Understanding Systematic Errors Through Modeling of ALMA Primary Beams

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by  Kara Kundert

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Understanding Systematic Errors Through
Modeling of ALMA Primary Beams
Kara Kundert Urvashi Rau Edwin Bergin Sanjay Bhatnagar
were previously easily ignored are beginning to make
theirwayabovetheeverfallingnoisefloor.Thesenewly
Abstract—Many aspects of the Atacama Large Millimeter
7 unearthed sources of error have yet to be characterized
Array (ALMA) instrument are still unknown due to its
1 and thoroughly understood, making them an especially
young age. One such aspect is the true nature of the
0
primary beam of each baseline, and how changes to the treacherous threat for the astronomers using the obser-
2
individualprimarybeamsaffectastronomicalobservations vatories. This is even more prevalent in interferometric

b when said changes are ignored during imaging. This systems which combine numerous telescopes, often of
e paper aims to create a more thorough understanding of
different sizes, across a variable distance scale between
F the strengths and weaknesses of ALMA through realistic
antennas.
modeling of the primary beams and simulated observa-
4
tions, which in turn can inform the user of the necessity
of implementing more computationally costly algorithms, Though there are many factors contributing to each
]
M suchasA-Projection,andwhensimpler,quickeralgorithms image, one that remains relatively unexplored is that of
will suffice. We quantify our results by examining the how the primary beam affects both the data collected
I dynamicrangeofeachobservation,alongwiththeabilityto
andthefinalimagesproduced[2].Asprimarybeamsare
.
h reconstructtheStokesIamplitudeofthetestsources.These
relatedtothebaselineaperturesthroughasimpleFourier
p tests conclude that for dynamic ranges of less than 1000,
- for point sources and sources much smaller than the main transform, the primary beam of each baseline offers a
o lobe of the primary beam, the accuracy of the primary uniqueinsightintotherelativestateoftheantennasinan
r beam model beyond the physical size of the aperture array.Inthispaper,wesimulateseveralprobableprimary
t
s simply doesn’t matter. In observations of large extended beambasedsystematicerrorsthatcouldbeintroducedto
a sources, deconvolution errors dominate the reconstructed
[ images and the individual primary beam errors were thedatacollectedatALMAinordertobetterunderstand

indistinguishable from each other. the effect it could have on the final images produced.
2
These effects include but are not limited to improper
v
2 Index Terms—Aperture antennas, submillimeter wave calibration of the secondary reflector, minor offsets in
0 propagation, radio interferometry, radio astronomy. the pointing during an actual observation, gravitational
4 distortion of the aperture, and ignoring parallactic an-
4
gle rotation during imaging. They also affect the final
0
images in varying ways. For example, an offset of the
. I. INTRODUCTION
1 receiver from the focus in the cryostat will change the
0
way the aperture of the dish is illuminated, leading to
7 Radio interferometry has been a key innovation in the
phase errors and diminished signal amplitude. Finally, a
1 fieldofmodernastronomy[1].Theabilitytosynchronize
: pointing offset during observation produces phase errors
v the observations of many separate antennas, along with
and amplitude calibration issues dependent on the scale
i dramaticallyimprovedbandwidthsandbackendtechnol-
X of the offset.
ogy, has led to increased observational sensitivities and
r
precision in resolution. These improvements have led to
a Without a thorough understanding of the health of the
the lowest systematic levels of noise ever seen in radio
individual antennas and their relationships to each other
astronomy. As the technology used to create the digital
while performing observations, new errors can prolif-
backends of observatories continues to see astounding
erate into the images through incorrect calibration of
rates of progress, new sources of systematic errors that
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2
the primary beam in the imaging process. Such errors question this simulation seeks to answer regards how
includebutarenotlimitedtothehidingoflow-amplitude much uncorrected changes in the primary beams of
astronomicalsourcesinthesidelobesofbrightersources ALMA affect the final images produced using standard
and an overall increase in the noise floor of an image CLEAN tool, which is deconvolution and image correc-
relative to the baseline thermal noise. tion software that operates by iteratively finding point-
sourcepeaksinanimageandsubtractingthescaleddirty
Intrinsically, there are two questions to be answered beam (also called the point-spread function1, or PSF)
about these revealed sources of error. What effects are from that point [4]. This method helps to minimize the
they imparting on astronomical observations, and how effects of the sidelobes of the PSF and to make a true
can those effects be corrected? The work done on this mapofthepointsources’locationsandtheiramplitudes.
