LEAST-SQUARES ADJUSTMENTS - GPS Satellite Surveying - page 117

∂ v 2
∂ w 2 =−

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P − 1
2

B 2 T (4.134)

By using, again, the law of variance-covariance propagation and (4.129), we obtain
th e cofactor for v 2 :

P − 1
2

B 2 TB 2 P − 1

Q v 2 = (4.135)

2

Th e estimated variance-covariance matrix is

Σ v 2 =σ

2
0 Q v 2

(4.136)

The variance-covariance matrix of the adjusted observations is, as usual,

[11

Σ a = Σ b − Σ v (4.137)

As for iterations, one has to make sure that all groups are evaluated for the same
ap proximate parameters. If the first system is iterated, the approximate coordinates
fo r the last iteration must be used as expansion points for the second group. Because
th ere are no observations common to both groups, the iteration with respect to the
ob servations can be done individually for each group.
Occasionally, it is desirable to remove a set of observations from an existing so-
lu tion. Consider again the uncorrelated case in which the set of observations to be
re moved is not correlated with the other sets. The procedure is readily seen from
(4 .118), which shows how normal equations are added. When observations are re-
m oved, the respective parts of the normal matrix and the right-hand term must be
subtracted. Equation (4.118) becomes

Lin
—
0.6
——
No
PgE

[11

=− A 1 M − 1 A 2 − 1 A 1 M − 1 w 2

A 2 M − 1 A 2 M − 1 x A 1 − w 1 −

1 2 1 2

(4.138)

=− A 1 M − 1 A 2 − M − 2 A 2 − 1 A 1 M − 1 A 2 − M − 2 w 2 A 1 + w 1 +

1 1

One only has to use a negative weight matrix of the group of observations that is
being removed, because

B 2 − P − 2 B 2

− M 2 = (4.139)

Ob servations can be removed sequentially following (4.116).
The sequential solution can be used in quite a general manner. One can add
or remove any number of groups sequentially. A group may consist of a single
ob servation. Given the solution for i

1 groups, some of the relevant expressions
th at include all i groups of observations are,

−

x i = x i − 1 + ∆ x i (4.140)

Q i − 1 A i M i + A i Q i − 1 A i − 1 A i x i − 1 + w i ∆ x i =− (4.141)