Synthetic Aperture Radar (SAR)
Basic Concept of SAR:
⇒ Angular resolution
θB ≅ λ / L
L
Consider
fixed array
v i (t )
Σ
Claim Equivalent to Moving Antenna:
θB
Assumes phase coherence in
oscillator during each image
Lec20.4-1
2/26/01
v
R1
Synthetic Aperture Radar (SAR)
Ideal point
reflector
Maintain
coherence
Typical CW pulse signal and echo, E e − jφ
v=x
t
x
φ(t)
Received phase:
2π
x, t
0
Process a "window" of data, e.g.
Received magnitude:
If scat ≠ f(angle):
Phase φ(t ) =
f( λ, point source range)
Magnitude E =
f(point source σ s , range)
E
θB • Ro
x, t
0
We seek 2-D source characterization: σs = f(azimuth, range)
Lec20.4-2
2/26/01
R2
Resolution of “Unfocused” SAR
Reconstruction of image: first select R, xo of interest
W(x)
↔
x0
L
x
λ
L
θSAR
0
W(x) • E(x) observed
ψx
ˆ ψ )
W(ψ ) ∗ E(ψ ) = E(
x
SAR angular resolution: θSAR ≅ λ LSAR ≥ λ RθB = D R
≅λD
Lec20.4-3
2/26/01
R3
Resolution of “Unfocused” SAR
SAR angular resolution: θSAR ≅ λ / LSAR ≥ λ / RθB = D / R
Lateral spatial resolution = θSAR • R ≅ D
(want small D, large θB , large L)
Range resolution ≅ cT / 2
≅ λ /D
Note: If antenna is steered toward the target, increasing
L, the lateral resolution can be < D.
Then repeat for all R, x 0
R
Rmax
Yields image:
∆R
Rmin
Lec20.4-4
2/26/01
0
∆x > D
cT
∆R ≅
2
x
R4
Phase-Focused SAR
ψ (x)
θB ≅ λ / D
R
D
x0
L
2
⎧
L
⎛ ⎞
2
2
2
+
=
+
δ
≅
+ 2δR
R
(R
)
R
⎪
⎜2⎟
⎝ ⎠
∼ λ / 2D
⎪
⎪
λ
λ
L2
δ<
⇒⎨
≤
δ≅
8R 16
16
⎪
2L2
⎪
⎪therefore R ≥ λ "farfield"
x
and focusing is
⎩
unnecessary
Phase focusing is required if δ > λ 16,
so the target is in the SAR near field,
i.e., if R < 2L2 / λ = 2λR2 / D2 ( where L = λ / θSAR = λR / D ) .
Lec20.4-5
2/26/01
S1
Phase-Focused SAR
Solution: W(x) → W '(x) with phase correction
φ(t)
Pulse
2π
xo
x, t
↑
PRF must be higher; to reconstruct φ(t)
we need at least 2 samples per phase wrap.
Note: With phase focusing, L increases and
spatial resolution in x ca be less than D,
using beamsteering (e.g. a phased array)
Also: L.O. phase drift can be corrected if bright
point sources exist in scene.
Lec20.4-6
2/26/01
S2
Required Pulse-Repetition Frequency (PRF)
Ι(x)W(x )
Ι ( x) ↔ Ι (ψx )
L′
L
λ / L′
0
∆ψ x
ψx
ˆ
W(x) • Ι(x) • E(x)
Observed
W(ψ x ) ∗ Ι(ψ x ) ∗ E(ψ x ) = Eˆ ( ψ x )
PRF = v / L′ (e.g., v = aircraft velocity)
Want ∆ψ x = λ / L′ >> λ / D, or D >> L′
Therefore want PRF >> v/D
Lec20.4-7
2/26/01
S3
PRF Impacts SAR Swath Width
Rmax
Swath being imaged
vx
Rmin
θBy
x
y
Don't want echoes to overlap for successive pulses.
D
⎛ 1 ⎞
′
<<
=
Therefore let 2 (Rmax − Rmin ) / c < ⎜
L
/
v
x
⎟
vx
⎝ PRF ⎠
D
Implies that large swath widths require large
vx
and yield poorer spatial resolution.
Lec20.4-8
2/26/01
S4
Envelope Delay Focusing
Envelope-delay focusing
required when δR > ∆R 3
Range Box
∆R
R0
θB 2
θBR0
Envelope delay
δR
vx
2Ro c
x0
(Target location)
x
Compensation for both envelope delay and phase
delay is needed for very high spatial resolution
[i.e., for small ∆R (large B)] and large θBR0 .
