ACCURATE OPTIMAL DOPPLER CENTROID ESTIMATION FOR SAR DATA

ACCURATE OPTIMAL DOPPLER CENTROID ESTIMATION FOR SAR DATA

IGARSS 2011Vancouver, Canada

July 27th, 2011

Pietro Guccione Politecnico di Bari

Andrea Monti Guarnieri Politecnico di Milano

Summary

Motivations and Rationale of the algorithm

Target Model

Problem Statement and the Maximum Likelihood estimation

Results on simulated and X-band real data

Conclusions

2IGARSS Symposium, Vancouver, CanadaJuly 27th , 2011

Motivations /1Radiometric calibration is a fundamental item in the processing of Synthetic Aperture Radar (SAR): it allows accurate measures of radar reflectivity.The determination of an accurate azimuth antenna pattern (AAP) and attitude -i.e. accurate Doppler centroid-, acquires particular relevance for calibration process.

AAP estimation is actually performed using transponders: they are high precision and geolocated devices that provide a fixed Radar Cross Section require maintenance, accurate and complex calibration: so are expensive. Provide the 1-way gain of the AAP and require a sufficient background/target contrast: interference with the mission

requirements.


Motivations /2We perform AAP (and AAP pointing) estimation using natural targets observation over a stack of SAR images. The natural targets we want to exploit are:1) Sparse nearly over all the images, allowing a possible extraction of the AAP from all the acquired dataset;2) Dense, i.e. many targets, especially in acquisition over cities or man-made objects, can be present in a single image, making the estimation robust;3) Stable in time, (stability is required for a robust AAP estimation in a multi-image context)

4

Mai

n-lo

be

wide-aperture

Side-lobes

Examples of persistent point scatterers (PPS)

[… we use amplitude and phase of impulse response]

IGARSS Symposium, Vancouver, CanadaJuly 27th , 2011

Target Model /1Synthetic Aperture Radar acquisition and focusing can be modeled as a complex source (the ground reflectivity) passed through a linear time-variant (LTV) system (the SAR impulse response) which smears out the energy of a single point scatterer. The inverse processing, said focusing, tries to focus the point scatterer response back to a single point. Model after focusing of SAR data:

5

)()(exp)()()( fWfjfAfSfZ antsc

SAR is pulsed in azimuth direction; this makes the spectrum folded.(Part of the clutter noise is then composed by the spectral replica that superimpose to the one of interest).

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Azimuth frequency [Hz]


Target Model /2Previous model is more accurate if high-SNR and isolated point targets are considered (to avoid alias over sidelobes).

For SAR data the effective azimuth bandwidth, which can be extended also over the main lobe, is greater than the actual sampling frequency.Since we want a proper representation of the antenna (i.e. without ambiguity), the sampling frequency must be increased.

Azimuth oversampling can be performed before azimuth focusing with usual oversampling methods but accounting for the time-variant nature of the phase history. With the upsampling the complete history of the point scatterer is reconstructed after focusing, but spectral replica are generated (digital spotlight, Prati 1991)

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frequency [Hz]

PS

D [d

B]

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azimuth [m]

ampl

itude

[dB

]

target impulse response: cut in azimuth


Problem StatementIn an AAP multi-image framework estimation, the problem can be stated as follows.

We have available a stack of N images: acquired by repeated geometry properly co-registered w.r.t. a master referencewhere a set of P targets have been properly selected on the basis of their stability (i.e. they are present in all the images of the stack) their whiteness (i.e. the spectrum is very similar to the ideal antenna shape, so they

result very “point-shape”)

Doppler centroid Maximum Likelihood estimationδfdc is estimated as the optimal spectral shift of the target Power Spectral Density.All the almost-point targets in the image are exploited, as we suppose that the residual centroid is the same for them (this is the major assumption, valid over quite flat areas).The estimate shows to be more accurate than the traditional correlation-based methods and near to the theoretical Cramer-Rao Bound of the estimate.

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)()(exp)()()( ,,,,, fWfjfAfSfZ pnpnantpnscpn


Algorithm Rationale

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Stable PointScatterer est

SpotlightAz Focusing

Antenna IdealModel

Quality FactorEvaluation

ClutterEstimation

Evaluation δfDC

Stable PointDB

Target spectralshape est

NormalizeImages

Target shapeDB

AAPestimation

Stable Point Scatterer Estimation

99

This procedure performs the selection of a set of puntiform targetsstarting from a SLC image. Targets are selected by passing the following tests: High contrast.

Their mean intensity is computed; then it is computed the power of the neighbor backstatterer. The higher is the ratio, the more probable is the target a point scatterer.

Similarity of its spectrum with the model, i.e. the Azimuth Antenna Pattern. A quality factor is computed and target are sorted and thresholded w.r.t. it:

Stable PointScatterer est

1

2,

2,

,

)()(ˆmin

)()(ˆ

mmidmnz

n

mmidmnz

pn

fAfS

fAfS

q

It is the normalized root mean square error of the target spectrum w.r.t. the ideal antenna, summed on all the frequencies.This similitude test is a sort of a Generalized Likelihood Ratio Test (GLRT).


