1 Introduction

Pulsars are highly magnetised, rapidly rotating neutron stars, which emit radiation in the form of two highly collimated beams, approximately originating above the pulsar’s magnetic poles (Lorimer and Kramer 2004). Due to the small sizes of the emission regions, pulsars are observed as point sources, mainly in the radio frequency band, with fast-periodic changes in flux as the star rotates and the emission beam sweeps across the observer’s line-of-sights (LoSs). Propagation of highly linearly polarised and broad-band pulsar emission through intervening media results in a number of physical effects which alter the observational properties of the radiation. Therefore, studying the modifications introduced in pulsar’s emission allows to derive properties of the intervening media, and this makes pulsars remarkable “cosmic beacons” for interstellar medium (ISM) and space weather monitoring (Cordes et al. 1991; Han et al. 2006; Tiburzi et al. 2021). In particular, magnetised plasma induces Faraday rotation, i.e. a frequency-dependent rotation of the plane of the linear polarisation of pulsar signals. From this, one can extract information on the electron densities \(n_e\) and projected magnetic field \({{\varvec{B}}}\) of the intervening media by measuring the rotation of the plane of the linear polarisation \(\Psi \). The Faraday rotation is parameterised by the so-called rotation measure (RM):

$$\begin{aligned}&\Psi = \Psi _0 + \lambda ^2 \text {RM}, \nonumber \\&\text {RM}\left[ \text {rad m}^{-2}\right] =0.81 \int _{\text {LoS}}n_e \left[ \text {cm}^{-3}\right] \Big ({{\varvec{B}}}[\mu \text {G}] \cdot \text {d}{{\varvec{l}}}[\text {pc}]\Big )\,. \end{aligned}$$
(1)

where \(\lambda \) is the radiation wavelength and \(\Psi _{0}\) is the initial angle of the polarisation plane. The in situ magnetic field \({{\varvec{B}}}\) projection on the displacement \(\text {d}{{\varvec{l}}}\) vector is expressed within the corresponding dot product. The integration is performed through the entire LoS from the source to the observer; therefore, the major contribution to the Faraday rotation is given by the magneto-ionic ISM. The amount of Faraday rotation, and hence of RM, is not stable in time and undergoes strong temporal variations. Temporal RM variations occurring on timescales less than several years are largely dominated by the terrestrial ionosphere (Oberoi and Lonsdale 2012). These variations are caused by both intrinsic variability in the electron density of the Earth ionosphere, and geometrical effects due to the motion of the LoS through the atmosphere. The magneto-ionic ISM is as well the source of temporal RM fluctuations due to its highly turbulent nature (Rickett 1977; Armstrong et al. 1995). However, the latter is exceeded by the former by at least 3 orders of magnitude for ordinary loss (Minter and Spangler 1996; Xu and Zhang 2016; Porayko et al. 2019), therefore, will be neglected for the rest of the analysis. All of this makes regular monitoring of the changes in pulsar’s Faraday rotation a powerful tool to examine the magneto-ionic content of the ionosphere, supplementing the data from other atmospheric probes. Additionally, pulsar data can be used to independently evaluate the performance of the available ionospheric models and provide avenues for their further improvement, complementing the similar role from dual-frequency altimeter and Global Navigation Satellite System (GNSS) measurements (Hernández-Pajares et al. 2017; Roma-Dollase et al. 2018) as well as ionosonde observations (Jerez et al. 2020). One should note that RM can be independently probed in the GNSS experiments by measuring the so-called second-order ionospheric effect, which is, however, currently too small to be detected in the L-band frequencies of GNSS (see, e.g. Hernández-Pajares et al. 2014).

As it can be seen from Eq. (1), the effect of Faraday rotation is intensified at lower radio frequencies, making low-frequency facilities preferable for these kinds of studies. In this work, we use the observations of PSR J0332+5434 obtained with the LOw Frequency ARray (LOFAR) telescope. LOFAR is the most sensitive instrument in Europe operating at the lowest boundary of the radio window, from 10 to 240 MHz. With these observations, we aim to investigate the performance of three models of the ionosphere: the single layer model (SLM), the upgraded SLM which incorporates the “thickness” of the ionospheric layer (IRI-Plas extended SLM), and the dual-layer voxel TOmographic Model of the Ionosphere (TOMION). In Sect. 2, we discuss our observations and data processing routines. We summarise the description of used ionospheric models in Sect. 3. The results and discussion are presented in Sects. 4 and 5, respectively.

