Geopotential model

The geopotential model represents the Earth's external gravitational potential as a sum of spherical harmonics, with coefficients that describe the departure from a reference ellipsoid. The potential is expressed as V(r,θ,λ) = (GM/r) Σ (a/r)ⁿ Σ (C_nmcos mλ + S_nmsin mλ) P_nm(cosθ), where C_nm and S_nm are the Stokes coefficients.¹ The degree n and order m determine the spatial resolution; higher degrees capture finer details of the gravity field. The model is often truncated at a finite degree, balancing resolution with computational cost and data availability.

Modern geopotential models are derived from a combination of satellite tracking, satellite altimetry, and terrestrial gravity data. The GRACE and GOCE missions provided unprecedented global coverage, enabling models like EGM2008 and EIGEN-6C4.² Satellite altimetry over oceans yields gravity anomalies that are merged with land-based surveys. The combination of these datasets requires careful weighting and error modeling to produce a consistent global field.

Geopotential models are essential for precise orbit determination of satellites, including those for navigation, remote sensing, and scientific missions. They also underpin the computation of the geoid, which serves as the reference for orthometric heights in national and global height systems.³ In geophysics, they help infer subsurface mass distributions, such as mantle convection and ice sheet changes. The models are also used in inertial navigation and in the definition of the International Gravity Formula.

Beyond the standard spherical harmonic expansion, geopotential models can be represented in alternative forms, such as ellipsoidal harmonics or mass concentration (mascon) blocks, which are used for regional studies. The first geopotential model, developed in the 1960s, had only a few dozen coefficients; today's models have thousands. A subtlety is the treatment of the permanent tide: models may be tide-free, mean-tide, or zero-tide, affecting the geoid by up to a few decimeters.⁴ The choice of reference ellipsoid (e.g., WGS84) also influences the coefficients. Additionally, the concept of 'geopotential number' is used in physical geodesy to define heights that are independent of the path of leveling.

Spherical harmonics

Mathematical functions used to represent variations on a sphere, forming the basis of geopotential models.

Stokes coefficients

The coefficients C_nm and S_nm in the spherical harmonic expansion of the gravitational potential.

Geoid

The equipotential surface of the Earth's gravity field that approximates mean sea level.

Mascon

Mass concentration blocks used to model gravity variations in a localized region.