array design david woody owens valley radio observatory presented at nrao summer school 2002

Array Design

David Woody

Owens Valley Radio Observatory

presented at

NRAO summer school 2002

Outline

• Statement of the problem• UV metrics• Imaging metrics• Beam metrics

– PSF and synthesized beam

– Sidelobe statistics

– Optimization

• ALMA configurations

Problem

• Given a collection of antennas, what is the the best array configuration?

• Considerations– Best science output

– Geography

– Flexibility

– Costs

• Need methods for evaluating configurations

UV metrics• Brightness = FT of complex visibility

• Resolution >/max_baseline

• Nyquist theorem– UV sample spacing < 1/FOV

• Simple DFT would seem to require complete sampling of the UV-plane

less than Nyquist sampling

0 200 400 600 800 1000 1200 1400 1600 1800 20001

0.5

0

0.5

1

Filled Disk

First Arrays

• Large telescopes on tracks– => linear, “Tee”, and star arrays

– Often with regular spacing

• Science was mostly small fields• Ryle strived for complete sampling

Westerbork

Minimum Redundancy

• Possible in 1-D– Earth rotation gives good 2-D coverage for polar

sources

• Not solved for 2-D snapshots

Early Configuration Design

• Layout antennas with some idea in mind• Look at UV coverage• Iterate• Seemed OK for small numbers of antennas

VLA

VLA Zenith Snapshot

VLA Tracks

VLBA

VLBA UV Coverage

More Recent UV Designs

• Optimize UV coverage• Good snapshot coverage

• Uniform UV coverage– Reuleaux triangles (SMA)

• Complete coverage– MMA circular configuration

• Match image visibility distribution– Gaussian UV coverage

Reuleaux Triangle

Reuleaux Triangle Tracks

Imaging Metrics

• Use the imaging tools we have to evaluate configurations

Image Evaluation

• Set of test pictures• Generate model visibilities• Process images• Compare images to pictures

– Dynamic range

– Fidelity

• Repeat for different configurations• Modify configuration?

• Iterate

General Results

• More UV coverage gives better images• Guassian UV coverage gives better images,

especially for wide fields

Why are the images so good?

• Small fields greatly relaxes the Nyquist sampling• Many algorithms are essentially model fitting• Complete Nyquist sampling is not necessary

Generalized Nyquist Theorem

• Uniform complete sampling not requiredaverage sampling exceeds Nyquist

0 200 400 600 800 1000 1200 1400 1600 1800 20001

0.5

0

0.5

1

• Just need the average number of samples to exceed Nyquist

Imaging Approach to Array Design

• Selection of test pictures prejudices the design• Multiple definitions of dynamic range and fidelity• Time consuming• Difficult to evolve to better configuration

• But imaging is the bottom line for what we expect to get out of an array

• Part of the ALMA configuration design process

• Can we quantify the effect of the missing UV data?

• Optical– Optical transfer function

• Autocorrelation of field

• Radio– Spectral sensitivity function

• Transfer function or UV sampling

20 15 10 5 0 5 10 15 200

0.5

1

Aperture Field

distanc e

espo

nse

20 15 10 5 0 5 10 15 200

1

2

3Spectral Sensitivity Function

U

resp

onse

Beam Metrics

• Optical– Optical transfer function

• Autocorrelation of field

– Point Spread Function• FT of optical transfer function

• Radio– Spectral response function

• Transfer function or UV samples

– Synthesized beam x Primary Beam• FT of spectral response function

• The PSF is the optical image generated by a point source. It is a measure of the instrumental or array artifacts that the imaging algorithms must deal with.

• A good PSF => good images

PSF and Radio Astronomy• Voltage pattern of array beam

nN

nnPSF rpkipU

NpU

exp)(1

)(1

.

.

• Point Spread Function = synthesized beam plus single antenna response

• Same as the power from the array used as a transmitter

N

mnm

N

nmn

PSF

rrpkipBN

pUpPSF

1 12

2

)(exp)(1

)()(

Synthesized Beam and PSF

1 0.5 0 0.5 10

0.5

1

Synthesized & Primary Beam

angle

1 0.5 0 0.5 10

0.5

1

Point Spread Function

angle

Sidelobes

• Near sidelobes from large scale distribution• Far sidelobes from small scale distribution and

gaps

• Visibility data can be weighted to improve the sidelobes at the expense of S/N

• At mm and sub-mm, sensitivity is a dominant design criteria and data weighting should be avoided

