Abstract
Abstract. Representation of data on the sphere is conventionally done using spherical harmonics. Making use of the Fourier series of the Legendre function in the SH representation results in a 2D Fourier expression. So far the 2D Fourier series representation on the sphere has been confined to a scalar field like geopotential or relief data. We show that if one views the 2D Fourier formulation as a representation in a rotated frame, instead of the original Earth-fixed frame, one can easily generalize the representation to any gradient of the scalar field. Indeed, the gradient and the scalar field itself are simply linked in the spectral domain using spectral transfers. We provide the spectral transfers of the first-, second- and third-order gradients of a scalar field in a local frame. Using three numerical examples based on gravity and geometrical quantities, we show the applicability of the presented formulation.
Highlights
The spherical harmonics are usually employed to represent geoscience data globally on the sphere
In the case of harmonic scalar fields like gravitational potential, since the spherical harmonics are the solutions of the Laplace equation in spherical coordinates, the most appropriate choice is the representation in terms of spherical harmonic series (Heiskanen and Moritz, 1967)
The representation of geodetic or geophysical quantities in spherical coordinates using a 2D Fourier series has been confined in the literature to a scalar field like geopotential
Summary
The spherical harmonics are usually employed to represent geoscience data globally on the sphere. An alternative formulation of a scalar field on the sphere is derived by expanding the Legendre function in the spherical harmonic representation as a series in sine and cosine terms (e.g., Schuster, 1903; Ricardi and Burrows, 1972; Colombo, 1981; Sneeuw and Bun, 1996). The problem of global spherical harmonic analysis of anisotropic functionals can not be dealt with and explicitly using spherical harmonics As an another example, the aliasing problem on the sphere is better solved and explained based on the Fourier series representation (Jekeli, 1996).
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