The third coordinate in the geographic system
is approximately parallel to a radial line from
Earth’s center and is referred to as elevation.
Often, elevation is referenced to a sea level datum,
that is mean sea level as determined by tidal
gauges. This is not an unreasonable choice, but it
is subject to perturbations caused by water temperature variations and currents, and it is limited
in applications to the coastline. So-called leveling
surveys are performed to determine elevations
away from the coast as heights above (or below)
another datum called the geoid. The geoid is a
physically defined surface (Fig. 2.2a) that is everywhere perpendicular to the local direction of the
acceleration of gravity, g, so it is everywhere perpendicular to the direction we call “down” from
observations of falling objects. Leveling instruments (for example a bubble level) are capable of
very precise determinations of the local direction
of gravity and therefore can determine the shape
of the geoid. For a fluid Earth, the geoid would be
a perfect ellipsoid, but the Earth’s geoid has a
very irregular shape. These irregularities mean
that the local direction of gravitational acceleration (“down”) does not point directly toward the
center of the Earth. Indeed, this acceleration
varies from place to place, because of topography
and variable rock density. In some applications
the particular geoid chosen as the datum is the
one that best approximates mean sea level at the
nearest coastline.
Thus, an important question to ask about elevations is: what is the datum? Is it a mathematically defined ellipsoid or a physically defined
geoid (see Fig. 2.2a)? This information should be
provided on any map that depicts elevation. On
many older maps in the continental United
States, for example, the datum used is the
North American Datum for 1927 (NAD-27). This
coincides with the ellipsoid that closely approximates the geoid in North America, but the center
of this ellipsoid is about 100 m from the actual
center of the Earth. Another familiar datum is the
North American Datum for 1983 (NAD-83) that
appears, for example, on many topographic and
geologic maps published by the US Geological
Survey. Today the datum used for the Global
Positioning System (GPS) is the World Geodetic
System datum for 1984 (WGS-84) and this is almost
identical to the ellipsoid called GRS-80. However,
this ellipsoid may be separated from the mean sea
level geoid by a considerable distance. For
example, the WGS-84 ellipsoid is about 32.5 m
above the mean sea level geoid at Stanford
University in central California (Fig. 2.2b). Thus,
the reported elevation relative to the WGS-84
datum for a location just outside the Geology
Building at Stanford University using a high-precision GPS receiver is Ϫ3.6 m, but the building
clearly is not below sea level! The elevation relative to the mean sea level geoid is about ϩ28.9 m.
This difference is not due to errors in measurement, but rather to use of the different datums.
A metric coordinate system for location on the
Earth’s surface is becoming popular because of
the accessibility of inexpensive receivers for the
GPS and the fact that a metric system is easier to
manipulate. The GPS provides locations by measuring the times of travel of radio signals from a
set of satellites to a receiving antenna held at the
desired location (Hofmann-Wellenhof et al., 1997).
The travel time measurement requires very
precise (atomic) clocks on the satellites and
sophisticated electronic techniques for synchronizing the clock in the receiver to the satellite
clocks. Using these clocks, the time of sending
and receiving a radio signal from a particular
satellite can be differenced to compute the travel
time, t, for the signal from that satellite. Knowing
the velocity of the radio signal, v, corrected for
atmospheric delays, and the travel time, t, the distance from the antenna to that satellite is computed as d ϭ vt. Knowing the positions of all the
satellites (provided by the government operators
of the system) and the respective distances from
the antenna to each satellite, the location of the
antenna can be computed. A minimum of four
satellite distances is required, three to determine
location and a fourth to synchronize the clocks.
Any redundant data are used to refine the precision of the location.
Locations can be determined to better than
Ϯ1 m using portable receivers, and these are an
excellent choice for mapping most geological
structures (Chapter 2 Frontispiece). The two geologists shown on the frontispiece for this chapter
2.1 GEOGRAPHIC COORDINATES AND MAP PROJECTIONS
29
is approximately parallel to a radial line from
Earth’s center and is referred to as elevation.
