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The Ionosphere and Interferometric/Polarimetric SAR
Tony FreemanEarth Science Research and Advanced
Concepts Manager
AF- 2
Faraday Rotation
PROPAGATION EFFECTS
• Troposphere - Refraction
- Absorption
- Phase Fluctuations
• Ionosphere
f = 100 MHz f = 1 GHz f = 10 GHzUnits Day Night Day Night Day Night
Faraday Rotation rad 94.25 9.42 0.94 0.09 0.01 -
Group Delay usec 12.00 1.2 0.12 0.01 - -
Phase Change rad 7500.00 750 75.00 7.50 0.75 0.08
Phase Stability rad 150.00 15 15.00 1.50 1.50 0.15
Frequency Stability Hz 0.04 0.004 - - - -
Absorption dB 0.10 0.01 - - - -
Refraction deg 0.05 0.005 - - - -
[Daytime Total Electron Content = 5 x 1017 m-2, Night-time TEC = 5 x 1016 m-2]- after Evans and Hagfors (1968)
• Ionospheric phase delay can be calibrated if simultaneous phase measurements areavailable at two widely separated frequencies
Calibrated phase, φc =φ2 –ν1ν2
φ1
AF- 3
Faraday Rotation
Surface Clutter ProblemRepeat-pass Interferometry
• Subsurface return has phase difference 1 due to ionosphere propagation (same as surface return from location O)
• Surface clutter return (from location P) has phase difference 2 due to ionosphere propagation
• So 1 - 2 is unknown - could be zero - depends on correlation length of ionosphere
• Is 1 - 2 variable within a data acquisition? (Probably)
Radar
i
rz r
O
Q
P
r1r2+r+2
r1+r
r2 +1
Ionosphere
AF- 4
Faraday Rotation
Ionospheric Effects
Two-way propagation of the radar wave through the ionosphere causes several disturbances in the received signal, the most significant of which are degraded resolution and distorted polarization signatures because of Faraday rotation. Except at the highest TEC levels, the 100 m spatial resolution of CARISMA should be readily achievable [Ishimura et al, 1999]. Faraday rotation in circular polarization measurements is manifested as a phase difference between the R-L and L-R backscatter measurements. If this phase difference is left uncorrected, it is not possible to successfully convert from a circular into a linear polarization basis – the resulting linear polarization measurements will still exhibit the characteristics of Faraday rotation. The need to transform to a linear basis stems from CARISMA’s secondary science objectives – and the requirement to use HV backscatter measurements, which have exhibited the strongest correlation with forest biomass in multiple studies.
As shown in [Bickel and Bates ,1965], [Freeman and Saatchi, 2004] and [Freeman, 2004] it is, in theory, relatively straightforward to estimate the Faraday rotation angle from fully polarimetric data in circularly polarized form, and to correct the R-L to L-R phase difference. Performing this correction will then allow transformation to a distortion-free linear polarization basis.
AF- 5
Faraday Rotation
Calibration/Validation
Calibration of the near-nadir radar measurements to achieve the primary science objectives of ice sheet sounding is relatively straightforward. The required 20 m height resolution matches the capability offered by the bandwidth available, and is easily verified for surface returns by comparison with existing DEMs. For the subsurface returns CARISMA measurements will be compared with GPR data, ice cores and airborne radar underflight data. The required radiometric accuracy of 1 dB is well within current radar system capabilities and can be verified using transponders and or targets with known (and stable) reflectivity. Radiometric errors introduced by external factors such as ionospheric fading and interference require further study.
Calibration of the side-looking measurements over the ice is a little more challenging but the primary science objectives can still be met. Validation that the differentiation between surface and subsurface returns has been successful, will be carried out by simulating the surface ‘clutter’ using DEMs and backscatter models.
