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Global versus local internal-external field separation on the sphere: a Hardy-Hodge perspective

X. Huang, C. Gerhards, Z. Ren

Abstract

Internal-external field separation is crucial for many aspects of geomagnetism, aiming at distinguishing contributions of the magnetic field generated within the Earth (or any other planet) from those produced in the exterior. When data is available on a full spherical observation surface, this separation is a standard, stable, and widely used procedure dating back to Gauss. However, when data is only available in a subdomain of the observation surface (as is the case for aeromagnetic and ground-based surveys), the situation changes. Here we show that, without prior assumptions, an internal-external field separation is not uniquely possible. Given the geophysically reasonable assumption that the exterior sources (e.g., ionospheric current systems) are located above a source-free spherical shell, we show that a unique separation becomes possible but that it is highly unstable. The results are based on a Hardy-Hodge decomposition of spherical vector fields and provide an explanation of the intrinsic difficulties of regional data-based internal-external field separation.

Global versus local internal-external field separation on the sphere: a Hardy-Hodge perspective

Abstract

Internal-external field separation is crucial for many aspects of geomagnetism, aiming at distinguishing contributions of the magnetic field generated within the Earth (or any other planet) from those produced in the exterior. When data is available on a full spherical observation surface, this separation is a standard, stable, and widely used procedure dating back to Gauss. However, when data is only available in a subdomain of the observation surface (as is the case for aeromagnetic and ground-based surveys), the situation changes. Here we show that, without prior assumptions, an internal-external field separation is not uniquely possible. Given the geophysically reasonable assumption that the exterior sources (e.g., ionospheric current systems) are located above a source-free spherical shell, we show that a unique separation becomes possible but that it is highly unstable. The results are based on a Hardy-Hodge decomposition of spherical vector fields and provide an explanation of the intrinsic difficulties of regional data-based internal-external field separation.
Paper Structure (19 sections, 8 theorems, 46 equations)

This paper contains 19 sections, 8 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Convention (sources vs. domains).
  3. Hardy--Hodge decomposition as a general framework for internal--external source separation
  4. Hardy--Hodge decomposition on closed Lipschitz surfaces
  5. The $B$-operators and Hardy spaces.
  6. Potential-theoretic interpretation (physical meaning).
  7. The divergence-free component.
  8. Hardy--Hodge decomposition as a topological direct sum (global stability).
  9. Spherical Hardy--Hodge decomposition and (Gauss) vector spherical harmonics
  10. (Gauss) internal/external poloidal fields.
  11. Connection to the $B$-operators.
  12. Identification of the spherical Hardy spaces.
  13. Nonuniqueness and uniqueness for patches
  14. Localized separation is non-unique in $$general
  15. An altitude/analyticity condition restores uniqueness
  16. ...and 4 more sections

Key Result

Theorem 2

The restriction operator has the nontrivial null space with $\tilde{H}^1(U^c)=\{f\in H^1(\mathbb{S}):f=\textnormal{const. on }\, U\}$ and $L^2(U^c)=\{f\in L^2(\mathbb{S}): f=0 \textnormal{ on }\, U\}$.

Theorems & Definitions (21)

  • Remark 1: Zero degree case
  • Theorem 2: Non-uniqueness for patch data
  • proof
  • Remark 3: Geophysical interpretation
  • Definition 4: External-source altitude class
  • Theorem 5: Uniqueness for patch data under an altitude/analyticity constraint
  • proof
  • Remark 6
  • Corollary 7: Bandlimited external-source fields
  • Remark 8: Geophysical interpretation
  • ...and 11 more