Post on 03-Feb-2022
Coherent Radar Imaging
Ronald F. WoodmanJicamarca Radio Observatory
Instituto Geofisíco del Perú
Acknowledgments:Jorge L Chau, David Hysell, Erhan Kudeki
Scope• Coherent Radar Imaging is the outgrowth of the more general Radar
Imaging technique• Radar Imaging includes:
– SAR Imaging– Planetary imaging– Georadar (underground)– Meteorological imaging– Coherent radar imaging
• Ionospheric irregularities• Upper atmospheric turbulence
• Radar Imaging is, in turn, part of a broader technique: Radio Imaging which includes radio imaging in Astronomy. All these techniques are similar.
• We will limit ourselves to the topic of the title.
Scope (continued)
• Nevertheless, the most fundamental problems are common. Thus, coherent radar imaging borrows from the advances made in the other applications, especially fromAstronomical Radio Imaging, which precedes radar imaging by a couple of decades.• We hope some of the advances made by coherent radars will benefit the other techniques
• We will try to answer the questions:• How is it done?• What can be used for?• What has produced? (Include a few examples)
Peculiarities of Coherent Radar
• Target is three (space) dimensional• Target changes in time in two scales
– short defining “color”(frequency spectrum) – long, the scale of non-stationary
• Target is a non-stationary non-homogenous 4-dimensional stochastic process
• Relatively small number of independent samples available for averaging
Coherent radar imaging techniques
• Scanning• Imaging with multiple-antenna arrays .
– Filled arrays– Sparsely populated apertures– Interferometers
Imaging by scanning• Scanner analog• Slit camera• Images by scanning (Jicamarca)
– Sp-F irregularities– E Region irregularities– 150 km echoes– Mesospheric echoes– Stratospheric and Tropospheric
Limitations
Slit Camera
Imaging by scanning• Scanner analog• Slit camera• Images by scanning (Jicamarca)
– Sp-F irregularities– E Region irregularities– 150 km echoes– Mesospheric echoes– Stratospheric and Tropospheric
Limitations
The Jicamarca Radio Observatory
ESF echoes(from Woodman and Chau [2001])
Equatorial Electrojet
18:45 18:50 18:55 19:00 Day: 19-Nov-2003
100
110
120
130
140
Ran
ge (
km)
(a) SNR E (dB)
-6.0
0.0
6.0
12.0
18.0
[Chau and Hysell, 2004]
150 km echoes
Mesosphere
Stratosphere and Troposphere
Slit-camera Analogy and Problems
used with permission
Imaging from multi-antenna arrays
• (Diffraction theory)
• Filled aperture array– Camera analog– Truncation (aperture not large enough)– Inversion of truncated visibility
• Sparsely populated aperture– Non redundant spacing– Sparsely sampled visibility– Inversion
• Interferometry
θ
L >> D
D
( )F q
( )f x
( ) ( ) iF f e d∞ −
−∞= ∫ xx xiqq
1 2
1 2
{sin ,sin }{ , }kx kx
==x
θ θq
( ) ( ) * ( )V f f≡ ⟨ + ⟩r x x r
( ) ( )F f xq ( ( )B V r) q
( ( ) * ( )B F F) ≡ ⟨ ⟩q q q
kerneli ie e− −←⎯⎯ ⎯⎯→i ix rθ θ
θ
L >> D
D
( )F q
( ) ( )F f xq
( ( )B V r) q
( )B q
( )V r( )f x
Where( ( ) * ( )
and( ) ( ) * ( )
( ) ( )and
( ( )
B F F
V f f
F f
B V
≡ ⟨ ⟩
≡ ⟨ + ⟩r x x r
x
r
)
)
q q q
q
q
Actually:
( ) ( , )i id d e e ′ ′− +′ ′ ′= ∫∫ i iθ,θ θ θx xx x x xB V
