3.1. Offset dependencies

As described in Section 2.5, we compared the positional offsets derived for each target source scan with the relative separation between the target scan and the selected calibrator scan in time, elevation, and angular distance to determine any potential dependence on these observational factors. This was done for each beam and both frequency bands. Since no significant differences were found between the offset distributions for the beams (see Section 4 and Figure 5) and the offset dependencies for each beam were consistent, we take beam 30 to be a representative beam. We find no significant trend in offset versus angular separation from the calibrator scan and so omit these plots.

Figures 1 and 2, respectively, show the mid- and low-band offset dependencies on the fractional separation in time (MJD) and elevation (degrees) relative to the calibrator scan for RA (top two panels) and Dec. (bottom two panels). We find some dependence on time and elevation in both bands, with the trends in Dec. more pronounced than those in RA, but these are generally weak in both directions with a lot of scatter.

This scatter is more significant for the low-band data. However, the frequency-dependent gain features in the bandpasses discussed in Section 2.2 possibly contribute to this increased scatter. While good beam weights will lead to a well-behaved PAF beam that closely approximates the desired Gaussian form, poor beam weights could potentially cause deviations from this ideal in a frequency-dependent way. If the bandpass is then taken at a fixed location (e.g., the nominal beam centre), the resulting gain as a function of frequency will be distorted (relative to that obtained from a more Gaussian PAF beam), which is true of many of the low-band scans. However, as this is true of both the data presented here and other typical ASKAP observations, our interpretation should be valid for real-world observations in general.

Notably, the largest offsets in elevation for the low-band data are those of J1557, which is the most northerly source in our sample. As discussed in Section 2.4, our positional fitting process neglects correlations between the right ascension and declination uncertainties. At low elevations, however, the synthesised beam becomes increasingly elongated, leading to both a larger major axis for the synthesised beam and an increasing covariance between the errors in these two coordinates (depending on the synthesised beam’s position angle). However, the dependence of the potential underestimation of offset on elevation due to the changing synthesised beam properties is expected to be smaller than the overall trend seen here, and so we conclude that there is some offset dependence on elevation.

As noted in Section 2.2, we also imaged the low- and mid-band visibilities using Briggs weighting with a robustness of 0.0. We detected no significant deviations in the offset trends (with time, elevation, or angular separation) found when using the Briggs versus naturally weighted images for either frequency band.

Figure 3 shows the frequency dependence in the beam 30 offsets versus time. As with the offset distributions, we found no significant differences between the beams and therefore take beam 30 to be representative. While the dependence appears to be non-linear in some scans, fitting the offsets versus wavelength in each scan showed that most scans are adequately described by a linear fit, with only a few scans (across all beams) being more consistent with a non-linear model. Linear growth in offset with wavelength is consistent with a frequency-independent phase error. In particular, the size of the synthesised beam grows linearly with wavelength, and so given a fixed fraction of the PSF (i.e., a constant phase error), the offsets would likewise grow linearly with wavelength.

3.2. Implications for future observations

Fundamentally, the method used to obtain astrometric corrections for the ASKAP frame requires the creation of a model of the field and the use of this model to determine the positional corrections to be applied. This can be accomplished either through self-calibration to a sky model formed from the data or via field source comparison to an external model known to have sufficient accuracy (i.e., the current method in use). In the case of the former, this requires a reasonably high astrometric registration accuracy. At present, the ASKAP hardware correlator data when fully processed has a known systematic astrometric offset of up to ${\sim}1$ arcsec, which is well above the statistical uncertainty in the position of a typical FRB ( ${\sim}100$ mas). However, this is expected to improve in the future, with a reasonable estimate of the accuracy limit attainable likely on the order of 0.05 arcsec—i.e., well within the uncertainty obtainable for a high S/N FRB and comparable to that of the Faint Images of the Radio Sky at Twenty-Centimeters (FIRST) catalogue (Becker et al. Reference Becker, White and Helfand1995).

As discussed in Section 2, the current method is largely limited by the number and brightness of field sources present, which vary stochastically from field to field. To that end, when employing this comparison method, improvements to the typical accuracy we can obtain in any given field using the CRAFT software correlator data must come from an increase in sensitivity, which is attainable via the longer integration times used for the ASKAP hardware correlator data. For example, a 60-second integration (i.e., 20x that of CRAFT) would result in $\sqrt{20}{\sim}5$ x higher sensitivity than the current CRAFT data products and would therefore reduce the uncertainty in a typical field to that below the statistical positional uncertainty for a typical ASKAP FRB. While the integration time of the ASKAP hardware correlator data (default 10 s) would depend on the configuration required for the observation with which CRAFT would run commensally, we would be able to reprocess a subset of the data suitable for use with the CRAFT voltages, including selecting a reasonable duration (i.e., longer slices of data for faint fields if needed) roughly centred on the temporal position of the FRB. The use of 5-minute scans in this work and the results obtained are therefore representative of what would ultimately be used and the typical corrections these scans would yield when using the hardware correlator data to conduct the field source comparison.

