title: A Synoptic VLBI Technique for Localizing Non-Repeating Fast Radio Bursts with CHIME/FRB
authors: Calvin Leung Juan Mena-Parra Kiyoshi Masui Mohit Bhardwaj P. J. Boyle Charanjot Brar Mathieu Bruneault Tomas Cassanelli Davor Cubranic Jane F. Kaczmarek Victoria Kaspi Tom Landecker Daniele Michilli Nikola Milutinovic Chitrang Patel Andre Renard Pranav Sanghavi Paul Scholz Ingrid H. Stairs Keith Vanderlinde
text: 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  Calvin Leung
  calvinl@mit.edu
  
  0000-0002-4209-7408]Calvin Leung
  MIT Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA
  Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA
  0000-0002-0772-9326]Juan Mena-Parra
  MIT Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA
  0000-0002-4279-6946]Kiyoshi Masui
  MIT Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA
  Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA
  0000-0002-3615-3514]Mohit Bhardwaj
  Department of Physics, McGill University, 3600 rue University, Montréal, QC H3A 2T8, Canada
  0000-0001-8537-9299]P.J. Boyle
  Department of Physics, McGill University, 3600 rue University, Montréal, QC H3A 2T8, Canada
  McGill Space Institute, McGill University, 3550 rue University, Montréal, QC H3A 2A7, Canada
  0000-0002-1800-8233]Charanjot Brar
  Department of Physics, McGill University, 3600 rue University, Montréal, QC H3A 2T8, Canada
  McGill Space Institute, McGill University, 3550 rue University, Montréal, QC H3A 2A7, Canada
  0000-0001-6630-7871]Mathieu Bruneault
  McGill Space Institute, McGill University, 3550 rue University, Montreal, QC H3A 2A7, Canada
  0000-0003-2047-5276]Tomas Cassanelli
  Dunlap Institute for Astronomy & Astrophysics, University of Toronto, 50 St. George Street, Toronto, Ontario, Canada M5S 3H4
  David A Dunlap Department of Astronomy & Astrophysics, 50 St George St, Toronto, Ontario, Canada, M5S 3H4
  0000-0003-2319-9676]Davor Cubranic
  Department of Physics and Astronomy, University of British Columbia, 325 - 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada
  0000-0003-4810-7803]Jane F. Kaczmarek
  National Research Council Canada, Herzberg Astronomy and Astrophysics Research Centre, Dominion Radio Astrophysical Observatory, PO Box 248, Penticton, British Columbia, V2A 6J9 Canada
  0000-0001-9345-0307]Victoria Kaspi
  Department of Physics, McGill University, 3600 rue University, Montréal, QC H3A 2T8, Canada
  McGill Space Institute, McGill University, 3550 rue University, Montréal, QC H3A 2A7, Canada
  
  National Research Council Canada, Herzberg Astronomy and Astrophysics Research Centre, Dominion Radio Astrophysical Observatory, PO Box 248, Penticton, British Columbia, V2A 6J9 Canada
  0000-0002-2551-7554]Daniele Michilli
  Department of Physics, McGill University, 3600 rue University, Montréal, QC H3A 2T8, CanadaMcGill Space Institute, McGill University, 3550 rue University, Montréal, QC H3A 2A7, Canada
  0000-0001-8292-0051]Nikola Milutinovic
  Department of Physics and Astronomy, University of British Columbia, 325 - 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada
  
  Dunlap Institute for Astronomy & Astrophysics, University of Toronto, 50 St. George Street, Toronto, Ontario, Canada M5S 3H4
  Department of Physics, McGill University, 3600 rue University, Montréal, QC H3A 2T8, Canada
  McGill Space Institute, McGill University, 3550 rue University, Montréal, QC H3A 2A7, Canada
  0000-0003-3463-7918]Andre Renard
  Dunlap Institute for Astronomy & Astrophysics, University of Toronto, 50 St. George Street, Toronto, Ontario, Canada M5S 3H4
  0000-0001-5504-229X]Pranav Sanghavi
  CSEE, West Virginia University, Morgantown, WV 26505, USA
  Center for Gravitational Waves and Cosmology, West Virginia University, Morgantown, WV 26505, USA
  0000-0002-7374-7119]Paul Scholz
  Dunlap Institute for Astronomy & Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada
  0000-0001-9784-8670]Ingrid H. Stairs
  Department of Physics and Astronomy, University of British Columbia, 325 - 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada 
  0000-0003-4535-9378]Keith Vanderlinde
  Dunlap Institute for Astronomy & Astrophysics, University of Toronto, 50 St. George Street, Toronto, Ontario, Canada M5S 3H4
  David A Dunlap Department of Astronomy & Astrophysics, 50 St George St, Toronto, Ontario, Canada, M5S 3H4
  
