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 dj >l/max_baseline
• Nyquist theorem
– UV sample spacing < 1/FOV
less than Nyquist sampling
1
0. 5
0
0. 5
1
0
200
400
600
800
1000
1200
1400
1600
1800
2000
• Simple DFT would seem to require complete sampling of
the UV-plane
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 required
average sampling exceeds Nyquist
1
0.5
0
0.5
1
0
200
400
600
800
1000
1200
1400
1600
1800
2000
• 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?
• Radio
• Optical
– Spectral sensitivity function
– Optical transfer function
• Autocorrelation of field
• Transfer function or UV sampling
Aperture Field
esponse
1
0.5
0
20
15
10
5
0
distanc e
5
10
15
20
10
15
20
Spectral Sensitivity Function
3
response
2
1
0
20
15
10
5
0
U
5
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
1

U PSF ( p) 
N
.
N


 
U n ( p) exp  i k p  rn  .
n 1
• Point Spread Function = synthesized beam plus
single antenna response

 2
PSF ( p)  U PSF ( p)
1
 2
N
N
N

  
 Bmn ( p) exp i k p  (rm  rn )
m 1 n 1
• Same as the power from the array used as a
transmitter
Synthesized Beam and PSF
Synthesized & Primary Beam
1
0.5
0
1
0.5
0
angle
0.5
1
0.5
1
Point Spread Function
1
0.5
0
1
0.5
0
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
 0  .01 1
  0 .01 10
Antenna distribution
Point Spread Function
1
1
0.5
0.1
0
0
0.2
0.4
0.6
0.8
1
radius
 0  .01  1.5
UV distribution
0.01
1
0.5
1 10
0
0
0.2
0.4
0.6
0.8
baseline
1
1.2
1.4
3
0.1
1
angle
10
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
– s > 1/N for Magnification >> N
– N = number of antennas
– Magnification = (primary beam width)/(synthesized beam width)
Standard Deviation vs. Magnification
x  1  1.02 3
minimum s for bell shaped and uniform UV distributions for N =64
Standard Deviation of PSF Sidelobes
Standard Deviation
0.1
1.6%
0.01
1 10
3
10
100
Array Magnification
1 10
3
Sidelobe Calculation
N
1