Calibration of the side-looking measurements over forested areas will be yet more challenging. The techniques described in [Freeman, 2004] will be used to generate calibrated linear polarization measurements. Data acquired over targets of known, stable RCS, such as corner reflectors and dense tropical forest will be used to verify the calibration performance. Validation of biomass estimates and permafrost maps generated from CARISMA data will be carried out by comparison with data acquired in the field.
AF- 6
Faraday Rotation
Ω (degrees)
C-Band (6 cm) 2.5o
L-Band (24 cm) 40o
P-Band (68 cm) 321o
• Faraday rotation is a problem that needs to be taken into consideration for longer wavelength SAR’s
• Worst-case predictions for Faraday rotation for three common wavebands:
Introduction and Scope
AF- 7
Faraday Rotation
• General problem:
€
Mhh
Mvh
Mhv
Mvv
⎛
⎝ ⎜ ⎞
⎠ ⎟= A(r, θ ) e
j φ1 δ
2
δ1
1
⎛
⎝ ⎜ ⎞
⎠ ⎟1 0
0 f1
⎛
⎝ ⎜ ⎞
⎠ ⎟
cos Ω sin Ω
− sin Ω cos Ω
⎛
⎝ ⎜ ⎞
⎠ ⎟S
hhS
vh
Shv
Svv
⎛
⎝ ⎜ ⎞
⎠ ⎟
cos Ω sin Ω
−sin Ω cos Ω
⎛
⎝ ⎜ ⎞
⎠ ⎟1 0
0 f2
⎛
⎝ ⎜ ⎞
⎠ ⎟
1 δ3
δ4
1
⎛
⎝ ⎜ ⎞
⎠ ⎟+
Nhh
Nvh
Nhv
Nvv
⎛
⎝ ⎜ ⎞
⎠ ⎟
• For H and V polarization measurements, and ignoring other effects, themeasured scattering matrix, M, can be written as M = RSR , i.e.
M hh M vh
Mhv M vv
= cosΩ sinΩ– sinΩ cosΩ
Shh Svh
Shv Svv
cosΩ sinΩ– sinΩ cosΩ
i.e.
M hh = Shh cos 2Ω – Svv sin2Ω + Shv – Svh sinΩ cosΩ M vh = Svh cos 2Ω + Shv sin2Ω + Shh+ Svv sinΩ cosΩ M hv = Shv cos 2Ω + Svh sin2Ω – Shh+ Svv sinΩ cosΩ M vv = Svv cos2Ω – Shh sin 2Ω + Shv – Svh sinΩ cosΩ
• This is non-reciprocal for Ω ≠ 0 , (i.e. Mhv ≠ Mvh , even though S hv = S vh )
Effects on Polarimetric Measurements
AF- 8
Faraday Rotation
• For an HH-polarization measurement (e.g. JERS-1) the expected value of theradar cross section in the presence of Faraday rotation is:
M hhMhh
* = ShhShh
* cos4Ω – 2Re ShhSvv
* sin 2Ω cos2Ω + SvvSvv
* sin4Ω
assuming ShhShv
* = ShvSvv* = 0 (azimuthal symmetry)
• For repeat-pass data, the decorrelation due to Faraday rotation will be:
ρ Faraday =
Shh
Shh
* cos2Ω – ShhSvv
* sin2Ω
Shh
Shh
* Shh
Shh
* cos4Ω – 2Re Shh
Svv
* sin2Ω cos2Ω + Svv
Svv
* sin4Ω
• Either of these two measures will depend on both the Faraday rotationangle, Ω, and the polarization signature of the terrain
Effects on Interferometric Measurements
AF- 9
Faraday Rotation
• Modeled L-Band ‘HH’ backscatter vs. Faraday rotation angle:
L-Band HH Backscatter vs. Faraday rotation
-25
-20
-15
-10
-5
0
0 20 40 60 80 100 120 140 160 180
Bare SoilPastureUpland ForestSwamp ForestPlantationConifers
• Backscatter drops off to a null at Ω = 45 egrees
• Depth o νull epeνs oν polarizatioν sigνature - but eects woul probablybe maske by νoise ( at -17 B) iν JERS-1 ata, or example
