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METHODS OF ESTABLISHING HORI- 
ZONTAL CONTROL BY SHORAN: 
1. Establishment of primary and secondary 
geodetic points by Shoran trilateration. 
For photogrammetric mapping of any large 
area, a net of primary geodetic points is 
required. Geodetic points can be established 
by the Shoran trilateration method which 
has replaced the classical triangulation 
method in the mapping of inaccessible 
areas. These points are usually in the order 
of 200 miles apart. In Shoran trilateration, 
distances are measured instead of angles as 
in precise ground geodesy. In triangula- 
tion, a triangle determined by angle-side- 
angle is the basic geodetic figure of a 
chain. In trilateration the triangle is solved 
by side-side-side, and the chain is made 
redundant to provide a system of rigid 
quadrilaterals (fig. 4). The distances 
Figure 4: Shoran trilateration 
between points are measured by using the 
horan "Line Crossings" procedure (fig. 5). 
The aircraft flies across the line to be 
measured at about its midpoint and records 
simultaneously at short intervals the dis- 
tance to each of the ground stations. The 
distance between two ground stations À and 
B can be determined as the sum of a plus 
b, when a + b is a minimum. The flight 
pattern followed resembles a  "figure- 
eight”, shown in fig. 6. To ensure that the 
cyclical error of the final goniometer of 
the phase advance circuit, which repeats 
itself in one-mile units, may be averaged 
out, the sum a + b is measured over a 
sufficient distance on each side of the min- 
imum, that à and À will each increase by at 
least a mile. Thirty-one observations, taken 
at 3 second intervals, on a flight path 
inclined to the perpendicular to the line 
AB by 8° to 14°, provide the required 
pattern. In trilateration work each line is 
determined by a minimum of 16 line 
crossings. These are flown in two separate 
groups of eight on two different days. The 
minimum sum of a plus 4 can be deter- 
mined graphically (fig. 7), or by using 
the least square method. In fig. 7, the 
abscissa represents equal time intervals 
between observations of the distance. The 
ordinate represents the sum of a plus b for 
each corresponding observation. These 
plotted values approximate a parabolic 
curve which can be expressed by the equa- 
tion y = ax? + bx + c. The minimum 
value of this curve is where dy/dx — 0. 
One of the trilateration nets being 
established by Canadian Aero Service in 
Northern Canada for photogrammetric 
mapping is shown in fig. 8. The trilatera- 
tion net is adjusted by the least square 
method. The observation equations are 
derived from the differential formula for 
geodetic line given by Helmert. 
  
  
Figure 5: Sboran line crossings