© The Authors 2024

1. Introduction

The Sun’s photosphere is subject to observation by a multitude of instruments, each endowed with distinctive characteristics, such as spectral and spatial sampling, and resolution. Independently, each instrument provides its unique particular view of the photospheric magnetic field. By combining instruments with different capabilities, it is possible to capture the same phenomena from different points of view, extracting more comprehensive information than using a single view from a single instrument. The successful combination of different instruments needs precise calibration procedures. Numerous calibration efforts have already been carried out for instruments on board space missions. For instance, Liu et al. (2012) conducted a comparison of line-of-sight (LoS) magnetograms obtained by the Helioseismic and Magnetic Imager (HMI; Scherrer et al. 2012; Schou et al. 2012) on board the Solar Dynamics Observatory (SDO; Pesnell et al. 2012) and the Michelson Doppler Imager (MDI; Scherrer et al. 1995) on board the Solar and Heliospheric Observatory (SOHO; Domingo et al. 1995). Similarly, Sainz Dalda (2017) presented a statistical comparison between magnetograms produced by the data-processing pipelines of the Solar Optical Telescope-Spectro-Polarimeter (SOT-SP; Tsuneta et al. 2008; Lites et al. 2013) on board Hinode (Kosugi et al. 2007) and SDO/HMI. Cross-calibrations can also be carried out between instruments located on the ground and on board space missions, such as was presented by Riley et al. (2014), the preliminary cross-calibration between the Global Oscillation Network Group (GONG; Harvey et al. 1996) of the National Solar Observatory (NSO) and SDO/HMI data by Plowman & Berger (2020), or the Kitt Peak Spectropolarimeter (Wenzler et al. 2004; Yeo et al. 2014). This paper presents a comparison between the Full Disc Telescope (FDT; Álvarez Herrero in prep.) of the Polarimetric and Helioseismic Imager (SO/PHI; Solanki et al. 2020) on board the Solar Orbiter spacecraft and SDO/HMI, as was done between SO/PHI-HRT and SDO/HMI by Sinjan et al. (2023).

Solar Orbiter (Müller et al. 2013, 2020) is a joint space mission of ESA and NASA. It was launched on 10 February 2020. Its overarching goal is the study of the Sun and the heliosphere as a single system. It carries a total of ten scientific instruments on board, four for in situ measurements, and six for remote-sensing observations. Among the latter, SO/PHI is the first magnetograph and tachograph to observe the Sun outside the Earth-Sun line. SO/PHI images the entire solar disk with the FDT or a fraction of the solar surface with the High Resolution Telescope (HRT; Gandorfer et al. 2018).

For the overarching goal of the Solar Orbiter mission, it is crucial to obtain a proper cross-calibration with other instruments. On 7 March 2022, Solar Orbiter was located almost exactly between the Sun and the Earth, providing a unique opportunity for cross-calibrating SO/PHI with other similar ground-based and space-borne instruments. Then, on 8 March, SO/PHI-FDT recorded observations along with SDO/HMI. SDO/HMI is particularly useful for a comparison because it observes the same spectral line as SO/PHI. It is of similar importance to perform a proper comparison between SDO/HMI and SO/PHI-FDT, particularly because both instruments provide full solar disk observations, and combining their data is of highest relevance for synoptic map generation using both instruments (Loeschl et al. 2024), and so on. The objective of this work is to perform a consistency check of the magnetic data products of SO/PHI-FDT and SDO/HMI. The comparison performed in this paper allows us among others to study the long-term evolution in the photospheric magnetic field, which extends beyond the capabilities of a single instrument at a single observation point. This also opens the door for implementing stereoscopy techniques on solar features and conducting diverse multi-angle studies through the combination of SDO/HMI and SO/PHI-FDT data (e.g., Albert et al. 2023a).

In Sect. 2 we present an overview of the SO/PHI and SDO/HMI instruments. In Sects. 2.1 and 2.2 we briefly describe the different pipelines used to process the data from the two instruments. In Sect. 3 the datasets used for this study and their characteristics are explained. The alignment of the regions chosen for the calibration and the pixel selection in the two instruments are presented in Sect. 3. Section 4 presents and discusses the results we obtained from the comparison. Our conclusions are drawn in Sect. 5.

2. Overview of the SO/PHI and SDO/HMI instruments

As part of the Solar Orbiter mission, SO/PHI observes the Sun using its two telescopes. It provides data that cover the entire solar disk via the FDT and obtains high-resolution data for a portion of the Sun with the HRT. The angular resolution of the FDT is 3$\stackrel{″}{.}$75, and that of the HRT is 0$\stackrel{″}{.}$5. The latter is approximately the same of that of the SDO/HMI, which observes the full disk of the Sun with an angular resolution of 0$\stackrel{″}{.}$5. The main difference between SO/PHI and SDO/HMI is that while SDO/HMI consistently provides a spatial resolution of approximately 725 km at the center of the solar disk, the spatial resolution of the SO/PHI-Telescopes varies throughout the orbit. During the closest approach to the Sun of Solar Orbiter, at a distance of nearly 0.28 au, the FDT achieves a resolution of 1450 km at the center of the disk. The main characteristics of the two instruments are summarized in Table 1.