However, directly applying corrections derived using the hardware correlator data products to the CRAFT data products is only feasible if there are no systematic differences resulting from the different data paths. These could, for instance, result from differences in the geometric models used in the two correlators or the effects of the requantisation of the CRAFT voltages. The datasets we present here allow us to place upper limits on the maximum size of any such systematic differences.

Figure 4 shows the offset probability distribution functions (marginalised over beam) obtained for both the CRAFT software correlator and ASKAP hardware correlator positions (with the former measured from the naturally weighted images) less the catalogue positions. The PDF formed from the difference of the CRAFT- and ASKAP-derived positions averaged over beam is also shown along with the 16th, 50th, and 84th cumulative percentiles derived from evaluating the normalised cumulative distribution function of this difference (see also Table 1).

Table 1. The 16th, 50th, and 84th percentiles of the CRAFT-ASKAP cumulative distribution functions for the low- and mid-band observations. Note that the top set of values were derived from the offset distributions made using naturally weighted CRAFT images while the bottom set are those from Briggs weighted CRAFT images.

The mean difference between the offsets derived using the CRAFT and ASKAP correlators should be zero in the case of identical inputs, centre times, calibration solutions, and geometric models. However, the inputs are not identical (e.g., the ASKAP scans are much longer than the CRAFT ones), the times of both scans are not precisely centred (see, e.g., Figures 1 and 2), the derived calibration solutions (in this simple case of bandpass and phase calibration only) differ (e.g., due to the larger CRAFT bandpass, which is also not centred at the ASKAP band centre), and there are potentially small differences in the geometric models used. These relative deviations can lead to a non-zero mean difference between the positions that is expected to change with each observation, as evidenced by the differential offset between the positional offsets derived for each scan (see Figures 1 and 2). Any mean measurement, then, is a function of the sources and the parameter space sampled. In order to sample this parameter space in a representative manner, we use a set of sources with a range of RA and Dec. positions and relative separations in time, elevation, and angular offset. For each of the frequency bands and the combined positional axes, we find that the central 68% of the sample spans the differential offset range of ${\sim}0.5-0.6$ arcsec (low-band) and ${\sim}0.2-0.3$ arcsec (mid-band) (i.e., taking the maximum absolute values of the 16th and 84th percentiles across position shown in Table 1). Given the distributions are not perfectly Gaussian, we conservatively estimate the residual systematic offset between the ASKAP and CRAFT frames to be the larger of the position-combined asymmetric percentiles. Accordingly, in the simple case of applying a bandpass and phase calibration and in the limit of high S/N (which will always be the case with the hardware correlator data), we estimate the systematic uncertainty of low-band and mid-band observations, respectively, when using hardware correlator-derived corrections will be ${\sim}0.6$ arcsec and ${\sim}0.3$ arcsec in RA and Dec.

Additionally, as shown in Figures 1 and 2, although the residual offsets between positions derived from the hardware and software correlator data products are not constant, there is no dependence on time or elevation in the systematic offset between the two frames. Likewise, we find no trend when comparing the offsets versus angular separation. We therefore conclude that we should be able to obtain good solutions when applying the hardware correlator-derived offsets to the software correlator data products regardless of differences in time, elevation, or angular separation.

As discussed in Section 2.2, in addition to natural weighting, images were made using Briggs weighting with a robustness of 0.0 to quantify any variation or improvement in the offsets and their uncertainties due to the resultant increase in resolution. In the high-S/N regime, for reasonable (u,v) coverage, and across a wide range of elevations, we find that Briggs weighting with a robustness of 0.0 yields improvement of 17% and 33% in the 68% confidence intervals we derive respectively for the low- and mid-band residual offsets between the CRAFT and ASKAP image frames (Table 1). This is unsurprising in the high-S/N case we’ve studied here since Briggs weighting will result in a closer approximation of the PSF when fitting the positions (Section 2.4). However, while Briggs weighting performs better for the parameter space we’ve tested here, future investigations should confirm that this result holds in the low S/N regime as well as for observations with different antenna arrangements or smaller sub-arrays. Typically, CRAFT field image sources have low-S/N, and so both the loss of sensitivity and higher resolution obtained when using Briggs weighting could result in reduced S/N and poorer approximations of the true PSF. Of note, once the use of the hardware correlator data is employed, these data will always be in the high-S/N regime, mitigating any issues arising from low-S/N sources. Further studies on the affects of array size and configuration as well as fitting low-S/N sources on the typical offsets and uncertainties will be conducted in a future work.

3.3. Modelling large-scale ionospheric effects

As described in Section 2.2, along with the datasets obtained by applying the nominal calibration solutions, we also produced datasets which additionally model the variations in the ionosphere—in particular, the dispersive delays caused by deviations in the total electron content (TEC) between sightlines—by including corrections derived using the aips task tecor. For this, we used the International GPS Service for Geodynamics (IGS) Global (IGSG) ionosphere maps in the The IONosphere Map EXchange (IONEX) format for each observing day to derive these solutionsFootnote e.