2 Observations and data reduction

LOFAR (van Haarlem et al. 2013) is a fully digitally steerable interferometer with a dense core located in the Netherlands and sparsely distributed international stations in Europe, that can be detached from the main interferometer and used as stand-alone telescopes. The data used in the presented study are part of a large observational campaign on more than 100 pulsars, carried out within the German LOng Wavelength Consortium (GLOW)Footnote 1. with the high-band antennae (HBA) fields of the six German stations of LOFAR since 2013. The total frequency coverage of the HBA is from \(\sim \) 110 to 240 MHz. After being recorded, the data are redirected to computing resources in the Max-Planck-Institut für Radioastronomie and at the Forschungszentrum Jülich in Bonn, where they are digitally corrected for the average time delay in the ISM (or ’de-dispersed’) and ’folded’ at the pulse period using dspsr software. Due to technical specification, the final observing bandwidth is limited to 71.5 MHz bandwidth with central frequency of 153.81 MHzFootnote 2 The final data product represents the data cube with 195.3125-kHz resolution in frequency, 10-sec resolution in time and 1024 phase bins. For more details on observation setup and essential data-taking pipeline LuMP,Footnote 3 see Tiburzi et al. (2019); Donner et al. (2019). In particular, for this work we used the data collected for PSR J0332+5434, which is one of the brightest sources in linearly polarised intensity of the whole observing campaign (signal-to-noise S/\(\hbox {N}_\text {L} \sim 800\) with 1 h long integration), so that RM can be measured with high precision. In order to demonstrate the effectiveness of different ionospheric mitigation techniques on a baselines of several years, we have used a 4-year observational dataset obtained with the international LOFAR station DE605, located near Jülich (hereinafter, “long-term dataset”).Footnote 4 The source was observed weekly with 1 to 2 h of integration time per epoch from 2014 to 2018. Besides the long-term dataset, we have used a 3-day observing run of PSR J0332+5434, which is a circumpolar source, performed between the 2nd and the 5th of February 2018, carried out with the international LOFAR station DE609, located near Norderstedt (hereinafter, “short-term dataset”). The continuity of the short-term dataset was interrupted by \(\sim 5\) hour-long observations of sources with higher priority.

The basic processing of the recorded data was performed with the psrchive software (Hotan et al. 2004; van Straten et al. 2012). All the collected observations underwent a procedure of radio-frequency interference (RFI) excision using a modified versionFootnote 5 of the “Surgical” algorithm from the coastguard python package (Lazarus et al. 2016). The projection effects and rotation of the parallactic angle were corrected thanks to the dreambeam python software package,Footnote 6 based on the Hamaker measurement equations (Noutsos et al. 2015).

Additionally, each observation was corrected for the dispersion propagation effect. Dispersion is a frequency-dependent delay in the arrival time of pulsar radiation, uniquely dependent on the column density of free electrons along the LoS, and it is parameterised by the “dispersion measure” (DM) parameter. While the average DM of a pulsar is usually known, dispersion also has a time-variable component because of its dependence on the LoS changes. As it was mentioned above, the average DM contribution is precisely removed at the pre-processing stage by coherently dedispersing the recorded data with the dspsr software (see van Straten and Bailes 2011). However, to properly study Faraday rotation, it is necessary to correct also for the aforementioned, additional DM fluctuations (that cannot be known a priori as the average DM component) in order to: i) properly align pulses in the frequency-vs-phase domain, so that the overall S/N is increased and ii) remove possible biases in the RM determination caused by the change in DM. Such a residual time-variable DM was computed post-dspsr using the frequency-resolved timing technique described in Donner et al. (2019); Tiburzi et al. (2019). This technique consists in cross-correlating each frequency-resolved observation with a frequency-resolved, time-invariable template, to estimate the arrival time of the pulsar signal in each frequency band at the epoch of the observation. Obtained arrival times are then fit to obtain the residual DM for the functional form: \(\Delta t = {\mathcal {D}} ~\Delta DM/f^2 + k\) with \(\Delta t\) being the difference between the expected (without an additional \(\Delta DM\)) and the calculated arrival time of an impulse at the frequency f, \({\mathcal {D}} \approx 4.1488 \times 10^3\) \(\hbox {MHz}^2\) \(\hbox {pc}^{-1}\) \(\hbox {cm}^3\) s being the dispersion constant and k being a constant offset.

