This section was edited by Kenneth Foote, Department of Geography, University of Texas Austin.

This unit is part of the NCGIA Core Curriculum in Geographic Information Science. These materials may be used for study, research, and education, but please credit the author, Peter H. Dana, and the project, NCGIA Core Curriculum in GIScience. All commercial rights reserved. Copyright 1998 by Peter H. Dana.

Your comments on these materials are welcome. A link to an evaluation form is provided at the end of this document.

Topics covered in this unit

• This unit provides an overview of concepts related to the shape of the earth, including:
  • Geodetic Datums
  • Geometric Earth Models
  • Reference Ellipsoids
  • Earth Surfaces

Learning Outcomes

• After learning the material covered in this unit, students should gain an appreciation for:
  • The various methods of describing the size and shape of the earth
  • The evolution of a flat earth model into an accurate spherical representation

Instructors' Notes

Full Table of Contents

Metadata and Revision History

1. Introduction to Geodetic Datums

• Geodetic datums define the reference systems that describe the size and shape of the earth, and the origin and orientation of the coordinate systems used to map the earth.
  • Hundreds of different datums have been used to frame position descriptions since the first estimates of the earth's size were made by Aristotle.
  • Datums have evolved from those describing a spherical earth to ellipsoidal models derived from years of satellite measurements.

• Modern geodetic datums range from flat-earth models used for plane surveying to complex models used for international applications which completely describe the size, shape, orientation, gravity field, and angular velocity of the earth.

• While cartography, surveying, navigation, and astronomy all make use of geodetic datums, the science of geodesy is the central discipline for the topic.

• Referencing geodetic coordinates to the wrong datum can result in position errors of hundreds of meters.

• Different nations and agencies use different datums as the basis for coordinate systems used to identify positions in geographic information systems, precise positioning systems, and navigation systems.

• The diversity of datums in use today and the technological advancements that have made possible global positioning measurements with sub-meter accuracies requires careful datum selection and careful conversion between coordinates in different datums.

2. The Figure of the Earth

• Geodetic datums and the coordinate reference systems based on them were developed to describe geographic positions for surveying, mapping, and navigation.

• Through a long history, the "figure of the earth" was refined from flat-earth models to spherical models of sufficient accuracy to allow global exploration, navigation and mapping.

• True geodetic datums were employed only after the late 1700s when measurements showed that the earth was ellipsoidal in shape.

2.1. Geometric Earth Models

• Early ideas of the figure of the earth resulted in descriptions of the earth as an oyster (The Babylonians before 3000 B.C.), a rectangular box, a circular disk, a cylindrical column, a spherical ball, and a very round pear (Columbus in the last years of his life).

• Flat earth models are still used for plane surveying, over distances short enough so that earth curvature is insignificant (less than 10 kms).

• Spherical earth models represent the shape of the earth with a sphere of a specified radius.
  • Spherical earth models are often used for short range navigation (VOR-DME) and for global distance approximations.
  • Spherical models fail to model the actual shape of the earth.
  • The slight flattening of the earth at the poles results in about a twenty kilometer difference at the poles between an average spherical radius and the measured polar radius of the earth.

• Ellipsoidal earth models are required for accurate range and bearing calculations over long distances.
  • Loran-C, and GPS navigation receivers use ellipsoidal earth models to compute position and waypoint information.
  • Ellipsoidal models define an ellipsoid with an equatorial radius and a polar radius.
  • The best of these models can represent the shape of the earth over the smoothed, averaged sea-surface to within about one-hundred meters.

2.2. Reference Ellipsoids

• Reference ellipsoids are usually defined by semi-major (equatorial radius) and flattening (the relationship between equatorial and polar radii).

