The Geoid and Figure of the Earth

The figure of the Earth is approximated by a rotational ellipsoid, but the true equipotential surface of mean sea level, the geoid, undulates above and below it in response to the planet's uneven mass distribution.

Definition

The figure of the Earth is its overall shape, conventionally modeled as a best-fitting rotational ellipsoid, while the geoid is the equipotential surface of the gravity field that coincides with undisturbed mean sea level and serves as the physical reference for heights.

Scope

This topic covers the geometric and physical description of the Earth's shape: the reference ellipsoid that captures the rotational flattening, the geoid as the equipotential surface defining mean sea level, and the geoid undulations measured relative to the ellipsoid. It treats normal gravity and the gravity formula, the relationship between geoid height and the disturbing potential through Stokes's theorem, and the distinction between ellipsoidal, orthometric, and geoid-referenced heights. The emphasis is on defining and computing the Earth's shape and its height reference.

Core questions

• Why is the Earth's figure modeled as a flattened ellipsoid of rotation?
• What is the geoid, and how does it relate to mean sea level?
• How are geoid undulations computed from gravity measurements?
• How do ellipsoidal, orthometric, and geoid heights differ?

Key concepts

• Reference ellipsoid and flattening
• Geoid as an equipotential surface
• Geoid undulation and height anomaly
• Normal gravity and the gravity formula
• Stokes's theorem and the disturbing potential

Key theories

Reference ellipsoid for the figure of the Earth

The Earth's rotation flattens it into an oblate spheroid, and a best-fitting reference ellipsoid with defined size and flattening provides the geometric datum against which the geoid and positions are expressed.

Stokes's determination of the geoid

Stokes's theorem relates the geoid undulation to a surface integral of gravity anomalies over the whole Earth, providing the classical means of computing the shape of the geoid from gravimetric data.

Mechanisms

Because the geoid follows surfaces of constant gravity potential, mass excesses pull it upward and mass deficits let it dip, so its undulations relative to the smooth reference ellipsoid mirror the large-scale density structure of the Earth; heights measured from the geoid (orthometric) differ from purely geometric ellipsoidal heights by the geoid undulation, which must be modeled to convert between them.

Clinical relevance

A precise geoid is essential for converting satellite-derived ellipsoidal heights into physically meaningful elevations used in surveying, hydrology, and engineering, and for unifying national height systems and monitoring sea level.

History

Newton argued the rotating Earth must bulge at the equator, eighteenth-century geodetic expeditions to Lapland and Peru confirmed the flattening, Stokes provided the integral linking gravity to geoid shape in 1849, and modern satellite gravimetry now resolves the global geoid to centimeter accuracy.

Key figures

• Isaac Newton
• George Gabriel Stokes
• Friedrich Robert Helmert

Seminal works

• hofmannwellenhof2006
• torge2012
• fowler2005

Frequently asked questions

What is the difference between the geoid and the ellipsoid?

The ellipsoid is a smooth mathematical surface approximating the Earth's flattened shape, while the geoid is the actual lumpy equipotential surface of gravity matching mean sea level; the geoid rises and falls relative to the ellipsoid by tens of meters because of uneven mass inside the Earth.

Why does GPS need a geoid model to give elevations?

Satellite positioning yields heights above the reference ellipsoid, which are geometric and not the elevations people use; subtracting the geoid undulation converts them into heights above mean sea level that correspond to how water flows and how surveys are referenced.