Spherical cap harmonics revisited and their relationship to ordinary spherical harmonics

Abstract The “global” representation of the geomagnetic field in terms of ordinary spherical harmonics (SHs) and its corresponding set {g,h} of coefficients has been studied extensively, but the “local” representation in terms of spherical cap harmonics (SCHs) and its corresponding set {G,H} of coefficients is not yet well understood. This paper clarifies some of the main properties of the SCHs and their proper use along with their relationship with the SHs. In particular, it shows that for the spherical cap part of a global field specified by spherical harmonics there is a strict relation between the ordinary Legendre functions of the global representation and the fractional functions of the local expansion; hence we can express the set of coefficients {G,H} in terms of the set {g,h}. Finally, some attention will be given to the role of the leading (n = 0, m = 0) term of the SCH expansion.

[1]  Cheinway Hwang,et al.  Fully normalized spherical cap harmonics: Application to the analysis of sea-level data from TOPEX/POSEIDON and ERS-1 , 1997 .

[2]  Robert A. Langel,et al.  International Geomagnetic Reference Field , 1992 .

[3]  Arthur D. Richmond,et al.  Mapping electrodynamic features of the high-latitude ionosphere from localized observations: technique , 1988 .

[4]  G. V. Haines.  Computer programs for spherical cap harmonic analysis of potential and general fields , 1988 .

[5]  F. Lowes,et al.  RESEARCH NOTE: Orthogonality and mean squares of the vector fields given by spherical cap harmonic potentials , 1999 .

[6]  Alicia García,et al.  New representation of geomagnetic secular variation over restricted regions by means of spherical cap harmonic analysis: application to the case of Spain , 1992 .

[7]  Robert A. Langel,et al.  A geomagnetic field spectrum , 1982 .

[8]  J. M. Torta,et al.  Spherical cap harmonic analysis: a comment on its proper use for local gravity field representation , 1997 .

[9]  A. De Santis,et al.  Translated origin spherical cap harmonic analysis , 1991 .

[10]  C. Falcone,et al.  Spherical cap models of Laplacian potentials and general fields , 1995 .

[11]  Joan Miquel Torta,et al.  A simple approach to the transformation of spherical harmonic models under coordinate system rotation , 1996 .

[12]  G. V. Haines.  Spherical cap harmonic analysis , 1985 .

[13]  J. M. Torta,et al.  On the derivation of the Earth's conductivity structure by means of spherical cap harmonic analysis , 1996 .

[14]  G. V. Haines.  Power spectra of sub-periodic functions , 1991 .

[15]  G. V. Haines,et al.  Determination of equivalent current sources from spherical cap harmonic models of geomagnetic field variations , 1994 .