• Other reference ellipsoid parameters such as semi-minor axis (polar radius) and eccentricity can computed from these terms.
  • Figure 1 - Reference Ellipsoid Parameters

• Many reference ellipsoids are in use by different nations and agencies.
  • Figure 2 - Selected Reference Ellipsoids
  • Table 1 - A More Complete List of Reference Ellipsoids

• Some geodetic datums are based on ellipsoids that touch the surface of the earth at a defined point. North American Datum 1927 (NAD27) is tangent to the mean sea level surface at Meades Ranch in Kansas. NAD27 is not a global datum. Others are "topocentric" datums with a reference ellipsoid that has its center at the center of mass of the earth. Word Geodetic System 1984 (WGS-84) is an example of a global datum. These global datums can be better fits to the gravity surface for the entire earth but can be less accurate in specific areas.

2.3. Earth Surfaces

• The earth has a highly irregular and constantly changing surface.

• Models of the surface of the earth are used in navigation, surveying, and mapping.

• Topographic and sea-level models attempt to model the physical variations of the surface, while gravity models and geoids are used to represent local variations in gravity that change the local definition of a level surface.
  • Figure 3 - Earth Surfaces
  • The topographical surface of the earth is the actual surface of the land and sea at some moment in time. Aircraft navigators have a special interest in maintaining a positive height vector above this surface.
  • Sea level is the average (methods and temporal spans vary) surface of the oceans. Tidal forces and gravity differences from location to location cause even this smoothed surface to vary over the globe by hundreds of meters.
  • Gravity models attempt to describe in detail the variations in the gravity field. The importance of this effort is related to the idea of leveling. Plane and geodetic surveying uses the idea of a plane perpendicular to the gravity surface of the earth, the direction perpendicular to a plumb bob pointing toward the center of mass of the earth. Local variations in gravity, caused by variations in the earth's core and surface materials, cause this gravity surface to be irregular.
    • National Geodetic Survey Geoid-96
  • Geoid models attempt to represent the surface of the entire earth over both land and ocean as though the surface resulted from gravity alone. Bomford described this surface as the surface that would exist if the sea was admitted under the land portion of the earth by small frictionless channels.
  • The WGS-84 Geoid defines geoid heights for the entire earth.
  • The U. S. Defense Mapping Agency publishes a ten by ten degree grid of geoid heights for the WGS-84 geoid.
    • By using a four point linear interpolation algorithm at the four closest grid points, the geoid height for any location can be determined.
    • Table 2 - A Ten by Ten Degree WGS-84 Geoid Height Model.
    • The same grid can be used to produce a contour map of geoid heights for the globe.
    • Figure 4 - WGS-84 Geoid Heights
    • The National Imagery and Mapping Agency (formerly the Defense Mapping Agency) publishes a 0.25 degree model of the WGS-84 Geoid (1441 by 721 grid points).
    • Figure 5 - Shaded Relief of NIMA 0.25° WGS-84 Geoid Height Model
    • Figure 6 - Shaded Relief of NIMA 0.25° WGS-84 Geoid Height Model

3. Geodetic Datums

3.1. Datum Types

• Horizontal
  • Datums that define the relationship between the physical earth and horizontal coordinates such as latitude and longitude. Examples include the North American Datum of 1927 (NAD27) and the European Datum 1950 (ED50)..

• Vertical
  • Datums that define level surfaces. Examples include the National Geodetic Vertical Datum of 1929 (NGVD29) and the North American Vertical Datum of 1988 (NAVD88). Some are based on sea-level measurements and leveling networks (NGVD29), others on gravity measurements (NAVD88).

• Complete
  • Datums that describe both vertical and horizontal systems. Some, such as World Geodetic System 1984 (WGS-84), also describe other parameters such as the rotation rate of the earth and various physical constants such as the angular velocity of the earth and the earth's gravitational constant.

3.2. Datums in Use

• Hundreds of geodetic datums are in use around the world.

• North American Datum of 1983 is used for United States marine, aviation, and topographic maps (based in the past on NAD 1927).

• The Global Positioning system is based on the World Geodetic System 1984 (WGS-84).