• P-Baν results are similar iν behavior
AF- 10
Faraday Rotation
Measure Ω = 3 Ω = 5 Ω = 10 Ω = 20 Ω = 40 Ω = 90
Dso( ) - HH dB 0 −0.1 −0.2 −> −0.5 −0.9 −> −1.9 −2.7 −> −7.2 −2.7 −> +1.7
Dso( ) - VV dB 0 −0.1 −0.2 −> −0.5 −0.9 −> −1.8 −1.8 −> −7.3 −1.7 −> +2.7
Dso( ) - HV dB +0.1 −> +0.5 +0.3 −> +0.7 +1.0 −> +3.7 +2.6 −> +7.6 +4.6 −> +10.8 0
Dr(HH1HH2*) 0 0 0 −> -0.01 0 −> -0.03 -0.15 −> -0.42 -0.24 −> -0.87
Dr( *)HHHV +0.11 −> +0.27 +0.18 −> +0.42 +0.32 −> +0.64 +0.43 −> +0.75 +0.17 −> +0.39 0
Dr( *)HHVV 0 −> −0.01 −0.02 −> +0.01 −0.06 −> +0.06 −0.21 −> +0.25 −0.13 −> +0.87 0φ ( - ) - HV VH deg 0 0 0 180or 0 180or 180 0φ ( - ) - HH VV deg −0.2 −> +2.2 −0.5 −> +6.4 −2.1 −> +31.6 −11.0 −> +102.4−143 −> +171.2 = - φ( - *)HH VV |Ω = 0
Summary of Model Results
• Spread of relative errors introduced into backscatter measurements across a wide range of measures for a diverse set of scatterer types
• Effects considered negligible (i.e. less than desired calibration uncertainty*) are shaded
*Radiometric uncertainty - 0.5 dB Phase error - 10 degrees Correlation error - 6%
• A Noise-equivalent sigma-naught of - 30dB is assumed
AF- 11
Faraday Rotation
1. (Freeman, 2004) Since speckle and additive noise may be present in the backscatter signatures, may be estimated from averaged 2nd-order statistics:
Zhv
= 0 . 5 Mvh
– Mhv
the n es timating from:
Ω =
1
2
tan
– 1
Zhv
Zhv
*
Mhh
Mhh
*
+ Mvv
Mvv
*
+ 2 Re Mhh
Mvv
*
2. (Bickel aν Bates, 1967 ) - or ully polarimetric (liνear polarize) spaceborνe SARs, it is straightorwar to estimate the Faraay rotatioν aνgle, , via a rotatioν to a circular basis:
Z11
Z12
Z21
Z22
=
1 j
j 1
Mhh
Mvh
Mhv
Mvv
1 j
j 1
iν which case
Ω =
1
4
arg Z12
Z21
*
Estimating the Faraday Rotation Angle, Ω
AF- 12
Faraday Rotation
NOISE
a) NE so -100 dB -50 dB -30 dB -24dB -18 dB
DΩ1 ( )deg 0 1.2 11 18 23.6
DΩ2 ( )deg 0 0 1.3 5.4 31.1
CHANNEL AMPLITUDE IMBALANCE
) |b f1|2 0.0 dB 0.1 dB 0.2 dB 0.3 dB 0.4 dB 0.5 dB 1.0 dB
DΩ1 ( )deg 0 0.4 0.9 1.3 1.8 2.2 4.4
DΩ2 ( )deg 0 0.1 0.3 0.4 0.6 0.7 1.4
CHANNEL PHASEIMBALANCE
) c (Arg f1) 0 deg 2 deg 5 deg 10 deg 20 deg
DΩ1 ( )deg 0 1.3 3.4 6.6 12.4
DΩ2 ( )deg 0 0.4 1.0 2.1 5.1
-CROSS TALK
) |d |2 -50 dB -30 dB -25 dB -20dB -15 dB
DΩ1 ( )deg 0 0.1 0.2 0.7 2.1
DΩ2 ( )deg 0.3 2.6 4.7 8.2 15.4
• Sensitivity to Residual System Calibration Errors (shaded cells represent errors in Ω > 3 degrees)
Estimating the Faraday Rotation Angle, Ω
AF- 13
Faraday Rotation
a) P-Band ||2= -30 dB ||2= -25dB
DΩ1 ( )deg 10.5 10.5
DΩ2 ( )deg 3.2 5.1
) -b L Band ||2= -30 dB ||2= -25dB
DΩ1 ( )deg 10.6 10.5
DΩ2 ( )deg 2.9 5.2
Estimating the Faraday Rotation Angle, Ω
