phase requirements, design and validation of phase

CEOS SAR 2010 – Zurich 25-27 August

Phase Requirements, design and validation of phase preserving processor for a spaceborne SAR

Michele Belotti, Davide D'Aria, Nuno Miranda,

Andrea Monti Guarnieri, Betlem Rosich, Silvia Scirpoli

OUTLINE

Defining a suitable focusing phase accuracy requirement for interferometric applications

Definition of the acquisition modes

Test definition

Test pertinence

2


Test pertinence

ω-k CZT algorithm

Phase preserving test applied at CZT algorithm

Conclusion

3Phase requirements for interferometric application

In interferometric application the important parameter to be optimized is the phase

standard deviation. The Cramér Rao Bound for N images, considering averaging on L

pixels is [1]:

Where

So it is equivalent to maximize the CNR.

We can identify 2 main contribution to CNR:

BAQ noise and atmospheric delay residual

error.


Using a Permanent scatter analysis we can

evaluate the phase residual error when the

atmospheric contribution is removed.

Simulating a set of PS scenario with an

estimated correlation and a density of 250

PS/Km2 we found a residual phase noise

dispersion of about 7°.

3 days ERS-1 correlation

error.

[1] Monti-Guarnieri, A.; Tebaldini, S Signal Processing Letters, IEEE , vol.14, no.12, pp.1012-1015, Dec. 2007.

4

Definition of the focusing process phase accuracy requirement

The interferometric phase error introduced by the

focuser must be at least smaller than the expected

error introduced by residual APS


error introduced by residual APS

5

To evaluate the performance of different focusing algorithm and to validate their efficacy and precision the definition of a series of phase

preserving test is necessary.

It’s also important to take into account the different nature of raw data to address suitable tests.

The future SAR mission introduce innovative acquisition geometries to

maximize target bandwidth and ground coverage.


In this work we take into account four different class of acquisition:

STRIPMAP

SCANSAR

TOPSAR

SPOT

6STRIPMAP ACQUISITION GEOMETRY

Stripmap is the conventional full

resolution Synthetic Aperture Radar

mode. The antenna footprint moves at

the satellite speed along its orbit.


•Continuous acquisition

•Single swath

7SCANSAR ACQUISITION GEOMETRY

Multi BURST Acquisition by switching the antenna

look angle.

Antenna is fixed in azimuth direction.


This acquisition scheme allows for the coverage of a

ground scene that is N times wider than that of

conventional SAR modes with a proportional loss in

the azimuth resolution and in radiometric precision.

8TOPSAR ACQUISITION GEOMETRY

Multi Burst acquisition by changing

the antenna look angle.

Antenna is steered in azimuth

direction during the acquisition of

each burst from backward to forward

achieving a larger bandwidth for all

the ground points.


9SPOT ACQUISITION GEOMETRY

Antenna is steered from forward to

backward during the acquisition, in

opposite direction to respect TOPSAR

acquisition.


Continuous acquisition with selected area

with increased resolution

TEST DEFINITION

The new acquisition geometries cannot always be validate with the existing tests so it is necessary to define some extension and new test to perform phase quality check of the incoming data.

10


We introduce two test classes:

Single image/ single burst test

Multi swath/ multi burst point target test

CEOS OFFSET

MODIFIED CEOS OFFSET

SIZE BLOCK

INTER BURST TEST

INTER SWATH TEST

11STANDARD CEOS OFFSET TEST

Process two SLCs from the same raw data set and with the same orbit, but offset by

l00 lines in azimuth and l00 sample in range. The interferogram formed from these

two properly coregistered SLCs should ideally have a constant phase of zero and

thus reveals processor induced artifacts.

Bamler, R. and Hartl, P., Inverse Problems, vol.14 1998 R1-R54.


Pass/Fail criteria:

•Mean of interferogram phase ≤ 0.1°

•Standard deviation ≤ 5.5°

•No discontinuity at block boundaries.