Fig. 1
figure 1

Upper panels: observed RMs for the short-term (left) and for the long-term (right) datasets. Regions with elevations lower than 30° are marked with blue shaded areas on the left panel. On the right panel the indicator of the Solar activity (the solar flux at 10.7 cm) is shown in green. Lower panels: corresponding RM residuals after the SLM (black), IRI-Plas extended SLM (orange) and TOMION dual-layer voxel model (blue) are subtracted from the observed values. For demonstrative purposes, additional shifts of +0.3 rad \(\hbox {m}^{-2}\) and \(-\)0.3 rad \(\hbox {m}^{-2}\) were introduced for the latter two models. Mean value of \(\Delta \)RM corresponds to the contribution of the ISM. The grey vertical lines on the right lower panel indicate daily boundaries. The puncture lines on the right lower panel correspond to the best-fit agnostic model (see the main text for details)

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Finally, each observation was split in 15-minutes segments, corresponding to the finest time resolution of the available ionospheric correction models. The sensitivity of LOFAR as well as any other non-steerable antenna decreases rapidly away from the zenith due to the projection effects (Noutsos et al. 2015). Therefore, S/N of a signal which directly translates in the accuracy of the measured RMs, tends to scale down accordingly. The behaviour is clearly visible on the right panels of Fig. 1. For some of the data chunks this decrease was so significant that RM could not be estimated reliably. These data were eliminated from the analysis. To increase the credibility of the analysis of the long-term dataset, observations which were carried out at zenith angles higher than 70° were discarded.

For each remaining 15-minute slice of data, we have calculated the RM using the RMcalc python packageFootnote 7 based on the Bayesian generalised Lomb-Scargle technique (Porayko et al. 2019). This method translates sinusoidal oscillations in Stokes Q and U as a function of observational frequency into a 1D posterior distribution of RM. A peak in the RM domain corresponds to the sought value, while the width of a peak characterises the RM uncertainty, which is obtained by fitting a Gaussian in the region of a global maximum. Additionally, in order to eliminate the under-/overestimation of the resultant uncertainties, they were normalised with the reduced \(\chi ^2\) of residual Stokes Q/U series.

3 Modelling the ionospheric Faraday rotation with GNSS data

The ionosphere is the part of Earth’s atmosphere that is significantly ionised by the solar radiation, and it is defined to be located from \(\sim \)100 to \(\sim \)1000 km from the Earth’s surface. The ionosphere significantly affects the propagation of radio waves; in particular, it introduces time-dependent Faraday rotation in the linearly polarised component of the passing radiation. These temporal variations encapsulate diurnal, seasonal and Solar cycle periodicities, and can be as high as 4 rad/\(\hbox {m}^2\) in amplitude (Oberoi and Lonsdale 2012). The uncompensated ionosphere is a common source of errors in radioastronomy at lower frequencies (Jelić et al. 2014; Mevius et al. 2016; Beser et al. 2022). In the context of pulsar observations, temporal changes in the RM due to the terrestrial plasma have been seen evidently in multiple studies (see, e.g. Sotomayor-Beltran et al. 2013; Porayko et al. 2019). One must subtract the ionospheric contribution (e.g., computing it with one of the available ionospheric models) to carry out several pulsar-based studies, such as accessing the astrophysical RM along LoS towards a pulsar, or performing polarimetry after correcting pulsar’s polarisation for the propagation effects. The problem can be reversed, and the residual between the observed pulsar RMs and the modelled ionospheric RM values can be used to estimate the predictive power of the ionospheric models themselves. To perform this kind of analysis, one needs to assume that the observed RM fluctuations are generated purely by the highly dynamic terrestrial plasma. This is a valid assumption, since the standard deviation of the RM fluctuations due to the ISM are expected to be \(\sim 10^{-4}\) rad \(\hbox {m}^{-2}\) on a 5 year timescale,Footnote 8 which is 3 orders of magnitude lower than the current accuracy of the ionospheric modelling (\(\sim 0.2\) rad \(\hbox {m}^{-2}\)). Moreover, the used pulsar has an ecliptic latitude of 34.26°, which implies that J0332\(+\)5434 does not arrive closer to the Sun than \(\approx 0.56\) astronomical units. This means that any RM contribution from both the magnetic field of the Solar wind and events like coronal mass ejections (CME) can be neglected: for example, the expected RM contribution from CME (Liu et al. 2007; Oberoi and Lonsdale 2012) and Solar wind (You et al. 2012) at about 0.6 astronomical units is \(<10^{-2}\) rad \(\hbox {m}^{-2}\) and \(<10^{-4}\) rad \(\hbox {m}^{-2}\), respectively, that is well below the magnitude of the considered systematics.

