A Measurement of Source Noise at Low Frequency: Implications for Modern Interferometers
PPublications of the Astronomical Society of Australia (PASA)doi: 10.1017/pas.2021.xxx.
A Measurement of Source Noise at Low Frequency:Implications for Modern Interferometers
J .S. Morgan, R. Ekers, , International Center for Radio Astronomy Research, Curtin University, GPO Box U1987, Perth, WA 6845, Australia CSIRO Astronomy and Space Science (CASS), P.O. Box 76, Epping, NSW 1710, Australia
Abstract
We report on the detection of source noise in the time domain at 162 MHz with the Murchison WidefieldArray. During the observation the flux of our target source Virgo A (M87) contributes only ∼
1% to thetotal power detected by any single antenna, thus this source noise detection is made in an intermediateregime, where the source flux detected by the entire array is comparable with the noise from a singleantenna. The magnitude of source noise detected is precisely in line with predictions. We consider theimplications of source noise in this moderately strong regime on observations with current and futureinstruments.
Keywords:
Techniques: interferometric – Instrumentation: interferometers – Radio continuum: general –Radio lines: general – Radiation mechanisms: general
Source noise (also known as self noise, wave noise orHanbury Brown Twiss noise) arises from the fact thatmost sources studied in radio astronomy are themselvesintrinsically noise-like: i.e. stochastic, ergodic, Gaussianrandom noise (Radhakrishnan, 1999; Thompson et al.,2017, 1.2). Since these natural sources are typically veryweak relative to other sources of noise (referred to hereand elsewhere as “system noise”) this contribution canalmost always be neglected. However, as telescopes arebuilt with increasing sensitivity we are more likely toapproach or even reach the strong source limit. In thislimit (i.e. where the signal strength dominates over othertypes of noise such that generated in the amplifier) thenoise becomes proportional to the signal strength andso the signal to noise ratio becomes independent of thesensitivity of the instrument: the only way to increase thesignal to noise is to obtain further independent samplesby increasing either the bandwidth or observing time.Larger collecting area or more antennas no longer help.For a single dish, source noise is straightforwardlyunderstood as a contribution to the total noise when thesource is in the field of view. For interferometry the inter-pretation is a little less straightforward and source noiseis better understood as distinct from other noise withdistinctive properties. This is because unlike other typesof noise, each interferometer element receives the same source noise from an unresolved source (i.e. the source noise is correlated between antennas). It is this corre-lation which is exploited in an intensity interferometer(Hanbury Brown & Twiss, 1954).This has two important consequences: first, just likethe single-dish case, source noise is only seen at the loca-tion of the source in either an aperture synthesis imageor a phased-array beam. More accurately, in a synthesisimage, the noise is distributed with the shape of theinstantaneous Point Spread Function (PSF). Secondly,for an interferometer with a large number of elements,source noise may contribute significantly to the imagenoise at the location of the source even if it is dwarfed bythe system noise of a single interferometer element, sinceunlike noise which is uncorrelated between elements, thenoise cannot be reduced by averaging over baselines .This intermediate regime, where the source is weakerthan the noise, but is not weak enough to be negligible,is the subject of this paper. We present a clear detectionof source noise (arguably the first in the intermediateregime) in data from the Murchison Widefield Array(MWA; a 128-element low-frequency interferometer; Tin-gay et al., 2013). We then examine the scenarios in whichthis may have a negative impact.The paper is organised as follows: in Sect. 2 we reviewthe literature on source noise in interferometry and use Note that coherence across baselines, a concept fundamentalto interferometry, should not be confused with temporal coherence.One does not imply the other a r X i v : . [ a s t r o - ph . I M ] J a n J. S. Morgan et al. the formalism obtained to design an experiment to detectsource noise with the MWA; in Sect. 3 we describe theobservations and analyse the results; in Sect. 4 we discussthe implications of our results.