Near Sidelobes for Cylindrical Antenna Distributions

r 0 .01 1

0 0.2 0.4 0.6 0.8 10

0.5

1

radius

Antenna distribution

b 0 .01 1.5

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

baseline

0 .01 10

0.1 1 101 10

3

0.01

0.1

1

angle

Point Spread Function

UV distribution

Far Sidelobes

• Caused by gaps in UV coverage• Average of PSF

– Including single dish measurements • PSF > 0• Average = 1/(N-1)

– Interferometric data only• PSF > -1/(N-1)• Average = 0

• Standard deviation– > 1/N for Magnification >> N– N = number of antennas– Magnification = (primary beam width)/(synthesized beam width)

Standard Deviation vs. Magnificationx 1 1.02 3

10 100 1 1031 10

3

0.01

0.1Standard Deviation of PSF Sidelobes

Array Magnification

Sta

ndar

d D

evia

tion

minimum for bell shaped and uniform UV distributions for N =64

1.6%

Sidelobe Calculation

• Sidelobes are sum of N unit vectors of different orientations

2

12

exp)( 1

)(

N

nnrpkipB

NpPSF

Vector rotation = rp

2

Vector Random Walkfr 0

1 0.5 0 0.5 11

0.5

0

0.5

1field vector sum

Vector Random Walkfr 1

1 0.5 0 0.5 11

0.5

0

0.5

1field vector sum

Vector Random Walkfr 2

1 0.5 0 0.5 11

0.5

0

0.5

1field vector sum

Vector Random Walkfr 100

1 0.5 0 0.5 11

0.5

0

0.5

1field vector sum

Vector Random Walkfr 101

1 0.5 0 0.5 11

0.5

0

0.5

1field vector sum

Statistical Distribution of Sidelobes

• Should have a Rayleigh distribution

NsNsg exp)(

)ln(2

max magN

s

• Expected maximum sidelobe

• Test against three configurations– UV coverage optimized circular array

– Random Gaussian array

– Systematic Gaussian array

Circularuvdata UV config( ) U Re uvdata( ) V Im uvdata( )

1 0.5 0 0.5 1

1

0.5

0

0.5

1

UV coverage

V

U psfplot PSFlog psf .05 .15( ) PSF plot

psfplot

LPlot Rbeam psf( ) i 0 1000

1 10 100 1 1030

0.05

0.1

0.15

Angle/PSF_HW HM

PSF

pea

k, a

vera

ge a

nd r

ms

X Re config( ) Y Im config( ) Antenna positions

0.5 0 0.5

0.5

0

0.5

Y

X

Peak sidelobe vs. radius

Antenna distribution UV distribution

PSF

Pseudo RandomX Re config( ) Y Im config( ) Antenna positions

0.5 0 0.5

0.5

0

0.5

Y

X

uvdata UV config( ) U Re uvdata( ) V Im uvdata( )

1 0.5 0 0.5 1

1

0.5

0

0.5

1

UV coverage

V

U psfplot PSFlog psf .05 .15( ) PSF plot

psfplot

LPlot Rbeam psf( ) i 0 1000

1 10 100 1 1030

0.05

0.1

0.15

Angle/PSF_HWHM

PSF

pea

k, a

ver

age

and

rms

Peak sidelobe vs. radius

Antenna distribution UV distribution

PSF

SystematicX Re config( ) Y Im config( ) Antenna positions

0.5 0 0.5

0.5

0

0.5

Y

X

uvdata UV config( ) U Re uvdata( ) V Im uvdata( )

1 0.5 0 0.5 1

1

0.5

0

0.5

1

UV coverage

V

U psfplot PSFlog psf .05 .15( ) PSF plot

psfplot

LPlot Rbeam psf( ) i 0 1000

1 10 100 1 1030

0.05

0.1

0.15

Angle/PSF_HWHM

PSF

pea

k, a

vera

ge a

nd r

ms

Peak sidelobe vs. radius

Antenna distribution UV distribution

PSF

Distribution of Sidelobesj 0 200

0 0.05 0.1 0.151 10

3

0.01

0.1

1

10

100

sidelobe magnit ude

num

ber

of

sam

ple

s, n

orm

aliz

ed

Histograms of the PSF sidelobe amplitudes for the initial circular array (dotted), pseudo-random (dashed) and systematic (dot-dash). The solid line is the theoretical distribution for pseudo-random configurations.

