TY - JOUR

T1 - Spherical cap harmonics revisited and their relationship to ordinary spherical harmonics

AU - De Santis, A.

AU - Torta, J. M.

AU - Lowes, F. J.

N1 - Funding Information:
This is a contribution under the Project “Mathematical models of the geomagnetic field in Europe” funded by the Italian-Spanish Joint Commission. We would like to thank David R. Barraclough and an unknown referee for their valuable suggestions. Two of the authors (JMT and FJL) thank the Istituto Nazionale di Geofisica for its financial support for a visit to the Italian institute. Part of the work was presented at the VIII Scientific IAGA Assembly, Uppsala, 4-15 August, 1997 and at the XXIII EGS General Assembly, Nice, 20-24 April, 1998 as an invited talk.

PY - 1999

Y1 - 1999

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0033387882&partnerID=8YFLogxK

U2 - 10.1016/S1464-1895(99)00138-6

DO - 10.1016/S1464-1895(99)00138-6

M3 - Article

AN - SCOPUS:0033387882

SN - 1464-1895

VL - 24

SP - 935

EP - 941

JO - Physics and Chemistry of the Earth, Part A: Solid Earth and Geodesy

JF - Physics and Chemistry of the Earth, Part A: Solid Earth and Geodesy

IS - 11-12

ER -