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Real-Time Calibration of the Murchison Widefield Array

D. A. Mitchell, L. J. Greenhill, R. B. Wayth, R. J. Sault, C. J. Lonsdale, R. J. Cappallo, M. F. Morales, S. M. Ord

arXiv:0807.1912v2astro-ph

TL;DR

The MWA’s crowded, polarized low-frequency sky and direction-dependent effects make calibration difficult, while its data stream is impractical to store for offline processing. The paper develops a real-time peeling approach and reports that it works in simulated crowded fields with strong galactic emission and ionospheric refraction.

  • Problem

    Direction-dependent antenna responses, ionospheric shifts, and a crowded sky complicate MWA calibration, while the data stream is impractical to store for offline processing.

    Source: §I · pp. 1–3

  • Method

    The calibrator loop peels strong catalogue sources by fitting ionospheric phase shifts and direction-dependent antenna gains, then subtracting their modeled contributions.

    Source: §I · p. 1, C. Instrumental Gain Measurements · p. 6

  • Results

    Simulated visibility data show that peeling works with crowded fields, strong galactic emission, and ionospheric refraction, and that it converges quickly.

    Source: §VI · p. 10

  • Takeaways & Limitations

    The approach provides a general way to measure strong point sources for calibration in wide-field, low-frequency arrays and has been adopted for the MWA.

    Source: §VI · p. 10

Abstract

from arXiv · show

The interferometric technique known as peeling addresses many of the challenges faced when observing with low-frequency radio arrays, and is a promising tool for the associated calibration systems. We investigate a real-time peeling implementation for next-generation radio interferometers such as the Murchison Widefield Array (MWA). The MWA is being built in Australia and will observe the radio sky between 80 and 300 MHz. The data rate produced by the correlator is just over 19 GB/s (a few Peta-Bytes/day). It is impractical to store data generated at this rate, and software is currently being developed to calibrate and form images in real time. The software will run on-site on a high-throughput real-time computing cluster at several tera-flops, and a complete cycle of calibration and imaging will be completed every 8 seconds. Various properties of the implementation are investigated using simulated data. The algorithm is seen to work in the presence of strong galactic emission and with various ionospheric conditions. It is also shown to scale well as the number of antennas increases, which is essential for many upcoming instuments. Lessons from MWA pipeline development and processing of simulated data may be applied to future low-frequency fixed dipole arrays.

I. INTRODUCTION

The MWA’s dense, polarized low-frequency sky and direction-dependent instrumental and ionospheric effects complicate calibration, while its data rate makes offline storage impractical. The paper motivates real-time calibration that fits source positions and antenna gains to support subsequent imaging.

I. INTRODUCTION

  • The MWA is an 80–300 MHz synthesis array planned with 512 antennas, and its visibilities encode spatial cross-correlations used to form sky images.The paper notes that MWA visibilities are approximately coplanar, allowing a two-dimensional Fourier relationship even for widefield images.Source: §I · p. 1
  • The polarized galactic signal is at least a few orders of magnitude brighter than the unpolarized EoR signal, motivating fully polarized calibration.The galactic synchrotron component varies with position and frequency.Source: §I · p. 2
  • The 19 GB/s data stream is impractical to store, so the paper proposes fitting ionospheric shifts and antenna gains toward bright catalogue sources to support imaging.The fitted measurements also aid deconvolution and resampling.Source: §I · pp. 1–2
  • Direction-dependent antenna gain and polarization, confusion, sidelobe contamination, and ionospheric refraction complicate calibration across the MWA’s wide field.The field contains compact and extended emission, including diffuse galactic synchrotron; antenna response patterns may change during an observation.Source: §I · pp. 1–2
  • For normal ionospheric conditions, the short maximum antenna separation supports modeling the ionosphere as a two-dimensional phase screen that shifts apparent source positions.The paper limits this discussion to unpolarized-source ionospheric calibration, which cannot calibrate polarization position angles.Source: §I · p. 3

II. THE VISIBILITIES

This section formulates the visibility data using a polarized matrix model and specifies the cadence and averaging used by the MWA calibrator loop. It also states the decorrelation trade-off associated with time and frequency averaging.

