Geoid und Ellipsoid

An ellipsoid is a smooth, mathematically defined shape created by rotating an ellipse around its minor axis, with the polar axis approximately 21 kilometers shorter than the equatorial axis. The geoid represents Earth's actual gravitational shape and aligns with mean sea level. Unlike the smooth ellipsoid, the geoid is irregular due to mass distribution variations, causing global mean sea level undulations of up to 100 meters.

The geoid is an equipotential surface because gravity acts perpendicular at all points on it. This means the gravitational potential is constant everywhere on the geoid surface. The geoid closely aligns with mean sea level over oceans when external influences like tides are excluded, making it the reference surface for vertical measurements in traditional surveying.

Traditional surveying used spirit levels to reference mean sea level or the geoid for local measurements. Modern GPS provides a global perspective by measuring elevation relative to an ellipsoid, such as the World Geodetic System 1984. GPS observations calculate Cartesian coordinates in an Earth-Centered, Earth-Fixed system, which are transformed into geographic coordinates like latitude, longitude, and height for precise mapping and field application of global positioning system.

The Earth-Centered, Earth-Fixed (ECEF) coordinate system is a Cartesian system with its origin at Earth's center of mass. The x-axis extends to the equator at the Greenwich meridian, the y-axis lies perpendicular on the equatorial plane, and the z-axis aligns with the polar axis. GPS calculates Cartesian coordinates in this system, which are then converted into latitude, longitude, and ellipsoidal height for practical surveying and mapping applications.

GPS measures elevation relative to the ellipsoid, but practical applications require heights relative to mean sea level, which the geoid represents. A geoid model converts GPS ellipsoidal heights to approximate mean sea level values. This conversion is essential because the geoid and ellipsoid differ by up to 100 meters globally due to Earth's irregular mass distribution.

Variations in Earth's mass distribution cause the geoid to be irregular rather than smooth. These mass variations create gravitational anomalies that result in global mean sea level undulations of up to 100 meters. The geoid remains an equipotential surface where gravity acts perpendicular at all points, but its shape reflects the actual gravitational field influenced by subsurface density variations.

The World Geodetic System 1984 (WGS 84) is the ellipsoidal model that GPS uses as its reference surface for measuring heights. Aligned with the Geodetic Reference System 1980, WGS 84 incorporates satellite data to refine measurements of Earth's size, shape, and gravitational forces. GPS observations based on WGS 84 are converted into geographic coordinates and state plane coordinates for accurate surveying and mapping.