Global Positioning System Reference
In-Depth Information
counted for by the smooth and simple ellipsoid. Gauss wrote: ''What we
call the surface of the Earth in a mathematical sense, is nothing else but
that surface, which everywhere intersects the direction of gravity at right
angles, and of which the surface of the ocean is a part. The direction of
gravity at every point is determined by the shape of the rigid part of the
Earth and its uneven density.''
15
What Gauss called his ''figure of the Earth'' was clearly the geoid, though
the term and idea would not become standard until later in the nineteenth
century. Friedrich Robert Helmert, writing in 1880 and 1884, laid the
foundations of modern geodesy on the framework of the geoid, a term he
coined. Today we know that the geoid is more than Gauss intended be-
cause we can measure the e√ects of lunar gravity and can see that the
earth's surface is continually moving, so we define the geoid as an aver-
age surface. It includes local gravity anomalies—a mountain range to the
north, for instance—but also other e√ects that are not necessarily under-
stood. For example, there has been an extended discussion in the technical
literature about whether an ellipsoid is the best theoretical shape—local
anomalies apart—to describe the earth. Perhaps the equator is not exactly
a circle; perhaps it is an ellipse. Perhaps the curvature at the South Pole
is less than the curvature at the North Pole, giving the earth a slight
pear shape.
These e√ects, if they exist, are due to the complex gravitational inter-
action of the earth with all the other bodies in the solar system. Because
they are not well established in the minds of theorists, these tiny gravita-
tional e√ects have not yet been applied to modify the reference ellipsoid,
but have been added to the geoid. To this extent, the geoid is a catchall for
everything we don't understand about the earth's shape. The di√erence
between the ellipsoid and the geoid is the di√erence between what we
understand about the shape of the earth and what we measure. That the
di√erence is small—less than a couple of hundred meters over the entire
surface of our planet—tells us that geodesists have a pretty good idea about
why our earth is the way it is.
16
Over a four-year period from 1821, Gauss conducted a detailed survey
15. Howarth (2007). Gauss was a mathematical genius of the first magnitude who
concerned himself with many of the applications of his math, including geodesy and
surveying. We will encounter him again in this topic.
16. For the history of modern geodesy, see, e.g., Hoare (2005), Smith (1997), Torge
(2001), or Wilford (2000). In addition, the USGS and NOAA maintain educational web-
sites on the subject.
