ALGAE is an algebraic procedure for jointly estimating terrain topography and interferometric phase offsets in space varying geometries from multi-baseline SAR interferometry. It handles cases where the normal baselines undergo large variations along the imaged swath. ALGAE casts the problem in an algebraic framework and uses a truncated singular value decomposition to solve issues caused by the problem's ill-conditioning. It was tested on P-band data from the BioSAR 2008 campaign, extracting ground contributions using algebraic synthesis before inputting phases to ALGAE for estimation of absolute terrain topography.

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
 Kznmz 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
  Kznmz  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  Kznmz P  Kznmz 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