SAR imaging for amplitude modulated signals
Faraday rotation for spaceborne SAR revisited
Semyon Tsynkov1
1 Department of Mathematics
North Carolina State University, Raleigh, NC
2017 AFOSR Electromagnetics Contractors Meeting
January 10–12, 2017, Arlington, VA
S. Tsynkov (NCSU)
SAR with amplitude modulation
AFOSR, 1/11/17, Arlington, VA
1 / 25
Collaborators and Support
Collaborators:
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Dr. Mikhail Gilman (Research Assistant Professor, NCSU)
Dr. Erick Smith (Research Mathematician, NRL)
Support:
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AFOSR Program in Electromagnetics, Dr. Arje Nachman
S. Tsynkov (NCSU)
SAR with amplitude modulation
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Motivation
Typical SAR waveforms are frequency modulated signals that
have constant amplitude.
Why may we be interested in signals with variable amplitude?
Windowing is used to suppress the sidelobes.
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This presentation is NOT about windowing.
For single polarization spaceborne SAR, the signals may acquire
a substantial variation of amplitude due to the Faraday rotation:
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Caused by anisotropy (gyrotropy) of the magnetized ionosphere;
Our goal is to explain the mechanism and discuss the implications:
☀
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The quality of the image may deteriorate due to the presence of FR;
For fully polarimetric SAR, amplitude variation is less of an issue.
S. Tsynkov (NCSU)
SAR with amplitude modulation
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Faraday rotation for SAR
1
Detection of Faraday rotation (FR).
▸
2
Reconstruction of the Faraday rotation parameters.
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3
Sub-band processing.
Image autocorrelation analysis.
Correction of image distortions due to Faraday rotation.
The effect of FR adds to that of “plain” temporal dispersion.
A broader context of anisotropy for SAR includes:
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Substantial variation of the geomagnetic field (∼20%);
FR for non-plane waves, in particular, spherical waves;
Going beyond gyrotropy, i.e., beyond chirality (e.g., ionic coupling);
Anisotropy of ionospheric turbulence;
Anisotropy of radar targets.
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SAR with amplitude modulation
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Governing equations
The Maxwell equations (with induced currents and charges):
1 ∂H
+ curlE = 0 ,
c ∂t
4π
1 ∂E
− curlH = − j ,
c ∂t
c
divH = 0,
divE = 4πρ.
The current density: j = −eNe ve ; Ne is the electron number density.
No motion of ions as they are much heavier than electrons.
The Newton’s second law for the electrons (no collisions):
me
dve
1
= −e(E + ve × (H0 + H )).
dt
c
Characteristic frequencies of the plasma:
√
4πe2 Ne
−e∣H0 ∣
ωpe =
(Langmuir) and Ωe =
(Larmor).
me
cme
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Time-harmonic electric field
Plane time-harmonic waves: {E , H , ve } ∝ ei(kr −ωt) .
Assumptions: H0 = [0, 0, ∣H0 ∣]T , ∣H ∣ ≪ ∣H0 ∣, and ∣ve ∣ ≪ c.
Equation for the electric field:
Λmn En ≡ [
k2 c2 km kn
( 2 − δmn ) + εmn ]En = 0.
ω2
k
ε ≡ {εmn } is the dielectric tensor:
⎡ε⊥ −ig 0 ⎤
⎢
⎥
⎢
⎥
ε = ⎢ ig ε⊥ 0 ⎥ ,
⎥
⎢
⎢0
0 ε∥ ⎥⎦
⎣
where
ε⊥ = 1 −
S. Tsynkov (NCSU)
2
ωpe
ω2
,
2
− Ωe
ε∥ = 1 −
2
ωpe
ω
,
2
SAR with amplitude modulation
and
g=
2
ωpe
ω2
− Ω2e
Ωe
.
ω
AFOSR, 1/11/17, Arlington, VA
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Double circular refraction
Let β denote the angle between k and H0 .
Parallel propagation (the simplest case): β = 0, k = [0, 0, ∣k ∣]T .
The dispersion relation det Λ = 0 yields:
k2 c2 2
) − g2 ] = 0.
ω2
Longitudinal (plasma) waves: ε∥ = 0 ⇒ ω = ωpe .
High frequencies ω ≫ ωpe correspond to transverse waves:
ε∥ [(ε⊥ −
2
2
ωpe
ωpe
Ωe
(k± )2 c2
=
1
−
±
.
ω2
ω2
ω3
The corresponding polarization vectors (eigenvectors):
⎡1⎤
⎢ ⎥
± ⎢ ⎥
E = ⎢±i⎥
⎢ ⎥
⎢0⎥
⎣ ⎦
define two circular polarizations with opposite direction of rotation.
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SAR with amplitude modulation
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Double circular refraction and Faraday rotation
Phase velocities:
v±ph
2
2
ωpe
ωpe
Ωe −1/2
ω
)
.
