Orthogonality of harmonic potentials and fields in spheroidal and ellipsoidal coordinates: application to geomagnetism and geodesy

Abstract

SUMMARY We investigate the orthogonality of the potential distributions that are the basis solutions of Laplace's equation appropriate to 3‐D ellipsoidal (including spheroidal) coordinate systems, and also the orthogonality of the corresponding vector gradient fields, both over the surface of the ellipsoid, and for integration over the volume of the annular shell between two confocal ellipsoids. The only situation for which there is orthogonality is for the vector gradients when integrated over the annular shell. In the other three cases (potential over surface or annulus, and field over surface) orthogonality can be restored by using an appropriate geometrical weighting factor applied to the integrand; it is therefore still possible to perform the equivalent of a classical spherical harmonic analysis. In the special case of the sphere, there is real orthogonality in all four cases; in effect the weighting factors are all unity. In geodesy, spheroidal harmonic analysis is done using a method that relies on a particular result valid only for potential; it cannot be extended to the corresponding vector field, or to ellipsoidal geometry. The lack of orthogonality over the surface means that care must be taken when interpreting conventional geomagnetic ‘power spectra’, and geodetic ‘degree variance’, as these no longer correspond exactly to the mean‐square values over the actual ellipsoidal surface. We illustrate some of the problems by comparing different versions of the power spectrum for a spheroidal analysis of the global lithospheric magnetic field. We use only simple vector algebra, and do not need to know the details of the actual basis solutions, only that they are the product of three functions, one for each coordinate and involving only that coordinate, and that they satisfy Laplace's equation. Similarly, our results do not depend on the normalization used in the basis functions.

Convergence and divergence in spherical harmonic series of the gravitational field generated by high‐resolution planetary topography—A case study for the Moon

“…Near or at the surface of the masses, series convergence must be considered an unstable property [ Krarup , ], whereby “ an arbitrarily small change ” of the mass distribution may “ change convergence to divergence ” [ Moritz , , p19]. Generally, divergence is thought to occur more likely, the higher the spectral resolution of the gravitational model, the more irregular the planetary body and the deeper the evaluation points are located inside the Brillouin sphere [ Wang , ; Lowes and Winch , ; Hu and Jekeli , ].…”

Spheroidal forward modelling of the gravitational fields of 1 Ceres and the Moon

“…194-195;Jekeli, 1981). This pragmatic procedure, however, is not without difficulties, as discussed by (Lowes and Winch, 2012), and cannot completely substitute an explicit spheroidal forward modelling in the spectral domain, which is what we present here.…”

“…The rationale for using the degree variances of a spheroidal harmonic spectrum is explained in (Lowes and Winch, 2012). The power spectrum σ e n (b) of Eq.…”

Prolate Spheroidal Harmonic Expansion of Gravitational Field

“…More precisely, it is well approximated by an oblate spheroid. Thus, the spherical and the oblate spheroidal harmonic expansions are extensively used in geodesy and geophysics (Heiskanen & Moritz 1967;Stacey & Davis 2008;Maus 2010;Pavlis et al 2012;Lowes & Winch 2012;Wang & Yang 2013). So is the situation in astronomy and planetary science when dealing with major planets as Mars, large satellites as the Moon, and massive asteroids as Ceres.…”