• Combining effects for a ‘typical’ set of system errors, we see that a cross-talk level < -30 dB is necessary to keep the error in Ω < 3 degrees using measure (2)
• P-Band case has channel amplitude imbalance of 0.5 dB, phase imbalance of 10 degrees and NE so = -30 dB
• L-Band case has channel amplitude imbalance of 0.5 dB, phase imbalance of 10 degrees and NE so = -24 dB
For Measure (1) error is dominated by additive noise
AF- 14
Faraday Rotation
• To correct for a Faraday rotation of Ω, use: S = R tMR t
where Rt ≡ R-1 . This can be written:
Shh Svh
Shv Svv
= cosΩ – sinΩsinΩ cosΩ
M hh M vh
M hv M vv
cosΩ – sinΩsinΩ cosΩ
• Since values of tan -1 are between ± π/2, values of Ω will be between ± π/4,which means that Ω can only be estimated modulo π/2
• This problem can be identified from the cross- pol terms by comparingmeasurements before correction and after, i.e.:
ShvSvh
* = – M hv′ M hv
′ *
and
0.25 Shv + Svh Shv + Svh
*≠ M hv
′ M hv′ *
• This test should readily reveal the presence of a π/2 error in Ω, provided thatthe cross- pol backscatter S hv (or S vh ) is not identically zero
Correcting for Faraday Rotation
AF- 15
Faraday Rotation
4. Channel Amplitude
and Phase Balance
1. Radiometric Correction
3. Symmetrization
Uncalibrated
Polarimetric SAR data
5. Estimate Ω
6. Check for p /2
error in Ω
7. Correct for Ω
C alibrated
Polarimetric SAR data
P - re determined
a , ntenna patterns
, range variation
, transmit power
, receiver gain
.etc
Prior estimates of
-cross talk
2. - Cross talk removal
( )if necessary
Signature of t arget
with reflection
symmetry
Signature of t arget
/ with known HH VV
or prior knowledge
/ of HH VV ratio
Signature of any
t arget
Signature of t arget
- with non zero HV
• Calibration Procedure for Polarimetric SAR data
(Cannot estimate cross-talk
from data)
(Use any target with reflection
symmetry to ‘symmetrize’ data)
(Trihedral signature or known
channel imbalance)
• Taking Faraday rotation and ‘typical’ system errors into account
AF- 16
Faraday Rotation
Polarimetric Scattering ModelDistributed scatterers
• Take a cross-product:
• Take an ensemble average over a distributed area:
• For uncorrelated surface and subsurface scatterers:
• Which leaves:
Mhh
Mvv
*
= Shh
1
Svv
1
*
+ Shh
2
Svv
2
*
+ e
+ j φ ( r )
Shh
1
Svv
2
*
+ e
− j φ ( r )
Shh
2
Svv
1
*
Mhh
Mvv
*
= Shh
1
Svv
1
*
+ Shh
2
Svv
2
*
+ e
j φ ( r )
Shh
1
Svv
2
*
+ e
− j φ ( r )
Shh
2
Svv
1
*
Shh
1
Svv
2
*
= Shh
2
Svv
1
*
= 0
Mhh
Mvv
*
= Shh
1
Svv
1
*
+ Shh
2
Svv
2
*
• Similar arguments hold for other cross-products
No interdependence between surface and subsurface returns
AF- 17
Faraday Rotation
Polarimetric Scattering ModelSurface-Subsurface correlation
• Why should the scattering from the surface and subsurface layers be uncorrelated?
