Ellipsoid vs Geoid: Two Reference Surfaces, Two Jobs

The Earth has no single “reference surface.” It has two: the ellipsoid and the geoid. Modern coordinate work uses both, for different purposes — the ellipsoid for satellite-tracking and GNSS arithmetic; the geoid for sea-level-relative work like engineering and hydrology. This article puts them side by side, derives the fundamental equation that connects them, and explains why both are needed.

The /learn/what-is-a-geodetic-datum pillar introduces both surfaces as components of a datum. /learn/the-geoid-explained goes deep on the geoid. This article focuses on the comparison.

The ellipsoid: smooth math

A reference ellipsoid — also called a spheroid — is a slightly flattened sphere defined by two parameters:

• Semi-major axis (a) — equatorial radius. WGS 84: 6,378,137 m.
• Flattening (f) — how much the polar radius is squeezed relative to the equatorial. WGS 84: 1/298.257223563.

From these, the polar radius b = a(1 − f) ≈ 6,356,752 m. The ellipsoid is a 21.4 km flattening at the poles — barely visible at human scales (the Earth looks like a sphere from space) but critical at the geodetic scale.

The ellipsoid is mathematical: a closed-form surface defined by an equation. Two ellipsoids are identical if their a and f values match. The ellipsoid's convenience is that every geodetic formula — Vincenty distance, UTM projection, ECEF conversion — has a known closed-form expression in terms of a and f. There's no “regional” or “local” ellipsoid; the same ellipsoid applies globally, and the math is the same everywhere.

Common reference ellipsoids:

| Ellipsoid | Semi-major axis | Flattening (1/f) | Used by | | ------------------ | ---------------- | -------------------- | ---------------------- | | WGS 84 | 6,378,137 m | 298.257223563 | GPS, GeoJSON, EPSG:4326 | | GRS 80 | 6,378,137 m | 298.257222101 | NAD 83, ETRS89, GDA2020 | | International 1924 | 6,378,388 m | 297.0 | Older European datums | | Clarke 1866 | 6,378,206.4 m | 294.978698 | NAD 27 | | Airy 1830 | 6,377,563.396 m | 299.3249646 | OSGB36 (Britain) | | Bessel 1841 | 6,377,397.155 m | 299.1528128 | Older Asian datums |

Modern global datums all use either WGS 84 or GRS 80 (and the two differ only in the eighth decimal of flattening — effectively identical at the metre scale).

The geoid: irregular physics

The geoid is the equipotential surface of Earth's gravity that best approximates mean sea level. At every point on the geoid, the gravitational potential is the same; the local gravity vector points perpendicular to the geoid surface.

The geoid is physical: it's defined by Earth's actual mass distribution, not by a formula. The /learn/the-geoid-explained support article goes into the details. The key facts for the comparison:

• The geoid is irregular. It has bumps and dips of up to ~85 m on top of the overall ellipsoidal shape, caused by uneven mass distribution inside Earth.
• The geoid is observed, not derived. Modern global models (EGM2008, GEOID2022) are computed from satellite gravity observations (NASA GRACE / GRACE-FO; ESA GOCE) plus terrestrial gravity surveys plus altimetry.
• The geoid changes — slowly. Mass redistribution from melting ice sheets and crustal motion changes the geoid at the centimetre-per-decade scale, captured by recent GRACE data.

Side-by-side comparison

| Aspect | Ellipsoid | Geoid | | --------------------- | ---------------------------------- | ----------------------------------- | | Definition | Two-parameter math: a, f | Equipotential gravity surface | | Smoothness | Perfectly smooth | Irregular (bumps and dips ≤ ±85 m) | | How derived | Mathematical idealisation | Observed: satellites + terrestrial gravity | | Global / regional | Global single formula | Global model + regional refinements | | Time evolution | Fixed at definition | Slowly changing (~cm/decade) | | Heights above | Ellipsoidal height h | Orthometric height H | | Natural use case | Satellite tracking, GNSS | Engineering, hydrology, sea level | | Computational cost | Closed-form, fast | Spherical-harmonic evaluation | | Visualisation | Smooth flattened sphere | “Lumpy potato” visual model |

The two surfaces serve different domains. The ellipsoid is the geometric reference; the geoid is the gravitational reference. Neither replaces the other; both are needed for full coordinate work.

The fundamental equation: h = H + N

The relationship between the three heights:

h = H + N

Where:

• h — ellipsoidal height (height above the reference ellipsoid). What GPS reports natively.
• H — orthometric height (height above the geoid). What “height above sea level” means in everyday speech.
• N — geoid undulation (height of the geoid above the ellipsoid). Positive where the geoid is above the ellipsoid; negative where the geoid is below.

To convert:

• GPS-reported ellipsoidal height to sea-level height: H = h − N.
• Sea-level height to ellipsoidal height: h = H + N.

The value of N comes from a geoid model. For US work, GEOID18 (soon GEOID2022); globally, EGM2008. Most modern GPS receivers apply the conversion internally and report H, but some — and most professional GNSS units — expose h directly so the user can apply a specific geoid model.

