LEAST-SQUARES ADJUSTMENTS - GPS Satellite Surveying - page 104

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value of y . The third line follows by substituting (4.33) for the expected value of y .
Equation (4.32) has been substituted in the third line for y , and, finally, the A matrix
has been factored out. Thus the variance-covariance matrix of the random variable y
is obtained by pre- and postmultiplying the variance-covariance matrix of the original
random variable x by the coefficient matrix A and its transpose. The constant term
a 0 cancels. This is the law of variance-covariance propagation for linear functions of
random variables. The covariance matrix Σ y is a full matrix in general.

For later reference, the expression for the covariance matrix (4.26) can be rewritten

as

E x

− µ x T − µ x x

Σ =

x

(4.35)

E x x T

x

T

= − µ x µ

[10

4. 4 MIXED ADJUSTMENT MODEL

Lin
—
7.6
——
Nor
*PgE

To simplify the notation, the tilde will not be used in this section to identify random
va riables. Observations or functions of observations are always random variables. A
ca ret is used to identify quantities estimated by least-squares, i.e., those quantities that
ar e a solution of a specific minimization. Caret quantities are always random variables
be cause they are functions of observations. To simplify the notation even further, the
ca ret symbol is used consistently only in connection with the parameters x .
In the mixed adjustment model, the observations and the parameters are implicitly
re lated. If

[10

a denotes the vector of n adjusted observations and x a denotes u adjusted
pa rameters (unknowns), the mathematical model is given by

f ( a , x a ) = o (4.36)

Th e total number of equations in (4.36) is denoted by r . The stochastic model is

2

0 Σ − 1

P = σ (4.37)

b

w here P denotes the n Σ b denotes the covariance matrix

of the observations. The objective is to estimate the parameters. It should be noted
th at the observations are stochastic (random) variables and that the parameters are
de terministic quantities. The parameters exist, but their values are unknown. The
es timated parameters, however, will be functions of the observations and therefore
ra ndom variables.

× n weight matrix, and

4. 4.1 Linearization

If we let x 0 denote a vector of known approximate values of the parameters, then the
parameter corrections x are