![An Algebraic Interpretation
The problem can be cast in general terms as: nP KznP z P nP
Unwrapped
PD at point P
Interferometric phase
Height to phase Target height at point P in track n
at point P in track n
conversion factor at w.r.t. the reference
point P in track n topography
Stacking all phase measurements we get:
z Topography
Gm G [NP × 1]
Data
[N∙NP × 1] Parameters describing
PDs in all tracks
Forward Operator [Nα × 1]
Forward Problem [N∙NP × (Nα+ NP)]
This problem admits the general solution:
z The null space determines the ambiguity of the problem
G NG c o Data fit is invariant w.r.t. to the choice of c
The constants c represent the degrees of freedom of the
Pseudo- Set of
arbitrary
problem, providing the proper access point to plug
Inverse of G Null Space
of G constants external information into the solution
Inverse Problem](https://image.slidesharecdn.com/algae-100729134844-phpapp01/85/TH1-L09-ALGAE-A-FAST-ALGEBRAIC-ESTIMATION-OF-INTERFEROGRAM-PHASE-OFFSETS-IN-SPACE-VARYING-GEOMETRIES-8-320.jpg)
TH1.L09 - ALGAE: A FAST ALGEBRAIC ESTIMATION OF INTERFEROGRAM PHASE OFFSETS IN SPACE VARYING GEOMETRIES
- 1. ALGAE: A FAST ALGEBRAIC ESTIMATION OF INTERFEROGRAM PHASE OFFSETS IN SPACE VARYING GEOMETRIES Stefano Tebaldini, Guido Gatti, Mauro Mariotti d’Alessandro, and Fabio Rocca Politecnico di Milano Dipartimento di Elettronica e Informazione IGARSS 2010, Honolulu
- 2. ALGebraic Altitude Estimation ALGAE is an algebraic procedure for the geometrical interpretation of interferometric measurements from a multi-baseline SAR system Features: • Joint estimation of terrain topography and phase offsets • Handles the case where the normal baselines undergo a large variation along the imaged swath • Compatible with different kinds of a-priori information to carry out absolute phase calibration • Fast • Robust
- 3. Problem Statement SAR Interferometry has repeatedly been shown to constitute a major tool for the retrieval of terrain topography at vast scales • SAR imaging provide information about the target to sensor distance Information from multiple (≥2) images can be merged to locate the target in 3D Track N • Interferometric phase differences provide high accuracy measurements, being sensitive to wavelength scale Track n variations of the target to sensor distance 4 r rm rN nm P P P Reference rn n Track θ r elevation P Pulse Envelope Wavelength ground range azimuth
- 4. Problem Statement SAR Interferometry has repeatedly been shown to constitute a major tool for the retrieval of terrain topography at vast scales • SAR imaging provide information about the target to sensor distance Information from multiple (≥2) images can be merged to locate the target in 3D Track N • Interferometric phase differences provide high accuracy measurements, being sensitive to wavelength scale Track n variations of the target to sensor distance 4 r rm drnP drm rN+drN P P P P Reference rn +drn nm n Track • Though, distance measurements are affected by Propagation Disturbances (PDs), arising from θ uncompensated delay of the Radar echoes elevation r + dr o Propagation through the atmosphere o Residual platform motion • PDs make the retrieval of absolute topography nearly an P Pulse Envelope ill-posed problem in absence of a-priori information Wavelength ground range azimuth
- 5. Problem Statement • Problem sensitivity is easily discussed considering phase variation w.r.t. target height: 4 nm P Kz 1 P Kznmz P z z P P nm sin nm nm Track m Track n Kznm = Height to phase conversion factor for tracks n and m θ • Height error due to PDs is readily obtained as: r elevation sin P ez P dr Δθ nm drP = Total propagation error at point P P ΔzP • For a typical InSAR configuration Δθ = 1/1000 – 1/100 ground range => error amplification is huge azimuth
- 6. Problem Statement • Problem sensitivity is easily discussed considering phase variation w.r.t. target height: 4 nm P z Kznm nm 1 Kznmz z P P P P Track n Track m nm sin Kznm = Height to phase conversion factor for tracks n and m θ • Height error due to PDs is readily obtained as: r elevation sin P ez P dr Δθ nm drP = Total propagation error at point P P R ΔzP • For a typical InSAR configuration Δθ = 1/1000 – 1/100 ΔzR ground range => error amplification is huge azimuth • However, PDs typically exhibit a large decorrelation length w.r.t. to the SAR resolution cell o Atmospheric delay is uniform within about 1 km o The dynamics of platform deviations from nominal trajectory is slow w.r.t. to platform velocity This information allows accurate topography retrieval w.r.t. to a reference point sin P nm R nm Kznm z z P R ezP ezR nm dr P dr R ezP
