Trigonometric leveling is a surveying method to determine vertical distances between two points by taking vertical angle measurements and known distances. It accounts for refraction and curvature of the Earth. Refraction makes points appear higher, while curvature makes them appear lower, so their combined effect is that points appear lower. Refraction corrections are negative for elevation angles and positive for depression angles. Curvature corrections are positive for elevations and negative for depressions. The total corrections combine these and consider the distance between points, Earth's radius, and coefficient of refraction.

1. Advance Surveying
Trigonometric Leveling
Prof. Rajguru R.S.
Civil Engineering Department
(rajgururajeshcivil@sanjivani.org.in)
Sanjivani College of Engineering,
Kopargaon,MH,India

2. Lecture Outline
 Definition
 Terrestrial refraction
 Angular correction for curvature and
refraction

3.  Definition of Trigonometric leveling:
Trigonometric leveling:
Trigonometric leveling is the branch of Surveying in which we find out
the vertical distance between two points by taking the vertical angular
observations and the known distances. The known distances are either
assumed to be horizontal or the geodetic lengths at the mean sea
level(MSL). The distances are measured directly(as in the plane
surveying) or they are computed as in the geodetic surveying.
 Due to curvature, the points appear to be lower than they actually
are; while due to refraction, they appear to be higher than they
actually are. The effect of curvature being greater than that
of refraction, the combined effect causes the points to appear to be
lower than they actually are.

4.  Terrestrial refraction:
• Station A & B located at very long
distance
• AA’ & BB’ tangent at A & B resp. to
curve line of sight AB
• BB1 & AA1 horizontal line at B & A
resp.
• α1 & α2 = Observed angle at A & B
resp.
• r = correction for refraction =
∠ 𝐵𝐴𝐴′
= ∠ ABB’
• R=Mean Radius of the earth =6370km
• d = horizontal dist. between A & B
• 𝜃 = Angle subtended at the center
• m= coefficient at refraction

5.  Terrestrial refraction:
• Corrected angle for refraction
Angle of elevation
∠ 𝐵𝐴O′ = α1 – r
(refraction correction is – ve )
Angle of depression
∠ A𝐵B2 = α1 + r
(refraction correction is + ve )
Coefficient of Refraction (m)
m =
𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑟𝑒𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛
Angle at center
=
𝑟
𝜃
r = m 𝜃
m = 0.06 to 0.09, adopt m =0.07

6.  Terrestrial refraction:
• Corrected angle for curvature
• Angle of elevation
• α1 Should be measured w.r.t. AA1
Correction for curvature = c =
𝜃
2
, +ve
• Angle of depression
• α2 Should be measured w.r.t. BB1
Correction for curvature = c =
𝜃
2
, -ve
From Fig. 𝜃 =
𝑑
𝑅
,
𝜃
2
=
𝑑
2𝑅
𝜃 =
𝑑
𝑅𝑠𝑖𝑛1"
sec.

7.  Combine correction:
• For angle of elevation,
Combine correction = c - r =
𝜃
2
- m𝜃 = (
𝑑
2𝑅𝑠𝑖𝑛1"
-
𝑚𝑑
𝑅𝑠𝑖𝑛1"
) =
(1−2𝑚)𝑑
2𝑅𝑠𝑖𝑛1"
sec.
For angle of elevation combine correction is + ve
similarly,
• For angle of depression,
Combine correction = - c + r = -
𝜃
2
+ m𝜃 = - (
𝑑
2𝑅𝑠𝑖𝑛1"
+
𝑚𝑑
𝑅𝑠𝑖𝑛1"
) = -
(1+2𝑚)𝑑
2𝑅𝑠𝑖𝑛1"
sec.
For angle of depression combine correction is - ve

8. Thank you