© The Authors 2024

1. Introduction

Synoptic charts are a well-established method for representing the entire solar surface in a single map and have been a cornerstone of discovery in solar physics since their first introduction in the 19th century (Peters 1856; Carrington 1858) and the start of their systematic production almost a century ago (D’Azambuja 1928). Further development in the late 1960s eventually led to magnetic synoptic maps, which typically depict meridian measurements of the line-of-sight (LoS) component of the magnetic field (see e.g. Howard et al. 1967; Schatten et al. 1969; Livingston et al. 1970). These maps are produced by remapping a series of full-disc observations into heliographic coordinates and combining slices around the central meridian of each image to form a map that covers an entire Carrington rotation (CR).

While the basic method has mostly stayed the same, numerous advances to improve this process have been made over the years. Early methods were limited by long observation cadences and were therefore prone to spatial smearing introduced by the evolution of the magnetic fields and differential rotation of the solar surface. Therefore, various forms of weighting function were used to minimise this effect (Harvey et al. 1980; Harvey & Worden 1998), before Worden & Harvey (2000) suggested the concept of evolving synoptic maps. These latter authors combined classic synoptic charts from the National Solar Observatory (NSO) at Kitt Peak with flux transport models based on the differential rotation, meridional flow, supergranulation diffusion, and random flux emergence in the form of a background magnetic field in order to provide a map that can be considered a snapshot of the whole surface rather than a mosaic of observations over a full Carrington rotation. Similar advances were made by Schrijver & De Rosa (2003) based on space-based data from the Michelson Doppler Imager (MDI) on board the Solar and Heliospheric Observatory (SOHO) satellite.

Later, the Air Force Data Assimilative Photospheric flux Transport (ADAPT; Arge et al. 2010) expanded on the work of Harvey & Worden and their evolving synoptic maps, introducing the Los Alamos National Laboratory data assimilation framework in order to account for data and model uncertainties and thus produce improved synoptic maps. Designed to work with all common solar data, including magnetic approximations from helioseismic far-side data (Arge et al. 2013; Lindsey & Braun 2000; Chen et al. 2022), ADAPT models globally instantaneous synchronic maps with consistent polar fields, which serve as input for solar wind or coronal modelling.

At the same time, classic synoptic maps saw advances in processing and observation techniques; nowadays, they are mainly produced from the NSO ground-based Global Oscillation Network Group (GONG; Harvey et al. 1996) and the space-based SDO/HMI instrument (Schou et al. 2012; Scherrer et al. 2012), which provide vector magnetograms at a cadence of 12 minutes. Combining this high-cadence data with corrections for the differential rotation as suggested by Ulrich et al. (2002), SDO/HMI now provides high-resolution magnetic line-of-sight (LoS) and vector synoptic maps with minimal smearing and good signal-to-noise ratio (S/N; Liu et al. 2017). Together with GONG data, these maps serve as the standard input for ADAPT synchronic maps.

Despite efforts to recreate the surface magnetic field for a specific instant in time, so far measurements of the solar magnetic field have been constrained to the Earth-bound view point, and therefore a full Carrington rotation of ∼27 days is required to build a synoptic map. This limitation has now been overcome with the launch of Solar Orbiter (SO; see Müller et al. 2020), which carries the Polarimetric and Helioseismic Imager (PHI; see Solanki et al. 2020). Being on a heliocentric orbit, SO/PHI provides magnetograms from outside the Earth–Sun line and thereby provides the very first opportunity to produce synoptic maps from multi-view observations. Depending on the orbital positions, the combination of for example the synoptic observations of SDO/HMI with SO/PHI can significantly reduce the observation time of a full Carrington rotation down to ∼15 days when the two spacecraft are close to superior conjunction. Given that active regions can evolve significantly on a timescale of weeks, this reduction represents a significant improvement; for example, for the early observation of emerging bipoles, which would otherwise stay undetected for days. Depending on the viewing angle, the reduction of the required observation time span can lead to significant improvements in numerous applications. These include, but are not limited to, global quantities, such as the surface flux and the total magnetic energy (Mackay et al. 2016), the open magnetic flux and its footpoint locations on the solar surface 2016ApJ...828..102W, the shape of the source–surface neutral line, and coronal hole patterns (Petrie et al. 2018). In turn, such information affects the results of numerical models of the corona, and may potentially lead to better predictions of the solar wind speed, for example with the WSA-ENLIL model (Pevtsov et al. 2020).

In this work, we present some of the challenges facing multi-view magnetic synoptic maps based on observations gathered by the SDO/HMI and SO/PHI instruments. We also show how these challenges can be overcome. This work therefore serves as the basis for a pipeline for multi-view magnetic synoptic maps. Starting with the necessary upgrades to the SDO/HMI synoptic map code base in Sect. 2, we then present our modelling efforts based on SDO/HMI data in Sect. 3 and conclude with an outlook on the future capabilities of this novel data product in Sect. 4. Please note that, while this data product is technically a diachronic map, we decided to instead use the term synoptic in accordance with the heritage SDO/HMI synoptic data products.

2. Method

Our multi-view synoptic maps are built from central meridian observations taken by different instruments that observe from different orbital positions at different times. Merging such data always comes with the challenge of data compatibility, as the resolution, observation cadences, and general data quality can vary dramatically between instruments.

Given that both instruments underlying this work are aboard space-based missions, and given the common operation on the Fe I 617.3 nm spectral line, the similar Milne-Eddington-based inversion technique (see Borrero et al. 2011), and the well-established and openly available data processing pipeline (Liu et al. 2017), SDO/HMI is an ideal instrument for such a joint observational program with SO/PHI. Therefore, we based our method to produce combined synoptic maps on the state-of-the-art SDO/HMI pipeline, which comes with the advantage of processing both data sets with identical routines. The SDO/HMI code is well designed and can be adapted to solve many of the additional challenges that come with SO/PHI being operated as a deep space mission. These mainly include a variable observation cadence and orbital distance for SO/PHI observations, as is described in Sects. 3.1 and 2.2. While the changing orbital distance to the Sun can entirely be managed with the provided world coordinate system, the intermittently very low observation cadence of the SO/PHI synoptic program, particularly when SO is on the far side of the Sun, reintroduces the smearing problems of early synoptic maps. For this reason, we had to introduce a form of weighted averaging to the method. This and other necessary changes to the synoptic map processing are discussed below.

