Three test cases, based on the Bogdan and Low (1986) approach. Contours show radial field component (red positive, blue negative), and greyscale shows azimuthal angle. The radial component is the same in all three cases; the case on the left is a potential field, and the field becomes more radial than potential from left to right (parameter in the Bogdan-Low model a=1.0, a=1.3 and a=2.0 from left to right).

This model has purely horizontal currents, which change the magnitude, but not the direction of the horizontal field at the surface. Thus, if one can construct the potential field whose radial component matches on the boundary, a potential field acute angle disambiguation will work very well. This is unlikely to be true for the Sun, so the potential field disambiguiation here is likely to be better than we can expect for real data.

Another way to view the importance of the differences: the greyscale shows the magnitude of the difference in the field between the model field and the results of the disambiguation, saturated at 300G.

As expected, the closer to radial the model field is, the better the most radial disambiguation does. However, note that, like the potential field, it produces swathes of incorrect results, which will lead to preferred directions to the field which are not real. Whether the potential field disambiguation or the most radial disambiguation will get a larger fraction of pixels correct for real data will depend on whether the solar magnetic is closer to radial or potential. I don't think we know the answer to that.

The main impact of this type of noise is to fuzz out the boundaries between correct and incorrect solutions. For the level of noise added, there is not a substantial change in the fraction of pixels which are correctly disambiguated.

To quanitfy these results, I ran the disambiguation code, including annealing of strong transverse field pixels (unlike above, where no pixels were annealed), for a set of random number seeds, and the various choices for what to do in weak transverse field regions. In addition to the "potential field acute angle" and "most radial" choices, I included a "random" determination. Four ways of quantifying the results are given in the following table, with an uncertainty estimate based on the variations among the random number seeds.

Note that there is a general trend for the results to be worse when the test case is farther from potential (left column is potential; right column is farthest from potential). This is at least partly a result of a few annealed pixels being incorrect. The results can be improved with a slower cooling schedule, but these are already somewhat slow to run, because the majority of pixels have strong transverse fields, unlike in the HMI data.

Now consider a different test case, also based on the Bogdan-Low solution, but which is hopefully more solar-like. In this case, a Green's function was used to produce small spatial scale features by placing point sources a small distance below the surface. The model field was used to generate spectra, to which Poisson noise was added, then the spectra were spatially binned and inverted. Because of the submerged point sources, the model field no longer is more radial (than the corresponding potential field) at every pixel, although in some average sense this is likely still true. Note that again, the potential field probably still performs better than one should expect it to in the solar case.

In the same format as above, here is one example from this test case. (The top row shows in black/white which pixels are correctly disambiguated; the second row shows in a greyscale the magnitude of the difference in the field.)

And here are the corresponding quantitative measures after annealing pixels in strong transverse field areas.