A Method to Extrapolate the Data for the Inverse Magnetisation Problem with a Planar Sample

Dmitry Ponomarev

REVIEW 4 major objections 5 minor 6 references

A Method to Extrapolate the Data for the Inverse Magnetisation Problem with a Planar Sample

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For ideal data, the vertical magnetic field measured on a finite patch over a planar sample has a unique full-plane extension, and a double-spectral algorithm constructs that extension with about 7% relative L2 error in a numerical test.

desk verdict New double-spectral idea, but the paper's central noise-stability claim is unvalidated; deserves review as a promising sketch. read the letter →

arxiv 2411.09331 v1 pith:LSPK6P3E submitted 2024-11-14 math.AP math-phmath.MP

Dmitry Ponomarev This is my paper · ORCID

classification math.APmath-phmath.MP MSC 35R3047A7565R32

keywords inversesourceproblemmagnetisationplanarsamplefieldextrapolationspectraldecompositionRiesztransformsPoissonkernelnetmagneticmoment

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

This paper tries to establish that the vertical component of a magnetic field measured on a finite planar patch can be extended uniquely to the whole plane, at least for noiseless data, and that a practically implementable 'double-spectral' algorithm can approximate that extension. The uniqueness claim rests on the observation that the field produced by a planar magnetisation is a real-analytic function of the horizontal variables; two different full-plane extensions that agree on the patch would differ by a function that is both analytic and annihilated on the patch by a positive convolution operator, forcing them to be identical. The algorithm solves this extrapolation problem by decomposing the measured data into eigenmodes of a compact integral operator built from the known forward kernel, then solving a coupled spectral problem for the two source quantities that determine the field. In the numerical illustration, the method recovers the field outside the measurement patch over a ten times larger region with about 7% relative L2 error. If the method scales to real measurements, it would remove a main practical limitation of magnetic moment estimation, which currently degrades when only a small noisy patch of field data is available.

What carries the argument

The machinery is a pair of compact integral operators on the measurement region: $K_{12}f = -\tfrac{1}{2} p_h \star_Q f$ (the Poisson kernel) and $K_3 f = -\tfrac{1}{2} \partial_h p_h \star_Q f$ (its height derivative), together with a finite-rank correction operator $S_J$ built from the eigenfunctions of $K_{12}$. Their coupled spectral problem is assembled into the self-adjoint matrix operator $L_J$, whose right-hand side in (2.9) is the measured data plus a remainder $r_J$ that tends to zero as the spectral truncation $J$ grows. The extrapolant is formed by evaluating the full-plane version of the same kernels against the recovered eigenmode components. The proof's load-bearing identity is that $K_{12}$ and $K_3$ are linked through $S_J$ by $\int\!\!\int_Q K_{12}(t-y)S_J(x,y)\,d^2y = K_3(t-x)+r_J(t,x)$, which makes the auxiliary operator invert the forward relation on $Q$ up to a controllable remainder.

What would settle it

Take the same four-rectangle magnetization and generate $B_3^{\mathrm{meas}}$ on $Q=[-1,1]^2$ with added noise at a realistic amplitude (say 1% of the signal), run the algorithm with $J=N=80$, and compare the extrapolation on $[-10,10]^2$ with the exact field; if the relative $L^2$ error fails to stay near the clean-data level or grows with noise amplitude, the central stability assumption is falsified.

