Localizing FRBs through VLBI with the Algonquin Radio Observatory 10 m Telescope

Introduction

Fast radio bursts (FRBs) are bright, millisecond-long flashes of radio emission that were first discovered by Lorimer et al. (2007) and have been observed to be distributed throughout the sky (Josephy et al. 2021). Their origins are among the key unresolved questions in astrophysics, despite hundreds of FRBs having been observed to date (Petroff et al. 2016; The CHIME/ FRB Collaboration et al. 2021). A significant challenge toward understanding FRBs lies in our inability to determine their precise on-sky positions, which is critical for identifying their host galaxies (Eftekhari & Berger 2017). Despite the fact that some facilities are able to localize FRBs in limited numbers, localization remains a challenging problem (with the exception of low-redshift nearby events; Bhardwaj et al. 2021). This is mainly due to the fact that most FRBs are non-repeating, and a large field of view (FOV) is required in order to increase the probability of detection. A few astronomical facilities are able to identify single-burst FRBs' host galaxies (Bannister et al. 2017(Bannister et al. , 2019;;Ravi et al. 2019), but their detection rates and fields of view are low. By achieving more precise localizations of non-repeating FRBs, we can constrain FRB populations (Marcote et al. 2017;Kirsten et al. 2021) and improve the classification for repeaters and non-repeaters, and also put them to use as probes for fundamental astrophysics (Petroff et al. 2019).

The Canadian Hydrogen Intensity Mapping Experiment (CHIME) is a radio telescope operating in the 400-800 MHz band, with an 8000 m 2 aperture area of semicylindrical paraboloid reflectors, located at the Dominion Radio Astrophysical Observatory (DRAO) near Penticton, British Columbia. Each of the four cylinders is instrumented with 256 cloverleaf feeds suspended along the cylindrical axis, and observes a >200 deg 2 FOV. The combination of the FOV, wide bandwidth, large aperture area, and a powerful correlator makes CHIME a great tool for FRB detection. The CHIME/ FRB Collaboration (CHIME/FRB Collaboration et al. 2018) has already published the detection of 21 repeating and 474 non-repeating FRBs (The CHIME/FRB Collaboration et al. 2021). CHIME/FRB and its baseband system are able to localize FRBs with a resolution higher than D l 27 , and as shown in (Masui et al. 2019), with a best-case limit of ∼1′ (Michilli et al. 2021). In order to determine the host galaxies of FRBs, however, localizations better than 1″ are required.

Very long baseline interferometry (VLBI) is a technique that combines observations from multiple separated telescopes, effectively turning them into one single telescope with an enhanced angular resolution that scales inversely with the projected baseline length. This technique is one method by which we can improve the diffraction-limited beamwidth of CHIME/FRB's localizations: by correlating baseband data between CHIME and another site, we can use VLBI to find fringes for a single FRB event and coherently delay reference to a nearby calibrator, hence improving the CHIME/FRB localizations by two to three orders of magnitude. So far, only repeaters have been localized with VLBI scheduled observations (Marcote et al. 2020) and not one-off events.

In this paper we describe VLBI between CHIME and a 10 m telescope at the Algonquin Radio Observatory (ARO). ARO is 260 km West of Ottawa in Algonquin Park, Ontario Canada. It was established in 1963 by the National Research Council Canada (NRC), and its major instrument was a 46 m radio telescope. The 10 m telescope, a prime-focus paraboloid, was constructed at the same time. The baseline between CHIME and ARO is over 3000 km, offering a diffraction-limited resolution within θ  30 mas. The Earth projection of the baseline is shown in Figure 1.

The CHIME/FRB Collaboration is actively developing a new set of CHIME Outrigger, composed of cylindrical telescopes at distances of one ∼20 to several thousand kilometers from the CHIME telescope. The ARO 10 m telescope dish and the CHIME Pathfinder (Leung et al. 2021) serve as testbeds for the CHIME/FRB Outriggers project. Along with CHIME, the Outriggers will perform an autonomous VLBI survey to localize over 1000 FRBs with 50 mas precision. Outriggers will cover mostly the same area that CHIME does (with up to three baselines), and correlate beams digitally while preserving coherence over the FRB sweep. Further, near in-beam calibration can be performed while digitally pointing to different locations on sky, with only an angular space interpolation (rather than over time; clock system), and eventually will be expanded to a grid of pulsar calibrators. The Outrigger project's aim is to localize one-off FRBs with a VLBI network particularly designed for this purpose, different from a classical VLBI imaging facility, where the pulse is only visible over ∼millisecond timescales. The ARO 10 m testbed is the initial effort to correlate FRBs for the upcoming project. This paper will proceed as follows. We provide an overview of ARO, the 10 m telescope, and our new instrumentation of the telescope in Section 2. In Section 3 we describe VLBI experiments between CHIME and the ARO 10 m telescope. Section 4 describes early science results, including the detection of VLBI fringes from one FRB. We discuss our results in Section 5 and present our conclusions in Section 6. Appendix A details the 10 m radio frequency (RF) chain, and Appendix B provides a detailed explanation of visibilities and cross correlation with baseband data.

The 10 m Telescope

The 10 m telescope (Figure 2) was inaugurated in 1964 (Medd & Broten 1961;National Research Council of Canada & Division 1969), with an equatorial mount and a manual mechanical engine with decl. and hour angle drive (or polar drive). The paraboloidal dish has a structure with four feedsupport struts to support the focus equipment, with a surface accuracy of 0.063 cm rms at the time of its construction (Dawson 1970). Coaxial cables with standard Bayonet Neill-Concelman (BNC) and N-type lines run through the feedsupport struts and connect the feed and first-stage amplifiers (power and analog cables). The telescope is equipped with an analog system for pointing and a polar axis drive unit located physically at the telescope exterior.

Table 1 gives specifications of the telescope and its location. The decl. and polar drive and their indicator systems were only partly functional at the beginning of our work, but we were able to point the telescope by applying external torque to the gearboxes. The telescope was set to the decl. of Taurus A to receive pulses from PSR B05321+21, which we used as a test source. The hour angle was set west of the ARO meridian so that area of sky seen by the main beam coincided with that seen by CHIME on the meridian at DRAO (see Figure 7). The azimuth and elevation of this position are 248 °and 50 °, respectively. We estimate the accuracy of this pointing to be ∼1 °, checked by the PSR B05321+21 transit through the telescope beam.

We installed a new analog system on the 10 m telescope, including a new feed and low-noise amplifiers (LNAs) at the focus, and second-stage amplification in the telescope pedestal. We used underground coaxial cables, installed in the 1960s, to connect to a new digital acquisition system in the control room of the 46 m telescope, 200 m away. See Section 2.3 and Appendix A for details of the analog system, and Section 2.4 for a description of the digital system.

With the completion of these upgrades, the telescope is fully operational and working 24/7 with occasional power interruptions during the winter months due to the site's remote location.

Its day-to-day operations and monitoring are done remotely over a satellite-based internet connection.

Radio Frequency Signal Chain

There are three amplifier stages:  1 ,  2 , and  3 . The firststage amplifiers are located right next to the cloverleaf feed (telescope's focus) with an SMA connection. The first-stage amplifiers are connected by dual polarization lines to the second-stage amplifiers in the telescope's pedestal. The pedestal also contains the power supplies for  1 and  2 . The final stage,  3 , is at the end of the underground connected lines from the telescope's pedestal to the control room. In the control room there is another set of amplifiers right before the digitizer ICE system (Bandura et al. 2016).