- 7. Problem Statement • Still, the problem changes when the spatial variation of the height to phase conversion factors is relevant. In this case: Track m nm nm Kznmz P Kznmz R P R P R Kznm z P z R Kznm Kznm z R Track n P P R θ r elevation • A new term arises, that depends on Δθ o The spatial variation of Kz o The true height of the reference point w.r.t. the P reference ground plane R ΔzP ΔzR • Impact on height estimation: ground range azimuth Kznm R R e e 1 P ez P R Space Variant Kznm z z Error • Even if PDs are perfectly compensated for by the phase locking operation, the error about the reference point height results in a space variant error propagating throughout the scene o Particularly relevant for airborne geometries, where Kz can undergo a variation by a factor 3 Handling space varying geometries require precise external information about the sensor to target distances for the reference point
- 8. An Algebraic Interpretation The problem can be cast in general terms as: nP KznP z P nP Unwrapped PD at point P Interferometric phase Height to phase Target height at point P in track n at point P in track n conversion factor at w.r.t. the reference point P in track n topography Stacking all phase measurements we get: z Topography Gm G [NP × 1] Data [N∙NP × 1] Parameters describing PDs in all tracks Forward Operator [Nα × 1] Forward Problem [N∙NP × (Nα+ NP)] This problem admits the general solution: z The null space determines the ambiguity of the problem G NG c o Data fit is invariant w.r.t. to the choice of c The constants c represent the degrees of freedom of the Pseudo- Set of arbitrary problem, providing the proper access point to plug Inverse of G Null Space of G constants external information into the solution Inverse Problem
- 9. An Algebraic Interpretation Remarks: • Linearization about a reference topography does not entail any loss of generality, the eventual ambiguity or ill-conditioning of the inversion being intrinsic to the nature of the problem • Linearization error can be recovered through iterative inversion methods (Newton-Raphson)
- 10. An Algebraic Interpretation Remarks: • Linearization about a reference topography does not entail any loss of generality, the eventual ambiguity or ill-conditioning of the inversion being intrinsic to the nature of the problem • Linearization error can be recovered through iterative inversion methods (Newton-Raphson) Remarks (II): z • How many parameters for the PDs? Gm G • Correct choice depends on o Extent of the imaged scene o Physics of PDs (i.e.: atmosphere, dynamics of platform motion…) o Precision of motion compensation procedures previously applied • In this work we assume a constant phase offset in each track nP n N N The assumption is simplistic, yet: o Valid on limited areas o Allows the discussion of the effects of space varying geometries
- 11. An Algebraic Interpretation For the case of constant phase offsets it may be shown that: • The forward operator allows a 1D null space if and only if the height to phase conversion factors undergo the same spatial variation in all tracks, i.e.: k , K n P s.t. : KznP kn K P (C1) • If C1 is fulfilled, the general solution for terrain topography is given by: z P zInversion c zNullspace zNullspace K P P P P 1 We distinguish three cases: Spaceborne InSAR Airborne InSAR - Ideal Airborne InSAR – Real Kz can be considered constant if the Assuming parallel trajectories C1 is C1 is not fulfilled swath is not too large fulfilled with => The problem is well posed => C1 is fulfilled with: kn bn / b1 => Retrieval of absolute terrain topography is theoretically kn Kzn K P K P r P , sin P possible KP 1 => Terrain topography is retrieved => Terrain topography is retrieved up to a space varying – up to a constant topography dependent – term z P zInversion c z P zInversion c K P P P 1
- 12. ALGebraic Altitude Estimation Spaceborne Absence of the null space is only apparently an Singular Values advantage 0 • The problem is extremely ill-conditioned, the last Airborne - Ideal Singular Value being very close to zero Singular Values • Consistent with the arising of a 1D null space in nominal 0 conditions Airborne - Real Singular Values 0.002 ALGAE • The forward operator is modified by zeroing the last Singular Value (Truncated SVD) Ill-conditioning problems are solved, at the expense of the arising of a 1D null space z P zInversion c zNullspace P P • The value of c is then determined according to the available a-priori information o Point match with an accurately measured reference point o Best global match with coarse reference DEM o Zeroing mean slope o Other….