= ± = c(1 − 2 ±
k
ω
ω3
Typical relation between frequencies: ω ≫ ωpe ≫ ∣Ωe ∣.
The contribution of H0 to v±ph is negligible (same for group speeds).
Each circular polarization propagates almost as if H0 = 0 .
The difference of v±ph is important only when leading terms cancel.
Let E field be linearly polarized in (x, z) plane at z = 0. Then,
E (t, z) = const ⋅ (E + ei(k
+ (ω)z−ωt)
+ E − ei(k
− (ω)z−ωt)
),
where
k± (ω)z ≈
ω 2 Ωe def
1√ 2
2 z (1 ± pe
ω − ωpe
) = k(ω)z ∓ ϕF .
c
2ω 3
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
k(ω)
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SAR with amplitude modulation
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Faraday rotation
The angle ϕF is the angle of rotation of the polarization plane:
E (t, z) = 2 ⋅ const ⋅ e
⎡cos ϕ ⎤
F⎥
⎢
⎥
⎢ sin ϕF ⎥ .
⎢
⎥
⎢ 0 ⎥
⎣
⎦
i(k(ω)z−ωt) ⎢
In the case of oblique one-way propagation (β =/ 0):
2
2
ωpe
Ωe cos β
1
z ωpe Ωe cos β
ϕF = − k(ω)z
≈−
.
2
ω3
2c
ω2
For round-trip propagation, the Faraday rotation angle doubles.
For the case of inhomogeneous plasma:
ϕF = −
1
2
∫ ω Ωe cos βds.
2ω 2 c S pe
Ωe can be taken out of the integral only if H0 is (nearly) constant.
S. Tsynkov (NCSU)
SAR with amplitude modulation
AFOSR, 1/11/17, Arlington, VA
9 / 25
Detrimental effect of Faraday rotation
The FR effect on single polarization imaging can be detrimental.
The FR angles can be large (∼several full revolutions):
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Received polarization can be (nearly) perpendicular to emitted one;
This may cause a substantial degradation of the image.
FR leads to a mismatch between the received signal and the filter:
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Caused by the variation of the FR angle within the radar pulse;
Adds to the mismatch due to “plain” temporal dispersion.
Polarimetric SAR channels may compensate for one another.
FR can be used for TEC reconstruction in polarimetric SAR:
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No variation is allowed of either H0 or the FR angle over the chirp;
The ambiguity due to full revolutions still remains.
Yet a fully polarimetric capability may not always be available:
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Design of a specific instrument;
Data transfer rate limitations.
S. Tsynkov (NCSU)
SAR with amplitude modulation
AFOSR, 1/11/17, Arlington, VA
10 / 25
Faraday rotation for SAR signals
Interrogating waveform — linear chirp:
P(t) = A(t)e−iω0 t ,
2
where A(t) = χτ (t)e−iαt ,
and χτ (t) is the indicator function of the interval t ∈ [−τ /2, τ /2].
Instantaneous frequency: ω(t) = ω0 + 2αt, t ∈ [−τ /2, τ /2].
Scattered field subject to FR:
u(t, x ) ≈ ∫ ν(z )A′2δ (t − 2Tgr (x , z , ω0 ))e−iω0 (t−2Tph (x ,z ,ω0 ))
⋅ cos ϕF (t − 2Tgr (x , z , ω0 ))dz ,
where A′2δ accounts for the variation of the chirp duration and rate.
The FR angle varies along the chirp because ϕF = ϕF (ω):
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The received linearly polarized signal is “twisted;”
Causes a nonuniform attenuation over the duration of the chirp.
S. Tsynkov (NCSU)
SAR with amplitude modulation
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Matched filter and data inversion
The plain non-dispersive filter is mismatched:
P(t − 2∣y − x ∣/c) = A(t − 2∣y − x ∣/c)eiω0 (t−2∣y −x ∣/c) .
The filter corrected for temporal dispersion is also mismatched:
Pcor (t, x , y ) = A′2δ (t − 2Tgr (x , y , ω0 ))eiω0 (t−2Tph (x ,y ,ω0 )) .
Requires ionospheric parameters that we assume known exactly.
The imaging kernel (generalized ambiguity function, or GAF):
WF (y , z ) = ∑ e2iω0 (Tph (x
n ,z ,ω
0 )−Tph (x
n ,y ,ω
0 ))
n
⋅ ∫ A′2δ (t − 2Tgr (x n , y , ω0 ))A′2δ (t − 2Tgr (x n , z , ω0 ))
χ
⋅ cos ϕF (t − 2Tgr (x n , z , ω0 )) dt.
The filter Pcor (t, x , y ) was used to build this GAF.
