Global Positioning System Reference
In-Depth Information
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We substitute (4.300) into the right-hand side of (4.301) and the matrix elements of
Q
i
of (4.292) into the left-hand side and compare the coefficient of
y
1
y
2
on both sides,
giving
2
q
x
1
,x
2
sin 2
ϕ
=
(4.302)
Q
1
Q
2
λ
− λ
Th
e eigenvalues follow directly from the characteristic equation
=
q
x
1
− λ
Q
i
− λ
Q
I
=
Q
Q
q
x
2
− λ
Q
−
q
x
1
− λ
q
x
1
,x
2
q
x
1
,x
2
=
0
Q
q
x
1
,x
2
q
x
2
− λ
(4.303)
Th
e solution of the quadratic equation is given in (4.304) to (4.308). The terms sin 2
ϕ
an
d cos 2
ϕ
determine the quadrant of
ϕ
.
[14
q
x
1
+
q
x
2
1
2
W
Q
1
λ
=
+
(4.304)
Lin
—
4.7
——
No
PgE
2
q
x
1
+
q
x
2
1
2
W
Q
2
λ
=
−
(4.305)
2
q
x
1
−
q
x
2
2
W
=
+
4
q
x
1
,x
2
(4.306)
2
q
x
1
,x
2
W
sin 2
ϕ
=
(4.307)
[14
q
x
1
−
q
x
2
cos 2
ϕ
=
(4.308)
W
Figure 4.6 shows the defining elements of the standard ellipse. Recall Equation
(4
.297) regarding the interpretation of the standard ellipses as a confidence region.
In
any adjustment, any two parameters can comprise
x
i
, regardless of the geometric
m
eaning of the parameters. Examples are the intercept and slope in the fitting of a
str
aight line or ambiguity parameters in the case of GPS carrier phase solutions. The
co
mponents
x
i
can always be interpreted as Cartesian coordinates for drawing the
sta
ndard ellipse and thus can give a graphical display of the covariance. In surveying
ne
tworks, the vectors
x
i
contain coordinates of stations in a well-defined coordinate
sy
stem. If
x
i
represents latitude and longitude or northing and easting, the horizontal
sta
ndard ellipse is computed. If
x
i
contains the vertical coordinate and easting, then
th
e standard ellipse in the prime vertical is obtained.
Because the shape of the standard ellipses and ellipsoids depends on the geometry
of the network through the design matrix and the weight matrix, the geometric inter-
pretation is enhanced if the network and the standard ellipses are displayed together.
Occasionally, users prefer to compute coordinate differences and their covariance
matrix and plot relative standard ellipses.