We note that SO/PHI offers data from both its High Resolution Telescope (PHI-HRT) and Full Disc Telescope (PHI-FDT). In order to avoid unobserved regions in the combined synoptic maps, we designed this method to exclusively rely on PHI-FDT and hereafter generally refer to them as SO/PHI observations. While SO/PHI is used to conceptually discuss the challenges that arise from the combination of two instruments in this section, we decided to not rely upon early SO/PHI calibration data to test this method in Sect. 3. Instead, PHI-FDT observations are simulated using archived SDO/HMI data, opportunely reduced in cadence or resolution where necessary.

2.1. Variable cadence

Unlike SDO, SO is an encounter mission and SO/PHI observations are not provided at regular time intervals. Although a synoptic program is carried out outside encounter periods, the highly variable distance from Earth and the resulting variable telemetry necessitate operational periods with sparse observation programs. During these times we expect no more than two to six magnetograms per day. While higher-cadence observations are possible in the so-called remote-sensing windows during encounter periods, SO/PHI will mainly carry out high-resolution observations and only a few full-disc magnetograms have so far been obtained during these periods. Typically, three such windows with a duration of 10 days each are placed at strategic positions along SO’s trajectory in every orbit (e.g. at or around closest approaches; see Zouganelis et al. 2020). The exact observation cadence is subject to changes and depends on the telemetry budget and the particular observation program. Considering this, SO/PHI data are available at a very variable cadence, from a few minutes to many hours, with even longer gaps possible during encounter and recalibration periods. An example of this variation can be seen in the right half of Fig. 1. Here, contributions representative of those from SO/PHI (red) are from single magnetograms every 6 to 24 h. These magnetogram slices strongly depend on the observation cadence and orbital speed, leading to stripes of between 7 and 14° longitude for a 24 h cadence (the variable distance in longitude comes from the highly elliptical orbit and the variable orbital velocity of the spacecraft). This is in stark contrast to the SDO/HMI portion of the map (yellow), which is constructed from averaging the 20 closest magnetograms for each coordinate, effectively leading to four-hour contributions to the map. Please note that the widths of the portrayed magnetogram strips in Fig. 1 are exaggerated for improved visibility.

Fig. 1. Example of a synoptic map created with data gathered with different cadences in the SDO/HMI (left) and SO/PHI (right) portion of the map. Data on the SDO/HMI side are averaged from the 20 closest 720s magnetograms to each coordinate, effectively forming contributions with a temporal width of 4 h (yellow grid). SO/PHI observations are only available at lower cadences. For this test case, we assumed an observation every 6–24 h (red grid). The blue region of interest shows the studied region around NOAA 12056 and NOAA 12059, which is investigated more closely in Sect. 3. Please note that the temporal widths of the contributing magnetogram stripes are not to scale. This synoptic map for CR2150 was entirely computed from SDO/HMI data.

2.2. Variable distance

SO/PHI operates at quasi-periodically changing solar distances of between 0.28 and ∼1 au. Consequently, SO/PHI observes a solar disc diameter that continuously changes between approximately 750 and 1800 pixels. At the closest solar distance, SO/PHI provides roughly half of the SDO/HMI spatial resolution, which has an average disc diameter of about 3800 pixels. A comparison of typically observed solar disc sizes is shown in Fig. 2.

Fig. 2. Variation in solar disc size along SDO and SO orbits in pixels. The horizontal line marks the reprojected vertical image size after the preprocessing steps.

In addition to the smaller disc size, SO/PHI, although currently nearly in the ecliptic plane, will in future observe from an inclined orbit with respect to SDO/HMI and the ecliptic plane. The inclination to the ecliptic will steadily increase to over 30° over the mission lifetime. Both of these effects are managed by the existing SDO/HMI modules that transform the helioprojective coordinates of the solar disc observations into the cylindrical equal-area Carrington maps in heliographic longitude and sine of latitude used in the synoptic maps (CEA; Thompson 2006). The effects of a variable horizon, that is the change in the visible area of the Sun with distance, are negligible and are therefore not considered.

2.3. SO/PHI data interface

In order to use the SDO/HMI pipeline for the multi-view synoptic maps, we developed an interface to integrate SO/PHI observations into our local clone of the Data Record Management System (netDRMS), which is the data-management system used to process, archive, and distribute SDO/HMI data products¹. This enables the processing of SO/PHI data directly alongside their co-observed SDO/HMI counterparts from 720 second LoS and vector-magnetogram data series, “HMI.M_720s” and “HMI.B_720s”. Where necessary, SO/PHI keywords are adjusted to match those used in the netDRMS. While the majority of these keywords are compatible by definition, we had to work around the SDO/HMI observation time slot T_REC to ensure proper data selection during the subsequent processing steps. Observations taken by SDO/HMI at observation times T_OBS are stored in netDRMS for specific record times T_REC at continuous 12 minute increments and decreasing observed Carrington longitude. In order to integrate SO/PHI observations at a different Carrington longitude but co-temporal with some existing SDO/HMI ones, we assign SO/PHI observing times to the future or past, that is, the times at which the observed Carrington longitudes would have been seen by SDO/HMI. Depending on the locations of the two spacecraft, this can lead to a combination of observations that would classically be assigned to different Carrington rotations for an earthbound observatory.

The choice of formally placing the SO/PHI observations in the past or in the future can be optimised by requiring that the overall time-period covered in building the map is the shortest possible. By default, SO/PHI data are always mapped to the nearest SDO/HMI time slot to reach a continuous evolution in time for the shortest observation period with a minimal number of transitions between the data products of the two observatories. This can be manually adjusted if a different combination of multi-view observation data is particularly beneficial for a given case.