simulation aims primarily to answer the first question. There is also a version of CLEAN which enables the
By creating a thorough and realistic model of ALMA, usertochoosemodelsourcesofmultiplesizesforlarger
the simulation can observe the propagation of errors objects. This tool is called MS-CLEAN [5].
generated by selectively introducing perturbations to the
primary beams. From this knowledge, software can be First, the mathematical relationship between the ob-
developed to target the largest sources of error in data servedvisibilitiesandtheskybrightnessmustbedefined,
analysis. along with how baseline apertures and primary beams
can affect these observations. In its most basic form,
Inthefollowingsection,themathematicsoftheproblem for one timestep, baseline, and polarization, we get this
are summarized. In the subsequent two subsections, Fourier pair of equations:
a brief description of the ALMA instrument and the
experimental model is provided. This is followed by the
resultsofourtestingandtheramificationswepredictfor V =V ∗A (1)
obs true
observations on the ALMA instrument.
I =I ×B (2)
obs true
II. BACKGROUND where V is the observed visibility, V is the true
obs true
visibility, I is the observed image, I is the true
obs true
In the case of the ALMA and similar instruments, the
image.Notethat∗denotestheconvolutionfunction,and
visibility function of a source is observed and converted
×ismultiplication.Fora source definedonthecelestial
into images via a Fourier transform. As the interferome-
sphere, the Fourier relationship between the visibility
ter is neither infinite in size nor sampling at every point
function and the image is given by the van-Cittert-
inspace,eachvisibilitymeasurementbecomesadiscrete
Zernike theorem, given in symbolic form in Eq. (3) [6].
linear weighted sum of a range of spatial frequencies
whicharedeterminedbythegeometryoftheinstrument.
This weighting is a function of the baseline apertures V(u,v)(cid:10)F I(l,m) (3)
of the interferometer, A. The baseline aperture is found
by correlating the aperture illumination functions of
Now let us build further complexity into equation (1)
a pair of antennas in an array. This can be Fourier
by accounting for the geometry of the array, which we
transformedtoprovidetheprimarybeam,B.BothAand
will mathematically describe as the sampling function
B determine how well a given baseline pair of antennas
S(u,v).
are able to see the true sky. In attempting to recreate
the true image, the estimated primary beam of a given
observationcanbefactoredoutinthefinalstagesofdata
corrections and reduction, or used to correct the image Vobs(u,v)=[Vtrue(u,v)∗A(u,v)]×S(u,v) (4)
using A-projection (an image processing algorithm fur-
therdescribedinSectionV)[3].Thebettertheestimated
where the sampling function S can be written as:
beam matches the true beam from the instrument, the
better the corrections on the data will be. The main 1Alsosometimesreferredtoastheimpulseresponsefunction.
3
7-m dishes [8]. There are also two 12-m antennas in the
(cid:88)
S(u,v)= δ(u−u )δ(v−v ) (5) TotalPowerArray,usedtomakesingledishobservations
k k
k in order to fill the central hole in the u-v place. Each
antenna is equipped with receivers to observe at bands
where each k represents a measurement from a single 3, 4, 6, 7, 8, and 9, which corresponds to wavelengths
baseline. of about 3.1, 2.1, 1.3, 0.87, 0.74, and 0.44 mm. There
are many configurations of the array, with maximum
Nowfinally,letusinvertequation(4)viaEq.(3)toyield baselines ranging from approximately 160 m to 1.5 km,
an image of the observed source brightness distribution, though the maximum baseline for bands 8 and 9 is
givingusamorecomplexandrealisticversionofEq.(2). approximately 1 km.
