# Tied-Array Beam Localisation of Radio Transients and Pulsars

M.C Bezuidenhout, C.J Clark, R.P Breton, B.W Stappers, E.D Barr, M

Caleb, W Chen, F Jankowski, M Kramer, K Rajwade, et al.

► To cite this version:

M.C Bezuidenhout, C.J Clark, R.P Breton, B.W Stappers, E.D Barr, et al. Tied-Array Beam Localisation of Radio Transients and Pulsars. RAS Techniques and Instruments, 2023, 2 (1), pp.114-128. 10.1093/rasti/rzad007. hal-04021921

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# Tied-array beam localization of radio transients and pulsars

M. C. Bezuidenhout<sup>1,2★</sup> C. J. Clark<sup>2,3,4</sup> R. P. Breton<sup>2</sup> B. W. Stappers<sup>2</sup> E. D. Barr<sup>2,5</sup> M. Caleb,<sup>6</sup> W. Chen<sup>2,5</sup> F. Jankowski<sup>2,7</sup> M. Kramer<sup>3</sup> K. Rajwade<sup>8</sup> and M. Surnis<sup>2,9</sup>

Accepted 2023 February 13. Received 2023 January 27; in original form 2022 May 20

## ABSTRACT

Multi-element interferometers such as *MeerKAT*, which observe with high time resolution and have a wide field of view, provide an ideal opportunity to perform real-time, untargeted transient and pulsar searches. However, because of data storage limitations, it is not always feasible to store the baseband data required to image the field of a discovered transient or pulsar. This limits the ability of surveys to effectively localize their discoveries and may restrict opportunities for follow-up science, especially of one-off events like some fast radio bursts. Here, we present a novel maximum-likelihood estimation approach to localizing transients and pulsars detected in multiple *MeerKAT* tied-array beams at once, which we call tied-array beam localization, as well as a Python implementation of the method named SEEKAT. We provide real-world examples of SEEKAT’s use as well as a Monte Carlo analysis to show that it is capable of localizing single pulses detected in beamformed *MeerKAT* data to (sub)arcsec precision.

**Key words:** data methods – fast transients – interferometry – localization – software.

## 1. INTRODUCTION

Starting with the discovery of the first radio pulsar in 1968 (Hewish et al. 1968), the advent of high-time resolution astronomical observations has allowed for the study of a broad class of astrophysical objects that are now collectively referred to as radio transients. A subset of radio transients, whose emission varies on time-scales of seconds or less, is classified as *fast radio transients* (Cordes 2007). The short light traveltimes suggested by this rapid variability point to very small emission regions. This, combined with the vast amounts of energy released, naturally leads to fast transients being associated with the extreme environments of compact objects like neutron stars and black holes.

One-off or irregularly repeating fast transients are most easily observed in single-pulse searches. These sources include Rotating RAdio Transients (RRATs), magnetars, and extragalactic fast radio bursts (FRBs). The degree to which fast transient single pulses are dispersed by their frequency-dependent propagation through the intervening matter, given by their dispersion measures (DMs), can serve as a probe of the extragalactic medium. For example, DM values of distant FRBs can be used in concert with independent distance measurements to estimate cosmic baryon densities (Macquart et al.

2020). Walters et al. (2019) showed that independent measurements of the redshift of ~100 FRB host galaxies are required to make a measurement of the diffuse gas fraction in the Galaxy, which would help alleviate the ‘missing baryon problem’ in the current Lambda cold dark matter model of the Universe. DM measurements of FRBs with host galaxy associations have also been used to constrain the properties of intervening galactic halo gas (Prochaska et al. 2019). To obtain independent distance measurements, FRBs must be unambiguously associated with host galaxies, which requires localization to a precision of ~1 arcsec (Eftekhari & Berger 2017).

Precise localization can also enable improved follow-up observations of pulsars and repeating fast transients with more accurate source targeting that maximizes the telescope gain at the source position, leading to higher measured signal-to-noise (S/N) ratios. Additionally, localization can help resolve the problem of the covariance of the pulse period derivative ($\dot{P}$) and source position. This can enable more robust $\dot{P}$ measurements in cases where observing time for follow-up timing is scarce, as well as for sparsely observed periodic sources like RRATs and radio-loud magnetars.

Typically, transients and pulsars can be localized through radio synthesis imaging using complex voltage data captured from the array elements (see e.g. Pearson & Readhead 1984; Bannister et al. 2019). Untargeted fast transient surveys, which make use of tied-array beamforming to perform single-pulse searches (often in real time), may employ transient buffers that store snippets of imaging

© 2023 The Author(s)

<footnote n="1"><sup>1</sup>Centre for Space Research, North-West University, Potchefstroom 2351, South Africa</footnote>

<footnote n="2"><sup>2</sup>Jodrell Bank Centre for Astrophysics, University of Manchester, Manchester M13 9PL, UK</footnote>

<footnote n="3"><sup>3</sup>Max Planck Institute for Gravitational Physics (Albert Einstein Institute), D-30167 Hannover, Germany</footnote>

<footnote n="4"><sup>4</sup>Leibniz Universität Hannover, D-30167 Hannover, Germany</footnote>

<footnote n="5"><sup>5</sup>Max-Planck-Institut für Radioastronomie, Auf dem Hugel 69, D-53121 Bonn, Germany</footnote>

<footnote n="6"><sup>6</sup>Sydney Institute for Astronomy, School of Physics, The University of Sydney, NSW 2006, Australia</footnote>

<footnote n="7"><sup>7</sup>LPC2E, Université d'Orléans, CNRS, 3A Avenue de la Recherche Scientifique, F-45071 Orléans, France</footnote>

<footnote n="8"><sup>8</sup>ASTRON, the Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, NL-7991 PD Dwingeloo, the Netherlands</footnote>

<footnote n="9"><sup>9</sup>Department of Physics, IISER Bhopal, Bhauri Bypass Road, Bhopal 462066, India</footnote>

<footnote>*E-mail: bezmc93@gmail.com</footnote>
data around candidate pulse times for the purposes of off-line localization. However, the memory demands of recording imaging data often becomes prohibitive, especially for surveys that form hundreds of tied-array beams (TABs) and search up to very high DMs. Such surveys must rely on alternative methods for localizing discovered transients.

It is also helpful to be able to localize a newly discovered transient or pulsar quickly with sufficient precision to trigger rapid follow-up observations at higher frequencies or with instruments requiring precise positional information. Rapid follow-up observations may reveal repeat pulses, multiwavelength counterparts, and fast-fading afterglows (Yi, Gao & Zhang 2014; Vedantham et al. 2016).

It is therefore valuable to have a method for localizing transients observed in beamformed data to a precision smaller than the size of the half-power beamwidth of a TAB. The first instance of such a method was used by the *Westerbork Synthesis Radio Telescope* 8GR8 survey, where they localized detected sources to the crossing point of fan-beams at different hour angles (Rubio-Herrera et al. 2013). Subsequently, Spitler et al. (2014) used spatial information about the gain of the *Arecibo* ALFA receiver at different frequencies to map out the instrument's spectral index, and based on the observed spectral index of an FRB 20121102A detection determined that it must have occurred on the rising edge of the first sidelobe.