An example of such a procedure is given Fig. 1, where simultaneous observations with the two instruments at opposite sides of the Sun result in SO/PHI observing the far side longitudes ahead of SDO/HMI, which can lead to identical (bold) observation times T_OBS. In order to avoid duplicate entries in the database, they are registered to T_REC slots that correspond to the Carrington time for each coordinate as observed by SDO/HMI 14 days earlier.

2.4. Preprocessing

Once the SO/PHI data are incorporated into netDRMS, each data set for the multi-view synoptic maps follows the preprocessing steps for the standard synoptic maps of SDO/HMI. Therefore, the full-disc observations are spatially interpolated to an evenly spaced Carrington grid in longitude and sine latitude. The Carrington coordinate of each pixel in the reprojected images is adjusted relative to their central meridian to account for the differential rotation in between observations (Ulrich & Boyden 2006). This reduces smearing when the central slices of multiple observations are combined and averaged in the final step of the synoptic map creation. The process is repeated for each magnetic field component. By default, SDO/HMI observations are subject to an oversampling-smoothing scheme to suppress the possible aliasing. For this, the reprojected images are initially super-sampled from 4096 × 4096 pixels to 5400 × 4320 pixels, which is followed by downsizing by a factor 3 –using Gaussian smoothing– to the final synoptic map resolution of 1800 × 1440 pixels. While this process is optimised for SDO/HMI observations, the intermediate and final rebinning dimensions are freely adjustable and can be adapted to better suit the various disc sizes observed by SO/PHI. This could result in maps at half or one-third of the original SDO/HMI synoptic map resolution and is determined individually for the available data products. A more detailed description of the Carrington grid projections can be found in Liu et al. (2017), Thompson (2006), and Calabretta & Greisen (2002).

2.5. Synoptic map processing

Following the preprocessing, the original SDO/HMI pipeline averages central meridian slices of the 20 closest magnetograms to each Carrington longitude into the final map. Outliers that deviate by more than 3σ from the 20 magnetogram average are excluded and replaced by the value of the next-closest magnetogram in order to maintain the S/N. This effectively creates a constant temporal width of 4 h at each Carrington longitude from SDO/HMI observations. Given the operational constraints of Solar Orbiter, SO/PHI data will only be available at a much lower cadence for most of the mission lifetime. Consequently, this temporal width cannot be maintained by SO/PHI. Moreover, using the same 20 magnetogram average on observations at much longer cadences of the order of multiple hours would introduce a large amount of smearing because of the evolution of the observed solar scene (see Sect. 3.1). For this reason, it was necessary to modify the magnetogram integration of the synoptic map creation process. With the return to observation cadences similar to those provided by ground-based observatories before the data-rich era of SDO/HMI and GONG, we decided to reintroduce the use of weight functions into the averaging process (see Sect. 2.6 for details). This not only prevents unwanted smearing effects, but also provides a mechanism to control the transition between the different data products.

In addition, the aforementioned outlier rejection is disabled for low-cadence SO/PHI observations. This procedure can mislead the pipeline, causing it to identify weak transient magnetic features as outliers if their lifetime is shorter than the available observation cadence. With SO/PHI at times only providing daily magnetograms, the outlier rejection would effectively discard any short-lived magnetic structures, which makes it incompatible with a low-cadence use case.

2.6. Adaptive weight functions

Weight functions in synoptic maps are typically used to smoothly merge adjacent contributions taken at different times into a continuous map. The exact shape of such a weight function mainly depends on the desired correction and the available observation cadence. In order to prevent the smearing of evolving magnetic features over different Carrington longitudes in observations separated by long time intervals, weight functions such as cos⁴(ϕ) (with ϕ being the Carrington longitude) or Gaussian functions (with full widths at half maximum corresponding to the temporal width) are commonly used by observatories such as the National Solar Observatory (NSO; see Harvey & Worden 1998). The weight function for the multi-view synoptic map pipeline was designed to meet the following requirements:

1. The SDO/HMI section of the multi-view synoptic map should be identical to its SDO/HMI standard synoptic map counterpart.

2. The averaging of low-cadence SO/PHI observations needs to be limited to nearby frames (i.e., closer in time) to prevent smearing.

3. The width of the weight function must be variable in order to adjust for the changing cadence of SO/PHI observations.

Consequently, the weight function serves a dual purpose. On the SDO/HMI side of the synoptic map, it must maintain the 20 magnetogram (≈4 h) average that is meant to improve the S/N. The contrary is necessary for the SO/PHI portion of the synoptic map, where the use of low-cadence observations and the thereby captured evolution of the solar scene requires that contributions from the previous and subsequent observations be kept to a minimum. Therefore, the weight function is designed around two components that separately control the contributions of the two instruments.

High-cadence SDO/HMI observations are managed by a rectangular weight function of value unity where any contributing magnetogram is equally weighted. The width of this plateau starts at a temporal equivalent of 4 h and spans halfway to the central meridian of the following observation for longer cadences. As a result, observations at cadences faster than 4 h will exclusively be combined within the overlapping central regions (i.e., flat tops) of their weight functions, which leaves the original average unaffected.

For lower cadences, the weight function splits the contribution to the synoptic map into two parts. With reference to Fig. 3, we consider first the orange curve, WF2, which represents the weighting function associated to one particular observation. The curve is composed of three parts, a plateau surrounding the central meridian (dashed line) of the considered observations, and two wings, decreasing from the plateau to zero. The plateau gives equal, large weight to data recorded close to the central meridian. Contributions to the averaging process from beyond the central plateau (orange wings) are intended to provide a continuous transition to the subsequent magnetograms in the synoptic map, which are represented by the curves WF1 (blue) and WF3 (grey). The wings are in the shape of the polynomial

(1)

Fig. 3. Weight function for the multi-view synoptic maps. The central plateau corresponds to a four-hour temporal width with a polynomial drop-off to a 5% contribution at the adjacent central meridian. Dashed black lines mark the central meridians and the solid lines mark the transition between adjacent magnetograms. The individual weight functions combine to smooth effective weight functions (red line, only central WF shown) during the weighting process. The effective weight function is the shown weight function divided by the sum of the weights.

