In photogrammetric data-reduction, a least squares estimation with
the complete functional model (with all the parameters for :
selection) is usually carried out first as a preliminary ad justment,
based on which the model will be revised for the subsequent
adjustment. Thus, the least squares estimators of all the unknown
parameters in the complete model are available after the preliminary
adjustment. Those estimators provide the information about the
magnitude of the gross and systematic errors in the current
photogrammetric realization. „Now denote the estimators by € and $ ;
and their variances by D($)=05 Que and D(s)= 05043 respectively.
X and $ are unbiased estimators of x and s respectively because the
complete functional model is adopted for the least squares estima-
tion. It is known from Statistics that
$i - 8:
—"hü tite
Dol asi
which expresses that the left side Statistic has a t-distribution
With n-m,-mp degrees of freedom. In the above relation (3) Si is
a element of S, its true value is denoted by s, itself, and oi.
denotes the dingonal element of Q§§ corresponding to s; .
Then the confidence interval for 5; with confidence probability B
ean be given as follows:
^ ^ A ^
(Si - 16/» 9o J dsj » Si + 18/2 0 J ds; )
which means
~~ t(n-m,-m>) (3)
^ M A ^ ^
Pr(s; zn 58/2 O5 ds; <5;< Si + 58/2 Oo 4 dsi) = é
In words, when $i has been obtained, the true value S, iS within the
interval with probability 8 . So the maximum value that sj can,
reach with probability ß when si has been realized, denoted by Vs; ,
is
Ÿs; = max Si
sie (Si: Q probable given 8i]
= |S1]+ 58/2 &% Us; « (4)
When s; is deleted in the final adjustment, fs, is just the maximum
absolute magnitude of the bias for s, which is possible in the,
Sense of probability in the result. From section 2 we know Vs; is
a description of the internal reliability concerning S, for the
result. As Vs; contains the posteriori estimator of 5; , it might
be called posteriori internal reliability (measure) concerning Si +
For a certain procedure of parameter Selection, there is a
corresponding rejection „region in which if $i falls, S4 is deleted.
AS is Shown.in Eq.(4), YS, is a function of- x . So it has a
maximum value or a supremum in the rejection region. Denote the
maximum value by V,8i and the rejection region by W . Then the
symbolized expression is
- 369: +
