A NEW TREATMENT FOR THE ADJUSTMENT OF TRILATERATION NETWORKS 
By 
Fouad Zaki, PH.D, Technical Advisor. 
Mona El Kady, PH.D, Director. 
Manal Ahmed, Eng. 
Survey Research Institute, 
Water Research Centre, Ministry of Public Works, Egypt. 
ISPRS Commission III 
Abstract: 
In the least-squares adjustment of a trilateration network composed of interlacing braced quadrilaterals and centered polygons, the current 
procedure is to compute approximate values of the angles forming the geometrical figures of the net. These values are then used to get the 
correction for the measured sides either as a least-squares triangulation adjustment with condition equations or by applying the least-squares 
method of variation of parameters. 
In either case, the question arises why then take the difficulty of measuring lengths, if at the end we are computing approximate values of the 
angles which could have been measured more easily and effectively with a theodolite. 
In this paper, a proposed least-squares adjustment of the trilateration problem is introduced which does not rely altogether on the 
computation of any angle. Besides, it reduces the number of conditions to only one for each geometrical figure (a braced quadrilateral or a 
central polygon) instead of four or more conditions. 
A computer program using the "Basic language" has been devised for such treatment with two different applications for the two common 
figures, together with a comparison with the current procedure applied, until now, even in the most advanced and newest treatises on the 
subject. 
KEY WORDS: #“justment. Trilateration, Software. 
INTRODUCTION However, if only four measurements are made, there will be 
no independent check, and there will be the possibility of an 
undetected mistake or blunder in the measurements. 
If additional measurements are taken as checks, they can also be 
used to obtain better estimates of the unknowns. 
ionally, objects in the field are located in the horizontal 
ON p e or a combination of both. A point 
can be located in relation to two other points by measuring two 
quantities: either a direction from each of the two points 
(triangulation), or a direction and a distance from one of the we, 
points (traverse), or a distance from each of the =o point 
(trilatcration) [1]. Thus in Fig.1, if À and B are too fixed poin 
with known plane rectangular coordinates, and C and D are new 
In the Framework of Fig.1 a, there are eight interior angles 
and five distances (AB is assumed fixed) giving thirteen possible 
measurements to determine four unknowns, i.e., nine "redundant 
measurements” while in Fig. 1b there are nine interior angles and 
  
points whose coordinates are to be determined, den four 
measurements are therefore nec 20 sufficient for e 
determination of the unknown points. ese measurements cou 
  
  
  
be: 
N 
B 
{ 
FiqWa) 
A 
N 
  
? 
E cma 
ere te 4 
e ——————————————————————————————————— 
Fig? 
(a) angles DA B, D P CA D andCD A 
distances A D, B D, , Or 
©) angles A B D, C A D and distances B D, A C, or any other 
suitable combination of four measurements. 
28 
five distances giving fourteen possible measurements to determine 
four unknowns, i.e.,ten redundant measurements. Whether or 
not all thirteen (fourteen) measurements are made is a matter to 
be decided taking into consideratiom the time taken to make the 
additional measurements and also the time necessary for 
computation (although this is not now of primary importance due 
to the introduction of large electronic computers, except for 
saving of storage). 
For such "overdetermined" problems, the most probable 
values (MPV) of the unknowns can be found from the 
measurements by using either the so-called direct method 
(condition equations), or the indirect method (observation 
equations or variation of parameters). Both methods make use of 
the Least Squares principle (LS), and both can be applied to the 
same measurements to obtain the same unknows. 
TRILATERATION 
The usual method of fixing points C and D (Fig.1) has long 
been the triangulation method, where the eight (nine) interior 
angles are measured by a theodolite. In this case, we have four 
(five) redundant measurements ending with four (five) condition 
equations. : : : : 
The introduction of EDM ECT nas Doe a passible i 
replace this usual ground triangulation by trilateration, in whic 
a five distances yy namely, AC, AD, BD and CD 
(AB is assumed fixed) for the fixation of points C and D. Thus 
we have one redundant measurement ending with one condition 
equation, which has to be satisfied before computing the 
horizontal coordinates of points C 
and D. 
Laurila, - 1983, seems to be the only geodesist who realized 
this fact in the adjustment of a trilaterated braced quadrilateral 
(Fig.1 a). However, his treatment for this one condition equation 
turned to be a geometrical relation between the four corner 
angles BAD, ADC, DCB, CBA, which he computed from the 
measured sides. 
All other geodesists adjusted the trilaterated quadrilateral 
either as a triangulated quadrilateral with four geometrical 
conditions relating the computed angles of the quadrilateral 
(Moffitt,1975.) or by the method of variation of parameters 
(Mikhail, 1981)[A]