Here, the strength of the weight w at a position d along the extent of the wing is modulated by the exponent x, with x = 1 giving linear wings and thus resulting in a trapezoidal weight function. In practice, x is automatically calculated by

(2)

that is, from the configurable weight, w_m, and the position, d_m, of the central meridian of the adjacent magnetogram.

The combination of the plateau with polynomial wings was chosen instead of a classical Gaussian or a cos⁴(ϕ) function to allow for individual algorithmic control of the two components, that is, the extent of the plateau for averaging SDO/HMI data and a smooth transition with limited overlap for low-cadence SO/PHI data. The exact shape will vary slightly depending on the input data and will therefore be different for LoS and vector synoptic maps.

2.6.1. Line-of-sight synoptic maps

Let us first consider LoS synoptic maps. By default, observations at cadences of longer than 4 h (typically the case for SO/PHI data) contributing to the synoptic map are weighted as mostly individual data sets. Figure 3 shows a simple case with a three-image weight function at a four-hour cadence. The weight along the wings polynomially decreases to a w_m = 5% contribution at the position d_m of the neighboring central meridians before it asymptotically reaches zero at the boundary. The main contribution of each individual data set is separated by solid lines that correspond to the extents of their central plateau. Dashed lines indicate the position of the central meridians that are used to control the shape of the wings in Eq. (2). Due to the low cadence, only the wings overlap with the central plateau of the adjacent frames, which strongly favors the contribution of data in the vicinity of the central meridian at the center of the plateau.

For instance, at the central meridian of WF2, the map is built from the contributions of three magnetograms, which are weighted according to the value of the corresponding weighting function at that longitude: the blue with relative weight w_b = 5%, the orange with relative weight w_o = 100%, and the grey with relative weight w_g = 5%. The map is then produced by taking the weighted average of the three magnetograms, which is divided by the sum of the weights.

Combining all contributing weights (blue, orange, grey) at the coordinates of the entire middle data set produces the effective weight function (red). Although the individual weight functions have sharp edges at the transition between the plateau and the polynomial wing, the effective weight function becomes smooth (red line). This is by design and is meant to prevent discontinuities in the resulting synoptic maps. The effective weight function for the high-cadence case of SDO/HMI is a boxcar function that results from the overlap of the plateaus of the 20 closest magnetograms. The sum of the effective weight functions at a specific coordinate gives unity.

Finally, the width of the central plateau is automatically adjusted to account for changes in the observation cadence. Therefore, the width of the weight function is calculated as the distance in longitude to the central meridians of neighboring observations. Local changes in the observation cadence can lead to the incorporation of magnetogram slices that are asymmetric in longitude around the central meridian, as shown in Fig. 4.

Fig. 4. Weight functions adapted to the variable observation cadence of SO/PHI. The width of each weight function is based on the distance to the next central meridian. The dashed black lines mark the central meridians and the solid lines mark the transition between adjacent magnetograms. Transition widths are limited for large changes in the observation cadence (light blue/orange, right).

Here, the observation cadence changes from an initial regular 4 h to 6, 12, and 24 h, which was followed by an operational gap of 36 h to the next observation. Such large changes in the observation cadence can lead to overly wide transitions spanning multiple magnetograms. Therefore, a maximum transition width can be set to limit the degradation of the surrounding high-cadence observations. An example can be seen in the right weight function (orange), which shows symmetrical wings despite the difference between the 12 h interval since the previous observation (left) and the 32 h gap until the next one (right).

2.6.2. Vector synoptic maps

Vector synoptic data need to be incorporated differently into the synoptic maps, because of the inherently higher noise levels in the transverse field. The projection of the radial, poloidal, and toroidal magnetic field components used for the synoptic map are computed from the inclination and azimuth angles of the inversion results, which in turn are based on observations in all polarised Stokes components, I, Q, U, and V. The uncertainty in the inclination and azimuth angles (with respect to the LoS) is dominated by the typically lower S/N in Stokes Q and U relative to that in Stokes V, especially for weaker magnetic fields. The noise levels at higher latitudes are further amplified during the projection to the Carrington grid (CEA; Thompson 2006), which is one of the reasons why SDO/HMI synoptic maps are averaged from 20 contributing magnetograms. This is not possible for low-cadence SO/PHI observations because of the significant solar evolution between observations. Instead, we extend the wings of the weight function over two or three adjacent observations and increase the level of averaging for the vector synoptic maps, which is meant to balance the reduction of noise and introduction of smearing. Unlike the narrow transitions with wings of one magnetogram in width, as used for the LoS synoptic map, these wider weight functions help with the reduction of noise levels while still limiting the introduction of smearing from magnetic evolution to some degree. Their configuration is a trade-off between the two effects and needs to be customised to the observational conditions of the respective SO/PHI synoptic campaigns. As the noise levels in the Carrington projections increase with the foreshortening effects towards the poles (see Sect. 3.5), the weight function width can further be configured to increase with latitude, δ.

An example weight function as used in the vector synoptic maps is shown in Fig. 5. Here, a narrow weight function similar to the LoS data standard case is used for equatorial latitudes up to δ = 20°. From there on, the width of the weight function gradually increases until it reaches a maximum width at δ = 70°. The maximum width of the weight function is limited to five contributing magnetograms.

Fig. 5. Latitude-dependent weight functions with increasing contribution between 20° and 70° Carrington latitude (see legend). The dashed black vertical lines mark the central meridians and the solid vertical lines mark the transition between adjacent magnetograms. The weights start at 5% contribution at the adjacent meridian for 20° latitude and linearly increase to 45% contribution for 70° latitude.

Figure 6 shows the combined effect of the variable cadence, latitude specific weight function widths, and the transition between SDO/HMI and SO/PHI data. The weight functions are configured to cover one adjacent observation on each side and increase from 5% to 45% contribution at the neighboring central meridians at higher latitudes. Starting on the left, the saturated yellow area in (a) represents the SDO/HMI region where 20 observations are available. This is followed by a drop in observations cadence for SO/PHI and is identical to the variable cadence presented in Fig. 4. The vertical beige lines indicate contributions from multiple frames and thus mark the transitions between SO/PHI observations. Panel b shows the number of contributing magnetograms for each pixel. While this involves up to three frames for the initially relatively high cadence portion (left) and one or two magnetograms in the low-cadence portion of the map (right), the cumulative weight stays close to 1 over most of the map (a). Therefore, the majority of the SO/PHI section is dominated by single observations.