Spotlight Azimuth Focusing /1

1010

SAR acquisition is pulsed and the resulting azimuth spectrum folded.The effective azimuth bandwidth is greater than the actual sampling frequency.

Azimuth oversampling is performed basically by adding N-1 zeros among each couple of samples in azimuth (after range compression) and by a time-variant filtering.With the upsampling the complete phase history of the point scatterer is reconstructed.

t

faz

PRF/2

-PRF/2

Toss

PRF/FM t

faz3PRF/2

-3PRF/2

Toss

filter


Folded spectral supportAzimuth t-v filtering



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azimuth [samples]

slan

t ran

ge [s

ampl

es]

20080919051557

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azimuth lines

rang

e sa

mpl

es

Point # 1

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azimuth lines

rang

e sa

mpl

es

Point # 1

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Doppler [samples]

rang

e sa

mpl

es

Point # 1

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x 104

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Extraction of slice of data

(3x, 5x)

PPS DB

Range Foc SLC

Selectivewindowing

Azimuth FFT &oversampling

selective windowing (on range compressed data to avoid interference of other targets nearby)

range samplesaz

imut

h lin

es

Zero-interleaved RGC point target, fDC=0

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x 104


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Am

plitu

de [d

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Point 20101218T174818. Theoric fDC=0 Hz, 5 Antenna Ap

Ideal Target psd, SNR=20dB, null fDC, simulated for 5 antenna apertures


Doppler [samples]

rang

e sa

mpl

es

Point # 1

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x 104

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Selective s-vfilter

selective space-variant filter is applied to avoid the effect of replica

Azimuth stripmapfocusing

Central stripextraction

Spotlight focusedPPS DB

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x 104

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Position of the ghosts of the central target

Starting from the model of target spectrum

we suppose that targets are white (we can weight this hypothesis by the quality factor) and initially we put also

Doppler Centroid Estimation /1

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For each image n, SAR data observations Zn,p are usually considered samples from a multivariate Gaussian; so their joint probability, conditioned to δf has a closed-form expression:

ffAfA idmant )(

z

zH

P CZCZfZZp1

15.0exp)|,...,(

m mz

mpn ffS

Zf

);(

2

,

The parameter estimation is carried out by maximizing the log-likelihood (LLH). Simplification: the covariance matrix is almost diagonal and its elements are basically the power spectrum density of the target. This leads to a very simple formulation of the LLH for each target:

i.e. the power of the target whitened spectrum.

)()(exp)()()( ,,,,, fWfjfAfSfZ pnpnantpnscpn


Doppler Centroid Estimation /2

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The CRB of the estimate can be found by numerical computation, since for jointly circular Gaussian process its assumes a simple form

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)()('tr

m mz

mzzzCRB fS

fSfC

fC

numerically solved by means of sinc4 shape for the antenna PSD.Result is also dependent on the background clutter level w.r.t. the target power (i.e. SNR-1).

The most relevant spectral contributions to the estimation are the ones with high derivative and low power, i.e. the spectral parts close to the nulls. In Stripmap case (aliased version of the antenna) we found that the limit for SNR→∞ is

sampCRB

NPRF2516.02

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plitu

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Point 20101218T174818. Theoric fDC=0 Hz, 5 Antenna Ap


in agreement with results of Bamler (TGRS 1991).

Results: fDC estimation /1

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-1.008x 10

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Residual Doppler centroid trial values [Hz]

Log

Like

lihoo

d of

the

spec

trum

SNR=20dB

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Residual Doppler centroid trial values [Hz]

Log

Like

lihoo

d of

the

spec

trum

SNR=30dB

Name Value UoMCarrier frequency 9,60E+09 HzWavelength 3,12E-02mAntenna length 5,6 mOff-nadir reference angle 26,55degPRF 3,06E+03 HzPulse length 40usecBandwidth 119,8242188MHzSampling frequency 146,25 MHzSWL 172,417094usecSimulated SNR 20, 30 dB

Antenna pointingRight, with yaw

steering Orbit propagator NORAD4TM

Number of simulated apertures 5Wrong fDC in spotlight focusing 10Hz

m mz

mpn ffS

Zf

);(

2

,

is solved by exhaustive search. High SNR case: covariance matrix is expected to be

very good and we use the average of the spectra to estimate it

Low SNR case: we take the ideal model, i.e. the target ideal PSD, as the measures are unreliable


Simulated targets caseML equation


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Cramer-Rao Bound resultsRepeated simulation allows to get the standard deviation of the estimate and compare it with the theoretical bound provided by the CRB.


