Geometry of the Sphere - Spatial Mathematics: Theory and Practice through Mapping - page 15

A

E B

C

Figure 1.12 A, B, and C represent seismic stations. The point E, calculated from circle
intersections, represents the estimated epicenter location.

the accuracy of the position of the epicenter. The coordinates may be quite
precise, such as to six significant digits to the right of the decimal point, but
the accuracy may not reflect this precision. Furthermore, the error produced
by the misreading of the timing of the seismic waves may be random, result-
ing in a lower precision in terms of the location of any aftershocks recorded.
Repeated measurements may not show the same results. The data are still
valuable, but like any data, they must be as fully understood as possible so
that their limitations can be recognized enabling the data user to make well-
informed choices.

A location generated using a Global Positioning System (GPS) is derived in a
similar manner. It offers another example of limitations that must be consid-
ered when using data. Each GPS satellite generates a signal that is received by
the receiver held by the operator on the surface of the Earth. The receiver uses
the messages it receives to determine the transit time of each message and
computes the distances to each satellite based on the formula: Distance = rate
× time. Because the signals travel at the speed of light, and because the GPS
satellites travel in near-circular orbits with radii at 26,000 kilometers, the time
it takes for the signals to travel from satellite to receiver is only a tiny fraction
of a second. Therefore, timing is everything. Determining the precise time
allows for precise positions to be calculated. The major difference between
the determination of seismic epicenters and the determination of a ground
position via GPS is that the former works in two dimensions while the latter
works in three dimensions. Therefore, in the GPS world, instead of circles, we
are working with spheres ( Figure 1.13 ) .

If only one satellite's signal is sensed by the GPS receiver, all the GPS receiver
knows is that it is somewhere on a sphere at a certain distance from that