Nonlinear force-free field (NLFFF) extrapolation of photospheric vector magnetograms is the standard tool for inferring the coronal magnetic field that powers solar flares and coronal mass ejections (CMEs), and the free magnetic energy it yields is a widely used eruptivity indicator. When the same full-disk snapshot (2011 November 9) was extrapolated from vector magnetograms taken by the Helioseismic and Magnetic Imager (HMI) on the Solar Dynamics Observatory (SDO) satellite and by the Vector Spectromagnetograph (VSM) of the Synoptic Optical Long-term Investigations of the Sun (SOLIS) project, Tadesse et al. (2013) found that the two NLFFF models agreed well in field-line topology but differed by 14.4% in free magnetic energy, a difference attributed to weaker transverse fields in HMI data. The authors stated that the significance of this difference for flare and CME processes higher in the corona "deserves a separate study", and that, pending error estimates from the inversions, their error-weighting scheme had to rely on an ad hoc mask. The open problem is to quantify how photospheric vector-data uncertainties and instrument-dependent systematics propagate through preprocessing and NLFFF optimization into coronal free-energy and current-density estimates, and to determine whether differences of this magnitude materially affect predictions of flare/CME onset.

Background

The solar corona is magnetically dominated, and its field cannot be routinely measured directly, so it is commonly reconstructed by extrapolating photospheric vector magnetograms. In a nonlinear force-free field (NLFFF), the current density is everywhere parallel to the magnetic field (∇×B=αB\nabla\times\mathbf{B}=\alpha\mathbf{B}), the field is divergence-free (∇⋅B=0\nabla\cdot\mathbf{B}=0), and the Lorentz force vanishes — an idealization expected to hold in the low-plasma-beta corona (β\beta being the ratio of gas pressure to magnetic pressure). The magnetic free energy, Efree=ENLFFF−EpotE_{\mathrm{free}}=E_{\mathrm{NLFFF}}-E_{\mathrm{pot}}, the excess of the NLFFF energy over that of the potential (current-free) field with the same boundary flux, is a central quantity: it is the reservoir tapped by solar flares and coronal mass ejections (CMEs) and one of the main indicators used in space-weather forecasting of active-region eruptivity.

Two complications dominate in practice. First, the photosphere is not force-free, so observed magnetograms must be preprocessed (smoothed and driven toward force and torque balance) before being used as boundary conditions. Second, the boundary data themselves carry uncertainties: noise in the transverse (horizontal) field component, pixels where the spectro-polarimetric inversion fails, and the 180° azimuth ambiguity inherent to Zeeman polarization measurements, which must be resolved by disambiguation algorithms. Full-disk vector magnetograms are provided by the Helioseismic and Magnetic Imager (HMI) on the space-based Solar Dynamics Observatory (SDO) satellite, which observes the Fe I 6173 Å line, and, over Solar Cycle 24, by the ground-based Vector Spectromagnetograph (VSM) of the Synoptic Optical Long-term Investigations of the Sun (SOLIS) project; the two instruments differ in spectral line, spatial sampling, resolution, and inversion scheme.

The optimization-method extrapolations of Tadesse et al. (2013) in spherical geometry incorporate an error-aware boundary term LphotoL_{\mathrm{photo}} (Wiegelmann & Inhester 2010; Tadesse et al. 2011): instead of enforcing an exact match to the observed field Bobs\mathbf{B}_{\mathrm{obs}}, the functional penalizes deviations weighted by a matrix WW whose entries are meant to be inverse squared measurement errors, with W=0W=0 for pixels with failed inversions (missing data). In practice, since error maps were not available from the inversions, the authors adopted an ad hoc mask W=(Bt/max⁡Bt)2W=(B_t/\max B_t)^2 favoring strong transverse fields BtB_t. Earlier work had already shown the sensitivity of NLFFF results to their inputs: a community code-comparison campaign (DeRosa et al. 2009) found that different NLFFF codes applied to the same active-region boundary data produce markedly different field-line configurations and widely varying free-energy estimates, and Wiegelmann et al. (2010) showed, using simulated noisy spectro-polarimetric (Stokes-profile) inversions of a magnetohydrodynamic (MHD) simulation snapshot, that noise, limited resolution, and filter effects degrade NLFFF reconstructions.

Against this background, Tadesse et al. (2013) compared full-disk NLFFF models of the same day (2011 November 9) built from near-simultaneous HMI and SOLIS/VSM vector magnetograms. The reconstructed field lines agreed well, but the HMI-based model contained more total magnetic energy, more free magnetic energy, higher longitudinal magnetic-pressure variation, and stronger surface electric current density; the free-energy difference was 14.4%, attributed by the authors to the weaker transverse fields measured by HMI. They concluded that it is not clear how important this 14.4% difference is for flare and CME processes originating in fields higher in the corona, explicitly leaving this "separate study" undone, and that a detailed error computation from inversion and ambiguity removal — the input a rigorous error-weighted extrapolation and error budget would need — remained unavailable. The broader open issue is that the propagation of photospheric measurement uncertainties and instrument-dependent systematics through preprocessing and the nonlinear optimization into final energy and current estimates has never been quantified end-to-end for real multi-instrument data, and the consequence of such uncertainties for eruption prediction has not been assessed.

Problem Statement

Quantify, end-to-end, how uncertainties and instrument-dependent differences in photospheric vector magnetograms propagate into nonlinear force-free field (NLFFF) extrapolations of the coronal magnetic field, and determine whether differences of the magnitude found by Tadesse et al. (2013) — a 14.4% difference in coronal free magnetic energy between full-disk SDO/HMI- and SOLIS/VSM-based NLFFF models of the same snapshot — materially affect predictions of solar flare and coronal mass ejection (CME) onset. Concretely, the required results are: (1) a defensible error budget for NLFFF-derived total and free magnetic energy and electric current density in the coronal volume, tracing the contributions of transverse-field noise, pixels with failed inversions, 180° azimuth disambiguation, instrument-dependent systematics (such as the weaker transverse fields measured by HMI), and the preprocessing and optimization choices, from the measured boundary data through to the final energy and current estimates; (2) an attribution of the observed HMI–VSM free-energy difference to these sources, distinguishing instrument systematics from modeling choices; and (3) an assessment of the consequences for eruptivity prediction, i.e. whether free-energy uncertainties of this order change the ranking, threshold-crossing behavior, or skill of free-energy-based (or related nonpotentiality-based) flare/CME onset predictions. The 2011 November 9 full-disk HMI/VSM comparison is the motivating instance; the problem is the general robustness question it instantiates, not the re-analysis of a single snapshot.

References

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