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

Abstract

Theoretically, spherical harmonic (SH) series expansions of the external gravitational potential are guaranteed to converge outside the Brillouin sphere enclosing all field‐generating masses. Inside that sphere, the series may be convergent or may be divergent. The series convergence behavior is a highly unstable quantity that is little studied for high‐resolution mass distributions. Here we shed light on the behavior of SH series expansions of the gravitational potential of the Moon. We present a set of systematic numerical experiments where the gravity field generated by the topographic masses is forward‐modeled in spherical harmonics and with numerical integration techniques at various heights and different levels of resolution, increasing from harmonic degree 90 to 2160 (~61 to 2.5 km scales). The numerical integration is free from any divergence issues and therefore suitable to reliably assess convergence versus divergence of the SH series. Our experiments provide unprecedented detailed insights into the divergence issue. We show that the SH gravity field of degree‐180 topography is convergent anywhere in free space. When the resolution of the topographic mass model is increased to degree 360, divergence starts to affect very high degree gravity signals over regions deep inside the Brillouin sphere. For degree 2160 topography/gravity models, severe divergence (with several 1000 mGal amplitudes) prohibits accurate gravity modeling over most of the topography. As a key result, we formulate a new hypothesis to predict divergence: if the potential degree variances show a minimum, then the SH series expansions diverge somewhere inside the Brillouin sphere and modeling of the internal potential becomes relevant.

Forward modelling of global gravity fields with 3D density structures and an application to the high-resolution (~ 2 km) gravity fields of the Moon

“…For the other three non-Brillouin spheres of radii = + ( ) = 1728.2 km, = = 1737.4 km, and = 1738.0 km (red, blue and black curves, respectively), the powers of the gravitational spectra amplify with increasing d/o. We note that the selection of the Brillouin sphere conforms with Hirt and Kuhn (2017), who also pointed out this issue by investigating the convergence of the gravitational potential spectra inferred by the lunar topography.…”

“…Their analyses indicated that gravity values were strongly influenced by the choice of the Brillouin sphere. However, the numerical restrictions of WP98 method investigated in this section are different from and complementary to those addressed by Hirt and Kuhn (2017).…”

“…3 and 4. Hirt and Kuhn (2017) performed an investigation of the WP98 method and its performance for the Moon. They studied how the spatial resolution of the lunar topography affects the gravitational potential spectra.…”

“…We confirm that the accuracy of these "standard approaches" may be strongly affected by the choice of input parameters (cf. Sun and Sjöberg 2001, Root et al 2016, Hirt and Kuhn 2017. In contrast, the RFM is more robust and, to simplify and disseminate its exploitation, we make the computer codes, user manual and scripts publicly available to potential users, see Appendix A.…”

Forward Gravity Modelling to Augment High-Resolution Combined Gravity Field Models

“…However, this statement is missing the fact that the convergence of the series cannot be guaranteed at or near the Earth's surface. Well-established theoretical works suggest that the series may be convergent or divergent at the Earth's surface (Hirt and Kuhn 2017;Hu and Jekeli 2015;Jekeli 1983;Krarup 1969;Moritz 1978Moritz , 1980. It is presented both theoretically and numerically that the spherical harmonic (SH) series of gravitational potential converges outside of the sphere that encompasses all field-generating mass (Hu and Jekeli 2015).…”