For the spectroscopic analysis, SO/PHI samples the Fe I spectral line at 617.3 nm across six different wavelength positions, [–14, –7, 0, 7, and 14] pm, plus the continuum at 30 pm, either to the blue or the red side of the line, depending on the Solar Orbiter velocity relative the Sun. To achieve this, the instrument is equipped with a tunable LiNbO₃ Fabry–Pérot etalon, enabling narrow-band imaging spectroscopy. At each of the six sampled wavelengths, four different linear combinations of the four Stokes parameters are imaged. Additionally, SO/PHI can account for the changing spacecraft velocity and keep the line centered by tuning the LiNbO₃ etalon. The transmission bandwidth of SO/PHI is about 10.6 pm. On the other hand, SDO/HMI uses a Lyot filter and two Michelson interferometers to sample the Fe I line at 617.3 nm. This sampling is performed in an interval of 6.9 pm, ranging from −17.25 pm to +17.25 pm around the rest wavelength position of the line. The transmission bandwidth of SDO/HMI is about 6.8 pm.

For the polarimetric analysis, both SO/PHI and SDO/HMI use distinct approaches. In SO/PHI, the polarization modulation package (PMP; Álvarez Herrero et al. 2018) is responsible for modulating and analyzing the polarization of light. This process is achieved through the use of two liquid-crystal variable retarders (LCVRs) in conjunction with a linear polarizer. SO/PHI-FDT achieves a polarization sensitivity better than 10⁻³ (in units of the continuum intensity I_c) in about one minute (Martínez Pillet 2007; Del Toro Iniesta & Martínez Pillet 2012; Campos-Jara et al. 2019; Albert et al. 2023b). This polarimetric sensitivity translates into a LoS magnetic field component (B_LoS) sensitivity of 7 G (Martínez Pillet 2007). The SO/PHI dataset-acquiring times vary from about 60 s to 120 s, depending on the acquisition scheme, which can be configured. On the other hand, SDO/HMI employs rotating waveplates together with a beam splitter. In its Doppler camera, SDO/HMI only carries out a longitudinal polarization analysis: I ± V are measured with a cadence of 45 s. In the vector camera (Borrero et al. 2007; Hoeksema et al. 2014), it measures all four Stokes parameters, I, Q, U, and V, in 135 s. As of 13 April 2016, HMI/SDO changed the collection scheme for the vector magnetic field measurement. The filtergrams needed to calculate the Stokes parameters are provided in 90 s using all filtergrams from both SDO/HMI cameras. With this new strategy, the noise is reduced by about 5% in the LoS magnetic field component compared to the previous collection scheme (Liu et al. 2016). The SDO/HMI vector camera data are weighted averages over ten datasets every 720 s to increase the signal-to-noise ratio (S/N). The reported sensitivity of the longitudinal magnetogram series taken with the Doppler camera is about 10.2 G (Liu et al. 2012), which decreases to approximately 5.3 G (Liu et al. 2016) for the 720 s LoS magnetograms. In contrast to Liu et al. (2012), we use G instead of Mx/cm⁻² because they are dimensionally equivalent and the assumed magnetic filling factor is unity.

Finally, because of its role in a deep-space mission with limited telemetry, SO/PHI is designed to autonomously process and conduct a full scientific analysis onboard. This functionality is achieved by a specifically tailored software (Albert et al. 2018, 2020) and a newly designed device, the electronic inverter of the radiative transfer equation (RTE) for polarized light (Cobos Carrascosa 2016; Cobos Carrascosa et al. 2016). The recorded and the processed data both undergo compression using conventional procedures before they are downlinked (Hernández Expósito et al. 2018).

2.1. SO/PHI-FDT data processing

For this study, we have used the three components of the vector magnetic field: strength, inclination, and azimuth (B, γ, ϕ), inferred from SO/PHI-FDT raw data processed on ground. The raw dataset consists of 24 raw polarized images, corresponding to four polarization modulation states and six wavelength samples. These images were corrected on ground for dark current, flat-field, prefilter transmission effects, as well as for ghost and fringe artifacts¹. Polarimetric demodulation converts the polarized images into Stokes I, Q, U, and V images. These were then normalized to the continuum intensity measured around disk center. Finally, the data were corrected for residual cross-talk from Stokes I to Stokes Q, U, and V, and it is ensured that the median value of the polarization continuum maps is zero. The S/N in Stokes Q, U, and V is about 1.3 × 10⁻³ I_c. The increased noise observed in the data can be attributed to the compression factor, which is limited to 6 bits-per-pixel for the dataset.

