The GNSS integer ambiguities: estimation and validation

2.2.9 The geometry-based observation equations
The geometry-based observation equations are obtained after linearization of equations
(2.19) with respect to the receiver position. The satellite-receiver range can be expressed
as function of the satellite and receiver positions as follows:
ρ s r = r s − rr (2.21)
where · indicates the length of a vector, rs = xs ys zsT the satellite position
vector, and rr = T xr yr zr the receiver position vector.
In order to linearize the observation equations, approximate values of all parameters are
needed. These approximate parameters will be denoted with a superscript 0 . It will be
assumed that the approximate values of all parameters are zero, except for the satellitereceiver
range, so that the observed-minus-computed code and phase observations are
given by:
∆p s r,j(t) = p s r,j(t) − ρ s,0
r (t)
∆φ s r,j(t) = φ s r,j(t) − ρ s,0
r (t)
(2.22)
The linearization of the increment ∆ρ s r is possible with the approximate satellite and
receiver positions r s,0 and r 0 r:
with
∆ρ s r = −(u s r) T ∆rr + (u s r) T ∆r s
u s r = rs,0 − r 0 r
r s,0 − r 0 r
the unit line-of-sight (LOS) vector from receiver to satellite.
This gives the following linearized observation equations:
∆p s r,j(t) = −(u s r) T ∆rr + (u s r) T ∆r s + T s r (t) + µjI s r (t)
+ cdtr,j(t) − cdt s ,j(t) + e s r,j(t)
∆φ s r,j(t) = −(u s r) T ∆rr + (u s r) T ∆r s + T s r (t) − µjI s r (t)
2.2.10 Satellite orbits
+ cδtr,j(t) − cδt s ,j(t) + λjM s r,j + ε s r,j(t)
(2.23)
(2.24)
(2.25)
In order to compute approximate satellite-receiver ranges, approximate values of the
satellite positions are required. Several sources are available in order to obtain these
approximate values. Firstly, the information from the broadcast ephemerides can be
used, which is available in real-time. Secondly, more precise information is made available
by the International GPS Service (IGS). Finally, Yuma almanacs are freely available, e.g.
from the website of the United States Coast Guard. These almanacs are especially
interesting for design computations, but are not used for positioning. Table 2.3 gives an
overview of the availability and accuracy of the different orbit products.
GNSS observation equations 13