In earlier publications (1-4) concerned with aerial triangulation the writer expounded the advantages of using groups of points near the positions of the minor control and pass points instead of single points at these positions. Within the limits imposed by the presence of correlation, this procedure increases the precision of evaluating the parameters of orientation and scale, and makes it possible to obtain more reliable estimates of this precision from considerations of the residuals at the observed points. In this paper the underlying theoretical reasoning is developed beyond the tentative stage at which it was then left.
Assuming that n groups of correlated observations having the following covariancc matrix of order g 91 ...~ are used to determine m parameters by means of the method of least squares, it is shown that the mean value or mathematical expectation E of the sum of the squares of the residuals (S.S.) is equal to
Denoting this quantity by to, and assuming that the variance of the observations is 0 2 instead of 1, then S.S. E -~'-'.
--)'()
A numerical investigation of one thousand specially-computed samples of three correlated values of unit variance and Q = .5 confirms this result. It further shows that the sum of the squares is distributed like Z22/2 which also describes the frequency distribution of half the sum of the squares when 9 = 0. This is suggestive that, in the more general case of determining m parameters from n groups of g correlated observations of equal variances and equal covariances, the quantity S.S./re is distributed like Zo2/J,o, where r o =ng --m. In other words, the ratio of the sum of the squares of the residuals to the reduced number of degrees of freedom ro = (ng --m) --m(g --1)o is a valid estimate of the variance ~2 based on the full number of degrees of freedom r 0 = ng --m.
An analysis of the frequency distribution of the residual want of correspondence in overlaps computed on pairs of minor control points is shown to be as predicted by the theory presented in the paper.