After preprocessing, the three components of the vector magnetic field and the line-of-sight velocity were inferred. First, these quantities were estimated by using the center-of-gravity technique and the weak-field approximation (Semel 1967, 1970; Rees & Semel 1979; Landi Degl’Innocenti & Landolfi 2004), which we call classical estimates. These are the initial guesses for the second step, which is the RTE inversion, because they are an optimum initial guess for any inversion of the RTE (Del Toro Iniesta & Ruiz Cobo 2016). This inversion was performed under the assumption of a Milne–Eddington (ME) atmosphere and using the MILOS code (Orozco Suárez & Del Toro Iniesta 2007). Then, B_LoS was derived as B cos γ, directly using the ME inversion results. In SO/PHI, the effective exposure time is 7.2 s. This encompasses the acquisition of 480 individual frames, 20 at each of the six wavelength positions and the four modulation states. The remaining time, a total acquisition period of 68 s, is spent in tuning the PMP and the etalon, and in the data-transfer process.

2.2. SDO/HMI data processing

SDO/HMI provides the raw spectropolarimetric data. All the reduction and analysis procedures are applied on ground. The SDO/HMI pipeline is implemented at the SDO Joint Science Operations Centers (JSOC) and was fully described in Hoeksema et al. (2014).

One of the SDO/MI products analyzed in this work are the LoS scientific products (v_LoS, B_LoS), which are provided every 45 s, are obtained with the Doppler camera. In addition, LoS products can be provided every 720 s, obtained from joint measurements of the two SDO/HMI cameras and a temporal weighted average to increase the S/N. They are only based on I ± V measurements. Every 45 s and 720 s SDO/HMI also delivers the Fe I line width, line depth, and continuum intensity. The algorithm implemented in the LoS pipeline is the Fourier tachometer technique (Title & Tarbell 1975), which was also used for data of MDI. SDO/HMI uses all the six wavelength samples across the selected spectral line, while MDI uses five.

The other SDO/HMI data products analyzed in this work are the vector magnetic field maps, which are provided with a 720 s cadence. These maps are inferred from data collected by the vector and Doppler camera. Together, they provide the four Stokes parameters every 90 s. According to Couvidat et al. (2016), these filtergrams are weighted averages over 1350 s in the pipeline to increase the S/N. B, γ, and ϕ are obtained through the inversion of the RTE by using the very fast inversion of the Stokes vector code (VFISV; Borrero et al. 2011; Hoeksema et al. 2014), which assumes the Milne-Eddington approximation, just as the MILOS code does. In this case, B_LoS was derived as B cos γ, directly using the ME inversion results.

3. SO/PHI-FDT and SDO/HMI data

To perform the comparison, we employed SO/PHI-FDT data acquired on 8 March 2022 at 07:00:09.875 UTC, close to the inferior conjunction of Solar Orbiter. On this date, SO/PHI was at a solar distance of 0.485 au, and the spatial resolution of the FDT was therefore about 2640 km at disk center. At the time of observation, the angular separation between Solar Orbiter and SDO was 3.4°. This small angle provided a unique opportunity of comparing data products from SDO/HMI and SO/PHI-FDT from nearly the same line of sight. Solar Orbiter was positioned at a Stonyhurst latitude of −4.32° and a Stonyhurst longitude of 1.75°. In the case of SDO/HMI, the Stonyhurst latitude was −7.24° and the Stonyhurst longitude was −0.01°. In the selection of the SDO/HMI data, we considered the initial time of the SO/PHI observation and took the light travel time between the two instruments into account, which was 4 min and 13 s. In particular, the 45 s SDO/HMI LoS products that we selected are from 8 March 2022 07:04:11.900 UTC, and the SDO/HMI full vector products are from 8 March 2022 06:58:34.400 UTC. At these specific times, the radial velocity of SDO was approximately 0.3 km s⁻¹. Consequently, the effect of the line shift near the maximum and minimum radial velocities can be considered negligible for the purposes of this study (Liu et al. 2012; Hoeksema et al. 2014).

The LoS magnetic field data from SO/PHI-FDT and SDO/HMI 45 s are presented in Fig. 1. To conduct our analysis, we selected four distinct areas, each corresponding to different active regions situated at various positions on the solar disk. The comparison of the magnetic field data of the two instruments, as detailed in Sect. 4, is primarily centered on the NOAA AR 12960.

Fig. 1. SO/PHI-FDT and SDO/HMI BLoS 45 s magnetograms (panels A and B) with their original orientations in the focal plane. The axes represent helioprojective-Cartesian coordinates in units of arcseconds. The blue, red, green, and yellow rectangles correspond to the selected areas. Panels C and D show zoomed-in magnetograms corresponding to the blue area. Panels E and F show more detailed zoom-in magnetograms up to the pixel resolution as purple squares. The cyan squares represent the SO/PHI-FDT pixel size in the SO/PHI-FDT and SDO/HMI magnetograms. The magenta square corresponds to the SDO/HMI pixel size. The arrows in panels A and B point to solar north.