II. THE VISIBILITIES

  • For the MWA, the calibrator loop receives visibilities averaged over 8 seconds and 768 channels of 40 kHz, a cadence chosen to oversample ionospheric fluctuations shorter than a minute.The model follows a matrix formulation for estimating each antenna’s polarized response.Source: §II · p. 3
  • A source contribution is modeled through each antenna’s 2 × 2 Jones matrix and the incident signal, with the incident radiation represented by a 2 × 2 polarization covariance matrix.The visibility model assumes a sum of unresolved calibrators with negligible smearing over the averaging intervals, plus additive noise.Source: §II · p. 3
  • At 140 MHz, the fractional bandwidth causes less than 1% decorrelation on the longest baselines for edge-of-map sources, while integration can cause tens of percent there.The paper says longer baselines will be processed at a 2-second cadence to reduce integration-time decorrelation to the percent level; most baselines are short and suffer less.Source: §II · p. 3
  • Without ionospheric effects and with constant Jones matrices during estimation, the baseline visibility matrix is expressed using the expected value of the measured signal.The model also includes thermal and confusion noise in the polarization products.Source: §II · p. 4
  • Ionospheric refraction is represented as a small shift from each source’s expected position, expressed as a corresponding phase-shift perturbation in the visibility model.A prime marks variables disturbed by the ionosphere.Source: §II · p. 4

III. THE CALIBRATOR MEASUREMENT LOOP

The calibrator measurement loop applies peeling to strong catalogue sources, estimating ionospheric refraction and direction-dependent antenna gains before subtracting each source. Its frequency-parallel design coordinates nodes for wideband refraction measurements and source-by-source processing.

III. THE CALIBRATOR MEASUREMENT LOOP

  • The loop tracks a few hundred strong point sources, using catalogue positions and fluxes to measure antenna primary beams and ionospheric refraction in each source direction.Flux bootstrapping is needed because no catalogue covers the full MWA frequency range; initially, only unresolved point sources are used as calibrators.Source: §III · p. 4
  • For each calibrator, the loop fits a model-dependent phase ramp for ionospheric refraction, estimates direction-dependent antenna gains, subtracts the modeled source, and repeats for the remaining strong sources.Further refraction and primary-beam measurements may follow without source subtraction.Source: §III · p. 4
  • Processing is parallelized across frequency channels, with loosely coupled threads sharing information so the full 32 MHz band can constrain ionospheric phase measurements.The design also allows potential pan-frequency averaging during antenna gain measurements.Source: §III · p. 4
  • Calibrators are ranked by expected received power, then pre-peeling subtracts estimates for all targets so strong sources can be measured before weaker ones.Ranking accounts for frequency-dependent beam shapes and is updated as sources move through antenna beams.Source: §III · p. 4
  • Each node rotates and averages visibilities, contributes intermediate sums to the pan-frequency refraction fit, then estimates antenna gains and subtracts a source if fit tests pass.The process loops over calibrators; low-bandwidth communication gathers node sums and returns fitted coefficients.Source: §III · pp. 4–5

A. Rotate Visibilities and Sum over Frequency

For each calibrator, the pipeline phase-centers the peeled visibility matrices and averages them across a frequency sub-band. The average accounts for baseline-dependent multiplicative factors and retains noise and residual sidelobes.

A. Rotate Visibilities and Sum over Frequency

B. Ionospheric Refraction Measurements

The ionospheric-refraction measurements use averaged, peeled calibrator visibilities to fit apparent source offsets. The approach combines frequency handling and a small-offset approximation, with short baselines used initially when larger deviations could invalidate the expansion.

B. Ionospheric Refraction Measurements

  • The four polarization products are reduced to Stokes I for fitting, with catalogued calibrator flux variations divided out across the observing band.The visibilities are treated as dominated by a single direction.Source: B. Ionospheric Refraction Measurements · p. 6
  • Jones matrices are assumed constant over ionospheric phase timescales of tens of seconds, and recent gain-calibrated visibilities provide a sky-visibility estimate.The instrument response is expected to vary in time on minute scales.Source: B. Ionospheric Refraction Measurements · p. 6
  • The ionosphere is modeled as apparent calibrator offsets α_cλ_0^2 and β_cλ_0^2 in the l and m directions.These offsets represent an apparent phase shift after the visibilities are phased toward the source.Source: B. Ionospheric Refraction Measurements · p. 6
  • The offset expansion requires (α_cu_jk,f0 + β_cv_jk,f0)λ_0^2 ≪ 1; short baselines are used initially because it breaks down for arcminute deviations on long baselines.The authors anticipate several-arcsecond-or-smaller RMS displacements over 8-second periods, then track the short-period deviations.Source: B. Ionospheric Refraction Measurements · p. 6
  • Simulated visibility sums appear close to normally distributed, and partial sums can be computed per node before gathering them to obtain α_c and β_c.The least-squares solutions estimate calibrator intensity and the two offset parameters.Source: B. Ionospheric Refraction Measurements · p. 6

C. Instrumental Gain Measurements

The instrumental-gain procedure estimates antenna responses toward each calibrator using matrix least squares. It revises an initially unstable formulation, provides an analytic per-antenna solution, and can incorporate known response information.

