Reference Ellipsoids & the Geoid

Overview#

The Earth is not a sphere. It is not even a simple ellipsoid. Its true physical surface -- mountains, valleys, ocean trenches -- is far too irregular to serve as a mathematical reference for positioning. To solve this problem, geodesists and surveyors work with three distinct surfaces, each serving a different purpose:

1. The Topographic Surface -- the actual physical surface of the Earth, including all terrain features. This is where we stand, where we measure, and where boundaries exist. It cannot be described by a simple equation.

2. The Geoid -- an equipotential surface of gravity that closely approximates mean sea level. It is the physical reference for heights. Water, if unaffected by tides and currents, would settle along the geoid.

3. The Ellipsoid -- a smooth mathematical surface (an oblate spheroid) that approximates the overall shape of the Earth. It is the reference for horizontal positions and the basis for coordinate systems like NAD 83 and WGS 84.

"Three surfaces must be clearly understood in studying geodesy and mapping. They are the Earth's surface itself, the geoid, and the reference ellipsoid." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 19, p. 540

Every geodetic computation a surveyor performs -- computing state plane coordinates, converting GPS heights, performing datum transformations -- depends on understanding how these three surfaces relate to one another. The ellipsoid gives us a computable geometry. The geoid gives us a physically meaningful "level" reference. The topographic surface is where we actually work.

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The Ellipsoid#

Definition and Geometry

A reference ellipsoid is an oblate spheroid -- a three-dimensional surface formed by rotating an ellipse about its minor (polar) axis. It is defined by two parameters:

• Semi-major axis ($a$) -- the equatorial radius
• Semi-minor axis ($b$) -- the polar radius

Because the Earth bulges at the equator due to its rotation, $a > b$. The amount of this bulging is expressed as the flattening:

$f = \frac{a - b}{a}$

Flattening is a very small number (roughly $1/298$ for the Earth), so it is conventionally expressed as the inverse flattening $1/f$. A sphere has $f = 0$; a completely flat disk has $f = 1$.

A related parameter is the first eccentricity squared:

$e^2 = 2f - f^2 = \frac{a^2 - b^2}{a^2}$

The eccentricity describes how much the ellipse deviates from a circle. For the Earth, $e^2 \approx 0.00669$, meaning the deviation is small but significant over long distances.

Why an Ellipsoid?

A sphere would be simpler, but it introduces errors of up to 21 kilometers in position. The ellipsoid captures the Earth's equatorial bulge (about 21.4 km difference between equatorial and polar radii) while remaining mathematically tractable. All standard geodetic formulas -- for distance, azimuth, latitude, longitude -- are built on ellipsoidal geometry.

"Since the Earth's equatorial radius is about 21 km longer than the polar radius, the oblate ellipsoid is a much better approximation of the Earth's shape than a sphere." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 19, p. 541

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Common Reference Ellipsoids#

Over the past two centuries, geodesists have defined many ellipsoids, each a best fit for a particular region or for the Earth as a whole. Three are of primary importance to surveyors working in the United States:

Clarke 1866

The Clarke 1866 ellipsoid was the foundation for the North American Datum of 1927 (NAD 27). It was a best fit for North America based on 19th-century survey measurements. NAD 27 used Meades Ranch, Kansas as its origin point.

GRS 80

The Geodetic Reference System of 1980 (GRS 80) is a geocentric ellipsoid adopted by the International Association of Geodesy. It is the ellipsoid for NAD 83. Unlike Clarke 1866, GRS 80 is not tied to a single surface station -- it is Earth-centered, Earth-fixed.

WGS 84

The World Geodetic System 1984 (WGS 84) ellipsoid is used by the Global Positioning System. Its semi-major axis is identical to GRS 80, and its flattening differs by an amount so small (roughly 0.1 mm in the semi-minor axis) that the two ellipsoids are effectively the same for all practical surveying purposes.

"For practical purposes, the GRS 80 and WGS 84 ellipsoids can be considered the same." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 19, p. 543

This shift in both the size and center of the ellipsoid is why coordinates changed significantly between NAD 27 and NAD 83 -- a point's latitude and longitude can shift by tens of meters.

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The Geoid#

An Equipotential Surface

The geoid is the equipotential surface of the Earth's gravity field that best fits mean sea level in a least-squares sense. "Equipotential" means that the force of gravity is everywhere perpendicular to the surface, and a free water surface at rest would coincide with it.