We found that the day the mid-band data were recorded had increased ionospheric activity relative to the day the low-band observations were conducted. This led to larger ionosphere corrections for the mid-band data. To determine the overall affect of these solutions, we differenced the offsets calculated for each scan, using the offsets derived from the data with the nominal calibration solutions applied as the reference. For the low-band data, we find typical differential offsets for beams 15, 28, and 30 in RA and Dec. of (0 mas, 0 mas), (0 mas, 0 mas), and (12 mas, 10 mas), with differential offsets up to 13 mas in RA across all beams and 10, 20, and 30 mas in Dec. for beams 15, 28, and 30, respectively. In contrast, for the mid-band data, we find differential offsets of (42 mas, 60 mas), (41 ms, 50 mas), and (41 ms, 60 mas) in RA and Dec. for beams 15, 28, and 30, respectively, with maximum values of 985, 938, and 879 mas in RA and 350, 320, and 290 mas in Dec. respectively for beams 15, 28, and 30.

Since the corrections in the model used are smoothed over approximately 2 h and roughly $2^{\circ}$ , this model is not well suited to small arrays due to this coarse sampling and the need to interpolate over a much larger spatial extent than the resolution of the array. This method therefore does not probe small-scale ionospheric effects but does provide an estimate of the large-scale effects of the ionosphere over a range of activity levels. Our results indicate that the ionosphere might contribute to the spatial and temporal offsets we see, but further investigation is required.

4.1. Optimising field source offset derivation

As described in Section 2, the current method used to correct the astrometry of the FRB image frame uses a simple weighted mean to derive the final mean image offsets in RA and Dec. (i.e., assuming any offsets to be simple translations of the image frame) along with the associated systematic uncertainties (i.e., either the error in the weighted mean of the estimated uncertainties or the scatter in the points about the mean in the case of scatter-dominated offsets, as discussed in Section 2). We note that, as we have done in this work (Section 2.4), this method uses positional uncertainties projected onto RA and Dec., which loses the ability to show covariances in the final systematic uncertainty between RA and Dec. In order to investigate if the current method is optimal and, if not, what a preferred model might be, we test various hypotheses to determine the most reasonable estimates of the true mean FRB image frame offsets ( $\mu^t_{\alpha}$ , $\mu^t_{\delta}$ ) and uncertainties ( $\sigma^t_{\alpha}$ and $\sigma^t_{\delta}$ ). We use the t and e superscript notation throughout to respectively indicate the true and estimated quantities.

For each FRB, the data consist of $\textrm{N}_{src}$ field source offsets in RA, $\alpha^e_i$ , and Dec., $\delta^e_i$ (i.e., taking the relative offsets of the $i^{th}$ field source from the nominal source positions as estimates of the image frame offsets in RA and Dec.) and estimates of the uncertainties in these individual offsets ( $\sigma^e_{\alpha,i}$ , $\sigma^e_{\delta,i}$ ). Table 2 lists the FRBs used for this analysis. FRB 180924 and FRB 181112 were not included in the sample because their field source comparisons were based on a single continuum source in their respective fields.

Our initial hypothesis ( $\textrm{H}_0$ ) assumes the provided estimated uncertainties ( $\sigma^e_{\alpha,i}$ , $\sigma^e_{\delta,i}$ ) on the measured field source offsets correctly estimate the true uncertainties in the image frame offsets as measured by each source. We also assume that all estimated uncertainties of the measured field source offsets are independent Gaussian random variables with a mean of zero and the stated deviation. In this case, the $i^{\textrm{th}}$ field source for some FRB with RA and Dec. offsets [ $\alpha^e_i$ , $\delta^e_i$ ] and estimated errors [ $\sigma^e_{\alpha,i}$ , $\sigma^e_{\delta,i}$ ] will be related to the true mean image offsets [ $\mu^t_{\alpha}$ , $\mu^t_{\delta}$ ] via

(1) \begin{align} [\alpha,\delta]^e_i &= \mu^t_{[\alpha,\delta]} + d[\alpha,\delta]^e_i\ \textrm{and} \end{align}

(2) \begin{align} d[\alpha,\delta]^e_i &\sim N(0,\sigma^e_{[\alpha,\delta],i}), \end{align}

where N denotes the Normal distribution and we use the square bracket notation throughout to indicate evaluation of equations using either RA or Dec. values.

Under the assumption that $\textrm{H}_0$ is true, the best estimates $\Delta[\alpha,\delta]$ of the true mean image offsets $\mu^t_{[\alpha,\delta]}$ are given by the weighted estimates

(3) \begin{equation} \begin{aligned} \Delta[\alpha,\delta] &= \frac{\sum_i w_i [\alpha,\delta]^e_i}{\sum_i w_i} \\[3pt] w_i &= \frac{1}{\Big({\sigma^e_{[\alpha,\delta],i}}\Big)^2}, \end{aligned}\end{equation}

where $w_i$ is the weight for either the RA or Dec. of the $i^{th}$ source. To test the validity of our hypothesis, we can use a chi-squared ( $\chi^2$ ) test to construct an unbiased estimator of the sample variance given by