One fact is evidently clear - ALMA is the foremost
leader in interferometric imaging at millimeter wave-
I (l,m)=[I (l,m)×B(l,m)]∗I (l,m) (6)
obs true psf
lengths. As image fidelity is most strictly limited by the
number and coverage of samples in the u-v plane [9],
where Ipsf is the Fourier transform of the sampling the sheer number of antennas along with the variation
function S(u,v). in baseline spacings gives ALMA superior imaging
capabilitiestoanyothermoderninterferometercurrently
In these equations for the simplified case of a single
online. This unprecedented ability means that errors
snapshotin asinglefrequency binanda singlebaseline,
generated by systematic errors in primary beam analysis
wefindthattheobservedimageisthetrueskymultiplied
can limit the imaging capabilities of ALMA.
bytheprimarybeamandthenconvolvedwiththepoint-
spread function. This is the mathematical foundation of
the simulation. B. The Model
Our goal is to model a realistic interferometric array,
A. The ALMA Observatory with as many of the known problems of the ALMA
primary beam as we know how to replicate compu-
InthecaseofALMA,thearrayisstillinitsearlyscience tationally. In its most basic form, the simulation does
phase, so some parameters are bound to change as the the calculations shown in Eq. (4) for a set of mutually
remainingcomponentsoftheinstrumentgraduallycome unique aperture models over a series of time steps. As
online. In its final design, ALMA will consist of a 12-m time progresses in the simulation, so does the relative
arraycomposedof50antennasoftwodesignsdenotedas geometry of the array to the celestial sphere. An indi-
DA and DV with baselines ranging from 15m to 16km, vidual simulation is composed of snapshots taken every
a 12-element compact 7-m array with baselines ranging 3 minutes over the course of 2 hours, for a total of
from 8.5m to 30m, and four 12-m antennas for single 40 snapshots per “observation”. Snapshots were chosen
dish(orTotalPower)observations,whichprovidespatial over integrations to conserve computational resources,
information equivalent to baselines of 0m to 12m. The thedurationwaschosentopreventshadowingofclosely-
7-m and total power arrays – with their overall shorter placed antennas.
baselines – aim to fill the hole in uv-coverage typically
seen in radio interferometers. ALMA will be capable of Thesimulationisdesignedtobeatestofthecapabilities
observingfrom31-950GHzwithfulllinearpolarization of ALMA in Cycle 2 of its early science testing. There-
(X, Y) [7]. fore, the unperturbed interferometer which acts as the
controlcaseofthesimulationhas34identicalapertures,
However, ALMA is not yet fully finished. In Early takingdataat100GHzinreceiverband3.Thenumerical
Science Cycle 2 - which the simulation aims to model - interferometer is built by generating a set of antennas in
the array has the following specifications: 34 12-m an- an array, using the specifications on Cycle 2 released
tennasofthreeseparatedesigns,alongwiththeAtacama by ALMA, including the reference location of the ob-
Compact Array (ACA), which consists of nine identical servatory on the earth, the placements of the antennas
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(a)AverageDAMeasuredAperture–Real (b)AverageDAMeasuredAperture–Imaginary
(c)RMSDAMeasuredAperture–Real (d)RMSDAMeasuredAperture–Imaginary
(e)AverageDVMeasuredAperture–Real (f)AverageDVMeasuredAperture–Imaginary
(g)RMSDVMeasuredAperture–Real (h)RMSDVMeasuredAperture–Imaginary
Fig.1. Theaveragerealandimaginarycomponentsofthe12mDA-andDV-typeapertures,picturedwiththeRMSvariationbetweenapertures,
normalizedbythemaximumrealresponse.Whiletheseaperturesarebasedontruemeasurementsoftheapertureillumination,theyneglectthe
effectsofsupportlegdiffraction.