"The geoid is defined as that particular equipotential surface that most closely corresponds to mean sea level." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 19, p. 540

If you could dig canals from the ocean to any point on the continents and let water fill them (with no currents, tides, or friction), the water surface would trace out the geoid.

Why the Geoid Is Irregular

The geoid is not smooth. It has bumps and depressions caused by the uneven distribution of mass within the Earth -- denser rock pulls the geoid up; less dense material lets it sag. Mountain ranges, ocean trenches, mantle convection patterns, and variations in crustal thickness all warp the geoid away from any simple ellipsoid.

The geoid departs from the best-fitting ellipsoid (GRS 80) by as much as:

• +85 meters near New Guinea (geoid above the ellipsoid)
• -106 meters in the Indian Ocean (geoid below the ellipsoid)

These departures, while small compared to the Earth's radius, are enormous in the context of surveying, where centimeter-level accuracy is routine.

The Geoid as a Height Reference

When a surveyor reports an elevation, they mean the distance above the geoid -- an orthometric height. Spirit leveling measures differences in orthometric height because the bubble aligns with the local gravity vector, which is perpendicular to the geoid.

GPS, on the other hand, measures positions relative to the ellipsoid. The height from GPS is an ellipsoidal height -- the distance above or below the mathematical ellipsoid surface along a line perpendicular to it. Converting between these two height systems is one of the most important practical applications of geoid models.

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Geoid Undulation ($N$)#

The geoid undulation (or geoid height) $N$ is the separation between the ellipsoid and the geoid at a given location. The fundamental relationship linking the three height types is:

$h = H + N$

where:

• $h$ = ellipsoidal height (from GPS)
• $H$ = orthometric height (elevation above the geoid)
• $N$ = geoid undulation (geoid height relative to the ellipsoid)

When $N$ is negative, the geoid lies below the ellipsoid. When $N$ is positive, the geoid lies above the ellipsoid.

Geoid Undulation in the United States

In the conterminous United States, geoid undulations relative to the GRS 80 ellipsoid (NAD 83) range from approximately $-8$ meters along the Atlantic coast to roughly $-53$ meters in the Rocky Mountain region. The geoid is everywhere below the ellipsoid across the continental U.S.

This means that for a given point, the ellipsoidal height $h$ is always less than the orthometric height $H$, because $N$ is negative: $h = H + N$ yields a smaller number when $N < 0$. For example, a benchmark with orthometric elevation $H = 1{,}500.00$ m in Colorado, where the local geoid undulation is $N = -16$ m, would have an ellipsoidal height of $h = 1{,}500 + (-16) = 1{,}484$ m. The ellipsoidal height is lower because the geoid sits below the ellipsoid at that location.

"In the conterminous United States, geoidal heights are negative, ranging from approximately $-8$ m to $-53$ m." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 19, p. 548

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Geoid Models#

Purpose and Application

Because the geoid is an irregular physical surface, it cannot be described by a simple formula. Instead, the National Geodetic Survey (NGS) publishes geoid models -- gridded datasets of geoid undulation values ($N$) covering the United States. These models allow surveyors to convert GPS ellipsoidal heights to orthometric elevations:

$H = h - N$

GEOID18

GEOID18 is the current hybrid geoid model published by NGS (released 2018). It provides geoid undulations at a resolution of $1' \times 1'$ (approximately 1.8 km) across the conterminous United States, Alaska, Hawaii, and U.S. territories.

The term "hybrid" means GEOID18 is constrained to fit published NAVD 88 orthometric heights at GPS-on-benchmarks stations, so it directly converts NAD 83 ellipsoidal heights to NAVD 88 elevations. This is distinct from a purely gravimetric geoid model, which is computed from gravity data alone.

Practical Workflow

A typical GPS-to-elevation workflow:

1. Observe ellipsoidal height $h$ from GPS (referenced to NAD 83 / GRS 80).
2. Look up geoid undulation $N$ from GEOID18 at the observation point.
3. Compute orthometric height: $H = h - N$.

Since $N$ is negative across the U.S., subtracting a negative number increases the value, so $H > h$ -- the orthometric elevation is greater than the ellipsoidal height. NGS provides free online tools (including the NGS Coordinate Conversion and Transformation Tool, or NCAT) to perform these lookups.

