The theory and formulas used by the calculation service of the ICGEM are described in the Scientific Technical Report STR09/02

(critics and comments are welcome)

The Report can be downloaded here.
The revised edition of the Report can be downloaded here.

Introduction

The intention of this report is to present the definitions of different functionals of the Earth's gravity field and possibilities for their approximative calculation from a mathematical representation of the outer potential. In history this topic has usually been treated in connection with the boundary value problems of geodesy, i.e. starting from measurements at the Earth's surface and their use to derive a mathematical representation of the geopotential.
Nowadays global gravity field models, mainly derived from satellite measurements, become more and more detailed and accurate and, additionally, the global topography can be determined by modern satellite methods independently from the gravity field. On the one hand the accuracy of these gravity field models has to be evaluated and on the other hand they should be combined with classical (e.g. gravity anomalies) or recent (e.g. GPS-levelling-derived or altimetry-derived geoid heights) data. Furthermore, an important task of geodesy is to make the gravity field functionals available to other geosciences. For all these purposes it is necessary to calculate the corresponding functionals as accurately as possible or, at least, with a well-defined accuracy from a given global gravity field model and, if required, with simultaneous consideration of the topography model.
We will start from the potential, formulate the definition of some functionals and derive the formulas for the calculation. In doing so we assume that the Earth's gravity potential is known outside the masses, the normal potential outside the ellipsoid and that mathematical representations are available for both. Here we neglect time variations and deal with the stationary part of the potential only.
Approximate calculation formulas with different accuracies are formulated and specified for the case that the mathematical representation of the potential is in terms of spherical harmonics. The accuracies of the formulas are demonstrated by practical calculations using the gravity field model EIGEN-6C2 (Förste et al. 2012).
More or less, what is compiled here is well-known in physical geodesy but distributed over a lot of articles and books which are not cited here. In the first instance this text is targeted at non-geodesists and it should be “stand-alone readable”.