Often, elevation is referenced to a sea level datum,
that is mean sea level as determined by tidal
gauges. This is not an unreasonable choice, but it
is subject to perturbations caused by water temperature variations and currents, and it is limited
in applications to the coastline. So-called leveling
surveys are performed to determine elevations
away from the coast as heights above (or below)
another datum called the geoid. The geoid is a
physically defined surface (Fig. 2.2a) that is everywhere perpendicular to the local direction of the
acceleration of gravity, g, so it is everywhere perpendicular to the direction we call “down” from
observations of falling objects. Leveling instruments (for example a bubble level) are capable of
very precise determinations of the local direction
of gravity and therefore can determine the shape
of the geoid. For a fluid Earth, the geoid would be
a perfect ellipsoid, but the Earth’s geoid has a
very irregular shape. These irregularities mean
that the local direction of gravitational acceleration (“down”) does not point directly toward the
center of the Earth. Indeed, this acceleration
varies from place to place, because of topography
and variable rock density. In some applications
the particular geoid chosen as the datum is the
one that best approximates mean sea level at the
nearest coastline.
Thus, an important question to ask about elevations is: what is the datum? Is it a mathematically defined ellipsoid or a physically defined
geoid (see Fig. 2.2a)? This information should be
provided on any map that depicts elevation. On
many older maps in the continental United
States, for example, the datum used is the
North American Datum for 1927 (NAD-27). This
coincides with the ellipsoid that closely approximates the geoid in North America, but the center
of this ellipsoid is about 100 m from the actual
center of the Earth. Another familiar datum is the
North American Datum for 1983 (NAD-83) that
appears, for example, on many topographic and
geologic maps published by the US Geological
Survey. Today the datum used for the Global
Positioning System (GPS) is the World Geodetic
System datum for 1984 (WGS-84) and this is almost
identical to the ellipsoid called GRS-80. However,
this ellipsoid may be separated from the mean sea
level geoid by a considerable distance. For
example, the WGS-84 ellipsoid is about 32.5 m
above the mean sea level geoid at Stanford
University in central California (Fig. 2.2b). Thus,
the reported elevation relative to the WGS-84
datum for a location just outside the Geology
Building at Stanford University using a high-precision GPS receiver is Ϫ3.6 m, but the building
clearly is not below sea level! The elevation relative to the mean sea level geoid is about ϩ28.9 m.
This difference is not due to errors in measurement, but rather to use of the different datums.
A metric coordinate system for location on the
Earth’s surface is becoming popular because of
the accessibility of inexpensive receivers for the
GPS and the fact that a metric system is easier to
manipulate. The GPS provides locations by measuring the times of travel of radio signals from a
set of satellites to a receiving antenna held at the
desired location (Hofmann-Wellenhof et al., 1997).
The travel time measurement requires very
precise (atomic) clocks on the satellites and
sophisticated electronic techniques for synchronizing the clock in the receiver to the satellite
clocks. Using these clocks, the time of sending
and receiving a radio signal from a particular
satellite can be differenced to compute the travel
time, t, for the signal from that satellite. Knowing
the velocity of the radio signal, v, corrected for
atmospheric delays, and the travel time, t, the distance from the antenna to that satellite is computed as d ϭ vt. Knowing the positions of all the
satellites (provided by the government operators
of the system) and the respective distances from
the antenna to each satellite, the location of the
antenna can be computed. A minimum of four
satellite distances is required, three to determine
location and a fourth to synchronize the clocks.
Any redundant data are used to refine the precision of the location.
Locations can be determined to better than
Ϯ1 m using portable receivers, and these are an
excellent choice for mapping most geological
structures (Chapter 2 Frontispiece). The two geologists shown on the frontispiece for this chapter
2.1 GEOGRAPHIC COORDINATES AND MAP PROJECTIONS
29