(4) \begin{equation} s^2_{[\alpha,\delta]} = \sum w_i ([\alpha,\delta]^e_i - \Delta[\alpha,\delta])^2,\end{equation}

since $s^2_{[\alpha,\delta]} \sim \chi^2_{\textrm{N}_{\textrm{src}}-1}$ , where $\chi^2_{\textrm{N}_{\textrm{src}}-1}$ is a $\chi^2$ distribution with ${\textrm{N}_{\textrm{src}}-1}$ degrees of freedom. Simultaneously checking both RA and Dec. yields

(5) \begin{equation} \begin{aligned} s^2_{\alpha,\delta} &= s^2_{\alpha} + s^2_{\delta} \\ &\sim \chi^2_{2(\textrm{N}_{\textrm{src}}-1)}. \end{aligned}\end{equation}

Using the data from 9 FRBs (see Table 2), the above procedure was performed on each FRB. Individual $s^2$ values were calculated, fitting mean $\Delta[\alpha,\delta]$ values. The results of all $\chi^2$ tests are given in Table 2. Only FRB 200430 and FRB 200906 have a sufficient number of sources to allow for a sensitive test of $\textrm{H}_0$ , and these reject the null hypothesis at high significance.

To perform a more accurate test, we also sum over the $s^2$ values of all FRBs to obtain the total variance

(6) \begin{equation} \begin{aligned} s^2_{\textrm{tot}} &= \sum_{j=1}^{\textrm{N}_{\textrm{FRB}}} s^2_j \\ &\sim \chi^2_{\textrm{NDF}_{\textrm{tot}}}\end{aligned}\end{equation}

where the total number of degrees of freedom (NDF) is defined as

(7) \begin{equation} \textrm{NDF}_{\textrm{tot}} \equiv \sum_{\textit{j}=1}^{\textrm{N}_{\textrm{FRB}}} 2(\textrm{N}_{\textrm{src},\textit{j}} - 1), \end{equation}

and j is the $j^{th}$ FRB. This yields a one-sided p-value of $2.19 \times 10^{-12}$ ; that is, the estimated errors are smaller than the true errors at 7.0 $\sigma$ significance.

Our findings could be due to the presence of systematic effects in the data. Potential systematics are likely to arise from effects such as unmodelled mismatched source structure (due to differences in the frequency and angular resolution in the compared observations), which would act to shift the fitted CRAFT field source centroid relative to that of the reference source; or directional dependencies in the offsets (e.g., due to a wedge in the ionosphere). Both would result in the additional error ( $d[\alpha,\delta]^e$ ) not being centred at zero. Thus, we would not expect $\textrm{H}_0$ to accurately model real-world observations.

We therefore discard $H_0$ and examine two further reasonable alternatives. Ordering by complexity, the hypotheses are:

• $\textrm{H}_1$ : All uncertainties are equal but unknown; i.e., $\sigma^t = C$ . In the limit in which the measurement uncertainty becomes negligible compared to unmodelled systematic effects (i.e., all field sources have a high S/N), we would expect $\textrm{H}_1$ to be satisfied, as we would reach a ‘floor’ in the attainable precision set by systematic contributions unrelated to the measurement S/N. The simplest possible form of such a systematic error floor would be a constant independent of the source.

• $\textrm{H}_2$ : The true uncertainties are proportional to the estimated uncertainties, with a constant of proportionality to be estimated from the data; i.e., $\sigma^t_i = C \sigma^e_i$ . This provides the simplest possible way to include the effects of unmodelled error contributions to the field source offsets and avoid the underestimation of the total uncertainty (on average) that would result from neglecting them.

In the following, we outline the results of testing each in turn.

Assuming $\textrm{H}_1$ to be true, we first calculate unweighted mean offsets. Each resulting residual offset r is calculated as

(8) \begin{equation} r_{[\alpha,\delta]} = [\alpha,\delta]_{i,j}^e - \Delta[\alpha,\delta]_j,\end{equation}

and then scaled to account for the residual-minimising effect of the procedure employed (i.e., the reduction in error due to the points being used to estimate the mean) when fitting the residuals versus estimated uncertainty

(9) \begin{equation} r^{\prime}_{[\alpha,\delta]} = r_{[\alpha,\delta]} \sqrt{\frac{\textrm{N}_{\textrm{src}}}{\textrm{N}_{\textrm{src}}-1}}. \end{equation}

We then determine if these scaled residuals show any dependence on the estimated uncertainties, which would reject $\textrm{H}_1$ (i.e., the assumption that the distribution of offsets is a constant). A complexity arises, however, because three or more sources per FRB are required to obtain any meaningful data. Two sources, for instance, will always result in the fitted mean being halfway between them, and so the result will be independent of the true uncertainty. Thus, only data from FRBs with $\textrm{N}_{\textrm{src}} > 2$ are included in the fit.