The LNAs in  1 are custom-built CHIME amplifiers (see the top panel in Figure 3). Each amplifier is bounded with an RF shield enclosure that also protects for weather conditions.

Due to the large distance spanned by the connection between the 10 m dish and the control room, an extra amplification was required from  2 to  3 . These are commercial LNAs placed consecutively and fed with the analog lines that come from the telescope focus in  1 (see the middle panel in Figure 3).

 3 contains the last set of amplifiers after the long underground lines (∼200 m) and before the signal digitalization. This last set of amplifiers is composed of custom CHIME Pathfinder amplifiers (Bandura et al. 2014), a set of commercial line equalizers, and 400-800 MHz bandpass filters. The  3 amplifiers' gain (bottom panel of Figure 3) compensates for the gain drop in  1 .

For a detailed description of components and the analog chain, see Appendix A.

Digital Processing

The digital processing at the ARO 10 m telescope is similar to that done by CHIME/FRB. The analog signal arrives at the analog-to-digital converters (ADCs) on the ICE, which are clocked by a hydrogen maser, and an initial time stamp for each acquisition is provided by a global positioning system (GPS) signal. The underground analog signals (two polarizations) come from the 10 m, and two extra SMA connections on the board are related to the maser and GPS time stamps. Following this, the same main configuration from CHIME/FRB is applied, i.e., we sample raw voltages at 8 bit, apply a polyphase filter bank (PFB; with an identical number of taps; Price 2016) at 18 bit, and then quantize to 4+4 bit resolution. After channelization is done, baseband data has 1024 frequency channels spanning from 400-800 MHz and a sample rate of 390.625 kHz.

The ICE system hardware (Bandura et al. 2016) at the ARO 10 m telescope is made of two ADC data-acquisition daughter boards attached to the field-programmable gate array (FPGA), and an ARM processor running Linux, which loads the firmware onto the FPGAs (see Figure 4) and provides monitoring functionality. The ICE reads data from the ADCs at 8 bit with a sample rate of 800 mega samples per second, resulting in a data rate of 6.4 Gbit s -1 for each of the ADCs. The data products from the ICE (after PFB and quantization) go to the main recording node on site, where the data are carried through two QSFP+ connectors on the motherboard, which then become eight connections with SFP+ ports. The total possible data rate on the recording node is therefore 51.2 Gbit s -1 , but the actual data rate used is 6.4 Gbit s -1 from two of the 16 ADC inputs (in the 51.2 Gbit s -1 ).

The main difference between the signal chains of CHIME/ FRB and the ARO 10 m telescope is the fact that CHIME/FRB uses a GPS disciplined crystal oscillator for its 10 MHz clock, thereby keeping the clock in sync with the GPS on long timescales. Separately, we digitize a maser signal (DRAO maser), in order to correct for timing variations of the GPS clock (on short timescales), i.e., CHIME/FRB has effectively a maser-precision. This combines the long-term absolute accuracy of the GPS clock with the short timescale relative precision of the maser (Mena-Parra et al. 2021).

On the other hand, the 10 MHz clock at ARO is provided by a hydrogen maser (which provides tempo), and the GPS unit only provides an initial reference GPS time stamp via an IRIG-B connection to the ICE system. The clock is therefore free running off the maser and not adjusted to keep time with GPS after the start of an acquisition. In VLBI, clock stability is crucial to understand and localize sources (due to its geometric relation; see Section 3), and its reliability is measured with the Allan variance (standard measure of frequency stability in clocks).

As in CHIME, the data processing system used at the ARO 10 m telescope is kotekanfoot_3 , which is run in the recording node. This is a framework for assigning blocks of processing components. The kotekan software works the same as in CHIME, but its configuration is appropriate for single-antenna recording.

There is currently no real-time processing of baseband data at the ARO 10 m telescope; instead, all data are stored in a set of 10 drives of 11 TB each, giving a buffer with a theoretical capacity of 30 hr. In reality, this time period is lower due to the limitation of the data writing to disk, the space used in the drives, and the constantly running disk-cleaner utility. Accounting for this, the actual buffer is roughly ∼ 24h. The recording depends on: maintenance, power outages, and other down times. This window is more than enough to establish a connection with CHIME/FRB and a potential triggering signal. The data are recorded to the set of hard drives simultaneously for fast and efficient storing and processing on the recording node. Data that have been on the disks for longer than the buffer time are eventually deleted by the disk-cleaner (which is a simple script used to remove data after a certain disk usage has been reached), unless the system is stopped manually or data have been hard-linkedfoot_4 by a CHIME/FRB triggering event.

Important elements in this chain such as the FPGAs, kotekan, disk-cleaner, and hard-linker are automatically serviced on the recording node and constantly monitored by users.

Other components of the digital chain are machines, including (1) a controller node, which is in charge of the intensity stream from kotekan and receiving trigger signals from CHIME, and (2) an analysis node, the main machine used to check data and early science results before data are transferred to a common location. To minimize the RFI from different components in the analog chain, all systems described in this section are enclosed in an RF-shielded rack; see Figures 5 and6.

Other network and power elements are the main switch and the power distribution unit (PDU).

Site communication is done via Xplornet satellite-based internet. The internet connection varies with weather with an average upload/download of 4.5 Mbit s -1 . Nevertheless, we only require checks on the telescope through Secure Shell (ssh), and once a day, pulses (potential FRBs or calibrators) are transferred over to the Compute Canada system SciNet (Ponce et al. 2019). This is a common location where the crosscorrelation process between the DRAO and ARO sites is carried out.

Triggering from CHIME

The ARO 10 m telescope data are not searched independently for astrophysical transients. Data are only saved upon receipt of a trigger from CHIME/FRB. Only FRBs that are detected in the shared FOV between CHIME/FRB and the ARO 10 m telescope cause a trigger to the ARO site. The overlapped FOV is centered at decl. +22°, and only beams in the same sky location as the 10 m are able to trigger events (roughly 36 beams at 60 MHz; see Figure 7). Baseband data for processed events are then triggered at CHIME (within the same CHIME/FRB pipeline as normal events; see CHIME/FRB Collaboration et al. 2018), and a trigger is sent to ARO, which activates hard-link storage and backs up the recently recorded data. For calibration and delay reference purposes, pulses from PSR B0531+21 (the Crab pulsar) are also dumped daily in the same way as an FRB.

CHIME/FRB detection happens in beamformed beams (Ng et al. 2017), which then alert the ARO 10 m telescope with a triggering signal. Figure 7 shows an up-to-scale comparison of CHIME/FRB beams and the single beam from the ARO 10 m telescope. With this current configuration, we expect to observe ∼4-7 FRBs per month (estimated from CHIME/FRB rates without considering down times). However, only the brightest pulses are detected in cross correlation.

The signal-to-noise ratio (S/N) is only computed at CHIME, which is because of the following reasons:

1. PSR B0531+21 pulses are observed through the Crab Nebula, which will increase the system temperature (irrelevant for FRBs). 2. Some pulses are not visible in intensity at the ARO 10 m telescope, but they cross correlate. 3. The ARO 10 m telescope pointing is only known within ∼1 deg, meaning that some of the detected pulses could fall in a side lobe. 4. System temperature and beam model are unknown for the ARO 10 m telescope.

