Global Positioning System Reference
In-Depth Information
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is zero because the residuals represent random errors for which positive and negative
errors of the same magnitude occur with the same probability. It follows from (4.41)
that
E(
w
)
=−
Ax
(4.63)
N
ote that
x
in (4.63) or (4.41) is not a random variable. In this expression,
x
simply
de
notes the vector of unknown parameters that have fixed values, even though the
va
lues are not known. The estimate
x
is a random variable because it is a function of
th
e observations. By using (4.35) for the definition of the covariance matrix (4.53)
an
d using (4.63), it follows that
E
ww
T
=
Σ
E(
w
)E(
w
)
T
+
w
(4.64)
[10
2
0
M
Axx
T
A
T
= σ
+
Substituting (4.64) into (4.61) yields the expected value for
v
T
Pv
:
E
v
T
Pv
= σ
Lin
—
2.7
——
Lon
PgE
0
Tr
r
I
r
−
M
−
1
A
A
T
M
−
1
A
−
1
A
T
(4.65)
2
= σ
0
(r
−
u)
Th
e difference
r
u
is called the degree of freedom and equals the number of
re
dundant equations in the model (4.36). Strictly, the degree of freedom is
r
−
R(
A
)
be
cause the second matrix in (4.65) is idempotent. The symbol
R(
A
)
denotes the rank
of
the matrix
A
. The a posteriori variance of unit weight is computed from
−
[10
v
T
P v
r
2
0
σ
=
(4.66)
−
u
Using (4.65), we see that
E
0
2
0
σ
= σ
(4.67)
Th
e expected value of the a posteriori variance of unit weight equals the a priori
va
riance of unit weight.
Finally, the estimated covariance matrices are
2
0
Q
x
Σ
=σ
(4.68)
x
2
0
Q
v
Σ
v
=σ
(4.69)
2
0
Q
Σ
a
=σ
(4.70)
a
With Equation (4.59) it follows that
Σ
a
=
Σ
b
−
Σ
v
(4.71)