C. Instrumental Gain Measurements

  • Antenna gains toward each calibrator are estimated by fitting matrices to the visibility data.The method follows a matrix form and uses the visibilities measured with an antenna to constrain its gain model.Source: C. Instrumental Gain Measurements · p. 6
  • A revised matrix least-squares formulation has more robust solutions than the earlier formulation when sidelobes from other sources are significant.The earlier solutions were unstable under those conditions.Source: C. Instrumental Gain Measurements · p. 6
  • Equation (10) has an analytic solution for each antenna.Source: C. Instrumental Gain Measurements · p. 6
  • Once the matrices converge, they can be updated for each calibrator or restricted to change more slowly to track gradual gain changes.Source: C. Instrumental Gain Measurements · p. 7
  • Known information in the Jones matrices can be absorbed into other matrices, leaving only unknown quantities to fit.The paper gives the sky-to-instrumental polarization transformation as an example of information that may be known accurately.Source: C. Instrumental Gain Measurements · p. 7

D. Source Subtraction

The expected large calibrator power range motivates peeling sources sequentially, from stronger to weaker. The paper estimates a finite peeling depth and notes that subtracting all of each source may add noise back into the visibilities.

D. Source Subtraction

  • A sidelobe noise floor of several hundred mJy is the preliminary estimated peeling limit, leaving remaining point sources below 5 sigma and suggesting a few hundred available calibrators.The large expected power range among the strongest calibrators motivates subtracting stronger sources before measuring weaker ones.Source: D. Source Subtraction · p. 7
  • Before subtraction, source models must be multiplied by the complex η_jk,c,f factors that were divided out during frequency averaging.Source: D. Source Subtraction · p. 7
  • Subtracting 100% of each source may be suboptimal because measurement noise is added back into the visibilities.The authors are considering CLEAN-like modifications for testing.Source: D. Source Subtraction · p. 7

Performance

The calibration loop scales quadratically with antenna count, but its full-array workload remains substantial within each 8-second cycle. The operation estimates are approximate and depend on how many sources each step processes.

Performance

  • Most compared gain-calibration techniques scale as N_a^3, while LOGLS is the least-computational alternative described and scales as N_a^2.The comparison assumes a hypothetical dual-polarization generalization of LOGLS.Source: Performance · p. 7
  • The CML uses O(24N_a^2) complex multiplications plus O(12N_a^2) for weighting, described as about as efficient as reasonably expected.The comparison estimates LOGLS at 2(2N_a)^2 multiplications for two polarized receptors, possibly doubled to include same-antenna receptor correlations.Source: Performance · p. 7
  • The CML processes sources through pre-peeling, unpeeling, visibility rotation, ionospheric-offset and gain solving, and peeling, with different steps potentially handling different source counts.Table I estimates floating-point operations per source for 512 antenna tiles and one frequency channel; the estimates are approximate.Source: Performance · p. 7, Table I · p. 7
  • A few trillion floating-point operations are anticipated for the full-array CML over each 8-second cycle, before memory-access overheads add a factor of a few.The estimate scales selected operations by 768 frequency channels or about 50 sub-bands, as applicable, and by the number of sources.Source: Performance · p. 7

Options for Reducing the Load

The paper considers reducing real-time processing by limiting visibility data and cycling through calibrators, while retaining measurements across the sky. It also describes a calibration-only option when peeling residuals could compromise EoR analysis.