Lec20.4-9
2/26/01
S5
SAR Intensity Estimates: Speckle
For each SAR pixel, the estimated cross-section is:
2
K = range-dependent constant
σˆ = K ∑ w ( xi ) Eij where i = pulse number
i, j
j = subscattering index
Many subscatters j per pixel
Simplifying:
σˆ =
N
∑ εk
k =1
2
=
N
∑ ak e
j( ωt +φk )
Scattering
centers
j
Lec20.4-10
2/26/01
= r
2
k =1
random variable
Ppixel
2
Ιm { ε}
uniformly over 2π typically
∑ Re {ε} = g.r.v.z.m.
k
∑ Ιm{ε} = g.r.v.z.m.
k
j+1
Re { ε}
v
U1
SAR Intensity Estimates: Speckle
p { ∑ εk } =
"r "
r
σ
2
e
−r 2 / 2 σ 2
p{r }
Ιm { ε}
p {r > ro } = e
− ( ro / σ ) / 2
2
r
σ
E[r ] = σ π / 2
Re { ε}
2
2⎤
2
⎡
E r = σ (2 − π / 2) ∝ ε
r
⎣ ⎦
Because we are adding (averaging) phasors, not scalars
0
2
E ⎡⎣r − r ⎤⎦ ≅ 2σ / 3 ≠ f(N)!
Thus raw SAR images, full spatial resolution,
STD deviation
2σ / 3
= 0.53 ⇒ very grainy images
have
Q≅
mean intensity
σ π/2
Lec20.4-11
2/26/01
U2
Reduction of SAR Speckle
Alternate ways to reduce "speckle" by averaging pixel intensities:
1) Blur image by averaging M2 pixels ⇒ Q = 0.53 / M
(using large Bτ signals yields high resolution, can be smoothed)
2) Average using spatial diversity
Q=
0.53
N
L
N = 3 or more looks
possible. Trade
against potential
resolution in x.
x
Lec20.4-12
2/26/01
U3
Reduction of SAR Speckle
Alternate ways to reduce “speckle” by average pixel intensities:
3) Average using frequency diversity, where each band
yields an independent image. Note that the same total
bandwidth could alternatively yield more range resolution
and pixels for averaging.
Image 1
Image 3
Image 2
Q=
f
Lec20.4-13
2/26/01
0.53
N
N Bands
U4
Reduction of SAR Speckle
4) Time averaging works only if source varies. For example a
narrow swath permits high PRF and an increase in N, but
unless the antenna translates more than ~λR/D between
pulses [R = range, D = pixel width (m)], then adjacent
pulse returns are correlated.
j
D
θp ≅ λ D
R
In general, reasonably smooth images need > 8 levels, so
Q¢ = S.D./M ≅ (1/3)/4 levels = 1/12, versus Q ≅ 1/2. To
reduce standard deviation by 6, need N ~
> 62 = 36 looks.
Lec20.4-14
2/26/01
U5
Alternative SAR Geometries and Applications
Case A.
Source Stationary
Case B.
Source Moves
Case C.
Source rotates
v
(e.g., geology)
v
Antenna
Moves
(e.g. moon or
spacecraft)
(e.g. satellite)
Antenna
Stationary
Antenna
Stationary
Velocity dispersion within inverse SAR (ISAR) source (e.g.,
a moving car) can displace its apparent position laterally.
Lec20.4-15
2/26/01
U6
ESTIMATION
Examples taken from Remote Sensing
Principles are Nonetheless Widely Applicable
Linear and Non-Linear Problems
Gaussian and Non-Gaussian Statistics
Professor David H. Staelin
Massachusetts Institute of Technology
Lec20.4-16
2/26/01
A1
Linear Estimation: Smoothing and Sharpening
Smoothing reduces image speckle and other noise;
sharpening or “deconvolution” increases it.
Sharpening can “de-blur” images, compensating for
diffraction or motion induced blurring.
Audio smoothing and sharpening are similar.
Consider antenna response example; we observe:
( )
TA ψ A
Lec20.4-17
2/26/01
(
) ( )
1
=
G ψ A − ψ s TB ψ s dΩ s
∫
π
4
4π
A2
Linear Estimation: Smoothing and Sharpening
Consider antenna response example; we observe:
1
TA ψ A =
G ψ A − ψ s TB ψ s dΩ s
∫
4π 4 π
1
TA ψ A ≅
G ψ ∗ TB ψ (small solid angles)
4π
( )
(
( )
( )
( )
( )
( )
( )
) ( )
1
G fψ • TB fψ
4π
↑
↑
Cycles/Radian
Antenna Spectral Response
T A fψ
=
− j2 π( ψ x fψ x +ψ y fψ y )
⎡
⎤
dψ x dψ y ⎥
⎢i.e., G fψ x , fψ y = ∫∫ G ( ψ x , ψ y ) e
⎣
⎦
(
Lec20.4-18
2/26/01
)
A3
Example: Square Uniformly-Illuminated Aperture
( )
( )
( )
1
TA ψ A ≅
G ψ ∗ TB ψ (small solid angles)
4π
1
=
G fψ • TB fψ
T A fψ
4π
↓
↓
Cycles/Radian Antenna Spectral Response
( )
Aperture
( )
( ) ( )
0
τ y , fψ
λ
Try:
( )
T̂B fψ =
Lec20.4-19
2/26/01
G(ψ)
R E τλ , G fψ
D
D
( )
y
Dλ
e.g.