5
in 3D space, the Stokes parameters, the rest frequency hetereogeneous quality of the array [2].
of the array, and its frequency resolution [8]. In this
simulation, the minimum baseline between antennas is Byconvolvingtheperturbedapertures,baselineaperture
approximately 15 m and the maximum is approximately functions are calculated for each pair of antennas, the
800 m. Fourier transform of which gives the primary beam of
that baseline. For each baseline, the real component of
In choosing the perturbation effects to study, two sets of this primary beam is multiplied by the true sky image.
simulationswereperformed.Thefirstwasapreliminary, Visibilities were calculated from the perturbed data to
exploratory version and a variety of apertures, both sim- produceasimulateddatasetusingEq.(4).Thestandard
ulated and measured, with handmade perturbations that CASA2 MS-CLEAN task is then run on these images
included blind and corrected pointing offsets, ellipticity, of the “observed sky”. The off-source rms-level of the
noise on the aperture, parallactic angle rotation, and the image is saved to disk. Image fidelity is also tested by
uncorrected combination of the DA/DV antenna types finding the amplitude of the cleaned source divided by
(which have a 45◦ offset of the support legs). Several the normalized amplitude value of the CASA model
combinationsoftheaboveeffectswerealsotested,inor- primary beam at that point. This test was done to see
dertobetterunderstandhowtheperturbationscompound how closely the primary beams need to match in order
with each other. These tests concluded that pointing to get desired fidelity in image reconstruction.
offsetsandilluminationoffsetswerethedominantsource
of imaging errors. A more thorough description of the Three main tests were run, a 1 Jy point source pointed
layout of this version and its results can be found in slightlyoff-center,asmallextendedsource(toemulatea
Appendix A. protoplanetarydiskorsmallcloud),andalargeextended
source to fill the whole primary beam, as can be seen in
The second version of the simulation (whose results are Fig.2.Thesourceinthepointsourcetestwaslocatedat
the focus of this paper) used solely measured apertures approximatethe75%powerpointinthemainlobeofan
similartotheaverageonesshowninFig.1,andfocused unperturbed primary beam. The small extended source
onlyonthestrongestperturbationeffectsthatwerefound was centered at approximately the 60% power level of
in the previous round of testing. The pointing offset the main lobe of the unperturbed primary beam. The
case used identical apertures modified only by the offset large extended source was centered.3 The simulated uv-
itself. The illumination offset case included a whole set coverageandantennaplacementcanbeseeninFigure3.
of different apertures to test the effect of the variations
depictedinFig.1.Parallacticanglerotationwasignored
because it was found to be a relatively small effect III. RESULTS
similar to combining the DA and DV antennas for this
short time range as can be seen in Table I, and because
Table II shows that the most prominent primary beam
it is very computationally expensive to run. Polarization
effects in this study were those involving pointing off-
was also largely ignored for the sake of time.
sets.Topreventtheintroductionofpointingoffseterrors,
ALMA creates pointing models for each antenna in the
Neglecting to correct for heterogeneous antenna com-
array. This is done in two ways. On a weekly basis, a
binations, such as using the ALMA Compact Array
shortpointingrunismade,conductedwith30-50sources
(ACA) with the 12-m array elements or varying designs
spread evenly between 20-85◦ elevation. This enables
of the 12-m antennas, was also chosen as a subject
the array to “blindly” find any source in the sky with
for investigation in both versions of the simulation.
an accuracy between 2-4” [10]. Beyond that, in order
We have elected to include these results in order to
both validate the results of our simulation, particularly
2CommonAstronomySoftwareApplication
in reference to previous work done on the CARMA 3The small sources were placed off-center in order to observe the
instrument, and to understand and probe the relative effects of primary beam corrections. A point source or small source
differences between errors due to various effects within that is perfectly or nearly perfectly centered in the beam would lead
toonlyverysmallcorrectionsfromtheprimarybeam,asitwouldbe
the simulation framework and improper handling of the
locatedatnearlymaximumpower.Primarybeamcorrectionerrorsare
generatedinthelower-powerregionsandsidelobesoftheimage.
6
(a)PointSource
(a)TestAntennaLocations
(b)SmallExtendedSource
(b)TestUV-Coverage
Fig.3. Thelocationsofthe34antennaarrayusedinallsimulations
intheexperiment(Fig.3a),andtheuv-coveragegeneratedfromeach
observation (Fig. 3b). A hole in the center of the uv-field is clearly
seen,asthecompactarraygeometrywasnotincludedinthesimulation.
