A method for global inversion of multi-resolution solar data

J. de la Cruz Rodr\'iguez

REVIEW 3 major objections 5 minor 27 references

A method for global inversion of multi-resolution solar data

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that spectropolarimetric data at different spatial resolutions can be inverted jointly by applying dataset-specific linear degradation operators to synthetic spectra, and shows numerically that the recovered continuum…

desk verdict Genuinely useful extension of van Noort's global inversion to multi-resolution data with a clean proof-of-concept; the main caveat is that real-data PSF misestimation is not tested. read the letter →

arxiv 1909.02604 v1 pith:DTFXULR3 submitted 2019-09-05 astro-ph.SR astro-ph.IM

J. de la Cruz Rodríguez This is my paper · ORCID

classification astro-ph.SRastro-ph.IM

keywords highangularresolutionradiativetransferpolarizationSun:magneticfieldschromospheremulti-resolutioninversionpoint-spreadfunctionLevenberg-Marquardt

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper is trying to establish that observations taken at different spatial resolutions do not have to be resampled to a common grid before inversion; instead, each dataset can constrain the spatial scales it actually contains. It generalizes the Levenberg-Marquardt inversion algorithm by inserting chains of linear operators—telescope PSF convolution, rebinning, shifts, rotations—that degrade the synthetic spectra of a finely gridded model atmosphere to match each observation. In numerical experiments with two simulated Fe I lines observed through different apertures and sampled on different grids, the joint inversion recovers the original granulation contrast almost exactly (14.8% RMS versus 14.9% in the reference) and keeps all derived quantities on the one-to-one line. If correct, this makes it feasible to combine ground-based and space-borne chromospheric diagnostics without sacrificing the resolution of the best dataset.

What carries the argument

The central object is a chain of linear degradation operators in sparse matrix form, D = D1D2...Dn, applied to the synthetic spectra of a model computed on a fine grid before comparison with each observational dataset. The merit function becomes a sum of per-dataset terms, the Hessian becomes a sum of sparse spatially coupled Hessians weighted by the PSF autocorrelation, and the correction step is solved with an iterative sparse linear solver; spatial Tikhonov regularization adds narrow bands to the Hessian and is used in a two-cycle scheme to stabilize convergence.

What would settle it

Run the paper's Experiment 3 with a PSF width deliberately wrong by 10%, or with a small artificial time offset between the two spectral lines, and check whether the retrieved B||, v_los, and continuum-contrast scatter plots still follow the one-to-one line; any systematic bias away from that line would show that the assumed known-operator premise is essential.

Watch

Extended reading notes

Core claim

The central claim is that a single model atmosphere can be fitted simultaneously to multiple observations with different spatial resolutions, provided each forward prediction is passed through a known linear degradation operator specific to that dataset. The paper demonstrates this with a Milne-Eddington-based proof of concept: a high-resolution Stokes I dataset in Fe I 6301 Å is combined with a lower-resolution full-Stokes dataset in Fe I 6302 Å, each degraded by a different telescope PSF and resampling, and the joint inversion recovers the continuum intensity contrast at 14.8% versus 14.9% in the reference model. Scatter plots of the longitudinal and transverse magnetic field, azimuth, line-of-sight velocity, and continuum intensity all follow the one-to-one line, indicating that amplitudes and contrasts are preserved rather than blurred away. Each dataset constrains only the spatial scales present in it, so the magnetic field is limited by the lower-resolution polarimetric data while the intensity and velocity maps retain the higher resolution.

Load-bearing premise

The load-bearing premise is that every difference between the datasets can be represented by a known, space-invariant linear operator applied to the synthetic spectra, and that the datasets are strictly co-temporal; if the real degradation, including time-varying seeing residuals, or the alignment differs from that operator, the inversion will fold the mismatch into the retrieved atmospheric quantities.