3.1. Pixel selection method and data alignment

As illustrated in Fig. 1, the SDO/HMI and SO/PHI-FDT data are clearly not initially orientated in the same direction within the detector frame. Moreover, the two instruments have different plate scales (see Table 1). To avoid rotations and interpolations, we therefore opted for a one-to-one pixel-matching method. The alignment method we employed relies on a pixel-to-pixel selection process, facilitated by using the World Coordinate System (WCS) information contained in the metadata of both datasets (Hanisch et al. 2001), and on the application of equations for transforming between different coordinate systems (Calabretta & Greisen 2002; Greisen & Calabretta 2002; Thompson 2006).

To facilitate a direct one-to-one pixel comparison between the two instruments, we first obtained the pixel coordinates of the selected area in SO/PHI-FDT. Then, we employed a coordinate system transformation that yielded the corresponding pixel coordinates of this selected area in SDO/HMI. This transformation identified several pixels of the SDO/HMI data that belonged to a single FDT pixel due to the difference in plate scales, which were averaged. When handling pixels that are not entirely encompassed within a SO/PHI-FDT pixel, they were considered in the averaging calculation when they occupied an area equal to or greater than 50% within the SO/PHI-FDT pixel. Instead of using SO/PHI-FDT pixel coordinates as input, as done in this work, SDO/HMI might also be selected as input because we transformed between well-defined coordinate systems. The key point of this method lies in using a coordinate system featuring a common origin, namely the heliographic Stonyhurst reference system. In this reference system, the origin is situated at the intersection of the solar equator and the central meridian as seen from Earth. This strategy was followed for each of the different parameters analyzed in this study: B, γ, B_LoS, and ϕ.

However, a small shift still exists between the selected areas in the two instruments due to inaccuracies in the calculation of the WCS keywords and to optical distortions, which cannot be corrected for with this pixel-matching. Hence, the data alignment was therefore corrected prior to comparison by applying an ad hoc alignment process based on the maximum correlation between the selected areas. The precision of the alignment routine was 0.01% of a pixel. After this two-step alignment of the areas, we plotted the SDO/HMI values against those of SO/PHI-FDT for each parameter to obtain scatter plots and linear fits for each area, with their respective fit parameters.

The analysis was made using the Python programming language provided by the Python Software Foundation². All procedures implemented for this study can be obtained by contacting the author³.

4. Comparison of the magnetic field products of SO/PHI-FDT and SDO/HMI

4.1. Comparison of the vector data products

Panels A, B, and C in Fig. 2, show the comparison between the inferred B, γ, and B_LoS from SO/PHI-FDT and SDO/HMI corresponding to NOAA AR 12960 (blue area in Fig. 1) and do not take into account the different spatial resolution of the instruments. We concentrated on this AR because it is closest to solar disk center, thus experiencing fewer perturbations from projection effects (the cosine of heliocentric angle is 0.95). Furthermore, it harbors the strongest fields and exhibits some fine-scale sunspot structure, which allows for a much better comparison of the magnetic field inclinations and azimuths.

Fig. 2. Pixel-to-pixel comparison of the blue area in Fig. 1 (NOAA AR 12960) for SDO/HMI (vector products) against SO/PHI-FDT. The original quantities are shown in the top row: B (panel A), γ (panel B), and BLoS (panel C). The middle row displays the same quantities after convolving the original SDO/HMI data with the SO/PHI-FDT PSF (see text): B′ (panel D), γ′ (panel E), and $B_{\text{LoS}}^{\prime }$ (panel F). The total number of pixels compared in these plots is 4306. The blue lines depict linear fits. The dashed gray lines represent the identity function. The fits and Pearson correlation coefficients are given in the insets. The histograms of the differences between SDO/HMI and SO/PHI-FDT values are shown in the bottom row (panels G, H, and I). The dashed lines are skewed Gaussian fits to the histograms, with μ, σ, and η the mean standard deviations and skewness of the fit, respectively. The plots corresponding to the other active regions can be found in the Appendix A.

In the comparison, SO/PHI-FDT pixels with negligible polarization signals,

$$\begin{array}{c}\hfill p=\frac{{\sum }_{\lambda }\sqrt{Q{(\lambda )}^2+U{(\lambda )}^2+V{(\lambda )}^2}}{6}<4\times {10}^{-3},\end{array}$$(1)

were discarded (see Fig. B.1). In the scatter plots (Fig. 2, first two rows), the blue line represents the linear fit, which is the mean value computed from two linear regression analyses, as done in Sinjan et al. (2023). The noise in the on-ground SO/PHI Stokes data is approximately 1.3 × 10⁻³ I_c, limited by the data compression. Hence, the noise in B_LoS is estimated to be about 30 G. The Pearson correlation coefficients, r, are also shown.

In panels A, B, and C of Fig. 2, the magnetic field strengths inferred from SDO/HMI tend to exceed those of SO/PHI-FDT, with a slope of 1.33, or about 33% stronger, and with an offset of −97 G. The field inclination also shows a trend with slightly more vertical fields in the SDO/HMI results, with offsets close to −14°. A tendency for the scatter to decrease when the field inclinations approach 90° is also seen. The reason might be that the Stokes Q and U signals around the sunspot penumbra are stronger than in the quieter areas.