Options for Reducing the Load

  • Initial investigations suggest ignoring more than 99% of short baselines for stronger-source ionospheric measurements, since long baselines offer higher angular resolution and see less extended Galactic structure.Each offset measurement can otherwise use data from every baseline, polarization, and frequency, amounting to over 1.5 gigasamples.Source: Options for Reducing the Load · p. 8
  • Measure the strongest sources every 8 seconds and cycle through subsets of others to retain sky-distributed gain measurements every few minutes while peeling only 20 or 30 sources at a time.This schedule exploits the slow variation of tile gains.Source: Options for Reducing the Load · p. 8
  • The authors propose fitting calibrator sources simultaneously and potentially reducing degrees of freedom with slowly varying dipole gains and phases, but present direct dipole fitting as future work.After field measurements of tile beams, source subtraction would rely on dipole fits rather than direct tile-gain measurements.Source: Options for Reducing the Load · p. 8

Foreground Subtraction

  • If peeling residuals might mimic the EoR signal, peeling can be used for calibration only and images formed from unpeeled visibilities.Otherwise, residuals from real-time peeling can be averaged into the resulting images; calibration data are stored for offline assessment and reduction of peeling errors.Source: Foreground Subtraction · p. 8
  • Offline foreground-removal approaches include exploiting spectral differences and rotation-measure synthesis, while direction-dependent deconvolution is also expected to be essential for imaging.The passage cites examples,,, and for these approaches.Source: Foreground Subtraction · p. 8

Radio Frequency Interference

The paper describes flagging contaminated time and frequency samples before imaging, then evaluates the calibration loop using simulated MWA visibilities. The examples show source peeling revealing diffuse foreground emission, while convergence and limits depend on the sources and simulated conditions.

Radio Frequency Interference

  • Contaminated data are flagged and ignored before imaging, with missing time or frequency samples accounted for in least-squares weighting.The stated RFI sources include satellite signals, reflected FM broadcasts, and lightning.Source: Radio Frequency Interference · p. 8

V. EXAMPLES

  • The simulations use MAPS with diffuse and point-source sky components, modeled ionospheric structure, and tile-receptor gain variations, but use unpolarized input.The modeled turbulent ionospheric power is about 2%; traveling ionospheric disturbances were not included in these simulations, and polarized-input testing was incomplete.Source: §V · p. 8
  • As 0, 10, and then 100 sources are peeled, weaker sources and diffuse Galactic foreground become apparent until the Galactic center and its sidelobes dominate.The displayed images cover 45° × 45°, but sources are peeled from anywhere in the sky according to apparent strength.Source: §V · p. 9, Figure 3 · p. 9
  • For the strongest calibrator, gain-error convergence proceeds at the same rate in full simulations and point-source-only simulations, while image RMS becomes galaxy-dominated after the first few strong sources are peeled.The gain-error comparison includes galactic emission and ionospheric effects in the full simulations; short-baseline visibilities were downweighted.Source: §V · p. 9, Figure 4 · p. 9
  • Mean gain errors for the five strongest sources decrease over iterations, with strong sources updated quickly and weaker ones more slowly, before convergence is limited by a local minimum.The strongest source could update by 50% per iteration; some weak sources were updated by only a few percent, and the common relative-error level was not investigated.Source: §V · p. 9, Figure 5 · p. 10
  • Typically a few tens of calibrators converge well, but some weaker sources can be significantly affected by sidelobes; several mitigation options remain under investigation.These include changing calibration bandwidth, using large pre-peels, and incorporating all-sky ionospheric and primary-beam models.Source: §V · pp. 9–10

VI. SUMMARY

The paper presents a strong-source measurement approach adopted for MWA calibration and reports that peeling works in simulations of crowded fields, strong galactic emission, and ionospheric refraction. Further convergence and tolerance testing remains necessary, including under severe conditions and possible source variability or dipole mutual coupling.

VI. SUMMARY

  • The paper describes a general approach for measuring strong point sources in wide-field, low-frequency array calibration.The approach has been adopted for the MWA, where software development and assessment of its benefits and limitations are ongoing.Source: §VI · p. 10
  • Simulated peeling works in crowded fields, strong galactic emission, and ionospheric refraction, with fast convergence and computationally efficient components.Some parts of the system also offer significant optimization potential.Source: §VI · p. 10
  • Detailed convergence analysis and tolerance tests remain the next steps for assessing algorithm behavior under expected observing conditions.Tests are to include weak emission masked by polarized diffuse foregrounds, high-dynamic-range observations near the Sun, and severe ionospheric conditions.Source: §VI · p. 10
  • Initial antenna data are expected to clarify how harmful source variability from ionospheric scintillation and dipole mutual coupling may be.These effects can then be incorporated into tolerance tests.Source: §VI · p. 10
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