Fig. 6. Cumulative weight functions (a) and number of contributing magnetograms in each pixel (b) for a test case with three-observation weight functions of latitude-specific weight.

This hybrid system of narrow and wide weight functions introduces ample fine control over the averaging process and allows us to include SO/PHI observations at a trade-off between magnetic evolution and good S/N. Ultimately, the automatic customisation of the weight function for each individual data set not only enables control over the precise integration of SO/PHI data into the SDO/HMI synoptic maps, but also serves as a simple but effective way to add flexibility to the configuration of the SDO/HMI synoptic map pipeline. It is noted that the configuration of the vector synoptic maps also supports the default weight functions without latitude-specific weights, as used for LoS synoptic charts.

2.7. Postprocessing

Following the successful completion of a synoptic map, the pipeline resizes the full-resolution map from 3600 × 1440 pixels to the SDO/HMI small synoptic map size of 720 × 360 pixels. This is achieved by rebinning the map by a factor 5 in longitude ϕ and a factor 4 in latitude δ. The 1.25 ratio in ϕ versus δ is used to compensate for the 1.25 aspect ratio in the heliographic projections. The result is the commonly used, so-called SDO/HMI small synoptic map format with a pixel size of 0.5° in Carrington longitude, which is provided in addition to the full-resolution map.

3. Results

In order to test the multi-view synoptic map pipeline in a controlled environment, we decided not to rely upon early SO/PHI calibration data but to use archived SDO/HMI observations instead. For this, SDO/HMI data for CR 2150 were chosen because of its ample magnetic activity. We selected the bipolar structure formed by active regions NOAA 12056 and NOAA 12059 seen in Fig. 1 as a test area to study the effects of different observation cadences and the changing orbital distance of SO/PHI. This subframe features a magnetically stable sunspot and a highly variable and still emerging magnetic pore, which enables us to study the effects of different temporal observation schemes. Overall, this approach allows us to simulate potential observation scenarios that will be encountered by SO/PHI. In this section, we present our studies on the variable observation cadence, the changing resolution caused by the elliptical orbit of SO/PHI and the effects of noise in low-cadence synoptic maps. Unless stated otherwise, only LoS synoptic maps are studied for the sake of simplicity.

3.1. Weighted low-cadence synoptic maps

Depending on the observational program, SO/PHI is often limited to observation cadences exceeding multiple hours. While the SDO/HMI synoptic map pipeline is in principle capable of handling long data gaps that would resemble the low SO/PHI cadences, the initial results were affected by massive smearing caused by the evolution of the magnetic field. This can be seen in Fig. 7d, whereas panel a shows the reference quality of the standard SDO/HMI synoptic map.

Fig. 7. Test region from CR 2150 observed for different cadences and averaging methods. (a) Reference case standard SDO/HMI synoptic map at 12 minute cadence. Each Carrington coordinate is averaged from the 20 closest ‘HMI.M_720s’ magnetograms within a 30° window. (b) Updated synoptic map pipeline, which incorporates a narrow weight function with 5% contribution at the central meridian of the adjacent magnetogram slice. (c) Same as (b) but a wider weight function with 55% contribution at the central meridian of the adjacent magnetogram slice. (d) Same as panel a but at a reduced cadence of 12 h. Panels e–g show a comparison of the magnetic field distribution between the reference synoptic map in (a) and the adjacent low cadence and the low-cadence variants in (b), (c), and (d). The resulting linear fit equation is provided in each panel.

Limiting the number of magnetograms that contribute to each Carrington coordinate with weight functions strongly reduces this smearing, as seen in panels b and c. Both cases are simulated for a typically expected observation cadence of 12 h with a narrow weight function of w_d = 5% in (b) and a wide weight function of w_d = 55% contribution at the adjacent central meridians in (c). The narrow case (Fig. 7b) shows a large reduction in smearing compared to the standard synoptic map pipeline result at 12 h in Fig. 7d. The peak magnetic field of 1300 G in the positive pore (ϕ = 254° and δ = 5°; panel b) is lower than the 1540 G in the SDO/HMI reference case (a), but is better preserved than for the more extensive averaging of case (c) and the unmodified averaging in panel d with 1080 G and 890 G, respectively. Differences in the peak magnetic field strengths compared to the SDO/HMI reference case in panel a can mainly be attributed to the evolution in the source magnetograms, as this particular region was quickly evolving and moving westwards during the observation period. The same is true for the difference between the simulations with the narrow (b) and wide (c) weight functions, as the latter is affected by a longer evolution to the synoptic map. A closer look at the weak magnetic fields of the reference case (a) and the result of the narrow weight function average in (b) shows a generally similar but smoother field spatial distribution for (a). This is a result of the fact that SDO/HMI synoptic maps are four-hour averages. Smoothing is less apparent in synoptic maps produced with the narrow weight function averages, because the weighted average mainly ensures a continuous transition between the frames.

The quantitative differences in the magnetic field between the synoptic map methods are further explored in panels e–g of Fig. 7. Here we plot the magnetic field of each pixel in the SDO/HMI reference synoptic map (a) against the low-cadence variants in the adjacent panels b–d. The narrow (5%) weight function in panel b produces a concentrated pattern with low scatter that closely resembles the SDO/HMI reference for the evolutionary stable negative field. The scatter in the positive magnetic field partly remains and can be attributed to the aforementioned evolution of the positive fields. Contributions to the weighted average are limited to the adjacent observations in this case. Panel c, with a weight function of 55% residual contribution at the next meridian, shows increased scatter for negative fields and a loss of the peak values on the positive side. Both are clearly superior to the scatter in the equally weighted SDO/HMI-like averaging shown in panel d.