( Diagonal[ ( , ) ( ) * ( )]B F F′ ′≡ ≡ ⟨Bθ θ θ θ θ)
and * ( ) ]where
+ ⟩x r( ) ( ) * ( ) DiagonalAverage[ ( , ) ( )
V f f f f′≡ ⟨ + ⟩ = ≡ ⟨
′= −
r x x r x x xr x x
i V
( , )θ θ ′B ( , )′x xV
Imaging from multi-antenna arrays
• (Diffraction theory)
• Filled aperture array– Camera analog– Truncation (aperture not large enough)– Inversion of truncated visibility
• Sparsely populated aperture– Non redundant spacing– Sparsely sampled visibility– Inversion
• Interferometry
(from Woodman [1997])
Analogy with an pin-hole cameraTrue
Brightness
True Visibility
Meassured Visibility
Estimated Brightness
)(θB
)(ˆ θB
( )V rˆ ( )V r
( ) (f F θ )x
ˆ ˆˆ( ) (f F θ )x
ˆ ( ) ( ) ( )f a f=x x x
( ) ( )B V
Object Plane
Aperture Plane
ImagePlane
θ r
ˆ ˆ( ) ( )B Vθ r
ˆ( ) ( ) ( ) ( )V a a V∗ ×= ∗r x x +r r2ˆ ( ) ( ) ( )B A Bθ θ θ= ∗
2 ( ) ( ) ( )A a aθ ∗ +x x r
(from Woodman [1997])
F
f
f
F
ˆ
ˆ
ˆˆ =F M fi
BAB 2 ∗=ˆ
Analogy with an optical camera
ˆ ∗= ∗ ×MiV a a B
In radar imaging a can be an arbitrary complex vector
,{ } { }i jii jM e θ−≡ = xM i
Butler MatrixButler and Lowe, 1961
FFT AlgorithmCochran et al., 1967
ˆ ∗= ∗ ×a a MiV Bˆor = ∗2B A B
Given:
Imaging Problem:
Find a Bestimate that “best” agrees with B under valid constrains. A typical inversion problem
Frequency spectra information (Color)
, ,
, ,
,
( ) is actually ( , )ˆ ˆ is Fourier transformed into
ˆ ˆ Cross-spectra, , is evaluated at (4) different -bands
Each band is treated independently as explained Four
t
f f t
f f
f f
B
ω
ω ω
ω
ω′
•
•
•
••
x x
x x
x
x x
, corresponding to four bands, are evaluated. 3 colors (BGR) are assigned to 3 center -bands Color saturation is an indication of narrow spectral width
ωω
−
••
ˆ ∗= ∗ ×a a MiV Bˆor = ∗2B A B
Given:
Imaging Problem:
Find a Bestimate that “best” agrees with B under valid constrains. A typical inversion problem
Inversion Techniques-A classification intent:
Let scale size of and the scale size of , we can then considerthe following cases:
1) Filled a
a cL r= ∗ =a a V
ˆperture Use as a good estimate of ˆ Divide by ( 0) and then (deconvolve).ˆ Divide by
a c
a c
L r
L r
−
> •
• ∗ ≠
≤ •
1
B B
V a a M
V ( 0) , extrapolate
and then (deconvolve). Extrapolate using MaxEnt 2) Sparse aperture
−
∗ ≠
•
1
a aM
th st nd
Use Capon, Clean, deconvolution and Multiblob models Use MaxEnt 3)Interferometer Evaluate 0 , 1 and 2 moments.a cL r
••
•
case action
Brightness and visibility used for the examples that follow
θF
rV
Inversion Techniques-A classification intent:
Let scale size of and the scale size of , we can then considerthe following cases:
1) Filled a
a cL r= ∗ =a a V
ˆ Use as a good estimate of ˆ Divide by ( 0) and then ( ).ˆ
perture
dec
onvolv
Divide by
e
a c
a c
L r
L r
−
> •
• ∗ ≠
≤ •
1
B B
V a a M
V ( 0) , extrapolate
and then ( ). Extrapolate using MaxEnt
deconvolve
Sparse aper 2) tur e
−
∗ ≠
•
1
a aM
th st nd
Use Capon, Clean, and Multiblob models Use MaxEnt 3)Interferometer Evaluate 0 ,
deconvoluti
1 and
o
2 momen
n
ts.a cL r
••
•
case action
Brightness and visibility used in the following examples
5 10 15 20 25 30 35
20
40
60
80
5 10 15 20 25 30 35
1
2
3
4
5
θF rV
Case a c
ˆB,B ˆV,V
L r>
5 10 15 20 25 30 35
2.5
5
7.5
10
12.5
15