S. Tsynkov (NCSU)
SAR with amplitude modulation
AFOSR, 1/11/17, Arlington, VA
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Data inversion
The SAR image: IF (y ) = ∫ ν(z )WF (y , z ) dz = WF ∗ ν.
Factorized GAF (the error of factorization is small):
WF (y , z ) ≈ WA (y , z ) ⋅ WRF (y , z ).
The azimuthal factor is unaffected by Faraday rotation:
WA (y , z ) ∝ Nsinc (π
y1 − z1
),
∆A
where
∆A =
πRc
.
ω0 LSA
Only the range factor is affected by ionospheric anisotropy:
2q
WRF (y , z ) = pWpR (y , z ) + WqR (y , z ),
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹τ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
“non-rotational”
due to FR
where cosine of the FR angle has been linearized over the chirp:
cos ϕF (ω(t)) ≈ p +
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2q
t,
τ
p = cos ϕF (ω0 ),
SAR with amplitude modulation
q=
τ d
cos ϕF (ω0 ).
2 dt
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Correction of ionospheric distortions
General approach to correction:
1
2
3
Use a non-corrected filter to obtain the image (or images);
Derive the required parameters of the ionosphere;
Correct the filter and evaluate the residual errors.
Isotropic parameters are the TEC and horizontal derivatives:
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Can be obtained by dual carrier probing;
For anisotropic case, are assumed to be reconstructed exactly;
Corrected filter uses the actual travel times and adjusted chirp rate.
The additional parameters that account for FR are p and q:
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Characterize the variation of cos ϕF along the chirp;
Can be expressed via the TEC if H0 varies insignificantly;
Otherwise, can be obtained by image autocorrelation analysis.
Correction of the filter in the anisotropic case proves delicate.
S. Tsynkov (NCSU)
SAR with amplitude modulation
AFOSR, 1/11/17, Arlington, VA
14 / 25
General idea of a matched filter
Received signal: u(t, x ) = ∫ ν(z )P (t − 2Rz /c) dz .
Image: Ix (y ) = ∫ ν(z ) ∫ K(t, y )P (t − 2Rz /c) dt dz .
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
W(y ,z ) — PSF
The PSF W(y , z ) must be as close to δ(y − z ) as possible.
∞
1
iω2Rz /c
P̃(ω)K̃(−ω, y )dω.
Fourier: W(y , z ) =
∫ e
2π −∞
In the simplest 1D setting (y = Ry ≡ y and z = Rz ≡ z):
∞
2 1
2
−iω2(Ry −Rz )/c
δ(y − z) = δ (2Ry /c − 2Rz /c) =
dω.
∫ e
c
c 2π −∞
Hence, P̃(ω)K̃(−ω, y) = e−iω2Ry /c for ∣ω − ω0 ∣ ⩽ B/2 (bandwidth):
−1
˜
iω2Ry /c
K̃(ω, y) = (P̃(−ω)) eiω2Ry /c = const ⋅ P(ω)e
,
Consequently, K(t, y) ∝ P(t − 2Ry /c) — matched filter.
S. Tsynkov (NCSU)
SAR with amplitude modulation
AFOSR, 1/11/17, Arlington, VA
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Matched filter for variable amplitude
The signal: Pa (t) = P(t)a(t), where, e.g., a(t) = cos(ϕF (t − tz )).
ω − ω0
Fourier (by stationary phase): P̃a (ω) ≈ P̃(ω)a(
).
2α
∣P̃a (ω)∣ varies and may even turn into zero within [ω0 − B2 , ω0 + B2 ].
̃a (ω, y) = reg[
Regularization required for inversion: K
1
eiωty ].
̃
Pa (−ω)
Back in the physical space:
Ka (t, y) =
ω0 +B/2
1
1
reg[
]eiω(t−ty ) dω.
∫
ω−ω0
2π −ω0 −B/2
P̃(ω)a(
)
2α
The integration is performed by the method of stationary phase:
Ka (t, y) ∝ reg[
S. Tsynkov (NCSU)
1
]P(t − ty ).
a(t − ty )
SAR with amplitude modulation
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Weighted matched filter
The range factor of the GAF with the weighted filter becomes:
WwR (y , z ) = ∫ reg[
χ
a(t − tz )
]A(t − ty )A(t − tz ) dt,
a(t − ty )
where ty = 2Tgr (x , y , ω0 ) and tz = 2Tgr (x , z , ω0 ).
Regularization may not be needed if the denominator =/ 0:
a(t − ty ) = cos ϕF (t − ty ),
ϕF (t − ty ) ≈ ϕF (ω0 )(1 −
4α(t − ty )
).
ω0
The central carrier FR angle ϕF (ω0 ) can be considered arbitrary.
What matters is the variation of ϕF over the chirp, ∣t − ty ∣ ⩽ τ /2:
∆ϕF = ϕF (ω0 )
2
2B
BR ωpe Ωe
=2
cos β.
