Yang Liu^{1}, J. Todd Hoeksema^{1}, Xudong Sun^{1}, & Keiji Hayashi^{2}

^{1 }Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305, USA

^{2 }Institute for Space-Earth Environmental Research, Nagoya University, Nagoya Aichi 464-8601, Japan

HMI vector magnetic field synoptic charts are now available through the JSOC website at hmi.b_synoptic. Here we describe the procedure to produce the synoptic charts and the disambiguation method we choose to solve the 180-degree ambiguity of the azimuth in vector magnetic field measurement. The procedure to produce vector magnetic field synoptic charts is described below.

The maps are constructed from HMI full-disk vector magnetograms that are now available for the entire *SDO* mission^{1}. The first step is to transform the components of the HMI 720-second vector magnetograms from field strength, inclination, and azimuth into three field components in heliographic coordinates: *B _{r}*,

*B*and

_{θ}*B*.

_{φ}Second, each of the three components undergoes a cubic interpolation onto a regular, evenly spaced Carrington grid in longitude and sine latitude. The Carrington coordinate for each observed point is adjusted to account for differential rotation relative to the time of central meridian crossing in order to minimize smearing when multiple observations are averaged, as suggested by ref. 2. The remapping is done in two steps. Initially the remapped grid retains the spatial resolution of the disk-center pixel, i.e., 0.03°. Then the resolution is reduced to 0.1° using a nine-pixel box-car average. This results in a rectangular longitude — sine-latitude map for each component with nominal dimensions 1800 × 1440.

The remapped, resized vector magnetograms are combined to produce synoptic charts. The vector data measured at each Carrington coordinate in the final map is an average of exactly twenty contributing magnetograms. The average is computed using the measurements from the 20 720-second magnetograms that contribute pixels observed closest to the central meridian. Outliers are excluded from the average if the value departs from the median by more than three times the rms. If a measurement is excluded, the values from another magentogram observed next-closest to central meridian are added. The effective temporal width of the HMI synoptic-map contribution is about 4 hours at each Carrington longitude, i.e., data are observed within 2 hours of central meridian passage. This means that the contributing pixels used are observed within about ±1.2° of central meridian. The final synoptic charts have a size of 3600 x 1440, with the x-axis referring to longitude in degrees, and the y-axis to equal steps in sine latitude.

Inversion of the vector field has an unavoidable 180-degree ambiguity in the azimuthal field direction^{3}. Assumptions about the field must be made to resolve the ambiguity. In strong-field regions the azimuth is determined using a minimum energy algorithm^{4-6}. The minimum-energy-method computation is time consuming in pixels where the signal is dominated by noise, so for weaker field regions in the quiet Sun (most of the disk where the field strength is below about 150 G) we resolve the 180-degree ambiguity using three simpler and quicker methods: the potential-field method (potential method hereafter), the radial acute-angle method (radial method), and the random method^{1}.

The potential method resolves the 180-degree ambiguity in such a way that the observed and potential transverse fields form an acute angle. The potential field is computed from the well-observed line-of-sight field. The radial method chooses the direction of the transverse field such that the field vector is closer to local radial direction. The random method assumes the there is not enough information to disambiguate the azimuth and so randomly assigns the direction of the transverse field for each pixel. All the three solutions are saved with the full-disk vector magnetic field data.

Figure 1 |Vector magnetic field synoptic charts ofB(top),_{r}B(middle), and_{φ}B(bottom) for CR 2145. Panel (a) shows the results of using the potential method to disambiguate the azimuthal field direction in weak-field regions; note the large-scale patterns surrounding strong-field regions. Panel (b) shows the radial method; note the difference in noise in the_{θ}Band_{φ}Bdirections. Panel (c) shows random method. The maps are longitude (3600 points) vs. sine latitude (1440 points)._{θ}

Results for the three weak-field solutions are compared to assess the impact on the vector field synoptic charts. Fig. 1 shows synoptic charts for Carrington Rotation (CR) 2145 (from December 18, 2013 to January 16, 2014) with disambiguation done using the potential method (a), radial method (b), and random method (c). The potential method (Fig. 1a) shows obvious large-scale shadow-like patterns surrounding stronger magnetic field patches that cannot be real. The radial method results (Fig. 1b) show another systematic error — there is significantly different noise in the *B _{φ}* and

*B*components. This arises because of a geometrical bias in the way the noisier transverse component of the field affects the disambiguation. The random method appears to produce maps without artificial patterns and the noise in the three components is comparable.

