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