Bernese GPS Software Version 5.0 - Bernese GNSS Software

7.5.3 Epoch-Parameters
7.5 Parameterization
Epoch-parameters are set up for each epoch. Examples are station and receiver clock corrections,
kinematic station coordinates, and stochastic ionosphere parameters. Due to their
large number it is generally necessary to pre-eliminate them epoch-wise (EVERY EPOCH,
see Section 7.5.5). This is possible because epoch-parameters, by definition, are not directly
correlated (physical correlations are neglected) but only through non-epoch parameters.
Clock parameters and kinematic coordinates may be recovered in a back-substitution step
after solving the main normal equation system in order to write the output file(s), see
Section 7.5.6 for more information. This mechanism is not yet implemented for stochastic
ionosphere parameters.
Back-substitution is suppressed if no output files for epoch parameters are requested, if
option “Suppression of output concerning epoch parameters” in panel “GPSEST 3.3: Extended Printing
Options” is activated, and if no residuals have to be written. This saves computing time and
reduces the size of the program output.
7.5.4 Constraining of Parameters
In general, the observations of a given type are not sensitive to all parameters in a theoretical
model. In this case the normal equations (NEQs) are singular. Additional information, or
constraints, must be introduced into the least-squares solution to make the normal equations
non-singular. Additional constraints may be useful also for parameters which would be
estimated with a very high rms. Let us introduce “exterior” information concerning the
parameters
where
Hp = h + vh with D(h) = σ 2 P −1
h
H r × u matrix with given coefficients with rank H = r,
r number of constraining equations with r < u,
p vector of unknown parameters with dimension u × 1,
h r × 1 vector of known constants,
vh r × 1 residual vector, and
P −1
h dispersion matrix of the introduced constraining equations with dimension r × r.
(7.23)
If the constraints are non-linear, a linearization has to be performed through a first-order
Taylor series expansion. We may interpret the constraints (7.23) as additional pseudoobservations,
or as fictitious observations. That leads us to the observation equations:
y
h
+
vy
vh
=
A
H
or to the associated NEQ system
p with D(
y
h
) = σ 2
P −1 ∅
∅ P −1
h
(7.24)
(A ⊤ P A + H ⊤ P hH)p = A ⊤ P y + H ⊤ P hh. (7.25)
Bernese GPS Software Version 5.0 Page 149