The B_LoS values are clearly correlated and have a small offset of −3.7 G, although, as in the case of the field strength, the SDO/HMI results tend to be larger. They are about 37% for large longitudinal fields. All in all, considering the different techniques used by the two instruments, the differences in spatial resolution, exact observation times, effective exposure times, and different inversion codes employed in either case, the agreement between SDO/HMI and SO/PHI-FDT can be considered satisfactory.

Further improvement can be achieved by convolving the SDO/HMI data with the SO/PHI-FDT spatial point spread function (PSF). This corrects for the different spatial resolutions of the two instruments. Efforts are currently still underway to obtain the SO/PHI-FDT spatial PSF using phase-diversity measurements. We therefore used the theoretical Airy function associated with the SO/PHI-FDT aperture. Before convolving the data, we adjusted the PSF to account for the different distances of the two instruments to the Sun. The difference in spatial resolution is substantial, approximately a factor of 7.5 (see Table 1), between SO/PHI at 0.485 au and SDO/HMI. We therefore assumed that the SDO/HMI spatial PSF can be considered negligible. Consequently, we applied the convolution solely to the SDO/HMI data using the SO/PHI-FDT PSF. The calibrated 24 Stokes images forming an SDO/HMI dataset (four Stokes parameters and six wavelength samples) were convolved separately with the SO/PHI-FDT PSF. Then the data were inverted with the VFISV code, from which we obtained a new set of SDO/HMI products, B′, γ′, and $B_{\text{LoS}}^{\prime }$, whose spatial resolution is comparable to that of the SO/PHI-FDT. The data were inverted using the corresponding SDO/HMI filter transmittances and the applicable calibration files. The results obtained after the convolution are shown in panels D, E, and F of Fig. 2.

Figure 3 shows longitudinal magnetic field maps of NOAA AR 12960 (top row) and a cropped region of the quiet Sun at the disk center (bottom row) as retrieved from SO/PHI-FDT data (panels A and D), from SDO/HMI (panels B and E), and SDO/HMI after convolution with the theoretical SO/PHI PSF (panels C and F). The morphological similarity of the less noisy $B_{\text{LoS}}^{\prime }$ maps with B_LoS from SO/PHI-FDT is clear.

Fig. 3. Active region (top row) and quiet-Sun (bottom row) longitudinal magnetic field maps. The active region corresponds to the blue square in Fig. 1 (NOAA AR 12960) and the quiet-Sun region has 50 × 50 pixels selected around disk center. We present SO/PHI-FDT retrievals (panels A and D), original SDO/HMI retrievals (panels B and E), and SDO/HMI results after convolution with the SO/PHI-FDT theoretical PSF (panels C and F). To facilitate comparison, the maps are rotated such that solar north points upward.

Panels D, E, and F of Fig. 2 show the pixel-to-pixel comparison after finding the one-to-one pixel correspondence following the same method as explained above, between the SO/PHI-FDT and the spatially degraded SDO/HMI maps. The dispersion in the scatter plots is clearly smaller than when we neglect the spatial PSF. The correlations remain very good, and the consistency between results from the two instruments is even better. The slopes for the comparison of SO/PHI-FDT with B′ and $B_{\text{LoS}}^{\prime }$ are close to 1.1, while SDO/HMI still shows more vertical fields. The offset of B′ is −107.5 G, slightly greater than for that of B.

An alternative representation can be found in the last row of Fig. 2, which display histograms of the difference between SDO/HMI and SO/PHI-FDT both without and with convolution of SDO/HMI with the PSF. Skewed Gaussian fits are overplotted. Means, μ, standard deviations, σ, and skewness, η are indicated in the insets. The widths of the skewed Gaussian fits decrease when the spatial PSF of SO/PHI-FDT is taken into account. The mean of the histogram is clearly displaced to more negative fields for the magnetic field strength and to more positive fields for the LoS magnetic field. Furthermore, the application of the SO/PHI-FDT PSF to the SDO/HMI data for the magnetic field strength and the line-of-sight magnetic field component results in a noticeable reduction in the skewness of the fits. This indicates that when we account for the differences in spatial resolution, the discrepancies observed in the inferences of stronger magnetic fields between SO/PHI-FDT and SDO/HMI diminish significantly. No significant changes are seen for the inclination after applying the spatial PSF.