  99(CHIME/FRB Collaboration)
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
      We demonstrate the blind interferometric detection and localization of two fast radio bursts (FRBs) with 2- and 25-arcsecond precision on the 400 m baseline between the Canadian Hydrogen Intensity Mapping Experiment (CHIME) and the CHIME Pathfinder. In the same spirit as very long baseline interferometry (VLBI), the telescopes were synchronized to separate clocks, and the channelized voltage (herein referred to as "baseband") data were saved to disk with correlation performed offline. The simultaneous wide field of view and high sensitivity required for blind FRB searches implies a high data rate—6.5 terabits per second (Tb/s) for CHIME and 0.8  Tb/s for the Pathfinder. Since such high data rates cannot be continuously saved, we buffer data from both telescopes locally in memory for ≈ 40 s, and write to disk upon receipt of a low-latency trigger from the CHIME Fast Radio Burst Instrument (CHIME/FRB). The ≈200 deg^2 field of view of the two telescopes allows us to use in-field calibrators to synchronize the two telescopes without needing either separate calibrator observations or an atomic timing standard. In addition to our FRB observations, we analyze bright single pulses from the pulsars B0329+54 and B0355+54 to characterize systematic localization errors. Our results demonstrate the successful implementation of key software, triggering, and calibration challenges for CHIME/FRB Outriggers: cylindrical VLBI outrigger telescopes which, along with the CHIME telescope, will localize thousands of single FRB events to 50 milliarcsecond precision.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § INTRODUCTION
   
  Fast radio bursts <cit.> are brief (∼ ms), usually nonrepeating radio transient events with dispersion measures in excess of that predicted by the electron column density of the Milky Way. Currently, their progenitors and production mechanism are unknown but their high luminosity and impulsive nature have generated significant interest in the astrophysics community <cit.>. In addition, due to their cosmological distances <cit.>, FRB pulses are strongly dispersed by the ionized intergalactic medium and have the potential to probe the large-scale structure of the universe <cit.>.
  
  The vast majority of FRBs are not observed to emit multiple bursts <cit.>[See  for the latest statistics on repeat bursts from known FRB sources.], and the handful of known repeaters are observed to do so stochastically with the notable exceptions of FRB 180916.J0158+65 <cit.> and possibly FRB 121102 <cit.>. This unpredictability makes localization and followup studies extremely challenging. Since the serendipitous detection of the first FRB in 2007 <cit.>, two repeating FRBs have been studied with very long baseline interferometry (VLBI): FRB 121102 <cit.>, with optical followup performed by <cit.>; and FRB 180906.J0158+65 <cit.>. The localization of seven others with sufficient precision to identify their respective host galaxies at redshifts between z = 0.1-0.6 (180924 , 181112 , 190523 , 190102, 190608, 190611, and 190711 ) demonstrated a modern-Universe measurement of Ω_b using FRBs, including the so-called “missing baryons”. This measurement is consistent with that of <cit.>, experimentally evaluating the possibility of using localized FRBs as cosmological probes <cit.>.
  
  Having detected over seven-hundred FRBs in its first year of operation <cit.>, the Canadian Hydrogen Intensity Mapping Experiment/FRB Project () has opened up a window for population-level studies of the properties of FRBs <cit.>. However, CHIME/FRB's real-time localization pipeline, which has a precision of arcminutes, does not yet always allow for unambiguous identification of an FRB's host galaxy. For
  very bright
  FRBs with very low dispersion measure (DM), it is sometimes possible to identify a host by imposing a prior on the host galaxy's maximum redshift.
  