A single pulse from PSR B0531+21 (J0534+2200; Lyne et al. 2015) is recorded daily at the 10 m and at CHIME. These are giant pulses (GPs), with a flux density of the order of 1-10 kJy, of dispersion sweep 1.1 s (dispersion measure, DM, approximately 56 pc cm -3 ) at 400-800 MHz, and unknown time of arrival (ToA). Only a single pulse per day is triggered because of limitations of the CHIME/FRB system and the short time over which the two FOVs overlap (roughly ∼10 minutes). The recording system in CHIME/FRB is limited to 100 ms duration snapshots and has a trigger cooldown time of ∼5 minute.

The simultaneous baseband dumps are used on a regular basis to check data quality and clock stability, study the ionosphere, and serve as a potential calibrator for an FRB event before or after its recording. The single pulse is selected via an S/N threshold at CHIME of S/N CHIME 30 and then triggered at the ARO 10 m telescope with the selected CHIME/FRB time stamp ±2.45 ms (geometric delay due to baseline b). An example of a pulse observed simultaneously at CHIME/FRB and ARO 10 m telescope is in Figure 8.

In addition to standard triggering using the CHIME/FRB back-end, CHIME is equipped with a VLBI tracking beam that can digitally point and save baseband data to disk. Observations with the CHIME VLBI tracking beam are a continuous stream of data (similar to the ARO 10 m telescope), and the CHIME VLBI tracking beam does not receive triggers. The CHIME VLBI tracking beam was used to observe PSR B0531 +21 in continuum mode (i.e., several minutes recorded) because the CHIME/FRB trigger system could not record raw voltage data from multiple pulses on such short timescales. These observations were used to study the clock stability over a single day in Section 3.5.1.

Baseband Data

As mentioned in Section 2.4, both CHIME and the ARO 10 m telescope return a similar type of baseband data as a product. The complex voltage is stored in an array of N = 1024 frequency channels, K time bin width of 2.56 μs, each of which is called a frame, and dual linear polarizations (in CHIME north-south Y and east-west X polarizations), hereafter called baseband data (after passing through the PFB). Figure 9 shows a representation of the baseband data, where the elements V X nk ( ) and V Y nk ( ) with k = 0,K,K -1 and n = 0,K,N -1, are complex numbers, which are part of the X and Y polarizations. Baseband data will be represented as V:

] refers to discrete dependency on frequency channels (n) and frames (k). The first channel n = 0 has a center frequency of 800 MHz, and the last channel n = 1023 is 400.390625 MHz.

At the ARO 10 m telescope and the CHIME VLBI tracking beam, data are stored to disk as a continuum, i.e., column n = 0 has the same start time for all its elements. In contrast, the CHIME/FRB back-end sends callback detections from intensity data and clips around baseband data on buffer; the clipping returns a block of baseband data where frequency channels have different start times and follow the dispersion of a pulsar or FRB.

A higher time resolution than 2.56 μs can be achieved by inverting the PFB (raw ADC data); this is a complicated procedure, but in principle is feasible. Alternatively, this can be achieved by studying the phase information in the complex number frames of the baseband data. In this analysis, we will only use the latter, baseband data space. For a more detailed description of the baseband and raw ADC data properties, see Appendix B.

VLBI between the ARO 10 m Telescope and CHIME

We are interested in finding the cross correlation that exists between the same wavefront detected on different sites. Once the wavefront reaches CHIME and the ARO 10 m telescope (hereafter site A and site B), it is digitized and transformed to baseband data. The telescopes A and B will observe delayed copies of the same signals. The nondispersive delays contain localization information. But also, other sources of nondispersive delay will exist, such as the troposphere and the clock error between stations. In addition, the dispersive delays given by the ionosphere and instrumental effects must be properly calibrated.

Then starting from V V , Figure 10 shows the basic components from the VLBI system. We are interested in finding the localization given by the baseline angle θ (angle between the baseline vector and the source vector), which is function of the R.A. and decl. θ = θ(α, δ) of the source, and in our case, most of the contribution will be given by the R.A. (Figure 1). The baseline angle is andn N 1 ¢ =were used. Each polarization has N × K complex numbers, where N = 1024 frequency channels and K is the number of frames. The quantities V X nk ( ) and V Y nk ( ) (with k = 0,K, K -1 and n = 0,K,N -1) are complex numbers np.cdouble. Each element V X nk ( ) or V Y nk ( ) has a GPS time stamp associated with a time bin of 2.56 μs called frame. r t will be maximum when the delay τ geo is applied.

The variable b represents the baseline (same as in Figure 1) Euclidian distance between sites A and B, and θ is the baseline angle that is a function of R.A. and decl., θ = θ(α, δ). Notice that the wavefront is only an approximation of a plane wave; the atmosphere and interstellar medium will modify its structure (see total delay, Equation (3)).

expressed simply as:

with c the speed of light, b the baseline (b ≈ 3000 km), and τ geo the geometric delay.

For the ARO 10 m telescope testbed, calibrators are pulsars (in particular, PSR B0531+21 pulses), since pulsars are easier to tell apart from the background as compared to standard VLBI (steady-source) calibrators. Pulsars, in addition, are sufficiently compact and with no confusion (time-domain separation from background) at our frequencies.

The CHIME and ARO 10 m telescope stations have systemequivalent flux densities (SEFDs; S sys ) of ∼40 kJy and ∼1.7 kJy. When Taurus A is in the beam (e.g., during a Crab GP trigger), the SEFD at the ARO 10 m telescope increases to 1 kJy (S TaurusA ≈ 1kJy; Perley & Butler 2017). The CHIME observation will be completely dominated by the Taurus A flux (Cordes et al. 2004). VLBI observations of the Crab pulsar baseline noise level will not be discussed, since the nebula is not correlated (Popov et al. 2017; nebula determines the noise level and does not lead to a correlation), with the exception of Crab's GPs. In addition, Crab GPs can easily exceed the Taurus A emission with fluxes above 1 kJy (Bij et al. 2021;Thulasiram & Lin 2021), i.e., S sys ≈ S GPs (Section 2.5; since only high enough S/N pulses are correlated).

Earlier tests were performed at CHIME frequencies with steady-source calibrators in VLBI, but correlations were not possible to achieve. Steady-source calibrators are known to be sufficiently unresolved at high frequencies and narrow bandwidths, which is not the case at CHIME and ARO 10 m telescope. Nevertheless, the future CHIME/FRB Outrigger project is taking steady-source VLBI calibrators into consideration, as Low-Frequency Array has been successful in doing so at low frequencies (Moldón et al. 2015). The cross correlation using pulsars is in principle similar to that for an FRB (with the exception of longer dispersion times, as will be discussed in detail in Section 5.6).

We must find and isolate the geometric delay τ geo from the total (observed) delay τ total . We can then decompose it into its different components:

In Equation (3), left to right, the delays are: total, clock error (constant), geometrical (baseline angle and time dependent, which also includes tropospheric delay), ionospheric (frequency-dependent), instrumentation (frequency-dependent), and noise errors.

A general introduction to VLBI and how to treat and remove delays from Equation (3) is described by Reid & Honma (2014).

