The GNSS integer ambiguities: estimation and validation

The first term on the right-hand side of the last equation is independent of g(x). If the
second term on the right-hand side is called K, the following can be derived:
K = {2g(x) T Q −1
â (x − a) + g(x)2 }fâ(x)dx
=
{2g(x) T Q −1
â (x − a) + g(x)2 }fâ(x)dx
z∈Zn Sz
=
=
z∈Zn S0
{2g(y) T Q −1
â (y + z − a)fâ(y + z) + g(y) 2 fâ(y + z)}dy
{2g(y) T Q −1
â
(y + z − a)fâ(y + z) + g(y)
z∈Z n
S0
F (y)
F (y)
= {g(y) +
G(y)
S0
2
F (y)
−
G(y)
≥0
2 }dy
independent of g(y)
2
z∈Z n
fâ(y + z) }dy
G(y)>0
where Sz are arbitrary pull-in regions, and use is made of the fact that if x ∈ Sz then
y = x − z ∈ S0 and g(y + z) = g(y).
This shows that K is minimized for g(y) = −
F (y)
G(y)
is periodic.
F (y)
G(y)
, which is an admissible solution since
If fâ(x + a) is symmetric with respect to the origin, then ˆ S(−x) = − ˆ S(x) from which
it follows that ˆ S(â) is unbiased. In that case the estimator ˆ S(â) is of minimum variance
and unbiased.
In practice, it is assumed that the float ambiguities are normally distributed. The BIE
ambiguity estimator, ã follows then from equation (C.3) as:
z∈Z
ã =
n
z exp{− 1
2x − z2Qâ }
exp{− 1
2x − z2 Qâ }
(C.4)
z∈Z n
This shows that the BIE ambiguity estimator can be written in a similar form as equation
(3.6):
ã =
zwz(â) (C.5)
z∈Z n
The weights wz(â) are given by equation (C.4), from which it follows that:
0 ≤ wz(â) ≤ 1 and
wz(â) = 1
z∈Z n
Best integer equivariant unbiased ambiguity estimation 155