  To routinely pinpoint the host galaxy of FRBs detected by CHIME/FRB, the CHIME/FRB collaboration is developing CHIME/FRB Outriggers, a set of cylindrical telescopes at distances of one hundred to several thousand kilometers from the CHIME telescope. Along with CHIME, the Outriggers will perform a blind VLBI survey to localize thousands of FRBs with 50 milliarcsecond precision. To our knowledge, there has only been one previous attempt to blindly localize FRBs with VLBI. V-FASTR was a campaign to search
  for FRBs in archival data taken by the Very Long Baseline Array <cit.>. None was found, highlighting the difficulty of detecting FRBs with traditional radio telescopes. In contrast, the CHIME/FRB Outriggers program will combine CHIME/FRB's high discovery rate with the localization precision afforded by continental baselines, allowing astronomers to conduct detailed population-level studies of FRB host environments.
  
  We report here on the development of a voltage recording backend as a testbed for CHIME/FRB Outriggers that was deployed on the CHIME Pathfinder, itself a reduced-scale testbed for the CHIME telescope <cit.>. We demonstrate a synoptic VLBI calibration technique for CHIME/FRB outriggers, and demonstrate the performance of our technique on automatically triggered single-pulse detections of the bright pulsars B0329+54 and B0355+54. We also localize two FRBs detected during two observing campaigns using CHIME and the Pathfinder in October and December 2019. Our east-west baseline allows for localization of each source in the RA direction on the sky with arcsecond-level statistical uncertainties for bright FRBs.
  
  
  
  § INSTRUMENTATION
   
  CHIME <cit.> is a beamforming <cit.> interferometer located at the Dominion Radio Astrophysical Observatory (DRAO) near Penticton, British Columbia, Canada. It consists of four stationary 20-m × 100-m parabolic cylindrical reflectors oriented north-south, each of which houses 256 dual-polarization feeds which are uniformly spaced on the focal line of each reflector. Operating as a phased array over the frequency range 400 to 800 MHz, each reflector has a primary beam of 2.6 to 1.3 degrees East-West, directable to any north-south direction from horizon to horizon, with north-south beamwidth increased by the cosecant of zenith angle.
  
  
  The telescope backend is built with an FX correlator architecture. The first correlator stage, the F-engine, digitizes the analog voltage inputs and spectrally divides the incoming data into 1024 frequency channels over the 400 - 800 MHz frequency band using a polyphase filter bank <cit.>. It is synchronized to a GPS-disciplined ovenized crystal oscillator. The channelized voltage data, hereafter referred to as “baseband” data, are passed to the second stage of the correlator
  (the X-engine) <cit.> at 4 real + 4 imaginary bit depth, for all 1024 frequencies and 2048 signal chains, every 2.56, for an overall rate of 6.5 Tb/s. In addition to performing real-time processing, the X-engine buffers the baseband data in memory in a 36-s long ring buffer. If the real-time FRB search pipeline <cit.> detects an FRB candidate, the ring buffer saves the appropriate ≈100 segment of data to disk, with the exact duration being determined by the uncertainty in the dispersion measure (DM) estimated by the real-time searc pipeline.
  
  The CHIME Pathfinder was built prior to CHIME and is used for ongoing technology development for projects such as CHIME/FRB Outriggers. It has approximately one eighth of the collecting area of CHIME and operates on an independent clock. The effective baseline of Pathfinder is approximately 385.42 m due East, 50.43 due South, and 5.17 m lower than that of CHIME. It consists of two 20-m × 40-m cylinders which have the same field of view as CHIME, and have 64 dual-polarization antennas per
  cylinder for a total of 256 correlator input channels. The Pathfinder shares the same F-engine architecture as CHIME, and runs on an independent GPS-disciplined crystal oscillator from that of CHIME. However, in contrast to a full FX correlator, the Pathfinder F-engine feeds baseband data to a baseband recorder backend. This backend, shown in Fig. <ref>, was built to demonstrate the technique of triggered VLBI observations for CHIME/FRB Outriggers. Using four server-grade network
  cards which each provide 80 Gb/s of bandwidth, the recorder stores baseband data in RAM for a quarter of CHIME/FRB's 1024 frequency channels, spaced approximately evenly across the band, at an input data rate of 204.8 gigabits per second (Gb/s) (for details, see Appendix <ref>). Our ring buffer architecture is implemented in [], a flexible and efficient software framework written in C++ for real-time data processing for digital radio astronomy <cit.>. 
  