Editorial extensions

If this is right

A joint multi-resolution inversion keeps each dataset's native resolution: the high-resolution intensity dataset constrains velocity and temperature at fine scales while the low-resolution polarimetric dataset constrains the magnetic field at its own scales.
Because the model grid is decoupled from the observation grids, the model can be sampled denser than the data, reducing pixel discretization and high-frequency damping near the Nyquist cutoff.
The two-cycle regularization strategy, starting with an overestimated smoothness weight and then reducing it, lets the global inversion converge in fewer than about ten iterations per cycle.
Any degradation that can be written as a linear operator—PSF convolution, rebinning, shifts, rotations—can be chained and handled in a single inversion, so the method applies broadly to multi-facility data.
The method opens a practical path for combining ground-based and space-borne chromospheric diagnostics, which is increasingly relevant as the resolution gap grows with future large-aperture telescopes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the method is applied to real ground-based data, residual seeing not captured by a static PSF will be absorbed into the retrieved model; a testable prediction is that retrieved Doppler widths and field strengths will be biased in exactly the spatial regions where the seeing PSF is worst.
The same operator-chain formalism could absorb temporal misalignment by inserting time-interpolation operators between datasets, extending the method beyond the strictly co-temporal assumption the paper states as a limitation.
The power-spectrum comparison suggests that weak, small-scale magnetic fields remain the hardest part to recover; adding a third full-Stokes dataset at the highest available resolution should push the recovered frequency band higher, an experiment the paper's setup could run directly.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take on the de la Cruz Rodríguez paper. The key thing to know: this is a genuine, useful extension of van Noort's global inversion, and the proof-of-concept is solid. The main caveat, which the stress-test note correctly identifies, is that the experiments assume the degradation operators are known exactly. That's a real limitation for ground-based data, but it's not a reason to dismiss the method.

What's new: the paper generalizes van Noort (2012) from a single PSF on a common grid to arbitrary chains of linear operators—PSF convolution, rebinning, interpolation, rotation—and decouples the model grid from the observation grids. That's a real step forward. Adding spatial Tikhonov regularization is useful, and the two-cycle approach (heavy regularization first, then light) is a practical way to avoid local minima. The numerical experiments are well designed: the forward model is STiC with LTE, the inversion is Milne-Eddington, so the test is not circular. The comparison with two 1D experiments is illuminating: naive resampling of the low-res dataset into a common grid degrades the continuum contrast to 9.2%, while the multi-resolution method recovers 14.8% against the reference 14.9%. The power-spectrum analysis in §4.2 is a nice addition, showing that the high-res Stokes I dataset constrains small-scale magnetic fields when Zeeman splitting is detectable.

The soft spots are in proportion. The stress-test concern about known degradation operators is the main one. The paper does not test what happens when the PSF is misestimated, which is the typical situation for ground-based CHROMIS data after MOMFBD. The author acknowledges this in §5 but doesn't quantify it. Also, the regularization weight is hand-tuned in two stages; no automated selection is given. The proof-of-concept code is not released, which limits reproducibility. No error bars on retrieved parameters are reported. These are all minor-to-moderate for a method paper; they don't undermine the central claim, but a careful referee should ask for a sensitivity test on PSF mismatch and ideally a code release.

Who would benefit: anyone doing multi-instrument inversions, especially chromospheric studies combining IRIS with ground-based telescopes. I'd cite it if I worked in that area. The writing is clear, the math is straightforward, and the figures support the conclusions.

My recommendation: definitely send it to peer review. The method deserves serious referee attention, with the expectation of revisions to address the PSF-mismatch question and reproducibility. It's a solid contribution, not a finished product.

Referee Report

3 major / 5 minor

Summary. The paper presents a generalization of van Noort's spatially-coupled Levenberg-Marquardt inversion to datasets acquired at different spatial resolutions and on different spatial grids. The key idea is to represent all instrumental degradation steps—telescope PSF, rebinning/resampling, interpolation, shifts, and rotation—as a chain of linear operators applied to the synthetic spectra before comparison with the observations, with the merit function written as a sum over windows that share a common model grid. Spatial Tikhonov regularization is added, and the resulting sparse Hessian system is solved with BiCGStab. The method is tested in three numerical experiments using ME inversions of STiC LTE synthetic spectra computed from a Rempel (2012) rMHD snapshot: a 1D inversion of only the high-resolution Stokes I dataset, a 1D inversion of the high-resolution I plus a low-resolution full-Stokes dataset naively resampled to the high-resolution grid, and the proposed multi-resolution global inversion. The global inversion recovers a continuum intensity RMS contrast of 14.8% against a 14.9% reference and shows one-to-one scatter for the magnetic field, velocity, and intensity, while the 1D benchmarks show degraded contrast and amplitude. Limitations regarding co-temporality, time-varying residual seeing, and the assumption of known degradation operators are acknowledged in Section 5.