In this letter, we focus on comparing the performance of three GNSS-driven models of the ionosphere, namely the SLM, the IRI-Plas extended SLM and the TOMION dual-layer voxel model, by analysing the RM residuals of PSR J0332+5434. The first two models make active use of global ionospheric maps (GIMs) which comprise the information on the integrated electron densities in the direction of zenith, also known as vertical total electron content (VTEC). For the current work, we have used the UQRGFootnote 9 GIMs (described later in Sect. 3.3) which as demonstrated in Porayko et al. (2019), outperformed other GIMs when tested on the RM datasets of five pulsars. The latter ionosphere model utilises the full electron density tomography which is solved for when constructing the UQRG GIMs. A summary description of the models is provided below.

3.1 Single layer model

The SLM (Wild 1994; Schaer 1999) is one of the most conventional and straightforward ways to estimate the column electron density towards an arbitrary direction from the GIM-provided VTECs. Within this model, the ionosphere is approximated by an infinitesimally thin sheet of plasma fixed at an arbitrary altitude, known as the effective ionospheric height, that is typically assumed to lie between 350 and 650 km (Hernández-Pajares et al. 2005). The exact choice of the effective ionospheric height depends on the objectives. The recommended value of the effective height to be used for the SLM mapping function with the UQRG GIMs, is 450 km (Mannucci et al. 1998; Hernández-Pajares et al. 2009). Within this approximation, the integral in Eq. (1) can be reduced to a simple product of the LoS integrated electron density, also known as slant total electron content (STEC), and the LoS projection of the magnetic field in the ionosphere piercing point (IPP), \(B_\text {IPP}\) in which the LoS intersects the thin layer of the ionosphere:

$$\begin{aligned} \text {RM}\Big [\text {rad m}^{-2}\Big ]=2.6\times 10^{-17}\times \text {STEC} \Big [\text {m}^{-2}\Big ] \times B_{\text {IPP}} [\text {G}].\nonumber \\ \end{aligned}$$
(2)

The STEC values are recalculated from the GIM VTECs with the help of the mapping function. Although the construction of the UQRG GIMs, assumes dual-layer model in its core, the simple SLM mapping function with fixed effective height has been used to perform vertical-to-slant TEC conversion.

VTEC values provided in GIMs are epoch-specific and defined on a regular grid with finite spatial resolution. Recalculation to an arbitrary location is performed using the inverse-distance weighted-interpolation of the four nearest IPP datapoints (Gurtner 1998). In time, the simple linear interpolation between two consecutive maps is used. As for the terrestrial magnetic field, we used the world magnetic model (WMM) (NCEI Geomagnetic Modeling Team and British Geological Survey 2020).Footnote 10 The SLM RMs were estimated using the publicly available rmextract python package (Mevius 2018).

3.2 IRI-plas extended SLM

Although the ionospheric electron density profile has a sharp maximum at a height typically ranging from \(\sim \)250 km to \(\sim \)450 km depending on, e.g. local time, season and Solar cycle, the ionised matter in the terrestrial atmosphere has in fact a wide profile. Moreover, the ionosphere is followed by the plasmasphere, i.e. a layer of dilute plasma which extends from 1000 km to as far as 20,000 km above the Earth’s surface. The typical electron densities above 1000 km are at least 1–2 orders of magnitude smaller than the peak value. However, given that vertical and slant TECs are integrated observable, the contribution of the plasmaspheric layer alone is between 10% and 60% depending on the time of a day (Yizengaw et al. 2008). Calculated from the GNSS data, VTEC values published in GIMs completely accounts for the integrated electron density throughout all the layers of the atmosphere. Utilisation of these VTECs in Eq. (1) introduces an overestimation of the resultant modelled RM, as the TEC is multiplied by a magnetic field strength at an altitude of 450 km, which is significantly larger than in the upper layers of the atmosphere. In order to account for the spread of the ionic matter and gradient of the magnetic field in the atmosphere, we have performed a numerical integration of the original expression in Eq. (2), using the semi-empirical electron density profile obtained with International Reference Ionosphere (IRI) model (Bilitza et al. 2017). The plasmaspheric electron density profiles have been derived thanks to IRI-Plas, which is an extension of the IRI model to plasmaspheric altitudes (Bilitza and Tamara 2012; Gulyaeva et al. 2011; Sezen et al. 2013). As in the case of SLM, the magnetic field profiles were modelled with WMM. Given that plasma in the terrestrial atmosphere is very dynamic and both IRI and IRI-Plas provide only monthly averaged parameters, the obtained electron density distributions were combined with GNSS measurements as follows. The IRI-Plas profile was determined on a 10 km grid between the height of 60 and 20,200 km. This was then normalised such that the total integral would match the value from the UQRG GIMs.