Below, we draw on the literature to describe the proper-ties of source noise. We refer the reader to other articleswhich discuss the effect of source noise in other specificcases such as pulsar scintillometry (Johnson & Gwinn,2013), spectral line observations (Liszt, 2002), pulsartiming (Osłowski et al., 2011), polarimetry (Deshpande,1994) or spectropolarimetry (Sault, 2012). The effectof scattering on source noise is discussed in Codona &Frehlich (1987).Measurements of the statistics of source noise on veryshort time scales have also been used to try to detectcoherent emission from pulsars (Smits et al., 2003, andreferences therein). In the current work we focus on thetypical synthesis imaging case where the a large numberof Nyquist samples are averaged in the correlator, andthe integration time is far longer than any timescaleon which source noise could conceivably be correlated.In this case the central limit theorem will ensure thatsource noise is highly Gaussian. We return briefly to thistopic in Sect. 4.3.Both sources and noise can be described either with acharacteristic temperature, or as a flux density. We followKulkarni (1989) and McCullough (1993) and use thesource flux density S and Noise Equivalent Flux Density N throughout. The latter, more commonly known asSystem Equivalent Flux Density (SEFD) is the fluxdensity of a source that would double the power receivedby a single element. Thus, at the output of a singleelement there is the sum of a power proportional to S and power proportional to N , where the constant ofproportionality is the product of the collecting area ofthe antenna, the bandwidth of the receiver, and theantenna gain (and various efficiency parameters). Notethat N includes not only receiver noise but any noiseother than S . This is justified below.Throughout, we assume that all interferometer ele-ments are identical, and that both N and S are station-ary, stochastic and ergodic and so 2 Bτ samples of base-band data are independent. For an unpolarised source,the number of independent samples can be doubled byobserving two orthogonal polarisations, so an implicit n pol can be assumed alongside B and τ in the equationsbelow. Polarisation of any kind implies correlation ofsome kind between orthogonal polarisations (Radhakr-ishnan, 1999) which will complicate the picture. We referthe reader to the references given above for more detailson the source noise associated with polarised sources orin polarimetric images.In the usual case that S (cid:28) N , the RMS noise in a synthesis map is given by σ = N √ n b Bτ (1)where N is the SEFD in jansky, n b is the number ofbaselines given by n ( n − / n is the number ofantennas (we assume throughout that all cross correla-tions are used), B is the bandwidth in Hz, and τ is theobserving time in seconds.A similar expression can be derived for σ in the casethat one of the antennas is used as a “single dish”. Inthis case σ will be √ √ σ = S √ Bτ , (2)replicating the single-dish result. This is intuitive sinceall antennas see the same noise, and so unlike the weak-source case, the noise cannot be reduced by averagingover baselines.Anantharamaiah et al. (1989) derive the properties ofsource noise in various other regimes to draw a numberof important qualitative conclusions:Firstly, if a strong source is fully resolved on all base-lines, then it will be uncorrelated between antennas andso will behave like system noise. This is relevant forlow-frequency interferometers such as the MWA where N is actually dominated by Galactic synchrotron emis-sion, which is almost completely resolved on all but theshortest baselines, and totally absent from the longer > λ baselines typically used for continuum imaging.This provides the justification for lumping sources suchas synchrotron radiation from our Galaxy, the cosmicmicrowave background, atmospheric noise, etc. in with N ; and provides a more precise definition of S as noisewhich is correlated between antennas.Secondly, the noise only appears in the image wherethe source does; therefore, for a point source, the sourcenoise level in the map is shaped like the instantaneousPSF of the array.Finally, they note that for an interferometer with n elements, the source noise in the image will be compara-ble with the system noise when N ∼ nS . In table 1 welist N/n for a number of interferometers to emphasise
Measurement of Source Noise Table 1
N/n for various radio interferometers, built andplanned. All come from the SKA Baseline Design table 1except the MWA figure which is derived from Tingay et al.(2013). Instrument
N/n
JyMWA 400LOFAR 45.2ASKAP 42.5Meerkat 8.6SKA-low 2.8SKA-mid 1.7 that this “moderately strong source regime” can easilybe reached.Anantharamaiah et al. (1991) derive a very simpleexpression for the noise (system noise and self noise) atany location in the map for a total power image. Wemodify this slightly so that it simplifies to equations 1&2in the appropriate limits: σ = 1 √ Bτ (cid:18) S + N √ n b (cid:19) (3)where S is the apparent source brightness (source bright-ness distribution convolved with the PSF) at any pointin the image.This expression is only approximate for an interferom-etry image because it does not account precisely for howthe noise on different baselines are correlated in this“moderately strong source” regime. A more completetreatment was given by Kulkarni (1989), who consideredthe correlation between each pair of baselines . Threeclasses of baseline pairs were identified: the autocorrela-tions of each baseline, those where the pair of baselineshave an element in common, and those where the pairof baselines is composed of 4 different antennas. Sourcenoise correlates between the baselines differently in eachcase. For a point source at the phase centre, the visibilityon each baseline is identical, and the on-source noise is σ = S + N √ n b Bτ p n − S + [1 + ( n −
1) ( n − S (4)where S = SS + N . (5)Here we use the form of the equation given by McCul-lough (1993) who corrected a minor error in Kulkarni(1989). The three terms reflect the three classes of base-line pair: the autocorrelations will correlate perfectlyregardless of whether S or N dominates and the othertwo classes scale with S and S respectively.Equation 4 reduces to equation 1 or 2 in the appropri-ate limits. It is also well-approximated by equation 3 for SKA-TEL-SKO-DD-001 -2 -1 nS/N -2 -1 D y n a m i c r a n g e / p B τ on sourceoff source Figure 1.
