Geoid – Surface of Constant Gravitational Potential Approximating Mean Sea Level

The geoid is a concept fundamental to modern geodesy and surveying, representing the shape that the surface of the Earth’s oceans would take under the influence of Earth’s gravity and rotation, if not disturbed by other influences such as winds and tides. It is the hypothetical, continuous surface that best fits global mean sea level (MSL), making it the true “level” surface used for defining elevations and heights everywhere on Earth.

What Is the Geoid?

The geoid is defined as the equipotential surface of the Earth’s gravity field which, on average, coincides with mean sea level and its continuation under the continents. In other words, it is the surface where the gravitational potential (including centrifugal effects from the Earth’s rotation) is everywhere the same. Water at rest would settle on the geoid, and the geoid is always perpendicular to the local direction of gravity (the “plumb line”) at every point.

Unlike a sphere or ellipsoid, the geoid is an irregular, undulating surface, shaped by variations in Earth’s internal density and mass distribution—such as mountains, ocean trenches, and mantle convection. These undulations can reach over ±100 meters when compared with a simple reference ellipsoid.

The reference ellipsoid is a mathematically smooth surface, while the geoid is an irregular, physically defined surface that closely follows mean sea level and deviates from the ellipsoid due to Earth’s mass anomalies.

Why Does the Geoid Matter?

The geoid is the physical zero-height reference for orthometric heights—what is commonly meant by “height above sea level.” It underpins all modern vertical datums and is crucial for:

• Surveying and engineering: Ensuring consistent, accurate height measurements.
• Navigation and mapping: Providing a common elevation reference.
• Oceanography: Understanding dynamic ocean topography.
• Geophysics: Studying Earth’s internal structure and mass redistribution.

All measurements of elevation, whether from GNSS, leveling, or altimetry, are ultimately referenced to the geoid for physical meaning and consistency.

Theoretical Background

Gravity, Equipotential Surfaces, and the Geoid

Earth’s gravity field is neither uniform nor simple, being affected by the planet’s rotation and complex internal structure. An equipotential surface is a surface where the gravitational potential (including centrifugal effects) is constant everywhere. The geoid is the unique equipotential surface that best coincides with global mean sea level.

The Reference Ellipsoid

For mapping and satellite navigation, geodesists use a reference ellipsoid—a smooth, mathematically defined oblate spheroid. The reference ellipsoid (e.g., WGS84 or GRS80) simplifies calculations but does not account for the Earth’s mass irregularities. The geoid, by contrast, is a physically meaningful surface that reflects these irregularities.

The separation between the geoid and ellipsoid at any location is called the geoid undulation or geoid height (N). This value is positive where the geoid is above the ellipsoid and negative where it is below.

Geoid Undulation: The Key to Height Conversion

The relationship between a point’s height above the ellipsoid (h, measured by GPS), its orthometric height (H, above the geoid), and the geoid undulation (N) is:

[ h = H + N ]

or rearranged,

[ H = h - N ]

This relationship is fundamental in converting GPS heights into elevations referenced to mean sea level.

Causes of Geoid Undulations

The geoid’s irregularities are due to:

• Mass concentrations (mountain roots, mantle upwellings): Make gravity stronger, raising the geoid.
• Mass deficits (deep ocean trenches, crustal thinning): Weaken gravity, causing the geoid to dip.
• Large-scale topography and density variations: Influence the gravity field and thus the geoid’s shape.

Globally, geoid undulations can range from about -106 meters (Indian Ocean region) to +85 meters (near Papua New Guinea).

Tidal Variations and Geoid Definitions

The geoid is affected by the gravitational pull of the Moon and Sun, which deform the Earth and sea surface. To standardize measurements, geodesists use specific tidal systems:

• Tide-Free (Nontidal) Geoid: Removes all permanent tidal effects. Used in most global geoid models.
• Zero-Tide Geoid: Removes the direct permanent tide but retains the Earth’s elastic response. Recommended for orthometric heights by the International Association of Geodesy.
• Mean-Tide Geoid: Includes all permanent tidal effects; matches observed mean sea level.