(c)LargeExtendedSource
Fig.2. Thesimulatedsourcesusedforeachtest.Thesearethepoint
source (Fig. 2a), the small extended source (Fig. 2b), and the large
extended source (Fig. 2c). The smaller sources are offset from the
centeroftheimageinordertointroduceprimarybeameffectstothe
image, as a small centered source will not experience primary beam
related perturbations. Also shown in each image are contours of the
main lobeof theprimary beam,drawn at80%, 60%, 40%,and 20%
power. The innermost contour indicates the largest image scale that
thesimulationissensitiveto,assetbytheminimumuv-spacing.
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offsetsfora100GHzobservation,dynamicrangeswere
foundtobeapproximately4750forthepointsourcecase
and 3850 for a small extended source, or approximately
21−25% of the blind pointing offset case.
Illuminationoffsets,whichoccurwhentheopticalaccess
on the cryostat offsets the collected light slightly from
thereceiver,fellshortlybelowblindpointingoffsetsand
slightly above the corrected pointing offset case, with
a dynamic range near 3500 in the point source case
and 2150 for the small extended source, or between
approximately29−47%thestrengthoftheblindpointing
offset case.
Results from the small extended source case indicate
generallythesametrendinthehierarchyofperturbation
Fig.4. Thepoint-spreadfunction(PSF)ofthesynthesizedbeamused
inalltests,overlaidwithcontoursofthemainlobeoftheprimarybeam of data, though at slightly higher noise levels in all
at80%,60%,40%,and20%power.Testswererunwithanarrayof cases.Resultsfromthelargeextendedsourcearealmost
34antennas,observing40snapshotsoverthecourseof2hours.
entirely flat, indicating that deconvolution errors are
dominating over the problems in the primary beam.
to ensure that an observed source can be held stable However, the largest error came from changing the
at the center of the beam, each scientific observation is antennasizes,withdynamicrangesfromthesourcepeak
preceded by an offset pointing which enables the array tothenoisefloorofaround300inthesinglepointsource
to track any source to within a Gaussian RMS of 0.6” testand175inthesmallextendedsourcetest.Thiskind
of its true position [10]. We refer to this case as a of data error would be generated by using the ALMA
“corrected pointing offset”. While these seem like very 7m and 12m arrays in conjunction with each other, with
high accuracies, these numbers are a reflection of tele- the small baselines of the 7m array filling in the center
scope geometry, which makes them frequency invariant. of the uv-coverage plane, without correcting for the use
Given the primary beam’s scaling with frequency, this of three different combinations of antenna pairs leading
means that the imaging effect of pointing errors will get to three different types of primary beams. This matched
progressively worse at higher frequencies – a corrected the expectations set up by previous work performed by
pointingoffsetat400GHz(orthemiddleoftherangeof Stuartt Corder on the CARMA instrument, which found
ALMA’s observing capabilities) could be just as bad as that even an uncorrected beam size differential of 3%
a blind pointing at 100 GHz, the lowest band simulated wouldresultinreductioninimagefidelitybyafactorof
here. As such, the numbers presented for blind pointing two [2].
offsets at 100 GHz can also be considered a test of
Asetofpreliminaryteststookplacein2013todetermine
corrected pointing offsets at 400-800 GHz, which is a
which perturbations merited more thorough investiga-
relevant case to a typical ALMA observer.
tion.Thesetestsinvestigatedamuchbroaderselectionof
Simulated observations took place at 100 GHz, giving perturbation effects with less precision - the simulation
the primary beam a FWHM of approximately 45 arc- used an array of only 10 antennas and 4 time steps
seconds. At worst, the blind pointing offsets could shift over a simulated two-hour observation. Tests were run
the beam by almost 10% of its width. Tests for this on point source observations, with one point source and
case gave dynamic ranges of approximately 1000 for onemulti-sourcetestperperturbationeffect.Theinverse
both the point source and small extended source cases. dynamic ranges for these tests are given in Table I.