Following Kulkarni (1989) Fig. 2, Maximum achievabledynamic range achievable off-source (dashed line; equation 1) andon-source (solid line; equation 3) for as a function of source strengthfor a large- n interferometer. This is generalised by measuringsource flux density in units of N/n and dynamic range as a fractionof its maximum ( √ Bτ ). large n as predicted by Anantharamaiah et al. (1991)(the fractional error is <
1% for n >
N/n (see tables 1). On- and off-source noisestart to diverge at ∼ N/ n – just a few hundred mJy forthe SKA; and the on-source dynamic range has almostreached its maximum value by ∼ N/n .Table 2 shows the maximum dynamic range obtainablefor various common MWA observing parameters. Fora typical snapshot, the dynamic range will be > before source noise effects become noticeable. This levelof source noise is not easily measured since noise due toconfusion noise or calibration errors are likely to limitthe dynamic range far more.In contrast, if an image is made with the smallestpossible bandwidth and observing time with the cur-rent MWA correlator, the maximum achievable dynamicrange is ∼ ∼
1% acrosstimesteps, spectral channels and polarisations regardlessof any intrinsic change in brightness. However, rapidintrinsic changes in solar emission (which may be po-larised) in both time and frequency would complicatemeasurements of source noise in observations of the Sun.Both spectral line and 0.5 s snapshot imaging will startto show the effects of source noise at dynamic ranges ∼ J. S. Morgan et al.
Table 2 √ Bτ for various MWA observing modes. All band-widths take into account discarded band edges where appro-priate. Maximum resolutions in time and frequency refer tothe original online MWA correlator (still standard at thetime of writing). B τ √ Bτ UseMHz s26.88 120.0 56794 typical snapshot observation13.44 0.5 2592 IPS observing parameters (see § 3)0.01 120.0 1095 maximum spectral resolution0.04 0.5 141 minimum time–bandwidth product number of spectral channels or timesteps the RMS ofpixels on and off source would be expected to show ameasurable difference. The former is the approach takenby McCullough (1993) in measuring source noise in thestrong regime. We take the latter approach which hasmuch in common with the procedure used for makingIPS observations with the MWA (Morgan et al., 2018).
As part of our Interplanetary Scintillation (IPS) ob-serving campaign (Morgan et al., 2019) we observed aseries of calibrators in high time resolution mode with 5-minute observing time. These observations pre-date theupgrade of the MWA to Phase II (Wayth et al., 2018).Virgo A (M87), at our observing frequency of 162 MHzwas observed at 2016-01-16T21:04:55 UTC when it was53 degrees above the horizon and 112 degrees from theSun. The solar elongation is relevant since Virgo A hasa compact core and jet several janskys in brightness atGHz frequencies (e.g. Reid et al., 1982) which wouldbe compact on IPS scales. This means that Virgo Awill also show interplanetary scintillation. However thisvariability should be fully resolved with 2 Hz sampling.Throughout, we assume a flux density of 1032 Jy forVirgo A (based on values gleaned from the literatureby Hurley-Walker et al., 2017). We do not include anyuncertainty on the flux density of Virgo A in our errorsbelow (such an error would cause an equal scaling errorin all of our measured and derived quantities). Note thatno direction-dependent flux scaling was carried out.Each individual 0.5 s integration of the full observationwas imaged separately for each instrumental polarisation(XX&YY) using normal weighting and excluding base-lines shorter than 24- λ (a negligible fraction of the totalnumber of baselines). Throughout this paper we onlyrefer to results derived from the XX images, howeverusing the YY images produces qualitatively identicalresults. Fig. 2 shows the lightcurves for selected pixelsin the image: the pixel corresponding to the brightestpoint in Virgo A and several pixels distributed throughthe image a few resolution units from Virgo A. Addi-tionally we show a pixel corresponding to the brightestpoint in 3C270, a resolved double with a peak appar- ent brightness approximately 3% of that of Virgo A.After standard flagging of the start and end of the ob-servation, 573 timesteps remained. A number of clearlydiscrepant points were easily discernible in all lightcurves(the same in each) and these have been interpolated overfor all lightcurves. The on-source lightcurve shows clearvariability on a timescale consistent with ionosphericscintillation.Fig. 3 shows power spectra for each of the lightcurvesshown in Fig. 2. These were generated using Welch’smethod (Welch, 1967) with 17 overlapping FFTs of 32samples with a Hanning window function. The strongsignal that dominates at low frequencies is consistentwith ionospheric scintillation and shows the characteris-tic Fresnel “Knee” at about 0.1 Hz. Above this frequencythe power drops steeply. The same ionospheric scintil-lation can be seen for all off-source lightcurves due tothe sidelobes of Virgo A, however the sidelobes level islow enough that the variance is suppressed by at least 3orders of magnitude. 3C270 scintillates independentlyof Virgo A and also has a slightly lower noise level .Above 0.4 Hz the variance from ionospheric scintilla-tion is negligible, and the power spectrum consists onlyof white noise. We conclude that all variability due tocalibration errors such as those introduced by the iono-sphere are restricted to lower frequencies as we wouldexpect. This white noise is clearly at very different lev-els on-source and off-source. We can use the average ofall points above 0.4 Hz to measure the off-source noiseto be 0.410 ± ± N ) is125000 ± B = 13 .
