Basic GPS Concept - Fundamentals of Global Positioning System Receivers: A Software Approach - page 24

easier to show graphically. In order to solve a position in a two-dimensional case,
three satellites are required considering the user clock bias. In this discussion,
it is assumed that the user position can be uniquely determined as discussed in
Section 2.3. If this assumption cannot be used, four satellites are required.
Figure 2.8a shows the results measured by three satellites on a straight line,
and the user is also on this line. Figure 2.8b shows that the three satellites and
the user form a quadrangle. Two circles with the same center but different radii
are drawn. The solid circle represents the distance measured from the user to
the satellite with bias clock error. The dotted circle represents the distance after
the clock error correction. From observation, the position error in Figure 2.8a is
greater than that in Figure 2.8b because in Figure 2.8a all three dotted circles are
tangential to each other. It is difficult to measure the tangential point accurately.
In Figure 2.8b, the three circles intersect each other and the point of intersection
can be measured more accurately. Another way to view this problem is to measure
the area of a triangle made by the three satellites. In Figure 2.8a the total area
is close to zero, while in Figure 2.8b the total area is quite large. In general, the
larger the triangle area made by the three satellites, the better the user position
can be solved.
The general rule can be extended to select the four satellites in a three-
dimensional case. It is desirable to maximize the volume defined by the four
satellites. A tetrahedron with an equilateral base contains the maximum volume
and therefore can be considered as the best selection. Under this condition, one
satellite is at zenith and the other three are close to the horizon and separated by
120 degrees. ( 8 ) This geometry will generate the best user position estimation. If
all four satellites are close to the horizon, the volume defined by these satellites
and the user is very small. Occasionally, the user position error calculated with
this arrangement can be extremely large. In other words, the δv calculated from
Equation (2.11) may not converge.

2.15 DILUTION OF PRECISION ( 1,8 )

The dilution of precision (DOP) is often used to measure user position accuracy.
There are several different definitions of the DOP. All the different DOPs are a
function of satellite geometry only. The positions of the satellites determine the
DOP value. A detailed discussion can be found in reference 8. Here only the
definitions and the limits of the values will be presented.
The geometrical dilution of precision (GDOP) is defined as

σ x

1
σ

+ σ b GDOP = + σ y + σ z ( 2 . 58 )

where σ is the measured rms error of the pseudorange, which has a zero mean,
σ x σ y σ z are the measured rms errors of the user position in the xyz directions,
and σ b is the measured rms user clock error expressed in distance.