Bernese GPS Software Version 5.0 - Bernese GNSS Software

9.2.2 Sequential Least-Squares Adjustment
9.2 Sequential Least-Squares Estimation
In a first step the sequential LSE treats each observation series independently. An estimation
is performed for the unknown parameters using only the observations of a particular
observation series. In a second step the contribution of each sequential parameter estimation
to the common estimation is computed.
Starting with the same observation equations as in the previous section, Eqns. (9.1), we
may write
or, in more general notation:
y 1 + v1 = A1 p 1 with D(y 1) = σ 2 1P −1
1
y 2 + v2 = A2 p 2 with D(y 2) = σ 2 2P −1
2
(9.5)
yi + vi = Ai pi with D(yi) = σ 2 −1
i P i , i = 1,2 (9.6)
where the vector p i denotes the values of the common parameter vector p c satisfying observation
series y i only.
First, we solve each individual normal equation. The normal equations for the observation
equation systems i = 1,2 may be written according to Eqn. (7.4) as
A ⊤ i P iAi
p i = A⊤ iP
i yi D(p i) = σ 2
i A ⊤ iP iAi
−1
(9.7)
= σ 2 i Σi with i = 1,2. (9.8)
In the second step, the a posteriori LSE step, we derive the estimation for p c using the
results of the individual solutions (9.7) and (9.8). The parameters p 1 and p 2 are used as
pseudo-observations in the equations set up in this second step. They have the following
form
or more explicitly:
p1
p 2
+
vp1
vp2
y p + vp = App c with D(y p) = σ 2 cP −1
p
=
I
I
p c with D(
p1
p 2
) = σ 2 c
Σ1 ∅
∅ Σ2
.
(9.9)
The results of the individual estimations p i and Σi are thus used to form the combined LSE.
The interpretation of this pseudo-observation equation system is easy: Each estimation is
introduced as a new observation using the associated covariance matrix as corresponding
weight matrix. The corresponding normal equation system may be written as:
or more explicitly
I⊤
⊤ ,I
Σ −1
1 ∅
∅ Σ −1
2
A ⊤ pP pApp c = A ⊤ pP p y p
I
I
p c =
I ⊤
,I
⊤
Σ −1
1 ∅
∅ Σ −1
2
p1
p 2
(9.10)
. (9.11)
Bernese GPS Software Version 5.0 Page 185