44 MHz, τ = 0 . n = 118 i.e.10 antennas flagged). This is about a factor of 2.4 higherthan that given in table 1, which is not surprising giventhe off-zenith pointing and the fact that Tingay et al.(2013) assume a nominal sky temperature and field ofview.Using the source brightness and off-source noise wecan calculate what the on-source noise should be usingequation 4. This prediction (0.807 ± σ ; however even this smallexcess can be explained as a small leakage of source noiseinto the off-source pixels due to the sidelobes of Virgo A.Alternatively, this may be due to Virgo A being slightlyresolved on some MWA baselines.This does not leave any variance due to IPS. IPS isextremely variable on the night-side, and the scintillationindex would only be a few percent. Furthermore, mostof the variability is on timescales longer than 0.4 Hz.IPS may be responsible for the slight increase in power There is no obvious reason for the lower noise level in thevicinity of 3C270, and we believe it is due to some combinationof factors, none of which are consequential enough to affect ouranalysis or conclusions.
Measurement of Source Noise F l u x D e n s i t y / J y b e a m On source (Virgo A)Flagged and interpolatedOff-sourceOn Source (3C270)
Figure 2.
Timeseries (lightcurve) showing brightness of various pixels in the image as a function of time. Black line shows the on-sourcelightcurve; dot-dashed black line shows a much weaker source: the (Western lobe of) 3C270; grey lines show a selection of off-sourcepixels. 21 points in the lightcurve had to be flagged due to clearly discrepant points, and these have been linearly interpolated over in alllightcurves; red circles denote these flagged points for the on-source pixel. around 0.3 Hz.3C270 does not exhibit measurable source noise. Thisis expected since, like the sidelobes of Virgo A, its bright-ness puts it well within the weak regime.
We have detected source noise at a level which is firmly inline with the predictions of equation 3 (no empirical testfor the existence of the subtle effects that differentiateequations 3&4 are possible with our data since the twoare practically identical for n >
Figure 4 plots the ratio of on-source noise to systemnoise where these are calculated using equations 3&1respectively. To understand the effect this has off-source,consider a narrowband snapshot image. In this case,the source noise will have the shape of the snapshot,monochromatic
PSF. The sidelobes of this PSF can becharacterised by their RMS relative to the central peak;for example, the proposed SKA-low configuration has a SKA-SCI-LOW-001 http://indico.skatelescope.org/event/384/attachments/3008/3961/SKA1_Low_Configuration_ sidelobe level of 1% in the instantaneous monochromaticcase. Therefore for S = N/n = 2 . ∼ × the weak-noise level, andoff-source this 100% increase in noise will become just1%.As the bandwidth and integration time are increased,both the system noise and the source noise will reducewith √ Bτ ; the ratio between the two will remain con-stant, and the excess noise due to the source noise willremain at 1%. With appreciable fractional bandwidthand/or Earth rotation synthesis this source noise willbe spread smoothly over the image.For the example given, the effects of source noiseoff-source are extremely low; however there will be 5 × ∼ V4a.pdf
J. S. Morgan et al. freq / Hz10 Sp e c t r a l P o w e r / J y b e a m On source (Virgo A)Off sourceAverage Off sourceOn Source (3C270)System noiseSource noise
Figure 3.
