A Minimum Curvature Combination Method for the Generation of Multi-Platform DInSAR Deformation Time-Series Antonio Pepe (1) , Giuseppe Solaro (1) , Claudio Dema (2) (1) IREA-CNR, via Diocleziano 328, 80124 Napoli (Italy) (2) Università degli Studi della Basilicata, Viale dell’Ateneo Lucano 10, 85100 Potenza (Italy)
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A Minimum Curvature Combination Method for the Generation of Multi-Platform DInSAR Deformation
Time-Series Antonio Pepe(1), Giuseppe Solaro(1), Claudio Dema (2)
(1) IREA-CNR, via Diocleziano 328, 80124 Napoli (Italy)
(2) Università degli Studi della Basilicata, Viale dell’Ateneo Lucano 10, 85100 Potenza (Italy)
Summary • A DInSAR-based technique for the retrieval of 3-D (Up-Down, East-West, North-
South) deformation time-series through a multi-angle/multi-platform combination strategy is presented.
• A post-processing strategy, relying on a Minimum-Curvature assumption, is applied to sequences of LOS-projected time-series to recover inherent displacement time-series components.
• It does not require the simultaneous processing of large SAR data archives …
• The algorithm can be seen as a further development of the M-SBAS* processing algorithm, however …
• … and can be applied with no restrictions at all concerning the DInSAR tool (SB or PS-oriented) used for the retrieval of LOS displacement time-series.
• Information on the reconstruction quality of used LOS time-series as well as external data (GPS/leveling) can also be used to better constrain the solution to data.
• S. Samsonov, N. d’Oreye , “Multidimensional time-series analysis of ground deformation from multiple InSAR data sets applied to Virunga Volcanic Province”, Geophysical Journal International 191, pp. 1095-1108, 2012.
Algorithm Overview First SAR Data Set
DInSAR Tool1
LOS1 Filtered Deformation
time-series
Residual Topography
Γ1 Reconstruction Quality
Factor
M-th SAR Data Set
DInSAR ToolM
LOSM Filtered Deformation
time-series
Residual Topography
ΓM Reconstruction Quality
Factor
Minimum Curvature SVD-
based LOS Displacement Time-Series
Combination
East-West Time-Series
Up-Down Time-Series
North-South Time-Series
External Data
Algorithm Overview
Time
The algorithm allows the generation of “long-term” times-series of the 3-D components of deformation by combining M (independently retrieved) radar LOS-projected deformation time-series.
Algorithm Overview
Time
The algorithm allows the production of “long-term” times-series of the 3-D components of deformation
We extend the M-SBAS strategy to a wider scenario where:
-LOS measurements, potentially retrieved using different DInSAR tools, are combined in a post-processing phase;
-Additional data (GPS/levelling) can also be jointly exploited;
-LOS time-series (instead of unwrapped interferograms) are combined;
-At variance with M-SBAS, a minimum-curvature combination is used.
Acquisition Geometries
EAST
UP
NORTH
Desc
Near Polar-Orbit Trajectories
small LOS sensibility to N-S components is small
Mathematical Framework
(*)
Similarly to SBAS, the system of equations (*) is reformulated (for each pixel) with respect to the unknowns, representing the displacement velocity vectors between consecutive time epochs
Mathematical Framework
This linear system is underdetermined with more unknowns than independent equations.
1. Berardino P. et alii: “A new Algorithm for Surface Deformation Monitoring based on Small Baseline Differential SAR Interferograms”, IEEE Trans. Geosci. Remote Sensing, Vol. 40, No. 11, pp. 2375-2383, November 2002.
Equations
Unknowns
Mathematical framework On a pixel-by-pixel basis, a quality index for the obtained LOS-projected time-series is now available (e.g., the temporal coherence Γ).
Depending on the number of SAR acquisitions N, DInSAR time-series accuracy empirically decrease as:
1. Berardino P. et alii: “A new Algorithm for Surface Deformation Monitoring based on Small Baseline Differential SAR Interferograms”, IEEE Trans. Geosci. Remote Sensing, Vol. 40, No. 11, pp. 2375-2383, November 2002.
with .To solve this linear system, the SVD-method is applied1.