Alessandro Ferretti, Andrea Monti-Guarnieri, Claudio Prati, Fabio Rocca, Didier Massonnet InSAR Principles: Guidelines for SAR Interferometry Processing and Interpretation 2007 European Space Agency

12Modified CEOS

The original CEOS as it is defined, is

applicable only to STRIPMAP data

In order to define an extended CEOS test suitable also for SCAN/TOPS/SPOT we have to introduce some changes to the original CEOS procedure:

1) Zeroing the undesired raw data echoes in the shifted data in order


1) Zeroing the undesired raw data echoes in the shifted data in order to avoid spurious noise contribution in the final SLC data

2) Update the focuser Doppler centroid input in order to ‘simulate’ the different antenna steering induced by the cutting of the raw data

13UPDATE THE DC

FOR EXAMPLE FOR TOPSAR ACQUISITION IS NECESSARY TO ADJUST THE

STARTING DC DUE TO THE CHANGE IN THE STEERING LAW


14MODIFIED CEOS OFFSET TEST

1. Zeroing M (100) lines in range and N

(100) in azimuth

2. Cut and zero padding the second

dataset

3. Process the first with original DC

4. Process the second dataset with

updated DC according to steering law

change

First dataset Second dataset

∅

∅

∅

∅


change

5. Evaluate the mean and std for the

proper coregister interferogram

between the two SLC

AREA TO BE TESTEDPass/Fail criteria:




15SIZE BLOCK TEST

Process two SLCs from the same raw data set and with the same orbit. One of SLCs

has scaled dimensions with respect to the other one: 30% of range samples, 30% of

azimuth lines. The interferogram formed from these two properly co-registered SLCs

should ideally have a constant phase of zero and thus reveals processor induced

artifacts.


Pass/Fail criteria:




16INTER-BURST POINT TARGET TEST

Process two SLCs from the same simulated raw data set, with the same orbit and

with a single point target. The SLCs will contain different parts of the point scatterer

Doppler history. The phase of the point target should ideally be the same in the two

images. Phase differences thus reveal processor induced artifacts.


Pass/Fail criteria:

• Phase difference ≤ 5.5°.

17INTER-SWATH POINT TARGET TEST

Process two SLCs from the same simulated raw data set, with the same orbit and with

a single point target. The SLCs will contain different parts of the point scatterer

Doppler history. The phase of the point target should ideally be the same in the two

images. Phase differences thus reveal processor induced artifacts.


Pass/Fail criteria:

• Phase difference ≤ 5.5°.

18TEST PERTINENCE FOR DIFFERENT ACQUISITION MODES

Stripmap ScanSAR TopSAR Spot

Intra burst/swath

Inter burst/swath

Intra burst/swath

Interburst/swath

CEOS


Extended CEOS

Scaled Block

Inter burst PT test

Inter swath PT test

19TEST APPLICATION

We want now to apply this test to verify the phase preservation of a new implementation of omega-k algorithm.

This new implementation is based on a chirp Zeta transform to avoid the


This new implementation is based on a chirp Zeta transform to avoid the 2D domain Stolt interpolation

20IRF OF THE SAR ACQUISITION

The impulse response function of the SAR acquisition, for a point-like target, is

expressed as follows:

( )

−

−=

c

rRj

c

rRtprarths

);(2exp

);(2;);,( 0

τω

τττ

p(t) is the transmitted pulse

after range compression


( ) ),(),;( 0000 ttR ττττ PS −=

after range compression

R(ττττ;r) is the target-to-satellite

distance (hodograph)

21OMEGA-K FOCUSING (I)

The ω−k approach provides an efficient implementation of the range-varying

convolution:

In the 2D frequency domain the convolution becomes:

);,();,(),( 00 rthrttdta ss −−∗∗−=∗

∧

τττ

);,()exp();,();,( ,rkHrjrkDrkA x

tx

ssxx ωωω Ω−=∧


));,(exp();,();,();,( rkjrkHrkDrkA xrrefxrefxx ωψωωω =∧

Using MSP (Method of Stationary Phase) we can write the kernel as

));,(exp();,();,( rkjrkDrkA xxx ωψωω =∧

Phase ψ is split into a range variant and range invariant term:

Range independent Range dependent

22SPECTRAL SUPPORT OF DATA

The system filters the angular pulsation frequency in the interval:ω=ω0±2*π*B/2

Illumination pattern bounds antenna angular interval at:

