Global Positioning System Reference
In-Depth Information
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It follows from the law of variance propagation (4.34) and (4.53) that
P
−
1
B
T
M
−
1
M
A
A
T
M
−
1
A
−
1
A
T
M
−
1
BP
−
1
=
−
Q
v
(4.56)
The adjusted observations are
a
=
b
+
v
=
b
+
P
−
1
B
T
M
−
1
A
A
T
M
−
1
A
−
1
A
T
M
−
1
(4.57)
P
−
1
B
T
M
−
1
w
−
Be
cause
P
−
1
B
T
M
−
1
A
A
T
M
−
1
A
−
1
A
T
M
−
1
B
∂
a
P
−
1
B
T
M
−
1
B
b
=
I
+
−
(4.58)
∂
[10
it follows that
Lin
—
*
1
——
Lon
PgE
Q
=
Q
b
−
Q
v
(4.59)
a
where the inverse of
P
has been replaced by
Q
according to (4.4)
b
4.
4.4 A Posteriori Variance of Unit Weight
The minimum of
v
T
Pv
follows from (4.49), (4.50), and (4.52) as
[10
w
T
M
−
1
M
−
1
A
A
T
M
−
1
A
−
1
A
T
M
−
1
]
w
v
T
Pv
=
−
(4.60)
Th
e expected value of this random variable is
E
v
T
Pv
=
E
Tr
v
T
Pv
E
Tr
w
T
M
−
1
M
−
1
A
A
T
M
−
1
A
−
1
A
T
M
−
1
w
=
−
E
Tr
M
−
1
M
−
1
A
A
T
M
−
1
A
−
1
A
T
M
−
1
ww
T
(4.61)
=
−
Tr
M
−
1
M
−
1
A
A
T
M
−
1
A
−
1
A
T
M
−
1
E
ww
T
=
−
Th
e trace (Tr) of a matrix equals the sum of its diagonal elements. In the first part
of
(4.61), the property that the trace of a 1
1 matrix equals the matrix element
its
elf is used. Next, the matrix products are switched, leaving the trace invariant. In
th
e last part of the equation, the expectation operator and the trace are switched. The
expected value
E(
ww
T
)
can be readily computed. Per definition, the expected value
of the residuals
×
E(
v
)
=
o
(4.62)