Accordingly, the system can be reformulated as:
(**)
The system (**) is regularized introducing additional equations.
Minimum Curvature Method
In our case, we impose that acceleration between consecutive time acquisition is minimal; this is done adding following equations:
We remark that in the M-SBAS case, the regularization conditions imposed the solution had Minimum-velocity Norm (MN).
Minimum Curvature Method
with λ being a regularization factor.
The regularized system is solved in a LS sense with respect to the v unknowns, which temporally integrated lead to the 3-D time-series of displacement. While additional measurements (coming from GPS/levelling campaigns, or Pixel-Offset-based N-S measurements) are available, they can be treated as further sets to be used in the optimization.
Then:
However, since the LOS sensitivity to North-South components of deformation is rather small:
with:
Minimum Curvature Method
Let us consider two sets of SAR data collected (for instance) over ascending/descending orbits
Joint exploitation of Ascending / Descending data
ENVISAT Simulated Case
Linear Deformation
Time [day]
Time [day]
Disp
lace
men
t [cm
] Di
spla
cem
ent [
cm]
UP
EAST
Estimated
Simulated
MN
Linear Deformation
Time [day]
Time [day]
Disp
lace
men
t [cm
] Di
spla
cem
ent [
cm]
UP
EAST
Estimated
Simulated
MC
Linear Deformation + Jumps
Time [day]
Time [day]
Disp
lace
men
t [cm
] Di
spla
cem
ent [
cm]
UP
EAST
Estimated
Simulated
MN
Linear Deformation+Jumps
Time [day]
Time [day]
Disp
lace
men
t [cm
] Di
spla
cem
ent [
cm]
UP
EAST
Estimated
Simulated
MC
Quadratic Deformation
Time [day]
Time [day]
Disp
lace
men
t [cm
] Di
spla
cem
ent [
cm]
UP
EAST
Estimated
Simulated
MN
Quadratic Deformation
Time [day]
Time [day]
Disp
lace
men
t [cm
] Di
spla
cem
ent [
cm]
UP
EAST
Estimated
Simulated
MC
Effect of Noise: Large Signals
Time [day]
Time [day]
Disp
lace
men
t [cm
] Di
spla
cem
ent [
cm]
UP
EAST
Simulated
MN
Estimated Gaussian σ=0,75 cm
Effect of Noise: Large Signals
Time [day]
Time [day]
Disp
lace
men
t [cm
] Di
spla
cem
ent [
cm]
UP
EAST
Estimated Gaussian σ=0,75 cm
Simulated
MC
Piton de La Fournaise Area
Piton de La Fournaise
ASCENDING 48 ASAR/ENVISAT IMAGES DESCENDING
35 ASAR/ENVISAT IMAGES >20
<-20
cm/y
ear
>10
<-10
cm/y
ear
>20
<-20
cm/y
ear
Piton de La Fournaise
ASCENDING 11 ALOS-1 IMAGES
>10
<-10
cm/y
ear
UP-DOWN WEST-EAST
Piton de La Fournaise
UP-DOWN WEST-EAST
Piton de La Fournaise
ALOS ENV. ASC ENV. DES
Time [year]
Disp
lace
men
t [cm
] UP EAST
Time [year]
Disp
lace
men
t [cm
]
April 2, 2007
Conclusion • A DInSAR-based approach for the retrieval of the 3-D (Up-Down, East-West, North-
South) deformation time-series through a multi-angle/multi-platform LOS displacement combination strategy has been presented.
• Experiments conducted on simulated and real SAR data sets demonstrate the validity of the approach.
• A Minimum-Curvature (MC) combination approach has been implemented.
• The method can be used to combine LOS displacement time-series independently achieved also by different DInSAR tools (SB- and/or PS-like).
• External data (GPS/levelling measurements) and/or Pixel-Offset/MAI-based N-S time-series can be simply integrated in the proposed scheme.
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