A worked example

A surveyor in Denver, Colorado reads h = 1,650 m from her GPS receiver. The local geoid undulation from GEOID18 at her location is approximately N = −19.4 m (the geoid is 19.4 m below the WGS 84 ellipsoid).

H = h − N
H = 1,650 − (−19.4)
H = 1,650 + 19.4
H = 1,669.4 m

The surveyor is at an orthometric height of 1,669.4 m above mean sea level — within metres of the published Denver elevation of 1,609–1,675 m depending on which part of the metro area. The GPS reading alone, without the geoid correction, would understate the elevation by about 19 m.

In Reykjavík (Iceland) with h = 100 m and N ≈ +66 m:

H = 100 − 66 = 34 m

GPS reads 100 m at a place that's only 34 m above sea level — because the geoid is significantly above the WGS 84 ellipsoid in this region. Both numbers describe the same physical point; the difference is the choice of reference surface.

Why both surfaces exist

Different jobs:

• The ellipsoid is the satellite-friendly reference. GNSS satellites orbit Earth, and their positions and signal-propagation models are most naturally expressed in geocentric Cartesian coordinates (XYZ) referenced to the ellipsoid. The math is closed-form, fast, and globally consistent. The ellipsoid is the computational backbone of every modern GPS, GLONASS, Galileo, and BeiDou receiver.
• The geoid is the gravity-friendly reference. Water flows perpendicular to gravity, which means perpendicular to the geoid. Engineering systems that move water (sewers, irrigation, flood-control), measure flow (rivers), or that need consistent “up” direction (foundations, surveys, levelling networks) all reference the geoid. Without the geoid, an engineering project could grade a sewer line “downhill” on the ellipsoid that actually runs uphill in real water-flow terms.

The two domains coexist. A modern coordinate workflow typically takes ellipsoidal positions from GNSS at the input boundary, converts to orthometric heights using a geoid model, and stores and processes orthometric heights internally. The conversion is invisible to most users but foundational to every sea-level- relative claim.

When to use each

Use the ellipsoid (ellipsoidal height h) when:

• Working directly with raw GPS receiver output.
• Doing satellite-tracking arithmetic.
• Programming GNSS algorithms (orbit determination, signal propagation).
• Storing height data in an EPSG:4979 (WGS 84 3D geographic) CRS.

Use the geoid (orthometric height H) when:

• Reporting heights to non-specialist users (“how high is Mount Everest above sea level?”).
• Designing infrastructure that depends on water flow direction.
• Working with civil-engineering, surveying, or hydrology conventions.
• Comparing heights with topographic maps, tide-gauge records, or flood-zone references.

For coordinate work in Coordinately, the convention is orthometric output. The /tools/elevation tool applies the appropriate geoid model (USGS 3DEP for the US, OpenTopoData SRTM30m for the rest of the world) and returns orthometric elevation. The provenance — which dataset was used, and what accuracy band it falls in — is surfaced alongside every result.

Common misconceptions

“The geoid is below the ellipsoid everywhere.” False. The geoid is below the WGS 84 ellipsoid in some regions (e.g., the Indian Ocean, the Caribbean, CONUS) and above it in others (Europe, Iceland, Indonesia). The sign of N varies by location; the global range is roughly −106 m to +85 m.

“GPS gives sea-level height.” GPS gives ellipsoidal height natively. The conversion to sea-level (orthometric) height requires a geoid model. Most consumer GPS devices apply a global model (often EGM96 or EGM2008) and display H, but the conversion isn't always exact and isn't always disclosed. Survey-grade work specifies the geoid model used.

“The ellipsoid is more accurate because it's mathematical.” Accuracy is a property of measurement, not of mathematical idealisation. The ellipsoid is a smooth surface that doesn't match physical reality precisely. The geoid does match physical reality (it's defined by gravity) but is modelled to limited precision (current best models are accurate to ~5–10 cm globally, ~1 cm regionally). Both representations have finite accuracy bounds.

“You can compute orthometric height from coordinates alone.” You can't. You need a geoid model evaluated at those coordinates to produce N, then combine with the ellipsoidal height via H = h − N. Without a geoid model, the best you can do is the ellipsoidal height — which is not sea-level-relative.

“The geoid undulation is the same as elevation.” Geoid undulation N is the offset between two reference surfaces at a point — typically ±tens of metres. Elevation is the height of a physical surface above the geoid — typically zero to many thousand metres. The two concepts are distinct: N is a model parameter; elevation H is a property of the terrain.

“Modern GIS doesn't need to know about the distinction.” Modern GIS implicitly handles it — most tools convert between heights automatically using built-in geoid models. But the conversion is happening; just because it's hidden doesn't mean it's gone. For sub-decimetre work (survey monumentation, infrastructure as-builts, subsidence monitoring), explicit awareness of which model produced N for a given height value is essential. The geoid-model metadata is part of the height's provenance and should be recorded alongside the value.