Correlation with Baseband Data

In a traditional XF correlator framework (Price 2016), raw ADC sampled data are cross correlated with efficient fast Fourier transform (FFT) algorithms without any hierarchical separation of time or frequency scales, with the peak of the one-dimensional cross-correlation function yielding a measurement of the geometric delay, τ geo . One drawback of this approach is that the problem is further complicated by the introduction of nanosecond-scale systematic drifts. All of these drifts change on minute-to day-long timescales, and they exhibit different frequency dependencies. It is desirable to correlate our baseband data in a more hierarchical manner so we can characterize and separate these different shifts by their differing frequency dependence and time dependence. We break down the problem into two parts: first we determine the integer delay, measured in multiples of the baseband period of 2.56 μs (number of k to shift), followed by the fractional (or subframe) part of the delay (phase correction), according to

where geo 0 t is given by a prior source localization, which is accurate at the arcminute level (Section 3.2), and clock 0 t is a prior measurement of the constant delay.

If we have measured the frame delay correctly, k shift , the visibility matrix can be constructed:

, 6

ẃith  P P A B the matrix of frames and frequency channels with

) elements, which is simply an array element-byelement multiplication. The complex conjugate of all elements in matrix V is expressed as V . Equivalently we can rewrite Equation (6) in terms of its phase:

Arg , 7

P P P P P P i

with f a matrix with each element f nk (frames and frequency channels), and .   the norm of each complex element. The subframe delay of Equation (6) shows up in the phase of each complex number. One approach is to take the Fourier transform of  P P A B over the frequency axis (n). This is approximately equivalent to lag-correlating our baseband data in many small 2.56 μs integrations, ignoring spectral leakage effects. If the delay were purely geometric, finding a peak in subframe cross correlation, as a function of τ would perfectly determine the subframe part of the delay. However this approach does not allow us to separate out the different contributions to the delay, as we discussed earlier. Instead, we sum (integrate) over frames, measuring the phase as a function of frequency channel, as shown in Figure 11, left panel. Then the integrated visibility is defined as:

, 8

¢ integrated over a time t k 2.56 s m = ¢ ´proportional to a number of frames, k¢. We are left with a sequence of N = 1024 complex visibilities, which contain all of the subframe delays.

We now define the subframe cross-correlation function in terms of the integrated visibility of baseband data as:

A B n N P P nk un

where the norm of the FFT is over the frequency axis with n channels, and it returns the cross-correlation strength as a function of lag with u data points, cross-correlation strength:

Notice that the maximum value of the cross-correlation strength is:

with τ u the lag-correlation or first estimate of the constant delay that exists between the two data sets. To separate the different contributions to the total subframe delay, we must fringe-fit the visibilities  P P t A B á ñ from several sources to a delay model (dispersive and nondispersive) using the differing frequency and time dependencies.

Geometric Correction

Due to the large value of the geometric delay (several frames) and its considerable variation over time τ geo (t, θ), an initial guess for τ geo is required. The geometric delay model, geo 0 t , is used to re-align phase and frames of baseband data for correlation, but since it depends on the precision of the initial localization guess given by CHIME (or the precision of the baseline), a residual will be left to compute geo t here, where:

. 1 1

This residual calculation will be done in the wideband fringefitting algorithm (Section 3.4).

Each baseband frame is tagged with a local atomic clock (or close to atomic performance). As such, we need to know the proper time delay between the arrival of the wavefront at one antenna and the arrival at the second. Computing this is complicated by the reality that the object under observation is only stationary in a frame (of reference) in which the antennas are moving. For short baselines, this motion is negligible compared to the time it takes light to cross the baseline, but for VLBI, it can become significant. Geometric delays will be computed using difxcalc11 (Gordon et al. 2016), where by giving an observation time, sky coordinates, and Earth locations, a geo 0 t model can be calculated. The software first computes the exact site locations, taking into account crust deformations due to a high-order tide model (and if ocean coefficients are available, it will include those effects). Once it has the baseline, it applies the Consensus (Eubanks et al. 1991) relativity model, which accounts for the baseline motion in the solar system barycenter frame and for gravitational and special time dilation effects. Other additional delays are added afterwards such as the troposphere refractive delay of the source in each site. Geometric delay models are intensively studied in geodetic VLBI and tend to be described by the same models as in astrometry VLBI (Titov et al. 2020). The model at the CHIME and ARO 10 m telescope baseline using difxcalc11 is able to achieve picosecond precision (Soffel et al. 1991), which is more than sufficient for our purposes.

Coherent Delay Referencing

In order to localize a target (FRB) given a reference (calibrator), we use delay referencing. All astrometric information will be contained only in the phase of the two sets of visibilities, and by computing their phase difference, a good estimate of their sky angular difference (Δθ) can be computed. Other frequency-dependent effects such as the instrumentation phase error (given by τ inst (ν)) will also cancel off. We then calculate the angle of the visibility ratio, where in the ideal case τ total = τ geo , the ratio becomes:

where θ t is the baseline angle of the target, θ r is the baseline angle of the reference, and f represents the angle of the complex visibility. Notice that to reach Equation (12), we need to first account for all terms from Equation (3) in the phase f (delays smaller than a frame) and by finding the right alignment of the two baseband data matrices, k shift , at the time to form

In practice, the difference Δf geo can be related to a sky angular difference Δθ, which will have the reference and target locations, and it will be a function of (α, δ). The steps and assumptions to reach Equation (12) are in Sections 3.4 and 3.6. Finally in Section 5, we will discuss the limitations of the method.

Wideband Fringe Fitting

Fringe fitting is the method used to find a likelihood distribution that represents the most accurate parameters for geometric delay and time degeneracies in order to maximize the cross-correlation strength (or equivalently minimize χ 2 ). As mentioned earlier, the ARO 10 m telescope site is only intended as a coherent delay-reference telescope; hence, we will not take into consideration other source properties for this ] . Left and top panels are the integrated (summed) visibility angle over time and over frequency. The integration is done along the strongest section of the pulse width t w ( ), taking into consideration the scattering tail profile ∝ ν -4 (curved black dashed line). For these strong pulses (the same from Figure 8), its phase wrapping is close to 0 rad. Other conjugates of polarization pairs can be seen in Appendix B, Figure 28. Black horizontal lines are dead frequency channels.

analysis. The fringe-fitting algorithm will only be performed over the visibility as a function of frequency and normalizing by a strong (calibrator) visibility, as compared to the classical approach done at higher observational frequencies and narrow bandwidths (where delay and delay rate may be slowly varying quantities and can be computed with a two-dimensional Fourier transform; see Schwab & Cotton 1983).

In a traditional VLBI experiment, the coverage of the uvplane is a radial line whose length is related to the telescope bandwidth obs n n D . Here the track would show curvature due to the changing baseline projection while the pulse arrives as a function of frequency depending on its DM. In the uv-plane, this would be represented as shown in Figure 12. For a lower DM comparable to PSR B0531+21, the effect is linear, but for an FRB, it may not be. Figure 12 shows the Crab pulsar and an exaggerated simulation of an FRB in the uv-plane.