Obrocka, Stappers & Wilkinson (2015) and Obrocka (2015)<fnref n="1" /> expanded on this idea by applying it to multibeam detections with interferometric arrays. This method stipulates that, for a given pair of TABs, possible source positions are defined as those where both the observed ratio of S/N values and difference of spectral indices match the expected values derived from modelling of the TABs' spatially dependent and frequency-dependent gain within $1\sigma$ errors. The final localization is then taken as the intersection of the localization contours derived for each pair of TABs in which a pulse was detected above the detection threshold. Obrocka et al. (2015) demonstrated the ability of this method to localize a single pulse observed by the *MeerKAT* telescope to ~arcsec precision. Petroff et al. (2019) used an adapted version of this method to pinpoint FRB 20110214A detected by the *Parkes* telescope to one of three ~10-arcmin<fnref n="2" /> regions.

A similar method was used to constrain a single pulse from FRB 20170107A detected by the *Australian Square Kilometre Array Precursor (ASKAP)* to a region 8-arcmin across with 90 per cent confidence (Bannister et al. 2017). In this case, the authors assumed a simple Gaussian beam shape to model the expected flux density of the pulse over space, and compared this to the measured flux density in the primary beam and eight adjacent TABs. They applied a Bayesian methodology to sample the posterior positional probability. Shannon et al. (2018) and Qiu et al. (2019) have used the same method to localize several *ASKAP*-detected FRBs to ~arcmin to ~10 arcmin precision.

The Canadian Hydrogen Intensity Mapping Experiment (CHIME) have also used a variation of this method to constrain some of their FRB discoveries to polygonal regions of several arcmin to degrees (CHIME/FRB Collaboration 2019a, b; Michilli et al. 2021). They perform post-facto re-phasing of the received radio waves to various trial positions surrounding a detected pulse, and measure one S/N value for each formed beam. The beams are modelled using two-dimensional (2D) sinc<fnref n="2" /> functions, and the expected response ratios are fit to the derived S/N values using $\chi^2$ minimization. Confidence regions are taken corresponding to contours of equal $\chi^2$ value.

In this paper, we present a new localization method, named tied-array beam localization (TABLo), developed to rapidly localize sources discovered by the MeerTRAP real-time fast transient and pulsar search using *MeerKAT* (Stappers et al., in preparation; Sanidas et al. 2018; Bezuidenhout et al. 2022), as well as the TRAnsients and PUsars with *MeerKAT* (TRAPUM) survey (Stappers & Kramer 2016). TABLo combines the Obrocka et al. (2015) approach of utilizing the distribution of S/N values of a detected pulse in adjacent beams with a maximum-likelihood estimation (MLE) approach similar to that of the CHIME collaboration. This method eschews the spectral index difference fitting of Obrocka et al. (2015) for reasons explored in Section 5.5, as well as the phase-referencing grid approach described in Michilli et al. (2021) for CHIME FRBs. We show that we can obtain (sub)arcsec-level localizations with robust uncertainty estimation quickly enough for real-time use and rapid follow-up triggering.

Section 2 illustrates how the spatially dependent gain of TABs can be accurately modelled, Section 3 shows how TABLo uses those beam models for localization and describes a software implementation of the method, and Section 4 presents selected results from its application to *MeerKAT* observations. Finally, Section 5 outlines possible future developments of this method working towards improved localization precision.

## 2. MEERKAT BEAMFORMING

For parabolic-dish-based interferometers like *MeerKAT* and *ASKAP*, each steerable dish collects electromagnetic signals from a portion of the sky contained within the so-called primary beam, the extent of which is determined by the dish size. If a dish of diameter *D* observes signals with a wavelength $\lambda$, then its primary beam's relative sensitivity to those signals will take the form of an airy disc function over the sky, with the main lobe having a half-power point given, to first order, by $\theta_{PB} \sim \lambda/D$ in radians (see e.g. Burke & Graham-Smith 2014). The signals from all elements of the array can be combined coherently to form an aperture array. By introducing delays to each element's response before weighted addition, TABs can be formed with sizes determined instead by the maximum baseline<fnref n="2" /> *d*, so that $\theta_{TAB} \sim \lambda/d$. Since *d* $\gg$ *D*, $\theta_{TAB} \ll \theta_{PB}$, thus achieving much finer angular resolution. Through controlling the delays added to each dish's signal, hundreds of TABs can be formed at different positions within the primary beam at once, thus attaining a collective field of view (FoV) comparable to that of an individual dish with a much better angular resolution and sensitivity (see e.g. Chen et al. 2021).

Beamforming observations with *MeerKAT* can be performed using the Filterbank and BeamForming User-Supplied Equipment compute cluster developed by the Max-Planck Institute for Radio Astronomy, which coherently sums the digitized signals received from the *MeerKAT* dishes. Each dish receives a signal with a geometric delay,

$$ \tau_n(\alpha, \delta, t) = \bar{u}(\alpha, \delta, t) \cdot \bar{r}_n/c, \qquad (1) $$

where $\bar{u}(\alpha, \delta)$ is the pointing direction, $\bar{r}_n$ is the location of the antenna, and *c* is the speed of light. The channelized complex voltages $\bar{f}_n(v, t)$ observed by each dish can be coherently combined by accounting for the geometric delays with a complex phase shift,

<footnote n="1">¹PhD thesis available online at: https://ethos.bl.uk/OrderDetails.do?uin=uk.b1.ethos.647424.</footnote>

<footnote n="2">²In practice, the actual size of the beam also depends on the distribution of the dishes and the weighting scheme used to combine their signals.</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 1** Images from a beamforming simulation performed using the MOSAIC package for a MeerKAT observation at 11:30:52 UTC on 2020-07-16. During the observation, 384 TABs were tiled centred at RA 04:52:34.11 and Dec −18:04:23.4 to overlap at their 25 per cent gain level at 1.284 GHz. Left-hand panel: The instantaneous simulated PSF of one TAB at 1.284 GHz. The colour scale indicates the relative gain of the TAB. The white contour is an ellipse fit to the 25 per cent level of the PSF, with its semiminor, semimajor, and position angle indicated by a, b, and θ, respectively. Right-hand panel: The tiling pattern of TABs within the primary beam, where the inner ellipses represent the 25 per cent response level of the TABs, all of which are assumed to be identical. The outer grey circle indicates the edge of the tiling pattern (not the edge of the PSFs). In this case, the TAB tiling pattern extends to approximately the 55 per cent gain level of the MeerKAT primary beam at 1.284 GHz.</figcaption></figure>

to produce a (channelized total intensity) TAB, $F(\alpha, \delta, \nu, t)$, via

$$F(\alpha, \delta, \nu, t) = \left| \sum_n \tilde{f}_n(\nu, t) e^{-2\pi i \nu \tau_n(\alpha, \delta)} \right|^2 \quad (2)$$

For an idealized point source at a location $(\alpha_0, \delta_0)$, $\tilde{f}_n(\nu, t) = Ae^{2\pi i \nu \tau_n(\alpha_0, \delta_0, t)}$, where A is a constant amplitude across all dishes. The TAB point-spread function (PSF) is the (fractional) response of equation (2) to this idealized point source as a function of the angular offset from the TAB position:

$$PSF(\Delta\tilde{u}, \nu, t) = \left| \sum_n e^{2\pi i \nu (\tau_n(\alpha_0, \delta_0, t) - \tau_n(\alpha, \delta, t))} \right|^2 \\ = \left| \sum_n e^{2\pi i \nu \Delta\tilde{u}(t) \tilde{\tau}_n} \right|^2, \quad (3)$$

i.e. it is the 2D Fourier transform of the antenna locations.

