PHOTOGRAMMETRIC ENGINEERING 
nized as the expected standard of terrain 
model flatness when the vertical measure- 
ments in a stereoscopic model are made with 
a first-order stereoplotting instrument. 
The pattern of residual errors was observed 
on numerous strips which had been adjusted 
by the general equation that has been pre- 
viously described, namely, Equation (2). 
Specifically, thirty-five stereophotogrammet- 
ric extensions ranging in length from 7 to 59 
stereoscopic models were examined. The total 
amount of control utilized in each extension 
varied from 10 to 39 control points; control 
was randomly distributed in 15 of the strips 
and in three to ten bands on the remaining 
20 strips. Flying heights ranged from 10,000 
to 30,000 feet; the mean height of all flights 
tested was approximately 18,500 feet. It 
should be noted that the general equation 
corrected for a parabolic pattern of error, 
propagated in the longitudinal direction, 
and permitted one reversal in the direc- 
tion of lateral tilt accumulation. Any de- 
viation from this pattern of error propaga- 
tion would be reflected in the residual error 
pattern after adjustment. It was hoped, by 
the utilization of this technique, that a con- 
sistent error pattern would be revealed and, 
as a result, an algebraic equation of the re- 
quired degree could be written which would 
satisfy all extensions. After examination of 
these patterns, it concluded that the 
equation would have to be tailored, as to de- 
gree and terms, according to the distribution 
of the control. Obviously, the residual errors 
from “curve fitting" were random: the next 
consideration was to make certain that the 
accuracy of the adjusted pass points improved 
with the 
was 
to the control on 
which the solution was based. 
“goodness of fit" 
Two specific extensions, containing a super- 
abundance of control, were adjusted several 
times. The solutions were based respectively 
on 20 and 21 control points distributed in six 
bands on each strip. Furthermore, these strips 
contained approximately the same number of 
withheld control which occurred on alternate 
bands and were used subsequently to check 
the accuracy of the resulting solutions. The 
mean square errors both for the control used 
in the least squares solutions, and also on the 
withheld control are shown numerically and 
graphically in Figure 1. 
From the above referenced graph, it can 
be seen that although the mean square error 
on the withheld control was always larger 
than the standard error on the control used 
in the respective solution, the accuracy of the 
adjusted points will generally improve di- 
4 
rectly with the ‘‘goodness of fit" to the con- 
trol on which the solution is based. The lack 
of significant improvement between the 
fifth & sixth degree equations has not been 
satisfactorily explained; although it possibly 
results from the limitation of computing 
precision (this condition is currently being 
checked). Tests with other strips have shown 
that there is present a “maximum’ degree 
equation depending upon the amount and 
distribution of the control. Thus, if excessive 
flexibility is permitted in fitting to the con. 
trol, this same flexibility resulting from total 
compensation for the random error will have 
an adverse effect in '' 
pass points. 
In practice, the ‘rule of thumb” has been 
to include one (A;x?) term for each band of 
vertical control or isolated control point and 
one (B;x'y)term for each band of control 
starting with 7=0. The general algebraic 
equation is merely an extension of the poly- 
nomial in equation (2): that is, 
Ao = Boy + A IX + Bixy + A»x? + B» vy 
+... + Agx$ + Bexty + H, = H. 
over correcting" the 
(3) 
where all symbols are the same as defined for 
equation (2). 
This vertical adjustment (equation 3) has 
been programmed for the UNIVAC. The 
program has the following features: 
l. The terms of the general equation are 
selected by indicating the highest 4 ;x‘ and 
independently Bx'y 
(4 «6). 
2. With the average scale for the extension 
terms to be deleted 
or a changing scale coefficient predetermined, 
all stereo-photogrammetric elevations (Z 
coordinates) are converted to feet or meters. 
3. Earth curvature correction is optional. 
When it is significant, as from high altitude 
photography, the correction is applied as a 
function of the lateral or y distance (y;— 5); 
that is, 
(vi — $)*S 
= 
7 (304.8)(5280) (8000) 
where 
Ze=the correction for earth curvature 
(mm) 
vi= photogrammetric coordinate of point 
(2) 
Vy=mean photogrammetric coordinate of 
all the control points on the extension 
$ — denominator of the scale ratio for the 
average scale of the extension. 
The error accumulation due to earth curva- 
ature along the line of flight (x-direction) is, 
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