Bernese GPS Software Version 5.0 - Bernese GNSS Software

9.3.5 Changing the Validity Interval
9.3 Manipulation of Normal Equations
Changing the validity interval usually precedes parameter stacking (see below). In case of
a (piece-wise) linear function this is performed by a parameter transformation p → ˜p (see
Figure 9.1). The time-dependent value p(t) may be expressed in two ways:
x
x 1
t 1
t i
x i
x i+1
t i+1
Figure 9.1: Changing the validity interval for the linear function.
p(t) = pi
p(t) = ˜p1
ti+1 − t
ti+1 − ti
t2 − t
t2 − t1
+ pi+1
+ ˜p2
t − t1
t2 − t1
t − ti
ti+1 − ti
t 2
x 2
(9.25)
. (9.26)
Comparing both expressions we obtain the transformation matrix for the interval [ti,ti+1]:
p =
⎛
⎜
⎝
t2 − ti
t2 − t1
t2 − ti+1
t2 − t1
ti − t1
t2 − t1
ti+1 − t1
t2 − t1
⎞
⎟
⎠
˜p = C ˜p . (9.27)
This matrix may be extended for an arbitrary number of subintervals. Using Eqn. (9.27) program
ADDNEQ2 transforms the normal equation system according to Eqn. (9.22). Changing
the validity interval has an influence on the normal equation matrix N and on vector b
(right-hand side of normal equation system).
9.3.6 Parameter Stacking
A simple, but basic operation of the program ADDNEQ2 is the stacking of parameter. This
important operation is, e.g., performed if station coordinates from different normal equation
Bernese GPS Software Version 5.0 Page 189