Looking Ahead: NAPGD2022

NGS is developing a modernized National Spatial Reference System that will replace NAVD 88 with a geopotential datum defined by a gravimetric geoid model. Under this framework, orthometric heights will be derived directly from a high-resolution geoid model rather than from the existing network of leveled benchmarks. This represents a fundamental shift in how heights are determined in the United States.

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Radii of Curvature#

The ellipsoid is not a sphere, so its radius of curvature varies with latitude and direction. Two principal radii matter for geodetic computations:

Radius of Curvature in the Meridian ($M$)

The meridional radius of curvature $M$ describes the curvature of the ellipsoid along a north-south line (a meridian) at geodetic latitude $\phi$:

$M = \frac{a(1 - e^2)}{(1 - e^2 \sin^2\phi)^{3/2}}$

$M$ is smallest at the equator and largest at the poles. At the equator ($\phi = 0$), $M \approx 6{,}335{,}439$ m for GRS 80. At the poles ($\phi = 90°$), $M \approx 6{,}399{,}594$ m.

Radius of Curvature in the Prime Vertical ($\mathcal{N}$)

The prime vertical radius of curvature $\mathcal{N}$ (sometimes written $N$, though this conflicts with the notation for geoid undulation) describes the curvature of the ellipsoid along an east-west line at latitude $\phi$:

$\mathcal{N} = \frac{a}{(1 - e^2 \sin^2\phi)^{1/2}}$

$\mathcal{N}$ is also smallest at the equator ($\approx 6{,}378{,}137$ m, equal to $a$) and largest at the poles ($\approx 6{,}399{,}594$ m, where it equals $M$).

Comparison of Radii at Selected Latitudes

At the equator, the east-west curvature radius exceeds the north-south radius by nearly 43 km. This difference shrinks with latitude and vanishes at the poles, where the ellipsoid's cross-section is circular.

Why Surveyors Need Radii of Curvature

These radii appear in many geodetic formulas, including:

• Arc-to-distance conversions -- converting an angular difference in latitude or longitude to a ground distance
• Geodetic inverse and direct problems -- computing azimuths and distances between points on the ellipsoid
• Map projection formulas -- Lambert conformal conic and Transverse Mercator projections use $M$ and $\mathcal{N}$ in their defining equations
• Reduction of observations -- correcting measured distances and angles to the ellipsoidal surface

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Key Takeaways#

• Surveyors work with three surfaces: the topographic surface (physical), the geoid (physical/gravitational), and the ellipsoid (mathematical). Each serves a distinct purpose.
• A reference ellipsoid is defined by its semi-major axis $a$ and flattening $f$. The GRS 80 and WGS 84 ellipsoids are virtually identical and underpin NAD 83 and GPS, respectively.
• The geoid is the equipotential surface of gravity that defines the meaning of "elevation." It is irregular, reflecting the Earth's internal mass distribution.
• Geoid undulation $N$ connects the two height systems: $h = H + N$. In the conterminous U.S., $N$ ranges from about $-8$ m to $-53$ m.
• GEOID18 is the current NGS model for converting GPS ellipsoidal heights to NAVD 88 orthometric heights.
• The ellipsoid's radii of curvature ($M$ and $\mathcal{N}$) vary with latitude and are foundational to geodetic distance, azimuth, and projection computations.
• Understanding the separation between the ellipsoid and the geoid is essential for any surveyor working with GPS/GNSS-derived heights.

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References#

• Ghilani, C. D., & Wolf, P. R. (2012). Elementary Surveying: An Introduction to Geomatics (13th ed.). Pearson. Chapters 19--20.
• National Geodetic Survey. (2018). GEOID18 Technical Details. National Oceanic and Atmospheric Administration.
• Torge, W., & Muller, J. (2012). Geodesy (4th ed.). De Gruyter.
• National Geodetic Survey. NGS Coordinate Conversion and Transformation Tool (NCAT). https://www.ngs.noaa.gov/NCAT/
• Defense Mapping Agency. (1984). Department of Defense World Geodetic System 1984. DMA Technical Report 8350.2.
• Moritz, H. (1980). Geodetic Reference System 1980. Bulletin Geodesique, 54(3), 395--405.