1: Brigham Young University, Electrical and Computer ...bjeffs/publications/Jeffs_SKA_05.pdf · Analysis of the Peeling Algorithm Brian D. Jeffs1 and Sebastiaan van der Tol2 1: Brigham

Analysis of the Peeling Algorithm

Brian D. Jeffs1 and Sebastiaan van der Tol2

1: Brigham Young University, Electrical and Computer [email protected]

2: TU Delft, Circuits and Systems, [email protected]

The Bad News:

LOFAR is uncalibratable ...

The Good News:

with conventional algorithms.

Next Generation WidefieldInstrument Calibration Challenges

• Larger apertures.• Many more array elements.• Wider range of frequencies.

• Ionospheric interaction. • Calibration may be source

direction dependent.• Calibrated UV data may not

be possible.

Each station antenna sees the entire sky.

7200 dual-pol antennas.

Multiple simultaneous beams are formed in different directions.

~6˚ beam mainlobe

© ASTRON

LOFAR is a Widefield Instrument

LOFAR Geometry

• 72 stations.

• 100 km aperture.

• Significant ionospheric variation across the array complicates calibration.

• Nonisoplanatic iono-sphere across calibration sources and stations.

• Very low frequencies: 30 - 240 MHz.

© ASTRON

The LOFAR Calibration Problem

• At low frequencies the ionosphere perturbs phase and gain.

• Calibration terms must be estimated for each bright source & station.

• Calibration for other objects is interpolated.

• Physical constraints must be applied.

Full array aperture

Station beamfield of view

Ionosphere

LOFAR stationLOFAR station

Objects in field of viewsee different ionosphericphase and gain

Calibration is Direction Dependent

• Each station sees a differentdirection-dependent blur.

• Calibration on several brightpoint sources in the field of view is required.

x

G

K

m=1 m=M

Matrix Form Data Model

V: visibility matrix, computed over a series of time-frequency intervals. Observed.

G: calibration complex gain matrix. One column per calibrator source. Unknown.

K: Fourier kernel, geometric array response. sq is source direction vector. rm is station location. Known.

B: Calibrator source intensity. Known.

D: Noise covariance. Unknown.

=

=

⋅=

=

=

+⋅⋅=

=

MQ

mqcf

qm

QMM

Q

QMM

Q

d

d

b

b

ikkk

kk

gg

gg

nnE

OO

L

MM

L

L

MM

L

oo

11

2,

,1,

,11,1

,1,

,11,1

H

H

,

}exp{,

)()(]}[][{

DB

rsK

G

DKGBKGxxV

π

The Single Snapshot Calibration Ambiguity

• For conventional arrays without direction dependent ionospheric phase perturbation calibration is possible with a single Vk,n observation.

• Not so for LOFAR, there is an essential ambiguity.

• Calibration is Impossible with a single visibility snapshot!

1

H

HH

))(()(~

matrixunitary a ,

)()(

21

21

21

21

−•− ⋅⋅=

=

+⋅⋅=

KBUBKGUG

IUU

DKGBUUBKGV

oo

oo

H

Rotation by U is invisible in V

Each U produces a different calibration

Solutions to Calibration Ambiguity

• Time-frequency diversity• Fringe rotation over time and frequency changes

visibilities while calibration gains are nearly constant.• Low order polynomial fitting over time-frequency.• Peeling.

• Single snapshot calibration• Compact core.• Deterministic Frequency dependence.• Known gain magnitudes, |G|.

The Direct Least Squares Solution

• Problems:Direct optimization is computationally intractable.Too many parameters.Requires good initialization. Where do you get it?Does not exploit known smoothness structure over k,n.Due to ambiguity, solution is not unique if θk,n has same degrees of freedom as Gk,n.

[ ]2

,,,,,,,

,1,1

)}{()}{(minargˆ

,,

,1,1

∑∑ ⋅⋅−=Θ

=Θ

k n

Hnknknknknk

TTNK

T

NK

KGBKGV ooL

L

θθ

θθ

θθ

The Peeling Approach

• Current proposed LOFAR calibration method.• Replace joint estimation of G for Q sources with a series

of single source calibration problems.• Exploit relative fringe rotation rates among calibrator

sources.• Assume calibration gains are constant over a t-f cell. • Computationally efficient.• References:

1. J.E. Noordam, “LOFAR calibration challenges,” Proceedings of the SPIE, vol. 5489, Oct. 2004.2. J.E. Noordam, “Peeling the Visibility Onion, the optimum way of self-calibration,” ASTRON tech. report MEM-078 June 2003.