Numbers were also given for ALMA’s ability to self-
RMS-levels for the 34 antenna test case can be found
correct for pointing offsets, which have a maximum of
in Table II, which we have defined as being the noise
0.5arcseconds[7].Inthebestcaseofcorrectedpointing
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floorofthesimulatedobservation.Asthepeakamplitude will be to the high frequency demands of ALMA. If
wasnormalizedto1.0Jy,thesenumbersalsoindicatethe a method could be devised to correct pointing offsets
inverse dynamic range in each case. at the time of observation to the 0.01 – 0.05 arcsecond
level,eventhehighestfrequencieswouldretaindynamic
ranges of 10,000 or higher.
IV. CONCLUSIONS
With all other contributors having much smaller effects
on the residuals in standard imaging procedures, effort
Initialresultsshowthatunderstandingtheprimarybeam
should be directed towards correcting for these larger
and how it affects observations is critical to the science
effectsbeforetheystartseriouslyimpedingthescientific
being done at these instruments. The strongest effect in
goals of the ALMA instrument. Efforts should also be
the simulation was found to be blind pointing offsets,
made to improve deconvolution algorithms, in order to
with dynamic range from source peak to noise floor
improve imaging of extended objects to the point that
limited to less than 1000 in the test case of a single
primary beam perturbations can begin to affect the data.
2 hour observation of a point source. The next largest
effects were illumination offsets and corrected pointing Finally, the lack of proper calibration and clean-up of
offsets,withdynamicrangeslimitedtolessthan5000in mixed-array data was found to be a disastrous source of
thebestcases.Preliminarytestingdonein2013indicated excess image noise, limiting the image dynamic range
that other varieties of primary beam perturbations, e.g. to less than 300 in the best case of an unresolved
beam rotation, ellipticity, combination of the three kinds point source over one 2-hour observation. This was
of 12m antennas, were relatively minor effects, with the expected result which affirmed the accuracy of our
dynamicrangesatthe10,000levelorhigherforasingle ALMA model relative to previous similar tests done at
point source observation. CARMA [2]. The 7m and 12m arrays are designed to
be used together, with the 7m array filling in the uv-
The first effect to be concerned about would be the case
coverageholethatcouldn’tbefilledwiththelarger12m
of blind pointing offsets and high frequency corrected
dishes.Ifthealgorithmsandsoftwareusedforfull-beam
pointing offsets. At low frequencies, ensuring that all
imagingdonotaccountforthevalidphysicaldifferences
observations are done with use of pointing correction
in the apertures (and hence their primary beams), our
would certainly help. However, observations above 400
testing indicates a starkly limited dynamic range on
GHzareliabletoseethesesamelimitsevenwiththehelp
observations. This kind of algorithm is already available
of the pointing guides, as pointing offsets have an effect
in the ALMA Science Pipeline, and therefore shouldn’t
thatisrelativetothesizeofthebeam.Asfrequencygoes
affect further imaging efforts.
up and the beam size shrinks, even a corrected pointing
offset could have severe consequences on image quality.
Previous work from the MMA as described in ALMA A. Notes for ALMA Users
Memo #95 could potentially help to mitigate the effects
of pointing errors, though only in the case of pointing These tests aimed to quantify the dynamic ranges at
errorsthatareknown[11]. Sincemostcasesofpointing which different errors begin to affect the quality of
offsets are random and hence difficult to characterize, ALMA images when left ignored during the imaging
this algorithm will likely do little to mitigate the effects process. This information is useful to decide when one
of pointing offsets. However, an algorithm such as the needs to invoke computationally expensive algorithms,
Pointing SelfCal algorithm, which attempts to solve for such as A-Projection, when simpler methods and algo-
the unknown pointing errors and incorporate those into rithms will clean data without sacrificing image quality.
the aperture illumination models to be used in the A-
Projection algorithm during imaging, could potentially TakingthefullcomplexityofALMAbeamsintoaccount
drasticallyreducethemagnitudeoftheerrorspropagated via A-Projection is computationally expensive because
into final images through pointing offsets [5]. However, with N different apertures, there are N(N-1)/2 separate
thisalgorithmhasonlybeentestedforimageswithmuch primary beams (one per unique baseline), each of which
lower frequencies, so it is unclear how beneficial they hastobeprecomputedandcachedatdifferentparallactic
9
(a)NoPerturbation (b)IlluminationOffset
(c)CorrectedPointingOffsets (d)BlindPointingOffsets
(e)AntennaSizeDifference (f)AllEffects
Fig.5. Theseimagesshowtheprogressioninoverallnoiseasrelatedtotheperturbingeffects,correspondingtosuccessiverowsinTable II.