  
  
  
  § INTERFEROMETRIC LOCALIZATION
  
  
  
   §.§ Detection at CHIME
  
  
  CHIME/FRB features a real-time processing pipeline which coarsely estimates the DM, time of arrival, and signal-to-noise ratio of dispersed radio transients <cit.>. Upon detecting a sufficiently bright transient, a classification algorithm filters out false positives from radio frequency interference and known pulsars. Successful classification of a dispersed radio transient as an FRB triggers the dump of ≈ 100 ms of baseband data to disk at both telescopes with subsecond latency.
  
  Prior to data transfer and cross correlation, the baseband data from just the CHIME/FRB instrument are processed to estimate the FRB's dispersion measure and sky position. This is done by beamforming baseband data from CHIME/FRB's 2048 correlator inputs towards a grid of sky positions around the detection position, calculating the signal-to-noise ratio of the burst detection in each beam, and then fitting a 2D Gaussian model to the resulting intensity map of the signal. Finally, we perform
  coherent dedispersion to the optimal dispersion measure maximizing the burst signal-to-noise ratio and form a tied-array beam to the refined coordinates provided by this so-called “baseband pipeline” <cit.>. From here on we denote the beamformed baseband data from CHIME as F^C_ν b t. Here, C stands for CHIME, while ν represents the frequency channel (N_ν = 1024) ranging from 400-800 MHz. The integer b is the “beam number”, reflecting the fact that a single dump of full-array baseband
  data can be beamformed to multiple sky positions in both polarizations (north-south and east-west, hereafter NS and EW); b ranges from 1,2,…,N_b where N_b = 2 N_p and where N_p is the number of unique sky positions. Finally, t is the time index, measured in units of 2.56. We calculate the flux as a function of frequency channel, polarization, and time block, albeit a lower time resolution indexed by T:
  
      S^C_ν b T = ∑_t = T^t = T+t_int|F^C_ν b t|^2.
  
  Setting the integration time t_int = 40.96 yields the plots in Fig. <ref>. 
  
  
  
   §.§ FRB Cross Correlation Pipeline
  
  Our cross correlation pipeline picks up where the baseband pipeline leaves off. Due to the reduced sensitivity of the Pathfinder, we only cross-correlate the baseband data from bright FRBs. We calculate beamformed baseband at both telescopes (F^C_ν b t and F^P_ν b t), and divide the baseband data into segments of 40.96. For each segment we calculate the complex temperature-normalized visibility V^CP_ν b T as a function of frequency,
  polarization/beam, and time
  block T as we did previously for the flux. 
  
      V^CP_ν b T = ∑_t = T^t = T + t_int F^C_ν b tF^P_ν b t√(∑_t' = T^t' = T + t_int  || F^C_ν b t' ||^2 ∑_t” = T^t” = T + t_int || F^P_ν b t” ||^2)
   
  
  The quantity V^CP_ν b T, like the baseband data, is complex-valued. For geometric delays shorter than 2.56 the information about the geometric delay is completely encoded in the phase of the numerator of V^CP_ν b T. The denominator ensures that increasing the system temperature (i.e. scaling any of the F_ν b t by a constant factor) does not affect |V^CP_ν b T|. Hence, |V^CP_ν b T| as plotted in Fig. <ref> measures the strength of the
  cross-correlation independently of the system temperature. The morphological similarity of |V^CP_ν b T| in Fig. <ref> and S_ν b T in Fig. <ref> allows us to unambiguously interpret our cross-correlated baseband data as a genuine FRB detection. We cross-correlate the NS polarizations and EW polarizations at both telescopes separately; since the two telescopes' polarization axes differ by only ≈ 2 degrees, this approach is close to optimal. 
  
  While the above visibilities are sufficient for assessing a detection, for astrometric precision it necessary to minimize the uncertainty on the phase of the visibility. To do this, we formed a set of visibilities in which we integrated over the entire ≈100 baseband dump to reduce statistical uncertainty of the visibility phase. In addition, for the beams with pulsed emission, we perform the integration with the help of a real-valued time-domain matched filter, h_t, constructed from the pulse's intensity profile as detected in CHIME autocorrelation (i.e. the curves shown in the top panel of Fig. <ref>).
  
  
      V^CP_ν b = ∑_t F^C_ν b t h_t F^P_ν b t√(∑_t' || F^C_ν b t' ||^2 ∑_t” || F^P_ν b t” ||^2)
  
  The filter is normalized to have ⟨ h_t ⟩ = 0 and ⟨ h_t^2 ⟩ = 1. The former constraint enables optimal rejection of steady sources of correlated voltage signals other than the pulse of interest, and the latter constraint ensures that the noise variance of the data is preserved.
  