Number of sidelobes vs. magnitude

Optimization• Minimize peak sidelobe

– Find peak sidelobe

– Adjust a single antenna to reduce this peak

– Check that no other peak now exceeds new peak

– Repeat

fr 100

1 0.5 0 0.5 11

0.5

0

0.5

1field vector sum

Sequential Optimization• Rotation of the antenna unit vectors is• • Near sidelobes require large antenna shifts • Farther sidelobes require small shifts that don’t effect the near sidelobes• You can start by minimizing the near sidelobes and then progress outward without degrading nearer

sidelobes• Expected maximum sidelobe after optimization is

)ln()ln(211

max, NmagN

s opt

rp

2

Optimized Arrays

X Re config( ) Y Im config( ) Antenna positions

0.5 0 0.5

0.5

0

0.5

Y

X

X Re config( ) Y Im config( ) Antenna positions

0.5 0 0.5

0.5

0

0.5

Y

X

X Re config( ) Y Im config( ) Antenna positions

0.5 0 0.5

0.5

0

0.5

Y

X

X Re config( ) Y Im config( ) Antenna positions

0.5 0 0.5

0.5

0

0.5

Y

X

X Re config( ) Y Im config( ) Antenna positions

0.5 0 0.5

0.5

0

0.5

Y

X

X Re config( ) Y Im config( ) Antenna positions

0.5 0 0.5

0.5

0

0.5

Y

X

Antenna distribution

UV distribution UV distributionUV distribution

Antenna distributionAntenna distribution

Optimized Sidelobe DistributionLPlot Rbeam psf( ) i 0 1000

1 10 100 1 1030

0.05

0.1

0.15

Angle/PSF_HWHM

PSF

pea

k, a

ver

age

and

rms

LPlot Rbeam psf( ) i 0 1000

1 10 100 1 1030

0.05

0.1

0.15

Angle/PSF_HWHM

PSF

pea

k,

aver

age

and

rm

s

LPlot Rbeam psf( ) i 0 1000

1 10 100 1 1030

0.05

0.1

0.15

Angle/PSF_HWHM

PSF

pea

k,

aver

age

and

rm

s

j 0 200

0 0.05 0.1 0.151 10

3

0.01

0.1

1

10

100

sidelobe magnit ude

num

ber

of

sam

ple

s, n

orm

aliz

ed

Peak sidelobe vs. radius Peak sidelobe vs. radius

Peak sidelobe vs. radius Number of sidelobes vs. magnitude

Add more short spacingsX Re config( ) Y Im config( ) Antenna pos itions

0.5 0 0.5

0.5

0

0.5

Y

X

uvrdist Nrdist uvdata nbins 1.5( ) j 0 nbins

0 0.2 0.4 0.6 0.8 10

1

2

UV radius

UV

den

sity

LPlot Rbeam psf( ) i 0 1000

1 10 100 1 1030

0.05

0.1

0.15

Angle/PSF_HWHM

PSF

pea

k, a

vera

ge a

nd

rms

psfp lot PSFlog psf .001 .05( ) PSF plot

psfp lot

Peak sidelobe vs. radius

Antenna distributionRadial UV distribution

PSF

Results for several design studies

0

0.05

0.1

0.15

0.2

0.25

10 100 1000

Magnification

Sid

elob

e P

eak

Circle

random #1

Random #2

Systematic

ln(mag)*2/N

Circle, opt

Random #1, opt

Random #2, opt

Systematic, opt

Opt for 2.5%

[2*ln(mag)-ln(N)]/N

[1+2*ln(mag)-ln(N)]/N

Earth Rotation

plot st

10 100 1 1031 10

3

0.01

0.1Standard Deviation of PSF Sidelobes

Array Magnification

Sta

ndar

d D

evia

tion

Lower limit to the standard deviation for bell shaped (red curves) and uniform UV coverage (blue curves) for 64 antenna arrays for snapshot (solid), 6 min (dashed) and 1 hr tracks (dotted).

PSF Metric

• Simple to calculate• Gives quantifiable results• Can be compared to idealized expectations• Directly related to imaging artifacts• Easy to evolve into better configurations

– Can easily incorporate physical constraints

• Major part of ALMA configuration design

ALMA configurations

• Geography and boundaries are significant constraints

• Design approach– Large scale Gaussian UV distribution – Logarithmic spiral for zooming– Finally sidelobe optimization

• Multiple configurations– Start in compact, close packed– Transition to self similar logarithmic spiral– Largest configuration circle or star

• Final design in progress

Conway: Mag=14, XYscale=168m

psfplot

PSF plotpsfplot PSFlog psf .02 .06( )

psf PSFin aconfig samp grdext mag( )

samp 2 maggrdext 2 N3dB *N3dB 1mag 14

1 0 1

1

0

1

UV coverage

normalized U [m/Dant*mag]

norm

aliz

ed V

[m

/Dan

t*m

ag]

0.5 0 0.5

0.5

0

0.5

Antenna positions

normalized X [E/Dant*mag]

norm

aliz

ed Y

[N

/Dan

t*m

ag]

