The GNSS integer ambiguities: estimation and validation

2.2.6 Multipath
Ideally, a GNSS signal should arrive at the receiver antenna only via the direct path from
satellite to receiver. In practice, however, the signal often arrives at the antenna via
two or more paths because the signal is reflected on nearby constructions or the ground.
This effect is called multipath. It affects the phase and code measurements differently.
The resulting measurement error depends on the strength of the direct and the reflected
signals, and on the delay of the reflected signal.
Typically, the error induced by multipath on the code measurements varies between 1
and 5 meters. The effect on the phase measurements is 1-5 cm, and it will never be
more than a quarter of a cycle provided that the amplitude of the direct signal is larger
than the amplitude of the reflected signal.
Multipath is a systematic error and is very difficult to model. Although it is acknowledged
that it is one of the most important issues in the field of high precision GNSS positioning,
the multipath effect will be ignored here, i.e.
dm s r,j = δm s r,j = 0 (2.17)
2.2.7 Random errors or noise
The GNSS code and phase observations are of a stochastic nature, so that random
errors or observation noise must be taken into account in the observation equations.
In equations (2.7) and (2.12) the noise was denoted as e and ε for code and phase
respectively. The random errors are assumed to be zero-mean:
E{e s r,j(t)} = 0
E{ε s r,j(t)} = 0
2.2.8 The geometry-free observation equations
(2.18)
The observation equations (2.7) and (2.12) are parameterized in terms of the satellitereceiver
ranges ρ s r and are therefore referred to as non-positioning or geometry-free
observation equations. The final non-positioning observation equations are obtained by
inserting equations (2.13), (2.16), (2.17) into equations (2.7) and (2.12):
p s r,j(t) = ρ s r(t) + 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) = ρ s r(t) + T s r (t) − µjI s r (t) + cδtr,j(t) − cδt s ,j(t) + λjM s r,j + ε s r,j(t)
(2.19)
For notational convenience the receiver-satellite range is denoted as ρ s r(t) instead of
ρ s r(t, t − τ s r ). Furthermore, the initial phases of the signal are lumped with the integer
phase ambiguity, since these parameters are not separable. The resulting parameters in
M s r are real-valued, i.e. non-integer:
M s r,j = φr,j(t0) + φ s ,j(t0) + N s r,j
Here, they will still be referred to as ambiguities.
(2.20)
12 GNSS observation model and quality control