The PSF is both frequency- and time-dependent, as the pointing direction ($\tilde{u}$) towards a fixed celestial position changes over time due to Earth's rotation. However, here we are interested in localizing rapid transient events, with short time duration, that are detected in band-integrated data. For simplicity, we therefore drop the time and frequency dependencies and use instantaneous PSFs at the central observing frequency in the following sections (however, see Section 5 for how these effects can be accounted for).

Thus, using the known terrestrial coordinates of the antennas used for a certain MeerKAT observation, the instantaneous PSF can be recovered. The Python package MOSAIC<fnref n="3" /> (Chen et al. 2021) derives the MeerKAT TAB PSF given the observation time, target coordinates, antennas included in the array, and their relative

weighting. The left-hand panel of Fig. 1 shows the result of a MOSAIC PSF simulation using 40 core MeerKAT dishes.

Additionally, MOSAIC can use these simulated beam maps to determine the positions of individual TABs such that the desired number of them are optimally hexagonally packed. The right-hand panel of Fig. 1 shows the circular tiling pattern of 389 TABs formed using the PSF shown in the left-hand panel. The beams are specified to overlap at 25 per cent of their maximum gain,<fnref n="4" /> and the semi-axes a and b and position angle θ marked in the left-hand panel correspond to an ellipse that is a best fit to this level of the PSF.

### 3. TIED-ARRAY BEAM LOCALIZATION

The method described by Obrocka et al. (2015) models the apparent flux density $S_v$ of a single pulse over space as an intrinsic brightness modulated by the instrumental response (i.e. the instantaneous PSF). Since the intrinsic brightness is a constant across TABs, taking the ratio of flux densities in a pair of TABs leaves only the PSFs of those TABs to be modelled using equation (2). The modelled PSF ratios can then be used to predict the observed S/N in adjacent TABs assuming different positions of the source. This idea is illustrated in the top panel of Fig. 2, following Obrocka et al. (2015), in which sources placed at different positions relative to two adjacent TABs' centres have different apparent flux densities. In the bottom panels of Fig. 2, the relative gains of these beams are shown in grey-scale, along with coloured contours marking different values of their ratio.

<footnote n="3">³Wherever 'overlap' levels are quoted in this text, unless specified otherwise, they refer to the response level of the TAB PSFs at the centre frequency of the MeerKAT L band, 1.284 GHz.</footnote>

<footnote n="4">⁴https://github.com/wchenastro/Mosaic</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 2** Top panel: The spatially dependent gain, in normalized units, of two MeerKAT TABs that overlap at the 25 per cent level, modelled using MOSAIC. A source at the positions marked A, B, and C would have different observed flux densities in the two beams, as indicated by the red and black markers. Bottom: Two adjacent TABs, with the background grey-scale showing the superposition of their respective gains at a given position, and coloured contours shown at different values of their ratio. The right-hand panel is zoomed in on the centres of the beams. Red, dashed ellipses indicate the 25 per cent levels of the TABs.</figcaption></figure>

Assuming that all TABs are equally sensitive,<fnref n="5" /> the measured S/N values will all be proportional to the apparent $S_ν$ of a detected source.<fnref n="6" /> For a source at position $(α, δ)$, the ratio of S/N values in two TABs in directions $\vec{u}_1$ and $\vec{u}_2$ and with spatially dependent gain $PSF_1(α, δ) = PSF(\vec{u}(α, δ) - \vec{u}_1)$ and $PSF_2(α, δ) = PSF(\vec{u}(α, δ) - \vec{u}_2)$, respectively, is thus predicted to be

$$ \frac{S/N_1}{S/N_2} = \frac{(S_ν)_1}{(S_ν)_2} = \frac{PSF_1(α, δ)}{PSF_2(α, δ)}. \qquad (4) $$

For each pair of TABs in which a single pulse is detected, Obrocka et al. (2015) defines possible positions as those where equation (4) is true within $1σ$ errors on the S/N measurements. Obrocka et al. (2015) then uses power-law spectral index fitting of the PSFs as an additional constraint on the possible positions.

TABLo differs from the Obrocka et al. (2015) approach in two important ways. Firstly, we fit only the ratios of observed S/N values, and not the differences of spectral indices. Primarily, this is because complex TABs formed over a wide frequency band, such as those of MeerKAT, are generally poorly fit by simple power laws, introducing additional fitting errors that are difficult to account for and often lead to poor localizations. This is discussed further in Section 5.5. Secondly, we follow an MLE approach via linear regression analysis. This allows us to derive a positional probability distribution that can be maximized to find the most likely source position, as well as rigorously define standard errors. In contrast to CHIME, which has TABs that span ~0.3 degrees in the North–South direction, MeerKAT’s ~10-arcsec angular resolution is sufficient that the ‘gridding’ procedure described in Michilli et al. (2021) is not required. Instead, we can make use of S/N values already calculated in formed beams in the process of the transient search; this significantly saves on computing requirements, and allows for real-time localization. It also enables localization for surveys that do not have re-phasing capacity. Finally, because TABLo fits the ratio of S/N values in beam pairs, calibrated flux density measurements are not needed,

<footnote n="5">⁵In practice, this is not true for TABs with large angular separation because of the effect of the primary beam.</footnote>

<footnote n="6">⁶Technically, any observable proportional to the apparent $S_ν$ can be used in this analysis rather than the S/N value, provided the error on the measurement is also known.</footnote>
as is the case for the ASKAP method outlined in Bannister et al. (2017).

## 3.1 Deriving the probability distribution

TABLo uses a forward modelling approach: The PSF model produced by Mosaic is used to predict the set of S/N ratios that should be observed between pairs of TABs, as a function of the position of a putative source. These predicted ratios are then compared to the observed S/N ratios. This generates likelihood values for each combination of RA and Dec that can be maximized to find the most likely position, and integrated to calculate uncertainties.

Starting from a list of Q detections of a single pulse above the S/N threshold, we define a vector S containing all independent ratios of their S/N values. Since S is a ratio distribution, it can only be assumed to be Gaussian if the numerator and denominator are themselves independent and normally distributed with a positive mean, and if the error on the denominator is much smaller than its mean (Díaz-Francés & Rubio 2013). We therefore only use the ratio of the S/N value in each beam to the highest overall S/N (say, S/N₁). Hence S will be a vector Q − 1 elements in length:

$$S_i = \frac{S/N_{i+1}}{S/N_1}, \qquad (5)$$

with i = 1,..., Q − 1. Thus, the highest S/N value, S/N₁, is always in the denominator in order to prevent deviation from the assumption that the S/N ratios follow a multivariate Gaussian distribution in the limit where S/Nᵢ + 1 → 0.