SDO/HMI and SO/PHI-FDT inversions also provide the 180° ambiguous azimuth, ϕ, of the magnetic field vector. SDO/HMI also provides the disambiguated azimuth. As explained in Hoeksema et al. (2014), the disambiguation algorithm implemented is a variant of the minimum energy method presented by Metcalf (1994). SO/PHI can apply the same method, but also a novel stereoscopic disambiguation method (see Valori et al. 2023). However, as the SO/PHI instrument team has not released standard disambiguated data products, they are not considered here. In Fig. 4 we present the pixel-to-pixel comparison of the azimuth derived from both instruments and the corresponding maps. For this comparison, we used the SDO/HMI azimuth, ϕ, and the azimuth ϕ′ obtained after applying the spatial SO/PHI-FDT PSF. SO/PHI-FDT pixels with negligible polarization signals (see Eq. (1)) are excluded in the comparison. Furthermore, due to the intrinsic 180° ambiguity of the azimuth, pixels where |ϕ_HMI − ϕ_FDT|> 90° have been discarded. To ensure that both datasets have a consistent definition for the azimuth, we corrected the roll angle of each spacecraft. In contrast to B, γ, and B_LoS, for which we present the linear fit parameters and their formal errors of the comparison for the three selected areas together (see Table 2), we just present the parameters for the sunspot NOAA AR 12960 because it is the only sunspot in which the azimuthal structure is clearly resolved.

Fig. 4. Pixel-to-pixel comparison of the blue area (NOAA AR 12960) in Fig. 1 of the azimuth of SDO/HMI and SO/PHI-FDT. Panel A shows the original azimuth of SDO/HMI, ϕ, and the panel D displays the one inverted with the VFISV code using the Stokes maps convolved with the SO/PHI-FDT PSF, ϕ′. In both panels the azimuths are compared pixel-to-pixel against the SO/PHI-FDT azimuth. The roll angles have been corrected to establish a common origin for azimuth measurements in the SO/PHI-FDT and SDO/HMI datasets. The blue lines show linear fits. Points where |ϕHMI − ϕFDT|> 90° are excluded due to the intrinsic azimuthal ambiguity. The fits and Pearson correlation coefficients are given in the insets. Panels B and E show the same area as the SO/PHI-FDT azimuth without and with the polarimetric mask, respectively. Panels C and F show the SDO/HMI azimuth, ϕ and ϕ′, respectively. The total number of pixels compared are 1784 for panel A and 3417 for panel D.

The azimuth values of SDO/HMI and SO/PHI-FDT show a strong correlation in both cases, that is, taking the SO/PHI-FDT spatial PSF into account (panel D) or neglecting it (panel A), although a few points do not follow the linear trend. For panel A, the slope of the correlation is 1 and the offset is almost 0°. In panel D, where we take the SO/PHI-FDT spatial PSF into account, the correlation is worse because the spatial organization of the azimuth values in the SDO/HMI at full spatial resolution becomes distorted when we apply the SO/PHI-FDT spatial PSF. Furthermore, the fit yields a slope of 1.02 and the offset is very small, close to one degree. This suggests that SO/PHI-FDT and SDO/HMI estimate similar magnetic field azimuth values on average for both cases.

4.2. Comparison of LoS data products

The comparison of SO/PHI-FDT LoS magnetic fields with those of the SDO/HMI Doppler camera (45 s) and vector + Doppler cameras (720 s) is presented in panels A and B of Fig. 5, respectively. We did not take the spatial PSF effects into account here. The correlation between the two instruments is excellent for the two datasets with different cadences. The slopes are close to unity, exhibiting a slight deviation of 14%. Additionally, the offsets are close to 6 G in both cases. The data with a cadence of 720 s show slightly less scatter than the 45 s products because the effective exposure time is longer. Because the trend is lower for SO/PHI-FDT than for SDO/HMI B_LoS by some 14%, we conclude that it is also underestimated by the Doppler camera compared to the vector camera of SDO/HMI. This is a well-known effect for classical estimates, such as the center-of-gravity or the Fourier tachometer techniques (e.g., Orozco Suárez 2008).

Fig. 5. Pixel-to-pixel comparison of the area within the blue rectangle in Fig. 1 (NOAA AR 12960) of SO/PHI-FDT BLoS (RTE inversion mode) and SDO/HMI BLoS (MDI-Like algorithm) for 45 s (panel A) and 720 s cadence (panel B) magnetograms. The total number of pixels compared in both panels is 4306. Both panels show SDO/HMI-averaged quantities to match the SO/PHI-FDT pixel size. The blue lines depict linear fits. The dashed gray line displays the identity function. The Pearson correlation coefficients are given.

4.3. Results from all active regions

Figure 1 displays four active regions at different positions on the solar disk. Similar to the procedure for active region 1 (NOAA AR 12960) in Sects. 4.1 and 4.2, the same pixel-by-pixel comparison was conducted for the other three active regions for B, γ, and B_LoS. The results are provided in Appendix A.

Remarkably similar results are observed in active regions 3 (NOAA AR 12957/12959, see Fig. A.2) and 4 (NOAA AR 12962, see Fig. A.3) compared to active region 1, although they are located at different positions on the solar disk and exhibit distinct magnetic field distributions, including lower magnetic field strength values and a very low negative polarity in B_LoS. On the other hand, there are discrepancies between the three regions and area 2 (NOAA AR 12963, see Fig. A.1). The latter exhibits noticeably higher scatter in the pixel-by-pixel comparison of SO/PHI-FDT and SDO/HMI, indicating a weaker correlation and significant changes in the parameters of the linear fit. Upon inspecting the Stokes maps, we noted an artifact within the area of this active region, which was not adequately corrected for by the flat-field images in all the FDT datasets acquired on 8 March 2022⁴. For this reason, we excluded this active region from the results that combine all the regions.