Power spectrum for each of the lightcurves described in Fig. 2. The thin black line shows on-source power spectrum; thedot-dashed line shows a much weaker source (3C270); the grey lines show off-source power spectra, with the thick black line showing theaverage of the grey lines. Power spectrum parameters are described in the text. The single error bar shows the 95% confidence intervalfor a single point with these parameters. The dotted line is the mean of all off-source power spectrum points above 0.4Hz. The dashedline is the estimate of the on-source power based on the on-source brightness and off-source noise.
Measurement of Source Noise -2 -1 nS/N E x c e ss N o i s e Figure 4.
Ratio of on-source to system noise (equations 3&1)as a function of source strength for a large-N interferometer (thisis the ratio of the dashed line to the solid line in Fig. 1). of source noise assumes natural weighting of baselines.More uniform weighting will increase the weighting ofcertain baselines (the longer ones) by a large amountcompared to the shorter spacings, particularly for instru-ments like the SKA-low with very high concentrationsof collecting area at the centre. This will reduce theeffective number of antennas and therefore the sourcenoise relative to the system noise.
Anantharamaiah et al. (1989) note that source noise canbe subtracted perfectly from a snapshot by deconvolu-tion, while noting the computational effort required todo so. Here we explore the limitations on how cleanlythis can be done.Subtracting source noise requires that the field beimaged with sufficient time and frequency resolutionfor changes in the PSF to be insignificant from one im-age to the next. The required resolution will dependon the array and imaging parameters, however the re-quirements are similar to those required to minimisetime-average and bandwidth smearing (e.g. Thompsonet al., 2017, 6.3-6.4) and are likely to be demanding.Here we concentrate on a more fundamental issue thatsuch a subtraction poses: namely that if the source noiseis to be perfectly subtracted, the source brightness mustbe allowed to vary as a function of time and frequencyand no spectral smoothness can be assumed. Converselyif spectral smoothness is strictly imposed, as is normallythe case when subtracting continuum from spectral linecubes, no source noise will be subtracted. Clearly anynumber of compromise schemes between these to ex-tremes could be devised; however the principle remainsthat source noise can only be subtracted to the extentthat it can be separated from any spectral or temporalsignal that needs to be preserved.
The properties of source noise lead us to the strikingconclusions that the SKA will not more accurately char-acterise a 1000 Jy source than a 10 Jy source, and thatthe standard equations underestimate the noise on themeasurement of a 2 Jy source by a factor of two. Wehave demonstrated that even the MWA—which is ordersof magnitude less sensitive than the SKA—can measurethese effects at the 12- σ level (albeit it in an observationof a 1000 Jy source contrived for the purpose).Our observations show that the magnitude of the effectis precisely in line with predictions, and so source noisecan be predicted very precisely from easily measurableparameters (the ratio of source brightness to systemnoise). This is in contrast to other sources of variabilitysuch as the ionosphere. Source noise is also expected tohave stochastic behaviour as a function of frequency andtime, and we exploit this to separate source noise frommuch stronger ionospheric effects.Although the effects on standard continuum imagingappear to be benign in all but the most extreme cases,source noise could become problematic wherever weaksignals (including polarised signals, see Sault, 2012) needto be measured in the presence of strong emission; es-pecially where the signals involved have fine frequencystructure or vary on short timescales. This situationwill occur for any spectral line observations where thecontinuum has to be subtracted to see the much weakerline emission. In this situation the noise will be higherand not uniform across the image. Noise estimates madefrom regions of the image with no continuum will under-estimate the noise.Fast transients such as Fast Radio Bursts (FRBs) andpulsars will also reach the strong source limit, especiallysince their signals occupy only a narrow sloping band ina dynamic spectrum. For ASKAP, a few of the detectedFRBs (Shannon et al., 2018) already have peak fluxdensity in the strong source limit and for SKA sensi-tivity many will be in this regime. Thus measurementsof source noise of FRBs should be possible, and suchmeasurements may be valuable, since they probe theextent to which the emission is coherent (Melrose, 2009). ACKNOWLEDGEMENTS
We would like to thank J-P Macquart, Adrian Sutinjoand Wasim Raja for useful discussions. This scientificwork makes use of the Murchison Radio-astronomy Ob-servatory, operated by CSIRO. We acknowledge theWajarri Yamatji people as the traditional owners of theObservatory site. Support for the operation of the MWAis provided by the Australian Government (NCRIS), un-der a contract to Curtin University administered by As-tronomy Australia Limited. We acknowledge the PawseySupercomputing Centre which is supported by the West-
J. S. Morgan et al. ern Australian and Australian Governments.
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