Surveying - The Science of Navigation - page 78

ment, improved farm planning and management, and more e√ective disas-
ter relief.
The accuracy of the system depends mostly upon the quality of the
receiver. We have seen that the satellite transmissions are tied to very
accurate clocks, but such clocks are very expensive, and the clocks in GPS
receivers are much less accurate. The degree to which clocks are accu-
rate influences the accuracy of position fixes, which are attained from
trilateration calculations. (I will unpack trilateration for you later.) Re-
ceiver clock error can be cancelled out, leaving an error that depends upon
the sophistication of the receiver software. 14 The accuracy for entry-level
GPS receivers is about 5 m in latitude or longitude. This can be improved
upon by fancy (and more costly) signal processing, so that the best GPS
fixes can position a receiver to within a few cubic centimeters. (The error
in altitude estimation averages about 1.6 times the error in latitude and
longitude.)
From the physicist's point of view, GPS is very interesting because it is
perhaps the only everyday technology that requires its designers to take
cognizance of Einstein's theory of relativity. There are two strands to rela-
tivity, and both of them influence GPS: e√ects that arise from the speed of a
GPS satellite relative to a receiver, and e√ects that arise from the mass of
the earth. The first strand is called special relativity ; it is the simpler conse-
quence of relativity and was discovered by Einstein as a young man (aged
26) in 1905. It is responsible for that equation, and it would probably have
been discovered within a few years anyway, even without Einstein. The
second strand is called general relativity . It was the brainchild of Einstein
alone, and it is altogether more complicated. He discovered it a decade or
so after special relativity, once he had learned the language of di√erential
geometry that is spoken by general relativity.
A GPS satellite travels at a speed of 14,000 km hr -1 relative to a receiver
14. Three satellite signals provide a fix, but the receiver clock error can be such that this
fix is o√ by hundreds of meters. This is because the distance of the satellite from the receiver
(required for trilateration, as we will see) is determined by timing the signal as it travels
between satellite and receiver. A receiver clock error of, say, 1 microsecond, means a
position error of 300 m. Because there are four satellites within range at any given instant, a
second fix (involving data from the fourth satellite) can be used to cancel the error; the
di√erence in arrival times of the signals from the satellites cancels the receiver's clock error.
This form of GPS signal processing, known as di√erential GPS , leads to increasingly accurate
positioning data. Other errors—notably the varying time taken for a signal to traverse the
ionosphere—can also be mitigated by other di√erential GPS processing techniques.