Bernese GPS Software Version 5.0 - Bernese GNSS Software

2.3 Observation Equations
How do we approximate the solution? Euler, in his simple algorithm, approximated each
component of the solution vector by a polynomial of degree q = 2 by asking the approximating
solution to have the same initial values as the true solution and by enforcing the
approximating solution to satisfy the differential equation system at the left boundary epoch
t0. Let us illustrate Euler’s principle using the original initial value problem (2.14), (2.15),
(2.16):
r(t) = r0 + (t − t0) · v0 + 1
2 · (t − t0) 2 · f(t0,r0,v0,...) (2.20)
The above solution vector may of course be used to compute the velocity vector, too, just
by taking the time derivative of the formula for r(t):
v(t) = v0 + (t − t0) · f(t0,r0,v0,...) (2.21)
Let us point out that the Eulerian formulae may be used to compute position and velocity at
any point in the vicinity of the initial epoch t0. A collocation method has exactly the same
property. The only difference lies in the fact that instead of using polynomials of degree 2,
we use higher degree polynomials in the case of general collocation methods:
q
r(t) = (t − t0)
i=0
i · r0i
where q is the degree of the polynomials, r0i are the coefficients.
(2.22)
How are the coefficients r0i determined? Well, this is the nucleus of numerical integration
using collocation methods. The principle is very simple to understand and very closely
related to Euler’s method: the coefficients are determined by asking that the above approximation
passes through the same initial values as the true solution, and that the differential
equation system is satisfied by the approximating function at exactly q − 1 different time
epochs within the subinterval considered. The resulting condition equations are non-linear
and in general have to be solved iteratively. Needless to say that the integration algorithm
was programmed with efficiency in mind.
2.3 Observation Equations
The basic GPS and GLONASS observation equations are discussed in this section, focusing
on the most important aspects only. For further information the reader is referred to, e.g.,
[Rothacher, 1992], [Mervart, 1995], or [Schaer, 1999]. The implications of satellite-specific
GLONASS frequencies are addressed in section 2.3.7.
The following notation is used throughout this section:
t signal reception time (GPS system time),
τ signal traveling time (from satellite to receiver),
tk reading of receiver clock at signal reception,
δk error of receiver clock at time t with respect to GPS time. The signal reception
time may be written as
t = tk − δk , (2.23)
Bernese GPS Software Version 5.0 Page 35