Global Positioning System Reference
In-Depth Information
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4.5 OBSERVATION AND CONDITION EQUATION MODELS
Often there is an explicit relationship between the observations and the parameters,
such as
a
=
f
(
x
a
)
(4.79)
This is the observation equation model. Comparing both mathematical models (4.36)
and (4.79), and taking the definition of the matrix
B
(4.42) into account, we see that
the observation equation model follows from the mixed model using the specification
B
≡−
I
(4.80)
[11
≡
w
=
f
(
x
0
)
−
b
=
0
−
b
(4.81)
It is customary to denote the discrepancy by
instead of
w
when dealing with the
Lin
—
1.0
——
Nor
*PgE
observation equation model. The symbol
0
equals the value of the observations
as computed from the approximate parameters
x
0
. The point of expansion for the
linearization is
x
0
; the observation vector is not involved in the iteration because of
the explicit form of (4.79). The linearized equations
n
v
1
=
n
A
uu
x
1
+
n
1
(4.82)
are the
observation equations
. There is one equation for each observation in (4.82).
If the observations are related by a nonlinear function without use of parameters,
we speak of the condition equation model. It is written as
[11
a
)
=
f
(
o
(4.83)
By comparing this with the mixed model (4.36), and applying the definition of the
A
matrix (4.43) we see that the condition equation model follows upon the specification
A
=
O
(4.84)
The linear equations
r
B
nn
v
1
+
r
w
1
=
o
(4.85)
are called the condition equations. The iteration for the model (4.85) is analogous to a
mixed model with the added simplification that there is no
A
matrix and no parameter
vector
x
.
The significance of these three models (observation, condition, and mixed) is that
a specific adjustment problem can usually be formulated more easily in one of the
models. Clearly, that model should be chosen. There are situations in which it is
equally easy to use any of the models. A typical example is the adjustment of a level
