5_1555_TH4.T01.5_suwa.ppt

A Study on The Resolution Limit of Polarimetric Radar

and The Performance of Polarimetric Bandwidth Extrapolation

Technique

Kei Suwa, Toshio Wakayama, Masafumi Iwamoto

Mitsubishi Electric Co. Information Technology R&D Center

Outline

• Background -- SAR resolution, PRF, and Polarimetry --

• Polarimetric Bandwidth Extrapolation (PBWE)• The Signal Model• Statistical Resolution Limit (SRL)• The Cramér-Rao Bound (CRB)• SRL and The Performance of PBWE• Conclusion

Range resolution is determined by the signal bandwidth.

Azimuth resolution is determined by the synthetic aperture length.

Background -- SAR resolution, PRF, and Polarimetry --

Range resolution : signal bandwidth / Azimuth resolution : synthetic aperture lengthRange resolution : signal bandwidth / Azimuth resolution : synthetic aperture length

orbit

freq. range

cr =2B

FT

2 2

Ra

L

L

R

Strip map mode

D

222

DDa

DHigher resolution with smaller aperture size .

Spotlight mode

: Signal Bandwidth

: Speed of light

c

B

rB

: wave length

: Synthetic Aperture length

LR : Range

Background -- SAR resolution, PRF, and Polarimetry

orbit

BeamIlluminationarea

Platform trajectory

Doppler Freq.

TimePRF

Due to PRF limitation, Polarimetric SAR often give up the resolution.Due to PRF limitation, Polarimetric SAR often give up the resolution.

Conditions of the PRF PRF must be higher than the signal

Doppler bandwidth, which is determined by the azimuth beam width.

PRF must be low enough so that the range ambiguity is sufficiently low.

The problem with Polarimetry Polarimetry requires sending H-polarization

pulse and V-polarization pulse. PRF gets lower for each polarization

channel, so polarimetric SAR often need to give up the resolution.

It would be nice if polarization information helps enhancing

the resolution

Background -- SAR resolution, PRF, and Polarimetry

Dr. Mihai Datcu “Semantic Content Extraction from High Resolution Earth Observation Images”

Background -- SAR resolution, PRF, and Polarimetry

… resolution refers to separate two or more nearby targets, i.e., to determine that there are two instead of one.

~Radar Handbook (1970) 4-2~

… resolution refers to separate two or more nearby targets, i.e., to determine that there are two instead of one.

~Radar Handbook (1970) 4-2~

What is “resolution”?

Background -- SAR resolution, PRF, and Polarimetry

range

1. Flat plate and dihedral corner reflector case

2. Two flat plates case

Polarizationvector

range

HH

VV

HH

VV

Flat plateDihedral corner reflector

101

10

-1

polarization vectors

Shh

Shv

Svv

=

Two vectors are orthogonal to each other.

Polarization information would not help

It seems to be possible that polarization information helps improving the “resolution”.It seems to be possible that polarization information helps improving the “resolution”.

Polarimetric Bandwidth Extrapolation (PBWE)

FFT

IFFT

Polarimetric Linear Prediction Model Estimation

range

range

i ii

iii

ivv

2B

c

2B’

c

freq.HH

HVVV

B

freq.

B’

HHHV

VV

HHHV

VV

HHHV

VV

• A polarimetric linear prediction model is fitted to the spectral data. • Then the HH,HV,VV spectral data is extrapolated up to B’.

• A polarimetric linear prediction model is fitted to the spectral data. • Then the HH,HV,VV spectral data is extrapolated up to B’.

Polarimetric Bandwidth Extrapolation (PBWE)

http://www.emi.dtu.dk/research/DCRS/Emisar/emisar.html

EMISAR : dual frequency (L- and C-band) polarimetric Synthetic Aperture Radar (SAR) system developed at Technical University of Denmark.

Storebaelt : the great belt

range

azimuth

We have empirically shown that Polarization information does contribute tothe resolution enhancement

We have empirically shown that Polarization information does contribute tothe resolution enhancement

Resolution

“Nominal” Resolution does not reflect polarization properties

Fourier analysis would give the nominal resolution. Parametric spectrum analysis would achieve higher resolution.

--- e.g. PBWE, MUSIC, ESPRIT, MVM, MEM, etc

cr =2B

2 2

Ra

L

Quantitative analysis on the influence of polarization information onthe resolution is provided in this presentation.

