PENN STATE SURVEYING PROGRAM

Geodetic Models for the Earth

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Surfaces on the Earth

The Earth is a physical object rotating around its polar axis. In laymen's terms, we think of the Earth as a sphere. However it is obvious with the mountains, valleys, oceans, and other physical features that the earth is not a true sphere. There are three surfaces that surveyors must be aware of when performing their jobs. Understanding these surfaces is essential to understanding the proper formation of a map. The surfaces are

  1. Topographic Surface: The physical surface of the earth. This surface undulates, has abrupt elevation changes, and is the surface that we map when developing a topographic map. It is also this surface used by boundary surveyors when writing legal descriptions. This surface is represented by the green line in Figure 1.
  2. Geoid: This surfaces conforms to the pull of gravity. It is an equipotential gravitational surface. It undulates with changes in gravitational pull caused by orbiting bodies such as the moon and planets, and by local density changes in the surface of the earth. This irregular surface closely approximates mean sea level. This surface is shown in red in Figure 1. It
    1. varies from MSL by as much as 1 meter.
    2. is the surface to which we level our instruments
  3. Ellipsoid: This is a mathematical surface that generally approximates the geoid. As shown in Figure 1, it is an ellipse that is rotated about its semi-major axis. There are many defined ellipsoids which approximate the Earth. For instance in the U.S., the Clarke 1866 ellipsoid was used for the North American Datum of 1927. Although this ellipsoid was developed for the continent of Africa, it is the one that best approximated the geoid for the U.S. Currently, the North American Datum of 1983 uses the GRS80 ellipsoid. With the advent of GPS, control stations are now being defined with International Terrestrial Reference Frame of 1996 (ITRF96).


Geometric Properties of an Ellipsoid

An ellipse/ellipsoid can be defined by only two unique parameters. As is shown in Figure 1, the length of the semimajor axis, a, and the length of the semiminor axis, b, can define the ellipse. It is more common in practice to define the ellipse with the semi-major axis and the flattening factor of the ellipse. The flattening of the ellipse, f, is defined by the mathematical expression

Flattening factors for the Earth range from 1/300 to 1/295. Another important defining parameter is eccentricity. The square of the eccentricity term is mathematically defined as

e2 = 2f - f2

Parameters for commonly used ellipsoids are:

Definitions

Meridians are great circles that pass through the Polar axis of the Earth. In 1884, the International Meridian Conference established the meridian passing through the transit instrument at the Royal Observatory in Greenwich, England as the prime meridian. The prime meridian is also known as the Greenwich Meridian, and has a longitude of 0°. The locations of all other meridians are based on time. Meridians to the west of Greenwich are assigned values of 0° to 180° West longitude. Meridians to the east of Greenwich are assigned East longitudes. Longitudes are the arc distance (angle) in the equatorial plane from the Greenwich meridian to a meridian passing through the observer's station. Longitudes are considered

Parallels of Latitude are approximately same distance apart and run in a north-south direction. The are defined as the arc distance in the plane of the meridian from the equatorial plane to the prime vertical.

Prime vertical is a radius of the great circle that is perpendicular to the meridian passing through an observer's station. The plane that this great circle describes is known as the normal section, and the length of this radius is known as the normal. The angle that the normal make with the equatorial plane is the latitude of the observer. The length of the normal is given as

where f is the latitude of the station.

Radius in the Meridian is the radius of the great circle that passes through the observer's station and the Earth's Polar axes. Its magnitude is given as

Elevation is the vertical distance from a reference surface to the observer's station. The following are common datums used for elevations

COORDINATE SYSTEMS

There are several coordinate systems that can be used to describe the location of a point on the Earth. The most commonly used are



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Created by: Charles D. Ghilani
Copyright © 1999, Penn State Surveying Program