The Ionosphere and
Interferometric/Polarimetric SAR
Tony Freeman
Earth Science Research and Advanced
Concepts Manager
Faraday Rotation
PROPAGA TION EFFECTS
• Tr oposphe re
- Refraction
- Absorption
- Phase Fluctuations
• Ionosphere
Faraday Rotation
Group Delay
Phase Change
Phase Stability
Frequency Stability
Absorption
Refraction
Units
rad
usec
rad
rad
Hz
dB
deg
f = 100 MHz
Day
Night
f = 1 GHz
Day
Night
f = 10 GHz
Day
Night
94.25
9.42
0.94
0.09
0.01
-
12.00
1.2
0.12
0.01
-
-
7500.00
150.00
750
15
75.00
15.00
7.50
1.50
0.75
1.50
0.08
0.15
0.04
0.004
-
-
-
-
0.10
0.05
0.01
0.005
-
-
-
-
[Daytime Total Electron Content = 5 x 1 017 m-2, Night-time TEC = 5 x 1016 m-2]
- after Evans and Hagfors (1968)
• Ionosphe ric pha se delay can be calibra ted if simultaneous pha se mea surements are
available at two widely separated frequen cies
Calibrated pha se,


c = 2 –  1 1
2
AF- 2
Faraday Rotation
Surface Clutter Problem
Repeat-pass Interferometry
• Subsurface return has phase
difference 1 due to ionosphere
propagation (same as surface
return from location O)
Radar
Ionosphere
• 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)
r1
r2 +1
r2+r+2
r1+r
i
O
z
P
r r
Q
AF- 3
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- 4
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- 5
Faraday Rotation
Introduction and Scope
• 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:
(degr ees)
C-Band (6 cm)
2.5o
L-Band (24 cm)
40o
P-Band (68 cm)
321o
AF- 6
Faraday Rotation
Effects on Polarimetric Measurements
• General problem:
M hh

Mhv
M vh 
1
 A(r,  ) e j  
Mvv 
1
2 
1

1 0
0  cos sin Shh


f1sin  cosShv
Svh  cos sin 1


Svv sin  cos0
0  1

f 2 4
3 
N
  hh

1  N hv
N vh 

N vv 
• For H and V polarization measurements, and ignoring other effects, the
measured scattering matrix, M, can be written as M = RSR, i.e.

M hh M vh
= cos sin
M hv M vv
– sin  cos
Shh S vh
S hv Svv
cos sin
– sin  cos
i.e.

M hh = S hh cos 2 – S vv sin 2 +

M vh = S vh cos 2 + S hv sin 2 +

M hv = S hv cos 2 + S vh sin 2 –

M vv = S vv cos 2 – S hh sin 2 +
S hv – Svh sin cos
S hh+ S vv sin cos
S hh+ Svv sin cos
S hv – S vh sin cos
• This is non-reciprocal for  ° 0 , (i.e. Mhv ° Mvh , even though Shv = Svh )
AF- 7
Faraday Rotation
Effects on Interferometric Measurements
• For an HH-polarization measurement (e.g. JERS-1) the expected value of the
radar cross section in the presence of Faraday rotation is:

M hhM hh* = S hhShh* cos 4 – 2Re S hhS *vv sin 2 cos 2 + S vvS vv* sin 4 
assuming
S hhS hv* = Shv S *vv = 0
(azimuthal symmetry)
• For repeat-pass data, the decorrelation due to Faraday rotation will be:

S hhS *hh cos2  – S hhS vv* sin 2 
 F araday =
*
S hhShh
*
S hhShh
cos4 – 2Re S hhS *vv sin 2  cos2 + Svv Svv* sin 4
• Either of these two measures will depend on both the Faraday rotation
angle, , and the polarization signature of the terrain
AF- 8
Faraday Rotation
• Modeled L-Band ‘HH’ backscatter vs. Faraday rotation angle:
L-Band HH Backscatter vs. Faraday rotation
0
-5
Bare Soil
-10
Pas ture
Upland Forest
Sw amp Forest
Plantation
-15
Conifers
-20
-25
0
20
40
60
80
100
120
140
160
180
• Backscatter drops off to a null at  = 45 degrees
• Depth of null depends on polarization signature - but effects would probably
be masked by noise ( at -17 dB) in JERS-1 data, for example
• P-Band results are similar in behavior
AF- 9
Faraday Rotation
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
Measure
=3
=5
 = 10
 = 20
 = 40
 = 90
 (HH) - dB






o (VV) - dB






 (HV) - dB






(HH 1 HH 2 *)
0
0
0  -0.01
0  -0.03
-0.15  -0.42
-0.24  -0.87
(HHHV*)




(HHVV*)