ϕ=ϕ0±∆ϕ/2


23FIRST ORDER STOLT APPROXIMATION

We use a first order approximation of frequency interpolation:

( ) vvrv kc

kkgc

uk 10

1

0

1

010

2)(

2)( ββ

α

α

α

αωωααω +=+−≈=+≈=

We can obtain kt stretching the ω axis modulated of –β0


To implement first order frequency interpolation we use Chirp Z Transform in

the time-Doppler domain (t,kx)

We can obtain kt stretching the ω axis modulated of –β0

tk10 ββω =−

24ALGORITHM FLOWCHART (Azimuth compression)

Chirp Z Transform

Apply range invariant kernel


Apply range invariant kernel

Residual phase correction

25TOPSAR FOCUSER ALGORITHM

Frequency domain unfolding & resampling

Azimuth compression block

RGCdata

(1 burst)

To process TOPSAR and SPOT data further step are needed

1) A preprocessor to unfold the frequency domain


Time domain unfolding & resampling

SLC data

(1 burst)

2) Fine azimuth compression block

3) An azimuth 1D ‘unfolding’ processing block

26EXTENDED CEOS OFFSET TEST

Simulated uniform distributed scenario SENTINEL IW TOPSAR mode.

Symbol Meaning Value

λ Wavelength 0.055 m

f0 Carrier frequency 5.4 GHz

fs Range sampling frequency 64.35 MHz

W System bandwidth 54 MHz

La Antenna length 12 m

vs Platform mean velocity 7599 m/s

ϕ Max Squint Angle ±1.29°

RAW Data simulated with High DC 6 KHz

SLC Proper corregistered.


ϕ Max Squint Angle ±1.29°

Rc Orbit local radius of curvature 7068 km

Focused DATA Raw data Spectrogram

Azimuth

Range

200 400 600 800 1000 1200 1400

200

400

600

800

1000

1200

2000

40001

2

3

27EXTENDED CEOS OFFSET TEST (II)

Mean = -0.0345[deg]

Std = 2.4681[deg]

3.5

4x 10

6 Angle distribution

INTERFEROGRAM ON COMMON REGION


200 400 600 800 1000 1200

6000

8000

10000

12000

-3

-2

-1

0

1

-1.5 -1 -0.5 0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

Angle [rad]

28BLOCK SIZE TEST

Azimuth

Range

∠ ( Dato1 * conj( Dato2 ) )

1000 2000 3000 4000 5000 6000 7000 8000 9000

100

200

300

400

500

600

700

800

900

1000

1000 2000 3000 4000 5000 6000 7000 8000 9000-0.05

0

0.05

Azimuth

Mean ∠

100 200 300 400 500 600 700 800 900 1000-0.05

0

0.05

Range

Mean ∠

Mean ∠

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05


Azimuth

Range

∠ ( F(Dato1) * conj( F(Dato2) ) )

1000 2000 3000 4000 5000 6000 7000 8000 9000

100

200

300

400

500

600

700

800

900

1000-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

1000 2000 3000 4000 5000 6000 7000 8000 9000-0.05

0

0.05

Azimuth

Mean ∠

100 200 300 400 500 600 700 800 900 1000-0.05

0

0.05

Range

Mean ∠

Mean = -0.00032177[deg]

Std = 1.7804[deg]

29INTER BURST TEST

TOPSARBurst 1

TOPSARBurst 2

PT

40

45

50

55

60

65

70

75

80

85

90

Focused target

Worst case at the burst edge

phase difference: < 4 [deg]


45 50 55 60 65 70 75 80 85

20 40 60 80 100 120

20

40

60

80

100

120

20 40 60 80 100 120

20

40

60

80

100

120

Point target 1° burst Point target 2° burst

phase difference: < 4 [deg]

30CONCLUSION

We discuss the phase requirements accuracy for a focuser processor

We have presented new test and extension of existing ones to

evaluate phase performance of focusing algorithm and validate

efficiency and precision.

The applicability of this new tests to the existing acquisition

geometries is discussed.


We propose a new implementation of the ωωωω-k focusing algorithm

The phase preservation has been verified for this algorithm.

