REPRESENTATION OF GRADIENTS OF A SCALAR FIELD ON THE SPHERE USING A 2D FOURIER EXPRESSION

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