The Radial Integral of the Geopotential

Abstract

In Newtonian theory of gravitation, used in Earth’s and planetary sciences, gravitational acceleration is standardly regarded as the most fundamental parameter that describes any vectorial gravitational field. Considering only conservative gravitational field, the vectorial field can be described by a scalar function of 3D position called gravitational potential from which other parameters (particularly gravitational acceleration and gravitational gradient) are derived by applying gradient operators. Gradients of the Earth’s gravity potential are nowadays measured with high accuracy and applied in various geodetic and geophysical applications. In geodesy, the gravity and gravity gradient measurements are used to determine the Earth’s gravity potential (i.e., the geopotential) that is related to geometry of equipotential surfaces, most notably the geoid approximating globally the mean sea surface. Reversely to the application of gradient operator, the application of radial integral to gravity yields the gravity potential differences and the same application to gravity gradient yields the gravity differences. This procedure was implemented in definitions of rigorous orthometric heights and differences between normal and orthometric heights (i.e., the geoid-to-quasigeoid separation). Following this concept, we introduce the radially integrated geopotential, and provide its mathematical definitions in spatial and spectral domains. We also define its relationship with other parameters of the Earth’s gravity field via Poisson, Hotine, and Stokes integrals. In numerical studies, we investigate a spatial pattern and spectrum of the radial integral of the disturbing potential (i.e., difference between actual and normal gravity potentials) and compare them with other parameters of gravity field. We demonstrate that the application of radial integral operator smooths a spatial pattern of the disturbing potential. This finding is explained by the fact that more detailed features in the disturbing potential (mainly attributed to a gravitational signature of lithospheric density structure and geometry) are filtered out proportionally with increasing degree of spherical harmonics in this functional. In the global geoidal geometry (and the disturbing potential), on the other hand, the gravitational signature of lithosphere is still clearly manifested—most notably across large orogens—even after applying either spectral decompensation or filtering.