3.3 TOMION dual-layer Voxel model

Another possible way of relaxing the unrealistic assumption of the ionosphere being confined in an infinitesimally thin layer at a constant effective height is to use a series of GNSS-measured STECs to perform a tomography and resolve the anisotropic distribution of the electron density in the terrestrial atmosphere. Given that the GNSS satellites probe only a limited number of LoSs, the ionospheric layer should be split in a feasible number of cells, or voxels within which the electron density is assumed to be constant. This idea has been implemented in the TOMION dual-layer voxel model of the ionosphere (Hernández-Pajares et al. 1997, 1999). In particular, this model assumes two layers of voxels with sizes of 2.5°\(\times \) 5° in longitude \(\times \) latitude. Both layers have a thickness of 680 km and median altitudes of 450 and 1130 km, respectively, above the Earth’s surface. The tomographic model is then solved with a Kalman filter.

TOMION is able to process both ground-based GNSS ionospheric data, GPS LEO radio occultation data (Hernández-Pajares et al. 1998, 2000a), and ionospheric corrections for precise user positioning (Wide Area Real-Time Kinematic, Hernández-Pajares et al. (2000b)). This modification demonstrates a significant advantage with respect to SLM in removing the VTEC distortions when ground-based data only is used (Figure 2 in Hernández-Pajares et al. 1999). Moreover, the dependence of the classical TEC estimation on pseudorange measurementsFootnote 11 (extensively affected by multipath) and its dependence on the associated interfrequency instrumental delays, significantly affect the STEC calibration (see Figure 3 in Juan et al. 1997). This second source of error is avoided in TOMION, by using only the geometry-free combination of carrier phase measurements, whose ambiguities are properly estimated in the Kalman filter. Thanks to the continuous change of the geometry along the phase arcs, the ambiguity estimation can be properly done simultaneously with the mean electron densities within the illuminated voxels.Footnote 12

The interpolation of the individual VTEC estimates, computed from the tomographically calibrated STECs, was improved from a two hour resolution spline-based interpolated GIM (UPCG) to a fifteen-minutes resolution kriging-based interpolation (UQRG, Orús et al. 2005).

The UPC GIM product was incorporated in the SLM framework for ionospheric RM calculation in Sect. 3.1. In order to account for the varying effective ionospheric height, the TOMION dual-layer voxel model of RM makes use of the complete tomographic output of the TOMION software, which comprises the electron density values per voxel. The electron densities were recalculated to VTEC through a trivial integration, i.e. \(n_e \times w\), where \(w=680\) km is the voxel height. The VTECs of the illuminated voxels are converted to STECs with the SLM mapping function for each layer separately. The mapping function has the same functional form, but different effective heights of 450 km and 1130 km, which correspond to the median layer altitudes, are used for the two layers. The RMs per layer obtained from Eq. (2) are summed up to get the total ionospheric RM along the LoS. As in the case of the SLM, we have performed a linear interpolation in time and an inverse-distance weighted interpolation in space, incorporating the 4 nearest voxels to the illuminated one per layer. In total, 16 datacells are used for the complete interpolation procedure (temporal and spatial). As before, WMM was used to reconstruct the magnetic field profiles.

4 Results

Models under consideration were tested on the RM time series of PSR J0332+5434 described in Sect. 2. Given the reasoning in the previous section, throughout the analysis, it is assumed that the contribution of the magneto-ionic ISM is constant in time over the considered timescales and RM variations due to the interplanetary plasma are negligible. The RM time-series derived from the long-baseline and the short-baseline datasets were investigated separately and are shown in Fig. 1 (upper panel). The corresponding RM residuals (\(\hbox {RM}_\text {obs}\)-\(\hbox {RM}_\text {iono}\)) are in the lower panel of Fig. 1. Additionally, the Lomb-Scargle periodogram of the long-term dataset is provided in Fig. 2.

