Bernese GPS Software Version 5.0 - Bernese GNSS Software

9. Combination of Solutions
9.2 Sequential Least-Squares Estimation
In this section we review the concept of sequential least-squares estimation (LSE) techniques.
The result of a LSE using all observations in one step is the same as when splitting
up the LSE in different parts and combining the results later as long as these parts are independent.
To prove the identity of both methods we first solve for the parameters according
to the common one-step adjustment procedure. Thereafter, we verify that the same result
is obtained using a sequential adjustment.
Let us start with the observation equations (for the notation we refer to Section 7.2):
y 1 + v1 = A1 p c with D(y 1) = σ 2 1P −1
1
y 2 + v2 = A2 p c with D(y 2) = σ 2 2
P −1
2
. (9.1)
In this case we divide the observation array y c (containing all observations) into two independent
observation series y 1 and y 2. We would like to estimate the parameters p c common
to both parts using both observation series y 1 and y 2. We assume furthermore, that there
are no parameters which are relevant for one of the individual observation series, only. This
assumption is meaningful if we pre-eliminate “uninteresting” parameters according to Section
9.4.4. The proof of the equivalence of both methods is based on the assumption that
both observation series are independent.
The division into two parts is sufficiently general. If both methods are leading to the same
result, we might derive formulae for additional sub-divisions by assuming one observation
series to be already the result of an accumulation of different observation series.
9.2.1 Common Adjustment
In matrix notation we may write the observation equations (9.1) in the form:
which is equivalent to
y 1
y 2
with D(
+
y 1
y 2
v1
v2
=
) = σ 2 c
A1
A2
P −1
1
p c
∅
∅ P −1
2
(9.2)
y c + vc = Acp c with D(y c) = σ 2 cP −1
c . (9.3)
The matrices y c, vc, Ac, p c, and P −1
c may be obtained from the comparison of Eqn. (9.3)
with Eqn. (9.2). The independence of both observation series is given by the special form
of the dispersion matrix (zero values for the off-diagonal matrices). Substitution of the
appropriate values for y c, Ac and p c in Eqn. (7.4) leads to the normal equation system of
the LSE:
A ⊤ 1 P 1A1 + A ⊤ 2 P 2A2
p c
= A⊤ 1P 1 y1 + A⊤ 2P 2 y2
. (9.4)
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