Bernese GPS Software Version 5.0 - Bernese GNSS Software

9. Combination of Solutions
Substituting the results for Σ −1
i
we obtain
A ⊤ 1 P 1A1 + A ⊤ 2 P 2A2
p c =
A ⊤ 1 P 1 y 1 + A ⊤ 2 P 2 y 2
(9.12)
which is identical to Eqn. (9.4). This simple superposition of normal equations, also called
stacking of normal equations, is always possible if the individual observation series are
independent (indicated by a dispersion matrix in the form of Eqn. (9.2)).
Let us emphasize that this condition is not fulfilled in the case of the superposition of doubledifference
solutions based on different clusters of baselines. If the clusters are connected by
baselines (and only in this case stacking of the solutions yields an advantage due to common
parameters), non-zero elements in the off-diagonal parts of the weight matrix P p originating
from mathematical correlations in the double-difference mode are neglected. Clusters of
baselines have, therefore, to be selected carefully in order to minimize the effect of this
small incorrectness. In the same way clusters of stations in a zero-difference analysis are not
independent if satellite clocks or other common parameters are estimated.
9.2.3 Computation of the Combined RMS
In the previous section we only considered the combined parameter estimation. Sequential
LSE leads to identical results for the a posteriori estimate of the variance of unit weight:
Ωc =
m
y
i=1
⊤ m
iP i yi − y
i=1
⊤ iP iAip c
σ 2 c = 1
m
y
fc i=1
⊤ m
iP i yi − y
i=1
⊤
iP iAip c
(9.13)
(9.14)
Where fc is the degree of freedom of the combined solution. The importance of the “third
normal equation part” y ⊤ P y (see Eqn. (7.12)) is clearly seen in this formula. We refer to
[Brockmann, 1997] for a complete discussion.
9.3 Manipulation of Normal Equations
Normal equations can be manipulated in a number of different ways. The keywords are:
normal equation rescaling, expansion of normal equations, transformation of parameters or
of a priori parameter information, stacking of parameters, reduction of number of parameters
and introduction of additional parameters, parameter constraining, and parameter preelimination.
These manipulations are described in more detail in the following sections.
9.3.1 Changing Auxiliary Parameter Information
Parameters are always accompanied by auxiliary information such as, e.g., station name,
as well as epoch or time interval of validity. Changing of auxiliary parameter information
is a very simple operation which does not have any influence on the system of normal
equations. It includes renaming of stations and changing receiver and antenna names. In
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