Some aspects of harmonic analysis of data gridded on the ellipsoid

Abstract

In producing the next Earth Gravity Model (EGM), one task is to compute the spherical harmonic spectrum of gravity anomalies for which a global grid of band-limited area-means has been computed on the ellipsoid. For EGM96, a solid spherical harmonic formulation was used to compute the spectrum to degree 359, directly from 30′×30′ gridded anomalies using block-diagonal least squares [BDLS]. For the new EGM, an identical approach would use 5′×5′ gridded anomalies and BDLS to degree 2159. In the latter case, this BDLS approach exhibits minor instabilities in recovering zonal harmonics. Contrary to expectation, these instabilities result from converting gridded equiangular geodetic latitudes to non-equiangular geocentric latitudes, as this increases the nonorthogonality of the discretized Legendre polynomials and so causes the recovered zonal coefficients to be almost totally correlated. An alternative is to first compute the ellipsoidal harmonic spectrum to degree 2159, since the non-equiangular spacing of the reduced latitudes does not correlate the recovered zonal ellipsoidal harmonics. Jekeli’s (1988) transformation is then used to transform the ellipsoidal harmonic spectrum to spherical harmonics. Importantly, the final spherical spectrum must be truncated above degree 2159, otherwise the omission errors at polar latitudes become greatly amplified when continued down to the ellipsoid.