Global Positioning System Reference
In-Depth Information
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Because the diagonal elements of all three covariance matrices in (4.71) are positive,
it follows that the variances of the adjusted observations are smaller than those of the
original observations. The difference is a function of the geometry of the adjustment,
as implied by the covariance matrix
Σ
v
.
4.
4.5 Iterations
Because the mathematical model is generally nonlinear, the least-squares solution
must be iterated. Recall that (4.36) is true only for
(
a
,
x
a
)
. Since neither of these
quantities is known before the adjustment, the initial point of expansion is chosen as
(
b
,
x
0
)
. For the
i
th iteration, the linearized model can be written
B
x
0
i
,
0
i
v
i
+
A
x
0
i
,
0
i
x
i
+
w
x
0
i
,
0
i
=
o
(4.72)
[10
0
i
,
x
0
i
)
represents the previous solution. The symbols
where the point of expansion
(
ai
and
x
ai
denote the adjusted observations and adjusted parameters for the current
(
i
th) solution. They are computed from
Lin
—
0.0
——
Lon
PgE
v
i
=
ai
−
0
i
(4.73)
x
i
=
x
ai
−
x
0
i
(4.74)
once the least-squares solution of (4.72) has been obtained. The iteration starts with
01
x
0
. If the adjustment converges properly, then both
v
i
and
x
i
converge to zero, or, stated differently,
=
b
and
x
01
=
a
and
x
a
,
respectively. The quantity
v
i
does not equal the residuals. The residuals express the
random difference between the adjusted observations and the original observations
according to Equation (4.39). Defining
ai
and
x
ai
converge toward
[10
v
i
=
ai
−
b
(4.75)
it follows from (4.73) that
v
i
=
v
i
+
(
b
−
0
i
)
(4.76)
Substituting this expression into (4.72) gives
B
x
0
i
,
01
v
i
+
A
x
0
i
,
01
x
i
+
w
x
0
i
,
+
B
x
0
i
,
b
−
0
i
)
=
o
(4.77)
01
(
01
The formulation (4.77) assures that the vector
v
i
converges toward the vector of
residuals
v
. The last term in (4.77) will be zero for the first iteration when
0
i
=
b
.
The iteration has converged if
v
T
Pv
i
−
v
T
Pv
i
−
1
<ε
(4.78)
where
ε
is a small positive number.
