Trigonometrical levelling is used where difficult terrain, such as mountainous areas, precludes the use of conventional differential levelling. It may also be used where the height difference is large but the horizontal distance is short such as heighting up a cliff or a tall building. The vertical angle and the slope distance between the two points concerned are measured. Slope distance is measured using electromagnetic distance measurers (EDM) and the vertical (or zenith) angle using a theodolite. When these two instruments are integrated into a single instrument it is called a ‘total station’. Total stations contain algorithms that calculate and display the horizontal distance and vertical height, This latter facility has resulted in trigonometrical levelling being used for a wide variety of heighting procedures, including contouring. However, unless the observation distances are relatively short, the height values displayed by the total station are quite useless, if not highly dangerous, unless the total station contains algorithms to apply corrections for curvature and refraction.
From Figure it can be seen that when measuring the angle
h = S sin α
When using the zenith angle z
h = S cos z
If the horizontal distance is used
h = D tan α = D cot z
The difference in elevation (H) between ground points A and B is therefore
H = hi h − ht
= h hi − ht
hi = vertical height of the measuring centre of the instrument above A
ht = vertical height of the centre of the target above B
Trigonometric levelling – short lines
This is the basic concept of trignometrical levelling. The vertical angles are positive for angles of elevation and negative for angles of depression. The zenith angles are always positive, but naturally when greater than 90◦ they will produce a negative result.