3.2. Variable cadence simulation

So far, the effects of magnetic evolution have only been discussed for the scenario of a 12 h observation cadence with increasingly spread-out weight functions. In order to assess the combined effect of weight function and observation cadence, we calculated the standard deviation σ of the pixelwise difference in absolute value between the original SDO/HMI and the weighted synoptic maps. This was done for a number of possible SO/PHI observation scenarios with cadences ranging from 2 to 24 h and different parameters for the weight function, which can be seen in Fig. 8.

Fig. 8. Standard deviation, σ, of the pixel-by-pixel evolution between the classic SDO/HMI and the weighted synoptic maps for different weight functions from wd = 5% to 55% residual contribution at the adjacent central meridians. The dashed line shows the result for equally weighted HMI-like averaging of the low-cadence observation (see first paragraph of Sect. 2.5).

The resulting deviation from the reference maps does not represent an error but can rather be understood as a difference in the observed magnetic evolution with respect to SDO/HMI. Clearly, having a weighting function greatly reduces the difference with respect to high-cadence measurements for cadences of between 2 and less than 24 h. The width of the weighting function itself appears to play a much less important role, with narrower weighting functions producing more similar results to high-cadence measurements.

The cadence of the observations also strongly influences the difference between the weighted synoptic maps and the original SDO/HMI maps. While a low-cadence synoptic map will capture an equally correct but different state of the magnetic evolution, higher observation cadences provide averages over intervals that are similar to the classic 4 h of SDO/HMI synoptics, which in turn produces a more homogeneous magnetic field measurement over the combined synoptic map. If operational limitations do not allow high-cadence observations, the 6–8 h cadence range appears to be a good trade-off to achieve a temporal average that is still close to the SDO/HMI part of the map. Depending on the cadence, the observed difference in evolution is dominated by the width of the weight function (high cadence) or the observation cadence itself (medium to low cadences).

In summary, the weighted low-cadence synoptic maps are well suited to reproducing a field geometry similar to the four-hour averages in the classic SDO/HMI synoptic maps. Differences arise from the fact that the weighted low-cadence versions represent a more instantaneous snapshot in time, which portrays a slightly different state from the original SDO/HMI synoptic map. We note that observation cadence was fixed to assess its effects in a controlled environment. Depending on the operational constraints of SO/PHI, this cadence can vary between observations in real world applications. Reducing the number of observations that contribute to the magnetic field average also inevitably increases the noise levels in the map, which is a subject that we investigate further in Sect. 3.5.

3.3. Variable distance simulation

The second challenge facing combined synoptic maps with SO/PHI and SDO/HMI data is the highly elliptical orbit of Solar Orbiter. In order to simulate the combination of data observed at different spatial sampling and resolution, we downsized SDO/HMI data to typical solar disc sizes as observed by SO/PHI. In this section, we explore how the inclusion of SO/PHI data at different resolutions affects the combined synoptic map. Therefore, we chose four cases: The first one is at the closest approach of 0.28 au, where the SO/PHI FDT operates at half of the SDO/HMI resolution. We then increase the distance in 0.14 au steps and consider distances of 0.42 au, 0.56 au (where one-quarter of the SDO/HMI resolution is reached), and 0.70 au. Figure 2 compares the disc size (in pixels; left coordinate) with the distance from the spacecraft to the Sun (right coordinate). While the difference in resolution between SO/PHI and SDO/HMI might appear dramatic at a first glance, it becomes less of a problem once the observations are remapped onto the lower resolution of 1800 × 1440 pixels of the Carrington grid, as discussed in Sect. 2.4.

The same subdomain as in Fig. 7 can be seen for the synoptic map simulated at various distances in Fig. 9.

Fig. 9. Synoptic maps for the test region in CR 2150 as they would be observed from various distances. Panel a shows the synoptic map as would be observed from (a) 0.28 au, (b) 0.42 au, (c) 0.56 au, and (d) 0.70 au orbital distance. All four cases are constructed from observations at 12 h cadence and were processed with a narrow weight function of 5% residual contribution.

Panels a–d show 12 h cadence synoptic maps for the four orbital distances in increasing order. While these again show a slightly different state of the solar magnetic field from the reference, the striking difference can be observed when looking at the smoothness of the field distribution. The map in panel a still features a finer granularity, even though it is constructed at half of the resolution of the SDO/HMI reference case with its four-hour averages (see Fig. 7a). Moving further out, the synoptic maps at 0.42 au and 0.56 au in panels b and c start to show a smoother magnetic field distribution because of the loss of resolution at these distances. This effect is especially pronounced in simulations (c) and (d) for 0.56 au and 0.70 au as both are upsampled from their initial disc sizes.

Overall, the difference from the SDO/HMI reference in Fig. 7 is much smaller than one would expect for such a reduced resolution. Also, the field distribution is almost identical to the previous results at full resolution in Figs. 7e-g and is therefore not shown separately. This is mainly due to the fact that the resolution of SO/PHI is similar to the downsized heliographic projection over most of the orbit. This mostly compensates for the difference in resolution between the two instruments without the introduction of any major detrimental effects into the synoptic map. Therefore, we expect that segments of SO/PHI-FDT data taken at close approaches to the Sun provide a level of detail similar to the SDO/HMI part of the combined synoptic map. The resemblance becomes even closer once we consider –in the following section– that commonly used synoptic map formats are strongly binned.

3.4. Small synoptic maps

Above, the analysis mainly focuses on the full-resolution synoptic maps. Similar to the results for the high-latitude noise, rebinning to the small synoptic format (720 × 360 pixels) improves the agreement between the magnetic field distribution in the low-cadence maps and that in the 12-min-cadence maps by averaging out both the noise and differences in small-scale evolution.

This is particularly visible for the positive magnetic fields, as can be seen in Fig. 10.

Fig. 10. Comparison of the magnetic field distribution between the default SDO/HMI synoptic map at 12 minutes cadence and both (a) the 12 h cadence at full resolution and (b) the 5 × 4 rebinned small synoptic format.

Here, the reduction in resolution seems to correct for the previously seen differences from observing slightly different states of the evolution. Together with the close resemblance of the field structure between standard SDO/HMI small synoptic map and the 12 h low-cadence version in Fig. 11, the low-resolution synoptic maps appear to be mostly unaffected by the use of low-cadence observations.