Basic Peeling Algorithm Steps

• Over a time-frequency cell of nearly constant gains, rotate all visibilities to phase center the brightest remaining source.

}{diag}{diag~,,,,,, nkqnknkqnk kVkV =

“Image plane” equivalent



• Average centered visibilities to suppress non-centered sources.

∑∑=k n

nkq KN ,~1ˆ VV





• Solve as a conventional single source calibration.

Hqqq b ggVg

g−= ˆminˆ






• Subtract the calibrated source from visibilities.

}diag{ˆˆ}diag{ ,,,,,, nkqHqqnkqqnknk b kggkVV −=






• Subtract the calibrated source from visibilities.

• Repeat for next brightest source.


2-D Polynomial Model over time-frequency for Ionospheric Variation

• Variations in G are smooth over time and frequency

• Estimating the smaller parameter set, p, improves performance.

• Over a large window p does not depend on k,n.

{ })(exp

)(}{

112

02012

201000

112

02012

201000,

nknnkk

nknnkknk

tfttffi

tfttff

ΦΦΦΦΦΦ

ΓΓΓΓΓΓ

+++++

+++++=

⋅oθG

[ ]TT11

T00

T11

T00 }diag{,}vec{,,}vec{,}vec{,,}vec{ Dp ΦΦΓΓ LL=

Challenges

• Phase centered averaging does not completely remove non-centered sources.

• “Contamination” from non-centered sources biases estimate of .

• Solution: Multiple passes of peeling.

• Computational burden eliminates some candidate approaches.

• Many local minima in polynomial optimization makes a good initial solution critical.

qg

Why Study the Cramer-Rao Bound?

• Array calibration is a statistical parameter estimation problem.

• The CRB reveals the theoretical limit on estimation error variance.

• Absolute frame of reference: no algorithm can beat the CRB.

• Answers the questions:

• Is the existing algorithm adequate?

• Is there hope for finding a better solution, “How close are we?”

• Permits trading off performance and computational burden.

• Can be computed even if no algorithm exists yet.

• The BIG question: Can LOFAR be reliably calibrated?

CRB Definition

• Notationx: a vector of random samples with joint probability

density p(x |θ ).any unbiased estimator for θ. covariance matrix for .

M: Fisher information matrix.

• The Cramer-Rao theorem:

• Error variance is lower bounded by !

:θ:θC

1

T

21

ˆ)|(ln

−

−

∂∂∂

−=≥θθ

θxMC pEθ

}diag{θC

θ

A Simple Example

• Estimate a constant in additive white Gaussian noise:

+w[n] x[n]

θ

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

8

9

10

n

x[n]

θ

.)ˆvar(2

Nσθ ≥CRB result:

T]]1[,],0[[ −= Nxx Lx

This is the well known variance for the sample mean!

A Second Simple Example

• Line fitting in additive white Gaussian noise:

][][ 21 nwnnx ++= θθ

+w[n] x[n]

θ1+θ2n

0 10 20 30 40 50 60 70 80 90 1003

4

5

6

7

8

9

10

11

n

x[n]

θ1

θ2 We now have no intuition on estimation error for θ1 and θ2!

• Variance on constant term θ1 is now higher. → estimating more parameters increases error.

• Variance of slope term, θ2, drops more rapidly with N.→ θ2 is easier to estimate. → x[n] is more sensitive to θ2 due to multiplication by n.

A Second Simple Example: Insights

[ ] [ ])1(

12)var(,)1(

)12(2)var(

,

6)12)(1(

2)1(

2)1(

1

2

2

2,21

2

2

1,11

1

2

−==

+−

==

−−−

−

=

−−

NNNNN

NNNNN

NNN

σθσθ

σ

MM

M

3

2

2

2

112)var(lim4)var(lim

NN NN

σθσθ ≥≥∞→∞→

Now it Gets a Little Messy

O

Constraint Jacobian for packed central core

Block Fisher information

Blockclosedforms

Now it Gets a Little Messy

The important points:• Closed form CRB expressions have been derived for

most important LOFAR calibration models.• Though expressions are complex, computer codes

have been developed to evaluate them.• These solutions exist now and could be made available

for astronomers to predict calibration performance for a given observations.

