TEM 
SERIE 
METHOD 
Associate Professor University of Nairobi, 
Department of Surveying and Photogrammetry, 
4 
P,ü,Box 30197, NAIRGRI,KENYA 
INTERNATIONAL SOC 
AND REMOTE S 
ABSTRACT 
frcuracy improvement is often a desired requi- 
rement in many applications of photogrammetry. 
An extension of this study into the topic of 
variance reduction in data analyses hy use nf 
multi-control variate method among other Monte- 
Cario techniques is presented. Variance reduc- 
tion and the consequent increase of accuracy of 
the estimate is the motivation for the study. 
Straight single control variate,regression- 
adjusted control variate, concomitant control 
variate are some other variations in common use. 
The multiple-control variate method is particu- 
larly applicable in photogrammetry as it invel- 
vee pnoncon- trol object space coordinate para- 
meters estimation using simultaneously a set nf 
control object space coordinates. 
The 3-dimensional coordinates nf a point can 
be considered as functions of pholto-coordinates 
in at least twe photographe, interior orienta- 
tion parameters nf the camera and exterior B 
rientation parameters nf photographe 
used.Because of this functional relationship,the 
coordinates of adjacent points are dependent. 
Using this information in a judicipus way, it is 
possible to improve accuracy of computed 3-D 
coordinates of noncontroil object space points. 
Control point coordinates could thefore be used 
in variance reduction by multicontrol variable 
method. Comparison of resulting accuracy esti- 
mates would reveal efficiency and suitability 
of the selected number and locations of points 
for use as multiple control variates. Simpulation 
would be the basis of data formulation and  ana- 
lyses. 
KEY WORDS: 
ariance reduction, Monte Carlo, 
Close range, Si 
imulation, Accuracy improvement. 
y 
n 
a 
1. INTRODUCTION 
Monte Carle techniques are often used in many 
srien- tific disciplines and a considerable part 
nf this effort is in the area of variante reduc- 
tion in systems analyses. The applicability of 
some nf these techniques to the normal case of 
terrestrial and close range photogrammetry has 
been demonstrated in íNagaraja, 1990,91). Though 
at first sight it might seem that sampling pro- 
cedures only apply in case of simulation studie- 
z,further reflection should  indic- ate that 
conceptually and practically, it should be pos- 
sible to incorporate these ideas in reduction of 
practical data. However, this paper deals with 
a case study in simulation, 
The subject of variance reduction has received, 
outside photogrammetry,considerable attention 
Y DF PHOTOGRAMMETRY 
NG, CDMMISSI 
DN V 
and a number of methods have been developed. 
Therefore, there are a few techniques that help 
to increase accuracy and hence efficiency of 
zimulations, sometimes substantially, by produ- 
cing lest variable observations. Accordingly, 
there is a need to study such a possibility. 
Hillier and Liebermani1786) have given a tong 
list of successful applications of simulation 
studies. 
simulation are nnt new in photo- 
Applications of : ; 
lications are still possik- 
nrammetry but new app 
le.Variance reduction techniques seek either an 
increase in precisionidecreased variance) for a 
fixed sample size or a decrease in the sample 
size required to obtain a given degree of preci- 
sion. Several authors have cautioned us in using 
these techniques,I1f properly used, these techni- 
ques can provide tresendous increase in the 
efficiency of the model; however,if the intui- 
tion is faulty and analyst does nat use a 
reasonable design, the technique can also be 
very unpredictable and actually increase varia- 
nce tor some technigues.fecause nf this reason,a 
systematic and thorough study of the selected 
method is essential in adapting it for any spe- 
tific application. 
The control variables technique applies very 
well when there is an equivalent to the process 
we are simulating that can be treated theoreti- 
cally, For example,in the normal case of close 
range photogrammetry we have an eguivalence 
between computation of coordinates of a noncon- 
trol object space point and that nf a control 
object space point, We can then simulate the Y- 
coordinates {say} of a selected object space 
point and that of the known control point simul- 
taneously, using same random numbers in both 
computations, Then the difference in the known 
and computed coordinate of the control object 
space point is an estimate of the correction to 
be applied to the computed results of the selec- 
ted object space point. This procedure,therefore 
tuts put the variance due to common parameters 
in the two processes leaving only the variante 
nf the error of approximation. Obviously, this 
should be of a lower order of magnitude. In 
photogrammetry, as control point information is 
generally available, it can be used as a controi 
variate. This same ides can be further extended 
te a multi-control variablgiMDCV) technique and 
an attempt is made to present this extended 
application study in this paper. 
2. À SIMULATION EXPERIMENT BASED DN NORMAL CASE 
in order to understand the capabilities of the 
HCY technigue,it is necessary to set up a frame- 
work for the simulation study. This aspect is 
reported in this section. The assumed/true ob- 
jectt space coordinates ite nearest mm), and the