Fig. 11. Test region from CR 2150 SDO/HMI small synoptic map resolution. (a) Reference case standard SDO/HMI synoptic map at 12 min cadence and (b) 12 h cadence synoptic map using a narrow weight function.

3.5. Noise at high latitudes

Moving to higher latitudes beyond 50°, the lack of available data to counteract the increasing noise towards the limb becomes apparent in vector synoptic maps, as they heavily rely on the 20 image average of the original pipeline to remove this. While the use of wider weight functions can improve the noise to some degree (see Sect. 2.6.2), increased noise at high latitudes will generally be present in low-cadence vector synoptic maps.

In order to quantify the detrimental effect of low-cadence observations on the synoptic map, we calculated the increase in noise levels at different latitudes for observations at a simulated 12 h cadence. For this, the RMS was determined by fitting a Gaussian function to the distribution of the quiet Sun data in latitude windows of 10° in width for a range from δ = 20° to 90° located above the reference window of the CR 2150 synoptic map. The FoV starts at 20° to avoid contamination from the activity belts as seen in Fig. 1. The results for B_R, B_ϕ, and B_θ are presented in Fig. 12, with Fig. 13 showing the same region in B_LoS for comparison.

Fig. 12. Latitude-specific noise for the vector components Br, Bϕ, and Bθ for full-resolution low-cadence synoptic maps. The RMS (left) is measured in 10° increments from 20° to 90° in a quiet-Sun area. Solid lines show the result of the full-resolution synoptic maps at 3600 × 1440 px and the dashed lines for the small synoptic versions at 720 × 360 px. The three panels on the right show the respective region, starting north of the magnetic activity of the equatorial latitudes.

Fig. 13. Same as Fig. 12 but for BLoS.

The low-cadence maps for the radial, toroidal, and poloidal vector components B_R, B_ϕ, and B_θ clearly show an increase in noise over the B_LoS version of the map, with the latter staying close to the noise levels of the SDO/HMI synoptic counterpart. Unlike in the low-cadence vector maps, no obvious noise patterns are visible for higher latitudes in B_LoS. For B_R, starting at 20° latitude with less than 10 G, the RMS rapidly increases towards the 40 G range for medium and high latitudes (solid blue). This effect is caused by the noisier azimuthal component, which dominates the measurement of B_R at high latitudes and is further amplified by the disambiguation process. Here, the use of a wider weight function (solid orange) partly resolves the issue, when compared to the results with a narrow weight function (solid blue). A similar trend can be seen for B_ϕ, where the low-cadence results follow the general trend of the SDO/HMI results at significantly increased noise levels. The toroidal component at high latitude becomes dominated by the linear polarization, which has a better S/N, and therefore B_θ, contrary to B_R and B_ϕ, shows a decreasing noise with latitude for all curves.

Combining this with the latitude-dependent weight functions as introduced in Sect. 2.6.2 will produce results between these two limits. With the latitude-dependent weight functions, we utilise the averaging of data as much as necessary, while limiting it to as little as possible. However, further tests showed that the full-resolution maps might need to be created from higher-cadence data (i.e. every 2 h) or must employ additional processing of the polar noise before the data are suitable for high-level applications; for example, as input for pole-filling algorithms like the ones currently applied to SDO/HMI maps (Sun et al. 2011). The noise situation is considerably improved by rebinning the maps to the SDO/HMI small synoptic format (dashed lines).

4. Conclusions

We present a method to produce combined maps from data taken from multiple vantage points, with the two instruments having different cadences and resolution (including variable cadence and resolution). This is implemented as a modification of the SDO/HMI standard pipeline for the production of synoptic maps (Liu et al. 2017). The method, while being relatively general, is specifically useful for combining SDO/HMI and PHI-FDT data. Simulations were carried out including the effect of variable orbital distance and observation cadence and have shown the feasibility of the concept, even though, given its deep-space environment, SO/PHI operations are substantially limited compared with SDO/HMI.

The introduction of weight functions to the averaging process of the SDO/HMI synoptic map pipeline enables magnetograms to be incorporated more selectively into the high-cadence SDO/HMI and low-cadence SO/PHI segments of the synoptic map. Using this to limit the averaging process for the low-cadence observations significantly reduces the smearing that is introduced by magnetic evolution. While a lower observation cadence can lead to a decrease in the S/N at higher latitudes of vector synoptic maps, we find that rebinning to the small synoptic format is very efficient in reducing this effect. Even though it is self-evident that incorporating low-cadence SO/PHI observations has negative effects on the temporal and spatial resolution of the synoptic maps compared to SDO/HMI-only maps, the significant reduction in observing time for a full map is expected to outweigh them. This will be most effective for orbital positions that allow SO/PHI to observe the far side of the Sun. Such an alignment will occur for about 3–4 months each year.

Figure 14 shows the number of days that it takes to observe the full surface of the Sun with SDO/HMI and SO/PHI for each point in time of the SO mission. The observation time reaches local minima of 13–15 days in every second orbit. Ideally, these periods will be dedicated to high-cadence full-disc observations with SO/PHI.

Fig. 14. Variation of the observation time for the full Carrington longitude range when SDO/HMI and SO/PHI data are combined, calculated over the Solar Orbiter mission timeline. SO/PHI will periodically spend extended periods of time on the far side of the Sun, thus providing yearly opportunities for multi-view synoptic maps.

4.1. Outlook

This work was entirely built on simulations from SDO/HMI data. The first nearly science-ready PHI-FDT data are now available, allowing the first SO/PHI and SDO/HMI combined synoptic map to be produced (Loeschl et al. 2024).

One complication that has not been investigated in the present study is that, from 2025 onward, Solar Orbiter will follow an orbit that is inclined relative to the ecliptic. With time, this inclination will increase, providing SO/PHI with the opportunity to observe the solar poles when at highest latitude, thus providing important information that is missing or only sparse in current synoptic maps. The disadvantage of this is that at such times, the high latitudes around the other pole will not be sensed by SO/PHI at all. Working out how to combine magnetograms taken from different latitudes is beyond the scope of this paper and will be the subject of a future publication.

