LEAST-SQUARES ADJUSTMENTS - GPS Satellite Surveying - page 106

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∂ φ

∂ k =

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Bv + Ax + w = o (4.47)

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∂ φ

∂ x =−

A T

k

= o (4.48)

Th e solution of (4.46) to (4.48) starts with the recognition that P is a square matrix
an d can be inverted. Thus, the expression for the residuals follows from (4.46):

P − 1 B T k

v = (4.49)

Su bstituting (4.49) into (4.47), we obtain the solution for the Lagrange multiplier:

k

M − 1 ( Ax =− + w ) (4.50)

[10

with

Lin
—
* 1 ——
Lon
PgE

r M r = r B nn P − nn B r

(4.51)

Finally, the estimate x follows from (4.48) and (4.50)

=− A T M − 1 A − 1 A T M − 1 w

x (4.52)

The estimates x and v are independent of the a priori variance of unit weight. The
first step is to compute the parameters x from (4.52), then the Lagrange multipliers k
from (4.50), followed by the residuals v (4.49). The adjusted parameters and adjusted
observations follow from (4.38) and (4.39).
The caret symbol in v , k , and x indicates that all three estimated values follow from
minimizing of v T Pv . However, as stated earlier, the caret is only used consistently for
the estimated parameters x in order to simplify the notation.

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4.4.3 Cofactor Matrices

Eq uation (4.44) shows that w is a random variable because it is a function of the
ob servation b . With (4.37), the law of variance-covariance propagation (4.34), and

th e use of B in (4.42), the cofactor matrix Q w becomes

BP − 1 B T Q w = = M (4.53)

Fr om (4.53) and (4.52) it follows that
Q x = A T M − 1 A − 1

(4.54)

Combining (4.49) through (4.52) the expression for the residuals becomes

= P − 1 B T M − 1 A A T M − 1 A − 1 A T M − 1 P − 1 B T M − 1 w

v − (4.55)