In summary, the results exhibit consistency in the comparison, regardless of the varying positions of the observed areas on the solar disk. This suggests that our findings are compatible regardless of the heliocentric angle and that there are no gradients for magnetic fields which exceed the polarization signals of 4 × 10⁻³ across the entire solar disk. The results combining active regions 1, 3, and 4 are summarized in Table 2 and are displayed in Fig. C.1.

5. Conclusions

A comparison of the magnetic field products inferred from SDO/HMI and SO/PHI-FDT data has been presented in this work. The results of the comparison are summarized in Table 2, where data from three of the four framed regions marked in Fig. 1 are included. When the different spatial resolutions are not taken into account, SO/PHI-FDT tends to underestimate B and B_LoS and to retrieve somewhat less inclined fields. The correlations are improved when the different spatial resolution is taken into account.

Nevertheless, discrepancies exist between SDO/HMI and SO/PHI-FDT, as shown in Table 2. These slight deviations can primarily be attributed to the fact that a one-to-one pixel correspondence cannot be achieved because of two factor: First, the use of a theoretical spatial PSF that may not precisely match the one affecting the data. Second, it could be due to neglecting a spectral PSF in the analysis of the SO/PHI-FDT data. This latter effect is likely to be the primary cause of the discrepancies in the presence of intense magnetic fields.

These results are compatible with those presented by Sinjan et al. (2023) for the SO/PHI-HRT, the high-resolution telescope of SO/PHI. In both cases (FDT and HRT), the magnetic field products, that is the field strength, inclination, the azimuth and the LoS magnetic field, show similar correlation coefficients and slope values. The only noticeable difference is a slight increase in the scatter in the LoS magnetic field for SO/PHI-HRT for high negative values compared to those of SDO/HMI. This difference may to first order be due to the higher spatial resolution of SO/PHI-HRT.

It is important to bear in mind that the results presented in this article represent a consistency check rather than a practical cross-calibration. The unique orbit of Solar Orbiter affects the comparison of the two instruments because the proximity to the Sun of the spacecraft directly impacts the spatial resolution of SO/PHI-FDT. Additionally, the spacecraft angle in relation to SDO/HMI may introduce projection effects in the observed regions. Moreover, the spacecraft velocity, among many other things, plays an important role. Therefore, care must be taken when combining observations from the two instruments. Finally, we remark that the data used from SO/PHI-FDT in this study do not yet include the spectral PSF in the inversion, because although it is currently implemented in the pipeline, it is still in the process of being validated.

Acknowledgments

Solar Orbiter is a space mission of international collaboration between ESA and NASA, operated by ESA. We are grateful to the ESA SOC and MOC teams for their support. The German contribution to SO/PHI is funded by the BMWi through DLR and by MPG central funds. The Spanish contribution is funded by AEI/MCIN/10.13039/501100011033/(RTI2018-096886-C5, PID2021-125325OB-C5) and ERDF “A way of making Europe”; “Center of Excellence Severo Ochoa” awarded to IAA-CSIC (SEV-2017-0709, CEX2021-001131-S). The French contribution is funded by CNES. The HMI data are courtesy of NASA/SDO and the HMI science team. This research used version 4.1.3 of the SunPy open source software package (Barnes et al. 2020).

References

Appendix A: Scatter plots of active regions 2, 3, and 4

Fig. A.1. Pixel-to-pixel comparison of the different magnetic field products for NOAA AR 12963 (red area in Fig. 1). The top three rows show the same as Fig. 2, and the bottom row displays the same as Fig. 5, but for NOAA AR 12963 (red area in Fig. 1). The total number of pixels compared in the panels is 1989

Fig. A.2. Same as Fig. A.1, but for NOAA AR 12957/12959 (green area in Fig. 1). The total number of pixels compared in the panels is 2441.

Fig. A.3. Same as Fig. A.1, but for NOAA AR 12962 (yellow area in Fig. 1). The total number of pixels compared in the panels is 3625.

Appendix B: Polarimetric masks calculated for the active regions

Fig. B.1. Polarimetric masks generated for active regions 1, 2, 3, and 4, as described in equation 1. The white pixels in the masks correspond to the discarded pixels from the scatter plots in Figs. 2, A.1, A.2 and A.3.

Appendix C: Merged scatter plots for active regions 1, 3, and 4

Fig. C.1. Same as Fig. A.1, but for NOAA AR 12957, NOAA AR 12957/12959, and NOAA AR 12962 combined (areas within the blue, green and yellow rectangles, respectively, in Fig. 1). The total number of pixels compared in the panels is 10342.