After an FRB (target) and its calibrator (reference) have been observed (i.e., baseband data have been acquired on sites A and B), we can proceed with forming visibilities and fringe fitting. First we independently form two (or more) visibilities (Equation ( 8)): the target and the reference. The steps to form a visibility are as follows:

1. Select a single polarization (either linear or circular basis) from sites A and B, i.e., V P A and V P B . If the orientation of linearly polarized feeds is known, then one could take the expected pair that maximizes the cross correlation (e.g., highly polarized source). In our baseline, we do not have this information, and it is a trial and error process. 2. Compute geometric correction (using difxcalc11) for a given (α, δ); time stamp on site A, t A ; and Earth locations of sites A and B. The process is repeated for each frame of the baseband data; hence,

V exp i , 13

with geo 0 f the geometric phase correction and ν the vector of all N = 1024 frequency channels (and elements ν n ). Equation (13) contains a matrix, V P A , where its phase is rotated by the vector

Equivalently this correction can be applied to site B with a positive phase term. 3. Coherently de-disperse (Hankins & Rickett 1975) pulse to an optimum DM (which is the same for both polarizations, V P geo A ( ) and V P B ), either by optimizing by structure or S/N. A precision of roughly 1 × -3 pc cm -3 is needed in this step. The correction needed for a frequency channel is given by the dispersion time:

where k DM = 1/(2.41 × 10 -4 ) sMHz 2 pc -1 cm 3 is the dispersion constant (see Lorimer & Kramer 2012). In practice, we need an algorithm that de-smears frequency channels (intra-channel alignment) and lines up the pulse to a reference frequency (inter-channel alignment). In this case, the natural kernel for coherent de-dispersion is:

with f r the reference frequency, f the frequency, and Δψ the phase amplitude of the transfer function (see Lorimer & Kramer 2012). After the transfer function has been applied, time stamps will only be relevant in the referenced channel, since the pulse itself will move on the baseband frames. Note that the geometric correction needs to be applied prior to the coherent de-dispersion, and this is only the case because τ geo is time dependent. After coherent de-dispersion, the geometric delay is a function of the frame number and the frequency channel. Most of the dispersion t dd will be removed in this operation leaving a small (but significant) fraction due to the ionospheric electron content (τ iono ). 4. Align pulses up to a frame precision ⎢ ⎣ ⎥ ⎦ ;

2.56 s total t m in practice, it is only the geometric and clock error delays, k shift in Equation (5), that are adding an integer frame delay to V P geo A ( ) with respect to V P B . The value for τ clock is a changing quantity, varying significantly over days 33 , and depends on the rate of the two masers (of site A and B). This value will change more than a frame over a timescale of weeks, but in principle it is a small and known value from previous correlations and an estimate can be chosen clock 0 t . 34 5. Form integrated visibility (same as in Equation (6)), and complex sum along the pulse width t w (in the case of (step 5) will not be constant at 400-800 MHz (see Section 5.6). 33 In our baseline of the order of 0.1 μs day -1 , see Section 3.5.2. 34

with t 0 the pulse center, t t 0 1 2 w  is an integer number of frames, and  P P t A B w á ñ (from now on dropping (geo) superscript) will only depend on frequency channels. Note that if strong scattering is present, then we may need to adjust the pulse profile with a frequency-dependent broadening function to increase S/N. Then each frequency channel will have a pulse profile as t w ∝ ν -4 .

After the two visibilities are formed, we can start fringe fitting. We first reference the source to the calibrator by making the ratio of visibilities (equivalent to Equation (12)),

with  norm the normalized (integrated) visibility (vector size N), and each element  n norm ( ) (i.e., the sum or integration of each frequency channel n). Notice that a Fourier transform over the frequency channels axis of  norm (Equation ( 17)) represents the lag-correlation, discussed in Section 3.1.

By taking the ratio in Equation (17), we are effectively computing Equation (12); the ratio will remove other delay terms such as frequency-dependent instrumentation effects (τ inst (ν)) and repetitive errors (ξ). By doing so we are just left with the residuals for τ iono , τ geo , and τ clock . We will define residuals as: δτ iono ionospheric residual and δτ for all other constant residuals left. The ratio will be most effective (or the residuals will be the smallest) when the source and calibrator are observed closer in time and in sky angle. The former is due to the fact that there will be less clock residual to correct, and the latter because there will be fewer delay contributions from the ionosphere. Now by finding the residual terms, we can optimize a likelihood function for the visibility model (Pearson 1999),

with F the fringe model to be fitted, n 2 s the variance 35 of the complex random variable  norm , and N the total number of frequency channels (ν n channel n). In principle, the localization is only present in the phase of the visibility and not in its

where j(ν n ; δτ, δτ iono ) is a phase model independent of the amplitude, flux of the source, and time t. Notice that the real part of Equation ( 18) can be marginalized without losing information, since

with δ always expressing target (t; FRB) minus reference (r; calibrator). The procedure presented above follows the same approach developed in Leung et al. (2021) but is expanded to an additional degree of freedom for χ 2 , where the differential delay contribution from the ionosphere is non-negligible at this baseline b.

The solution to the presented method can be achieved by either a two-dimensional grid search or by a gradient search (nonlinear least-squares minimization), although the latter needs high precision in the initial guessed values (see results 35 The variance of a complex number is expressed as:

36 Total electron content and DM have equivalent units: 1TECu ≡ 1 × 10 16 electron m -2 ≡ 3.24 × 10 -7 pc cm -3 . from Section 3.6 in Figure 18). A good initial estimate is the FFT over the normalized (integrated) visibility,

which returns the cross-correlation strength (lag-correlation in Equation ( 9)) prior to residual corrections being applied. Further, if dispersive delays were not present in the phase, n [ ] j , the maximum likelihood solution for fringe fitting would be the constant value obtained from the Fourier transform. Then the observable shift over the lag-axis (Equation ( 10)) is the lag-correlation, a good initial value for δτ or equivalently the center of a grid search.

Clock Stability Single Day

During 2020 May 9, six pulses of PSR B0531+21 were observed and baseband data collected at both sites with the CHIME VLBI tracking beam (defined in Section 2.5) and the ARO 10 m telescope, with each pulse separated by minutes. The cross correlation follows the fringe-fitting procedure explained in Section 3.4, but instead of a pair of visibilities from the FRB and calibrator, the six pairs of pulses are referenced to the same pulsar, a single pulse (and the strongest) visibility of the same dayfoot_6 . The PSR B0531+21 (reference) localization was used from the Australian Telescope National Facility pulsar data setfoot_7 (Manchester et al. 2005) Figure 13 shows the wrapping phases of the normalized (integrated) visibilities:

= a vector of N channels. The best model fit can be achieved with Equation (19). The quantity in Equation ( 24) is computed six times for each pair of target pulses  Y X t A B . The matrix of complex numbers  Y X r A B represents the strongest visibility (selected by measuring the amplitude of its FFT, lag-correlation Section 3.1), and for this observation, it is the third pulse observed on 2020 May 9. Figure 13 also contains the best fit from fringe fitting and lag-correlation phase models (colored lines).

Since pulses are separated only by a few minutes, the visibility normalization will remove most of the unknown delays (Equation (3)), leaving only a small fraction of δτ, and δτ iono residuals. It's important to note that the ionospheric delay will depend on the zenith angle at each location (see Thompson et al. 2017), τ iono = τ iono (z), and hence δτ iono ≠ 0ns. This implies that even if ionospheric turbulences are stable over an hour, there will still be a nonzero ionospheric delay since the differential zenith angle is changing over time.