  
  
  
  
  
  
   §.§ Synoptic Calibration Technique
  
  
  Our calibration technique fundamentally relies on in-field steady sources to keep the two telescope backends synchronized over the ∼ 10 second duration of the dispersed burst.
  Each array only needs to be individually synchronized once per day during the transit of a bright radio calibrator, to re-compensate for the slow thermal expansion of cables between the antennas and the correlator. However, since CHIME and the Pathfinder are each synchronized to independent ovenized crystal oscillator clocks, the time difference between the two arrays jitters on timescales of minutes. Clock jitter and differences in the telescopes' analog chains introduce an unknown instrumental phase between the two telescopes which must be calibrated near or during the time of observation. 
  
  To solve for the instrumental phase, we used the fact that the primary beams of CHIME and Pathfinder completely overlap and that their large size virtually guarantees that there will be ∼ 5-10 bright NVSS <cit.> calibrators (S_1.4GHz > 1.5 Jy) detectable with a high signal-to-noise ratio in 100 of integration time. For each observation, we selected seven of the brightest NVSS calibrators within 1.8 degrees from the local meridian. In total, we formed 16
  beams from each triggered baseband dataset: one per polarization (north-south and east-west) per source (one transient and seven steady-source calibrators) towards catalogued positions of the calibrators as well as our initial estimate of the transient's position from the CHIME/FRB baseband pipeline. We calculated the visibility between the two telescopes as a function of beam and frequency as described in Eq. <ref> and we fit a delay model.
  
  
  
   §.§ Delay Model
  
  For each formed beam (indexed by b) and each frequency channel (indexed by ν), our general delay model (more generally, a phase model) can be written as:
  
  
      Φ^i_ν b = ϕ^i_ν + u⃗^i(t) ·n̂_b + KΔ DM(n̂_b)ν
  
  where ϕ^i_ν is a free function representing the instrumental phase for the ith telescope, u⃗^i(t) is the (time dependent) position of the ith telescope, n̂_b is the sky position of a source in the bth formed beam, and where the dispersive delay due to the ionosphere is a free function Δ DM(n̂_b) and where the dispersion measure constant is taken to be K = 1/(2.41×10^-4) s MHz^2 pc^-1 cm^3. This simple model takes into account the
  time-variable geometric delay and ionospheric delays; for simplicity we neglect small corrections such as tidal deformation that become necessary over long baselines. From here on, we suppress the time dependence of the telescope positions u⃗^i(t). Also, since CHIME and Pathfinder are approximately co-located, the ionospheric delay only varies as a function of sky angle (n̂_b) and not of position (u⃗^i).
  
  While Eq. <ref> could in principle be fitted directly to the visibilities with a least-squares algorithm, in practice it is helpful to slow down, or “fringestop”, the rapid phase variation of the visibility versus frequency to no more than a few radians over the telescope bandwidth using fiducial estimates for u⃗^i and n̂_b. This improves the robustness and convergence of the fit especially in the presence of noise. We denote these estimates with an additional subscript 0. First, we remove the geometric delay due to the nominal baseline (u⃗_0^C - u⃗_0^P), an estimate which is accurate to within a meter. We calculated the (uncalibrated) visibilities V^CP_ν b, reducing our dataset to a set of ∼ 10^4 complex numbers, one per frequency channel per formed beam. The phase of the uncalibrated visibilities after fringestopping can be modeled as
  
      ϕ^CP_ν b = Φ^C_ν b - Φ^P_ν b = ϕ^CP_ν + (u⃗^C - u⃗^P) ·n̂_b - (u⃗^C_0 - u⃗^P_0) ·n̂_b,0
  
  where ϕ^CP_ν represents the differential instrumental phase between CHIME and Pathfinder, where (u⃗^C - u⃗^P) is the true baseline, where n̂_b are the true positions, and where the last term encodes our fringestopping using nominal estimates of the sky positions and baseline. Note that the ionosphere term in Eq. <ref> is identical for each telescope and does not appear in Eq. <ref>.
  Since the differential instrumental phase is independent of sky pointing, we designate two reference beams (B) to use as phase references for the NS and EW polarizations of the telescope. We remove the differential instrumental phase by calculating 𝒱_ν b≡ V_ν b / V_ν B. We define σ_ν b to be the uncertainty on 𝒱_ν b, and denote the amplitude and phase of 𝒱_ν b as A_ν b and φ_ν b≡ϕ_ν b - ϕ_ν B respectively.
  