_{θ}

Figure 2 |Panels show the rms ofB(top),_{r}B(middle), and_{φ}B(bottom) as a function of sine latitude for the synoptic charts from the potential method (blue), radial method (green), and random method (black). The top panel also shows_{θ}Binferred from the standard line-of-sight-magnetogram synoptic map in red. The rms is computed at each latitude using all longitudes. Pixels with field greater than 23 G (about 3 sigma) are excluded._{r}

Fig. 2 gives a quantitative comparison of the rms for *B _{r}* (top),

*B*(middle), and

_{φ}*B*(bottom) as a function of sine latitude for the three types of synoptic charts; the top panel also shows the rms for

_{θ}*B*(in red) inferred for a synoptic map constructed using only the standard line-of-sight magnetograms. The rms is computed for all 3600 pixels at one longitude for each fixed latitude. Pixels with field strength greater than 23 G, about 3 times the field sigma, are excluded. As expected, the noise in the three components is generally much higher in the potential-method charts, significantly different in the radial-method charts, and comparable in the random-method charts. This suggests that the random method is better for synoptic maps. Thus, the vector field synoptic charts provided by the JSOC are produced with random method.

_{r}

Figure 3 |Median of absolute values of the three components of vector field from Stanford (black) and NSO (red) HMI vector field synoptic charts (CR 2145). The disambiguation is done using the random method. The synoptic charts in this comparison have a size of 360 × 180. The NSO version has much higher median values at active latitude, indicating effect of misalignment and feature evolution of the contributed pixels from different individual magnetograms.

The NSO group has constructed lower-resolution (360 × 180) HMI vector field synoptic charts processed in the same way that the SOLIS/VSM maps are made^{7}. To reduce the computational load, that version uses just two HMI vector magnetograms observed at 00:00 UT and 12:00 UT each day (rather than all 120) and for each longitude makes a weighted average of remapped magnetograms collected over several days. Compared to the high-resolution HMI maps, each data point in these synoptic charts comes from fewer magnetograms and includes observations taken much farther away from central meridian and observed at larger separations in time. This can lead to smearing of small magnetic features due to evolution and to peculiar motions of features relative to differential rotation. We note that UT noon and midnight are near the daily extremes in *SDO*-Sun radial velocity, but by averaging over a broader time range at each longitude, the NSO maps may show less daily variation in the background noise level. Fig. 3 shows the median values vs. latitude for CR 2145. For the comparison, the resolution of the HMI maps was reduced to 360 × 180. Reduced-resolution (720 × 360) HMI synoptic maps are available in the data series hmi.b_synoptic_small.

### References

[1] Hoeksema, J. T., Liu, Y., Hayashi, K. et al., 2014, *SoPh*, **289**, 3483.

[2] Ulrich, R. K., Evans, S., Boyden, J. E., & Webster, L. 2002, *ApJS*, **139**, 259.

[3] Centeno, R., Schou, J., Hayashi, K. et al., 2014, *SoPh*, **289**, 3531

[4] Metcalf, T. R. 1994, *SoPh*, **155**, 235

[5] Metcalf, T. R., Leka, K. D., Barnes, G., et al. 2006, *SoPh*, **237**, 267

[6] Leka, K. D., Barnes, G., Crouch, A. D., et al. 2009, *SoPh*, **260**, 83

[7] See ftp://solis.nso.edu/HMI/README.HMIserrmaps