Bernese GPS Software Version 5.0 - Bernese GNSS Software

7. Parameter Estimation
7.5.2 Piece-Wise Linear Parameters
A piecewise linear parameter x is modeled in each particular time interval [t1,t2] by a linear
function according to one of the two representations shown in Figure 7.2. Both types of
parameterizations
⎛ ⎞
˜x =
x1
x2
and x =
⎝ x1
dx
dt
⎠ (7.21)
are equivalent. For the transition between these two parameter sets we may use the following
transformation equation
x =
⎛
⎜
⎝
1 0
−1
t2 − t1
1
t2 − t1
⎞
⎟
⎠ ˜x = C˜x . (7.22)
In the Bernese GPS Software we use the first of the two parameterizations and piecewise
linear parameters are defined by their values at equidistant nodal points (except for Earth
rotation parameters in GPSEST). The nodal points ti are defined by a time offset from the
beginning of the session and a time interval. The time offset is common for all piecewise linear
parameters and is specified in panel “GPSEST 5.2: Setup of Parameters and Pre-Elimination 2”
with option “TIME OFFSET FOR PARAMETER INTERVALS”. The value, specified in hours,
minutes, and seconds, refers to the beginning of the session. The parameter spacing can be
defined individually for all piecewise linear parameters.
Note that nodal points of the parameterization have to coincide as soon as you want to
combine such parameters in program ADDNEQ2. This makes it necessary to carefully plan
the setup of a parameter. See Chapter 9 for more details on stacking of piecewise linear
parameters.
In Bernese GPS Software Version 4.2 and earlier the troposphere parameters were parameterized
as piecewise constant parameters. Program ADDNEQ2 in Version 5.0 is able to deal
with this parameterization if old normal equation files are processed. If old and new normal
equations are accumulated, problems with writing of troposphere output files may occur.
x
t 1
x 1
t 2
x 2
Figure 7.2: Modeling of time-dependent parameters by x1,x2 resp. x1, ˙x1.
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x
x 1
t 1
dx
dt