Three types of systematics arise in the datasets: i) prominent jumps associated with the offset between the GIMs on two consecutive days; ii) long-term linear trends; and iii) diurnal and annual periodic variations. The first systematic emerges when analysing the short-term dataset (see lower left panel of Fig. 1) and was inspected visually. In particular, the TOMION model successfully deals with the daily GIM jumps, while both SLM and IRI-Plas extended SLM show prominent leap in the RM residual plot at 00:00 on the 3rd and 4th of April which are indicated with grey color. The daily runs of TOMION model are performed considering the dual-frequency GNSS carrier phases of a global network of typically >200 worldwide distributed ground-based receivers from the given day, and from the latest 4 h of the previous day as well. This can explain the stability in the transition of the calibrated TEC values with the tomography across the day boundaries. At the same time, VTEC GIMs are produced using the input data from a specific date.

Fig. 2
figure 2

Lomb-Scargle periodograms of the RM residuals for the day-averaged long-term dataset. The dashed vertical line indicates 0.0027 \(\hbox {day}^{-1}\) (1 \(\hbox {yr}^{-1}\)) in frequency. The seasonal/diurnal variations and linear trend are visible as peaks at 1 \(\hbox {yr}^{-1}\) and excess of power at the low-frequency part of the periodogram, correspondingly

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Other two types of systematics were investigated more thoroughly using the long-term dataset. The linear trend, which is clearly visible on the right lower panel of Fig. 1, manifests itself as an excess of power at low frequencies of the Lomb-Scargle periodogram. This artefact indicates strong correlation with the Solar cycle which is shown in green on the right upper panel of Fig. 1. However, a data coverage of at least one whole Solar cycle should be reached to draw the final robust conclusion.

Periodic systematic shows up as a peak at 1 \(\hbox {yr}^{-1}\) in the Lomb-Scargle periodogram. Mismodelled seasonal effects, e.g. unaccounted annual change in the effective height of the ionosphere, can be a cause for these variations. Given that the source was observed at approximately the same zenith angle each day, the local time of a day was changing on an annual cycle. Therefore, along with the seasonal effects, the 1-\(\hbox {yr}^{-1}\) peak can contain possible diurnal footprints. With the current dataset, two contributions are completely aliased.

Table 1 The best-fit parameters of the agnostic model described by the Eq. (3) consisting of the linear trend and the sine wave with the fixed period of 1 year
Full size table

In order to quantitatively estimate the magnitude of the residual artefacts of the long-term dataset, we fit an agnostic model consisting of 1-yr sine wave with the amplitude \(A_{\text {yr}}\) and phase \(\phi \) and linear trend characterised by the slope \(A_{\text {trend}}\):

$$\begin{aligned} {\textbf {RM}}_{\text {mod}}=A_{\text {yr}} \sin (2\pi {{\textbf {t}}}/\text {yr} + \phi ) + A_{\text {trend}}{} {\textbf {t}} + \text {const}, \end{aligned}$$
(3)

where \({\textbf {RM}}_{\text {mod}}\) is the vector of RM residuals measured at time epochs \({\textbf {t}}\). The results of the least-square minimisation are presented in Table 1 and are in agreement with the qualitative results of the Lomb-Scargle periodogram.

As shown in the table and figures, none of the considered models can fully account for the ionospheric contribution. Two types of residual systematics persist when SLM is used for ionospheric reduction in the long-term dataset, which is largely in agreement with the results of Porayko et al. (2019). The IRI-Plas extended SLM significantly absorbs the long-term linear trend which is 1.5-2 times weaker than for the TOMION model and SLM, respectively. However, the periodic systematic remains and gets marginally exacerbated in comparison to the SLM. On the contrary, the TOMION model considerably reduces the excess of power at 1 yr (by a factor of 1.5 in comparison to the other two models). This improvement is especially visible in Fig. 3, where the RM residuals are plotted against time of a day. The diurnal variations have visually the same magnitude for all three models on the left panel, where we show the results for the first half of the long-term dataset (till 11.05.2016). At the same time, for the second half of the dataset the diurnal pattern persist for the SLM and IRI-Plas extended SLM, and is barely noticeable for the TOMION model (left panel of Fig. 3), which confirms the flattening of the RM residuals after May-June 2016 in the bottom panel of Fig. 1 (blue color). This signifies that the quality of the modeling correlates with the phase of the Solar cycle.

