Consistent, filter-independent helioseismic inference of sunspot subsurface thermal and magnetic structure (the open issues of the NOAA AR 9787 case study)

Gizon et al. (2009) applied every major technique of local helioseismology to the sunspot of NOAA Active Region 9787 and found that the inferred subsurface structure depends strongly and unexplainedly on the data-analysis protocol: inversions of phase-speed-filtered and ridge-filtered time-distance travel times are mutually inconsistent, ring-diagram and time-distance wave-speed profiles disagree, and Born-approximation kernels computed for the quiet Sun predict the wrong sign for the moat-flow signature in short-skip (small phase-speed) travel times. The underlying open problem is to place seismic inference of sunspot interiors on a physical footing: a validated framework in which travel times and amplitudes are predicted by forward models that treat the strong, inclined magnetic field, the Wilson depression, and modified wave excitation and damping, and in which the recovered thermal structure, flows, and magnetic field are invariant, within quantified uncertainties, to the choice of filtering and measurement method. Seventeen years of follow-up work — realistic MHD simulations as testbeds, magnetized reference models, Lorentz-stress kernels, magnetic-field-insensitive geometries — has produced substantial partial progress, but reviews through 2023 and primary work through 2026 still identify the inversion of strongly magnetized regions as an unresolved challenge, so the problem remains open.

Background

Sunspots are long-lived regions of the solar surface where a magnetic field of order kilogauss suppresses convection and cools the plasma; how they are structured below the visible surface (the photosphere) cannot be observed directly, and helioseismology — the analysis of the Sun's seismic waves — is the only available probe. The Sun oscillates in global acoustic modes, the p-modes (pressure modes), and a surface gravity mode, the f-mode; local helioseismology exploits these waves to image localized subsurface perturbations in flows, sound speed, density, and magnetic field (Gizon and Birch 2005). In time-distance helioseismology, the Doppler velocity signal recorded at two surface points is cross-correlated and fitted to measure the wave travel time between them; the travel-time perturbation relative to the quiet Sun is then inverted, using sensitivity kernels (functions relating subsurface perturbations to travel-time shifts, usually computed in the Born single-scattering approximation about a quiet-Sun model), to infer subsurface structure. Because solar oscillations are dispersive, measurements must be filtered in the two-dimensional wavenumber-frequency plane: phase-speed filters select waves whose phase speed $\omega/k$ lies in a chosen window, thereby selecting a family of ray paths with a fixed skip distance between bounces, while ridge filters select a single mode ridge (f, p1, p2, ...) spanning a range of phase speeds; narrow frequency bands are a further choice. Ring-diagram analysis instead fits the local oscillation power spectrum over a patch of the surface to obtain a local dispersion relation and hence a wave-speed profile, and helioseismic holography regresses the observed wave field onto the wave field expected from a surface source position. Each method commonly parametrizes the sunspot's subsurface effect on waves by an equivalent wave speed $c_{\mathrm{w}}$, a scalar combining sound-speed and density perturbations, which by construction absorbs the direct magnetic (Lorentz-force) effect on the waves into a pseudo-thermal quantity. Several strong-field phenomena complicate all of these measurements: the Wilson depression (the sunspot's visible surface, defined where the atmosphere becomes optically thick, lies several hundred kilometers deeper than in the quiet Sun, shifting the height at which waves are observed and excited); mode conversion (in the region where the sound speed and Alfvén speed are comparable, acoustic waves partially convert to magnetic slow and Alfvén waves, especially where the field is inclined, as in the penumbra); and source suppression and altered damping (convective excitation of the waves is reduced inside the magnetic region — emissivity reduction — which by itself produces spurious travel-time shifts of up to about 10 s). The data used in the source study came from SOHO/MDI, the Michelson Doppler Imager on the Solar and Heliospheric Observatory; its successor SDO/HMI (the Helioseismic and Magnetic Imager on the Solar Dynamics Observatory) provides the current data, and realistic radiative magnetohydrodynamic (MHD) and magnetohydrostatic (MHS) simulations of entire sunspots now serve as controlled testbeds for helioseismic methods.

Problem Statement

Establish a validated, physics-based helioseismic determination of the subsurface structure and dynamics of a sunspot, in which the seismic inference no longer depends on the data-analysis protocol. Concretely, for a sunspot such as NOAA AR 9787: (i) explain and remove the observed sensitivity of travel-time perturbations to the choice of frequency-wavenumber filtering — phase-speed filters, ridge filters, and narrow frequency bands — so that inversions performed with different protocols (time-distance with either filtering, ring-diagram analysis, holography) recover subsurface structure that agrees within quantified uncertainties, including in the small-phase-speed (short-skip) regime where Gizon et al. (2009) found qualitative contradiction between protocols; (ii) construct sensitivity kernels and forward-model power spectra that predict the measured travel times (and amplitudes) consistently across filter choices, by treating wave propagation in the strong, inclined magnetic field on its own terms — Lorentz-force effects, mode conversion, the Wilson depression, and the altered wave excitation and damping (with radiative-transfer effects) inside the sunspot — rather than absorbing them into an equivalent wave-speed perturbation $c_{\mathrm{w}}$ linearized about the quiet Sun; and (iii) deliver, with an uncertainty budget, the sunspot's subsurface thermal structure, flows, and magnetic field, validated against realistic MHD/MHS sunspot simulations used as testbeds and consistent with surface spectropolarimetric constraints. These three components jointly constitute the unified helioseismic description of sunspot subsurface structure and dynamics that the source study identified as its principal unfinished objective; progress on any one component (for example, a magnetized kernel or a bias estimate) is only partial progress toward the stated problem.

References

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