A multiscale power spectrum for the analysis of the lithospheric magnetic field
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A multiscale power spectrum for the analysis of the lithospheric magnetic field C. Gerhards
Received: 15 October 2013 / Accepted: 20 October 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract Degree variances and the corresponding power spectrum form an important tool in analyzing the geomagnetic field. While they describe the contribution of a fixed spherical harmonic degree (i.e, the contribution of a certain “wavelength/frequency”) to the total power of the magnetic field, the aim of this paper is to introduce multiscale variances as a spatially oriented generalization. Multiscale variances can be designed such that they represent the contribution of features of a certain scale-dependent spatial extend to the total power. We present different examples to illustrate their interpretation and capabilities. Keywords
Geomagnetic field · Power spectrum · Multiscale analysis
Mathematics Subject Classification
42C25 · 65D15 · 86-08 · 86A99
1 Introduction Degree variances of the geomagnetic field have been studied as early as Langel and Estes (1982), Lowes (1974), and Mauersberger (1956). They reflect the contribution of a fixed spherical harmonic degree to the total power of the magnetic field and find use, e.g., in the separation of the magnetic field into core and lithospheric contributions and in the identification of noise levels and systematic errors. The degree variance at spherical harmonic degree n is given by
Dedicated to Willi Freeden’s 65th Birthday. C. Gerhards (B) Geomathematics Group, University of Kaiserslautern, PO Box 3049, 67653 Kaiserslautern, Germany e-mail: [email protected]
123
Int J Geomath
2n+4 n ∧ R U¯ (n, k)2 , Rn (r ) = (n + 1) R r
(1.1)
k=−n
where R =6,371.2 km denotes the mean Earth radius. The Fourier coefficients U¯ R∧ (n, k) of degree n and order k (with respect to Schmidt semi-normalized spherical harmonics) of a harmonic potential U are typically known from a geomagnetic field model, noting that we assume that the investigated magnetic field can be represented by b = ∇U in the exterior of the sphere R = {x ∈ R3 : |x| = R}. By the power spectrum of b we generally mean the entire set of corresponding degree variances (although in Maus 2008 it is argued that the term power spectrum is more suitable for a slightly modified quantity). Being very useful in global interpretations, the degree variances (1.1) based on spherical harmonics do not provide information on the geographic origin of the main contributions. An alternative arises by the use of localizing basis functions such as splines (e.g., Freeden 1981 and Shure et al. 1982), spherical cap harmonics (e.g., Haines 1985 and Thébault et al. 2006), or spherical Slepian functions (e.g., Simons et al. 2006 and Simons 2010). Latter have been used in Beggan et al. (2013), e.g., to approximate the continental lithospheric magnetic field separately from the oceanic field and to derive degree variances that reflect this split-up. In other words, two power spectra are obtained that take into account contr
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