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