S/N values in neighbouring beams will be covariant proportional to the degree to which the TABs are designated to overlap. This effect is small for typical tiling patterns employed in MeerKAT observations. However, the S/N ratios (equation 5) are always covariant with one another, as they all share S/N₁ as their denominator. In order to compute the covariance of S/N ratios caused by this, we first generate T iterations of randomized S/N values within normally distributed 1σ errors of the observed S/N values.<fnref n="7" /> For each iteration, we then find Q − 1 S/N ratios to produce a (Q − 1) × T matrix of S/N ratios. The covariance of S in two beam pairs, Sᵢ and Sⱼ, is then defined as

$$\operatorname{cov}(S_i, S_j) = \frac{1}{T-1} \sum_{t=1}^{T} (S_{i,t} - \bar{S}_i) (S_{j,t} - \bar{S}_j). \qquad (6)$$

The covariance matrix, C, is then a (Q − 1) × (Q − 1) matrix defined as

$$C_{i,j} = \operatorname{cov}(S_i, S_j). \qquad (7)$$

Fig. 3 shows an example covariance matrix for the ratio of S/N values in seven beams, using 1000 iterations of normally distributed random errors. The diagonal elements indicate the variance within distinct beam pairs, while the off-diagonal elements indicate how different pairs of beams covary.

Next, we model the PSF of each beam using MOSAIC, and shift the beam centres to match the listed coordinates as exemplified in Fig. 2. From this, we compute the Q − 1 predicted ratios between PSF₁ and every other beam as a function of RA and Dec:

$$\psi_i(\alpha, \delta) = \frac{PSF_{i+1}(\alpha, \delta)}{PSF_1(\alpha, \delta)}. \qquad (8)$$

<figure><img src="image_5.png" /><figcaption>Figure 3 Example of covariance matrix of the ratio of the S/N values of a signal measured in seven adjacent TABs at once.</figcaption></figure>

We then compute a residual vector R of length (Q − 1) as a function of position:

$$R_i(\alpha, \delta) = S_i - \psi_i(\alpha, \delta). \qquad (9)$$

The generalized least-squares method is a technique for performing a linear regression analysis in the case that the model residuals (such as those defined in equation 9) are correlated to a certain degree (see e.g. Draper & Smith 1998). This approach defines the probabilistic χ² metric as the weighted sum of squared deviations, which can be written as a matrix product of the covariance matrix and residuals defined above:

$$\chi^2(\alpha, \delta) = R^T(\alpha, \delta) C^{-1} R(\alpha, \delta). \qquad (10)$$

Inserting the residuals from equation (9) and covariance matrix from equation (7) into equation (10) allows us to compute the positional likelihood distribution function L defined by

$$\ln \mathcal{L}(\alpha, \delta) = -\frac{1}{2} \chi^2(\alpha, \delta), \qquad (11)$$

which can be maximized over the RA and Dec parameters to obtain the most likely coordinates.

In practice, we evaluate the log-likelihood of equation (11) over a grid of sky positions. Assuming uniform priors, the probability Pr of the source existing within a given pixel of this grid is approximated simply by normalizing L(α, δ) so that its sum over all parameter space equals unity, i.e.

$$\operatorname{Pr}(\alpha, \delta) = A \mathcal{L}(\alpha, \delta), \qquad (12)$$

where A is a constant such that

$$A \sum_{m,n} \mathcal{L}(\alpha_m, \delta_n) = 1. \qquad (13)$$

## 3.2 Error estimation

The likelihood can then be used to estimate the statistical uncertainty on the most likely position. If the likelihood follows a 2D Gaussian distribution, it will have the probability density function, in polar coordinates,

$$\mathrm{PDF}(r) = \frac{1}{2\pi\sigma^2} e^{-r^2}, \qquad (14)$$

<footnote n="7">⁷The covariance matrix could be computed analytically, but the derivation is complicated, and this simulation method is computationally inexpensive.</footnote>
where $\sigma$ is the standard deviation and $r$ is the Mahalanobis distance, i.e. the distance from the mean in units of $\sigma$ (see e.g. Bensimhoun 2009).<fnref n="8" /> The cumulative distribution function can then be obtained by integrating over $r$ and the azimuth $\theta$, such that

$$ 
\begin{aligned}
\text{CDF}(r) &= \int_{0}^{2\pi} \int_{0}^{r} \frac{r'}{2\pi\sigma^2} e^{-\frac{r'^2}{2}} dr' d\theta \\
&= 1 - e^{-\frac{r^2}{2}}.
\end{aligned}
\qquad (15) $$

Points corresponding to integer multiples of the Mahalanobis distance from the mean lie on ellipsoids corresponding to $n\sigma$ error levels. For example, CDF(1) = 39.35 per cent of the total probability must be contained within the $1\sigma$ uncertainty region. Similarly, the $2\sigma$ uncertainty region must contain CDF(2) = 86.47 per cent of all probability. The 2D probability $P_{m,n}$ in equation (12) is thus flattened and sorted in descending order of value to form an ordered list $P_q$ with $P_1 \ge P_2 \dots \ge P_{M \times N}$. Finding the index $d$ where

$$ \sum_{q=1}^{d} P_{q} = 39.35 \text{ per cent,} \qquad (16) $$

the $1\sigma$ uncertainty is specified to include all values $m$ and $n$ for which $P_{m,n} \le P_d$. Similarly, the $2\sigma$ uncertainty is taken so that the sum in equation (16) equals 86.47 per cent, while for the $3\sigma$ uncertainty it equals 98.89 per cent.

Note that if the distribution of likelihood deviates from a 2D Gaussian distribution (see Section 4 for examples of such cases), the calculated uncertainties will not strictly correspond to $n\sigma$ levels, but will instead include equal likelihood as if the distribution were indeed Gaussian.

It should also be noted that the statistical uncertainty figures calculated as described in this section are only those arising from the errors on the S/N measurements. Additional errors may stem from imprecise modelling of the PSF. We assume, for instance, that all the antennas have equal gain, and that the phasing is 100 per cent efficient. Furthermore, the PSF is generated for one instant at one frequency, while the S/N values are for a signal averaged over a wide band (or sub-band; see Section 5.1), and may be averaged over time as well (see Section 5.2). Such errors are not taken into account by the SEEKAT likelihood estimation. For single pulses observed across a relatively narrow band, these errors should be small compared to the error due to the S/N measurement, but other use cases may necessitate a correction coefficient.

# 4. EXAMPLES AND TESTS

SEEKAT<fnref n="9" /> is a Python implementation of the TABLo method developed to localize single pulses found with the MeerTRAP single-pulse detection pipeline (see Sanidas et al. 2018; Jankowski et al. 2020; Malenta et al. 2020; Rajwade et al. 2021). In this section, we present ideal-case and real-world tests of the TABLo method using this software.

## 4.1 Idealized test cases

To verify that SEEKAT is working correctly given an ideal set of S/N measurements, we tested the code with a simulated bright pulse. First, we generated the TAB PSF for a certain observation along with

the beam positions using MOSAIC, and then determined the gain of each beam at a particular point. We then multiplied the gain values by the assumed intrinsic S/N to simulate the expected brightness pattern for a pulse at that position. Using the test S/N values derived in this manner as inputs to SEEKAT along with the generated PSF should therefore result in a localization probability map with a maximum exactly matching the specified position. Any deviation of the most likely coordinates from this position would indicate that the S/N values are not being fit correctly.