- 13. Experimental Results The boreal forest in the Krycklan catchment, northern Sweden, has been investigated in the framework of the ESA campaign BioSAR 2008 Scene: o Boreal forest o Hilly topographic, height variations up to 200 m Data has been acquired by the airborne system E-SAR, flown by DLR Data focusing, calibration, and co-registration have been performed by DLR Data-set under analysis o P-Band o Look Directions: South West and North East (6 + 6 tracks) o Nominal look between 25° and 55° o 100 MHz pulse bandwidth o 1.6 m azimuth resolution o 2.12 m slant range resolution o 40 m horizontal baseline aperture o Imaged area is 2.3 × 9.5 km2
- 14. Pre-Processing Pre-Processing Operations • The Algebraic Synthesis technique is used to extract contributions from ground-only scattering, thus avoiding being affected by vegetation bias • The retrieved ground-only contributions are processed with the Phase Linking algorithm to estimate the ground phases with respect to a common Master • Ground phases are unwrapped, and used as input for ALGAE Multi-Baseline, Phase Multi-Polarimetric Linking Unwrapped Data Algorithm interferometric ALGAE ground phases Best Estimate of the Ground Phases Algebraic with respect to a Algebraic Synthesis: Synthesis Common Master “Algebraic Synthesis of Forest Scenarios from Multi- baseline PolInSAR data” –Tebaldini – TGARS vol 47, no 12, Dec.2009 2-Dimensional Phase Ground-only Phase Linking: Unwrapping Contributions “On the Exploitation of Target Statistics for SAR Interferometry Applications” – Monti Guarnieri and Tebaldini – TGARS vol 46, no 11, Nov.2008
- 15. Algebraic Synthesis Algebraic Synthesis (AS) technique for the decomposition of Ground and Volume scattering basing on multi-polarimetric and multi-baseline SAR surveys Track n y n wi Polarization wi Track N Re{yn(w1)} Re{yn(w2)} Re{yn(w3)} Track n HH HV VH VV Im{yn(w1)} Im{yn(w2)} Im{yn(w3)} Track 1 HH HV VH VV Volume Contributions HH HV 60 50 40 VH VV 30 20 10 0 -10 200 600 1000 1400 1800 2200 Algebraic Synthesis Ground Contributions Ground 60 50 40 30 20 10 0 -10 200 600 1000 1400 1800 2200
- 16. Algebraic Synthesis Algebraic Synthesis (AS) technique for the decomposition of Ground and Volume scattering basing on multi-polarimetric and multi-baseline SAR surveys Track n y n wi Polarization wi Track N Re{yn(w1)} Re{yn(w2)} Re{yn(w3)} Track n HH HV VH VV Im{yn(w1)} Im{yn(w2)} Im{yn(w3)} Track 1 HH HV VH VV Volume Contributions HH HV 60 50 40 VH VV 30 20 10 0 -10 200 600 Further details in: POLARIMETRIC AND 1000 1400 1800 2200 Algebraic STRUCTURAL PROPERTIES OF FOREST Synthesis SCENARIOS AS IMAGED BY LONGER Ground Contributions WAVELENGTH SARS Ground 60 50 40 Poster Session: THP2.PA.3 - Radar Mapping 30 20 10 Thursday, July 29, 14:55 - 16:00 0 -10 200 600 1000 1400 1800 2200
- 17. Phase Linking Retrieving the set of the ground interferometric coherences does not solve the problem of retrieving the ground phases, since ground scattering can be affected by decorrelation phenomena such as: • Temporal decorrelation • Superficial decorrelation • Thermal noise This problem is solved by the Phase Linking algorithm • Multi-baseline Maximum Likelihood estimation of the phases associated with the optical path lengths from a target to the N SAR sensors, accounting for target decorrelation phenomena Interferogram phases: Track N n y ref y 4 r ref r n ref n + Phase Noise Track n n rN Linked phases: Reference rn Minimum Variance Phase Track n 4 r ref r n ref n + Noise given N tracks rref elevation rn : distance to the n-th SAR sensor αn : Propagation Disturbance in the n-th acquisition ground range