Watch

Extended reading notes

Core claim

Proposition 1 states that for ideal data satisfying (1.1) the extrapolant is unique: any two magnetisations that produce the same vertical field on the measurement region $Q$ must produce the same field on all of $\mathbb{R}^2$. The proof rewrites (1.1) as $B_3 = -\tfrac{1}{2}(\partial_h p_h \star f_M)$ with $f_M = R_1M_1+R_2M_2+M_3$; if two sources differ, their difference produces a real-analytic field vanishing on an open set, hence vanishing everywhere, and positivity of the Fourier multiplier $\pi|k|e^{-2\pi h|k|}$ forces the source difference to be zero. The paper then proposes the double-spectral algorithm (steps (i)-(vi), equations (2.2)-(2.6)): compute the dominant eigenfunctions of the truncated Poisson kernel $K_{12}\star_Q$, build a correction operator $S_J$ from them, solve the two-by-two self-adjoint spectral problem (2.3) for the vertical component and the in-plane divergence, and form the extrapolant by applying the full-plane kernels to the resulting eigenmodes. The numerical section reports that with $J=N=80$ the extrapolant reproduces the true field on $[-10,10]^2$ with about 7% relative $L^2$ error. Stability under noise is not proven; truncation of the spectra is intended to serve as low-pass filtering, with analysis of the truncation parameters postponed.

Load-bearing premise

The load-bearing premise is that the algorithm remains accurate and stable when the measured field contains realistic noise, because the paper proves uniqueness only for ideal noiseless data, tests only clean synthetic data, and leaves the analysis of the spectral truncation parameters to future work.

Editorial extensions

If this is right

The vertical field on a finite patch determines the full-plane field uniquely, so any ambiguity in extrapolation comes from noise or truncation, not from the inverse problem itself.
Because the method uses only the vertical component and the known planar support, existing paleomagnetic scan protocols already supply the needed input.
Moment-estimation formulas that assume full-plane data can be applied to patch measurements after extrapolation, extending the usable region from $[-1,1]^2$ to at least $[-10,10]^2$ in the tested configuration.
The remainder term $r_J$ vanishes as $J$ grows, so the single-operator spectral approximation is a convergent step; the remaining open question is the quantitative choice of $J$ and $N$ for a given noise level.
Low-pass spectral truncation is the designed stabiliser, so the method is expected to admit noise-controlling variants once the truncation analysis is completed; the paper does not yet supply that analysis.

Reading between the lines

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

A direct next experiment is to contaminate the synthetic data with sensor noise and track relative $L^2$ error as a function of $J$ and $N$; absent such a test, the 7% clean-data figure should not be read as a noise robustness claim.
The same two-by-two spectral construction could be adapted to use $B_1$ or $B_2$ components instead of $B_3$, or to recover the two source quantities separately rather than their combined field; the paper does not explore these variants.
Because the proof identifies $f_M = R_1M_1+R_2M_2+M_3$ as the recoverable quantity, the method implicitly sets up a route to estimate the net magnetisation without enforcing unidirectionality; that step is not carried out here.
The real-analyticity argument suggests the extrapolant is stable only if high-frequency components are controlled; the paper's spectral truncation is the concrete mechanism for that control, and its limits define the practical domain of validity.

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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

The genuinely new thing here is the double-spectral construction: build an auxiliary operator S_J from eigenfunctions of the truncated Poisson kernel, then solve a matrix-operator eigenproblem whose eigenvectors give the extrapolant. I have not seen that in the cited literature, and it is a clever way to turn the geometry into an algebraic object. The paper also proves uniqueness of the extrapolant for ideal data (Proposition 1), and that proof is essentially sound. One nitpick: the text says the operator is positive, but its Fourier multiplier is only nonnegative and has a zero at k=0. For L^2 functions the conclusion is still correct, so it is a wording issue, not a mathematical gap.

The numerical illustration is honest but thin: one noiseless example, J=N=80, about 7% relative L2 error on a larger box. No code, no data, no baseline. That could be acceptable if the analytic part were rigorous, but it is not. The derivation of the algorithm is heuristic. The key remainder r_J is asserted to go to zero without a rate or a proof, and the step from a self-adjoint compact operator to the spectral truncation is sketched in a few lines. More importantly, the paper's stated motivation is noisy measurements, and the only experiment is noiseless. The stress-test concern about S_J amplifying noise is concrete and lands: the coefficients b_n contain S_J B_meas, and S_J includes 1/mu_j factors where the mu_j are eigenvalues of a compact smoothing kernel. Unless J is chosen very carefully, additive noise enters as S_J delta, and there is no bound on ||S_J||. The paper says low-pass filtering should stabilize things and defers the choice of J and N to a future paper. That is the load-bearing gap. It is not a contradiction, but it means the method's central promise is unverified.