Peeling Simulation Results

• Polynomial fit over a 10 s by 100 kHz “snippet”window.

• Two sources.• Plot is for zero order

coefficient, 16th station, 1st

source.• 30 station array with 100

km aperture.• Performance is typical of

all other parameters.• Three peeling passes. -40 -35 -30 -25 -20 -15 -10 -5 0

-60

-50

-40

-30

-20

-10

0

10

20

SNR in dB for brightest calibrator

Par

amet

er e

rror v

aria

nce

in d

B

Sample error and CRB for parameter no. 76

sample err varcheat err varCRB

CRB Analysis for a realistic Scenario

• Point LOFAR beam in arbitrary direction.• Model station beam pattern and noise level accurately.• Calibrate on 5 brightest sources in beam mainlobe and

5 brightest in sidelobes.• Use 2-D 1st order polynomial fitting over a “snippet” of

10 seconds and 500 kHz.• Calculate CRB for polynomial coefficients.• Project coefficient CRB to corresponding complex gain

error variance.

EWI Circuits and Systems

Full Sky Map

3C461

3C144

3C405

3C147

3C274

Full Sky Map


Field of View

6667686970717254

54.5

55

55.5

56

56.5

57

4C+55.08

4C+54.06

4C+56.09

4C+56.10

4C+55.09

Field of View

Right Ascension (deg)

Dec

linat

ion

(deg

)


Station Beam Pattern

Station Beam Pattern

−20

−18

−16

−14

−12

−10

−8

−6

−4

−2

0

Multiple Source CRB for 2-D Polynomial Coefficients

Time (s)

Fre

quen

cy (

kHz)

Relative gain error − Antenna m=2 / Source q=3 (4C+55.08)

1 2 3 4 5 6 7 8 9 10 11

50

100

150

200

250

300

350

400

450

500

10

20

30

40

50

60

70

80

90

100

• Coefficient error variance relative to conventional single source calibration.

• Note that calibra-tion fails without at least two seconds and 150 kHz of diversity.

Multiple Source Complex Gain Error CRB

Relative gain error − Antenna m=2 / Source q=3 (4C+55.08)

1

2

3

4

5

6

7

8

9

10

• Use estimated polynomial coefficients to calculate complex gains.

• Calculate error variance CRB over full time frequency range.

• Largest error is near domain edges.

Conclusions

• The BIG answer: Yes, LOFAR can be calibrated.

• Given a range of time-frequency observations and compact core geometry: there are no theoretical roadblocks to achieving useful calibration estimates.

• Does peeling work for direction dependent calibration?

• Yes, so far so good.

• Ongoing progress in reducing cross-source contamination: multiple pass peeling & demixing.

• Low SNR performance improvements being studied.

Optional Slides

• Use these only if (by some miracle) there is extra time.

LOFAR Calibration with Compact Central Core

• The full array can be calibrated given a compact central core.• The wisdom of the LOFAR design is confirmed by CRB analysis

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x 105

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 105

-3000 -2000 -1000 0 1000 2000 3000-3000

-2000

-1000

0

1000

2000

3000

Full array Compact core

LOFAR Calibration with Compact Central Core

• Calibration succeeds for Q ≤ Mc+1. Q calibrator sources and an Mc element core.

0 10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

12 CRB for 10 element array,4 element subarray and 6 sources

Mea

n S

quar

ed E

rror

Parameter Index

CRBstart of phasesstart of noise powers

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5x 10

−3 MSE for 10 element array,4 element subarray and 3 sources

Mea

n S

quar

ed E

rror

Parameter Index

ML Monte CarloCRBstart of phasesstart of noise powers

×1012×10-3

Documents

1: Brigham Young University, Electrical and Computer ...bjeffs/publications/Jeffs_SKA_05.pdf · Analysis of the Peeling Algorithm Brian D. Jeffs1 and Sebastiaan van der Tol2 1: Brigham