0
 (HV-VH) - deg
0
0
0 or 180
0 or 180
180
 (HH-VV) - deg





0
= - (HH-VV*) | = 0
o
o


*Radiometric uncertainty - 0.5 dB
Phase error - 10 degrees
Correlation error - 6%
• A Noise-equivalent sigma-naught of - 30dB is assumed
AF- 10
Faraday Rotation
Estimating the Faraday Rotation Angle, 
AF- 11
Faraday Rotation
Estimating the Faraday Rotation Angle, 
• Sensitivity to Residual System Calibration Errors
(shaded cells represent errors in  > 3 degrees)
NOISE
a) NE o
-100 dB
-50 dB
-30 dB
-24dB
-18 dB
1 (deg)
0
1.2
11
18
23.6
2 (deg)
0
0
1.3
5.4
31.1
CHANNEL AMPLITUDE IMBALANCE
2
b) |f1|
1 (deg)
0.0 dB
0.1 dB
0.2 dB
0.3 dB
0.4 dB
0.5 dB
1.0 dB
0
0.4
0.9
1.3
1.8
2.2
4.4
2 (deg)
0
0.1
0.3
0.4
0.6
0.7
1.4
CHANNEL PHASE IMBALANCE
c) Arg (f 1)
1 (deg)
0 deg
2 deg
5 deg
10 deg
20 deg
0
1.3
3.4
6.6
12.4
2 (deg)
0
0.4
1.0
2.1
5.1
-50 dB
-30 dB
-25 dB
-20dB
-15 dB
1 (deg)
0
0.1
0.2
0.7
2.1
2 (deg)
0.3
2.6
4.7
8.2
15.4
CROSS-TALK
d) ||
2
AF- 12
Faraday Rotation
Estimating the Faraday Rotation Angle, 
• 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  < 3
degrees using measure (2)
2
1 (deg)
10.5
|| = -25
dB
10.5
2 (deg)
3.2
5.1
a) P-Band
2
|| = -30 dB
2
1 (deg)
10.6
|| = -25
dB
10.5
2 (deg)
2.9
5.2
b) L-Band
2
|| = -30 dB
For Measure (1)
error is
dominated by
additive noise
• P-Band case has channel amplitude imbalance of 0.5 dB, phase
imbalance of 10 degrees and NE o = -30 dB
• L-Band case has channel amplitude imbalance of 0.5 dB, phase
imbalance of 10 degrees and NE o = -24 dB
AF- 13
Faraday Rotation
Correcting for Faraday Rotation
• To correct for a Faraday rotation of use:
S = R t MR t
where Rt  R-1 . This can be written:

S hh S vh
=
Shv S vv
cos – sin  M hh M vh
sin cos M hv M vv
cos – sin 
sin cos
• Since values of tan -1 are between ± /2, values of  will be between ± /4,
which means that  can only be estimated modulo /2
• This problem can be identified from the cross- pol terms by comparing
measurements before correction and after, i.e.:

S hvS vh = – M hvM hv*
*
and

0.25
S hv + S vh S hv + S vh
*
 M hv M hv *
• This test should readily reveal the presence of a /2 error in , provided that
the cross- pol backscatter Shv (or Svh ) is not identically zero
AF- 14
Faraday Rotation
• 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- 15
Faraday Rotation
Polarimetric Scattering Model
Distributed scatterers
• Take a cross-product:
Mhh Mvv  Shh1 Svv1  Shh2 Svv 2  e
*
*
*
 j( r)
 j ( r )
Shh1 Svv2  e
*
*
Shh2 Svv1
• Take an ensemble average over a distributed area:
Mhh Mvv*  Shh1 S*vv1  Shh2 Svv* 2  e j( r ) Shh1 S*vv2  e  j (r ) Shh2 S*vv1
• For uncorrelated surface and subsurface scatterers:
Shh1 Svv* 2  Shh2 Svv* 1  0
• Which leaves:
Mhh Mvv*  Shh1 S*vv1  Shh2 Svv* 2
No
interdependence
between surface
and subsurface
returns
• Similar arguments hold for other cross-products
AF- 16
Faraday Rotation
Polarimetric Scattering Model
Surface-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 2   / 1'
AF- 17
Faraday Rotation
Geometry
Radar
r
r+r
i
mv(r), s, l, , i
P
O
z
rr
loss tangent,
tan 
Q
mv(r), s, l, , i
AF- 18
Faraday Rotation
Inversion?
Empirical Surface Scattering Model
• For each layer we have 5 unknowns: mv(r), s, l, , i
• For the surface return we know , and can estimate i if we know
the local topography and the imaging geometry
• For the subsurface scattering, the wavelength  is a function of the
2   / 1'
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
AF- 19
Faraday Rotation
Interferometric Formulation
B
• Correlation coeff:
  1
1
2   r cos i
 sin  i

2
Bcosi
r2
?
r1

Correlation coefficient
Baseline Decorrelation
0.8
()  = 4.0
0.6
Surface
r
Subsurface (1)
Subsurface (2)
0.4
z
0.2
0.0
Baseline,
B (m)
r
()  = 2.5
1.0
600
1350
2100
2850
3600
4350
’
5100
Baseline (m)
AF- 20
Faraday Rotation
Interferometric Formulation
B
• Correlation coeff:
  1
1
2   r cos i
 sin  i
~ invariant, as are
 , r

2
Bcosi
r2

tan  i
r1

Correlation coefficient
Baseline Decorrelation
1.0

0.8

()  = 2.5
r
()  = 4.0
Surface
Subsurface (1)
Subsurface (2)
0.6
0.4
0.2
r
z
’
0.0
Baseline, B (m)
600
1350
2100
2850
3600
4350
5100
Baseline (m)
AF- 21
Faraday Rotation
Surface Clutter Problem
Subsurface scattering from i = 2.9 deg
Radar
Surface clutter from i = 20 deg
r1
r2+r
r1+r
i
O
Baseline Decorrelation
Correlation coefficient
r2
P
1.0
0.8
0.6
Subsurface (1)
Surface (2)
z
rr
0.4
Q
0.2
0.0
Baseline,
B (m)
600
1350
2100
2850
3600
4350
5100
Baseline (m)
AF- 22
Faraday Rotation
2-Layer Scattering Model
Conclusions
• 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)
AF- 23