- 18. Preliminary Analysis LIDAR DEM Ground Range Slope Ground Range Slope 350 300 10 250 200 0 150 -10 • Ground coherence is extremely high Ground Coherence Ground Coherence excellent ground visibility excellent temporal stability • A slight trend w.r.t. the incidence angle can be (correctly!) appreciated in both directions 1 • The spatial variation of ground coherence is clearly correlated 0.75 with terrain slope, regardless of heading direction 0.5
- 19. Preliminary Analysis Height to Phase Conversion Factors Track 1 Track 2 (Master) • South-West Data-set slant range • Large range variation range and look angle variation within the imaged swath • Large azimuth variation Track 3 Track 4 Platform motion slant range Track 5 Track 6 slant range azimuth azimuth
- 20. Preliminary Analysis Ground Phases Track 1 Track 2 (Master) • South-West Data-set • LIDAR DEM removed slant range • Fringes are high quality, consistently with the observed high ground coherence Track 3 Track 4 • Fringes are correlated slant range with the Kz non-perfect compensation of platform deviation from nominal trajectories Track 5 Track 6 slant range azimuth azimuth
- 21. ALGAE - Results South West data-set Least Square Solution Null Space Look Direction
- 22. ALGAE - Results South West data-set Null Space - High Pass Component ALGAE Solution • Null space = along range trend + azimuth oscillations • Along-range errors recovered • The null space is added so as to zero the mean range slope • Along-azimuth errors partly recovered
- 23. ALGAE - Results South West data-set Least Square Solution Null Space Look Direction • Same reference point as in the SW data-set brings to a large trend in the LS solution
- 24. ALGAE - Results South West data-set Null Space - High Pass Component ALGAE Solution • Null space = along range trend + azimuth oscillations • Along-range errors recovered • The null space is added so as to zero the mean range slope • Along-azimuth errors partly recovered
- 25. Conclusions Space varying geometries result in space variant topographic errors that can not be fixed by evaluating terrain topography w.r.t. one reference point ALGAE defines an algebraic framework where space variance and propagation disturbances are explicitly accounted for o Propagation disturbances represented as constant phase offsets and estimated along with topography o The degree of freedom provided by the concept of null space is compatible with different kinds of a-priori information, either local or global o Fast o Robust Residual DEM errors: o characterized by a large spatial decorrelation length o correlated with the flight direction => Presence of residual phase terms due to residual platform motion Insufficiency of the constant phase offset model Future researches: • Incorporate more sophisticated models to describe propagation disturbances, taking into account both the physics of atmospheric propagation and platform motion