Basic GPS Concept - Fundamentals of Global Positioning System Receivers: A Software Approach - page 15

F. Compare δv with an arbitrarily chosen threshold; if δv is greater than the
threshold, the following steps will be needed.
G. Add these values δx u , δy u , δz u , δb u to the initial chosen position x u 0 , y u 0 ,
z u 0 , and the clock bias b u 0 ; a new set of positions and clock bias can be
obtained and they will be expressed as x u 1 , y u 1 , z u 1 , b u 1 . These values will
be used as the initial position and clock bias in the following calculations.
H. Repeat the procedure from A to G, until δv is less than the threshold. The
final solution can be considered as the desired user position and clock bias,
which can be expressed as x u , y u , z u , b u .

In general, the δv calculated in the above iteration method will keep decreasing
rapidly. Depending on the chosen threshold, the iteration method usually can
achieve the desired goal in less than 10 iterations. A computer program (p2 1)
to calculate the user position is listed at the end of this chapter. In this topic,
some lines in the programs are too long to be listed in one line; however, it
should be easily recognized.

2.8 USER POSITION IN SPHERICAL COORDINATE SYSTEM

The user position calculated from the above discussion is in a Cartesian coordinate
system. It is usually desirable to convert to a spherical system and label the
position in latitude, longitude, and altitude as the every-day maps use these
notations. The latitude of the earth is from − 90 to 90 degrees with the equator at 0
degree. The longitude is from − 180 to 180 degrees with the Greenwich meridian
at 0 degree. The altitude is the height above the earth's surface. If the earth is
a perfect sphere, the user position can be found easily as shown in Figure 2.4.
From this figure, the distance from the center of the earth to the user is

x u +

r = y u + z u ( 2 . 17 )

The latitude L c is

L c = tan − 1 z u x u + ( 2 . 18 )

y u

The longitude l is

l = tan − 1 y u
x u

( 2 . 19 )

The altitude h is

h = r − r e ( 2 . 20 )

where r e is the radius of an ideal spherical earth or the average radius of the
earth. Since the earth is not a perfect sphere, some of these equations need to
be modified.