Global Positioning System Reference
In-Depth Information
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where
I
is the
u
u
identity matrix. This is followed by
u
solutions of the type (A.103),
using the columns of
C
for
c
u
, to obtain the respective
u
columns of the inverse of
N
.
The Cholesky factor
L
can be used directly to compute uncorrelated observations.
From (A.94) it follows that premultiplying
N
with
L
−
1
and postmultiplying it with
the transpose gives the identity matrix. Therefore, the Cholesky factor
L
can be used
in ways similar to the matrix
D
in (A.51). A frequent application is the decorrelation
of observations. In this case the inverse
L
−
1
is not required explicitly. For example, if
we let
L
now denote the Cholesky factor of the covariance matrix of the observations
Σ
b
, then the transformation (4.235) can be written as
×
L
−
1
v
L
−
1
Ax
L
−
1
=
+
(A.107)
Denoting the transformed observations by a bar, we get
[35
Ax
+
¯
v
=
(A.108)
L
¯
=
Lin
—
0.3
——
Nor
*PgE
(A.109)
L A
α
=
A
α
(A.110)
A
and the vector
¯
The matrix
can be computed directly with
L
using (A.110)
and (A.109), respectively. The subscript
in (A.110) indicates the column. Upon
completion of the adjustment the residuals follow from
α
L v
=
v
(A.111)
[35
It is at times advantageous to work with decorrelated observations. Examples are
horizontal angle observations or even GPS vectors. Decorrelated observations can
be added one at a time to the adjustment, whereas correlated observations should be
added by sets. See also Section 4.10.6 for a discussion of decorrelated redundancy
numbers.
A.4 LINEARIZATION
Observations are often related by nonlinear functions of unknown parameters. The
adjustment algorithm uses a linear functional relationship between the observations
and the parameters and uses iterations to account for the nonlinearity. To perform an
adjustment, one must therefore linearize these relationships. Expanding the functions
in a Taylor series and retaining only the linear terms accomplishes this. Consider the
nonlinear function
y
=
f(x)
(A.112)
which has one variable
x
. The Taylor series expansion of this function is
