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

--
Variance Component Analysis
measurements be input in a specific order. The instrument station which is used to
measure to all other target stations has to be the first data set. For each instrument station
the distance measurements have to be input in ascending order.
The output file shows the date and time when the data was processed, which input file
was used, the header line of the input file and indicates if the calculations where done with
or without the addition constant It also shows the number of baseline monuments, the
value for the exponent H used and the start values for a, and %. The next section
contains the results after n iterations starting with the mean square error a postetiori, the
variance components 012 , CJ,~ and their estimated variances, the adjusted distances with
their r.m.s., and the’ addition constant if calculated. The variable SO represents the
approximate distance and dS the addition to obtain the adjusted distance. Finally, the last
section contains each measured distance with its residual and the estimated mean square
error M(s) according to equation (2-4). The weights ~/M(s)~ are also calculated.
2.3. Extensions to the mathematical and stochastic model
The previous section describes all the unknowns and variance components which are
calculated by the program VARCOM used at SLAC for the analysis of baseline
measurements. In a more recent paper Koch [7] expanded the mathematical and stochastic
model to include the case of distance dependent addition constants and cyclic errors which
can be determined if absolute distances are known. It also deals with the situation in which
an instrument with superior accuracy is used as reference for an instrument with inferior
accuracy. The mathematical model (2-l) can be expanded as follows:
~1; - xlj + C + K = dg + r with (2-15)
C=c,+cld,+czd;+od: and (2-16)
K = kll cos@> + k12 cos(2p) + kzl sin + k~ sin (2P) and
(2-17)
mod(du,F)
p=27t F (2-18)
Equation (2-16) represents in polynomial form the distance-dependent addition constant
C of the instrument to be calibrated. Equation (2-17) shows the frost terms of a Fourier
polynomial to describe the cyclic phase error K. The unknowns c0 through c3 and k,,
through k22 of the instrument to be calibrated are summarized in the vector of unknowns xc,
while the unknowns of the superior instrument are summarized in the vector xs Likewise
the matrix of coefficients is split into a section containing the coefficients of the instrument
to be calibrated and a section containing the coefficients of the instrument which is used as
reference. Therefore the following observation equations in matrix form can be derived.
44