Figure 14 shows the lag fringe fitting and lag-correlation (using Equation (23)), found in each of the phases of Figure 13. Figure 14

, in a single day of observations for six different pulses from PSR B0531+21. The third panel has zero phase since the normalization is its own visibility. Wrapping lines are the phase of the models from fringe fitting and lagcorrelation. Fringe fitting has an improved result since it takes into account the ionospheric contribution.

to the same day (Cary et al. 2021). The largest difference between them is roughly 3 ns. The bottom panel in Figure 14 shows the difference between the first-order delay with lagcorrelation and with fringe fitting. There exists a better estimation of the delay in the latter, since the δDM has also been fitted.

In addition, the fringe-fitting method (described in Section 3.4) was applied to each normalized (integrated) visibility, and a Gaussian error distribution was assumed. To find the set of two parameters (δτ, δDM), we calculated χ 2 over a grid in expected DM and delay values giving lag-correlation as the search start point. The search was conducted over ± 1 × 10 -6 pc cm 13 (centered at zero DM) and ± 2.5 ns centered at the peak of the lag-correlation. The minimum over the grid is selected with a confidence region of 68%,

Clock Stability over Multiple Days

Using the same techniques as Section 3.5.1, we now examine clock stability over multiple days using PSR B0531 +21 observations of one pulse per day. Observations were performed with the CHIME/FRB baseband back-end (defined in Section 2.5) and the ARO 10 m telescope dish. All pulses are referenced,  Y X r A B , to the first day of observation, following the fringe-fitting algorithm and forming visibilities. Figure 15 shows the FFT norm of the normalized (integrated) set of visibilities, where the strongest delay component (lag-correlation) is clearly visible as a peak. The phase graph of those visibilities is omitted because it will not add any helpful visualization, due to the fact that a lag on the order of 1 μs wraps the phase over 400 times.

The same (lag-correlation) delays from Figure 15 are then plotted in Figure 16. In the top panel, it can be seen that for every day of observation, there is a continuous time drift of ∼0.1 μs day -1 (model sloped solid line), the source of which is mainly due to the combined rate between the CHIME GPS crystal oscillator clock and ARO maser relative frequency offset. The slope has been adjusted to lag-correlation (circles) and fringe-fit (squares) independently. Both clocks should be 10 MHz, but it is not precise over the timescale of days; hence, they drift apart. The middle panel shows the difference between the best-fitted model (sloped solid line top panel) of the lagcorrelation trend (over several days) and the data points. The same procedure for fringe-fitting data points was used (where  or equivalently the cross-correlation strength, over several days for PSR B0531 +21 pulses. The top panel is the reference pulse, and the dashed line is the zero lag position. The lag-axis, τ u elements, is of a frame size with limits ±1.28 μs. The observed drift in the cross-correlation peak is analyzed in Figure 16. errors in the best-fitted line are included). The DRAO maser independent measurements with respect to the CHIME clock are also plotted (stars). Finally, in the bottom panel, the difference between the DRAO maser and the two methods is computed. By comparing stars and the other two sets of points (circles and squares), we see that there is a clear improvement in fringe-fit for earlier days, and only a few corrections need to be applied to the phase of the normalized (integrated) visibilities. The data points from 2020 October 28, 29, 31, and November 1 are less well constrained due to higher noise in the observations (as seen in the Figure 15 noise floor) and expected clock inaccuracies over more than five days. Hence, a clear value in χ 2 is hard to achieve (higher degeneracy in the χ 2 surface).

The other χ 2 degree of freedom is the δDM. This is not only the difference between sites, but since normalization took place (Equation ( 24)), the computed value from minimizing χ 2 is a differential ΔDM:

where ΔDM r is the crossed ionosphere from the reference pulse, and j represents all other individual pulses' ΔDM (targets). We can then compare the results from the fringefitting routine with the International Global Navigation Satellite System (GNSS) Service (IGS; Noll 2010), which is able to compute TEC values to a precision of 1 × 10 -6 pc cm 13 (or 2.3 TECu). The results are shown in Figure 17. TEC values interpolated with the ionospheric model take into consideration the zenith angle; this is referred to as slant TEC (sTEC), i.e.,

The IGS published maps have a fiducial ionosphere height of 450 km, and we assume a thin layer approximation in translating the published vertical TEC maps to sTEC, which estimates the TEC along the line of sight for each Earth location to the source.

Toward Localization

For CHIME and the ARO 10 m telescope, we have shown that the combined clock systems will not drift apart more than 10-100 ns over 24 hr. In addition, in the case where no calibrators are available over several days, the expected drift can be accounted for and removed. Nonetheless, limitations of the current coherent delay-referencing method (discussed in Section 5) will not provide a full VLBI diffraction-limited resolution.

Of utmost importance is the unknown contribution from the ionosphere, which depends on the Earth location and zenith angle of the observation. More generally, the ionospheric density along the lines of sight to the calibrator may be different from that along the lines of sight to the FRB.

From the analysis in Sections 3.5.1 and 3.5.2, the fringe fitting finds an estimated ionospheric contribution provided that our observations have a strong S/N. An example of a single point is presented in Figures 18 and19, which shows the χ 2 surface over δDM and lag space (grid search), and a zoom-in or slice section of χ 2 . The correlated pair corresponds to the first burst in Figure 14 observed on 2020 May 9, which has the largest scatter among all points (mainly due to the error in the CHIME GPS crystal oscillator) in a single-day observation.

The minimum χ 2 in Figure 18 is located at the center with cyan dashed lines. It is not surprising that the surface is highly irregular with neighboring values that could reach an equal level of degeneracy (as seen in the nearby lobes from Figure 19), but a high enough S/N and a good prior in lag and DM can solve this problem (no prior information was used in fringe fitting; Sections 3.5.1 and 3.5.2).

Figure 16. Clock stability between CHIME and ARO 10 m telescope over multiple days. The top panel shows the delays referenced to the first point at (0, 0) (peak respect to the center in Figure 15), and the sloped solid line is the best fit for the time drift. The middle panel shows the DRAO maser with respect to CHIME clock (stars) compared to the residual from the top panel (lagcorrelation and fringe fitting). The marker sizes represent the S/N strength in the lag-correlation with respect to the reference visibility (0, 0) point (dashed vertical line). The last observations have a lower S/N, and the fringe-fitting algorithm becomes less constrained in finding DM and lag. The bottom panel is the difference between the CHIME GPS crystal oscillator with respect to DRAO maser (independent from correlations) and the two methods.

Figure 17. Ionospheric values found from fringe fitting compared to those same days in the IGS database (Noll 2010). The DM values are the δDM over the referenced day (first day, vertical dashed line, 2020 October 22), see Equation (25). Error bars in fringe-fitting values are larger on the low-S/N cross-correlation cases, i.e., 2020 October 28, 29, 31, and November 1. The TEC difference between sites was computed by using the slant TEC (sTEC). The fringe-fit δDM agrees within errors compared to IGS, even after multiple days away from the referenced pulse.