Allimagesweresaturatedtothesamelevelinordertomakethebackgroundnoisevisible.
10
angles.Takingtheexampleofthefinal,fullALMAarray byAA† isanextremelydangerouscomputationtomake,
(with 66 antennas) and sky rotations calculated at 5◦ as it has the potential to fill the u-v plane with divide-
increments, over 25,000 separate convolution functions by-zero errors and create an entirely erroneous image.
would have to calculated over a 4 hour observation. Instead, the Fourier transform relationship between the
Computing one baseline beam and rotating it incurs apertureandtheprimarybeamisusedtorewrite(7)and
errors,soeachonemustbecomputedseparately.Theuv- (8) as
resolution at which this must be calculated also depends
on the particular setup of an individual imaging run. All
in all, it is clear that it is of immense practical value to V =A†∗V (9)
cor obs
understand precisely when this full detail is needed and
when various approximations will suffice. B2 (cid:10)F AA† (10)
Our results provide the following guidelines: for dy-
namic ranges of less than 1000, the accuracy of the B2 is later factored out of the final image to recover
primary beam model (beyond physical aperture size) Itrue, minimizing the divide-by-zero errors while main-
simplydoesnotmatter.Atdynamicrangesbeyond1000, taining the integrity of the A-projection relationship.
pointing offsets will begin to affect final image quality, This method allows for a great deal of instrumental
followed by changes in beam shape due to illumination image correction prior to actually entering the image
offsets beyond dynamic ranges of 5000. Finally, paral- domain, assuming an accurate model of the aperture
lactic angle rotation and the beam differences between functions A is used.
theDA/DVantennaswillbecomeappreciableatdynamic
Another subject for further exploration would be the
ranges beyond 10,000.
utilization of the full complex beam in imaging in order
to better understand the propagation of primary beam
errors through the Stokes polarization planes, such as
V. FUTUREWORK
the effect of beam squint, which would appear in the
Stokes Q plane in ALMA data. While our simulation
FuturetestscouldinvestigatethebenefitsofA-projection
used the full complex apertures to construct complex
inimagecorrection[3].A-projectionusesamodelofthe
beams, only the real component of the primary beam
apertureintheinitialstepsofimaginganddeconvolution
was used in imaging as we were focusing solely on the
in order to remove its effects from the final image. The
Stokes I plane. Furthermore, the complicated complex
idea goes that the baseline apertures A have contributed
structure of the aperture functions and their significant
various effects into the visibilities as they are observed,
antenna-to-antenna variations (as seen in the real and
as described in (1). In order to remove those effects, the
imaginary RMS apertures presented in Fig. 1) will lead
observed visibilities must be convolved with the inverse
of the baseline aperture function A−1, as seen in (7). toresidualphasestructureinthecomplexprimarybeam
which will affect Stokes I imaging as well.
The baseline aperture and its inverse will then form a
unitymatrix,returningthetruevisibilities,whichcanbe
Since the computational cost of A-Projection increases
Fourier transformed into the true image.
significantly if antenna-to-antenna variation is included,
it is of interest to determine tolerance levels at which,
V =A−1∗V (7) if correctable in the hardware, antenna-to-antenna varia-
true obs
tionsmaybeignoreduptosomehighdynamicrange.In
A† the results presented in this study, the effect of aperture
A−1 = (8)
AA† illumination offsets on the real part of the primary
beam is already the most significant factor beyond un-
Thisprocesstakesplaceduringtheregriddingofthevis- corrected pointing offsets and neglecting to correct for
ibilities,sothatA−1 replacestheweightingfunctionthat heterogeneous array elements. The effects of aperture
evenly samples the u-v plane. However, AA† may con- illumination are likely to become more significant when
tain zero valued components. As such, formal division the residual phase structures in the beams are also