Fig. 4 shows the localization of such simulated pulses with S/N 20 and 50 that are detected in TABs that overlap at 95 per cent sensitivity. The pulses are localized using three to nine TABs, successively; gains in localization precision using more beams than nine were increasingly marginal. We also generated S/N values for a pulse at the edge of the nine beams, 37 arcsec from the boresight, the localizations resulting from which are shown in Fig. 5. In both cases, the localization probability distribution is centred exactly on the specified coordinates as expected. The sizes of the $1\sigma$ error regions for these pulses using a given number of TABs are listed in Table 1.

These examples illustrate clearly the three main factors that determine the localization uncertainty region: the number of TABs in which the pulse is detected, the intrinsic S/N of the pulse, and its position relative to the TAB centres. It is important to note that this case is close to ideal for the purpose of localization, with an intrinsically bright source detected in multiple very closely spaced beams. For surveys that arrange TABs further apart, the fortuitous placement of the source within the tiling pattern becomes even more significant; a pulse close to the centre of a TAB with no close neighbours may not be detected in enough TABs to be well localized.

## 4.2 Monte Carlo analysis

To further test the accuracy of the fitting process, We used the idealized S/N values as in Figs 4 and 5 with the addition of Gaussian distributed random perturbations in S/N as inputs for SEEKAT. The noise terms in different TABs were uncorrelated. A number randomly drawn from a Gaussian distribution with a mean of zero and standard deviation of one was added to each of the nine S/N values, and the most likely position calculated by SEEKAT was compared to the position found using the unperturbed values. This process was repeated for 10 000 iterations, and the distribution of results compared to the original probability distribution.

The right-hand panel in Fig. 6 shows a heat map of the 10 000 best-fitting coordinates determined by SEEKAT using S/N values with a mean of 20 for a pulse at the centre of the tiling pattern as in Fig. 4. The best-fitting positions are normally distributed around that of the unperturbed case, with standard deviations closely matching those of the original fit, shown in red.

In the left-hand panel of Fig. 6, offsets of the perturbed fits along the major (top panel) and minor (bottom panel) axes of the original fit are plotted. The offsets along both axes are normally distributed around a mean of zero. These distributions also have one and two standard deviations (indicated by solid and dashed vertical lines, respectively), that align closely with the $1\sigma$ and $2\sigma$ uncertainties for the original fit indicated on the horizontal axes. The probability distribution generated by SEEKAT therefore agrees well with that predicted by a Monte Carlo iterative approach. Fig. 7 shows the result of the same analysis for the case where the source is at the edge of the tiling pattern.

The offsets between the probability distributions predicted using SEEKAT and the Monte Carlo analysis are compiled in Table 2. The quoted offsets are defined as the root sum squared of differences in the

<footnote n="8">⁸Available online at: https://upload.wikimedia.org/wikipedia/ commons/a/a2/Cumulative_function_n_n-dimensional_Gaussians.12.2013.pdf</footnote>

<footnote n="9">⁹https://github.com/BezuidenhoutMC/SeeKAT.</footnote>
<figure><img src="image_6.png" /><figcaption>**Figure 4** Localization probability distribution (grey-scale) for a simulated ideal case detection in three to nine TABs, assuming a pulse with intrinsic S/N of 20 (top panel) and 50 (bottom panel) located at the centre of the middle beam. In each case, the solid red contour represents the $1\sigma$ uncertainty level and the dashed red contour represents the $2\sigma$ uncertainty level. The S/N values were simulated to correspond to a 20 S/N pulse at the exact centre of the tiling pattern, with beams overlapping at 95 per cent gain. The blue numbers in the final panels indicate the S/N in each TAB.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 5** Similar localization probability distributions as shown in Fig. 4, but with the source located at the edge of the tiling pattern.</figcaption></figure>

minor and major axis directions, in units of $\sigma$. While in all cases, the offsets are smaller than unity, SEEKAT's $1\sigma$ and $2\sigma$ estimates for the edge pulse are mildly underestimated. An important factor is that the localization probability shown in Fig. 5 is somewhat asymmetrical,

particularly in the direction of the major axis of the unperturbed uncertainty region, and therefore not exactly normally distributed. An ellipse is therefore unlikely to be well-fit to the $1\sigma$ error on the distribution, leading to errors on the measured offsets.
<table><caption>**Table 1.** Areas of the $1\sigma$ uncertainty regions, in arcsec$^2$, for test pulses with different positions relative to the tiling centre and different intrinsic S/N values.</caption><thead><tr><th rowspan="2">No. beams</th><th colspan="2">S/N = 20</th><th colspan="2">S/N = 50</th></tr><tr><th>Centred pulse</th><th>Edge pulse</th><th>Centred pulse</th><th>Edge pulse</th></tr></thead><tbody><tr><td>3</td><td>2100</td><td>74</td><td>1000</td><td>10</td></tr><tr><td>4</td><td>32</td><td>35</td><td>4.7</td><td>5.1</td></tr><tr><td>5</td><td>17</td><td>27</td><td>2.8</td><td>4.2</td></tr><tr><td>6</td><td>9.2</td><td>26</td><td>1.5</td><td>3.8</td></tr><tr><td>7</td><td>8.3</td><td>20</td><td>1.1</td><td>3.1</td></tr><tr><td>8</td><td>6.9</td><td>20</td><td>0.9</td><td>3.0</td></tr><tr><td>9</td><td>5.5</td><td>19</td><td>0.8</td><td>2.7</td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>**Figure 6** Distribution of localizations for 10 000 iterations of a Gaussian random perturbation of the input S/N values. The left-hand panel shows the distribution of offsets along the major (top panel) and minor (bottom panel) axis of the localization region obtained using unperturbed S/N values (i.e. those in Fig. 4). The right-hand panel contains a heat map of these localizations, with the shading corresponding to $1\sigma$, $2\sigma$, and $3\sigma$ levels. The cyan cross and red ellipses correspond to the most likely position, $1\sigma$, and $2\sigma$ uncertainty regions of the localization using unperturbed S/N values, respectively. The solid and dashed vertical lines indicate one and two standard deviations from the mean, respectively.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 7** Similar to Fig. 6, but for a source located at the edge of the tiling pattern, as shown in Fig. 5.</figcaption></figure>

## 4.3 Tests with sparse spatial sampling

The previously described scenarios correspond to a situation where the approximate coordinates of an already discovered source is re-observed with very tightly packed TABs specifically to maximize

the potential for a precise SEEKAT localization. However, in the normal operation of an untargeted transient and pulsar search, surveys are unlikely to tile their TABs so closely, and instead will tend to maximize spatial coverage of the TABs while ensuring sufficient
<table><caption>**Table 2.** Offsets, in units of σ, of the probability distribution determined using SEEKAT compared to that found by Monte Carlo analysis, for two 20 S/N pulses at the centre and edge of the tiling pattern, respectively.</caption><thead><tr><th></th><th>Mean</th><th>1σ error</th><th>2σ error</th></tr></thead><tbody><tr><td>Centred pulse</td><td>0.11</td><td>0.07</td><td>0.12</td></tr><tr><td>Edge pulse</td><td>0.08</td><td>0.17</td><td>0.38</td></tr></tbody></table>

sensitivity to detect a reasonably bright pulse anywhere within the tiling. For the MeerTRAP real-time transient and pulsar search, for instance, TABs are normally arranged to intersect at to overlap at 25 per cent, while for the (mostly) targeted TRAPUM MeerKAT transient and pulsar search project (Stappers & Kramer 2016), they are variably tiled with overlaps in the 30 per cent to 80 per cent range depending on the target.

