Performance of Stand-Alone GPS - Understanding GPS: Principles and Applications - page 324

the foghorns are located approximately at right angles with respect to the user loca-
tion. In Figure 7.5(b), the angle between the foghorns as viewed from the user is
much smaller. In both cases, portions of the error-free range rings are indicated and
intersect at the user's location. Additional ring segments that illustrate the variation
in range ring position resulting from ranging errors to the foghorns are included.
The error range illustrated in both figures is the same. The shaded regions indicate
the set of locations that can be obtained if one uses ranging measurements within
the illustrated error bounds. The accuracy of the computed location is very differ-
ent for the two cases. With the same measurement error variation, geometry in
Figure 7.5(b) gives considerably more error in the computed user's location than
in Figure 7.5(a), as is evident from comparison of the shaded regions. Geometry in
Figure 7.5(b) is said to have a larger dilution of precision than geometry in Figure
7.5(a). For comparable measurement errors, geometry in Figure 7.5(b) results in
larger errors in the computed location.
A formal derivation of the DOP relations in GPS begins with the linearization of
the pseudorange equations given in Section 2.4.2. The linearization is the Jacobian
relating changes in the user position and time bias to changes in the pseudorange
values. This relationship is inverted in accordance with the solution algorithm and is
used to relate the covariance of the user position and time bias to the covariance of
the pseudorange errors. The DOP parameters are defined as geometry factors that
relate parameters of the user position and time bias errors to those of the
pseudorange errors.
The offset x in the user's position and time bias relative to the linearization
point is related to the offset in the error-free pseudorange values by the relation

Hx = (7.30)

The vector x has four components. The first three are the position offset of the user
from the linearization point; the fourth is the offset of the user time bias from the
bias assumed in the linearization point. is the vector offset of the error-free
pseudorange values corresponding to the user's actual position and the pseudorange
values that correspond to the linearization point. H is the n × 4 matrix

aaa
aaa

1
1

x 1 y 1 z 1

x 2 y 2 z 2

H = (7.31)

M M M M

a a a 1

xn yn zn

( a xi , a yi , a zi ) are the unit vectors pointing from the linearization point to
the location of the ith satellite. If n

and the a i =

4 and data from just four satellites are being
used, and if the linearization point is close to the user's location, the user's location
and time offset are obtained by solving (7.30) for x (i.e., if the linearization point is
close enough to the user position, iteration is not required). One obtains

=

−1

xH = (7.32)