The Use and Calibration of the Kern ME5000 Mekometer - SLAC ...

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Variance Component Analysis
Here od represents the variance of the distance d, q the constant, and 0, the distance-
dependent term of the equation. Equations 2-3 and 2-4 can not be directly compared: 2-3
is an empirical function while 2-4 is derived through error propagation. The exponent H
reflects the distance dependency of the variances. A positive exponent indicates that the
variance of the measured distances increases with increasing distance. A negative
exponent indicates that the variance of the measured distances decreases with increasing
distance. It is usually sufficient to limit the choice of the exponent to the values +l.O, -1.0,
+0.5, or -0.5.
Normally the variance - covariance matrix y is assumed to be known with the
exception of the variance o, which is estimated through the least squares process.
D(r) =dV (2-5)
In the case that variance components are to be estimated the stochastic model has to
be expanded. Suppose the vector of residuals is composed of a linear combination of the
two independent but not directly obtainable residual vectors rl and rz, with rI representing
the constant and r2 the distance proportional parts. Then the associated variance -
covariance matrices can be described as follows:
D(n) = 0: a:
1 0 0 . . .
0 1 0 . . .
0 0 1 . . . Db-2) = d 2 cd
. . . . . . . . . . . . I
2H
dlz 0 0 . . .
0 d;nf 0 . . .
0 0 dtiH . . .
. . . . . . . . . . . .
where or and oz represent the unknown variance components to be estimated and a, and
a, their approximate values. The values describing the accuracy of the instrument which
are usually supplied by the manufacturer can be used as approximate values (see for
example equation 2-3a). The total variance - covariance D is a linear combination of both
parts and is calculated as follows:
G-6)
D =& +o:v, P-7)
where & is the product of CX~~ and the associated matrix and ok are the variance
components for k = 1,2.
Using the method of least squares adjustment with additional constraints one can
calculate the unknowns and their associated statistical values as follows [5]:
2 = (ATD-lA)-lATD+f
-_- --- (2-W
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