  
   §.§ Fringe Fitting
   
  After applying this calibration procedure, the phase of the fringestopped and calibrated visibilities which we fit to our delay model is
  
      φ^CP_ν b = (u⃗^C - u⃗^P) · (n̂_b - n̂_B) - (u⃗^C_0 - u⃗^P_0) · (n̂_b,0 -n̂_B,0)
  
  
  
  
  With a good guess of the baseline offset, Eq. <ref> varies slowly as a function of frequency and can be fitted to extract sky localizations and baseline information, as shown in Fig. <ref>. First, using ∼ 10 auxiliary 100 snapshots similar to those shown in Fig. <ref>, each targeting ≈ 7 sufficiently-bright NVSS calibrators (for which n̂_b = n̂_b,0) at a wide range of sky positions, we determine the remaining baseline offset δu⃗≡ (u⃗^C - u⃗^P) - (u⃗_0^C - u⃗_0^P). Next, fixing δu⃗, we can determine the unknown sources' offsets from their nominal positions, denoted by δn̂_̂b̂≡n̂_b - n̂_b,0. Note that our approximately east-west baseline make us insensitive to the declination of sources in the sky, and that the sky positions of sources we are observing (all close to the local meridian) make our data insensitive to east-west baseline errors.
  
  The parameters δu⃗ and δn̂_b are estimated by maximizing the likelihood ℒ using an expression that does not depend on the intrinsic emission spectra of any of the sources. Since only the phase of the visibility is sensitive to astrometric quantities, we can analytically marginalize over the amplitude A_ν b of the calibrated visibilities without losing phase information. We suppress the superscript in Eq. <ref>, treating it as a free function φ_b ν of sky positions and baseline parameters which we collectively refer to as λ.
  Assuming a uniform prior and applying Bayes's theorem we can write the posterior distribution of λ with a χ^2 maximum likelihood estimator. Integrating over the amplitude of the visibility A_ν b simplifies our full χ^2 likelihood to its form in Eq. <ref>.
  
      P(λ | 𝒱_ν b) ∝   P(𝒱_ν b | λ)
      ∝   exp(-12∑_ν b||𝒱_ν b - A_ν bexp(iφ_ν b(λ))||^2σ^2_ν b)
      ∝   exp(-12∑_ν bIm[𝒱_ν bexp(-iφ_ν b(λ)) / σ_ν b]^2).
      logℒ∝    -12∑_ν,bIm[𝒱_ν bexp(-iφ_ν b(λ)) / σ_ν b]^2.
  
  
  Intuitively, this can be understood as follows. If the delay model allows us to perfectly derotate the 𝒱_ν b to the real axis of the complex plane, the imaginary part of 𝒱_ν b, normalized by its standard deviation, will be minimized and will be a zero-mean, unit-variance Gaussian random variable. Hence, the sum of squares follows a χ^2 distribution with N_b × N_ν degrees of freedom, and minimizing χ^2 allows us to recover the best fit parameters λ without ever explicitly fitting any spectra. Our FRB localizations are summarized in Table  <ref>.
  
  Statistical uncertainties are estimated by jack-knifing our data over frequencies: we can divide our calibrated visibilities 𝒱_ν b into 9 different “frequency combs”, spaced evenly across our band. By leaving out one comb at a time and repeating our χ^2 analysis, we can inspect the resulting likelihood curves and reject frequency-local RFI, which would show up as a discrepancy between different jack-knifed realizations of our analysis. We estimate the statistical error on our localizations using our jack-knifed samples in accordance with <cit.>.
  
  
  
  
  
  
  
   §.§ Systematic Errors
  
  Over short baselines, most radio sources remain unresolved and there is no shortage of calibration sources in the sky. While our database of NVSS sources serves as an abundant calibrator network, it also means that the probability of having two sources within a formed beam (FWHM ∼ 0.3 deg^2) is non-negligible. While we filtered out bright calibrator candidates that are too close to each other, we are forced to assume that the remaining calibrators are true point sources. For example, we cannot eliminate the possibility that the emission of sources at low frequencies (400-800 MHz) is offset from the catalogued survey position at 1.4. Angular offsets of the calibrator's emission region from its catalog position would severely impact our measurements in two major ways.
  