5 10 15 20 25 30 35
1
2
3
4
5
5 10 15 20 25 30 35
10
20
30
40
ˆ ∗= ×∗ ia a MV B
Deconvolved image
5 10 15 20 25 30 35
20
40
60
80
5 10 15 20 25 30 35
1
2
3
4
5
θF rVˆ f
ˆ
or 0∗∗
∗
= ≠∗∗
∗
i
i-1
aa
M
a
a
Ma
a
V
V
B
B =
Case a c
ˆB,B ˆV,V
L r<
not sufficientDivide by for x's such that 0
Extrapolate deconvolve)
ˆ
ˆ
∗
∗ ∗
= ×
≠
∗
∗ ∗
ia aa a a
Ma
V
V
B
(
5 10 15 20 25 30 35
1
2
3
4
5 10 15 20 25 30 35
1
2
3
4
5
5 10 15 20 25 30 35
2
4
6
8
10
Case: sparse array
ˆB,B ˆV,V
not sufficientUse band-limited and positive-definite properties of
ˆ ∗ ×∗= ia a M B VB
5 10 15 20 25 30 35
1
2
3
4
5 10 15 20 25 30 35
1
2
3
4
5
5 10 15 20 25 30 35
2
4
6
8
10
Case: sparse array
ˆB,B ˆV , V
5 10 15 20 25 30 35
2
4
6
8
10
5 10 15 20 25 30 35
1
2
3
4
5
Starting point
5 10 15 20 25 30 35
1
2
3
4
5
Divide by a*a
5 10 15 20 25 30 35
2
4
6
8
10
ˆ ′B,B ˆ ′V , V
5 10 15 20 25 30 35
2
4
6
8
10
Force to zero out-of band and negative B’s
5 10 15 20 25 30 35
1
2
3
4
5
Transform back
5 10 15 20 25 30 35
1
2
3
4
5
Correct measured values
5 10 15 20 25 30 35
2
4
6
8
10
Inverse transform to a new B’Iterate !
5 10 15 20 25 30 35
2
4
6
8
10
5 10 15 20 25 30 35
1
2
3
4
5
After some iterations
Inversion Techniques-A classification intent:
Let scale size of and the scale size of , we can then considerthe following cases:
1) Filled a
a cL r= ∗ =a a V
ˆperture Use as a good estimate of ˆ Divide by ( 0) and then (deconvolve).ˆ Divide by
a c
a c
L r
L r
−
> •
• ∗ ≠
≤ •
1
B B
V a a M
V ( 0) , extrapolate
and then (deconvolve). Extrapolate using MaxE 2) Sparse apert
nure
t
−
∗ ≠
•
1
a aM
th st nd
Use Capon, Clean, deconvolution,Multiblob models 3)Interferometer Evaluate 0 , 1 and 2 m
Use MaxEns
toment .a cL r
•
•
•
case action
• Maximize Entropy,
With the following constrains:
– The normalized measured visibility, for each antenna pair, conforms with the FT of the normalized brightness distribution,
plus an estimation error, ..
– Errors ( ) are allowed, but bounded to their estimated theoretical value, in the maximization process. Correlation between errors (After Hysell&Chau, submitted 2005) is taken into account.
ln( / ),S B B Bθ θ θθ θ
=
MaxEnt∑ ∑
/ ,B Bθ θθ∑
, ,ˆ / ,V P′ ′x x x x
,x xe ′
,x xe ′
Inversion Techniques-A classification intent:
Let scale size of and the scale size of , we can then considerthe following cases:
1) Filled a
a cL r= ∗ =a a V
ˆperture Use as a good estimate of ˆ Divide by ( 0) and then (deconvolve).ˆ Divide by
a c
a c
L r
L r
−
> •
• ∗ ≠
≤ •
1
B B
V a a M
V ( 0) , extrapolate
and then (deconvolve) or use . Extrapolate using MaxEnt 2) Sparse
Capon
ap
−
∗ ≠
•
1
a aM
th st nd
erture Use , Clean, deconvolucion, Multiblob models Use MaxEnt 3)Interferometer
Capon
Evaluate 0 , 1 and 2 moments.a cL r
••
•
case action
Capon’s method(from Capon [1969]
{ }, DFTw ≠θx
†
ˆ ˆGiven { }Capon's { } is given by:
where { } is such that is a minimum, under the constrain 1 for every .Here,
f fB
wB
′≡≡
≡
=
c
c
B
B = i i
i
V
w V w
w
e w
e
θ
θ,
θ
θ
x x
x
{ }are the sampled values, at , of a unitary plane wave coming from The constrains are satisfied by:
1 .
Advantage: Solution involves a single Matrix inversion
ie−≡
=cB
i
i i-1e V e
θ
θ.
x x
!
