[Paper Review] Locally Supported Wavelets for the Separation of Spherical Vector Fields with Respect to their Sources

Christian Gerhards|arXiv (Cornell University)|Jun 3, 2015

Geophysical and Geoelectrical Methods32 references19 citations

TL;DR

This paper introduces a space-domain multiscale method for separating spherical vector magnetic fields into internal and external source contributions using locally supported wavelets. By regularizing convolution kernels derived from spherical vector harmonics and Green's functions, the approach enables localized, multiscale decomposition of CHAMP satellite data, revealing high-resolution crustal field anomalies while filtering out ionospheric and magnetospheric signals with improved spatial localization compared to traditional spherical harmonic methods.

ABSTRACT

We provide a space domain oriented separation of magnetic fields into parts generated by sources in the exterior and sources in the interior of a given sphere. The separation itself is well-known in geomagnetic modeling, usually in terms of a spherical harmonic analysis or a wavelet analysis that is spherical harmonic based. In contrast to these frequency oriented methods, we use a more spatially oriented approach in this paper. We derive integral representations with explicitly known convolution kernels. Regularizing these singular kernels allows a multiscale representation of the internal and external contributions to the magnetic field with locally supported wavelets. This representation is applied to a set of CHAMP data for crustal field modeling.

Motivation & Objective

• Address the limitation of global spherical harmonics in modeling localized crustal magnetic fields.
• Develop a space-domain alternative to frequency-oriented wavelet methods for magnetic field separation.
• Enable multiscale decomposition of satellite magnetic data with localized spatial support for improved resolution of crustal anomalies.
• Filter out non-crustal contributions (e.g., ionospheric, ring current) from CHAMP data using source-based separation.
• Provide a numerically stable, multiscale representation using regularized convolution kernels in the space domain.

Proposed method

• Derive integral representations of internal and external magnetic field components using vector spherical harmonics and surface operators.
• Construct regularization of singular convolution kernels (Green's functions and single-layer kernels) to enable multiscale wavelet representation.
• Define scaling and wavelet functions in the space domain via regularized kernels, ensuring local spatial support.
• Implement a tree-structured multiscale algorithm where coarse trends are captured at initial scale J₀, and finer features are added via wavelet transforms.
• Use equiangular grid integration with M-estimation and Huber weighting to process real CHAMP data on a 180×180 grid.
• Apply the decomposition to separate poloidal (internal/external) and toroidal field components using operators ˜o(1), ˜o(2), ˜o(3).

Experimental results

Research questions

• RQ1 Can a space-domain multiscale method with locally supported wavelets outperform traditional spherical harmonic-based approaches in resolving localized crustal magnetic anomalies?
• RQ2 How can singular convolution kernels in the magnetic field separation problem be regularized to enable stable multiscale representation?
• RQ3 To what extent can the decomposition distinguish between internal crustal fields and external sources (e.g., ionospheric currents) in real satellite data?
• RQ4 What is the spatial resolution and localization performance of the wavelet-based separation when applied to CHAMP data?
• RQ5 How do the wavelet contributions evolve across scales, and what does this reveal about the spatial extent of crustal field anomalies?

Key findings

• The method successfully separates internal, external, and toroidal magnetic field components using a space-domain wavelet framework with locally supported basis functions.
• At scale J = 9 (Jmax), the internal field approximation captures strong crustal anomalies over Central Africa, Eastern Europe, North America, and Australia, with minimal contribution in oceanic regions.
• Wavelet contributions at scales J = 5, 6, 7 focus on the strongest crustal anomalies, indicating high-resolution localization of localized sources.
• The difference between input data and the internal field approximation at J = 9 reveals polar field structures and equatorial bands, consistent with ionospheric and ring current contributions.
• The wavelet transforms show minimal structural change beyond scale J = 5, suggesting that the dominant crustal field anomalies are resolved at scales J ≤ 5.
• The approach effectively filters out non-crustal signals, such as those from magnetospheric ring currents and polar ionospheric systems, as evidenced by residual patterns in the difference map.