  First, astrometric discrepancies for sources used as a delay center could directly lead to localization errors, creating a systematic offset in the measured RA. This effect should be smaller than the formed beam size and should be independent of the calibrator's angular distance from the transient of interest.
  
  The more serious impact of astrometric discrepancies is on baseline determination. Eq. <ref> implies that an inaccurate determination of the baseline translates to a systematic localization offset proportional to the on-sky distance between the target of interest (n̂_b) and the delay center (n̂_B).
  
  To quantify the systematic offsets in our RA measurements, we conducted triggered observations of pulsars, which are also summarized in Table <ref>. We added rules to the event classifier in the real time FRB detection pipeline to allow bright pulses from known pulsars to trigger a baseband dump, in the same way that an FRB would. In this way, we collected baseband data for three bright single pulses from PSR B0329+54 and one from PSR B0355+54, and localized the pulsars as if they were FRBs. We estimated the systematic errors in our localization analysis using the discrepancy between our results and the pulsars' known position, corrected for their proper motion.
  
  We phase reference the pulsar position to the 7 in-beam NVSS calibrators, whose sky positions are as far as 60 degrees away from the pulsar. We plot the astrometric localization error against the angular distance between the pulsar and the delay center in Fig. <ref>. We find that the astrometric discrepancy is roughly linearly proportional to the on-sky distance to the calibrator, and that using the nearest on-sky calibrator minimizes discrepancies from the catalogued positions of pulsars even with truly simultaneous phased-array observations through the same ionosphere. We attribute this discrepancy chiefly to a static baseline determination error corresponding to time delays of less than a nanosecond. To estimate the magnitude of systematic uncertainty in our FRB localizations, we find the intersection of the upper edge of the shaded area in Fig. <ref> with the on-sky distance to the nearest calibrator to each FRB. 
  
  In addition to an unknown static baseline error, the effective phase center of a beamforming telescope drifts slightly every day. The effective phase center position is the centroid of active antenna positions weighted by their sensitivity, and the centroid drifts from day to day on the order of ∼cm because a slightly different set of antennas are flagged (i.e. nulled) every day due to factors like rain causing increased noise in certain antennas. We take this effect into account during tied-array beamforming, but the current baseline positions are not yet constrained at a level to measure this day-to-day drift in astronomical data. Using a larger sample of pulsars at a wide range of declinations for baseline determination, not just validation, will reduce our systematic error floor and improve our ability to phase reference our observations to calibrators far away on the sky.
  
  
  
  
  § DISCUSSION AND CONCLUSION
  
  We have developed baseband recording hardware and software capable of handling the high data rate of wideband, multi-element radio interferometers such as CHIME for VLBI observations (Section <ref>). Also, we have demonstrated a calibration technique that exploits CHIME's wide field of view to localize several radio transients detected by CHIME/FRB and the CHIME Pathfinder in the same spirit as VLBI (Section <ref>). In an automatically triggered ≈100 duration baseband capture at CHIME and Pathfinder, we can simultaneously detect a single FRB in cross correlation between CHIME and Pathfinder, as well as multiple calibrators for phase referencing our telescopes. 
  
  We have developed efficient maximum likelihood estimators to perform fringe fitting in the absence of knowledge about the FRB spectrum(Section <ref>), and have localized FRB 20191021A and FRB 20191219F with statistical uncertainties of 1.6 and 25 arcseconds respectively along one direction in the sky (Table <ref>). Using single pulses from bright pulsars we have characterized the systematic errors on our FRB localizations (18 arcseconds and 3 arcminutes respectively) which are dominated by errors in baseline determination using NVSS calibrators (Section <ref>). 
  
  Eventually, CHIME/FRB Outriggers will include stations at baselines of thousands of kilometers to achieve an astrometric precision of ∼ 50 milliarcseconds. This precision is roughly matched to that of the best optical telescopes, and will allow for detailed followup studies of FRB host environments within their host galaxies. To achieve our goal, we anticipate a very different set of challenges from those presented here. Over long baselines, the ionospheric phase shift can vary by as much as ΔDM∼ 10^-5 (corresponding to a time delay of ∼200 as a function of sky position at sub-gigahertz frequencies). Achieving high astrometric precision will require removing this effect with observations of calibration sources close to the FRB on the sky. The relatively uncharted territory of low-frequency VLBI calibrators poses a major challenge for scaling CHIME/FRB VLBI observations to continental baselines.
  