The scaled residuals versus estimated uncertainty ( $\sigma^e$ ) results were fit using standard linear regression (using the linregress function in SciPy) using all data as well as RA and Dec. independently. These fitted lines are also unweighted since the scatter should be independent of $\sigma^e$ under the assumption of $\textrm{H}_1$ . We find

(10) \begin{equation} \begin{aligned} r^{\prime}_{\alpha,\delta} &= 1.76\sigma^e_{\alpha,\delta} - 0.3 \\[3pt] r^{\prime}_{\alpha} &= 0.88\sigma^e_{\alpha} + 0.2 \\[3pt] r^{\prime}_{\delta} &= 2.34\sigma^e_{\delta} - 0.7. \end{aligned}\end{equation}

Under $\textrm{H}_1$ , the true slope is zero. Testing for consistency with this hypothesis, the (two-sided) p-values for our fitted slopes are then $3.5 \times 10^{-3}$ and $2.6 \times 10^{-6}$ for the RA- and Dec.-only fittings, respectively. Simultaneously fitting RA and Dec. yields a p-value of $1.0 \times 10^{-8}$ . Therefore, there is very strong information that $\sigma^t$ and $\sigma^e$ are positively correlated, which is as expected. Therefore, we discard $\textrm{H}_1$ and proceed to examine $\textrm{H}_2$ .

According to $\textrm{H}_2$ , we calculate the weighted mean offsets in RA and Dec. using Equation (3). It can be shown that for a weighted mean, $\bar{x}$ , the expected deviation of the $i^{th}$ data point $x_i$ from that mean is given by

(11) \begin{eqnarray}\langle (x_i - \bar{x})^2 \rangle & = & \frac{1}{w_i} - \frac{1}{\sum_k^{N_{\textrm{src}}} w_k} \nonumber \\[3pt] & = & \sigma_i^2 - \frac{1}{\sum_k^{N_{\textrm{src}}} \frac{1}{\sigma_k^2}}.\end{eqnarray}

That is, the expected variance between a point and its estimated mean is not reduced by the usual $(\textrm{N}_{\textrm{src}}-1)/\textrm{N}_{\textrm{src}}$ factor of Equation (9) but rather a factor $\eta$ :

(12) \begin{equation} \eta_i = 1 - \frac{1}{\sigma_i^2 \sum_k^{N_{\textrm{src}}} \frac{1}{ \sigma_k^2}}.\end{equation}

The residuals defined according to Equation (8) must therefore be multiplied by $\eta^{-0.5}$ :

(13) \begin{equation} r^\prime_{[\alpha,\delta]} = \frac{1}{\sqrt{\eta}} r_{[\alpha,\delta]}, \end{equation}

in order for them to be unbiased estimates of their standard deviation. This results in a factor of between 1 and 2 in this work.

We use linear regression to obtain weighted fits of the residuals $r^{\prime}$ as a function of their estimated uncertainty and find

(14) \begin{equation} \begin{aligned} r^{\prime}_{\alpha,\delta} &= 1.43\sigma^e_{\delta} \\ r^{\prime}_{\alpha} &= 1.40\sigma^e_{\delta} \\ r^{\prime}_{\delta} &= 1.67\sigma^e_{\delta}. \end{aligned}\end{equation}

The best-fit value of C is 1.43. If $\sigma^e$ is correct, then the best-fit value of the slope of a fit to the residuals is expected to be:

(15) \begin{eqnarray} \langle C\,\rangle& = & \frac{2 \int_0^{\inf} r p(r) dr}{2 \int_0^{\inf} p(r) dr}, \end{eqnarray}

(16) \begin{eqnarray}p(r) & = & \frac{1}{\sigma^t \sqrt{2 \pi}} \exp\left\{-\frac{r^2}{{2(\sigma^t})^2}\right\}.\end{eqnarray}

Here, p(r) is the assumed Gaussian distribution of the errors. However, since the fits are against $|r|$ , the integrals in Equation (15) are evaluated from 0 to infinity. The denominator evaluates to unity since it is a normalised probability distribution. The numerator of Equation (15) is

(17) \begin{eqnarray}&2& \int_0^{\inf} r \frac{1}{\sigma^t \sqrt{2 \pi}} \exp \left\{-\frac{r^2}{{2(\sigma^t})^2}\right\} dr \nonumber \\[3pt]&=& \sigma^t \sqrt{\frac{2}{\pi}}.\end{eqnarray}

In other words, while under $\textrm{H}_2$ , using the weighted mean yields the correct best-fit position, but we expect the mean of the errors to underestimate $\sigma^t$ by a factor of $\sqrt{\frac{2}{\pi}}\approx 0.80 $ .

It therefore appears that treating $\sigma^t = C \sqrt{\pi/2} \sigma^e \approx 1.25 C \sigma^e$ is correct. Given we find a best-fit of $C=1.43$ , the best-fit true uncertainty is $\sigma^t = 1.79 \sigma^e$ .

While our current dataset fails to reject $\textrm{H}_2$ and found reasonable consistency with its predictions, this model can result in either over- or underestimating the true uncertainties, depending on the characteristics of the continuum source sample in a given field. Additionally, we can only confirm cases in which over- or underestimation has occurred in fields with sufficient background sources; with only a handful or sources, it is impossible to determine if this has taken place and to what degree.