 
PSF( p)  2 B( p)  exp i k p  rn 
N
n 1
2
• Sidelobes are sum of N unit vectors of different
orientations
2  
Vector rotation =
pr
l
fr  0
Vector Random Walk
field vector sum
1
0.5
0
0.5
1
1
0.5
0
0.5
1
fr  1
Vector Random Walk
field vector sum
1
0.5
0
0.5
1
1
0.5
0
0.5
1
fr  2
Vector Random Walk
field vector sum
1
0.5
0
0.5
1
1
0.5
0
0.5
1
fr  100
Vector Random Walk
field vector sum
1
0.5
0
0.5
1
1
0.5
0
0.5
1
fr  101
Vector Random Walk
field vector sum
1
0.5
0
0.5
1
1
0.5
0
0.5
1
Statistical Distribution of Sidelobes
• Should have a Rayleigh distribution
g ( s )  N exp  Ns 
• Expected maximum sidelobe
2
s max  ln( mag)
N
• Test against three configurations
– UV coverage optimized circular array
– Random Gaussian array
– Systematic Gaussian array
X  Re( config ) Y  Im( config )
Circular
uvdata  UV( config )
Antenna positions
Antenna distribution
U  Re( uvdata )
V  Im( uvdata )
UV coverage
UV distribution
1
0.5
0.5
V
0
0
0.5
0.5
1
0.5
psfplot  PSFlog( psf
 .05 .15)
0
PSF plot
0.5
LPlot  Rbeam( p sf )
1
0.5
X
0
i  00.5 1000
1
U
PSF
Peak sidelobe vs. radius
PSF peak, average and rms
Y
0.15
0.1
0.05
0
psfplot
1
10
100
Angle/P SF_HW HM
1 10
3
X  Re( config ) Y  Im( config )
Pseudo Random
uvdata  UV( config )
Antenna positions
Antenna distribution
U  Re( uvdata )
V  Im( uvdata )
UV coverage
UV distribution
1
0.5
0.5
V
0
0
0.5
0.5
1
0.5
psfplot  PSFlog( psf
 .05 .15)
0
PSF plot
0.5
LPlot  Rbeam( p sf )
1
0.5
X
0
i  00.5 1000
1
U
PSF
Peak sidelobe vs. radius
PSF peak, average and rms
Y
0.15
0.1
0.05
0
psfplot
1
10
100
Angle/P SF_HWHM
1 10
3
X  Re( config ) Y  Im( config )
Systematic
uvdata  UV( config )
Antenna positions
Antenna distribution
U  Re( uvdata )
V  Im( uvdata )
UV coverage
UV distribution
1
0.5
0.5
V
0
0
0.5
0.5
1
0.5
psfplot  PSFlog( psf  .05 .15)
0
PSF plot
0.5
LPlot  Rbeam( p sf )
1
0.5
0
i  0.5
0  1000
1
U
X
PSF
Peak sidelobe vs. radius
PSF peak, average and rms
Y
0.15
0.1
0.05
0
psfplot
1
10
100
Angle/P SF_HWHM
1 10
3
j  0  200
Distribution of Sidelobes
100
number of samples, normalized
Number of sidelobes vs. magnitude
10
1
0.1
0.01
1 10
3
0
0.05
0.1
sidelobe magnit ude
0.15
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.
Optimization
fr  100
• Minimize peak sidelobe
field vector sum
–
–
–
–
Find1 peak sidelobe
Adjust a single antenna to reduce this peak
Check that no other peak now exceeds new peak
0.5
Repeat
0
0.5
1
1
0.5
0
0.5
1
Sequential Optimization
2  
• Rotation of the antenna unit vectors is
pr
l
•
• 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
1
s max, opt  1  2 ln( mag )  ln( N )
N
 Re( config ) Y  Im( config )
Optimized Arrays
X  Re( config ) Y  Im( config )
Antenna positions
Antenna distribution
Y
0.5
0
Y
0
0.5
0.5
0.5
0
X  Re( confi g ) Y  Im( confi g )
0.5
0.5
0.5
0
X  Re( confi g ) Y  Im( confi g )
X positions
Ante nna
UV distribution
0.5
X positions
Ante nna
0.5
Y
0.5
0.5
0
Y
0.5
0.5
0
X
0.5
0.5
X positions
Ante nna
UV distribution
0.5
0
0
X  Re( confi g ) Y  Im( confi g )
UV distribution
0.5
Antenna positions
Antenna distribution
0.5
0
Y
X  Re( config ) Y  Im( config )
Antenna distribution
0.5
Y
Antenna positions
0
0.5
0.5
0
X
0.5
0.5
0
X
0.5
LPlot  Rbeam( p sf )
Optimized Sidelobe Distribution
LPlot  Rbeam( p sf )
i  0  1000
Peak sidelobe vs. radius
PSF peak, average and rms
PSF peak, average and rms
Peak sidelobe vs. radius
0.15
0.1
0.05
0
1
Plot  Rbeam( p sf )
10
100
i  0  1000
i  0  1000
1 10
0.15
0.1
0.05
0
3
1
10
j  0  200
Angle/P SF_HWHM
1 10
3
100
Angle/PSF_HWHM
100
number of samples, normalized
PSF peak, average and rms
Peak sidelobe vs. radius
0.15
0.1
0.05
0
1
10
100
Angle/PSF_HWHM
1 10
3
Number of sidelobes vs. magnitude
10
1
0.1
0.01
1 10
3
0
0.05
0.1
sidelobe magnit ude
0.15
Add more short spacings
X  Re( config ) Y  Im( config )
Antenna pos itions
uvrdist  Nrdist ( uvdata  nbins  1.5)
2
Antenna distribution
Radial UV distribution
UV density
0.5
Y
0
1
0
0
0.2
0.4
psfp lot  PSFlog( psf  .001  .05)
0.5
LPlot  Rbeam( p sf )
0.5
0
X
i  0  1000
0.5
PSF peak, average and rms
Peak sidelobe vs. radius
0.15
0.1
0.05
0
1
10
100
Angle/PSF_HWHM
j  0  nbins
1 10
3
0.6
UV radius
PSF plot
PSF
0.8
1
Results for several design studies
0.25
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
Sidelobe Peak
0.2
0.15
0.1
0.05
0
10
100
Magnification
1000
Earth Rotation
Standard Deviation of PSF Sidelobes
Standard Deviation
0.1
0.01
1 10
3
3
100
1 10
Array Magnification
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).
10
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
file  "Conways configs ascii.txt"
ifig  20
X  Re( config ) Y  Im( config )
mag  14
uvdata  UV( config )
U  Re( uvdata)
Antenna positions
1
normalized V [m/Dant*mag]
normalized Y [N/Dant*mag]
0.5
0
0.5
0
1
0.5
mag  14
V  Im( uvdata)
UV coverage
0
0.5
normalized X [E/Dant*mag]
N3dB  1
grdext  2 N3dB *
psf  PSFin ( aconfig  samp  grdext  mag )
psfplot  PSFlog ( psf  .02  .06)
psfplot
PSF plot
1
samp  2 mag
0
normalized U [m/Dant*mag]
1
Conway: Mag=14, peak sidelobe=5.8%
file  "Conways configs as cii.txt"
ifig  20
His tPlot  PSFhis to 
 ps f  1.5 grdext  1 grdext 
mag

