Global Positioning System Reference
In-Depth Information
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Figure 4.1
Accuracy and precision.
[96
4.2 STOCHASTIC AND MATHEMATICAL MODELS
Lin
—
-0.
——
Nor
PgE
Least-squares adjustment deals with two equally important components: the stochas-
tic model and the mathematical model. Both components are indispensable and con-
tribute to the adjustment algorithm (see Figure 4.2). We denote the vector of ob-
servation with
b
, and the number of observations by
n
. The observations are ran-
dom variables; thus the complete notation for the
n
×
1 vector of observations is
˜
b
. To simplify the notation, we do not use the tilde in connection with
b
. The true
value of the observations, i.e., the means of the populations, are estimated from the
sample measurements. Since each observation belongs to a different population, the
sample size is usually 1. The variances of these distributions comprise the stochastic
model. It introduces information about the precision of the observations (or accuracy
if only random errors are present). The variance-covariance matrix
[96
Σ
b
expresses the
stochastic model. In many cases, the observations are not correlated and the variance-
covariance matrix is diagonal. Occasionally, when so-called derived observations are
used which are the outcome from a previous adjustment, or when linear combina-
tions of original observations are adjusted, the variance-covariance matrix contains
off-diagonal elements. Because in surveying the observations are normal distributed,
the vector of observations has a multivariate normal distribution. We use the notation
N
T
,
Σ
b
b
∼
(4.2)
where
T
is the vector mean of the population. The cofactor matrix of the observations
Q
b
and the weight matrix
P
are defined by
1
σ
Q
=
0
Σ
b
(4.3)
b
Q
−
1
b
2
0
Σ
−
1
b
P
=
= σ
(4.4)
