Global Positioning System Reference
In-Depth Information
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cases that require a new mathematical model. Finding the right mathematical model
can be a challenge.
Much research has gone into establishing a mathematical formulation that is gen-
eral enough to deal with all types of globally distributed measurement in a unified
model. The collection of observations might include distances, angles, heights, grav-
ity anomalies, gravity gradients, geopotential differences, astronomical observations,
and GPS observations. The mathematical models become simpler if one does not deal
with all types of observations at the same time but instead uses additional external
information. See Chapter 2 for a detailed discussion on the 3D geodetic model.
A popular approach is to reduce (modify) the original observations to be compat-
ible with the mathematical model. These are the model observations. For example, if
measured vertical angles are used, the mathematical model must include refraction
parameters. On the other hand, the original measurements can be corrected for re-
fraction using an atmospheric refraction model. The thus reduced observations refer
to a simpler model that does not require refraction parameters. The more reductions
are applied to the original observation, the less general the respective mathematical
model is. The final form of the model also depends on the purpose of the adjustment.
For example, if the objective is to study refraction, one needs refraction parameters
in the model. In surveying applications where the objective typically is to determine
location, one prefers not to deal with refraction parameters explicitly. The relation
between observations and parameterization is central to the success of estimation
and at times requires much attention.
In the most general case, the observations and the parameters are related by an
implicit nonlinear function:
[98
Lin
—
0.0
——
Nor
PgE
[98
f
(
x
a
,
a
)
=
o
(4.5)
Th
is is the mixed adjustment model. The subscript
a
is to be read as “adjusted.” The
symbol
1 vector of adjusted observations, and the vector
x
a
contains
u
adjusted parameters. There are
r
nonlinear mathematical functions in
f
. Often the
ob
servations are explicitly related to the parameters, such as in
a
denotes the
n
×
a
=
f
(
x
a
)
(4.6)
This is the observation equation model. A further variation is the absence of any
parameters as in
a
)
=
f
(
o
(4.7)
This is the condition equation model.
The application usually dictates which model might be preferred. Selecting an-
other model might require a mathematically more involved formulation. In the case
of a leveling network, e.g., the observation equation model and the condition equation
model can be applied with equal ease.
The observation equation model has the major advantage in that each observation
adds one equation. This allows the observation equation model to be implemented
