Uncertainties associated with integral-based transforms of measured potential gradients

The geoid model over dry land areas is determined from data observed at or above the Earth’s surface. Observable quantities include various functional of the disturbing potential defined as the difference between the real and model (normal) gravity potential. Transformation of the measured data into the sought-after, but directly unobservable gravity potential is often carried out using mathematical tools of the potential theory. An example of such a transformation is the well-known Hotine integral transform that transforms disturbing gravity, i.e., the first-order vertical gradient of the disturbing potential (Stokes formula is applied to anomalous gravity which is still often encountered in geodesy). Higher-order gradients of the Earth's gravitational potential have been collected by sensors on board aircraft or low-orbiting satellites. This advancement in data availability has resulted in the formulation of new tools based on Green's integral transforms and equations. Associated deterministic models have been well developed, tested, and successfully implemented. However, stochastic models for estimating uncertainties in sought values have only been partially developed. These uncertainties should reflect the inevitable implementation and approximation errors, the propagation of formal errors, as well as external accuracy estimation if relevant independent reference values are available. This contribution discusses mathematical models that can be used to estimate various types of uncertainties related to integral-based transformations of the measured potential gradients into the disturbing potential.