LEAST-SQUARES ADJUSTMENTS - GPS Satellite Surveying - page 116

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The second set of observations contributes to all residuals. From (4.98), (4.102),
and (4.114) we obtain

v 1 + ∆ v 1 = v 1

(4.124)

B 1 M − 1 A 1 x ∗ + w 1 − P − 1

1

P − 1
1

B 1 M − 1 =− A 1 ∆ x

1

Th e expression for v 2 follows from Equations (4.99) and (4.117):

T A 2 x ∗ + w 2 P − 1

2

B 2

v 2 =− (4.125)

where

= M 2 + A 2 − 1 A 2 N − 1

T (4.126)

1

[11

The cofactor matrices for the residuals follow, again, from the law of variance-
covariance propagation. The residuals v 1 are a function of w 1 and w 2 , according to
(4.124). Substituting the expressions for x ∗ and

Lin
—
0.4
——
Nor
PgE

∆ x , we obtain, from (4.124)

B 1 M − 1 I A 1 M − 1 (4.127)

∂ v 1
∂ w 1 =−

P − 1
1

A 1 N − 1 A 1 M − 1 A 1 N − 1 A 2 TA 2 N − 1

− +

1 1 1 1

∂ v 1
∂ w 2 =−

P − 1
1

B 1 M − 1

1

A 1 N − 1 A 2

T (4.128)

1

Applying the law of covariance propagation to w 1 and w 2 of (4.91) and knowing that
the observations are uncorrelated gives

[11

M 1 O
OM 2

Q w 1 ,w 2 = (4.129)

By using the partial derivatives (4.127) and (4.128), Expression (4.129), and the law
of variance-covariance propagation, we obtain, after some algebraic computations,
the cofactor matrices:

Q v 1 = Q v 1 + ∆ Q v 1 (4.130)

where

B 1 M − 1 P − 1 B 1 T − P − 1

1

A 1 N − 1 P − 1 A 1 T P − 1

1

B 1 M − 1 B 1 M − 1

Q v 1 = (4.131)

1 1 1 1

Q v 1 = P − 1 A 2 T P − 1 A 2 T B 1 M − 1 A 1 N − 1 B 1 M − 1 A 1 N − 1 ∆ (4.132)

1 1 1 1 1 1

The partial derivatives of v 2 with respect to w 1 and w 2 follow from (4.125):

∂ v 2
∂ w 1 =

P − 1
2

B 2 TA 2 N − 1 A 1 M − 1 (4.133)

1 1