Geoid: The Mathematical Shape of Earth's Gravity
While the physical surface of the Earth is defined by rugged mountains and deep ocean trenches, the geoid represents a much smoother, more fundamental concept. Often described by Carl Friedrich Gauss as the "mathematical figure of the Earth," the geoid is the shape that the ocean surface would take if the influence of winds and tides were removed, leaving only the effects of Earth's gravity and its rotation.
Because mass is not distributed evenly within the Earth, gravity varies from place to place. This unevenness causes the geoid to undulate, creating a surface that is irregular compared to a perfect sphere but significantly smoother than the actual terrain. Understanding this surface is essential for modern science, particularly for the accuracy of satellite-based global positioning systems (GPS).
![Map of the undulation of the geoid in meters (based on the EGM96 gravity model and the WGS84 reference ellipsoid).[1]](/images/d3/ef/d3ef6c21dc8565df49f1a8a2b75ffd20e45accdab95952383fa906987cc434c6.png)
Key Facts
- The geoid is an equipotential surface, meaning all points on it share the same geopotential.
- It represents the free surface of water at rest under the influence of gravity and rotation.
- Geoid undulations typically range from +85 m to -106 m relative to a reference ellipsoid.
- The Indian Ocean Geoid Low is one of the most significant depressions, reaching 106 meters below average sea level.
- Modern geoid models rely heavily on satellite geodesy and data from missions like GOCE and GRACE.
The Relationship Between the Geoid and the Ellipsoid
To map the Earth, scientists use a reference ellipsoid—a mathematically idealized, slightly flattened sphere that accounts for the planet's equatorial bulge caused by rotation. However, the geoid does not perfectly match this ellipsoid.
The difference between the geoid and the ellipsoid is known as geoid undulation (or geoidal height). Generally, the geoid rises in areas where the Earth's material is locally denser, exerting a stronger gravitational pull. Conversely, it dips where there is a mass deficit. This relationship is critical for GPS users; because satellites orbit relative to a geocentric ellipsoid, a raw GPS reading must be corrected using a geoid model (such as EGM96) to determine an accurate orthometric height (height above sea level).
| Feature | Physical Surface | Geoid | Reference Ellipsoid |
|---|---|---|---|
| Description | Actual terrain (mountains/trenches) | Equipotential surface of gravity | Mathematical idealized sphere |
| Regularity | Highly irregular | Smooth but undulating | Perfectly smooth |
| Variation | Thousands of meters | Less than 200 meters total | Zero (by definition) |
Gravity Anomalies and Mass Distribution
The shape of the geoid is a direct reflection of the Earth's internal composition. A gravity anomaly occurs when the observed gravity at a location differs from the predicted normal gravity. These anomalies are caused by variations in crustal and lithospheric thickness.
When there is a mass excess, the geoid is displaced away from the mass. When there is a mass deficit, the geoid moves toward the deficit. For example, the North Atlantic Geoid High was influenced in part by the weight of ice cover during the Late Cenozoic Ice Age. These fluctuations allow scientists to study the Earth's internal structure and even the viscosity of the mantle.
Mathematical Determination
Calculating the geoid is a complex mathematical challenge. Historically, methods like Bruns' formula and Stokes' formula were used to relate geoid undulation to gravity anomalies. Modern advancements, such as the solutions developed by Petr Vaníček, have improved accuracy to the millimeter or centimeter level.
Today, scientists use spherical harmonics to approximate the geoid. This involves using complex mathematical coefficients to describe the gravitational potential. Models like EGM96 and the more advanced EGM2008 use thousands of these coefficients to provide high-resolution maps of the Earth's gravity field.
Monitoring Temporal Changes
The geoid is not static. Recent satellite missions, including the European Space Agency's GOCE and the GRACE mission, have allowed researchers to observe time-variable geoid signals. By monitoring these changes, scientists can track global hydrologic cycles, the mass balance of polar ice sheets, and the effects of postglacial rebound.
Frequently Asked Questions
How does the geoid differ from a geode?
A geode is a physical rock containing crystal growth, whereas the geoid is a mathematical model representing the Earth's gravitational surface.
Why does a ship's GPS show height variations at sea level?
GPS satellites measure height relative to a mathematical ellipsoid. Because the geoid (the actual sea level) undulates, a ship traveling across the ocean will experience changes in its height relative to the ellipsoid, even if it remains at a constant sea level.
What is an equipotential surface?
An equipotential surface is a surface where the sum of gravitational and centrifugal potential energy is constant. On the geoid, the force of gravity acts perpendicular to the surface at every point.
What causes the Indian Ocean Geoid Low?
The Indian Ocean Geoid Low is a significant depression in the geoid, sitting 106 meters below the average sea level, caused by specific mass distributions within the Earth.Who was instrumental in developing high-fidelity geoid models?
Mathematician Gladys West was the first person to synthesize a high-fidelity geoid from satellite data, which became a foundational component for global positioning systems.