MaxEnt Examples
• Jicamarca– Spread F– Electrojet
• Aurora• QPE• Trospheric Turbulence
SpF, Jicamarca Observatory
[Hysell et al., 2004]
± 50 m/s
SpF, Jicamarca Observatory
[Hysell et al., 2004]
±150 m/s
Imaging at Jicamarca:2D Imaging – EEJ at Twilight
[from Chau and Hysell,2004]
±50 m/s
Daytime Electrojet over Jicamarca
[Hysell, Chau et al., PC]
Puerto Rico
Arecibo
QP Echoes over Puerto Rico
[Hysell et al., 2004]±300 m/s
Hysell, p.c., 2005
Aurora, Alaska
[Bahcivan et al., 2005]
Hysell, p.c., 2005
Aurora, Alaska
[Bahcivan et al., 2005]
Imaging at Jicamarca (9):2D Imaging – 150-km echoes
[Chau, Kudeki, Hysell, PC, 2005]
Imaging at Low Latitudes (1): Piura (14± Dip)
[Chau, PC, 2005]
QP echoes over Piura (14± Dip)
[Chau, PC, 2005]
Capon MethodTropospheric Imaging at MU
(from Palmer et al. [1998])
Images of rain obtained through Doppler sorting
TroposphericImages
Fourier Capon
• Irregularities are “almost” isotropic.
• Use of Capon method
• Images of the aspect sensitivity (5 min integration).
• Brightness intensity is color-coded.
Improved resolution!
Tropospheric imaging with the Provence ST VHF radar
(from Hélal et al. [2000])
Fourier images [SNR( ) and Doppler ( )]
• Use of very wide tx/rx beam widths ( 60o) and 8/16 rx channels. Although, only one physical rx is used.
• Rx channels are multiplexed with a high-communication rate switch.
• Observations of very wide horizontal stratified structures using “Sequential” PBS, Capon and MUSIC.
3D Imaging at Jicamarca(from Chau and Woodman [2000] and poster)
TroposphericImages
ReceivingConfiguration
3D Equatorial Electrojet image
•Note a meteor echo in the 3D image.
•The tropospheric images do not show continuity with height.
•Not much gain in information is obtained by using more than 3 antennas, at least at the time of the experiment and/or with the “narrow”field of view employed.
Inversion Techniques-A classification intent:
Let scale size of and the scale size of , we can then considerthe following cases:
1) Filled a
a cL r= ∗ =a a V
ˆperture Use as a good estimate of ˆ Divide by ( 0) and then (deconvolve).ˆ Divide by
a c
a c
L r
L r
−
> •
• ∗ ≠
≤ •
1
B B
V a a M
V ( 0) , extrapolate
and then (deconvolve) or use Capon. Extrapolate using MaxEnt 2) Sparse ap
−
∗ ≠
•
1
a aM
th st nd
erture Use Capon, Clean, Multiblob models Use MaxEnt 3) Evaluate 0Inter ,ferom 1 and 2 momeete nts.r a cL r
••
•
case action
Interferometryx x’θ
2 22
(0) where Power (
( ) / Mean angle of arrival (
2(1 ( ) /Width ( ) (
These properties can be generalized to 2-Dimensions.*(
d d
d
d
Woodman and
P V P B d
r r B d
V r PB d
r
θ θ
θ φ θ θ θ θ
σ σ θ θ θ θ
= ≡ ≡ )
= ≡ ≡ )
−= ≡ ≡ − )
∫∫
∫
,1974) Guillen
In any FT pair, like ( ( ), the derivatives,
evaluated at the origen of one are proportional to the moments of the same order of the other. If the distance , where is the cara
d c
c
B V r
r rr
θ )
′= −x x
( ),
,
cteristic size of
= ( ) . Then, can be expanded in a Taylor series,
and it can be shown that:
i rV V r eV
φ−′
′
∗
x x
x x
rd
Cross-correlator
,V ′x x
Coherence
Interferometer Applications
• Meteor head echoes• Aspect sensitivity• Magnetic field inclination• First electrojet images• Electrojet drifts
Interferometer : Meteor heads radians&velocities
[Chau and Woodman, 2004]
Interferometry at Jicamarca (6)Aspect Sensitivity Configuration
Receive signal onVarious 64ths
Receive commonSignal in 1/64th
Transmit on East andWest Quarters
[e.g. F. Lu, 2005; Ph.D. Thesis]
Interferometry at Jicamarca (7)EEJ Aspect Sensitivity Measurements[Kudeki and Farley, 1989]
[e.g. F. Lu, 2005; Ph.D. Thesis]
Woodman, 1971
Interferometry at Jicamarca (2)Measurements of Magnetic Field Inclination
•Using Incoherent scatter theory, combined with N-S Interferometer, Woodman [1971] was able to tome measure the inclination of the magnetic field above Jicamarca with 1 min of arc accuracy. At the time, models were off by ~1o.
[e.g. Woodman, 1971]
Thank you