To coherently find the delay-referenced localization of a target pulse with respect to the reference pulse, we calculate it as a function of the residual baseline angle δθ (defined in

where δθ(α, δ) = θ t -θ r (difference from target pulse with respect to reference pulse on sky) is a function of R.A. and decl. (see Thompson et al. 2017), but higher restrictions will be given to R.A. due to the east-west baseline. The angle of the normalized (integrated) visibility j ≈ j geo is the same as defined in Equation ( 12), but now the small angle approximation has been used for δθ. The reference angle, θ r , is the known position of the calibrator, and (as with the geometric delay) it will depend on: Earth locations, observation time (t A ), and R.A. and decl. of the reference source. Then we simply use Equation (2) and compute geo geo 0 t t = (difxcacl11) since there will be no residual using the assumed reference pulse. For the single-day observations (Section 3.5.1), this value corresponds to θ r = 1.29 rad. Equation (27) will only be true when and the lag-correlation. In Figure 19 the degeneracy line or slice with the most degeneracies is plotted with their confidence levels, and Figure 20 shows the same lobe in a corner plot using the obtained χ 2 weights (likelihood).

Figure 19. Degeneracy line, one-dimensional slice from the χ 2 grid in Figure 18. The slice shows a zoomed-in section of the grid search along the most degenerated lobes. Horizontal dashed lines are the confidence levels from the min 2 c value, 68%, 90%, and 99%. Vertical dashed lines are the lagcorrelation and χ 2 minimum. The plot corresponds to the most deviated point in a single-day observation (Section 3.5.1), and the one that has most degeneracies.

the residual of the clock delay δτ clock is very small, viz., δτ ≈ δτ geo (same as in Equation (2); Section 3).

The top panel top axis from Figure 18 shows the baseline angle, δθ, with respect to the delay δτ (centered panel bottom axis), and for this case, the localization error is 180 mas. Fringe-fitting errors can also be estimated. Figure 20 shows the main lobe from Figure 18 with a subnanosecond error, and a 1 × 10 -8 pc cm -3 DM. The 180 mas error in localization represents the largest scatter (as seen in the first point of Figure 14). In contrast to single-day observations, the worst case over multiple days (October 31) is in Figure 21, which shows the largest scatter from all points mainly due to the low S/N in the lag-correlation and calibration after several days. In particular, days 2020 October 28, 29, 31, and November 1 do not have a clear Gaussian lobe, and their error bars were estimated from the 99% ( 9.21 min 2 c + ) confidence region (Figures 16 and17).

In general, errors in fringe fitting over a single day and even over a few consecutive days are within ± 10 ns from the expected CHIME clock drift (with the exception of low S/N over multiple days running clock). Beyond that, the combination of clocks is not reliable. Careful measurements from clocks at each station can improve the performance beyond a 50 mas localization by adjusting the delay from the known CHIME GPS crystal oscillator jitter.

Early Science Results

As mentioned earlier, CHIME/FRB statistics predict that we should observe ∼4-7 FRBs per month in the ARO 10 m telescope and CHIME shared FOV. However, this does not take into account the up time from CHIME and the ARO 10 m telescope combined, plus the estimated altitude, azimuth, and Earth location of the 10 m telescope; i.e., the total portion covering the CHIME FOV may not be complete. Nevertheless, FRBs have been recorded in simultaneous dumps at CHIME and ARO 10 m telescope, but unfortunately their S/N (S/N CHIME < 25) on the ARO site was insufficient for both visualization in data and cross correlations. One of the most prominent observations produced thus far is of the magnetar SGR 1935+2154 (CHIME/FRB Collaboration et al. 2020), where the ARO 10 m telescope recorded baseband data but CHIME unfortunately did not.

Lastly, observations from the FRB 20210603A (not known to repeat) were recorded simultaneously at CHIME and ARO 10 m telescope on 2021 June 3, and initial results show a strong cross correlation after delay referencing to a calibrator (PSR B0531+21 pulse). Figure 22 shows the cross-correlation strength (Equation ( 23)), where the reference pulse is in the top panel. The known drift between the CHIME and ARO 10 m clocks can again be seen in the PSR B0531+21 pulses (similar to Figure 15), but not in the FRB since its sky location is different.

The FRB burst, not visible at the ARO 10 m telescope, was triggered and successfully correlated hours after the event happened. The FRB 20210603A had a high S/N at CHIME of  100 and DM = 500.162 ± 0.005 pc cm -3 (with a dispersion time t dd = 9.728s over the band).

The phase of the normalized (integrated) visibility  Arg norm [ ] can be seen in Figure 23. No clear signal is visible since phases are wrapping fast (see Section 3.5.2) due to the CHIME clock and ARO maser combination, and due to the difference in sky location of the FRB (only applicable in the second panel from top to bottom).

By taking the estimated delay from the lag-correlation (Figure 22), we can partially fringe stop the phases in Figure 23. This is:

with τ lc the solution obtained from the subframe crosscorrelation function, the lag-correlation (Equation ( 10)). There will be four τ lc , one for each panel. Figure 24 shows the angle of the normalized (integrated) visibility after being corrected by The quantiles (10% and 90%) show a rough estimate required for a good cross correlation in δDM space of 1 × 10 -8 pc cm -3 . Contour lines correspond to 68%, 90%, and 99% confidence levels. The lobe is also shown in Figure 19, where it is compared to the closer and most degenerated lobes in the degeneracy line from the entire χ 2 grid (Figure 18). The example corresponds to the first burst (left to right) from Figure 14. Multiple lobes are formed in the case where a true optimum cannot be determined and non-Gaussian uncertainties are high. Black dashed lines correspond to the 10% and 90% quantiles, the center line is the obtained minimum χ 2 , and off-center dashed lines are the grid search center given by the lag-correlation. Contour lines correspond to 68%, 90%, and 99% confidence levels.

the delay found in Figure 22. This early study shows a clear detection of an FRB referenced to a calibrator as seen in Figure 24 (second panel from top to bottom).

A more in-depth analysis will be carried out in a separate study showing the proper localization results as well as the ionospheric and long dispersion corrections required at these frequencies.

Clock Stability

The CHIME GPS crystal oscillator is a GPS disciplined clock, which returns absolute time stamps but lacks the nanosecond precision required for VLBI. In contrast, the DRAO maser fulfills the nanosecond precision but it does not support absolute GPS time stamps. A VLBI GPS disciplined maser is expensive and not feasible for the scale of the ARO 10 m telescope testbed. Nevertheless, our analysis has demonstrated that such deviations can be, in principle, corrected (e.g., by measuring pulsars; Figures 14 and16) and measured using the CHIME GPS crystal oscillator with respect to the DRAO maser (Mena-Parra et al. 2021;Cary et al. 2021). Such measurements can measure uncertainties and correct clock variations to achieve nearly ∼50 mas for the future CHIME/ FRB Outriggers project. 15, where we computed the cross-correlation strength (norm of the subframe crosscorrelation function; Equations (9) and ( 23)). The plot shows three pulses from PSR B0531+21 and the single FRB pulse. The PSR B0531+21 (referenced to 2021 May 29) has a similar trend to that seen in Figure 15, but only over the pulsar pulses, since the FRB burst is expected to be at a different R.A. sky location. The normalized (integrated) visibility phase can be seen in Figure 23. [ ] from PSR B0531+21 pulses and FRB 20210603A. Phases have been referenced to the PSR B0531+21 pulse on 2021 May 29, and there is no correction applied on them. Figure 24 shows the same phases but corrected by the estimated lag τ lc Equation (28), a partial fringe stop.