ω0
c ω03
This quantity is O(1), so there is roughly equal chance of
becoming or not becoming singular.
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SAR with amplitude modulation
1
a(t−ty )
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Other filters
Unweighted matched filter (plain complex conjugate):
WuR (y , z ) = ∫ a(t − ty )A(t − ty )a(t − tz )A(t − tz ) dt.
χ
No FR at all (dispersion-compensated with exact reconstruction):
WR (y , z ) = ∫ A(t − ty )A(t − tz ) dt.
χ
Standard dispersion-compensated filter (exact reconstruction)
applied to an amplitude modulated chirp:
WsR (y , z ) = ∫ A(t − ty )a(t − tz )A(t − tz ) dt.
χ
Performance will be compared numerically for specific choices of
model parameters.
S. Tsynkov (NCSU)
SAR with amplitude modulation
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Weighting with no regularization, cos ϕF ⩾ > 0
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Weighting with no regularization, cos ϕF ⩾ > 0
The horizontal axis is dimensionless range: ξ = B(Ry − Rz )/c.
Standard filter has zeros raised; they become local minima.
The unweighted filter performs even worse.
The weighted filter performs well around ξ = 0, but has an
undesirable increase of return for large ξ.
The effect of FR on resolution is not significant. At -3dB we have:
2.8 for no FR and weighted, 3.1 for standard, 3.5 for unweighted.
A commonly used measure of image contrast is the ISLR:
⎡
−1 ⎤
Bτ /2
π
⎢
⎥
2
2
⎢
∣W(ξ)∣ dξ)( ∫ ∣W(ξ)∣ dξ) ⎥⎥ .
ISLR = 10 log10 ⎢( ∫
0
⎢ π
⎥
⎣
⎦
We have: -10dB, -8.2dB, -9.8dB, and -8.1dB, respectively.
Altogether, it is not obvious whether correction of the filter brings a
real improvement in the case where no regularization is required.
▸
In the “eyeball norm,” the weighted filter performs better.
S. Tsynkov (NCSU)
SAR with amplitude modulation
AFOSR, 1/11/17, Arlington, VA
20 / 25
Weighting with regularization, cos ϕF = 0 possible
Linearization of the amplitude within the chirp, t ∈ [− τ2 , τ2 ]:
a(t) ≡ cos ϕF (t) ≈ p +
2q
t.
τ
Works only if the variation ∆ϕF is sufficiently small.
Regularization (a simple intuitive choice):
reg[
1
Bτ
1
]=
reg[
]
a(t − ty )
2q
B[t − ty + τ p/(2q)]
B[t − ty + τ p/(2q)]
Bτ
=
.
2
2q d + B2 [t − ty + τ p/(2q)]2
Other types of regularization are obviously possible.
S. Tsynkov (NCSU)
SAR with amplitude modulation
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Weighting with regularization, cos ϕF = 0 possible
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SAR with amplitude modulation
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Weighting with regularization, cos ϕF = 0 possible
Standard filter performs poorly.
The weighted and unweighted filters offer better resolution yet
accompanied by the increase of the side lobes.
Resolution at -3dB: 2.8 (no FR), 2.6 (w), 6.2 (std), 3.3 (u).
ISLR: -10dB, -3.2dB, -5.3dB, -2.6dB.
No single measure seems to capture all of the important features.
The choice of regularization may play a very significant role.
Linearization is a convenient way to reduce the overall number of
parameters, as the regularization adds (at least) one more.
A practically more important case is the one for which the interval
∆ϕF is large and contains zero of cos ϕF .
▸
This case does not allow for linearization of cos ϕF .
S. Tsynkov (NCSU)
SAR with amplitude modulation
AFOSR, 1/11/17, Arlington, VA
23 / 25
Discussion and future work
Taking into account the Faraday rotation for SAR requires:
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▸
▸
Identifying FR as a significant contributing factor (detection);
Reconstructing the parameters of FR from the existing image;
Correcting the matched filter in order to improve the image.
The remaining issues include:
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The most difficult case of large ∆ϕF (no linearization) & cos ϕF = 0;
Reconstruction of the FR parameters with no use of linearization;
An automated procedure for detecting cos ϕF = 0;
An analytical study of the matched filters corrected for FR;
The choice of regularization for the weighted matched filter;
Better metrics for analyzing the GAF (than the resolution and ISLR);
Combined reconstruction of the TEC and the parameters of FR;
A more general approach to designing the matched filter.
Fully polarimetric framework (“mixing” channels for processing).
Other manifestations of anisotropy.
S. Tsynkov (NCSU)
SAR with amplitude modulation
AFOSR, 1/11/17, Arlington, VA
24 / 25
New research monograph (March 2017)
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SAR with amplitude modulation
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