Thus, while $\textrm{H}_2$ is a reasonable alternative to the currently used simple weighted mean method of estimating the mean image frame offsets and uncertainties and is testable given the current sample size of localised FRBs, the model should be further refined as the number of usable FRBs grows. One such refinement would effectively combine $\textrm{H}_1$ and $\textrm{H}_2$ —i.e., introducing a systematic term (A) in addition to the random estimated uncertainty: $\sigma^t_i=\sqrt{A^2+C^2 (\sigma^e_i)^2}$ . This might more accurately capture the systematic effects known to potentially exist in the data (e.g., source structure and directional dependencies in the individual offsets). In addition, future models could be parameterised over a given range appropriate to a particular catalogue or instrument, thereby accounting for the characteristics of the reference used for the comparison (e.g., the astrometric accuracy limits of a catalogue). In addition, the astrometric measurements outlined in this work could be performed on random fields with several sources (rather than on single sources), which would increase the number of degrees of freedom and facilitate multiple individual tests of a given model. This would, for example, enable consistency crosschecks of the scale factor derived for $\textrm{H}_2$ .

Until this becomes possible, we take $\textrm{H}_2$ as our working hypothesis, namely that the best estimate of the offset is given by a weighted mean according to the estimated errors $\sigma^e$ but that the magnitude of these errors—and consequently the estimated error in the weighted mean—should be scaled up by a factor of 1.79.

4.1.1. Deriving updated positional information

Table 3 lists the published positions, offsets, and uncertainties for each FRB in our sample along with the updated uncertainties obtained when using the above derived scale factor. While the scale factor only changes the systematic uncertainty estimation, we have also updated the positions and offsets for the published FRBs that were affected by the formerly incorrect weighting scheme detailed in Macquart et al. (Reference Macquart2020), which used $w_i = 1/\sigma^e_{[\alpha,\delta],i}$ as the weights in Equation (3) rather than these values squared.

Table 3. Published FRB positions ( $[\alpha,\delta]_{\textrm{pub}}$ ), weighted mean offsets ( $\Delta[\alpha,\delta]_{\textrm{pub}}$ ), and systematic uncertainties ( $\sigma_{[\alpha,\delta]_{\textrm{pub}}}$ ) and their revised values (where updates are required and denoted by the subscript rev) as per the work in Section 4.1. Finally, we list the total revised uncertainty in RA and Dec., $\sigma_{[\alpha,\delta],\textrm{tot}}$ (i.e., the quadrature sum of the statistical [not shown] and updated systematic uncertainties). We note that the precision of the uncertainties is given such that it matches that reported in the references listed in Table 2, with the RA precision including an additional significant figure to mitigate round-off errors when converting to seconds.

^†Published positions are from the same references listed in Table 2.

^‡For offsets consistent with zero, the updated uncertainties are simply $1.79 \times \sigma_{[\alpha,\delta]_{\textrm{pub}}}$ . Otherwise, they have been re-derived, if applicable, with both the updated weighting scheme in Equation (3) and the use of the scale factor.

${}^{\ast}$ The values for FRB 191228 and FRB 200906 (Bhandari et al., Reference Bhandari2021) were derived using the method described in Section 4.1. For these reasons, the updated value columns for these FRBs are intentionally left blank.

This change only affects FRBs that have both more than one field source used for the comparison (i.e., those for which this weighted mean method can be used) and offsets that were not originally consistent with zero. Therefore, the positions and offsets for FRBs 180924, 181112, and 190102 are unchanged, while the uncertainties are scaled by the factor derived in Section 4.1. Conversely, the positions, offsets, and uncertainties for FRBs 190608 (no change in position to the quoted precision), 190611, 190711, and 190714 have been updated using both the correct weights (affecting all positional information) and the scale factor (further modifying the uncertainties). The FRB 191001 position has been updated both to account for this new weighting scheme and to rectify an error in the original RA offset correction reported in Bhandari et al. (Reference Bhandari2020b).

FRB 200430 (Heintz et al., Reference Heintz2020) includes both the offsets and uncertainties derived via the nominal method along with an additional offset and uncertainty in Dec. to account for frequency dependence in the position (Section 4.2), and so the nominal components of the offsets and uncertainties are updated, including the use of the scale factor derived in Section 4.1), resulting in an overall change in the position and the estimated offsets and uncertainties. Since the positional information for both FRB 191228 and FRB 200906 (Bhandari et al., Reference Bhandari2021) was obtained by using the updated weights and the scale factor, this information is unchanged.

We also provide the revised total astrometric uncertainties (i.e., the quadrature sum of the statistical and systematic uncertainties) in RA and Dec. for each FRB in our sample.

4.2. FRB 200430: The case of frequency-dependent offsets

FRB 200430, which was detected at a central frequency of 863.5 MHz, shows a frequency dependence in its offsets, which has not been seen in any other FRBs to date. Heintz et al. (Reference Heintz2020) reported the detection and briefly outlined the steps taken to account for the bias introduced by this frequency dependence when using the field data to estimate the offsets and uncertainties in RA and Dec. for this burst. We expand this description here and compare the data presented in this work to that of FRB 200430.

