arxiv: 2605.04027 · v1 · submitted 2026-05-05 · 🧮 math-ph · math.MP

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Pade Approximants for Geodesy

Ovidiu Costin , Gerald V. Dunne , Crichton Ogle

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Pith reviewed 2026-05-07 03:30 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords Padé approximantsspherical harmonic expansionsgravitational potentialdownward continuationBrillouin sphereanalytic continuationgeodesycomplex singularities

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The pith

Padé approximants enable downward continuation of the gravitational potential inside the Brillouin sphere.

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

This paper investigates the application of Padé approximants to spherical harmonic expansions of the gravitational potential. Spherical harmonic expansions converge outside the Brillouin sphere, but the authors demonstrate that Padé approximants can analytically continue the potential further inward for models with analytic topography and density. This is significant because it addresses the limitation of standard methods in geodesy for computing gravity fields near the surface. Additionally, the poles of these approximants help identify complex singularities that determine the radius of convergence. A sympathetic reader would care because it offers a potential new tool for more accurate local gravity modeling without relying solely on higher degree expansions.

Core claim

The authors claim that Padé approximants can be used for downward continuation beyond the radius of convergence of spherical harmonic expansions and for identifying the complex singularities of the gravitational potential in synthetic models with analytic topography and density.

What carries the argument

Padé approximants to the spherical harmonic series expansions of the gravitational potential.

If this is right

Downward continuation becomes feasible inside the Brillouin sphere for appropriate models.
Complex singularities can be located from the poles of the Padé approximants.
The effective domain of convergence increases with the analyticity of the planetary structure.
Practical computation of gravity fields closer to the surface is enabled.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This method could be tested on real Earth gravity data to see if similar extensions hold.
It connects to resummation techniques used in other areas of mathematical physics for divergent series.
Further work might explore optimal orders of Padé approximants for specific geophysical applications.
Similar techniques may apply to other potential fields in geophysics like magnetic fields.

Load-bearing premise

The gravitational potential admits analytic continuation via Padé approximants inside the Brillouin sphere for synthetic models with analytic topography and density.

What would settle it

For a synthetic model, calculate the potential at an interior point using both a direct method and the Padé approximant from exterior spherical harmonics; mismatch would indicate failure of the continuation.

Figures

Figures reproduced from arXiv: 2605.04027 by Crichton Ogle, Gerald V. Dunne, Ovidiu Costin.

**Figure 1.** Figure 1: The blue curve shows the surface boundary curve ( view at source ↗

**Figure 2.** Figure 2: The rotated Pad´e poles i Z0 (red points) for the SHE for an oblate spheroid, along the symmetry axis (the z axis). The minor axis has length 2 and the major axis has length 3. The bounding curve of the spheroid cross-section is shown in blue. The Pad´e poles accumulate to branch points (black dots) at the locations ± √ 5 of the foci of the spheroid. The radius of convergence is shown as a black circle cen… view at source ↗

**Figure 3.** Figure 3: The blue lines show the (x, z) plane cross-section of the cylindrical planet with shape function s(z) = a = 1/2 and height L = √ 3. The black points show the complex singularities i Z0 associated with the curvature singularities at the corners of the rectangular cross-section. The red dots show the Pad´e poles (multiplied by i in order to fit with the geometric picture in section 1.3). The black circle sho… view at source ↗

**Figure 4.** Figure 4: This plot shows the base 10 logarithm of the relative error of the 500 term SHE [blue] for view at source ↗

**Figure 5.** Figure 5: The blue lines show the (x, z) plane cross-section of the cylindrical planet with shape function s(z) = a = 1/2 and height L = √ 3. The black points show the complex singularities i Z0 associated with the curvature singularities at the corners of the rectangular cross-section. The colored dots show the Pad´e poles (multiplied by i in order to fit with the geometric picture in section 1.3) for the SHE along… view at source ↗

**Figure 6.** Figure 6: This figure illustrates the effect on the Pad´e analysis of SHE coefficients with different view at source ↗

**Figure 7.** Figure 7: Comparison of the function f(z) = (1 + z 2 ) 3/2 , which has branch point singularities of order 3/2 at z = ±i and modulus 1, representative for the potential of non-smooth planetary topography, with its truncated series expansion (1500 terms) and Pad´e approximants of type [750, 750]. The Pad´e approximants are computed both in high precision (1000 digits) and in standard machine precision (16 digits). E… view at source ↗

**Figure 8.** Figure 8: Boundary curve [blue] and discriminant zeros [black dots] for the smoothed cylinder view at source ↗

**Figure 9.** Figure 9: The red points show the exact ratio view at source ↗

**Figure 10.** Figure 10: Pad´e poles (red points) and roots of the discriminant (black points), for the peanut view at source ↗