31OMEGA-K FOCUSING (I)

The ω−k approach provides an efficient implementation of the range-varying

convolution:

In the 2D frequency domain the convolution becomes:

);,();,(),( 00 rthrttdta ss −−∗∗−=∗

∧

τττ

);,()exp();,();,( ,rkHrjrkDrkA x

tx

ssxx ωωω Ω−=∧


– with

);,()exp();,();,( rkHrjrkDrkA xssxx ωωω Ω−=

( )c

02 ωω +=Ω

We need to estimate the 2D IRF spectrum Hsx,t (kx,ω;r):

( ) ( )( )rxRjrxHt

s ;exp;, Ω=ω

( ) ( )( ) ( )∫ −Ω= dxxjkrxRjrkH xx

tx

s exp;exp;,, ω

32OMEGA-K FOCUSING (2)

Using MSP (Method of Stationary Phase):

where

( ) ( )( )( )fxfx

tx

s xkrxRjrkH −Ω≈ ;exp;,, ω

Ω=⇒=−

∂

∂Ω

=

xx

xx

kk

x

rxR

f

ξ0);(

We obtain:


( )( )( )));,(exp();,(

;exp)exp();,();,(

rkjrkD

xkrxRjrjrkDrkA

xx

fxfsxx

ωψω

ωω

=

−ΩΩ−=∧

Phase ψ is split into a range variant and range invariant term:

));,(exp());,(exp());,(exp( refxrefxrx rkjrkjrkj ωψωψωψ ⋅=

Range dependent Range independent

33OMEGA-K FOCUSING (3)

Focusing becomes

));,(exp();,();,();,( rkjrkHrkDrkA xrrefxrefxx ωψωωω =∧

For each wave number kx it is assumed that there is separable decomposition:

);();();,( xxxr krvkurk ωωψ =


));();(exp();,();,();,( xxrefxrefxx krvkjurkHrkDrkA ωωωω =∧

We make a change of variable r - v and transform along stretched range

with

( )( )vxxrefxvx kkgkHkDkkA ,),(),(),( −∗=∧

ωδωω

)(),( vv kguk == ωω

34CHIRP Z TRANSFORM

k

k AWz−=

22and

πφπθ jjeWWeAA ==

With CZT we can compute the Z

transform on a more general contour of the form:

with:


By A we can control frequency shift (-β0)

By W we can perform frequency stretch (β1) selecting an appropriate frequency

interval: (fs * β1)

00 2

0

2

0 andπφπθ jj

eWWeAA ==

35Atmospheric correlation model


Assumption phase stable across time and using this stability we estimate the additional phase shift sue to atmosphere.

PS density of 250 /km2 affected by noise with standard deviation of 0.75 rad.

36ωωωω-k FOCUSING ALGORITHM

The impulse response function of the SAR acquisition, for a point-like target, is expressed as follows:

In ωωωω-k approach the range variant convolution operator

p(t) is the transmitted pulse after range compression

R(ττττ;r) is the target-to-satellite distance (hodograph) ( ) ),(),;( 0000 ttR ττττ PS −=


Can be expressed in frequency domain as

And the operator H can be written as:

Using the stationary phase method MSP we can express the integral as :

Satisfying the condition:

37Algorithm implementation (I)

The hodograph can be fitted with a 4 grade polynomial curve

And also its slope can be expressed as a polynomial form

HODOGRAPH DETERMINATION


This series can be inverted to determine the stationary point

And the focusing operator phase can be so expressed as

FOCUSING OPERATOR PHASE

38Algorithm implementation (II)

The focus kernel can be expressed as:

Range independent

Range dependent


Range dependent

The range dependant term can be treated with a Chirp Z transform, if we apply the following approximation :

The transform coefficient can be directly obtained:

39MODIFIED CEOS OFFSET TEST

Consider two SLCs from the same raw data set and with the same orbit, but offset by

100 lines in azimuth and 100 samples in range.

Process the first SLCs after resetting to zero the first 100 lines in azimuth;

process the second image after resetting the last 100 lines and updating the processor

initial Doppler centroid according to the cut steering law.

The interferogram formed from these two properly coregistered SLCs should ideally

have a constant phase of zero and thus reveals processor induced artifacts.


Pass/Fail criteria:




phase requirements, design and validation of phase

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