Stefano Tebaldini, Guido Gatti, Mauro Mariotti d’Alessandro, and Fabio Rocca

Politecnico di Milano

Dipartimento di Elettronica e Informazione

ALGAE: A FAST ALGEBRAIC ESTIMATION OF

INTERFEROGRAM PHASE OFFSETS IN

SPACE VARYING GEOMETRIES

IGARSS 2010, Honolulu

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

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

• Interferometric phase differences provide high accuracy measurements, being sensitive to wavelength scale variations of the target to sensor distance

Reference

Track

Track n

Track N

ground range

elev

atio

n

azimuth

r

rn

rN

Pulse Envelope

Wavelength

P

m

P

n

P

nm rr

4

P

θ

Problem Statement

Reference

Track

Track n

Track N

ground range

elev

atio

n

azimuth

rn +drn

rN+drN

Pulse Envelope

Wavelength

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

• Interferometric phase differences provide high accuracy measurements, being sensitive to wavelength scale variations of the target to sensor distance

P

m

P

n

P

m

P

n

P

nm drdrrr

4

• Though, distance measurements are affected by Propagation Disturbances (PDs), arising from uncompensated delay of the Radar echoes

o Propagation through the atmosphere

o Residual platform motion

• PDs make the retrieval of absolute topography nearly an ill-posed problem in absence of a-priori informationP

r + dr

θ

Problem Statement

• Problem sensitivity is easily discussed considering phase variation w.r.t. target height:

Track nTrack m

ground range

elev

atio

n

azimuth

rθ

PΔzP

Δθ

• For a typical InSAR configuration Δθ = 1/1000 – 1/100

=> error amplification is huge

P

nmnm

P Kzz 1PnmP

nm

P

nm zzKz

sin

4

P

nm

P

zdre

sin

• Height error due to PDs is readily obtained as:

drP = Total propagation error at point P

Kznm = Height to phase conversion factor for tracks n and m

Problem Statement

Track nTrack m

ground range

elev

atio

n

azimuth

rθ

PΔzP

Δθ

• For a typical InSAR configuration Δθ = 1/1000 – 1/100

=> error amplification is huge

RΔzR

RP

nm

R

nm

P

nm zzKz P

z

RP

nm

R

z

P

z edrdree

sin

• 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

• Problem sensitivity is easily discussed considering phase variation w.r.t. target height:

• For a typical InSAR configuration Δθ = 1/1000 – 1/100

=> error amplification is huge

PnmP

nm

P

nm zzKz

sin

4

Kznm = Height to phase conversion factor for tracks n and m

• Height error due to PDs is readily obtained as:

drP = Total propagation error at point P

P

nmnm

P Kzz 1

P

nm

P

zdre

sin

Problem Statement

• Still, the problem changes when the spatial variation of the height to phase conversion factors is relevant. In this case:

RR

nm

P

nm

RPP

nm

RR

nm

PP

nm

R

nm

P

nm

zKzKzzzKz

zKzzKz

Track n

Track m

ground range

elev

atio

n

azimuth

rθ

PΔzP

Δθ

RΔzR

• A new term arises, that depends on

o The spatial variation of Kz

o The true height of the reference point w.r.t. the reference ground plane

• Impact on height estimation:

R

zP

nm

R

nmR

z

P

z eKz

Kzee

1

Handling space varying geometries require precise external information about the sensor to target distances for the reference point

• 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

Space Variant Error

The problem can be cast in general terms as:

An Algebraic Interpretation

P

n

PP

n

P

nzKz

Unwrapped Interferometric phase at point P in track n Height to phase

conversion factor at point P in track n

Target height at point P w.r.t. the reference

topography

PD at point Pin track n

Stacking all phase measurements we get:Topography

[NP × 1]

Forward Operator

[N∙NP × (Nα+ NP)]

Data

[N∙NP × 1]

zGmG

Parameters describing PDs in all tracks

[Nα × 1]

This problem admits the general solution:

cNGz

G

Pseudo-Inverse of G Null Space

of G

Set of arbitrary constants

The null space determines the ambiguity of the problem

o Data fit is invariant w.r.t. to the choice of c

The constants c represent the degrees of freedom of the problem, providing the proper access point to plug external information into the solution

Forward Problem

Inverse Problem

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)

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)

z

GmGRemarks (II):

• How many parameters for the PDs?

• 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

The assumption is simplistic, yet:

o Valid on limited areas

o Allows the discussion of the effects of space varying geometries

n

P

n NN

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.:

:.., tsKk P

n

P

n

P

nKkKz

• If C1 is fulfilled, the general solution for terrain topography is given by:

(C1)

P

Nullspace

P

Inversion

P zczz 1

PP

NullspaceKz

We distinguish three cases:

Spaceborne InSAR Airborne InSAR - Ideal

Kz can be considered constant if the swath is not too large

1

P

nn

K

Kzk

czz P

Inversion

P

=> Terrain topography is retrieved up to a constant

=> C1 is fulfilled with:

Assuming parallel trajectories C1 is fulfilled with

PPPP

nn

rKK

bbk

sin,

/1

1

PP

Inversion

P Kczz

=> Terrain topography is retrieved up to a space varying –topography dependent – term

Airborne InSAR – Real

C1 is not fulfilled

=> The problem is well posed

=> Retrieval of absolute terrain topography is theoretically possible

ALGebraic Altitude Estimation

Absence of the null space is only apparently an

advantage

• The problem is extremely ill-conditioned, the last

Singular Value being very close to zero

• Consistent with the arising of a 1D null space in nominal

conditions

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

• 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….

