The GNSS integer ambiguities: estimation and validation

Theory of BIE estimation C
In chapter 4 the theory of Best Integer Equivariant estimation was developed. In this
appendix a more detailed description of the derivations is given based on (Teunissen
2003f).
C.1 Integer equivariant ambiguity estimation
Starting point is the integer estimator of the unknown integer ambiguities a ∈ Z n . Let
this estimator be S(â) with S : R n ↦→ Z n . The new class of integer estimators to be
defined here will be larger than the class of admissible integer estimators of definition
3.1.1. Only the third condition of that definition will be considered, i.e. S(â − z) + z =
S(â), ∀z ∈ Z n . This means that the result of integer estimation should not change
when first an arbitrary integer z is removed from the float solution, then apply the
integer estimator, and then restore the integer. So, the requirement is equivalent to:
S(x + z) = S(x) + z ∀x ∈ R n , z ∈ Z n
Estimators that satisfy this property will be called integer equivariant (IE).
Lemma C.1.1
(C.1)
S(x + z) = S(x) + z ∀x ∈ R n , z ∈ Z n ⇐⇒ S(x) = x + g(x) (C.2)
with g(x) periodic: g(x + z) = g(x).
Proof:
If S(x) = x + g(x) with g(x + z) = g(x), then S(x + z) = S(x) + z.
If S(x + z) = S(x) + z, then g(x) = x − g(x) is periodic and S(x) = x + g(x).
C.2 Integer equivariant unbiased ambiguity estimation
It follows that:
a = E{S(â)} = E{â + g(â)} = a + E{g(â)}
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