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SNR [dB]

std

of

f dc [H

z]

CRB for stripmap case, varying the SNR

CRB for spotlight case, varying the SNR

Experimental results on simulated targets, varying the SNR


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Real targets case

CSK X-band data on ‘Milan lakes’ datasetThe 12 best targets have been chosen: high quality factor (relative mean of MSE ≈0.043) and average SNR of 20.8dB.Joint LLH is achieved by a weighted sum, the weights being the quality factor values.

p

pnpntot fqf ,,


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Estimated fdc [Hz]

Spe

ctru

m L

LH

δfdc=40.75Hz

Mission CSKS1_SLC_B_HI_03_HH_RA_SF_20080903051606

Results: fDC estimation /4Real targets caseValidation of the fDC residual correction obtained by inspection on the CSK image: peakness of point targets improved

12 targets -3dB resolution [m] PSLR [dB] ISLR [dB]

Range cut 1.640 11.93 10.72

Azimuth cut 3.658 12.71 12.02


Range cut 1.440 12.49 10.85

Azimuth cut 3.535 12.29 11.72


Range cut 1.254 12.76 10.43

Azimuth cut 3.406 12.70 11.58


Range cut 1.197 12.18 9.70

Azimuth cut 3.157 12.71 10.96


Range cut 1.176 12.55 10.23

Azimuth cut 3.147 12.74 10.52


Range cut 1.142 12.87 10.23

Azimuth cut 3.184 12.88 10.94

Without correction

Correction of δfdc= 40.75Hz-3dB res ≈ +10%PSLR ≈ +3 %

IGARSS Symposium, Vancouver, CanadaJuly 27th , 2011 18

ConclusionsRobust (LLH based) estimation of Azimuth Antenna Pattern Pointing (Doppler centroid) for stripmap SAR data presented.

Based on natural point targets, with high SNR Use of few targets, as long as they own good SNR Antenna pointing estimation achieved with an accuracy of few Hz Antenna pointing estimation better than traditional (stripmap) methods

Further works: Implement azimuth antenna pattern estimation after correction of residual fDC in a multi-

image framework Exploit the AAP estimation for a better radiometric calibration

AcknowledgementsThis work has been supported in the framework of ASI AO-1080, under the conditions as in ASI document DC-OST-2009-116.


Appendix 1: spectral shape estimation (preliminary)


Preliminary steps: Correction of δf to align targets’ spectra Time-domain fine registration and phase alignment through the stack of images. [Phase

alignment needed for fine azimuth peaks registration and possible residual focusing operator errors, or residual offset].

Target spectrum model:

)()(exp)()()( ,,,,, mpnmpnmantmpnscmpn fWfjfAfSfZ

We exploit the measures of target spectrum over the stack, to average out the effect of the noise.

Determination of the spectral shape performed by a LMS polynomial approximation of the residual in the frequency domain, once that a model for the antenna is available and has been removed.

Lacking any kind of information (at initial step of iteration), we use as model for the antenna the weighted average of all the targets in all the images:

n ppn

n pmpnpn

midq

fZq

fA,

,, )(

)()(

)(

)(,,

mid

nmpnpn

m fA

fZq

fY

Appendix 2: antenna pattern estimation (preliminary)


Estimation of Azimuth Antenna is performed frequency by frequency on

The join PDF, conditioned to the antenna as parameter

z

zH

mantP CZCZ

fAZZp1

15.0exp

))(|,...,(

The best value of the antenna model at that frequency is the one that maximizes the log-likelihood,

nzz

H

Aant

Amant CZCZAfA

antan t

logmaxargmaxarg)(ˆ 1

We approximate the log-likelihood of N output as the sum of the log-likelihoods. For a high number of targets and images we have the same level of output accuracy of the strict ML formulation.


Appendix 3: results (preliminary)


Simulated set of targetsTarget spectral shape have been achieved by distortion of ideal spectrum (value of 1 at all the f) with 3rd order polynomial with random uniformly distributed coefficients.Since target amplitudes have been normalized, we express the target spectral shape as a deviation from the nominal value of 1 (or 0dB).At the first iterate the ideal antenna model has been used.

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Target Spectral Shape

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Target spectral shape estimation error

Model Std # Targets # Images

Sensor speed Gaussian 0.005 16 64

Spectral shape Polynomial

Spectral shape coeff. Uniform 0.15

Spectral shape polyn. Order 3

Clutter value Gassian 0.08

Appendix 4: results (preliminary)


We simplified the LLH estimation by exploiting

instead of

to avoid the estimate of the covariance matrix of the observations.The problem has been divided in a subset of frequencies: for each subset we coherently summed (1) after correction for the estimated spectral shape, and looked for the most probable value of by exhaustive way.


nzz

H

Aant

Amant CZCZAfA

antant

logmaxargmaxarg)(ˆ 1

(1)

(2)

)(ˆmant fA

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Estimated Antenna Pattern. Mse w.r.t. (real) antenna power: -51.79dB

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Estimated Antenna Pattern vs. real. Detail

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ACCURATE OPTIMAL DOPPLER CENTROID ESTIMATION FOR SAR DATA

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