The GNSS integer ambiguities: estimation and validation

The bias-affected success rate is a useful diagnostic tool in order to study the impact of
biases on ambiguity resolution. For example one could study the impact of unmodeled
ionospheric errors or multipath, see (Teunissen et al. 2000; Joosten et al. 2002).
3.3 The ambiguity residuals
The ambiguity residuals are defined as:
ˇɛ = â − ǎ (3.47)
These residuals can be used as a measure of the inconsistency between the data and the
model. In order to measure the significance of the inconsistency also their probabilistic
properties are needed. In practice it is often assumed that the fixed estimator ǎ is
deterministic, so that the residuals simply take the same probability distribution as the
float ambiguities, i.e. a normal distribution. This assumption, however, is not correct.
It should only be used if the PDF of the float ambiguities is sufficiently peaked, so that
the probability mass in Sa is very close to one.
3.3.1 Parameter distribution of the ambiguity residuals
In Teunissen (2002) and Verhagen and Teunissen (2004c) it was shown how the PDF
of the ambiguity residuals can be constructed. Here a somewhat different approach will
be followed.
Let R ⊂ R n be an arbitrary subset. Then
P (ˇɛ ∈ R) = P (â − ǎ ∈ R)
=
P (â − ǎ ∈ R, â ∈ Sz)
z∈Z n
=
P (â − z ∈ R, â ∈ Sz)
z∈Z n
=
P (â ∈ {R + z} ∩ Sz)
z∈Z n
=
z∈Zn {R+z} T Sz
=
=
z∈Zn R T S0
R
z∈Zn fâ(x)dx use change of variable y = x − z
fâ(y + z)dy
fâ(x + z)s0(x)dx
with s0(x) the indicator function, see equation (3.7). Since the above equation holds
46 Integer ambiguity resolution