The paper is candid about its limitations, and the author is not overselling. But a reader in paleomagnetism or inverse problems who wants to use this needs evidence that the algorithm survives noise and some idea of how to pick J and N.

Who is this for? Researchers working on inverse magnetisation from planar scans, particularly those who want to extend patch data before moment estimation. It is a useful sketch and a reasonable starting point for a fuller paper. It deserves a serious referee: the idea is new, the uniqueness result is correct, and the gaps are addressable by adding noisy experiments, deriving a perturbation bound, and justifying the truncation choice. I would send it to review, expecting major revision.

Referee Report

4 major / 5 minor

Summary. The paper studies a data extrapolation problem arising in the inverse magnetisation problem for a planar sample. Given measurements of the vertical component B3 on a bounded patch Q at height h, the aim is to extrapolate this field to the whole plane. The paper proves a uniqueness result (Proposition 1) for ideal noise-free data: any field satisfying the governing equation and matching the measured data on Q has a unique extension to R^2. It then proposes a 'double-spectral' algorithm, based on the spectral decomposition of an auxiliary self-adjoint matrix operator L_J that encodes the geometry and the support assumption, and illustrates the algorithm with a single numerical example using noiseless data, reporting a relative L2 error of about 7% on [-10,10]^2. The derivation of the algorithm relies on a remainder bound for a truncated projection r_J, which is stated without proof, and the stability of the method with respect to measurement noise is not analysed or tested numerically.

Significance. If the proposed extrapolation method is stable under noise, it would be a practically relevant tool for paleomagnetic data processing, since it addresses a bottleneck in estimating net magnetisation: the limited spatial extent and contamination of measured field maps. The uniqueness result for ideal data, while not surprising, is a useful clarification that the extrapolant is well-defined independently of the non-unique magnetisation distributions that produce the data. The algorithmic idea of reducing the extrapolation to a spectral problem for a self-adjoint matrix operator is novel and plausible. However, the paper's central claim that the method extrapolates realistic noisy measurements is not yet supported by the evidence: the key remainder estimate is unproved, no noisy experiments are reported, and no perturbation bounds are given. The paper is therefore an interesting announcement of a potential method, but it is not a complete justification of the method as presented.

major comments (4)

[Section 2, 'Towards justification'] The key assertion that ||r_J(·, x)|| -> 0 as J -> infinity uniformly for x in Q is stated without proof. This estimate is load-bearing: it is exactly what allows the replacement of the original system (2.8) by the spectral equation (2.9) for L_J. The eigenfunctions φ_j are those of K12⋆Q, not of K3⋆Q, so the uniform convergence of the projection of the continuous kernel K3(t-x) in this basis is not a standard consequence of Mercer's theorem. A proof, or at least a precise reference, is required before the derivation of the algorithm is valid.
[Section 2, Algorithm steps (iii) and (v)] The stability with respect to noise is not established. The operator S_J is defined through coefficients 1/μ_j, where μ_j are eigenvalues of the smoothing operator K12⋆Q; for a smooth positive definite kernel these eigenvalues decay rapidly, so ||S_J|| can be very large for moderate J. Additive noise δ in Bmeas enters the reconstruction coefficients b_n in (2.5) through ⟨S_J Bmeas, φ_n^3⟩, so noise is amplified by S_J without any bound on ||S_J||. The paper motivates the method by the presence of noise and states that low-pass filtering should stabilise it, but it provides no error estimates and the only numerical test (Section 3) uses noiseless data. A perturbation analysis or a noisy numerical experiment, together with a discussion of how to choose J and N, is essential to support the central claim that the method extrapolates contaminated measurement data.
[Section 3, Numerical illustration] The numerical evidence is very thin: a single test case with ideal data, with no details on the discretisation of the eigenvalue problems (ii) and (iv), no convergence study with respect to J and N, no noise contamination, and no comparison with a baseline. The reported 7% relative error is not enough to judge the practical utility of the method, especially because the motivating scenario involves measurement noise. At a minimum, the authors should report the dependence of the error on J and N, present a noisy-data experiment, and specify how the eigenfunctions are computed.
[Proposition 1] The statement of Proposition 1 is tautological as written: if the extrapolant must satisfy (1.1) on all of R^2 with the same fixed M1, M2, M3 that generated the data, then existence and uniqueness are immediate. The proof actually establishes the stronger and intended statement: any two magnetisations producing the same field on Q give the same f_M and hence the same field on all of R^2. This should be stated clearly in the proposition. In addition, the regularity assumption in the proposition (M1,M2 in L2) is inconsistent with the integration-by-parts step used later in (2.7), which requires W^{1,2} for M1,M2.

