The GNSS integer ambiguities: estimation and validation

can be derived:
L = {2h(x) T Q −1 (y − b) + h(x) 2 }fˆb|â (y|x)fâ(x)dxdy
z∈Z n
= {2h(q) T Q −1 (y − b) + h(q) 2 }
fˆb|â (y|q + z)fâ(q + z)dqdy
=
S0
S0
{2h(q) T
−1
Q
= {2h(q) T Q
z∈Z n
z∈Z n
(y − b)fˆb|â (y|q + z)dy
+ h(q) 2 }
fˆb|â (y|q + z)dy}fâ(q + z)dq
−1
z∈Z n
z∈Z n
(E{ ˆb|q + z} − b)fâ(q + z) + h(q)
S0
F (q)
F (q)
= {h(q) +
G(q) − b2 F (q)
−
G(q) − b2 }G(q)dq
S0
F (q)
L is minimized for h(q) = − G(q)
periodic.
2
z∈Z n
fâ(q + z) }dq
G(q)
+ b, which is an admissible solution since F (q)
G(q) is
Note that ˆ T in (C.10) is the BIE unbiased estimator if fâ(x + a) is symmetric with
respect to the origin.
If the float estimators are normally distributed the BIE baseline estimator, ˜ b, follows
from equation (C.10) as:
˜ b = ˆ b + b −
= ˆ b + b −
E{ ˆb|â + z} exp{− 1
2x − z2Qâ }
z∈Zn exp{− 1
2x − z2 Qâ }
z∈Z n
z∈Z n
b + Qˆ bâ Q −1
â (â − z) exp{− 1
2 x − z2 Qâ }
z∈Z n
exp{− 1
2 x − z2 Qâ }
= ˆb − Qˆbâ Q −1
â (â − ã) (C.11)
where the BIE ambiguity estimator of equation (C.4) is used.
Best integer equivariant unbiased baseline estimation 157