• Parameters for simple XYZ conversion between many datums and WGS-84 are published by the Defense mapping Agency.
  • Figure 7 - Selected Geodetic Datums and Three Parameter Conversion Constants
  • Table 3 - A More Complete List of Geodetic Datums

3.3. Datum Shifts

• Coordinate values resulting from interpreting latitude, longitude, and height values using the wrong datum can cause position errors in three dimensions of up to one kilometer.
  • Figure 8 - Horizontal Position Shifts from Datum Differences

3.4. Datum Conversions

• Datum conversions are accomplished by various methods.

• Complete datum conversion is based on seven parameter transformations that include three translation parameters, three rotation parameters and a scale parameter. These conversions are only accurate in the region for which the parameters were computed. These can result in local accuracies of less than one meter.

• Multiple regression formulas can be used with parameters that represent fits to large contiguous land areas. Multiple regression equations can model datum shifts with accuracies of two meters over large (continental) areas. When coupled with tables of locally measured shifts these can provide better than one meter accuracies.

• Simple three parameter conversion between latitude, longitude, and height in different datums can be accomplished by conversion through Earth-Centered, Earth Fixed XYZ Cartesian coordinates in one reference datum and three origin offsets that approximate differences in rotation, translation and scale. These conversions can have accuracies ranging from two to twenty-five meters.
  • Figure 9 - Conversion from ECEF XYZ to Latitude, Longitude, and Height
  • Figure 10 - Conversion from Latitude, Longitude, and Height to ECEF XYZ.
  • Figure 11 - XYZ Three Parameter Datum Conversion

• The Standard Molodensky formulas can be used to convert latitude, longitude, and ellipsoid height in one datum to another datum if the Delta XYZ constants for that conversion are available and ECEF XYZ coordinates are not required.
  • Figure 12 - Standard Molodensky Datum Conversion

4. Review and Study Questions

4.1. Essay and Short Answer Questions

• Essay Questions
  • In what ways does the existence of hundreds of local and regional geodeitc datums limit the possibility of for international cooperation in GIS projects?
  • To what extent is the problem of georeferencing a major obstacle to the creation of global GIS?

• Short Answer
  • Why is it important to know the datum used for a given map?
  • Which datum are you most likely to encounter for maps used in the US?
  • Why is the use of the correct map datum important in determining location?

4.2. Multiple-Choice Questions

• What is a geodetic datum?
  • The latitude and longitude of Meades Ranch, Kansas, the NAD 27 origin.
  • The data point that defines the location of Greenwich, England.
  • A theoretical map project designed to provide accurate scale over the entire surface of an oblate spheroid.
  • The set of parameters that define the size and shape of the earth and the origin of coordinate systems that describe positions on the earth.
• Which of the following datums can be used anywhere in the world for accurate mapping?
  • WGS 1983 and NAD 1927
  • WGS 1972 and NAD 1984
  • NAD 1927 and NAD 1983
  • There are no global datums
  • None of the above
• Which of the following datums is the best representation of the shape (or gravity field) of the entire earth?
  • NAD 27
  • NAD 83
  • WGS 84
  • WGS 27
  • None of the above

5. Reference Materials

5.1. Print References

• Bomford, G. 1980. Geodesy. Oxford: Clarendon Press.

• Burkard, Richard K. 1983. Geodesy for the Layman. Washington, DC: NOAA.

• National Imagery and Mapping Agency. 1997. Department of Defense World Geodetic System 1984: Its Definition and Relationships with Local Geodetic Systems. NIMA TR8350.2 Third Edition 4 July 1997. Bethesda, MD: National Imagery and Mapping Agency.

• National Oceanic and Atmospheric Administration. 1986. Geodetic Glossary. Rockville, MD: National Geodetic Information Center.

• Schwarz, Charles R. 1989. North American Datum of 1983. Rockville, MD: National Geodetic Survey.

• Torge, Wolfgang. 1991 Geodesy, 2nd Edition, New York: deGruyter.

5.2. Web References

• National Imagery and Mapping Agency
• National Geodetic Survey Geoid-96 Page

5.3. Glossary

Evaluation

Citation

Dana, Peter H. (1997) The Shape of the Earth, NCGIA Core Curriculum in GIScience, http://www.ncgia.ucsb.edu/giscc/units/u015/u015.html, posted (today).