Open AccessGlobal spherical harmonic analysis by least‐squares and numerical quadrature methods in historical perspective
Author(s) -
Sneeuw Nico
Publication year - 1994
Publication title -
geophysical journal international
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0956-540X
DOI - 10.1111/j.1365-246x.1994.tb03995.x
Subject(s) - mathematics , quadrature (astronomy) , mathematical analysis , spherical harmonics , discretization , computation , context (archaeology) , notation , gauss–kronrod quadrature formula , gaussian quadrature , nyström method , integral equation , algorithm , physics , optics , paleontology , arithmetic , biology
SUMMARY Methods of global spherical harmonic analysis of discrete data on a sphere are placed in a historical context. The paper concentrates on the loss of orthogonality in the direction of latitude, due to the transition from continuous to discretized functions. Special attention is paid to Neumann's (1838) solution to this problem. By recasting the formulae of spherical harmonic analysis into matrix‐vector notation, both least‐squares solutions and quadrature methods are represented in a general framework of weighted least squares. It is also shown that the two‐step formulation of global spherical harmonic computation was applied already by Neumann (1838) and Gauss (1839). Computational modifications to Neumann's method are reviewed as well.