  One option is to use bright pulsars for phase referencing observations with CHIME/FRB Outriggers, especially for hour angles close to the Galactic plane where pulsars are most abundant. Pulsars have the advantage of being transient and compact point sources, helping to eliminate confusion noise and the effect of uncertain calibrator morphology on our astrometric precision. Though the astrometric positions of some pulsars are known at the 10-20 milliarcsecond level, including less precisely localized pulsars in the calibrator network of CHIME/FRB Outriggers will improve astrometric localizations of those pulsars as observations accumulate over time.
  
  For hour angles where pulsars are sparse, phase referencing after the real-time detection of an FRB can be done by using a dense network of steady-source VLBI calibrators all over the northern sky, particularly near the celestial pole in the constant-coverage area of CHIME's primary beam.
  
  Following pioneering low-frequency VLBI surveys by <cit.> and  <cit.>, the advent of the International LOFAR Telescope has made systematic surveys of the low-frequency sky possible. The LOFAR Snapshot Calibrator Survey <cit.> has demonstrated that high quality, compact VLBI calibrators at low frequencies tend to be bright at 328 MHz (S = 0.1-1 Jy) and have a flat low-frequency spectrum. Recent results from the ongoing LOFAR Long-Baseline Calibrator Survey <cit.> project the density of high-quality VLBI calibrators over long baselines to be ∼ 1 deg^-2. While the LBCS covers even lower frequencies than those relevant for CHIME/FRB Outriggers, an understanding of promising low-frequency calibrators on long baselines will be crucial for future VLBI observing campaigns with CHIME/FRB Outriggers. The instrumentation and analysis techniques developed in this paper, combined with a dense network of pulsars or compact low-frequency VLBI calibrators, will pave the way for transformative studies of FRB host environments and of the intergalactic medium over long baselines with CHIME/FRB Outriggers.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   <cit.>,  <cit.>,  <cit.>
  
  We thank the CHIME Collaboration for use of the Pathfinder, and the staff at the Dominion Radio Astrophysical Observatory and Ev Sheehan for their hospitality and efforts to ensure the smooth deployment of our instrumentation. C. L. was supported by the U.S. Department of Defense (DoD) through the National Defense Science & Engineering Graduate (NDSEG) Fellowship. M.B. is supported by an FRQNT Doctoral Research Award. This research is funded in part by the Gordon and Betty Moore
  Foundation and the NEC Corporation Fund for Research in Computers and Communication. FRB research at UBC is supported by an NSERC Discovery Grant and by the Canadian Institute for Advanced Research. V.M.K. holds a Distinguished James McGill Chair and the Lorne Trottier Chair in Astrophysics & Cosmology and receives support from an NSERC Discovery Grant and Herzberg Award, from an R. Howard Webster Foundation Fellowship from the Canadian Institute for Advanced Research (CIFAR), and from the FRQNT Centre de Recherche en Astrophysique du Quebec. D. M. was supported by the Banting Postdoctoral Fellowships Program. P.S. is a Dunlap Fellow and an NSERC Postdoctoral Fellow. The Dunlap Institute is funded through an endowment established by the David Dunlap family and the University of Toronto. The CHIME/FRB baseband recording system was funded in part by a CFI John R. Evans Leaders Fund award to I.H.S.
  
  
  
  
  § BASEBAND RECORDER PARTS LIST
  
  Our recorder uses 1 terabyte of RAM to buffer approximately 40 seconds of baseband data corresponding to dispersion measures of up to ≈ 2000 pc / cm^3 upon receiving a trigger from CHIME/FRB's real-time detection pipeline. A photograph of the inside of the node is shown in Fig. <ref>, and a full parts list is given in Table <ref>. Future recorders may feature an auxiliary buffer or GPUs for real-time beamforming capabilities <cit.>, which will facilitate longer integration times on fainter calibrators, though this technical capability is not necessary for our bright calibrators.
  
  
  
  aasjournal
  
  
  
  