Figure 24. Normalized (integrated) visibility phase after phase correction (Equation ( 28)) from PSR B0531+21 pulses and FRB 20210603A. As in Figure 22, phases have been referenced to the pulse from 2021 May 29. In this early analysis, only a phase correction over a constant delay (partial fringe stop) has been applied, and the ionospheric correction has been set free (not corrected). The linear and nonlinear wrapping can be seen since the strong ionosphere has not been removed. The early study shows a clear detection of the FRB referenced to a calibrator (second panel top to bottom).

The limiting factor for this calibration will be given by the FRB position and its proximity to the Galactic Plane (higher pulsar density) as well as the S/Ns of both the calibrator and the source.

Ionospheric Contribution

The ionospheric delay is a problem that has been partially solved in the analysis sections. This is due to the fact that for FRB detections, it is highly unlikely that the event will cross the same portion of the ionosphere as the calibrator. In reality, the two will likely be off by a few degrees in the sky, and the calibrator will be observed minutes before or after the event. In contrast, the analysis presented here relied on single pulses from the same pulsar as both target and reference, and the only differences in ΔDM t and ΔDM r are cases given by the zenith angle over minutes (Section 3.5.1) and day-to-day ionospheric changes (Section 3.5.2). In the case of an FRB, we expect a higher degree of uncertainty since  norm will have a considerable fraction of residual ionospheric delay, δτ iono , in its phase after normalizing (Equation ( 24)). Ionospheric models such as IGS (Noll 2010) do not have the required TEC precision that we need to successfully correlate at a desired angular precision of 50 mas, and δDM ∼ 1 × 10 -8 pc cm -3 (Figure 20). However, high enough S/N levels can partially remove this restriction and return a clear minimum in the χ 2 domain. An additional resource to improve the search is to add a prior probability over the ionospheric delay, using time-series GPS data to estimate the scale and type of ionosphere fluctuations (∼10 TECu). Adding this prior in the χ 2 search can be expressed as: DM , DM , 29

This

effect becomes relevant at DM  200 pc cm -3 at comparable baselines. Figure 25 shows the effect of the geometric delay over the dispersion delay. The quantity Δτ geo is the difference of geometric delay (CHIME and ARO 10 m telescope baseline) at the start and end of the pulse dispersion (400 MHz bandwidth). The calculations were done on difxcalc11 centered at PSR B0531+21 coordinates, similarly as in Equation (31). A special treatment of the band may be required: for example, separating the band in two or more sections with a k n shift [ ], forming visibilities, and then concatenating phases. But in general, combining a coherent dedispersion with a time variable geometric delay is not a trivial problem to solve.

As explained in Section 3.4, we are interested in the differential DM between sites, which is more than just the ionospheric contribution at sites A and B. There will also be a Doppler effect given by the Earth's rotation respect to the solar system barycenter (SSB), which will impact the velocity of each telescope (Rankin et al. 1970;Pennucci et al. 2014). This results in DM DM  ~, a modified DM,

with c the speed of light, and v the velocity of one site. Equation (34) is equivalent to mapping the frequencies as

in the coherent de-dispersion kernel (Equation ( 16)). Doppler-shifting the DM during coherent dedispersion is equivalent to applying the time-dependent geometric phase correction (which is equivalent to Lorentz transforming the data to the CHIME rest frame), geo 0 f , followed by coherent de-dispersion to a reference DM (steps 2 and 3; Section 3.4). We take the reference DM to be that measured at CHIME site, averaged over the ∼1.1 s duration of the PSR B0531+21 sweep, and neglect the transformation to the SSB. This effect could be of an order of 1 × 10 -4 pc cm -3 (above the required δDM from fringe fitting), where other experiments such as the Pulsar Timing Array correct for it.

CHIME/FRB Outriggers Project

The CHIME/FRB Outriggers project will have multiple CHIME single cylinders with two baselines thousands of kilometers from CHIME in order to observe and localize FRBs down to a 50 mas precision. Confirmed sites are: Allenby BC, Canada, Green Bank Observatory WV, USA, and Hat Creek Radio Observatory CA, USA. For calibration, the project will make use of pulsars (which have well-known localizations of 10-20 mas) as an alternative compared to steady-source calibrators, and because of CHIME's FOV, the longest period of time without a pulsar on beam is roughly ∼1 h. These Outriggers will have tracking beams in each station, beamforming capabilities, and a triggering system, similar to the one implemented at the ARO 10 m telescope testbed. Correlation routines and methods described here will serve as a basis for the project. In particular, the Outriggers beamforming capabilities will have pulsars available for calibration (most of the time), with a nearly in-beam calibration scheme (Leung et al. 2021), by digitally pointing to target and references simultaneously (leaving only a sky angular distance interpolation), i.e., preserving coherence along the FRB dispersion time.

The project will also be interested in steady sources (VLBI calibrators) in order to account for the list of potential challenges described in this section. Such observations will enable us to characterize the VLBI network and measure our precision/uncertainties compared to the proposed pulsar calibration method.

Conclusions

In the presented work we have developed a testbed for the CHIME/FRB Outriggers project with the capability of studying FRB localizations, using pulsars to coherently delay reference, and demonstrated clock stability between an independent maser (ARO 10 m telescope) and GPS crystal oscillator (CHIME). The 1960s-era telescope at ARO was refurbished and updated to a modern system. The 10 m dish is able to receive triggers from CHIME/FRB, record baseband data, and transfer it back to the cross-correlation site in a semiautonomous process. We are currently recording single PSR B0531+21 pulses everyday (for potential calibration and clock stability), and in the future, this will be increased to several pulses per day as well as other potential sources. The 10 m dish has also demonstrated reliability over extreme weather conditions, poor internet connections (satellite), and low maintenance on site during the 2020 COVID-19 pandemic.

Our worst localization scenario is θ ≈ 200 mas (or equivalently -10-10 ns; baseline angle) in multiple-day observations, where only high-S/N bursts from the first five days in Figure 16 were considered for this estimate. Other sources of localization error are: the on-sky separation between FRB (target) and calibrator (reference), the simultaneity of the observations, whether or not the ionospheric contribution is known in the FRB and calibrator directions, the clock jitter (CHIME and ARO 10 m telescope clock combination), and lastly whether or not assumptions in our correlator model remain true (Equation ( 19)). Nevertheless, future calibrations (CHIME/FRB Outriggers project) will have access to pulsars (calibrators)  1 hr with respect to the observed target (with the possibility of a nearly simultaneous beamformed baseband recording), plus comparing CHIME clock and DRAO maser can add an extra set of corrections to the clock (CHIME end; Mena-Parra et al. 2021). Clock corrections can be done by constantly monitoring calibrators in VLBI, in addition to using measurements of the DRAO maser to correct the clock jitter to the expected delay (given by calibrators). This leaves a comfortable window for the requirement of 50 mas proposed by the CHIME/FRB Outriggers project.

On the other hand, testbed delays over a single day behaved as expected given the precision of the CHIME and ARO 10 m telescope clock system, yielding the required precision of DM ∼ 1 × 10 -8 pc cm -3 for a strong cross correlation (below Doppler effect, Section 5.6, and without any ionospheric prior in , Equation (18)) and eventual localization.

Lastly, we showed proof of an FRB candidate cross correlated with baseband data, providing enough S/N and phase information to be considered as a true VLBI correlation. A proper and more in-depth study of this event will be discussed in an upcoming paper.