Depth to magnetic source estimation using TDX contour

Hammed Oyekan

REVIEW 3 major objections 6 minor 17 references

Depth to magnetic source estimation using TDX contour

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read On a TDX magnetic map, the depth to a vertical contact equals half the distance between the 45-degree contours flanking the 90-degree edge contour.

desk verdict Correct but not new: the TDX contour rule is the tilt-depth method in disguise, and the Bishop validation is visual only. read the letter →

arxiv 2505.22780 v1 pith:DYLJH2GO submitted 2025-05-28 physics.geo-ph

Hammed Oyekan This is my paper

classification physics.geo-ph MSC 86A25

keywords TDXderivativemagneticdepthestimationphasetiltBishopmodelpotentialfieldedgedetectionverticalcontactassumptionbasement

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 proposes a fast, visual depth-estimation rule for magnetic data: after computing the TDX phase derivative, the contour where TDX equals 45 degrees lies one source-depth away from the source edge, which sits at the 90-degree contour. By measuring half the separation between the two 45-degree contours that flank an edge, an interpreter obtains the depth to the top of a vertical magnetic contact without inversion or modelling. The paper derives this from Nabighian's analytic expressions for a two-dimensional contact under vertical contact and vertical magnetization, reducing TDX to $\mathrm{TDX} = \tan^{-1}(z/h)$. Tests on a two-prism model with known depths of 4 km and 8 km reproduce those depths from the measured contour separations, and the method is applied to the 3D Bishop basement model with a claim of agreement with known depth.

What carries the argument

The central object is the TDX derivative, $\mathrm{TDX} = \tan^{-1}(\mathrm{THDR}/|\mathrm{VDR}|)$, a phase derivative that confines values to $0$--$90^\circ$ and peaks at source edges. Combined with Nabighian's 2D contact expressions under vertical contact ($d=90^\circ$) and vertical magnetization ($A=0$), it reduces to $\mathrm{TDX} = \tan^{-1}(z/h)$. This identity is what carries the argument: it maps contour values to the ratio of depth to horizontal distance from the edge, so the $45^\circ$ contour traces points one depth away from the edge, and half the $45^\circ$-to-$45^\circ$ contour spacing gives the depth.

What would settle it

Measure depth estimates from the 45-degree contour spacing on a synthetic model with known dipping contacts or non-vertical magnetization: if the estimated depth deviates systematically from the true depth as dip or magnetization direction departs from vertical, the universal rule fails. On the Bishop model, compute the RMS difference between the estimated depths along the edges and the true basement depths; a large mismatch where fault dips deviate from vertical would show the method does not survive realistic complexity.

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Extended reading notes

Core claim

The central discovery is the geometric equivalence $\mathrm{TDX} = \tan^{-1}(z/h)$ for a vertical contact under vertical magnetization, which turns the TDX contour map into a direct depth scale. Because TDX reaches $90^\circ$ at the edge ($h=0$) and reaches $45^\circ$ at horizontal distance $z$ from the edge, the separation between the two $45^\circ$ contours bracketing an edge is $2z$; half that distance is the depth. This gives a one-map estimator: locate edges at the $90^\circ$ contour and read depth as half the $45^\circ$-contour gap. The author demonstrates the rule on a two-prism model, where measured separations of about 8 km and 16 km match the known 4 km and 8 km depths, and argues the Bishop model results agree with the known basement depth despite dipping faults.

Load-bearing premise

The load-bearing premise is that each magnetic source edge behaves as a vertical contact with vertical magnetization, so the simplified relation $\mathrm{TDX} = \tan^{-1}(z/h)$ holds; the Bishop model's dipping faults violate this in places, and the paper's agreement claim rests on visual inspection rather than a quantitative error metric.

Editorial extensions

If this is right

On a TDX map, source edges and depths are visible simultaneously: the 90-degree contour marks the edge, and the flanking 45-degree contours supply depth.
Depth can be estimated directly by measuring contour separations, with no inversion, forward modelling, or assumption about magnetization strength.
Because TDX is normalized by the absolute vertical derivative, it is independent of magnetization amplitude, so weak and strong anomalies can be treated in the same map.
In the two-prism test, average 45-to-45 degree distances of about 8 km and 16 km correspond to the known depths of 4 km and 8 km, demonstrating the rule on ideal vertical-sided bodies.
The method extends from isolated prisms to a complex 3D basement, the Bishop model, where the author reports agreement with known depth despite fault structures.