Using the same observational parameters as in the previous section, we created idealized S/N values for TABs tiled at 25 and 50 per cent overlap. For a TAB configuration with 25 per cent overlap, a pulse situated exactly in the middle of three beams must have a minimum S/N value of about 50 for it to be detectable in all three above a cutoff S/N of eight. To perform a Monte Carlo analysis on these more sparsely sampled configurations, we therefore generate S/N values with a mean of 50.

The resulting SEEKAT localizations are shown in the top panels of Figs 8 (25 per cent) and 9 (50 per cent), respectively. The bottom panels in each figure show the results of a Monte Carlo analysis, as in the previous section. In Table 3, we list the sizes of the 1σ localization region for each localization. We also collate the offsets between the probability distributions determined using TABLo compared to the Monte Carlo analysis.

The probability distributions for these localizations are noticeably irregularly shaped. For the 25 per cent example, the distribution of offsets has a strong tail in the negative minor axis direction, causing the TABLo and Monte Carlo 2σ uncertainty estimates to differ by as much as 0.71σ. Note, however, that this comparison does not take into account the errors on the Monte Carlo estimates themselves, exaggerating the discrepancy. For the 50 per cent overlap case, meanwhile, the TABLo is close to normal in the minor axis direction, and therefore matches the Monte Carlo distribution well; the major axis direction, however, is significantly skewed in the positive direction, causing their mean values to vary by 0.44σ.

In contrast with the examples in the previous section, which illustrate how well the SEEKAT-determined likelihood matches the expected distribution in ideal or near-ideal circumstances with detections in many TABs, the examples in the 50 per cent and, especially, the 25 per cent overlap cases, show the performance of SEEKAT in the opposite edge case, i.e. the faintest single pulse detectable in three distant TABs. In both instances, the means of the localizations using perturbed S/N values are slightly offset from the most likely unperturbed position in one direction (although both are within 0.5σ). Additionally, the offsets in both cases have long tails that represent a departure from the normal distribution, which likely has a sizable impact on determining the 1σ and 2σ uncertainty regions.

Based on these results, we recommend using localizations from SEEKAT judiciously in the case of a few detections in disparate TABs. In such circumstances, users may be better served to quote the 2σ reported by SEEKAT rather than the 1σ uncertainty.

## 4.4 Real-world examples

### 4.4.1 PSR B0450−15

A MeerKAT observation on 16 July 2020 centred on the bright pulsar PSR B0450−15 produced detections above the S/N threshold in 10 adjacent TABs, which overlapped at 95 per cent gain. The brightest detection, with a S/N<fnref n="10" /> of 19, was in the central TAB of the tiling pattern. The SEEKAT localization, with best-fitting coordinates of RA 04:52:34.06 and Dec −17:59:24, is shown in Fig. 10. The 1σ and 2σ uncertainty regions are roughly circular, with half-widths of about 2 and 3 arcsec, respectively. The known coordinates for PSR B0450−18 are RA 04:52:34.1057(1) and Dec −17:59:23.371(2), which is an angular distance of 1.08 arcsec from the most likely coordinates determined using SEEKAT. The known coordinates are therefore within the 1σ error region predicted by SEEKAT.

### 4.4.2 PSR J1843−0757

Another example of SEEKAT in use is presented in Bezuidenhout et al. (2022). A new pulsar, PSR J1843−0757, was discovered by MeerTRAP’s real-time single-pulse detection pipeline. In a follow-up observation of the source with MeerKAT, the TABs were tiled to overlap at their 98-per cent level. Using a single pulse detected in 44 TABs at once, the source was localized to RA 18:43:33.01 and Dec −07:57:36 with 1σ statistical uncertainty of about 1 arcsec. A coherent timing solution for the source gives a best-fitting position of RA 18:43:33.06(2) and Dec −07:57:33(2), which is 3 arcsec from the SEEKAT position. Fig. 11 shows the SEEKAT localization of this source. In the right-hand panel, which is zoomed-in on the localization region, one can see that the SEEKAT and timing localizations are consistent to within the 1σ uncertainty level.

### 4.4.3 Globular cluster pulsars

A collaboration of the MeerTime (Bailes et al. 2020) and TRAPUM MeerKAT Large Survey Projects (LSPs) targeted the globular cluster (GC) NGC 6440 with MeerKAT to search for and time pulsars (Vleeschower et al. 2022). By tiling 288 TABs to overlap at 70 per cent gain, two new pulsars were discovered. For one pulsar, NGC 6440G, however, the MeerKAT timing solution could not initially be phase-connected with that obtained using archival Green Bank Telescope (GBT) observations. Because of the weak nature of this pulsar, an accurate localization of the source was needed for sufficiently accurate timing to phase-connect the time of arrival (ToAs).

Hence, SEEKAT was used with MeerKAT detections of NGC 6440G over two epochs in four and seven TABs, respectively, to localize the source to the coordinates RA 17:48:52.76 and Dec −20:21:38.45. This localization allowed for the GBT and MeerKAT ToAs to be phase-connected, and the resulting long-term timing solution gave a position of RA 17:48:52.6460(4) and Dec −20:21:40.63(1), 2.5 arcsec from the most likely SEEKAT position. The timing solution in this case is consistent with the SEEKAT localization within the 3σ uncertainty level. This relatively large

<footnote n="10">¹⁰Unless otherwise specified, all measured S/N values were determined using the SPYDEN Python package, available at https://bitbucket.org/vmorello/spyden/.</footnote>
<figure><img src="image_10.png" /><figcaption>**Figure 8** Localization of a single pulse located at RA 04:52:34 and Dec −17:59:23 and detected in three adjacent beams that intersect at 25 per cent gain. The top panels show the three TABs and the likelihood distribution determined using SEEKAT. The bottom-left panel shows a heat map of localizations resulting from a random Gaussian perturbation analysis, with the cyan cross and red ellipses indicating the best-fitting position, 1σ, and 2σ uncertainty regions determined using the unperturbed S/N values, respectively. In the bottom-right panels, the distributions of offsets of the localizations using perturbed S/N values in the minor axis and major axis directions, respectively, are plotted.</figcaption></figure>

offset is likely a reflection of the fact that the S/N values were
averaged over long observations of up to 4 h, during which time
the TAB orientation can change by up to 60°. See Section 5.2 for a
discussion of this problem.

SEEKAT has also been used to localize pulsars discovered in other *MeerKAT* GC surveys, e.g. NGC 6624 (Abbate et al. 2022) and NGC 1851 (Ridolfi et al. 2022). At the time of writing, however, no independent localizations of these sources are available for comparison.