Fig. 3
figure 3

The RM residuals of the long-term dataset after subtraction of the best-fit linear trend with respect to the time of a day. The residuals for the first half of the long-term dataset (till 11.05.2016) are shown on the left panel, while the second half of the dataset is displayed on the right. Color scheme is the same as in Figs. 1 and 2

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5 Summary and discussion

In this letter, we compare the performance of three models of the ionospheric Faraday rotation using the data of PSR J0332+5434 observed with two international LOFAR stations in Germany. To date, only SLM has been used to correct the ionospheric contribution in pulsar radio-astronomical data. However, as it was shown in Porayko et al. (2019), the simplified framework of SLM with fixed effective height is not capable of accounting for all the wealth of ionospheric physics. One of the ways to improve the performance of the modelling is to consider the thickness and dynamic behaviour of the ionosphere, which is the main idea behind the IRI-Plas extended SLM and the TOMION dual-layer voxel model. All three considered models are largely based on the GNSS-reconstructed maps of the ionospheric electron content and utilise WMM as the model of the Earth’s magnetic field. When applied to pulsar data, the models demonstrate varying degree of progress in accounting for the ionospheric RM. However, none of the models completely obliterates the residual systematics.

The classical and IRI-Plas extended SLMs exhibit the daily jumps attributed to the discontinuity between the publicly available GIMs these models are based on. The TOMION model successfully absorbs this artefact, which suggests that for the smooth TEC reconstruction, the input GPS data with at least several hour intersection window should be necessary interpolated. Additionally, the TOMION dual-layer voxel model significantly absorbs the diurnal/seasonal variations, especially at the epochs around the Solar minimum. However, the TOMION model fails to properly account for the long-term linear trend strongly correlated with the 11-year Solar cycle, giving up the preeminence to the IRI-Plas extended SLM. Being based on a complex highly resolved tomographic solution, TOMION model is expected to be superior to the other two considered methods. However, in practice the TOMION model assumes that the ionosphere spreads up to 1470 km only. Although, within such simplification one can carefully reconstruct the ionosphere, i.e. the evolution of its effective height and width, around the Solar minimum, it is clearly not sufficient for the epochs of the Solar maximum where the effective height of the ionosphere can significantly increase (Lei et al. 2005; Liu et al. 2009; Brunini et al. 2011). Such behaviour indicates the importance of accounting for the broad electron density profiles including upper ionosphere and plasmasphere when modelling the contribution of the terrestrial plasma to observed Faraday rotation of astrophysical sources (Lunt et al. 1999; Belehaki et al. 2004; Yizengaw et al. 2008). On the other hand, being based on monthly averaged electron density profiles, the IRI-Plas extended SLM cannot carefully reproduce the rapid changes in the ionospheric layer. These results suggest that in order to produce an accurate model of the ionospheric Faraday rotation, an instant change in plasma distribution (TOMION dual-layer voxel model) should be paired with a broad electron density profile resolved up to plasmaspheric altitudes (IRI-Plas extended SLM). The possibility of such combination and its effectiveness as well as other more complex models of the ionosphere (see e.g. Galkin et al. 2012; Lyu et al. 2018; Pignalberi et al. 2018, 2019) and plasmasphere are going to be investigated in future works.

The current manuscript is one in a series of works dedicated to ionospheric modelling in pulsar radio data. The regular monitoring of pulsars with the LOFAR stations within GLOW provides a unique opportunity of probing available ionospheric models and independently constructing new ones. At the next stage of our investigation, we will focus on replenishing the dataset with new sources and enlarging a sample of continuous daily runs as well as extending the observational timespan up to the next Solar maximum, to break the degeneracy between different types of systematics and strengthen our conclusions. The global objective of this study is derivation of the general algorithm to mitigate the ionospheric contribution in radio astronomical observations at low frequencies. The goal can be achieved by building the physically motivated model of the terrestrial plasma or at least deriving a mitigation technique using an empirically driven approach, e.g. by directly using pulsar Faraday rotation measurements. The results of this study are going to be instrumental for ionospheric calibration of astronomical data from the existing radio interferometers as well as from newly emerging MeerKAT and SKA radio antennas.