In determining the final statistical position (i.e., prior to any offset correction) for a given FRB, an optimal slice of the data roughly centred on the temporal position of the FRB (the ‘gated’ data) is correlated, and the subsequently calibrated visibilities are then re-weighted by a spectrum derived from the cube-imaged data (using pixels covering the peak FRB emission) to boost the S/N across the band (see, e.g., Bannister et al. Reference Bannister2019, for a full description). In doing so for FRB 200430, we found both a shift and larger statistical uncertainty in the fitted Dec. $>$ 1- $\sigma$ , both of which are indicative of phase/systematic errors in the Dec. as a function of frequency. In addition, we also measured significant offsets in Dec. (but not RA) in the background field sources relative to their FIRST counterparts, which likewise signals the presence of phase errors in the calibration data. We note that the same re-weighting is not used for the field data, and so, since the systematic offsets in the re-weighted FRB image frame cannot be corrected for with the field sources, this precluded us from using the re-weighted FRB position.

In order to confirm the presence of the suspected frequency dependence in Dec., we made a cube image of the optimally gated FRB data with a resolution of 56 MHz (i.e., 1/6th of the 336-MHz bandwidth), which appeared to show a drift in Dec. as a function of frequency. The FRB position in each of the six channels of the image was fitted via JMFIT in the manner described in Section 2.2 (see also, e.g., Day et al. Reference Day2020). We note that, since this FRB is brighter at lower frequencies, the statistical uncertainties on these positions also increase with frequency. While there was a slight, non-frequency-dependent offset in RA ( ${\lesssim}1\sigma$ and roughly accounted for by the increasing uncertainties), there was an ${\sim}7$ -arcsec offset in Dec. across the band (i.e., the the positions at the band edges were inconsistent at the 2.5- $\sigma$ level), with a clear dependence on frequency that cannot be accounted for with the increased uncertainties.

After confirming that the calibration data visibilities used for the FRB data showed the expected nominal properties (i.e., amplitudes $\sim$ 15 Jy [J1939] and phases centred around zero), the final calibrator scan ( $\sim$ 20 min after the first) was correlated to determine if it showed the same systematic offsets as a function of frequency. In performing the same steps as before, we found a similar non-frequency-dependent offset in RA and a slightly worse, frequency-dependent offset in Dec. (i.e., the the positions at the band edges were inconsistent at the 3- $\sigma$ level). Notably, when comparing the positions derived using the two scans to calibrate the FRB, the overall differential offset in each channel was stable as a function of frequency.

We also investigated if any differential phase offsets were present across the 20-minute time span between the two calibrator scans, where phase changes ${\gtrsim}1$ degree would give rise to non-negligible calibration errors. We calibrated the first calibrator scan with the last and then ran a gaincal in casa in the phase solution mode to determine a single phase offset (averaging the two polarisations) per antenna. Phase offsets of ${\lesssim}10$ degrees were found, with a mean of approximately 2 degrees. We therefore concluded that there were phase errors in one or both of these calibrator scans. However, since the better scan cannot be conclusively determined, we attempted to correct for this frequency dependence using the first scan as detailed in the following.

The typical spectral index of a continuum field source is ${\sim}-0.7$ (i.e., that of synchrotron radiation), while the FRB 200430 spectral index appeared to be much steeper from the initial spectrum. Since the offset in the FRB position is a function of frequency and given the differing spectral indices in the field sources versus the FRB, the field source centroids would not be affected in the same way by the frequency-dependent phase errors, and thus our nominal method of correction using the field sources would introduce a bias.

In order to account for this, we derived a coarse spectral index for the FRB and compared this to the typical field source spectral index. To calculate the FRB spectral index ( $\gamma_{\textrm{FRB}}$ ), using standard linear regression, we used the coarse cube of the FRB to fit the log of the extracted flux densities (S) versus frequency ( $\nu$ ), given by

(18) \begin{equation} \textrm{log} S = \textrm{log} A + \gamma \textrm{log} \nu,\end{equation}

where A is a constant of proportionality and $\gamma$ is the spectral index. We found $\gamma_{\textrm{FRB}} \approx -5.46$ .

The final images made from both the FRB and field datasets must be frequency averaged in order to maximise the S/N of source detections in all images and thereby the astrometric accuracy attainable. Given $S = A\nu^{\gamma}$ , the central frequency ( $\nu_{\textrm{cen}}$ ) is defined as the frequency at which the area under the curve is 50% of the total area from the lowest ( $\nu_1$ ) to highest ( $\nu_2$ ) frequencies, and this corresponds to the frequency at which the centroid of the averaged image positions will be located. These quantities are given by

(19) \begin{equation} \begin{aligned} \int_{\nu_1}^{\nu_{\textrm{cen}}} \nu^{\gamma} &= 0.5 \int_{\nu_1}^{\nu_2} \nu^{\gamma}, \\[3pt] \implies \nu_{\textrm{cen}} &= \big[0.5\big(\nu_1^{\gamma+1} + \nu_2^{\gamma+1}\big)\big]^{1/(\gamma+1)}.\end{aligned}\end{equation}

