On the use of spherical harmonic series inside the minimum Brillouin sphere: Theoretical review and evaluation by GRAIL and LOLA satellite data

Spherical harmonic expansions are the most popular parametrisation of the gravitational potential and its higher-order spatial derivatives in global geodetic, geophysical, and planetary science applications. The convergence domain of external spherical harmonic expansions is the space above the minimum Brillouin sphere, nevertheless, these series are commonly employed inside this bounding surface without correcting. Justification of this procedure has been debated for several decades, but conclusions among scholars are indefinite and even contradictory. In this article, we discuss the use of spherical harmonic expansion for the gravitational field modelling inside the minimum Brillouin sphere. In the theoretical part, we systematically summarise the mathematical apparatus of internal and external spherical harmonic series for the gravitational potential, gravitational gradient components, and second-order gravitational tensor components. We also derive analytical downward continuation errors for these quantities in the spectral form. In the experimental part, we evaluate the internal and external spherical harmonic series inside the minimum Brillouin sphere by employing the most recent LOLA topographic and GRAIL gravitational observations of the Moon. We first analyse line-of-sight gravitational accelerations from GRAIL and their forward-modelled counterparts. We next examine seven GRAIL-derived and four forward-modelled global gravitational field models. We further investigate in detail spectral and spatial (signal/error) characteristics from two forward-modelled global gravitational fields - one using the internal and the other employing the external spherical harmonic parametrisation. Notable findings of this study are: (1) GRAIL measurements taken at low altitudes, below the minimum Brillouin sphere, provide the observational evidence of the divergence in the existing spherical harmonic solutions of the global gravitational field models. (2) Power laws applied at high-frequencies of GRAIL-derived global gravitational field models are too strong. (3) Signal powers of the respective orthogonal components of the gravitational gradient and those of the second-order gravitational tensor are identical above the minimum Brillouin sphere, but may be different inside this bounding surface. (4) Analytical downward continuation of the external spherical harmonic series provides an inhomogeneous gravitational potential spectrum. (5) Vertical-vertical component of the second-order gravitational tensor inside the minimum Brillouin sphere depends on the density and may significantly differ from its equivalent neglecting the density. Most importantly, we unambiguously confirm that all lunar global gravitational field models based on the external spherical harmonic parametrisation do not correspond to the true counterpart inside the minimum Brillouin sphere. Also, analytically downward continued fields tend to diverge at high frequencies. Therefore, the present gravitational field models of the Moon using external spherical harmonic series must be applied only above the minimum Brillouin sphere (10.2 km above the mean lunar sphere). The theoretical part represents a rigorous methodological basis for the gravitational field modelling by the internal and external spherical harmonic series. This theory is complete up to the second-order gravitational tensor and holds for any planetary body. The experimental part reveals intricate aspects for the lunar gravitational field. Except for the Moon, however, these practicalities will have substantial implications on future gravitational field determinations of other planetary bodies.

Automated summary

This article discusses the use of spherical harmonic expansion for gravitational field modeling, highlighting limitations of external spherical harmonic parametrization and drawing clear conclusions. The study reveals issues in global lunar gravitational field models and provides important insights for future gravitational field determinations of other planetary bodies.