While we are currently focused on the integration of observations from PHI-FDT and the comparable SDO/HMI instrument, the concept of combined synoptic maps is not limited to this application. Future collaborations could involve data from the GONG observatory (Harvey et al. 1996), or the coming PMI instrument (Staub et al. 2020) on board the ESA Vigil mission. The latter is a direct evolution of the SO/PHI instrument, adapted for the new observational requirements around L5, and will therefore be a prime candidate for future multi-view observations.

Acknowledgments

We would like to thank Yang Liu, Art Amezcua and Zhi-Chao Liang for their endless patience and guidance on the SDO/HMI pipeline and DRMS. In addition, we also thank the anonymous referee who provided useful and detailed comments that significantly improved the manuscript. This work was carried out in the framework of the International Max Planck Research School (IMPRS) for Solar System Science at the University of Göttingen. 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 HMI data are courtesy of NASA/SDO and the HMI science team. The data were processed at the German Data Center for SDO (GDC-SDO), funded by the German Aerospace Center (DLR).

References

All Figures

Fig. 1. Example of a synoptic map created with data gathered with different cadences in the SDO/HMI (left) and SO/PHI (right) portion of the map. Data on the SDO/HMI side are averaged from the 20 closest 720s magnetograms to each coordinate, effectively forming contributions with a temporal width of 4 h (yellow grid). SO/PHI observations are only available at lower cadences. For this test case, we assumed an observation every 6–24 h (red grid). The blue region of interest shows the studied region around NOAA 12056 and NOAA 12059, which is investigated more closely in Sect. 3. Please note that the temporal widths of the contributing magnetogram stripes are not to scale. This synoptic map for CR2150 was entirely computed from SDO/HMI data.

In the text

	Fig. 2. Variation in solar disc size along SDO and SO orbits in pixels. The horizontal line marks the reprojected vertical image size after the preprocessing steps.
In the text

Fig. 3. Weight function for the multi-view synoptic maps. The central plateau corresponds to a four-hour temporal width with a polynomial drop-off to a 5% contribution at the adjacent central meridian. Dashed black lines mark the central meridians and the solid lines mark the transition between adjacent magnetograms. The individual weight functions combine to smooth effective weight functions (red line, only central WF shown) during the weighting process. The effective weight function is the shown weight function divided by the sum of the weights.

In the text

	Fig. 4. Weight functions adapted to the variable observation cadence of SO/PHI. The width of each weight function is based on the distance to the next central meridian. The dashed black lines mark the central meridians and the solid lines mark the transition between adjacent magnetograms. Transition widths are limited for large changes in the observation cadence (light blue/orange, right).
In the text

Fig. 5. Latitude-dependent weight functions with increasing contribution between 20° and 70° Carrington latitude (see legend). The dashed black vertical lines mark the central meridians and the solid vertical lines mark the transition between adjacent magnetograms. The weights start at 5% contribution at the adjacent meridian for 20° latitude and linearly increase to 45% contribution for 70° latitude.

In the text

	Fig. 6. Cumulative weight functions (a) and number of contributing magnetograms in each pixel (b) for a test case with three-observation weight functions of latitude-specific weight.
In the text

Fig. 7. Test region from CR 2150 observed for different cadences and averaging methods. (a) Reference case standard SDO/HMI synoptic map at 12 minute cadence. Each Carrington coordinate is averaged from the 20 closest ‘HMI.M_720s’ magnetograms within a 30° window. (b) Updated synoptic map pipeline, which incorporates a narrow weight function with 5% contribution at the central meridian of the adjacent magnetogram slice. (c) Same as (b) but a wider weight function with 55% contribution at the central meridian of the adjacent magnetogram slice. (d) Same as panel a but at a reduced cadence of 12 h. Panels e–g show a comparison of the magnetic field distribution between the reference synoptic map in (a) and the adjacent low cadence and the low-cadence variants in (b), (c), and (d). The resulting linear fit equation is provided in each panel.

In the text

	Fig. 8. Standard deviation, σ, of the pixel-by-pixel evolution between the classic SDO/HMI and the weighted synoptic maps for different weight functions from wd = 5% to 55% residual contribution at the adjacent central meridians. The dashed line shows the result for equally weighted HMI-like averaging of the low-cadence observation (see first paragraph of Sect. 2.5).
In the text

	Fig. 9. Synoptic maps for the test region in CR 2150 as they would be observed from various distances. Panel a shows the synoptic map as would be observed from (a) 0.28 au, (b) 0.42 au, (c) 0.56 au, and (d) 0.70 au orbital distance. All four cases are constructed from observations at 12 h cadence and were processed with a narrow weight function of 5% residual contribution.
In the text

	Fig. 10. Comparison of the magnetic field distribution between the default SDO/HMI synoptic map at 12 minutes cadence and both (a) the 12 h cadence at full resolution and (b) the 5 × 4 rebinned small synoptic format.
In the text

	Fig. 11. Test region from CR 2150 SDO/HMI small synoptic map resolution. (a) Reference case standard SDO/HMI synoptic map at 12 min cadence and (b) 12 h cadence synoptic map using a narrow weight function.
In the text

Fig. 12. Latitude-specific noise for the vector components Br, Bϕ, and Bθ for full-resolution low-cadence synoptic maps. The RMS (left) is measured in 10° increments from 20° to 90° in a quiet-Sun area. Solid lines show the result of the full-resolution synoptic maps at 3600 × 1440 px and the dashed lines for the small synoptic versions at 720 × 360 px. The three panels on the right show the respective region, starting north of the magnetic activity of the equatorial latitudes.

In the text

	Fig. 13. Same as Fig. 12 but for BLoS.
In the text

	Fig. 14. Variation of the observation time for the full Carrington longitude range when SDO/HMI and SO/PHI data are combined, calculated over the Solar Orbiter mission timeline. SO/PHI will periodically spend extended periods of time on the far side of the Sun, thus providing yearly opportunities for multi-view synoptic maps.
In the text