Combined Spherical Harmonic and Wavelet Expansion—A Future Concept in Earth's Gravitational Determination

Abstract

The basic theory of spherical singular integrals is recapitulated. Criteria are given for measuring the space-frequency localization of functions on the sphere. The trade-off between “space localization” on the sphere and “frequency localization” in terms of spherical harmonics is described in form of an “uncertainty principle.” A continuous version of spherical multiresolution is introduced, starting from continuous wavelet transform corresponding to spherical wavelets with vanishing moments up to a certain order. The wavelet transform is characterized by least-squares properties. Scale discretization enables us to construct spherical counterparts ofP(acket)-scale discretized andD(aubechies)-scale discretized wavelets. It is shown that singular integral operators forming a semigroup of contraction operators of class (C0) (like Abel–Poisson or Gauß–Weierstraß operators) lead in canonical way to pyramid algorithms. Fully discretized wavelet transforms are obtained via approximate integration rules on the sphere. Finally applications to (geo-)physical reality are discussed in more detail. A combined method is proposed for approximating the “low frequency parts” of a physical quantity by spherical harmonics and the “high frequency parts” by spherical wavelets. The particular significance of this combined concept is motivated for the situation of today's physical geodesy, 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.