Significance. If the method is robust beyond the idealized tests, it addresses a real and timely need: combining high-resolution ground-based data (CHROMIS, CRISP, and future DKIST/EST) with lower-resolution space-borne data (IRIS, Hinode) in a single self-consistent inversion. The linear-operator formalism is clean and plausibly generalizes the van Noort framework, and the test design is non-trivial because the forward model (STiC, LTE) differs from the inversion model (Milne-Eddington), so the good recovery is not a trivial self-inversion. The explicit quantitative contrast recovery (14.8% vs 14.9%) is a strong point, as is the paper's candid listing of limitations. However, the central demonstration depends on the exact knowledge of the degradation operators and on a hand-tuned regularization weight; without sensitivity analysis to PSF misestimation and an objective regularization-selection method, the practical claims are not yet fully supported. The paper is a solid proof-of-concept that, with the requested additions, would be a valuable contribution to the multi-resolution inversion literature.

major comments (3)

[Sections 3.2–3.3 and 5] The successful recovery in Experiment 3 presumes that the degradation operator used in the inversion is identical to the one that generated the synthetic data. Section 5 acknowledges that ground-based data are affected by residual, time-varying seeing after MOMFBD, which is not known to the precision assumed here. This is load-bearing because the method's advantage over simple resampling is precisely that it corrects a known linear operator; an incorrectly assumed PSF (or a misregistered shift/rotation) will be absorbed into the retrieved atmospheric parameters and bias the recovered contrasts and fields. Please add a numerical experiment that perturbs the assumed operator (e.g., a PSF with a different FWHM, a small unknown image shift, or a residual seeing term), and quantify the resulting bias in the recovered continuum intensity contrast and magnetic-field parameters.
[Section 3.3 and Section 4.1] The two-stage regularization schedule uses a 'largely overestimated (x50)' weight in the first cycle and then 'a final value' chosen because it 'allowed to converge the problem without showing artifacts from the deconvolution.' This is a subjective, hand-tuned free parameter. Since the central quantitative result (14.8% vs 14.9% contrast) and the one-to-one scatter plots may depend on this choice, the paper should either provide an objective recipe for selecting the regularization weight (e.g., L-curve or cross-validation) or demonstrate explicitly that the recovery is stable over a reasonable range of weights.
[Section 4, Figures 11 and 12] The claim that 'all derived quantities follow the one-to-one line' is supported only by qualitative density scatter plots and a single scalar RMS contrast value. No slopes, intercepts, correlation coefficients, or RMS differences are reported for B_parallel, |B_perp|, the azimuth, or v_los. Because the stated conclusion is that the amplitudes and contrast of all quantities are correct, please provide quantitative agreement metrics for each observable, and include uncertainties on the retrieved parameters (at least for the main experiment) so the reader can evaluate the precision of the recovery.

minor comments (5)

[Abstract and Section 2.3.1] The abstract contains a doubled article: 'for the the effects' should read 'for the effects.' In Section 2.3.1, 'the subindexes ji spam from' should read 'span,' and the heading of Section 2.3.2 spells 'Rebining' instead of 'Rebinning.'
[Section 4.2 and Figure 14] The text and the figure caption use 'Azimutal' instead of 'Azimuthal' in the description of the power-spectrum integration.
[Figure 13 caption] The caption contains the typo 'mutli-resolution' instead of 'multi-resolution.'
[Equation (7)] The normalization factor Nn in Eq. (7) is ambiguous: it appears both inside and outside the window sums, and it is not stated whether Nn is the total number of data points or the number of windows. Please define it explicitly.
[Section 3.3 and Table 1] The description of the two-stage regularization would benefit from giving the actual numerical values of the regularization weight used in the first and second cycles, rather than describing them only as 'largely overestimated' and 'correct value,' to aid reproducibility.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: the inversion is validated against an external simulation with known degradation operators; the cited prior work is incidental, not load-bearing.

