trigonometrical levelling

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 a tall building . The vertical angle and the slope distance between the two points concerned are measured .

slope distance is measured using electromagnetic using elecrtomagnetic distance measures (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 cintouring .

However , unless the observation distance are relatively short , the height values displayed by the total station are quite useless , if not highly dangerous , unless total station contains algorithms to apply corrections for curvature and refraction .

SHORT LINES :

it can be seen that when measuring the angle 

∆ h = S sin ∝

When using the zenith angle  Z 

∆ h = S cos Z

if the  horizontal  distance is used 

∆ H = hi +  ∆ h - h t

       =  ∆h  +  hi  -  ht

where  hi   =  vertical height of the measuring centre of the instrument above A

            hi   =  vertical height of the centre of the target above B

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 .

What constitutes a short line may be derived by considering the effect of curvature and refraction compared with the accuracy expected . 

the combined effect of curvature and refraction over 100 m = 0.7 mm over 200 m = 3 mm , over 400 m = 11 mm and over 500 m = 17 mm . IF we apply the standard treatment for small errors to the basic equation we have ,

∆ H = S sin a + hi  - ht

and then 

∂ (∆ H) = sin a . 𝜕 S + S cos a  𝜕 a + 𝜕   hi  -  ht

and taking standard errors :

THIS value is similar in size to the effect of curvature and refraction over this distance and indicates that short sights should never be greater than 300 m . it also indicates that the accuracy of distance S is not critical when the vertical angle is small . 

However the accuracy of measuring the vertical anhle is very critical and requires the of a theodolite , with more than one measurement on each face .