structural:reformatting
Determination

The undulation of the geoid N is closely related to the disturbing potential T according to Bruns' formula (named after Heinrich Bruns):

where γ {\displaystyle \gamma } is the force of normal gravity, computed from the normal field potential U {\displaystyle U}.

Another way of determining N is using values of gravity anomaly Δ g {\displaystyle \Delta g}, differences between true and normal reference gravity, as per Stokes formula (or Stokes' integral), published in 1849 by George Gabriel Stokes:

The integral kernel S, called Stokes function, was derived by Stokes in closed analytical form. Note that determining N {\displaystyle N} anywhere on Earth by this formula requires Δ g {\displaystyle \Delta g} to be known everywhere on Earth, including oceans, polar areas, and deserts. For terrestrial gravimetric measurements this is a near-impossibility, in spite of close international co-operation within the International Association of Geodesy (IAG), e.g., through the International Gravity Bureau (BGI, Bureau Gravimétrique International).

Another approach for geoid determination is to combine multiple information sources: not just terrestrial gravimetry, but also satellite geodetic data on the figure of the Earth, from analysis of satellite orbital perturbations, and lately from satellite gravity missions such as GOCE and GRACE. In such combination solutions, the low-resolution part of the geoid solution is provided by the satellite data, while a 'tuned' version of the above Stokes equation is used to calculate the high-resolution part, from terrestrial gravimetric data from a neighbourhood of the evaluation point only.

Calculating the undulation is mathematically challenging. The precise geoid solution by Petr Vaníček and co-workers improved on the Stokesian approach to geoid computation. Their solution enables millimetre-to-centimetre accuracy in geoid computation, an order-of-magnitude improvement from previous classical solutions.

Geoid undulations display uncertainties which can be estimated by using several methods, e.g., least-squares collocation (LSC), fuzzy logic, artificial neural networks, radial basis functions (RBF), and geostatistical techniques. Geostatistical approach has been defined as the most-improved technique in prediction of geoid undulation.