M. A. Holdaway
National Radio
Astronomy Observatory
949 N. Cherry
Ave.
Tucson, AZ
857210655
mholdawa@nrao.edu
Rick Perley
National Radio
Astronomy Observatory
1003 Lopezville
Rd
Socorro, NM
87801
rperley@nrao.edu
June 10, 1996
Cornwell
(1986) and Keto(1992) argued for
uniform Fourier plane coverage, which led them to ringlike array configurations.
Such arrays result in (u, v) coverage which attempts to sample as many
regions of the Fourier plane as possible.The
naturally weighted and uniformly weighted beams of these uniformFourierplane
distributionsareverysimilar,
sono noiseor
resolutionislosteitherway.
However,anotherwayto
overcomethe tradeoff between resolution
and noise for centrally condensed Fourier planecoverages
is through robust weighting (Briggs, 1995) .
As
there is an unfortunately large gap between the VLA and the inner VLBA,
the VLAupgrade plans to add a fewantennas
to provide continuous Fourier plane coverage from the shortest VLA spacings
to the longest VLBI spacings, permittingconstant
resolutionimaging capabilitiesfor
a widerange of frequenciesandresolutions,whichisessentialformanyscientific
problems. However, choosing optimal locations forthese
new antennas is a very tricky problem, aswe
must clarify how the antennas will be used with the VLA and the VLBA.
Theinner
VLBAantennasand
thenew upgradeantennas
willprobably be connected tothe
VLA correlator via fiberoptic cables,
and their signals willbe correlatedin
realtime, withoutobserving
with theother VLBA antennas. Wecall
this the A+ array. Hence,the first
requirement for the upgrade configurationis
that theVLA A+ array mustprovide
good imaging characteristics irea
standalone mode.We arbitrarily
choose to optimize the A+ array for baselines out to 300 km (about 8.5
times higher resolution than the VLA's A array).
The
newupgrade antennas and theVLA
(in phased arraymode or up to four
individual VLAantennas) willalso
be usedwith the entireVLBA,
so the secondrequirementfortheupgradeconfigurationisthatitmust
substantially enhance the imaging capability of the VLBA. We do not address
this issue here.
The
centrally condensed Fourierplane
distribution of both the VLA and the VLBAmake
theimplicitstatement
thatholes intheinner
partof the Fourier plane are more
damaging than holes in the outer part of the Fourier plane.The
visibilityfunction oftenchanges
morequickly inthe
inner Fourier plane, so holes in the inner Fourier plane result in more
damage to images thanholes in the
outerpart of the Fourierplane.
In considering the placementof theupgrade
antennas, weshould try tostick
with this philosophy ratherthan
attempt touniformly fill the gapsin
the Fourier plane.
Finally,
weare not attempting to generate
thefinal placement of the VLA upgrade
antennas.Rather, we are trying to
explorehow the problem may be solved.
There are anumber of parameters,
such as the size of the A+ array or the manner in which we weight the holes
as a functionof positionin
the(u, v)plane,
whichneed to berefined
in future work.Also, there are places
where itis clearly not acceptable
to placeantennas, suchas
inthe middleof
theGila Wilderness,and
the algorithms used have not attempted to deal with this problem. In any
event, this workmay inspire someone
to inventa better algorithm for
suggesting antenna placement.
In
orderto apply Keto's algorithmto
the VLA upgradeproblem, we had to
make a few changes:
·the
positions of the existing VLBA and VLA antennas are fixed. They contribute
to the (u, v)
coverage,
but do notparticipate in theiterative
antenna dance.
·because
of the very sparse (v, v) coverage in VLBI, we considered long (N12 hour)
tracks through
the
Fourier plane (Foster, 1996).(Keto's
original algorithm optimized for a snapshot at the zenith.)
·because
the VLA/VLBA system has a highly centrally condensed Fourier plane coverage,