Reading between the lines

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

This is essentially a contour-based variant of the tilt-depth idea; one could automate it by extracting contour lines and computing local half-widths, turning it into a fast reconnaissance depth-to-basement tool for regional magnetic surveys.
The depth rule is sensitive to the vertical-contact and RTP assumptions; a natural extension is to derive corrections for dip and magnetization direction from Nabighian's full expressions, which would quantify when the simple rule is reliable.
The Bishop-model test as reported lacks a quantitative error metric; computing an RMS difference between estimated and true depths would settle whether the method survives realistic non-vertical fault geometries or only works on ideal contacts.
Because TDX is related to TDR by $\mathrm{TDX} = 90^\circ - |\mathrm{TDR}|$, the 45-degree contour condition is the same as $|\mathrm{TDR}| = 45^\circ$, so the method's strengths and weaknesses relative to the established tilt-depth method could be compared directly on the same models.

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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

Quick take: the math is right, but the paper is not actually new. Equation (1) follows correctly from Nabighian under a vertical contact and RTP magnetization, and the half-distance rule reproduces the known depths on the two-prism test. But the paper itself states TDX = π/2 − |TDR|, which makes the 45° contour spacing the same information as the tilt-depth method of Salem et al. (2007) — cited but not credited as equivalent. The contribution is a relabeling, not a new capability.

What it does well: the derivation is clean, the two-prism example is quantitative (half the contour spacing gives 4 km and 8 km as expected), and the method is parameter-free and easy to apply by eye.

The soft spots are two. First, the Bishop model validation is asserted, not measured. The text says the results 'agree with the known depth' but gives no error statistic, no cross-section comparison, and no discussion of how the variable-dip 3D structure violates the vertical-contact assumption. Since the practical claim is that the rule works beyond idealized geometry, and the two-prism test cannot carry that weight, this is the load-bearing gap. Second, the theory section contains a garbled identity, TDX = tan⁻¹(tan⁻¹|TDR|), before the correct form appears; that needs fixing. The contour distance is also measured manually, which is fine for a demonstration but limits the reliability claim.

The citation pattern is fair but revealing: Salem et al. (2007) is listed, yet the paper doesn't flag that the two approaches are mathematically identical. That is a framing issue more than a scholarly integrity problem, but it dooms the novelty argument.

Who should read this: interpreters who want a quick visual depth estimate from TDX maps, and anyone interested in the TDR/TDX duality. It is a useful reminder, not a research advance.

Recommendation: send to peer review if the venue takes short methodological notes; the derivation is correct and the synthetic test is reproducible. The authors should be asked to acknowledge equivalence with tilt-depth and to add quantitative error analysis on the Bishop model. If the venue demands novelty, a desk reject is defensible.

Referee Report

3 major / 6 minor

Summary. The paper proposes using contours of the TDX derivative (the arctangent of the ratio of the total horizontal derivative to the absolute vertical derivative) to estimate the depth of magnetic sources. From Nabighian's two-dimensional contact formulas and assuming a vertical contact with vertical magnetization (RTP), the authors derive TDX = tan^-1(z/h) (Eq. 1), from which they infer that the depth equals half the distance between the 45-degree TDX contours on either side of the 90-degree contour. The method is tested on a two-prism synthetic model with known depths of 4 km and 8 km, and on the three-dimensional Bishop basement model, with the claim that the results agree with the known depths. The paper concludes that the contour-spacing rule is a fast, parameter-free depth estimator.

Significance. If the depth rule holds beyond idealized contacts, the method would be a simple, fast, and parameter-free visual tool for depth estimation along magnetic edges, requiring no inversion or fitting. The central derivation is sound: Eq. (1) follows correctly from Nabighian's expressions when the dip is 90 degrees and the magnetization is vertical, and the two-prism test quantitatively supports the rule (half the measured 8 km and 16 km contour spacings reproduces the known 4 km and 8 km depths). However, the method is algebraically identical to the published tilt-depth method of Salem et al. (2007), because TDX = pi/2 - |TDR|, so the conceptual novelty is limited. The practical significance hinges on the unquantified Bishop-model test, which is the only evidence for behavior when the assumptions are violated.

major comments (3)