Due to the frequency-dependent offset in Dec. and the differing spectral indices, the central frequencies in the two images differ. We therefore derived a typical deviation as a function of frequency, which could then serve as the uncertainty expected at a given frequency due to the introduced bias. We assumed a typical spectral index of $-0.7$ for the field sources and found $\nu_{\textrm{cen,field}} \approx 855.46$ MHz. Using the derived value of $\gamma_{\textrm{FRB}} = -5.5$ , we found a central frequency for the FRB of $\nu_{\textrm{cen,FRB}} \approx 806.38$ MHz. Thus, the frequency difference between the centroid locations in the two images is $\Delta\nu_{\textrm{cen}}{\sim}49$ MHz.

In order to determine the expected offset in the position for the offset in Dec., we used linear regression to obtain a weighted fit of the Dec. values measured in the 56-MHz resolution cube image versus frequency. We found offsets at the respective central frequencies of the FRB and field images of $\Delta\delta_{\textrm{cen,FRB}} = 1.39$ arcsec and $\Delta\delta_{\textrm{cen,field}} = 2.32$ arcsec. Thus, for a central frequency offset of $\sim$ 49 MHz, we found an offset of 0.93 arcsec (field to FRB). This was then added to the FRB Dec. position in addition to the offset derived via estimating the nominal offsets and uncertainties based on the scatter in the field source offsets. The originally reported position in Heintz et al. (Reference Heintz2020) used nominal weighted mean values of $-0.03 \pm 0.25$ arcsec and $3.19 \pm 0.47$ arcsec for RA and Dec., respectively. However, as detailed in Section 4.1.1, the weights used to derive them were incorrect, and so we have updated these values using the corrected weighting scheme to $-0.04 \pm 0.25$ arcsec for RA and $2.69 \pm 0.30$ arcsec for Dec. We conservatively estimated the uncertainty on the offset due to the frequency dependence of the FRB position to be equivalent to the offset correction (i.e., $0.93 \pm 0.93$ ). Summing the two sources of systematic offsets in Dec. and combining their uncertainties in quadrature, we found a total systematic Dec. offset and uncertainty of $3.62 \pm 0.98$ arcsec. These offsets were used to correct the FRB position, yielding a final RA $=$ 15h18m49.54s $\pm$ 0.021 (statistical; systematic: $\pm$ 0.011 s; $\pm$ 0.017 s) and Dec. $=$ 12d22m36.3s $\pm$ 1.01 (statistical; systematic: $\pm$ 0.24; $\pm$ 0.98). We note that the RA is unchanged from the previously published value (Table 3) at the quoted precision. Given the significant impact on the final position of the frequency-dependent offset observed for this FRB, all future FRBs, especially those at low frequencies, should be inspected to determine if such offsets exist in the data.

While we have not seen this frequency dependence in the position of any other FRBs, as discussed in Section 3, we do see a dependence on wavelength in the low-band data presented in this work. FRB 200430 was also detected in the lower frequency range observable with CRAFT, and while the gradient of this offset dependence is larger for this burst than that seen in Figure 3, it is not inconsistent with our data, in which we see offset changes of order a few arcsec across the band.

A possible contributor to the more extreme offset gradient exhibited by FRB 200430 is the ionosphere. We therefore investigated its likely contributions to the total systematic offset. The $\sim$ 7-arcsec shift in the Dec. across the band is about a quarter of the beam (i.e., ${\sim}90$ degrees), which is approximately 0.15 total electron content units (TECU) of difference in the differential ionosphere across the array. Mevius et al. (Reference Mevius2016) report measurements from the LOw-Frequency Radio interferometer ARray (LOFAR) of short timescale variations on ionospheric sightlines on of order km baselines. These show that we should not typically see variation ${\sim}0.15$ TECU across the array on baselines out to 6 km. However, variations roughly 5x smaller (corresponding to offsets of order 1 arcsec across the band at these frequencies) do occur. While these LOFAR measurements were taken at a Dec. of $+50$ , the ionosphere at declinations observable with ASKAP is not expected to be significantly different, and indeed, the mid-band data we present in this work shows offsets due to the ionosphere of up to ${\sim}1$ arcsec in RA and ${\sim}0.4$ arcsec in Dec., resulting from increased ionospheric activity during these observations (Section 3.3). Likewise, if extrapolating the mid-band results, similar conditions during the low-band observations would have led to offsets up to ${\sim}2$ arcsec in RA and ${\sim}1$ arcsec in Dec. Thus, the ${\sim}7$ arcsec shift in the FRB 200430 position across the band is much larger than the expected ionospheric contribution based on the tests presented in this work and the LOFAR measurements (Mevius et al. Reference Mevius2016). Moreover, if the ionosphere were the dominant component, the frequency dependence in the offsets would be better fit by a quadratic rather than a linear model. Thus, ionospheric effects cannot solely account for the observed offset across the band.