we do not seek uniform coverage but a Gaussian coverage. Actually, the
VLA/VLBA system (with only four VLA antennas)has
a Fourier plane density which varies radially approximately as r1.6, but
we use a Gaussian of half width 160 km because it wassimple
to program. We attempted to recover a Gaussian (u, v)distribution
by distributing the "random" points picked in the Fourier plane with a
Gaussian distribution (ie, a Gaussian "picking function").
After
experimenting with themodified Keto
algorithm, it became clear that thereareimplicitforces
operatingwhichbiasthe
solutionstowards placing antennason
some sort of ring.The upgrade antenna's
usually were placed ina partial,distorted
ring about150200 km awayfrom
the VLA, each separatedby about50
km. `Weobtained similar resultsfor
uniform picking functionsand Gaussianpicking
functions of avariety of widths.
These configurations produce (u, v) coverages which had some large patches
ofveryuniform
coverageadjacenttolarge
patchesat similarradial
distances which were verysparsely
covered. As we will see below, the Keto through other less elegant methods.
While
our experiencewith the Keto algorithm
was unsatisfactory, there are probablyfurther
modification thatcould bemade
to thealgorithm which could improve
its performance on this problem.
We
didnot eventry
the Cornwellalgorithm. Like theKeto
algorithm, it would have placed theupgrade
antennas in a ring specified by the circular region within which we were
willing to place antennas.
Again,
somemodification of the Cornwellalgorithm,
such as weighting the holes inthe
inner Fourier planemore highly
than theholes in the outer Fourier
plane, would probably improve its performance in this problem.
Weselected
randomlocations within250
kmof theVLA
forthe upgrade antennas. In order
toevaluate the success of these
random configurations, weinvestigate
boththe qualityof
theuniformly weightedpoint
spread function (PSF) generated fromthe
(u, v) data with a 300 km taper, and the Fourier plane coverage. Whilethere
is a general trend between the quality of thePSF
and the quality of theFourier plane
coverage (smaller Fourier holesgenerallyresultinsmallerPSFsidelobes),somelowhole
configurations (such as acompletely
uniform coverage with a sharp maximum baseline cutoff) will produce large
sidelobes.
The
qualityof thePSF
is gauged bythe rms sidelobe leveloutside
of a mask which covers themain beam.
The quality of the Fourier plane coverage isgauged
bythe sum ofthe
size ofthe holesin
the Fourierplane on baselines between
30 and 300 km on a 10 km grid, weighted by the radial (u, v)distance
tothe 1.5 power.In
orderto be computationallyfast,
we approximate the size of ahole
at a given (u, v) cell asthe number
of empty cells in the a direction times the number of empty cells in the
v direction. Since a similar numberwill
be repeated for each cell whichlies
in a given hole, we areactually
weighting the holes as the squareof
the hole area, which stronglydiscourages
largeholes. Aswe
mentionedabove, a holein
the center of the Fourier plane has a more sever affect on image reconstruction
than ahole ofthe
same size inthe outer Fourier plane.To
account for this, wehave weighted
the holesas the radial (u,v)
distance raised to the 1.5power.
Weneed to weightthe
holes bya powerless
than 1.0, sincethis powerjustcompensates
forthe biastowardslong
baselines which, through earth rotationsynthesis,
span more Fourier cells and hence reducemore
holesthana
shortbaseline would.Toarrive
ata single number, we integrate
the weighted hole image between the range of interest.Finally,
weweighted this (u, v) hole measure
suchthat it would be about twice
aslarge as the RMS sidelobe level,so
the sidelobe level basically distinguishesbetween
differentconfigurationsof
similarFourier plane coverage quality.
It
