Level 2.1 (Disambiguated SOT-SP Data)

Producing the SOT-SP Level 2.1 data involves resolving the 180° ambiguity that is present in the image-plane azimuthal angle in the inverted Level 2 data. This azimuthal direction ambiguity is inherent in the polarization signal, and hence is a feature of all inferences of the vector magnetic field by inverting Zeeman-effect polarization spectra.

The SOT-SP Level 2.1 data files also include the three components of the magnetic field vector after being transformed from the components described by the image plane coordinate system (described by the line-of-sight magnetic field \( B_{LOS} \), plane-of-sky azimuthal angle, and the inclination angle to the line-of-sight) into helioprojective components governed by Stonyhurst heliographic (longitude, latitude, radius) coordinates. Note, however, that while the components of the resulting \( B \) vector have been transformed, these transformed components are gridded on the original image-plane coordinate system.

The ME0 Algorithm

The disambiguation of the azimuthal magnetic field direction is computed using the ME0 code (Leka et al. 2009), which is based on the Minimum Energy algorithm by Metcalf (1994). This is the same algorithm that is operating in the SDO/HMI pipeline (Hoeksema et al. 2014). The ME0 code is publicly available.

The ME0 algorithm seeks to globally (across the field of view) minimize the "energy" \( \sum (|\nabla\cdot B| + \lambda |J_z|) \) that consists of two components: the divergence term involving \(\nabla\cdot B\), which provides a physical constraint, and a term involving the vertical current density \(J_z\), which provides a local smoothness constraint. The weighting factor \( \lambda\) determines the relative importance of the two components, and here is set to unity by default. All calculations are performed on the heliographic components although the ambiguity is in the image-plane azimuth. The vertical component of the divergence is estimated using a potential-field extrapolation.

ME0 uses simulated annealing to minimize the equation above. As such, the ME0 algorithm has several adjustable parameters that govern the "cooling schedule", i.e., the details of the optimization implementation. To create the SOT-SP Level 2.1 data files, the algorithm was run with the following attributes (recorded in AMB* header keywords, listed below, with their values):

• AMBTFCT0: Input factor to scale initial temperature [2.0];
• AMBTFCTR: Input factor that governs the magnitude of the steps taken to reduce temperature as the system cools (closer to 1.0 indicates slower cooling) [0.99];
• AMBNEQ: Number of reconfigurations performed at each temperature setting [100].

These settings result in considerably "slower" cooling (i.e., more computationally intensive but also more likely to find an optimal solution) than is used by the HMI pipeline, but because there are considerably fewer pixels in the SOT-SP scans than in HMI full-disk data, this approach is feasible.

The behavior of the disambiguation algorithm is governed by and recorded in the following additional FITS header keywords:

• AMBGFLG: Geometry flag [2: spherical geometry].
• AMBNPAD: Size of buffer (number of pixels to add to each side of the field of view), used to avoid ringing with Fourier transforms [200].
• AMBSEED: Input random number seed [1].
• AMBLMBDA: Weighting between divergence and vertical current density [1.0]
• AMBNAP: The apodizing-window width, in pixels, for computing the potential field [10.0]
• AMBNT[X,Y]: The tile size in \(x\) and \(y\) for computing the potential field (used in estimating vertical derivatives), pixels [10.0, 10.0]

The ME0 algorithm also supports the following keywords govern the threshold in the transverse field \(B_\perp\) below which annealing should stop and the potential-field acute-angle solution should be invoked. For SOT-SP Level 2.1 data, such screening is not invoked (i.e., all pixels are annealed), but the keywords are listed here for completeness:

• AMBPFLG: Flag set when the potential-field acute-angle method is used to disambiguate pixels with \(B_\perp <\) AMBBTHR [0.0]
• AMBATHR: Minimum transverse field strength for annealing [0.0]
• AMBBTHR: Field strength below which to use acute angle [0.0]
• AMBNEROD: Number of pixels by which to erode map above the threshold [1]
• AMBNGROW: Pixels by which to dilate (grow) the eroded map [1]

An evaluation metric of the the final solution is also included in the header. The AMBERAT keyword is the ratio of the final value to the initial value of the energy expression being minimized, \( \sum(\mid\nabla\cdot B\mid + \lambda \mid J_z \mid) \). Lower values of this energy are more optimal, however the exact number depends on the particulars of the scan. Unless scans are otherwise identical, these values should not be compared between scans. Nevertheless, a general rule of thumb is that values above approximately 0.5 may indicate that the result is suspect.

The optimization stopping condition includes two components: the fraction of pixels for which all flips of the azimuth are accepted, and the decrease in the energy from a baseline energy at which this occurs. The energy of the first iteration is used as this baseline energy.

The parameter set for the optimization process detailed above was chosen by examining the degree of variation in the solution as a function of field strength, according to the governing optimization parameters AMBNEQ and AMBLMDA. Cooling schedules each with 10 random number seeds (AMBSEED) were tested, and, as expected, it was found that slower cooling achieves a better answer, based on two factors: the variation in final energies between seeds decreased, and the value of \(B_\perp\) above which 99% of pixels agree consistently between seeds also decreased. Testing the various options requires its own optimization scheme for complete parameter-space exploration, however the significant variation in the characteristics of each scan makes achieving a perfect cooling schedule for all data impractical.

Level 2.1 Data File Format

Each Level 2.1 dataset is stored in a FITS file containing 4 extensions:

As mentioned above, the heliographic components are gridded on the (original) image-plane coordinate system. Additionally, the magnetic fill fraction and magnetic field have been multiplied together for these field-component outputs.

Known Issues with the Level 2.1 Data

The disambiguation algorithm makes assumptions about the nature of the input data. When the input data do not conform to these assumptions, this may lead to some questionable results. There are specific situations that can contribute to questionable results:

• Patches of \(B_\perp\) that apparently alternate sign from pixel to pixel ("checkerboarding") may appear when ME0 cannot reach a solution (see the bottom of Figure 15 in Hoeksema et al. 2014). There are various reasons for this effect (which a slower cooling schedule may mitigate), however we find that it often occurs in regions where MERLIN could not fit the Level 1 data well.

• Random-looking solutions in weak-\(B_\perp\) areas may occur. Because all SOT-SP pixels are annealed (recall that AMBATHR and AMBBTHR are both set to 0.0), flipping their azimuth angle by 180° does not significantly change the resulting energy. There lack of information in these pixels prohibits ME0 from coming to a solution.

• The reprojection of \(B\) from image-plane components to heliographic components relies on accurate pointing information in the SOT-SP headers. This pointing information is sometimes significantly off, especially for scans near the limb. Such errors can, for example erroneously place most of a scan off the solar limb, and negatively impact the annealing and the reprojection for heliographic components, or both. Disambiguated scans near the solar limb should be treated with care.

• Instances where SOT's correlation tracker resets in the middle of a scan will introduce significant discontinuities in the inferred field strength and angles that negatively impact the spatial derivatives required for ME0. Generally, while the disambiguation results in the immediate vicinity of these resets will be negatively impacted, the rest of the scan (and disambiguation results) should have little impact.

Finally, users should be aware that the 180° ambiguity is inherent in the image-plane components. Flipping the image-plane azimuth by 180° does not correspond to a 180° change in the heliographic components. Depending on the relative strengths of the \( B_{LOS} \) and \(B_\perp\) components, the viewing angle and other geometric factors, there can either be a significant or a minimal change in the resulting heliographic components.