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Alternative expressions for gravity gradients in local north-oriented frame and tensor spherical harmonics

The traditional expressions for gravity gradients in local north-oriented frame and tensor spherical harmonics have complicated forms involved with first- and second-order derivatives of spherical harmonics and also singular terms. In this paper we present alternative expressions for these quantities, which are simpler and contain no singular terms. The presented formulas are useful for those disciplines of geosciences which are involved with potential theory, tensor spherical harmonics and second-order derivatives of spherical harmonic series in the local north-oriented frame. A simple numerical test on the solution of the gradiometric boundary value problems presents the correctness of these new expressions and ability of the solutions to continue the gravity gradients from satellite level down to sea level using spherical harmonics.