[The Bishop Model; Conclusions] The central robustness claim, that the method 'agrees with the known depth' on the Bishop model despite violation of the vertical-contact and RTP assumptions, is supported only by a visual comparison of Figures 3e and 3f. No quantitative error metric (e.g., mean absolute error, correlation, depth-difference histogram, or scatter plot of estimated versus true depth) is reported. Because the Bishop test is the only evidence for performance when the assumptions of Eq. (1) are relaxed, this is a load-bearing omission that must be fixed before the reliability claim can be assessed.
[The Bishop Model] The algorithm that produced the depth estimates in Figure 3e is not described. The text states 'By differentiating equation 1 above with respect to h, we can relate the total horizontal derivative of the TDX to the horizontal derivative of the tilt. As such, the depth along the edge of the body can be estimated.' This is unclear: it does not explain how the half-distance contour rule is applied on a grid, how contour distances are measured, how interference between neighboring sources is handled, or what role differentiation plays. Without a reproducible description, the Bishop result cannot be independently checked.
[Theory] The paper presents the depth rule as a new development, but the method is algebraically identical to the published tilt-depth method: since TDX = pi/2 - |TDR|, the 45-degree TDX contours coincide with the +/-45-degree tilt contours whose half-separation is the standard tilt-depth estimator of Salem et al. (2007). The tilt-depth paper is included in the reference list but is not cited when the depth rule is introduced in the Theory section or elsewhere. The authors must acknowledge this equivalence and reposition the contribution as a variant or application of tilt-depth, or provide a demonstrable algorithmic difference.

minor comments (6)

[Theory] The identity 'TDX = tan^-1(tan^-1|TDR|)' is dimensionally inconsistent and is contradicted two sentences later by the correct 'TDX is effectively pi/2 - |TDR|'. The garbled identity should be removed or corrected.
[Theory] The reduction of the Nabighian expressions to Eq. (1) is asserted but not shown. Include the intermediate algebra for d = 90 degrees and A = 0 so readers can verify the cancellation of the magnetization term.
[Figure 1 caption] The caption contains a typo: 'prim blocks' should be 'prism blocks'.
[Figures 2g and 3e] The depth maps in Figures 2g and 3e lack defined color scales or units in the text; specify the color mapping and the depth units for quantitative interpretation.
[Introduction] The phrase 'Wenner method' is mentioned without a citation; either provide a reference or remove the name.
[Conclusions] The Conclusions repeat the Summary nearly verbatim; they should state the specific limitations of the method (two-dimensional vertical-contact assumption, RTP requirement, and the need for quantitative validation on realistic 3D models).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the TDX 45-degree-contour depth rule is a parameter-free consequence of Nabighian's contact equations, with only non-circular caveats around equivalence to tilt-depth and missing quantitative Bishop-model validation.

full rationale

The derivation chain is self-contained and non-circular. The paper starts from the standard definition TDX = arctan(THDR/|VDR|) (Cooper & Cowan 2006) and Nabighian's (1972) analytical expressions for dT/dh and dT/dz over a sloping contact. Substituting the stated assumptions d=90 degrees and A=0 (vertical contact and RTP) into Nabighian's expressions gives THDR proportional to z and |VDR| proportional to h, which immediately yields TDX = arctan(z/h); the 45-degree-contour depth rule follows algebraically from that derived relation. No parameter is fitted to the synthetic blocks or to the Bishop model, and the known depths are not used to calibrate the formula, so the validation is not a fitted-input-called-prediction situation. The two-prism test has vertical-sided prisms and vertical magnetization, which matches the derivation assumptions; this is a legitimate consistency test, not a circular reduction, because the depths are outputs rather than inputs. The Bishop-model validation is asserted visually as 'the results agree with the known depth' without a quantitative error metric, and the model contains non-vertical en echelon faults that violate the derivation's assumption, but this is a validation weakness and not circularity. The paper also states 'TDX is effectively pi/2 - |TDR|', which makes the proposed TDX contour rule algebraically identical to the tilt-depth method of Salem et al. (2007); that is a duplication/novelty concern rather than a circular reduction to the paper's own inputs. The garbled intermediate identity 'TDX = tan^-1(tan^-1|TDR|)' is a typographical error, not a load-bearing circular step. Overall, no load-bearing step reduces to its own input by construction, so the score is 1 rather than 0 only to register the non-circular caveats about duplication and missing validation metrics.

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

The method has no free parameters: Eq. (1) is a parameter-free consequence of Nabighian's standard 2D contact expressions once dip and magnetization direction are fixed. The derivation depends on the vertical contact and vertical magnetization (RTP) assumptions, plus the 2D infinite-strike geometry of the starting formulas; the paper itself calls the contact assumption restrictive. Validation is against synthetic models with known depths, so no constant is fit to the targets. No invented entities are introduced.

assumptions (4)

domain assumption Nabighian (1972) analytic expressions for vertical and horizontal derivatives over a sloping 2D magnetic contact
Adopted as the starting point in the Theory section; the derivation of Eq. (1) depends on these expressions being valid for the modeled sources.
domain assumption Vertical contact: dip d = 90 degrees
Substituted into Nabighian's expressions to obtain TDX = tan^-1(z/h); the paper itself calls the assumption restrictive. Violated by dipping faults in the Bishop model.
domain assumption Vertical magnetization with RTP (A = 0, I = 90 degrees)
Required together with the vertical dip to make the angle term (2I - d - 90) = 0 in the Nabighian expressions; real surveys usually have nonvertical magnetization, so RTP processing is assumed.
domain assumption Two-dimensional, infinitely striking contact geometry
Nabighian's expressions and Eq. (1) are 2D; the method is applied to a 3D basement model without an explicit correction for 3D geometry.

