Global Positioning System Reference
In-Depth Information
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be imposed sequentially to eliminate a singularity in the first group; e.g., conditions
should not be used sequentially to define the coordinate system. A one-step solution
is given by (4.118).
The a posteriori variance of unit weight is always computed from the final set
of residuals. The degree of freedom increases by 1 for every observed parameter
function, weighted parameter, or condition. In nonlinear adjustments the linearized
condition must always be evaluated for the current point of expansion, i.e., the point
of expansion of the last iteration (current solution).
The expressions in Table 4.2 and Table 4.5 are almost identical. The only differ-
ence is that the matrix
T
contains the matrix
M
2
in Table 4.2.
4.
8 MINIMAL AND INNER CONSTRAINTS
[12
Th
is section deals with the implementation of minimal and inner constraints to the
ob
servation equation model. The symbol
r
denotes the rank of the design matrix,
R(
n
A
u
)
u
. Note that the use of the symbol
r
in this context
is
entirely different from its use in the mixed model, where
r
denotes the number of
eq
uations. The rank deficiency of
u
=
R(
A
T
PA
)
=
r
≤
Lin
—
-1.
——
Lon
PgE
r
is generally caused by a lack of coordinate
sy
stem definition. For example, a network of distances is invariant with respect to
tra
nslation and rotation, a network of angles is invariant with respect to translation,
ro
tation, and scaling, and a level network (consisting of measured height differences)
is
invariant with respect to a translation in the vertical. The rank deficiency is dealt
w
ith by specifying
u
−
r
conditions of the parameters. Much of the theory of inner
an
d minimal constraint solution is discussed by Pope (1971). The main reason for
de
aling with minimal and inner constraint solutions is that this type of adjustment is
im
portant for the quality control of observations. Inner constraint solutions have the
ad
ditional advantage that the standard ellipses (ellipsoids) represent the geometry as
im
plied by the
A
and
P
matrices.
The formulation of the least-squares adjustment for the observation equation model
in
the presence of a rank deficiency is
−
[12
n
v
1
=
n
A
u
x
B
+
n
1
(4.167)
2
0
Σ
−
1
P
= σ
(4.168)
b
=
r
B
u
x
B
o
(4.169)
u
−
The subscript
B
indicates that the solution of the parameters
x
depends on the special
condition implied by the
B
matrix in (4.169). This is the observation equation model
with conditions between the parameters that was treated in Section 4.7. The one-step
solution is given by (4.106):
A
T
PA B
T
BO
x
B
−
A
T
P
−
=
(4.170)
k
2
o
