5, f4 = 1.25, f5 = 5.

Type of noise : random ∼ N (0, 1), spatially and spatio-temporally correlated

Type of gaps : random, correlated

SNR = [0.5,4.5]

Gaps : [0,80]%

Initialization value : spatial mean, spatial mean + random noise, spatial mean +correlated noise

Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 10 / 20

Page 14: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives

Numerical simulations : setup

Displacement fields with different complexity are computed :

g(r , t) Order

g1 (1− 0.5r)t + sin(2πf1t) cos(2πf1r) + 0.5 cos(2πf2t) cos(2πf3r) 3

g2 g1 + 0.1 sin(2πf4t) cos(2πf5r) 4

g3 A + b exp(−t/τe) + ct -

TABLE – f1 = 0.25, f2 = 0.75, f3 = 2.5, f4 = 1.25, f5 = 5.

Type of noise : random ∼ N (0, 1), spatially and spatio-temporally correlated

Type of gaps : random, correlated

SNR = [0.5,4.5]

Gaps : [0,80]%

Initialization value : spatial mean, spatial mean + random noise, spatial mean +correlated noise

Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 10 / 20

Page 15: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives

Numerical simulations : 3rd, 4th order and seismic deformation fields

Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 11 / 20

SNR and % of gaps are fixed :

SNR = 1.5

30% of gaps

Number of EOF modes vs. cross-RMSE :

→ Minimum of the error found at thesignal order

Page 16: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives

Sensitivity to SNR and % of gaps

Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 12 / 20

Cross-RMSE in function of % of gaps and SNR :

3rd order

4th order

The method is more sensitive toSNR than to the % of gaps

Random gaps affect more thereconstruction than correlated(seasonal) gaps

Initialization value affects the timeof convergence but not theestimation of M

Page 17: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

Context and motivation The EM-EOF method Numerical simulations Applications Conclusion and perspectives

Data and area of study

Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 13 / 20

Glacier Period Platform Data type Size [min, max]%missing

Missingimages

Gorner 11/2016-03/2017 Sentinel-1/A Interferometry 16 [11.8, 27.8]% 4Miage 12/2016-03/2017 Sentinel-1/A Interferometry 13 [11.4, 23.1]% 3Argentière 10/2016-12/2017 Sentinel-1/A/B Offset tracking 65 [2, 50]% 0

TABLE – Time series description.

Page 18: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

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.

Page 19: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

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.

Page 20: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

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

Page 21: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

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

Page 22: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

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

Page 23: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

Références

Worst case scenarii

Alexandre Hippert-Ferrer, Yajing Yan, Philippe Bolon MDIS-2019 Wednesday, October 16 19 / 20

Correlated gaps :

Random gaps :

Page 24: Gap-filling of InSAR displacement time serieseost.unistra.fr/fileadmin/upload/EOST/MDIS_presentation/MDIS2019_Hippert.pdf · Context and motivation The EM-EOF method Numerical simulations

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