The GNSS integer ambiguities: estimation and validation

C.4 Best integer equivariant unbiased baseline estimation
In practice the interest is not so much in the ambiguity solution, but in the baseline
solution. Let T (â, ˆ b) with F : R n × R p ↦→ R p be the baseline estimator. Two types of
restrictions will be put on the function T (x, y). The first requirement is that the if an
arbitrary constant ς ∈ R p is added to the float baseline ˆ b, the fixed baseline is shifted
by the same amount, i.e.
T (x, y + ς) = T (x, y) + ς ∀y, ς ∈ R p
It follows that if the constant is chosen ς = −y that
(C.6)
T (x, y) = y + f(x, 0) (C.7)
The second requirement is that the fixed baseline should not change if an arbitrary
integer z ∈ Z n is added to the float ambiguities:
T (x + z, y) = T (x, y) ∀x ∈ R n , z ∈ Z n
This is equivalent to the integer remove-restore condition of (C.1).
From (C.7) and (C.8) the following representation of the baseline estimator is found:
(C.8)
T (x, y) = y + h(x) with h(x) periodic (C.9)
Along similar lines as in section C.3 the BIE baseline estimator can be obtained.
Theorem C.4.1
ˆT (x, y) = y + b −
E{ ˆb|x + z}fâ(x + z)
fâ(x + z)
z∈Z n
z∈Z n
(C.10)
is the unique minimizer of T (x, y) − b 2 f â ˆ b (x, y)dxdy within the class of integer
equivariant estimators, T (x, y) = y + h(x) and h(x + z) = h(x)
Proof:
T (x, y) − b 2 fâˆb (x, y)dxdy
= y + h(x) − b 2 fâˆb (x, y)dxdy
= {y − b 2 + 2h(x) T Q −1 (y − b) + h(x) 2 }fâˆb (x, y)dxdy
The first term on between the braces on the right-hand side of the last equation is
independent of h(x). If the remaining part is called L and q = x − z ∈ S0, the following
156 Theory of BIE estimation