full rationale

The paper's strongest claim—recovering the continuum intensity contrast at 14.8% RMS against the reference 14.9%—comes from a synthetic proof-of-concept experiment. The ground truth is an independent 3D rMHD simulation (Rempel 2012), and the degradation operators (telescope PSF plus rebinning) are prescribed in the experiment and known to the inversion through the operator chain D = D1D2...Dn (Eq. 10). This is a consistency test of the algorithm, not a circular derivation: no target quantity from the simulation is used to set the method's constants. The regularization weight is chosen by convergence behavior ('we chose a final value for the second cycle that allowed to converge the problem without showing artifacts'), not fitted to the 14.9% contrast value. The method does build on van Noort (2012) for PSF-coupled inversions and cites de la Cruz Rodríguez et al. (2019) for the l-2 regularization derivation, and it uses the self-developed STiC code to synthesize the test spectra, but these are pointers to standard numerical machinery and previously published tools rather than imports of the paper's conclusion. The acknowledged limitations in Section 5—residual seeing aberrations in ground-based data and non-co-temporal datasets—mean the method could be misspecified in real applications, but that is a correctness/robustness concern, not circularity. No step reduces to its own input by construction.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central method introduces no new physical entities. Its main free ingredient is the hand-tuned spatial regularization weight; the domain assumptions are the linear-operator model of degradation, co-temporality of datasets, and adequacy of the Milne-Eddington parameterization.

free parameters (1)

Spatial Tikhonov regularization weight (two-stage) = not stated numerically; overestimated by factor 50 in first cycle, then a hand-chosen value in second cycle
Chosen by trial to balance convergence and smoothing; directly affects the recovered maps and the trade-off between deconvolution artifacts and noise.

assumptions (4)

domain assumption The instrumental degradation of each dataset can be represented as a known linear operator (PSF convolution, rebinning, interpolation, rotation) acting on the synthetic spectra.
Introduced in Section 2.2 and 2.3; the entire method depends on this representation. Real seeing residuals and unknown PSFs violate it.
domain assumption The datasets combined in one inversion are co-temporal and co-spatial (no significant evolution between them).
Stated as a limitation in Section 5: 'We have not considered the effects of small temporal inconsistencies that could appear when the datasets included in the inversion are not strictly co-temporal.'
domain assumption A Milne-Eddington model atmosphere with a linear source function is adequate to fit the observed spectral lines.
Used for the proof-of-concept inversions; the paper acknowledges that real chromospheric lines require NLTE and depth-stratified models (Sections 1 and 5).
domain assumption The telescope PSF is known a priori and is space-invariant over the field of view.
Assumed in Section 3.2 when constructing synthetic PSFs; Section 5 discusses residual aberrations from seeing as a possible application, implying the PSF is not perfectly known.

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Cite this review

Pith. "Pith review of A method for global inversion of multi-resolution solar data." pith.science (2026). https://pith.science/paper/DTFXULR3

@misc{pith2026190902604,
  author       = {Pith},
  title        = {Pith review of: A method for global inversion of multi-resolution solar data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DTFXULR3}},
  note         = {Machine review of arXiv:1909.02604}
}

read the original abstract

Understanding the complex dynamics and structure of the upper solar atmosphere benefits strongly from the use of a combination of several diagnostics. Frequently, such diverse diagnostics can only be obtained from telescopes and/or instrumentation operating at widely different spatial resolution. To optimize the utilization of such data, we propose a new method for the global inversion of data acquired at different spatial resolution. The method has its roots in the Levenberg-Marquardt algorithm but involves the use of linear operators to transform and degrade the synthetic spectra of a highly resolved guess model to account for the the effects of spatial resolution, data sampling, alignment and image rotation of each of the data sets. We have carried out a list of numerical experiments to show that our method allows extracting spatial information from two simulated datasets that have gone through two different telescope apertures and that are sampled in different spatial grids. Our results show that each dataset contributes in the inversion by constraining information at the spatial scales that are present in each of the datasets, without any negative effects derived from the combination of multiple resolution data. This method is especially relevant for chromospheric studies that attempt at combining datasets acquired with different telescopes and/or datasets acquired at different wavelengths, both limiting factors in the resolution of solar instrumentation. The techniques described in the present study will also help addressing the ever increasing resolution gap between space-borne missions and forthcoming ground-based facilities.