Spaceborne

Airborne - Real

Sin

gu

lar

Val

ues

Sin

gu

lar

Val

ues

Sin

gu

lar

Val

ues

0

0

0.002

Airborne - Ideal

P

Nullspace

P

Inversion

P zczz

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

Experimental Results

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

Unwrapped

interferometric

ground phases

Multi-Baseline,

Multi-Polarimetric

Data

Algebraic

Synthesis

Ground-only

Contributions

Phase

Linking

Algorithm

Best Estimate of the

Ground Phases

with respect to a

Common Master

2-Dimensional

Phase

Unwrapping

Pre-ProcessingPre-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

ALGAE

Algebraic Synthesis:

“Algebraic Synthesis of Forest Scenarios from Multi-

baseline PolInSAR data” –Tebaldini – TGARS vol 47, no 12,

Dec.2009

Phase Linking:

“On the Exploitation of Target Statistics for SAR

Interferometry Applications” – Monti Guarnieri and Tebaldini –

TGARS vol 46, no 11, Nov.2008

ynwi Track n

Polarization wi

Re{yn(w1)} Re{yn(w2)} Re{yn(w3)}

Im{yn(w1)} Im{yn(w2)} Im{yn(w3)}

Track 1 HH HV

VH VV

HH HV

VH VV

Track n

Track N

HH HV

VH VV

Algebraic

Synthesis

Volume Contributions

Ground Contributions Ground

200 600 1000 1400 1800 2200-10

0102030405060

200 600 1000 1400 1800 2200-10

0102030405060

Algebraic SynthesisAlgebraic Synthesis (AS) technique for the decomposition of Ground and Volume

scattering basing on multi-polarimetric and multi-baseline SAR surveys

ynwi Track n

Polarization wi

Re{yn(w1)} Re{yn(w2)} Re{yn(w3)}

Im{yn(w1)} Im{yn(w2)} Im{yn(w3)}

Track 1 HH HV

VH VV

HH HV

VH VV

Track n

Track N

HH HV

VH VV

Algebraic

Synthesis

Volume Contributions

Ground Contributions Ground

200 600 1000 1400 1800 2200-10

0102030405060

200 600 1000 1400 1800 2200-10

0102030405060

Algebraic SynthesisAlgebraic Synthesis (AS) technique for the decomposition of Ground and Volume

scattering basing on multi-polarimetric and multi-baseline SAR surveys

Further details in: POLARIMETRIC AND

STRUCTURAL PROPERTIES OF FOREST

SCENARIOS AS IMAGED BY LONGER

WAVELENGTH SARS

Poster Session: THP2.PA.3 - Radar Mapping

Thursday, July 29, 14:55 - 16:00

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 decorrelationphenomena such as:

ground rangeel

evat

ion

Reference

Track

Track n

Track N

rref

rn

rN

• Thermal noise• Superficial decorrelation • Temporal decorrelation

rn : distance to the n-th SAR sensor

αn : Propagation Disturbance in the n-th acquisition

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:

Linked phases:

n yref yn 4

rref rn ref n

n 4

rref rn ref n

+ Phase Noise

Minimum Variance Phase Noise given N tracks

+

Phase Linking

150

200

250

300

350

Ground Coherence

0.5

0.75

1

Ground Coherence

-10

0

10

Ground Range Slope Ground Range SlopeLIDAR DEM

Preliminary Analysis

• Ground coherence is extremely

high

excellent ground visibility

excellent temporal stability

• A slight trend w.r.t. the incidence

angle can be (correctly!)

appreciated in both directions

• The spatial variation of ground

coherence is clearly correlated

with terrain slope, regardless of

heading direction

Track 1 Track 2 (Master)

Track 3 Track 4

Track 5 Track 6

azimuth azimuth

slan

t ra

nge

slan

t ra

nge

slan

t ra

nge

Preliminary Analysis

Height to Phase Conversion Factors

• South-West Data-set

• Large range variation

range and look angle

variation within the imaged

swath

• Large azimuth variation

Platform motion

Track 1 Track 2 (Master)

Track 3 Track 4

Track 5 Track 6

azimuth azimuth

slan

t ra

nge

slan

t ra

nge

slan

t ra

nge

Ground Phases

• South-West Data-set

• LIDAR DEM removed

• Fringes are high quality,

consistently with the

observed high ground

coherence

• Fringes are correlated

with the Kz

non-perfect

compensation of platform

deviation from nominal

trajectories

Preliminary Analysis

ALGAE - Results

Look Direction

Least Square Solution Null Space

South West data-set

Null Space - High Pass Component ALGAE Solution

ALGAE - ResultsSouth West data-set

• Null space = along range trend + azimuth oscillations

• The null space is added so as to zero the mean range slope

• Along-range errors recovered

• Along-azimuth errors partly recovered

Look Direction

ALGAE - Results

Least Square Solution Null Space

South West data-set

• Same reference point as in the SW data-set brings

to a large trend in the LS solution

ALGAE - Results

Null Space - High Pass Component ALGAE Solution

South West data-set

• Null space = along range trend + azimuth oscillations

• The null space is added so as to zero the mean range slope

• Along-range errors recovered

• Along-azimuth errors partly recovered

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