COMBINED SPHERICAL HARMONIC AND WAVELET EXPANSION - A FUTURE CONCEPT IN EARTHS GRAVITATIONAL DETERMINATION

The basic theory of spherical singular integrals is recapitulated. Cri teria are given for measuring the space-frequency localization of func tions on the sphere. The trade-off between ''space localization'' on t he sphere and ''frequency localization'' in terms of spherical harmoni cs is described in form of an ''uncertainty principle.'' A continuous version of spherical multiresolution is introduced, starting from cont inuous wavelet transform corresponding to spherical wavelets with vani shing moments up to a certain order. The wavelet transform is characte rized by least-squares properties. Scale discretization enables us to construct spherical counterparts of P(acket)-scale discretized and D(a ubechies)-scale discretized wavelets. It is shown that singular integr al operators forming a semigroup of contraction operators of class (C- 0) (like Abel-Poisson or Gauss-Weierstrass operators) lead in canonica l way to pyramid algorithms. Fully discretized wavelet transforms are obtained via approximate integration rules on the sphere. Finally appl ications to (geo-)physical reality are discussed in more detail. A com bined method is proposed for approximating the ''low frequency parts'' of a physical quantity by spherical harmonics and the ''high frequenc y parts'' by spherical wavelets. The particular significance of this c ombined concept is motivated for the situation of today's physical geo desy, viz. the determination of the high frequency parts of the earth' s gravitational potential under explicit knowledge of the lower order part in terms of a spherical harmonic expansion. (C) 1997 Academic Pre ss.