The GNSS integer ambiguities: estimation and validation

have to be used. The most important approximations will be outlined here. An evaluation
is given at the end of this subsection.
Lower bound based on bootstrapping
The bootstrapped estimator has a close to optimal performance after decorrelation of
the ambiguities using the Z-transformation of the LAMBDA method. Therefore, the
bootstrapped success rate can be used as lower bound for the ILS success rate as was
proposed in Teunissen (1999a), since it is possible to give an exact evaluation of this
success rate, (Teunissen 1998d):
n
Ps,B = P ( |âi|I − ai| ≤ 1
2 )
=
=
=
i=1
n
P ([âi|I] = ai [â1] = a1, . . . , [âi−1|I−1] = ai−1)
i=1
1
2
n
i=1
− 1
2
n
i=1
1
√ exp{−
σi|I 2π 1
x
2 σi|I
2Φ( 1
) − 1
2σi|I 2
}dx
(3.36)
with Φ(x) the cumulative normal distribution of equation (A.7), and σ i|I the standard
deviation of the ith least-squares ambiguity obtained through a conditioning on the
previous I = {1, ..., i − 1} ambiguities. These conditional standard deviations are equal
to the square root of the entries of matrix D from the LDL T -decomposition of the
vc-matrix Qâ. In order to obtain a strict lower bound of the ILS success rate, the
bootstrapped success rate in equation (3.37) must be computed for the decorrelated
ambiguities ˆz and corresponding conditional standard deviations. Hence:
Ps,LS ≥
n
i=1
2Φ( 1
) − 1
2σi|I with σ i|I the square roots of the diagonal elements of D from Qˆz = LDL T .
Upper bound based on ADOP
(3.37)
The Ambiguity Dilution of Precision (ADOP) is defined as a diagnostic that tries to
capture the main characteristics of the ambiguity precision. It is defined as:
ADOP = 1
n
|Qâ| =
n
i=1
σ 1
n
i|I
(3.38)
Quality of the integer ambiguity solution 39