V Im uvdata( )U Re uvdata( )uvdata UV config( )Y Im config( )X Re config( )

mag 14ifig 20file "Conways configs ascii.txt"

Conway: Mag=14, peak sidelobe=5.8%

1 103

0.01 0.1 1 100

0.05

0.1

0.15

Array Beam maximum vs. angle

Angle/Primary Beam_HWHM

PS

F p

eak

an

d a

verag

e

i 0 1000

0 0.05 0.1 0.151 10

3

0.01

0.1

1

10

100Histogram of PSF sidelobes

sidelobe magnitude

nu

mber o

f s

am

ple

s,

no

rm

ali

zed

0 0.5 10

0.5

1

1.5UV radial distribution

UV radius

UV

den

sit

y

k 0 200j 0 nbinsuvrdist Nrdist uvdata nbins 1.5( )nbins 50

LargeSL LPlot grdext( ) 0.0581LPlot Rbeam psf( )HistPlot PSFhisto psf 1.5 grdext 1 grdextmag

2 100

nless 0fRan Dant 0random perturbation.mag 14ifig 20file "Conways configs ascii.txt"

Conway: Mag=70, XYscale=840m

psfplot

PSF plotpsfplot PSFlog psf .02 .15( )

psf PSFin aconfig samp grdext mag( )

samp 2 maggrdext 2 N3dB *N3dB 1mag 70

1 0 1

1

0

1

UV coverage

normalized U [m/Dant*mag]

norm

aliz

ed V

[m

/Dan

t*m

ag]

0.5 0 0.5

0.5

0

0.5

Antenna positions

normalized X [E/Dant*mag]

norm

aliz

ed Y

[N

/Dan

t*m

ag]

V Im uvdata( )U Re uvdata( )uvdata UV config( )Y Im config( )X Re config( )

mag 70ifig 9file "Conways configs ascii.txt"

Conway: Mag=70, peak sidelobe=13.2%

1 103

0.01 0.1 1 100

0.05

0.1

0.15

Array Beam maximum vs. angle

Angle/Primary Beam_HWHM

PS

F p

eak

an

d a

verag

e

i 0 1000

0 0.05 0.1 0.151 10

3

0.01

0.1

1

10

100Histogram of PSF sidelobes

sidelobe magnitude

nu

mber o

f s

am

ple

s,

no

rm

ali

zed

0 0.5 10

0.5

1

1.5UV radial distribution

UV radius

UV

den

sit

y

k 0 200j 0 nbinsuvrdist Nrdist uvdata nbins 1.5( )nbins 50

LargeSL LPlot grdext( ) 0.13248LPlot Rbeam psf( )HistPlot PSFhisto psf 1.5 grdext 1 grdextmag

2 100

nless 0fRan Dant 0random perturbation.mag 70ifig 9file "Conways configs ascii.txt"

Conway: Mag=240, XYscale=2,880m

psfplot

PSF plotpsfplot PSFlog psf .02 .15( )

psf PSFin aconfig samp grdext mag( )

samp 2 maggrdext 2 N3dB *N3dB 1mag 240

1 0 1

1

0

1

UV coverage

normalized U [m/Dant*mag]

norm

aliz

ed V

[m

/Dan

t*m

ag]

0.5 0 0.5

0.5

0

0.5

Antenna positions

normalized X [E/Dant*mag]

norm

aliz

ed Y

[N

/Dan

t*m

ag]

V Im uvdata( )U Re uvdata( )uvdata UV config( )Y Im config( )X Re config( )

mag 240ifig 2file "Conways configs ascii.txt"

Conway: Mag=240, peak sidelobe=16.0%

1 103

0.01 0.1 1 100

0.05

0.1

0.15

Array Beam maximum vs. angle

Angle/Primary Beam_HWHM

PS

F p

eak

an

d a

verag

e

i 0 1000

0 0.05 0.1 0.151 10

3

0.01

0.1

1

10

100Histogram of PSF sidelobes

sidelobe magnitude

nu

mber o

f s

am

ple

s,

no

rm

ali

zed

0 0.5 10

0.5

1

1.5UV radial distribution

UV radius

UV

den

sit

y

k 0 200j 0 nbinsuvrdist Nrdist uvdata nbins 1.5( )nbins 50

LargeSL LPlot grdext( ) 0.16048LPlot Rbeam psf( )HistPlot PSFhisto psf 1.5 grdext 1 grdextmag

2 100

nless 0fRan Dant 0random perturbation.mag 240ifig 2file "Conways configs ascii.txt"