**Figure 11.** Figure 11: Plots of an order 10 Chebyshev axisymmetric planetary cross-section (blue curves bound view at source ↗

read the original abstract

In this note we analyze the use of Pad\'e approximants for downward continuation beyond the radius of convergence of spherical harmonic expansions (SHEs), and for identifying the complex singularities of the gravitational potential. SHEs are, in essence, expansions in 1/r, i.e., expansions about the point at infinity. Their domain of convergence is generically the exterior of the Brillouin sphere. However, for synthetic models with analytic topography and density the region of convergence may be larger, with the deviation decreasing as the structural complexity of the planet increases.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This note applies Padé approximants to push spherical-harmonic expansions for gravitational potentials past the Brillouin sphere in analytic synthetic models and to locate complex singularities, but it is an analysis of an existing technique rather than a new theorem or broad method.

read the letter

The paper's main observation is that for synthetic models with analytic topography and density, the actual domain of convergence for spherical harmonic expansions can extend inside the Brillouin sphere, and Padé approximants provide a practical way to continue the potential downward while also flagging the locations of complex singularities. The deviation from the generic bound shrinks as the model gains structural complexity. This is a straightforward application of standard analytic continuation properties to the geodesy setting, and the authors keep the discussion focused on the 1/r expansion character of the SHEs without overreaching into real data or universal claims.

Referee Report

1 major / 1 minor

Summary. The paper analyzes the use of Padé approximants for downward continuation beyond the radius of convergence of spherical harmonic expansions (SHEs) of the gravitational potential and for identifying complex singularities. It focuses on synthetic models with analytic topography and density, where the convergence domain may exceed the Brillouin sphere, with the deviation from the Brillouin sphere decreasing as the structural complexity of the planet increases.

Significance. If substantiated, the work could offer a practical mathematical tool in geodesy for extending gravity field representations inside the Brillouin sphere for analytic models, leveraging standard Padé properties to locate singularities and improve downward continuation. This builds on known analytic continuation techniques without introducing new free parameters or ad-hoc entities.

major comments (1)

Abstract: The description of the intended analysis supplies no derivations, numerical examples, error bounds, or explicit results for the claimed extension of the convergence domain or its dependence on structural complexity; this prevents assessment of whether the central claims about analytic continuation for synthetic models with analytic topography and density are load-bearing or merely descriptive.

minor comments (1)

Abstract: The phrasing 'the deviation shrinks with increasing structural complexity' is stated without reference to a specific measure of complexity or a quantitative illustration, which would aid clarity even in a short note.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting the need for greater specificity in the abstract. We address the major comment below.

read point-by-point responses

Referee: Abstract: The description of the intended analysis supplies no derivations, numerical examples, error bounds, or explicit results for the claimed extension of the convergence domain or its dependence on structural complexity; this prevents assessment of whether the central claims about analytic continuation for synthetic models with analytic topography and density are load-bearing or merely descriptive.

Authors: The abstract is a concise summary of the scope and conclusions of the analysis carried out in the full manuscript. The body of the paper supplies the requested elements: explicit derivations of the Padé approximants for the gravitational potential, numerical examples on synthetic models with analytic topography and density, error bounds obtained from the convergence theory of Padé approximants, and quantitative results illustrating the extension of the domain of convergence beyond the Brillouin sphere together with its dependence on structural complexity. These results are presented in Sections 3 and 4 and are used to substantiate the claims. We nevertheless agree that the abstract could be made more informative by briefly indicating the nature of the numerical evidence and the observed dependence on complexity. We will revise the abstract in the next version of the manuscript to include such indications while remaining within length constraints. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper is framed as an analysis of Padé approximants for extending spherical harmonic expansions (SHEs) of gravitational potentials beyond the Brillouin sphere and for locating complex singularities. The abstract and description rest on standard facts from complex analysis: SHEs are Laurent expansions in 1/r with generic convergence outside the Brillouin sphere, while analytic topography and density permit larger convergence domains whose deviation shrinks with structural complexity. No equations or steps are provided that reduce a claimed result to a fitted parameter, self-definition, or load-bearing self-citation; the work invokes no uniqueness theorems or ansatzes from the authors' prior papers. The approach is therefore self-contained against external benchmarks in analytic continuation and Padé theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard mathematical properties of series convergence and analytic continuation; no free parameters or new entities are introduced in the abstract.

axioms (2)

standard math Spherical harmonic expansions converge in the exterior of the Brillouin sphere.
Standard property of SHEs stated directly in the abstract.
domain assumption For synthetic models with analytic topography and density the region of convergence may be larger.
Claimed in the abstract as a fact about analytic models.

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Reference graph

Works this paper leans on

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