• 3 reasons:
1. The scattering originates from different surfaces, with different roughness and dielectric properties
2. The incidence angles are very different (due to refraction)
3. The wavelength of the EM wave incident on each surface is also quite different, since l
2
= λ / ε1
'
AF- 18
Faraday Rotation
Geometry
Radar
mv(er), s, l, l, i
mv(er), s, l, l, ii
rz r
O
Q
P
rr+r
loss tangent, tan
AF- 19
Faraday Rotation
Inversion?
• For each layer we have 5 unknowns: mv(er), s, l, l, i
• For the surface return we know l, and can estimate i if we know the local topography and the imaging geometry
• For the subsurface scattering, the wavelength l is a function of the dielectric constant for the layer, i.e.
• In addition the attenuation of the subsurface return is governed by: tan , r
- This still leaves a total of 9 unknowns
• Under the assumptions of reciprocity (HV = VH), and that like- and cross-pol returns are uncorrelated, we can extract just 5 measurements from the cross-products formed from the scattering matrix
==> Inversion is not possible
Empirical Surface Scattering Model
l2
= λ / ε1
'
AF- 20
Faraday Rotation
Baseline Decorrelation
0.0
0.2
0.4
0.6
0.8
1.0
Baseline,B (m)
600 1350 2100 2850 3600 4350 5100
Baseline (m)
Correlation coefficient
Surface
Subsurface (1)
Subsurface (2)
Interferometric Formulation
r
z
r1
r2
B
Bcosθi
δθ
δθ ’
1 2
θ r
• Correlation coeff:
€
g=1−2δθ ρ r cosθ i
λ sinθ i
(1) e = 2.5
(2) e = 4.0
?
AF- 21
Faraday Rotation
Interferometric Formulation
r
z
r1
r2
B
Bcosθi
δθ
δθ ’
1 2
θ r
• Correlation coeff:
~ invariant, as are
Baseline Decorrelation
0.0
0.2
0.4
0.6
0.8
1.0
Baseline, B (m) 600 1350 2100 2850 3600 4350 5100
Baseline (m)
Correlation coefficient
SurfaceSubsurface (1)Subsurface (2)
€
tanθ i
€
g=1−2δθ ρ r cosθ i
λ sinθ i
€
rr
λ,
(1) e = 2.5
(2) e = 4.0
AF- 22
Faraday Rotation
Surface Clutter Problem
Baseline Decorrelation
0.0
0.2
0.4
0.6
0.8
1.0
Baseline,B (m)
600 1350 2100 2850 3600 4350 5100
Baseline (m)
Correlation coefficient
Subsurface (1)
Surface (2)
Subsurface scattering from i = 2.9 deg
Surface clutter from i = 20 degRadar
i
rz r
O
Q
P
r1r2+r
r1+rr2
AF- 23
Faraday Rotation
2-Layer Scattering ModelConclusions
• Polarimetry:– Model derived from scattering matrix formulation indicates that
there is no depth-dependent information contained in the polarimetric phase difference (or any cross-product)
– Unless - surface and subsurface returns are correlated– Inversion of polarimetric data does not seem possible– Stick with circular polarization for a spaceborne system?
==> Only problem is resolution, position shifts due to ray bending
• Interferometry: – Behavior of the correlation coefficients for surface clutter and
subsurface returns as a function of baseline length are different (assuming flat surfaces)