τ x , fψ
λ
x
↔
ψy
ψx
λ /D
Blurring Function
( ) This restores high-frequency components, but
G ( fψ ) ← goes to zero for fψ > D / λ !
x
4 πT A fψ
A4
Example: Square Uniformly-Illuminated Aperture
( )
T̂B fψ =
( )
Try: TˆB fψ =
( ) This restores high-frequency components, but
G ( fψ ) ← goes to zero for fψ > D / λ !
x
4 πT A fψ
( )
( )W f
(
ψ)
G ( fψ )
W fψ
4 πT A fψ
0
fψ
fψ , τ x
x
λ
D
cycles/radian
λ
y
( )
The window function W fψ avoids the singularity.
The "principal solution" uses a boxcar W (s).
Lec20.4-20
2/26/01
A5
Example: Square Uniformly-Illuminated Aperture
( ) ( )
Then TB ( fψ ) = 1, so TˆB ( fψ ) = 4 πT A ( f ψ ) W ( f ψ ) G ( f ψ ) = W ( fψ ) ,
since TA ( f ψ ) = G ( f ψ ) TB ( f ψ ) 4 π
T̂B ( ψ ) = 2-D sinc function for a uniformly illuminated rectangular
Consider the restored (sharpened) image of a point source, TB ψ = δ ψ :
ψy
antenna aperture.
( )
ψx
T̂B ψ =
Zero at λ / 2D
( )
Predicts TB ψ < 0!
Can clip, transform, iterate
( )
T̂B ψ
Lec20.4-21
2/26/01
0
↔
( )
T̂B fψ
fψ x
0
Can set to zero (apriori information)
A6
Sharpening Noisy Images
( )
( ) ( )
T̂ A fψ = T A fψ + N fψ
↑
↑
Truth ∆T
rms
( )
( )
T̂B fψ
N ( fψ )
0
D/λ
for point source
fψ
→ 0! Amplifier Noise
↓
−D / λ
/ G fψ ⎤ W fψ
⎦
( ) ( ) ( )
Tˆ B fψ = ⎡ 4πTˆ A fψ
⎣
( )
N fψ for small G fψ
TA ( fψ )
e.g.
( )
Amplifies white noise
0
D/λ
fψ x
( )
Therefore optimize W fψ
Lec20.4-22
2/26/01
A7
Sharpening Noisy Images
( )
Therefore optimize W fψ :
⎡
4 π W ( fψ )
Minimize E ⎢ TB ( fψ ) −
T A ( fψ ) + N ( fψ )
0
⎢
G ( fψ )
⎣
(
∂Q / ∂W = 0 yields:
( )optimum
W fψ
⎥ =∆ Q
⎥
⎦
2
∗
1
1 *
⎡
⎤
E T A ( f ψ ) + T A ( f ψ ) N ( fψ ) + T A ( f ψ ) N ( fψ )
o
o
⎢⎣ o
⎥⎦
2
2
=
2⎤
⎡
E T A ( f ψ ) + N ( fψ )
⎢⎣ o
⎥⎦
If E [ T A N] = 0, then
Lec20.4-23
2/26/01
)
2⎤
1 ⎞
⎛
≅
W optimum ( fψ ) =
⎜
2
N⎟
⎡
⎤
E N ( fψ )
⎢⎣
⎥⎦ ⎜⎝ 1 + S ⎟⎠
1+
2⎤
⎡
E T A ( fψ )
⎢⎣ o
⎥⎦
1
A8
Sharpening Noisy Images
( )
If E [ T A N] = 0, then
W optimum fψ
The weighting (1 + N / S)
−1
1 ⎞
⎛
=
≅
⎜
2
N⎟
⎡
⎤
E N fψ
1+ ⎟
⎢⎣
⎥⎦ ⎜⎝
S⎠
1+
2⎤
⎡
E T A fψ
⎥⎦
⎣⎢ o
( )
( )
is widely used
( )
W opt fψ
Wopt ( f ψ )
N≠0
for N = 0 :
0
( )
1
D/λ
fψ
0
fψ
W opt fψ ⇒ wider beam (lower spatial resolution), lower sidelobes.
Can be used for restoration of blurred images of all types:
( )
photographs TV, TA ψ A maps, SAR images, filtered speech, etc.
Lec20.4-24
2/26/01
A9