minor comments (5)

[Proof of Proposition 1] The Fourier multiplier π|k|exp(-2πh|k|) vanishes at k=0, so the operator is positive semidefinite, not strictly positive. The conclusion ~f_M - f_M = 0 follows because the difference lies in L2(R2); this nuance should be mentioned for precision.
[Equation (2.2)] The definition of S_J is given without derivation. A short explanation of how the kernel S_J is intended to approximate the inverse of K12⋆Q would help the reader follow the subsequent identity for ∫ K12(t-y)S_J(x,y)dy.
[Figures 3.1 and 3.2] The figures lack colorbars and axis labels for the plotted quantities. In particular, Figure 3.2 shows a 'pointwise error' but there is no indication of its magnitude scale or whether it is absolute or relative.
[Section 3] No information is given about the numerical implementation: the quadrature, the method for solving the eigenvalue problems, or the discretisation parameters. These details are necessary for reproducibility.
[Algorithm, step (ii)] The phrase 'J largest in modulus eigenvalues' is slightly confusing because the eigenvalues of K12⋆Q are negative; it would be clearer to say 'J eigenvalues of largest absolute value'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the extrapolant is validated against an independently generated ground-truth field, and the uniqueness proof and spectral construction are self-contained.

full rationale

Walking the derivation: Proposition 1 establishes uniqueness of B_ext^3 directly from real analyticity of the harmonic trace and positivity of the Fourier multiplier π|k|exp(-2πh|k|), without invoking the algorithm. The double-spectral construction (2.2)-(2.6) defines S_J and L_J from the known kernels K_12, K_3 and the domain Q; the coefficients b_n in (2.5) are inner products of the measured data with eigenfunctions, and the numerical test compares the resulting extrapolant with the true field produced from the known magnetization, so the outside-Q values are not forced to equal the input by construction. The approximation S_J K_12 ≈ K_3 is an analytic spectral-truncation statement, not a definition of the output. The only self-citation, [6], appears in a list of prior approaches and is not load-bearing. The unquantified effect of spectral truncation on noisy data is a genuine correctness/evidence gap, but it is not circularity: no fitted parameter is renamed as a prediction and no equation reduces to its own input.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central method rests on the stated physical model, the planar-support assumption, and several unproven convergence statements about spectral expansions. The truncation parameters J and N are chosen by hand without a selection rule.

free parameters (2)

J = 80 in numerical test
Number of eigenfunctions used to build the auxiliary operator S_J; no a priori selection rule given, chosen 'sufficiently large' and set to 80 in the experiments. The paper defers analysis of the choice to future work.
N = 80 in numerical test
Number of eigenpairs of L_J retained in the extrapolation series; no error bound or selection criterion provided, set to 80 in the experiments.