All Tables

All Figures

Fig. 1. SO/PHI-FDT and SDO/HMI BLoS 45 s magnetograms (panels A and B) with their original orientations in the focal plane. The axes represent helioprojective-Cartesian coordinates in units of arcseconds. The blue, red, green, and yellow rectangles correspond to the selected areas. Panels C and D show zoomed-in magnetograms corresponding to the blue area. Panels E and F show more detailed zoom-in magnetograms up to the pixel resolution as purple squares. The cyan squares represent the SO/PHI-FDT pixel size in the SO/PHI-FDT and SDO/HMI magnetograms. The magenta square corresponds to the SDO/HMI pixel size. The arrows in panels A and B point to solar north.

In the text

Fig. 2. Pixel-to-pixel comparison of the blue area in Fig. 1 (NOAA AR 12960) for SDO/HMI (vector products) against SO/PHI-FDT. The original quantities are shown in the top row: B (panel A), γ (panel B), and BLoS (panel C). The middle row displays the same quantities after convolving the original SDO/HMI data with the SO/PHI-FDT PSF (see text): B′ (panel D), γ′ (panel E), and $B_{\text{LoS}}^{\prime }$ (panel F). The total number of pixels compared in these plots is 4306. The blue lines depict linear fits. The dashed gray lines represent the identity function. The fits and Pearson correlation coefficients are given in the insets. The histograms of the differences between SDO/HMI and SO/PHI-FDT values are shown in the bottom row (panels G, H, and I). The dashed lines are skewed Gaussian fits to the histograms, with μ, σ, and η the mean standard deviations and skewness of the fit, respectively. The plots corresponding to the other active regions can be found in the Appendix A.

In the text

Fig. 3. Active region (top row) and quiet-Sun (bottom row) longitudinal magnetic field maps. The active region corresponds to the blue square in Fig. 1 (NOAA AR 12960) and the quiet-Sun region has 50 × 50 pixels selected around disk center. We present SO/PHI-FDT retrievals (panels A and D), original SDO/HMI retrievals (panels B and E), and SDO/HMI results after convolution with the SO/PHI-FDT theoretical PSF (panels C and F). To facilitate comparison, the maps are rotated such that solar north points upward.

In the text

Fig. 4. Pixel-to-pixel comparison of the blue area (NOAA AR 12960) in Fig. 1 of the azimuth of SDO/HMI and SO/PHI-FDT. Panel A shows the original azimuth of SDO/HMI, ϕ, and the panel D displays the one inverted with the VFISV code using the Stokes maps convolved with the SO/PHI-FDT PSF, ϕ′. In both panels the azimuths are compared pixel-to-pixel against the SO/PHI-FDT azimuth. The roll angles have been corrected to establish a common origin for azimuth measurements in the SO/PHI-FDT and SDO/HMI datasets. The blue lines show linear fits. Points where |ϕHMI − ϕFDT|> 90° are excluded due to the intrinsic azimuthal ambiguity. The fits and Pearson correlation coefficients are given in the insets. Panels B and E show the same area as the SO/PHI-FDT azimuth without and with the polarimetric mask, respectively. Panels C and F show the SDO/HMI azimuth, ϕ and ϕ′, respectively. The total number of pixels compared are 1784 for panel A and 3417 for panel D.

In the text

Fig. 5. Pixel-to-pixel comparison of the area within the blue rectangle in Fig. 1 (NOAA AR 12960) of SO/PHI-FDT BLoS (RTE inversion mode) and SDO/HMI BLoS (MDI-Like algorithm) for 45 s (panel A) and 720 s cadence (panel B) magnetograms. The total number of pixels compared in both panels is 4306. Both panels show SDO/HMI-averaged quantities to match the SO/PHI-FDT pixel size. The blue lines depict linear fits. The dashed gray line displays the identity function. The Pearson correlation coefficients are given.

In the text

	Fig. A.1. Pixel-to-pixel comparison of the different magnetic field products for NOAA AR 12963 (red area in Fig. 1). The top three rows show the same as Fig. 2, and the bottom row displays the same as Fig. 5, but for NOAA AR 12963 (red area in Fig. 1). The total number of pixels compared in the panels is 1989
In the text

	Fig. A.2. Same as Fig. A.1, but for NOAA AR 12957/12959 (green area in Fig. 1). The total number of pixels compared in the panels is 2441.
In the text

	Fig. A.3. Same as Fig. A.1, but for NOAA AR 12962 (yellow area in Fig. 1). The total number of pixels compared in the panels is 3625.
In the text

	Fig. B.1. Polarimetric masks generated for active regions 1, 2, 3, and 4, as described in equation 1. The white pixels in the masks correspond to the discarded pixels from the scatter plots in Figs. 2, A.1, A.2 and A.3.
In the text

	Fig. C.1. Same as Fig. A.1, but for NOAA AR 12957, NOAA AR 12957/12959, and NOAA AR 12962 combined (areas within the blue, green and yellow rectangles, respectively, in Fig. 1). The total number of pixels compared in the panels is 10342.
In the text