4.4.4 FRB 20210123

FRB 20210123 was discovered by MeerTRAP on 2021 January 23. A
single pulse from the source was detected in the *MeerKAT* incoherent
beam (IB) with S/N 11.8, as well as in two TABs, with S/N 22.8 and
S/N 8.83. The TABs were tiled to overlap at their 25 per cent level.
Details of this discovery will be presented in a forthcoming paper. In
order to localize the FRB with *SEEKAT*, the disparity in the gain of
the IB and TABs needed to be taken into account.

The ratio of the gain of the centre of a TAB $G_{\text{TAB}}$ to the gain of
the IB at that position $G_{\text{IB}}$ is given by

$$
\frac{G_{\text{TAB}}}{G_{\text{IB}}} = \frac{N_{\text{TAB}}}{\sqrt{N_{\text{IB}}}}, \qquad (17)
$$

where $N_{\text{TAB}}$ and $N_{\text{IB}}$ are the number of antennas that were used to
 synthesize the TABs and IB, respectively. For the observation during
 which FRB 20210123 was discovered, $N_{\text{TAB}} = 40$ and $N_{\text{IB}} = 60$.
 SEEKAT was modified such that equation (8) was replaced by

$$
\psi_{i,m,n} = \frac{G_{\text{IB}} N_{\text{TAB}}}{\sqrt{N_{\text{IB}}} \cdot P S F_{i+1,m,n}} \frac{P S F_{i+1,m,n}}{P S F_{\text{IB},m,n}}, \quad (18)
$$

where $G_{\text{IB}}$ is in each case the gain of the IB PSF at the centre of the
TAB.

The resulting localization is shown in the top panel of Fig. 12.
The most likely position was calculated along with an elliptical 1σ
uncertainty region with a half-width of 1.5 × 9 arcsec.
<figure><img src="image_11.png" /><figcaption>**Figure 9** Similar to Fig. 8, but for beams that intersect at 50 per cent of their maximum gain.</figcaption></figure>

<table><caption>**Table 3.** Area (in arcsec²) of the 1σ uncertainty region of localizations of a 50 S/N pulse in three TABs, as well as offsets (in units of σ) between the probability distribution determined using TABLo compared to that determined from a Monte Carlo analysis.</caption><thead><tr><th rowspan="2">Overlap</th><th rowspan="2">1σ area</th><th colspan="3">Offsets, TABLo versus Monte Carlo</th></tr><tr><th>Mean</th><th>1σ error</th><th>2σ error</th></tr></thead><tbody><tr><td>25 per cent</td><td>28.29</td><td>0.27</td><td>0.34</td><td>0.71</td></tr><tr><td>50 per cent</td><td>5.73</td><td>0.44</td><td>0.26</td><td>0.52</td></tr></tbody></table>

## 5. ADDITIONAL METHODS

The MLE approach to TABLo put forward in Section 3 opens up the possibility of incorporating additional information about the pulse detection to further improve the localization. In this section, we will list various ways that the approach to TABLo presented in this work may be improved upon, as well as the challenges or impediments of those elaborations on the method. Bezuidenhout (in preparation)<fnref n="11" /> contains a more detailed evaluation of these approaches.

### 5.1 Sub-band localization

A drawback of the approach described in the previous section is that the measured S/N is for a pulse integrated over a wide band, and that a single PSF is generated at a certain frequency despite the beam shape being highly frequency-dependent. It may thus be preferable to perform fits in various frequency sub-bands, and add the resulting log-likelihood distributions together to produce a combined localization. The pulse spectrum may also be used to weight the localization to reduce the effect of frequencies where the S/N is low. Since sidelobes are significantly frequency-dependent, this method may be better equipped to discriminate sidelobe detections. On the other hand, the signal in individual sub-bands may be diluted to the overall detriment of the fit. Preliminary testing has indicated that this approach may produce more precise localizations in some cases, but its accuracy depends strongly on the strategy used to divide the frequency band.

### 5.2 Time-averaged localization of repeating sources

While designed to localize single pulses, TABLo can be applied without much modification to time-averaged observations of repeating sources like pulsars and some fast transients. In this case, average

<footnote n="11">¹¹PhD thesis soon online; available upon request.</footnote>
<figure><img src="image_12.png" /><figcaption>**Figure 10** SEEKAT localization of PSR B0450−18 using S/N values measured during a MeerKAT observation centred on the source. The right-hand panel is zoomed-in on the localization region. The red cross shows the most likely position determined by SEEKAT, while the cyan cross shows the pulsar’s known coordinates, the uncertainties on which are too small to be seen.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Figure 11** SEEKAT localization of PSR J1843−0757 using S/N values measured in 44 MeerKAT TABs. The red cross indicates the most likely position determined by SEEKAT, with the red contours the $1\sigma$ and $2\sigma$ uncertainty on this position, and the cyan cross shows the localization derived through timing of the source. The right-hand panel is zoomed-in on the localization region. The cyan box approximates the $1\sigma$ errors on the timing localization. Note that the sides of the box correspond to the 1D marginalization of the true timing uncertainty, which is elliptical. The SEEKAT and timing localizations coincide within the $1\sigma$ uncertainty level.</figcaption></figure>

S/N values in each TAB can be used in concert with representative PSFs. However, there is the added complication of accounting for the sometimes drastic evolution of the PSF over the course of the observation (see the discussion around this in Chen et al. 2021). Time-averaged PSFs may be used, but they would become less representative for longer observations. A potentially superior approach is viable for sources that are visible over multiple subintegrations, in which case log-likelihood maps for individual subintegrations can be combined, similar to the sub-banding approach described above.

The same problem applies of optimal partitioning, and mitigating for subintegrations with low signal strength.

## 5.3 Combining single-pulse localizations

If multiple consecutive pulses from a transient or pulsar are detected in multiple beams, log-likelihood maps for individual pulses may be added together to produce a more precise aggregate localization. Our tests of this approach (see Bezuidenhout, in preparation) have
<figure><img src="image_14.png" /><figcaption>Figure 12 Localization probability FRB 20210123, which was localized with SEEKAT via a single pulse detected in the MeerKAT IB and two adjacent TABs.</figcaption></figure>

shown that combining many single-pulse localizations, each with a relatively large uncertainties but similar means, may produce a better result than any individual pulse. Conversely, one high-S/N pulse’s localization may only be deteriorated by combining it with those of lower S/N detections. This effect is partially mitigated by weighting the individual localizations by S/N, but the method’s utility remains to be judged on a case-by-case basis.

## 5.4 Incorporating non-detections in adjacent TABs

Real-time surveys like MeerTRAP often do not record S/N values in TABs where the S/N does not meet a certain threshold value. In those circumstances, it may be desirable to use the threshold S/N as an upper limit value in those TABs. Non-detections may then be used to further constrain the source position, especially if any non-detection TAB PSF has sensitive sidelobes coincident with a detection TAB. However, the MLE approach followed in this work does not allow for upper limits to be used.

Additionally, for non-detections to be successfully included in the localization, it will require that the single-pulse detection is complete above the specified S/N threshold. We caution that incorporating a false negative in the single-pulse detection process, for whatever reason, would be detrimental to the localization.