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Cite this review

Pith. "Pith review of Depth to magnetic source estimation using TDX contour." pith.science (2026). https://pith.science/paper/DYLJH2GO

@misc{pith2026250522780,
  author       = {Pith},
  title        = {Pith review of: Depth to magnetic source estimation using TDX contour},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DYLJH2GO}},
  note         = {Machine review of arXiv:2505.22780}
}

read the original abstract

Accurate depth estimation of magnetic sources plays a crucial role in various geophysical applications, including mineral exploration, resource assessments, regional hydrocarbon exploration, and geological mapping. Thus, this abstract presents a fast and simple method of estimating the depth of a magnetic body using the TDX derivative of the total magnetic field. TDX is a first-order derivative of the magnetic field that, in addition to edge detection, is less affected by noise, allowing for better depth resolution. The reduced sensitivity to noise enables a clearer estimation of depth and enhances the accuracy of the depth determination process. The TDX, as a variant of the phase derivative, is independent of magnetization and can be used to identify the edge of a magnetic body. In addition to excelling at edge detection, they can also estimate the depth of the magnetic source producing the anomalies. In this study, we explore the utilization of contour of the TDX derivative for estimating depth, assuming a vertical contact source. We demonstrate the effectiveness of the method using a two-prism block model and a simple bishop model with a uniform susceptibility of 0.001 cgs. The results agree with the known depth, providing evidence of the reliability of the method despite the restrictive nature of the assumption, especially for the Bishop model, where there are numerous fault structures.

Figures

Figures reproduced from arXiv: 2505.22780 by the authors.

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

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[2]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Cooper, G.R.J., and D.R. Cowan. 2006, Enhancing potential field data using filters based on the local phase: Computers & Geosciences 32 (10), 1585–91. https://doi.org/10.1016/j.cageo.2006.02.016

work page doi:10.1016/j.cageo.2006.02.016 2006

[2] [2]

Fairhead, J. D., S. E. Williams, and G. Flanagan. 2004 , Testing magnetic local wavenumber depth estimation methods using a complex 3 D test model : In SEG Technical Program Expanded Abstracts 2004, 742–45. Society of Exploration Geophysicists. https://doi.org/10.1190/1.1851313

work page doi:10.1190/1.1851313 2004

[3] [3]

Fairhead, J. Derek. 2015, Advances in gravity and magnetic processing and interpretation: Houten, EAGE

work page 2015

[4] [4]

Nabighian, Misac N. 1972 , The analytic signal of two‐ dimensional magnetic bodies with polygonal cross‐ section: its properties and use for automated anomaly interpretation, Geophysics 37 (3): 507 –17. https://doi.org/10.1190/1.1440276

work page doi:10.1190/1.1440276 1972

[5] [5]

1971, Automatic determination of depth on aeromagnetic profiles : Geophysics 36 (4), 717–22

Naudy, Henri. 1971, Automatic determination of depth on aeromagnetic profiles : Geophysics 36 (4), 717–22. https://doi.org/10.1190/1.1440207

work page doi:10.1190/1.1440207 1971

[6] [6]

Nettleton, L. L. 1940, Geophysical Prospecting for Oil, 1st ed.: McGraw-Hill book company Inc., New York

work page 1940

[7] [7]

1949 , The direct approach to magnetic interpretation and its practical application: Geophysics 14 (3), 290–320

Peters, Leo J. 1949 , The direct approach to magnetic interpretation and its practical application: Geophysics 14 (3), 290–320. https://doi.org/10.1190/1.1437537

work page doi:10.1190/1.1437537 1949

[8] [8]

Reid, A. B., J. M. Allsop, H. Granser, A. J. Millett, and I. W. Somerton. 1990 , Magnetic interpretation in three dimensions using euler deconvolution: Geophysics 55 (1), 80–91. https://doi.org/10.1190/1.1442774

work page doi:10.1190/1.1442774 1990