Figures

Figures reproduced from arXiv: 1909.02604 by the authors.

**Figure 1.** Chromospheric plage observation acquired with the CHROMIS instrument at the Swedish 1-m Solar Telescope and with NASA’s IRIS satellite on 2016-09-14 at 09:06 UT. The upper-left panel illustrates the CHROMIS field-of-view where the IRIS raster scan has been indicated. The three upper rightmost panels depict the overlapping FOV for a given time-step in the core of the Si iv 1400 Å line, the Mg ii k line and the Ca ii … view at source ↗

**Figure 2.** Structure of the global Hessian matrix without spatial contraints for a case with nx = ny = 5 and npar = 3. The color coding indicates the subspace containing parameters from the same pixel. The coordinates (y, x) of the pixel associated with each subspace are indicated next to each subspace. Each block has the dimension of the number of parameter per pixel, in this example 3 × 3. the cross product of py,x, two con… view at source ↗

**Figure 3.** Top: The Jacobian of the regularization functions. The regularization functions are ordered along the y-axis. The free parameters for all pixels are displayed along the x-axis. Bottom: The contribution to the Hessian from the spatial regularization functions. Article number, page 4 of 13 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: The spatially coupled Hessian resulting from applying a 3 × 3 pixels PSF of FWHM = 2 pixels. Each coloured square corresponds to a subspace of npar × npar where we have set all elements to 1 for illustration purposes. In reality the latter would also be scaled accordin…

**Figure 5.** Figure 5: Convolution operator using a 2D Gaussian PSF with FWHM = 2 pixels and a FOV of 10 × 10 pixels2 . 0 20 40 60 80 0 5 10 15 20 Drebin 0 20 40 60 80 0 5 10 15 20 Dtotal [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: Top: representation of a 2 × 2 rebinning operator for a 10 × 10 pixel FOV. Bottom: total operator resulting from the combination of a PSF degradation operator and a rebinning operator (Dtotal = DrebinDPSF). apply rotations or shifts. When written in matrix form, an int…

**Figure 8.** Figure 8: Temperature of the 3D rMHD simulation (Rempel 2012) extracted where log τ500 = −1 in each column. The black box indicates the subfield of the simulation that we have used in our tests. Our aim is recreating a situation where two co-temporal datasets that provide compl…

**Figure 7.** Figure 7: Bilinear interpolation operators. Top: constant image shift of (dy, dx) = (0.50, 0.35) over a FOV of 10 × 10 pixels2 . Bottom: 30◦ clockwise rotation for a FOV of 10 × 10 pixels2 . have not considered a resampling of the data in this case as in rebinning. 3. Numerical …

**Figure 9.** Figure 9: Degraded synthetic data at two wavelengths of the Fe i 6301 and of the 6302 Å lines. The former has been degraded assuming a telescope with d1 = 1 m and the latter with a telescope aperture d2 = 0.5 m. Stokes Q, U and V have been set to zero in the 6301 line dataset in…

**Figure 10.** Figure 10: Spatial degradation of the spectra computed from the rMHD simulation using two telescope apertures. Top: Illustration of the extent of the total degradation operator (PSF + rebinning) shown as a dark shade centered in a crop of the FOV for the 6301 line (left) and th…

**Figure 11.** Figure 11: From left to right: longitudinal component of the magnetic field, absolute value of transverse component of the magnetic field, magnetic field azimuth, line-of-sight velocity and continuum intensity. From top to bottom: original rMHD simulation (which has been resampl…

**Figure 12.** Figure 12: Density maps comparing the inversion results from [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]

**Figure 13.** Figure 13: Doppler-width inferred from the multi-resolution inversion (left) and from the 1D inversion including both spectral lines. comparison, the mutli-resolution inversion retrieves lower values over the entire FOV. 4.1. The effects of spatial regularization So far we have …

**Figure 14.** Figure 14: Azimutal integral of 2D power spectra computed from Bk maps: from the multires inversion (black), from the rMHD simulation with a frequency cut-off filter corresponding to d = 1 m (blue) and d = 0.5 m (red). The theoretical cut-off limits of 0.5 m and 1.0 m aperture t…

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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