is simple toinclude different source
declinations in the optimization.We
took *=
15° to be representative of the range 30°<*
< 0 °, with a weight of 0.37, *
=15° to be representative of the range 0° <*
< 30°, with a weight of 0.37,and *
= 45° to be representative ofthe
range 30° < *
< 60°, with aweight of 0.26.
We ignored the north polarcap since
it has a smallfraction ofthe
visiblesky. Theweights
are proportionalto the fraction
of sky which falls into each declination range.
As
weincrease the numberof
"random" iterations, wesample the
possible phase spaceof antennalocations
better. Inorder to samplethe
antenna plane of sizer _{max}
to a resolution of d with N_{ant} movable antennas, the number
of iterations N_{iter }must exceed
Niter
> ((r_{max}/d)^{2})!/ (((r_{max}/d )^{2}
 N_{ant})!N_{ant}!)(1)
With
N_{ant} = 4, this expression leads to about 2000 iterations to
sample to a resolutionof r_{max}/4.
Obviously, this will takeforever
to get any sort of resolution.However,
if we take a fewof the best configurations
which the"random" iterationshave
foundand wigglethe
antennasrandomly by order d, we
canproceed much more quickly. After
wiggling the antennas for many iterations,we
again select a few ofthe best configurations
and cut the sizeof the wiggle. Wename
these iterations the"wiggling" stage
of the algorithm.How many"wiggling"
iterations of wiggleamplitude d
will approximatelysample the antennasubspace
suchthat we candecrease
the
wiggle
by a factor of *?
Approximately
Niter
> (*^{2})N_{ant},(2)
or
about 256 iterationsfor *
= 2 and N_{ant},t = 4.It
is most efficient to decrease the wiggles bysmall
J factors and take more wiggle stages. Also, several wigglestages
are more effective atsampling the
space of antenna locations than the random stage, assuming that the wiggle
stages do not get trapped in a local minimum.
Being
trapped in a local minimum is a real concern. If we had simply made a uniform
grid of some coarse resolution and tried every possible combination of
antenna locations, thenreduced
the resolution and performed systematic (ie, nonrandom) antenna, "wiggles",
we would certainly navegotten stuckin
a localminimum. it isour
hope thatby randomizing the antenna
locations,running more iterations
than would seem required,and byexploringseveral
ofthe bestconfigurationsat
each stage, wewill jump out ofmany
of the localminima. However, the
bottom line isthat the "best" configurationis
not all thatmuch better than a "good"
configuration, as is demonstrated by our results below. Indeed, most of
the configurations with truly poor (u, v) coverage and PSF sidelobes are
the resultof placingone
or moreof the upgrade antennasright
next to another antenna.So, to first
order, the ruleof thumb "Don't
do anything dumb" is good enough to prevent you from making a bad configuration.
While
ouralgorithm really is dumb, and
does take agood deal of cpu time
to run, it wasquite simple to program.
It consists of a unix shell script which calls SDE tasks,and
took about 20 minutes to write, once we decided what wewanted
to do. Tradingcomputer time for
astronomertime isn't so dumb after
all.
The
three best configurations from each stage were usedas
startingconfigurations for the subsequentstages.
Figure 1 shows the best 10 configurations made with four upgrade antennas
at the end of therandom stage,the
80 kmwiggle, the 40km
wiggle, andthe 20 km wigglestages.
ExistingVLA andVLBA
stationsare marked witha
filled square while upgrade stationsare
marked with an empty square. The 10 best configurations fromeach
stage are laidon top of eachother
to show the range of possibilities ateach
stage. Initially, the upgrade antennas are widelydistributed,butas
thewiggleamplitudebecomes
smaller,the upgrade stations settle
down into variations on a single state, even though multiplestarting