Using consistent tidal systems is critical for accuracy—mixing systems can result in height errors of several centimeters.

Geoid, Ellipsoid, and Height Systems

Reference Ellipsoid vs. Geoid

• Reference Ellipsoid: A smooth, mathematically defined surface, used for GPS and mapping.
• Geoid: A physically defined, irregular surface matching mean sea level.

Their separation, the geoid undulation (N), is essential for height conversions.

Height Types

• Ellipsoidal Height (h): Vertical distance above the ellipsoid; provided by GNSS/GPS.
• Orthometric Height (H): Vertical distance above the geoid (mean sea level); used in engineering and mapping.

Conversion:
[ H = h - N ]

This allows GNSS heights to be turned into practical elevations for real-world applications.

Modeling the Geoid: Theory and Data

Data Sources

Geoid models are built from:

• Terrestrial gravity surveys: High density over land, but coverage may be uneven.
• Shipborne gravity: Key for oceans.
• Airborne gravity: Fills gaps in remote or difficult areas.
• Satellite missions: Like GRACE and GOCE, crucial for capturing global and long-wavelength features.

Mathematical Methods

• Disturbing Potential (T): Difference between actual and reference gravity potential.

• Bruns’s Formula: Relates geoid undulation to disturbing potential:

  [ N = \frac{T}{\gamma} ] where γ is normal gravity at the ellipsoid.

• Stokes’s Formula: Computes geoid undulation from gravity anomalies over Earth’s surface.

• Spherical Harmonics: The geoid is modeled as a sum of spherical harmonic coefficients, allowing global computation at various resolutions.

Geoid Model Construction

The remove-compute-restore method is widely used:

1. Remove known effects (terrain, reference field)
2. Compute residual geoid
3. Restore removed effects

Spatial resolution depends on gravity data density and elevation model quality.

Major Geoid Models

• EGM96: Global model based on 2.6 million measurements, referenced to WGS84, ~28 km grid.
• USGG2003: U.S.-focused, high-resolution, 1.8 km grid, includes terrain corrections.
• GRACE: Satellite mission for temporal and spatial gravity variations, ideal for large-scale features.
• GOCE: ESA mission, improved medium-wavelength accuracy.

Regional models (e.g., G99SSS for the U.S.) provide higher local accuracy by including dense local data and digital elevation models.

Applications of the Geoid

Surveying and Height Transformation

Geoid models are essential in converting GNSS-derived ellipsoidal heights to orthometric heights:

[ H = h - N ]

This enables precise, consistent elevations for construction, mapping, and engineering.

Floodplain Mapping and Disaster Management

Accurate heights above mean sea level are critical for identifying flood risk, planning coastal defenses, and managing disaster response. Geoid-based heights ensure all elevations refer to the same, physically meaningful surface.

Oceanography

To understand ocean currents and mean sea level, satellite altimetry measures sea surface height relative to the ellipsoid. Subtracting the geoid gives dynamic ocean topography (DOT), which reveals water movement and supports climate studies.

Geophysics and Geodynamics

Geoid anomalies expose information about Earth’s deep structure—such as mantle plumes, tectonic plates, and mass redistribution from ice melt or groundwater depletion. Monitoring geoid changes informs studies of sea level rise, earthquakes, and resource management.

Mapping and Vertical Datums

National agencies use geoid models to define vertical datums—the standard reference for all elevation data. This ensures all mapping and engineering projects use compatible, accurate heights.

Summary

The geoid is the Earth’s “level” surface—a complex, gravity-based reference that defines what we mean by “height above sea level.” It is essential for:

• Accurate surveying and engineering
• Consistent mapping and navigation
• Scientific understanding of Earth’s processes

Modern geoid models, built from gravity and elevation data, allow GNSS-measured heights to be reliably converted into practical, physically meaningful elevations for any location on the planet.

For further information or to learn how geoid models can support your surveying, mapping, or engineering needs, contact us or learn more about our solutions .