Quantitative analysis on the influence of polarization information onthe resolution is provided in this presentation.

range azimuth

We are interested in the resolution achievable by “polarimetric” parametric spectrum analysis method such as PBWE.

We are interested in the resolution achievable by “polarimetric” parametric spectrum analysis method such as PBWE.

Statistical Resolution Limit (SRL)

• “Statistical Resolution” : Δd

• “Statistical Resolution Limit” : min(Δd)

[1] S.T.Smith, “Statistical Resolution Limits and the Complexified Cramér-Rao Bound,” IEEE Trans. Signal Process., vol. 53, no. 5, pp1597—1609, May 2005

The estimate of target separation contains error due to noise. If the expected error is sufficiently small compared to the real separation, we can claim that these targets are resolvable.

true target separation = d

d̂

d̂

d

2

ˆ ˆˆs.t. , where

d dd d d d d E d d

2

ˆ ˆˆmin s.t. min , where

d dd d d d d E d d

SRL is a simple metric that provides the highest resolution achievableby any unbiased parametric spectral estimator.

SRL is a simple metric that provides the highest resolution achievableby any unbiased parametric spectral estimator.

The Signal Model

clutter

range

target

HH

HV

VV

power

scene

Polarization property

Polarimetricchannels

The signal at each resolution cell is represented by a polarization vector.The signal at each resolution cell is represented by a polarization vector.

The Signal Model

denotes the long column vector formed by concatenating the columns of X.

:z Va n

HE nn R I

11, exp : :

H

Nf z a z Va R z Va

R

n: Gaussian noiseV: matrix of steering vectorsa: complex amplitudes of

the targetsN: the number of samples

The probability density function of the data z is:

"X : "

1 1jj

a ae e

1

2

1

2

1

1

Nj

Nj

e

V

e

freq.

B

ranged

VVHH

11a

HHVV

target #2target #1

12a21a

22a

Two closely located point targets signal model is considered.Two closely located point targets signal model is considered.

The Cramér-Rao Bound (CRB)

2; ; log | ,a aG E f z a

1

|| | aVar a CRB a G

ˆ |2d

Nd CRB a

11, exp : :

H

Nf z a z Va R z Va

R

The CRB for the parameters are derived.The CRB for the parameters are derived.

• Likelihood function of the parameters

• Fisher Information Matrix

• Cramér-Rao Bound on the covariance of the estimator of the parameters

• Convert the CRB to the lower bound of the target separation d

The Cramér-Rao Bound (CRB)

The CRB for 2 polarimetric channels case is derived.The CRB for 2 polarimetric channels case is derived.

• The lower bound for the target separation

(Polarimetric / 2 channels / White Gaussian noise)

• cf) when ∆φ →large (two targets are far away)

0

2 2 20 1

[ ]2 cos / 2G

dNStd d Tr D D FRC

a d d

2

0 2 2

2

1 2 2

(0) '( )"(0)

(0) ( )

( ) '( )"( )

(0) ( )

s sd s

s s

s sd s

s s

sin2

( )1

sin2

N

s

1

2 "(0)

NStd d

a s

0 1"(0), 0d s d

SRL and the Performance of PBWE

The PBWE achieves CRL and SRL fairly well.The PBWE achieves CRL and SRL fairly well.

(a) ∆Ψ = 0 (b) ∆Ψ = π/6

Δd=0.4Δd=0.1

5

ranged

jaea jaea

HH

VV

• Monte Carlo iteration 1,000 times.• N=50 / SNR=40dB / order of the linear prediction filter = 6• If the polarization properties of the two targets are the same

polarization information does not help (a).

SRL and the Performance of PBWE

The SRL is minimum when the polarization properties of the two targets are orthogonalThe SRL is minimum when the polarization properties of the two targets are orthogonal

ranged

jaea jaea

HH

VV

(c) ∆Ψ = π/2 (d) ∆Ψ = π• The SRL is improved to ¼ by using polarization information,

when the polarization properties of the two targets are orthogonal (d).

• (d) corresponds to a case where two targets are trihedral and dihedral corner reflectors.

Conclusion

• The SRL of polarimetric radar is derived and compared with that of single polarization radar

• It has been shown analytically that the polarization information helps improving the resolution when the polarization properties of the two closely located targets are different, and thus on the average, the polarimetric radar outperforms single polarization radar. – e.g. If the polarization properties of the two closely located

targets are orthogonal to each other, the SRL is improved to ¼ by using polarization information.

• It has been shown that the resolution of the PBWE previously proposed by the authors almost achieves the SRL.