configurationsfromthe
previousstages werecarried
through to each stage.
Another
wayof looking at the results of
our algorithmis to plot the rms
sidelobelevelagainstthenetweighted(u,v)
holesizeforeach
configuration. Sincewe havecreated
thousands ofconfigurations, such
a plot isdifficult tolook
at andto print, sowe
havemade a schematic figurewhich
indicates theboundaries of(u,
v) holermssidelobe space within
whichthe configurationsin
each stageare found. Figure2
shows that thefour upgrade antenna
configurationsproduced by the random
stage spana largerange
inquality. Thevery
badconfigurations tendto
be redundant, placing upgrade antennasnext
to each other or next to existing antennas. Each stage of wiggle iterations
produces a small improvement over thelast
stage,andthe
levelofimprovement
decreasesas thewiggle
amplitude decreases.
So,
while we cannotbe certain that
we have found the "best" configuration (in fact, itis
clear that we haven't), we can statethat
we have found a goodconfigurationwhich
isconsiderablybetter
thanmany others.For
example, ourbest configuration
is considerablybetter than the
result of the modified neural networks configuration and Craig Walker's
configuration(Figure 2). Craig
Walker was not trying to solve the same problem as we are trying tosolve,
so it is not at allsurprising that
his configuration is not as good as our best configurations. Also, Walker's
principle motivation was to obtain good, uniform (u, v) coverage, which
explains the location of his configurationon
our plot, indicating good (u,v)
coverage but not so good PSF sidelobes.
Figure3
isaschematic
ofthe entirerange
coveredby thevery
best through thevery worstconfigurations
from all stages,displayed for the
four, five, and six upgrade antenna cases. The bottom left hand tips of
the regionsdip tolower
and lowerrms sidelobeand
integrated (u,v) hole levels as
we increasethe number of upgrade
antennas, but the improvements are incremental.Hence,
you wouldneed at least sevenantennas
to reduce the rms sidelobe and (u, v) hole level to be half of the level
for the best configuration producedwith
four upgrade antennas.On the other
hand, the best configurationwith
four upgrade antennasis considerably
better than many sixupgrade antenna
configurations, sowise placement
of antennas is worth many antennas.
The
tenbest configurations obtained
from thefinal stage (20 km wiggles)
for thefour, five, and six upgrade
antenna casesare shown in Figures
4.Table 1lists
the antenna longitude, latitude, andsite
name for the best configurations obtained for thefour,
five, and six upgrade antenna cases.Each
upgradeantenna can be moved 1020
km forbetter access to roads and
powerwithoutsignificantlyaffectingtheFourierplanecoverageor
properties ofthe point spread function.
A fewof the antenna sites, such
as ElkMountain in the Gila wilderness,
theGila Cliff Dwellings, and the
BlackRiver sitein
Arizona, areclearly notpossible.
It wouldnot be difficult to create
an antenna plane mask which prohibited placing antennas in wildernessareas
oron Indian reservations.Since
many configurationshaveverysimilar
beamand(u,v)
holecharacteristics,we
canfind excellent configurations
which are environmentally acceptable.
TheFourierplanecoveragesandbeamsfor15,15,and45degree
declinations forthe bestconfiguration
with four, five,and six upgrade
antennas are shown in Figures 5 through 7 .
Wepresent
anexample 100km
configurationmade by addingfive
upgrade antennas toPie Town and
theVLA A Array (Figures 8 and9)
. We have not gonethrough anyelaborate
optimizationprocess forthis
configuration, just laidthe antennas
down by eye. However,the Fourier
pane coverage is very good,with
mostgaps being thesame
size as thedistance between A array
antennasat the arm ends.The
few larger gapscould undoubtedly
be fixed
Four
Upgrade Antennas