assumptions (4)

domain assumption The forward model (1.1) exactly describes the vertical magnetic field produced by a planar magnetization distribution.
Taken from the physics of magnetostatics, cited to [2, Sec 3.2]; the method inherits any error in this model.
domain assumption The support of the magnetization is contained in the planar region Q which is also the measurement region.
The whole extrapolation scheme and the integration domains in (2.7)-(2.9) use supp M ⊆ Q; if the sample extends outside the measurement patch or outside the plane, the method's premises fail.
ad hoc to paper The kernel K12 (i.e., -1/2 p_h) induces a compact self-adjoint positive operator on L^2(Q) with complete eigenbasis, and the spectral expansion of K3 in this basis converges uniformly as asserted.
The algorithm's justification in Section 2 relies on the identity (K12⋆Q S_J)(x,t)=K3(t-x)+r_J(t,x) with r_J→0; the paper states this without a full proof and defers details.
standard math B3(x,h) is real-analytic on R2 as the trace of a harmonic function, so an open-set vanishing implies global vanishing.
Used in the proof of Proposition 1.

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Cite this review

Pith. "Pith review of A Method to Extrapolate the Data for the Inverse Magnetisation Problem with a Planar Sample." pith.science (2026). https://pith.science/paper/LSPK6P3E

@misc{pith2026241109331,
  author       = {Pith},
  title        = {Pith review of: A Method to Extrapolate the Data for the Inverse Magnetisation Problem with a Planar Sample},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LSPK6P3E}},
  note         = {Machine review of arXiv:2411.09331}
}

abstract

A particular instance of the inverse magnetisation problem is considered. It is assumed that the support of a magnetic sample (a source term in the Poisson equation in $\mathbb{R}^3$) is contained in a bounded planar set parallel to the measurement plane. Moreover, only one component of the magnetic field is assumed to be known (measured) over the same planar region in the measurement plane. We propose a method to extrapolate the measurement data to the whole plane relying on the knowledge of the forward operator and the geometry of the problem. The method is based on the spectral decomposition of an auxiliary matrix-function operator. The results are illustrated numerically.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

, Lima, E., Marmorat, J

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A., Weiss, B

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A., Weiss, B

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arXiv:220 5.14776, 31pp (2022)

Ponomarev, D.: Magnetisation moment of a bounded 3D samp le: asymptotic recovery from planar mea- surements on a large disk using Fourier analysis. arXiv:220 5.14776, 31pp (2022)

work page 2022

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

, Lima, E., Marmorat, J

Baratchart, L., Chevillard, S., Hardin, D., Leblond, J. , Lima, E., Marmorat, J. P .: Magnetic moment estima- tion and bounded extremal problems. Inverse Problems and Im aging 13 (1), 39–67 (2019)

work page 2019

[2] [2]

Silent and e quivalent magnetic distributions on thin plates

Baratchart, L., Chevillard, S., Leblond, J. Silent and e quivalent magnetic distributions on thin plates. Har- monic Analysis, Function Theory, Operator Theory, and thei r Applications 22, 17pp. (2018)

work page 2018

[3] [3]

P ., Lima, E

Baratchart, L., Hardin, D. P ., Lima, E. A., Saff, E. B., We iss, B. P .: Characterizing kernels of operators related to thin-plate magnetizations via generalizations of Hodge decompositions. Inverse Problems 29 (1), 29pp (2013)

work page 2013

[4] [4]

A., Weiss, B

Lima, E. A., Weiss, B. P ., Baratchart, L., Hardin, D. P ., Saff, E. B.: Fast inversion of magnetic ﬁeld maps of unidirectional planar geological magnetization. J. Geoph ys. Res.: Solid Earth 118 (6), 2723–2752 (2013)

work page 2013

[5] [5]

A., Weiss, B

Lima, E. A., Weiss, B. P ., Borlina, C. S., Baratchart, L., Hardin, D. P .: Estimating the net magnetic moment of geological samples from planar ﬁeld maps using multipole s. Geochemistry, Geophysics, Geosystems 24, 38pp (2023)

work page 2023

[6] [6]

arXiv:220 5.14776, 31pp (2022)

Ponomarev, D.: Magnetisation moment of a bounded 3D samp le: asymptotic recovery from planar mea- surements on a large disk using Fourier analysis. arXiv:220 5.14776, 31pp (2022)

work page 2022