nbins  50
mag  14
2
 100 


uvrdis t  Nrdis t ( uvdata  nbins  1.5)
random perturbation.
LPlot  Rbeam( ps f )
j  0  nbins
UV radial dis tribution
LargeSL( LPlot  grdext )  0.0581
His togram of PSF s idelobes
100
number of samples, normalized
UV density
nles s  0
k  0  200
1.5
1
0.5
0
fRan  Dant  0
0
0.5
UV radius
10
1
0.1
0.01
1 10
1
3
0
0.05
0.1
sidelobe magnitude
0.15
i  0  1000
Array Beam maximum vs . angle
PSF peak and average
0.15
0.1
0.05
0
3
1 10
0.01
0.1
Angle/Primary Beam_HWHM
1
10
Conway: Mag=70, XYscale=840m
file  "Conways configs ascii.txt"
ifig  9
X  Re( config ) Y  Im( config )
mag  70
uvdata  UV( config )
U  Re( uvdata)
Antenna positions
1
normalized V [m/Dant*mag]
normalized Y [N/Dant*mag]
0.5
0
0.5
0
1
0.5
mag  70
V  Im( uvdata)
UV coverage
0
0.5
normalized X [E/Dant*mag]
N3dB  1
grdext  2 N3dB *
psf  PSFin ( aconfig  samp  grdext  mag )
psfplot  PSFlog ( psf  .02  .15)
psfplot
PSF plot
1
samp  2 mag
0
normalized U [m/Dant*mag]
1
Conway: Mag=70, peak sidelobe=13.2%
file  "Conways configs as cii.txt"
ifig  9
His tPlot  PSFhis to 
 ps f  1.5 grdext  1 grdext 
mag  70
mag

nbins  50
2
 100 


uvrdis t  Nrdis t ( uvdata  nbins  1.5)
random perturbation.
LPlot  Rbeam( ps f )
j  0  nbins
UV radial dis tribution
LargeSL( LPlot  grdext )  0.13248
His togram of PSF s idelobes
100
number of samples, normalized
UV density
nles s  0
k  0  200
1.5
1
0.5
0
fRan  Dant  0
0
0.5
UV radius
10
1
0.1
0.01
1 10
1
3
0
0.05
0.1
sidelobe magnitude
0.15
i  0  1000
Array Beam maximum vs . angle
PSF peak and average
0.15
0.1
0.05
0
3
1 10
0.01
0.1
Angle/Primary Beam_HWHM
1
10
Conway: Mag=240, XYscale=2,880m
file  "Conways configs ascii.txt"
ifig  2
X  Re( config ) Y  Im( config )
mag  240
uvdata  UV( config )
U  Re( uvdata)
Antenna positions
1
normalized V [m/Dant*mag]
normalized Y [N/Dant*mag]
0.5
0
0.5
0
1
0.5
mag  240
V  Im( uvdata)
UV coverage
0
0.5
normalized X [E/Dant*mag]
N3dB  1
grdext  2 N3dB *
psf  PSFin ( aconfig  samp  grdext  mag )
psfplot  PSFlog ( psf  .02  .15)
psfplot
PSF plot
1
samp  2 mag
0
normalized U [m/Dant*mag]
1
Conway: Mag=240, peak sidelobe=16.0%
file  "Conways configs as cii.txt"
ifig  2
His tPlot  PSFhis to 
 ps f  1.5 grdext  1 grdext 
mag  240
mag

nbins  50
2
 100 


uvrdis t  Nrdis t ( uvdata  nbins  1.5)
random perturbation.
LPlot  Rbeam( ps f )
j  0  nbins
UV radial dis tribution
LargeSL( LPlot  grdext )  0.16048
His togram of PSF s idelobes
100
number of samples, normalized
UV density
nles s  0
k  0  200
1.5
1
0.5
0
fRan  Dant  0
0
0.5
UV radius
10
1
0.1
0.01
1 10
1
3
0
0.05
0.1
sidelobe magnitude
0.15
i  0  1000
Array Beam maximum vs . angle
PSF peak and average
0.15
0.1
0.05
0
3
1 10
0.01
0.1
Angle/Primary Beam_HWHM
1
10