Abstract

We explore a few possible algorithms for finding four to six antenna positions for the VLA Upgrade's A+ configuration, which will bridge the gap between the VLA's A configuration and the inner VLBA antennas.A modified neural networks approach (Keto, 1992) does not seem to work well for this problem in which most antennas are fixed and a centrally condensed (u, v) coverage is desired. We develop an algorithm based on random assignment of antenna positions and antenna position wiggles of decreasing size, with configurations differentiated by the size of holes in the (u, v) plane and the rms sidelobe level of the point spread function. We used this algorithm to generate 300 km configurations with visually appealing Fourier plane coverage and full track point spread functions with RMS sidelobes below 1%. The best configurations made with four upgrade antennas are significantly better than Walker's (VLBA Project Book, 1988) four antenna configurations (admittedly an unfair test since Walker's design criteria was to produce a better VLBA). While adding more antennas to the A+ array steadily improves the configuration quality, the improvements are incremental. We also find that a 100 km A+ array has only marginally better brightness sensitivity than a 300 km A+ array tapered to the same resolution.

1 Introduction

The arrangement of antennas in an interferometric array is very important for the quality of the images which the array will produce, but as yet, researchers cannot give an unambiguous answer asto whatthe arrangementshouldbe.TheVLA'santenna configurations were based onmisconceptions about how the imaging would be performed and on how images at different frequencies would be compared. The VLBA's antenna configuration was chosen based onthe desireto fill the Fourier plane reasonably well, with emphasis on the shorter baselines which were conspicuously absent from ad hoc VLBI arrays. (Holes in the inner part of the Fourier planecan affect image quality more than holes in the outer part ofthe Fourier plane.) Both ofthese arrays produce highly centrally condensedFourierplane distributions,whichresultin lowresolution naturally weighted images withoptimal noise, or high resolution uniformly weighted images with noise which is much higher than optimal.

Cornwell (1986) and Keto(1992) argued for uniform Fourier plane coverage, which led them to ring-like 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 stand-alone 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.

2 Keto's Neural Network Algorithm

Keto (1992) has written a neural network code which iteratively adjusts the antenna locationsuntil a uniform Fourierplane coverage is achieved. The algorithm picksa point in the Fourierplane randomly within some maximum (u,v) limitandpulls theclosest (u,v)sample towardsthat point, adjusting the coordinates ofthe antennas correspondingly. This process is repeated thousandsof times, and witheach iteration, the antennas become stifferand movesmallerdistances towardsthe randompoints.The end result is a Fourierplane coverage in which any point in the Fourier plane is close to a (u, v) sample.

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 r-1.6, 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 about150-200 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.

3 Cornwell's Simulated Annealing Algorithm

Cornwell (1986) wrote asimulated annealing algorithm which moves antennas about, calculatesthe "energy" of theresulting Fourier coverage (defined as thepotential energy ofthe system of charged,repelling particles at the antenna locations, each with 1/r potentials), subject to the constraint that the antennas lie within a circular region. Lower energy configurations were always accepted, and higher energy configurations were accepted with a probabilityofe-ElkT,where E isthe configurationenergy and Tis a fictitious temperature.As the iterations proceed,the antennas are moved by smaller amounts and the temperature slowly drops, hopefully resulting in a nearly minimum energyconfiguration. The possibility of accepting higher energy configurations permits escape from local minima in the energy.

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.

4 The "Modified Random" Algorithm

Sincebeing cleverdoesn't seem tohelp, perhapsits time totry being dumb. Simplysearching all possiblelocations for the fouror so upgrade antennasisprohibitively timeconsuming.However,algorithms suchas simulated annealingdon't aimat getting thevery best solution,but at getting one of manynearly optimal solutions. Typically, the minima of the functions we are trying to optimize are broad and shallow, so we don't need to find the very best solution to get comparable performance.

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),somelow-hole 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_itermust 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, non-random) 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.

5 Results of the Modified Random Algorithm

If youare confused about what exactlythe Modified Random Algorithm does or howit works, ignore the last section and lookat some of the results.For the four antennacase, we ran 2000 iterations in the random stage, and 500 iterations per starting configuration in each of the 80 km, 40 km and 20 km wiggle stages.

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) hole-rmssidelobe 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 10-20 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.

TheFourierplanecoveragesandbeamsfor-15,15,and45degree declinations forthe bestconfiguration with four, five,and six upgrade antennas are shown in Figures 5 through 7 .

6 Brightness Sensitivity

Itis somewhatarbitrary thatwe havechosen tooptimize for a300 km array.Suchanarraysizeisacompromisebetweenextendingthe capabilitiesoftheVLAand extendingthecapabilitiesof theVLBA.However, itcan be argued thata -100 km configurationwhich extends the VLA's resolutionby a factorof about 2.85 (theexpansion factor between adjacent VLA configurations) is the right way to spend extra antennas. Such an array would havesuperior brightness sensitivity, Fourier coverage, and imagingcapabilitiesfor bothstrongandweak sources,and shouldbe considered.

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 10-20 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.

7 Discussion

We will use thesection to round up all the loose ends we mentioned in the text. Whilewe cannot cutthem off yet, atleast we can putthem in one place:

·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 35-100 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.

References

Briggs, D.S., 1995, Thesis, "High Fidelity Deconvolution of Moderately Resolved Sources", New Mexico Institute of Mining and Technology, Socorro, NM.

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 Cross-Correlation Interferometers". Very Long Baseline Array Project Book, Version 71988, NRAO, Charlottesville VA, pp 1-1 to 1-20.