170 STANDARDIZATION OF EXPRESSIONS FOR ACCURACY, HALLERT
Assuming the standard error of unit weight s, in each of the measurements /, to I,
and that these measurements are mutually independent the special law of error propa-
gation means that the standard error of x and y can be expressed as
8, = 89 V [aa]
$, 7 89 V [BF]
The terms [aa] and [ff] are denoted weight numbers and are frequently written
Q,, and Q,, respectively. The definition is evident from the expressions (13) and (14).
(14)
6.2 The general law of error propagation. (Ref. 6, p. 182 and 9, p. 85).
Assume a linear function z of the quantities x and y
z= cx + dy (15)
x and y are assumed to be linear functions of the directly measured quantities
l ... L, for instance according to (13). x and y can be mutually dependent (correlated)
since they have been determined from the same measurements. Correlation is present if
the correlation number [ap] frequently written Q,, exists. The standard error of z can
be determined from the general law of error propagation which is expressed as
$, — sy V e2Q,, + d2Q,, + 2cdQ,, (16)
The expression under the root sign is frequently denoted Q., (the weight number
of 2).
yy
7. Expressions for the accuracy of instruments.
The accuracy of an instrument should preferably refer to the basic measurements in
the instrument. Consequently the accuracy should be expressed as the standard error of
unit weight of the actual observations. This means that the measurements should be
adjusted according to the method of the least squares and with a suitable number of
parameters. If possible the parameters should be chosen in such a way that they repre-
sent possible systematic errors. Evidently a large number of redundant observations is
desired in order to increase the precision of the standard error itself.
As examples of basic observations in photogrammetry can first be mentioned image
coordinates. x- and y-parallaxes can also be regarded as basic measurements although
parallaxes always are differences of image coordinates.
References.
[1] Kendall M. G. and Buckland, W. R, A Dictionary of Statistical Terms.
Oliver and Boyd, Edinburgh and London 1957.
[2] Rainsford, H. F.,, Survey Adjustment and Least Squares, Constable, London
1957.
[3] Cramér, H., Mathematical] Methods of Statistics. Uppsala and Princeton 1945-
1946.
Cramér, H., The Elements of Probability Theory. Stockholm 1954.
[5] v.d.Waerden, B. L., Mathematische Statistik. Berlin 1958.
[6] Helmert, F. R., Ausgleichungsrechnung nach der Methode der kleinsten Qua-
drate. Leipzig and Berlin 1907.
[7] Grossman, W., Grundzüge der Ausgleichungsrechnung. Berlin 1955.
[8] Jordan-Eggert, Handbuch der Vermessungskunde. Erster Band. Stuttgart
1935. P. 12-23.
[9] Tienstra, J. M. Theory of the Adjustment of Normally Distributed Observa-
tions. Amsterdam 1956.
SUMMARY.
First it is suggested that an
investigation of the terminology for the accuracy of
