A. B. Griñón-Marín^{1,2,3}, A. Pastor Yabar^{4}, Y. Liu^{1}, J. T. Hoeksema^{1}, & A. Norton^{1}

1 W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305-4085, USA

2 Institute of Theoretical Astrophysics, University of Oslo, NO-0315 Oslo, Norway

3 Rosseland Centre for Solar Physics, University of Oslo, NO-0315 Oslo, Norway

4 Institute for Solar Physics, Department of Astronomy, Stockholm University, Albanova University Centre, SE-106 91 Stockholm, Sweden

The Helioseismic and Magnetic Imager^{[1]} on board NASA’s Solar Dynamics Observatory^{[2]} mission has been observing the full Stokes vectors since 2010 May. These vectors are crucial for the inference of different atmospheric parameters of the observed areas. There are different techniques in to infer the physical information imprinted in the spectropolarimetric data, and one of such is the inversion codes of the radiative transfer equation for polarized data. These codes solve the radiative transfer equation to obtain synthetic Stokes profiles and compare them with the observed ones. After looking for the minimization of this difference, the inversion returns several model parameters, some of which have physical meanings.

Figure 1| Full-disk HMI vector magnetograms taken at 17:12 UT on 2015 November 17 (top) and 17:00 UT on 2015 November 20. The color table saturates at ±60 Mx cm

^{−2}. Images from left to right show three components of magnetic field, B_{r}(left; radial component), B_{t}(middle; north–south component), and B_{p}(right; east–west component). The vector field data are derived using the original version of VFISV. “A” and “B” denote two medium-strength field regions where the east– west field, B_{p}, changes sign after crossing central meridian.

There are many different inversion codes that include different degrees of radiative transfer complexity. This way, it is important to establish which code fits better the needs based on instrumentation, telescope, or speed/amount of data processing one must handle. In this case, the first version of the spectral line inversion code Very Fast Inversion of the Stokes Vector (VFISV^{[3]}) used to invert the Stokes parameters measured by HMI assumes a Milne-Eddington model for the solar atmosphere. This choice has shown to be nonoptimal for plage regions (regions from quiet Sun with intermediate magnetic field strengths) as it was found that it induces an instrumental hemispheric bias in the east–west component of the magnetic field^{[4]}. In particular, there is a change of the sign in the east-west component (B component) of the vector magnetic field (see Figure 1). The most likely reason behind this problem is that since the spatial resolution of the HMI instrument is not high enough to fully resolve these small-scale magnetic structures in the photosphere. Thus, the information from the various Stokes profiles is not internally consistent and a bias is introduced in the magnetic field.

Figure 2| Bayesian inference for the single-component (red) and two-component (blue) models for a plage pixel at heliocentric angle θ = 55°.25. Panels (1)–(36) show the two-dimensional probability density functions for the model parameter combinations. Panels (a)–(i) on the diagonal show the marginal posterior distributions for each model parameter independently. The color-coded values show the median value (color-coded vertical thin lines) for each model, as well as their 99.73% credible region. Panels (j)–(m) at the upper right depict the measured Stokes spectra (black crosses) with their uncertainties (vertical black bars), together with the synthetic profiles for the maximum likelihood case for the single-component model (red; S

_{0}= 6.91 kDN s^{−1}, S_{1}= 48.0 kDN s^{−1}, η_{0}= 1.48, Δλ_{D}= 40.0 mÅ, B = 695 G, γ = 47°, Ψ = 73°, and V_{los}= −1.27 km s^{−1 }) and two-component model (blue; S_{0}= 8.92 kDN s^{−1}, S_{1}= 45.77 kDN s^{−1}, η_{0}= 2.44, Δλ_{D}= 28.20 mÅ, B = 1383 G, γ = 29°, Ψ = 72°, V_{los}= −1.25 km s^{−1}, and α_{mag}= 0.41). The text box shows the actual evidences found (in natural logarithm scale), together with the Bayes factor.

We have modified the original VFISV inversion code so that the Stokes parameters inversion deals with this problem. In order to determine whether the new strategy is physically meaningful, we evaluate the inferred parameters inverted with the original version of the HMI inversion code and with the new approach. Since the number of free parameters is different and in order to fully address the behaviors of both methods, we use a Bayesian analysis to properly compare both methods (see Figure 2). In pixels with intermediate magnetic field strengths (such as the ones belonging to plages), not only does the fitting of the Stokes profile improve (in a Bayesian sense), but also the inference of the magnetic parameters and line-of-sight velocity is unique (narrow posterior distributions). The new strategy is proven to be effective for mitigating (if not completely solving) the anomalous hemispheric bias in the east–west magnetic field component in moderate field regions, as demonstrated in Figure 3 (see also Ref. [5]).

Figure 3| B

_{p}in a weak field region from the new two-component VFISV (left panels) and the original VFISV (right). The data are taken 3 days apart. The top panels show the data on November 17, when the region is in the east hemisphere; the bottom panels were observed on November 20, when it is in the west hemisphere.

### References

[1] Scherrer, P. H., Schou, J., Bush, R. I., et al. 2012, *Solar Phys*, **275**, 207

[2] Pesnell, W. D., Thompson, B. J., & Chamberlin, P. C. 2012, *Solar Phys*, **275**, 3

[3] Borrero, J. M., Tomczyk, S., Kubo, M., et al. 2011, *Solar Phys*, **273**, 267

[4] Pevtsov, A. A., Liu, Y., Virtanen, I., et al. 2021, *JSWSC*, **11**, 14

[5] Liu, Y., Griñón-Marín, A.B., Hoeksema, J.T., et al. 2022, *Solar Phys*, **297**, 17