## 5.5 Spectral index localization

Obrocka (2015) proposes that in order to constrain source positions, in addition to using the TAB PSFs as explained in Section 3, the source’s apparent spectral index as a function of RA and Dec may be used for the same purpose. In the formulation of Obrocka (2015), the observed single-pulse flux density in a given TAB can be assumed to be described by a power-law equation:

$$S(\nu) \propto \nu^{\alpha}, \qquad (19)$$

where $\nu$ is the frequency and $\alpha$ is the spectral index. The observed spectrum is assumed to be the product of two other power-law spectra, namely that intrinsic to the source, with index $\alpha_I$, and that imparted

by the frequency dependence of the TAB PSF,<fnref n="12" /> with index $\alpha_{TAB}$, such that

$$\alpha = \alpha_I + \alpha_{TAB}. \qquad (20)$$

Since the difference between spectral indices measured in two TABs is therefore independent of the intrinsic spectrum, the TABLo method described in Section 3 can easily be adapted to fit for $\alpha$ differences rather than ratios of S/N values. The method described in Section 3 can then be applied without any further changes. The resulting spectral index localization can either be combined with the S/N localization or used as a separate point of reference. The left-hand panel of Fig. 13 shows an example spectral index map for the PSF shown in Fig. 1.

It is important to note, however, that this localization method rests on the assumption that the TAB sensitivity spectrum is well-fit by a power law at all positions. This assumption may not hold, particularly away from the TAB centre beyond the main lobe, where the positions of sidelobes are strongly frequency-dependent. This effect can be seen in the right-hand panel of Fig. 13, which shows the $\chi^2$ values for the example power-law fit. Note also that the strength of a spectral power-law fit will depend strongly on the array configuration–arrays with non-Gaussian uv coverage will result in a complex TAB with plateaus, causing the spectrum to diverge from a power law.

Our tests (see Bezuidenhout, in preparation) have shown that power-law index fitting can provide a good localization of comparable precision to S/N fitting in the case that only closely overlapping TABs with centres nearby the source position are used. However, for source positions far away from the centre of an included TAB, the localization accuracy can become very poor. For spectral indices to be successfully incorporated into the localization procedure, more complicated spectral fitting would be required, and the accompanying errors would need to be accounted for.

# 6. DISCUSSION AND CONCLUSIONS

We have described TABLo, a novel transient localization method facilitated by the modelling of interferometer TAB PSFs. Section 4.1 illustrates the performance of TABLo under both favourable and unfavourable circumstances. In cases of bright detections in many TABs at once, the derived likelihood distributions coincide closely with what is expected assuming $1\sigma$ Gaussian variance of the observed S/N values. If the pulse was weakly detected in the minimum of three TABs, however, we see some significant departures from Gaussianity. The positional likelihood distributions are then noticeably skewed, and the errors are not very well described by the standard deviations predicted by SEEKAT. Hence, in the event that a pulse is weakly detected in a few TABs, and especially if the predicted likelihood distribution is visibly asymmetrical, we recommend that users report the $2\sigma$ uncertainty on SEEKAT localizations rather than the $1\sigma$ uncertainty.

We have also presented a number of real-world uses of SEEKAT for which the results could be compared to localizations using other methods. It is encouraging that the SEEKAT localizations agreed with the independent positions to within at least the $3\sigma$ uncertainty level. These represent the most precise non-image plane interferometric localizations of transient single pulses that have yet been achieved, suf-

<footnote n="12">¹²Technically, the response of the receiver imparts a third spectral component, but this is inseparable from the TAB response without calibration, and, assuming it is constant across TABs, drops out when taking the index difference. Hence, we ignore this component here.</footnote>
<figure><img src="image_15.png" /><figcaption>**Figure 13** Spectral indices (left-hand panel) and $\chi^2$ values (right-hand panel) for power-law fits to the TAB PSF displayed in Fig. 1 using eight sub-bands. The red contours indicate the 25 per cent, 50 per cent, and 95 per cent levels of the PSF at 1.284 GHz.</figcaption></figure>

ficient for rapid follow-up observations. The utility of TABLO is further illustrated by the use of SEEKAT to phase-connect the timing solution of a new GC pulsar, as described in Vleeschower et al. (2022).

Finally, we will list the major factors that will determine the efficacy of SEEKAT for the consideration of prospective users:

(i) The number of beams in which the pulse is detected. This is affected by the intrinsic brightness of the pulse, the position of the source relative to the TAB tiling, at what level of response the TABs intersect, and the chosen threshold for considering a detection real. We would recommend that projects who aim to make use of this or an analogous method for localization take this factor into account when deciding on their TAB tiling and single-pulse detection strategies.

(ii) The accuracy of the S/N measurements. These are inherently imprecise, and may differ significantly depending on how the data are processed, the method of radio frequency interference (RFI) excision, and how the value is determined. However, imprecisions should be mostly consistent across TABs, so that they are at least partially taken care of by dealing with the ratios of the values. Additionally, the higher the S/N value in a given beam the smaller the effect of any imprecision would be, so this method is certainly best suited for brighter pulses. It should be noted that instead of S/N values per se, theoretically any value proportional to the brightness of a source in a beam could be used to equal effect. If the noise factor differs significantly from TAB to TAB, then S/N may be too volatile a measurement, and another quantity proportional to the signal strength may give better results.

(iii) The accuracy of the beamforming simulation. Since the S/N is averaged over the full band or a subsection of the band, while the PSF is generated for a single frequency, there is an inherent error in comparing the ratios of PSFs to the ratios of S/N values. This error will be larger in the case of a sidelobe detection, since the sidelobes are much more frequency-dependent than the TAB's main lobe. This effect is partially mitigated in SEEKAT by setting values of the PSF below a certain value (by default 8 per cent of the maximum) to zero. We also recommend that users discard parts of the band where the pulse is not visible before the S/N calculation, as well as to generate the PSF for where the pulse is brightest rather than necessarily at the centre of the band. It is important to note that the error from this

imprecision, as well as any other possible error on the PSF model, is not included in the SEEKAT uncertainty calculation.

We have illustrated the value of beamforming simulations for localizing sources without the need for imaging. This is sure to prove useful considering the cost of storing large amounts of baseband data. We have also developed a suite of software<fnref n="13" /> for accomplishing this task, which has already been used by other *MeerKAT* projects for localizing newly discovered sources. This work should be easily adaptable to other interferometers, and could see increasing use in the coming Square Kilometer Array era of radio astronomy.

## ACKNOWLEDGEMENTS

MCB, BWS, FJ, KR, and MS thank the *MeerKAT* Large Survey Project teams for allowing MeerTRAP to observe commensally. The MeerTRAP project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 694745). KMR acknowledges support from the Vici research program 'ARGO' with project number 639.043.815, financed by the Dutch Research Council (NWO). CJC and RPB acknowledge support from the ERC under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 715051; Spiders). The *MeerKAT* telescope is operated by the South African Radio Astronomy Observatory, which is a facility of the National Research Foundation, an agency of the Department of Science and Innovation.

## DATA AVAILABILITY

The data will be made available to others upon reasonable request to the authors.

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