Bernese GPS Software Version 5.0 - Bernese GNSS Software

2.2 GNSS Satellite Orbits
because usually a priori weights (i.e., variances) are associated with these parameters. In
this sense the procedure is comparable to the ‘stochastic’ orbit modeling used by other
groups [Zumberge et al., 1994]. The attribute ‘pseudo’ is used because we are not allowing
the orbits to adjust themselves continuously at every measurement epoch (as it is the case
if Kalman filtering was used).
The use of pseudo-stochastic parameters proved to be a very powerful tool to improve the
orbit quality. Until about mid 1995 pseudo-stochastic parameters were set up at CODE
only for eclipsing satellites and for problem satellites, afterwards pseudo-stochastic pulses
in radial and in along-track directions were set up for every satellite twice per day (at
midnight and at noon UT). This clearly improved the CODE orbits. For more information
we refer to [Beutler et al., 1994].
2.2.2.5 Variational Equations
If the orbits of the GNSS satellites are estimated using the Bernese GPS Software, the
partial derivatives of the position and velocity vectors with respect to all orbit parameters
have to be computed by the program ORBGEN. Let us consider only the deterministic model
parameters at present:
p ∈ {a,e,i,Ω,ω,u0,p0,p1,...} (2.11)
We have to compute the partials
rp(t) = ∂r(t)
∂p
vp(t) = ∂v(t)
∂p
(2.12)
(2.13)
If the orbit were given by the Eqn. (2.7), it would be rather simple to compute the above
partials (at least for the osculating elements): we know the position and velocity vectors as
functions of the osculating elements and, therefore, may simply take the partial derivatives
of these known functions with respect to the orbit parameters. Since for longer arcs the
analytical approximation is not sufficient in all cases, all partials (2.12) and (2.13) are computed
in Version 5.0 using numerical integration. The procedure is very simple in principle.
We derive one set of differential equations, called variational equations, and one set of initial
conditions, for each orbit parameter p. Then we solve the resulting initial value problem by
numerical integration (see Section 2.2.3).
Although the procedure to derive variational equations is standard and may be found in
many textbooks, we include these variational equations for the sake of completeness. Let us
start from the original initial value problem (2.8) and the associated initial conditions:
¨r = −GM r
r 3 + a(t,r, ˙r,p0,p1,p2,...) = f(t,r, ˙r,p0,p1,...) (2.14)
r0 = r(t0;a,e,i,Ω,ω,u0) (2.15)
v0 = v(t0;a,e,i,Ω,ω,u0) (2.16)
Bernese GPS Software Version 5.0 Page 33