LongitudeLatitudeSite

109.0657133.77203Alpine,
AZ

105.9111932.95187High
Rolls, NM

106.1509833.44207Three
Rivers, NM

109.1067333.31532Glenwood,
NM


Five
Upgrade Antennas


LongitudeLatitudeSite

105.3257133.32017San
Patricio, NM

109.9434433.52937Black
River, AZ

110.0050534.14544Pine
Top, AZ

107.2121332.18726Akela,
NM

108.5191233.57188Elk
Mountain, NM


Six
Upgrade Antennas


LongitudeLatitude
Site

105.4193233.06540Elk
River, NM

110.4806533.37835San
Carlos, AZ

109.6397434.15905Green's
Peak, AZ

107.4428932.02733Akela,
NM

108.1121533.28230Gila
Cliff Dwellings, NM

108.7859833.15079Buckhorn,
NM


Table
1:Longitude and Latitude ofantenna
sites for four,five, and six upgrade
antennaconfigurations. These antenna
locationscould be moved by 1020
km without significantly affecting the quality of the configurations.
via
optimization. Figure 10 shows the Fourier plane coverages and beams for
comparison with the 300 km configurations.
Thesurface
brightnessnoise isproportional
tothe pointsource
noise divided bythe beam area. Sincethe
point source noiseis constant among
arrays ofdifferent sizes, asmall
configuration with alarger beam
will havelower brightnessnoise
(ie,the 100km
array haslower brightness noise
thanthe 300 km array).However,
to compare thesetwo arrays on an
equal footing, we must compare the brightness noise at the same resolution.As
wetaper inthe
Fourier plane toobtain a larger
beam,we throw away somelongbaseline
visibilitiesandour
pointsource noiseincreases.However,ourbeam
areaincreasesfasterthan
thepoints sourcenoise
increases, andtapering results in
higherbrightness sensitivity, but
not as high as if no tapering were required.
The
A_+array isin
an interestingsituation with regardsto
brightness sensitivity. Mostof
the sensitivity of the array isin
the center of the Fourier plane (ie, the VLA in its A array), andvery
little sensitivityis in theregion
beyond35 km. Inorder
to achieve goodresolution, wemust
use somesort of uniformweighting.
In this case pure uniform weighting drastically down weights the inner
Fourier planeand throws awaymuch
ofthe array's sensitivity.Robust
weighting (Briggs, 1995)is ahybrid
weighting schemewhich preserves
muchof the resolutionof
uniformweighting andmuch
ofthe sensitivityof
natural weighting,and something
likerobust weighting is
precisely whatthe A+ arrayrequires.
Robust weightingstill needsto
down weightthe central part of the
Fourierplane to get high resolution.
As we taper to get lower resolution,robust
weightingneedsto
downweight less,andthe
point sourcenoisecanactuallydecrease,thereforemaking
thebrightness sensitivity improve
more thanexpected. Unlike tapering
naturally weighted data, thebrightness
sensitivityimprovement which resultsfrom
tapering robustly weighteddata is
nota smooth function ofresolution.
Figure 11 shows the point source noise as a function of resolution for
the 300 km and 100 km arrays withfive
upgrade antennas. These point source sensitivities aregeneratedfor12hourtracksusing1GHzbandwidthintwo
polarizations,using current 8GHz
Tsysand27
VLA antennas,5 upgrade antennas,
and the PT,LA, FD, and KP VLBA stations,
tapered accordingly to achieve theplotted
resolution.Placing the antennas
intoa more compact (100 km) At
array results in surfacebrightness
sensitivity which is only marginally better than the 300 km A+ arrays.
The image fidelity of a 100 km A+ arraywill
probably be betterthan that of
a 300km array at the same resolution.
·What
should be the maximum (u, v) baseline over which the A+ array is optimized?
This work focused on 300 km arrays, but alsolookedat
one 100 km array.
·While
a 100 km array has a lot more 35100 km baselines than the 300 km array,
the brightness sensitivities of the two arrays at the same resolution are
actually not very different.
·How
should we evaluate the combination of the VLBA plus the VLA A+ array?
·How
should we weight the importance of (u, v) holes as a function of radial
(u, v) distance? We weighted our holes as the radial (u, v) distance raised
to the 1.5 power.
·How
many upgrade antennas do we need? While adding antennas incrementally produces
incremental improvements in the objective configuration quality measures,
the (u, v) coverages produced with six upgrade antennas subjectively look
a lot better than the four antenna (u, v) coverages.
·We
need to consider where we can and cannot realistically place antennas during
the optimization, rather than perform an unconstrained optimization and
then try to fit the resulting array to realistic antenna locations.
·What
other criteria should be used to gauge the effectiveness of a configuration?
Our "algorithm" is flexible enough that we could incorporate anything which
was not too cpu intensive.
·Should
we devise a more intelligent algorithm?
·Any
proposed configurations should be put to the test by simulating realistic
data and comparing the imaged simulation data with the true brightness
distribution from which the data were generated.
Cornwell,
T.J., 1986, MMA Memo 38, "Crystalline Antenna Arrays".
Foster,
Scott M., 1996, MMA Memo in preparation.
Keto,
Eric, 1992, SMA Memo, "Cybernetic Design for CrossCorrelation Interferometers".
Very Long Baseline Array Project Book, Version 71988, NRAO, Charlottesville
VA, pp 11 to 120.