2. Lecture Outline
Determination of Difference in elevation by
single observation
ο Introduction
ο Equation to determine the difference in
Elevation
3. ο Determination of Difference in elevation by single
observation:
β’ Introduction:
β’ Under many circumstances, it is not possible to occupy both the stations,
one of them being inaccessible, this method of determining the difference
in elevation between the two station is used.
β’ The vertical angle is corrected for curvature , refraction and axis signal
β’ Since the coefficient of refraction varies with temperature, this method does
not yield accurate results.
4. ο Derivation of an Equation to determine the difference in Elevation
: (Angle of elevation)
β’ β = Observed angle of elevation
β’ D = Horizontal distance = AAβ
β’ β1 = Corrected vertical angle for axis
signal
β’ β B AAβ =Ξ± +(C - πΎ) = Ξ± +
α
2
- m α
β’ β AAβ B = ( 90Β° -
α
2
) + α = 90Β° +
α
2
β’ β ABAβ =180Β°-(Ξ±+
α
2
- m α )-(90Β° +
α
2
)
β’ = 90Β° - (Ξ± +α- m α )
Fig.3.1
5. ο (Angle of elevation):
β’ Applying sine rule,
β’
BAβ
π ππβ BAAβ
=
AAβ
π ππβ ABAβ
β’
H
π ππ(Ξ±+α
2
β m α)
=
D
π ππ(90Β° β(Ξ± + α βm α))
β’ H = D
π ππ(Ξ± + α
2
β m α)
πΆππ(Ξ± + α β m α)
β’ But, D = R α, α =
D
π
=
D
π πππ1"
β’ H =D
π ππ(Ξ± + D
2π πππ1"
β πD
π πππ1"
)
πΆππ(Ξ± + D
π πππ1"
β πD
π πππ1"
)
β’ H =D
π ππ(Ξ± + D
2π πππ1"
β πD
π πππ1"
)
πΆππ(Ξ± + D
π πππ1"
β πD
π πππ1"
)
β’ H =D
π ππ (Ξ±+(1 β 2m) D
2π πππ1"
)
πΆππ (Ξ±+(1βm) D
π πππ1"
)
β’ Eq. can be modified by butting Ξ± = Ξ±1
β’ Fig.3.1
6. ο Derivation of an Equation to determine the difference in
Elevation : (Angle of depression):
β’ π = Angle of depression at B
β’ π1 = Corrected vertical angle for axis
signal
β’ H= Difference in elevation
β’ β ABBβ = π - C + πΎ = π -
α
2
+ m α
β’ β BBβA = ( 90Β° - α ) +
α
2
= 90Β° -
α
2
β’ β BβAB =180Β°-(90Β°-
α
2
)-(π -
α
2
+ m α)
β’ = 90Β° - (π - α + m α )
Fig:3.2
7. ο (Angle of depression)::
β’ Applying sine rule, ΞBABβ
β’
π΄Bβ
π ππβ ABBβ
=
π΅π΅β
π ππβ Bβ²AB
β’
H
π ππ(πβα
2
+ m α)
=
D
π ππ(90Β° β( π β α + m α))
β’ H = D
π ππ( π β α
2
+ m α)
πΆππ( π β α + m α)
β’ But, D = R α, α =
D
π
=
D
π πππ1"
β’ H =D
π ππ(π β D
2π πππ1"
+ πD
π πππ1"
)
πΆππ(π β D
π πππ1"
+
πD
π πππ1"
)
β’ H =D
π ππ(π β D
2π πππ1"
+ πD
π πππ1"
)
πΆππ(π β D
π πππ1"
+ πD
π πππ1"
)
β’ H =D
π ππ (π β(1 β 2m) D
2π πππ1"
)
πΆππ (π β(1 β m) D
π πππ1"
)
β’ Eq. can be modified by butting π = π1
Fig:3.2
8. ο Approximate Expression: Assuming AAβB as a plane right angle
β’ Ξ± = Angle of elevation
β’ πΎ = Angle of refraction
β’ C = Angle of Curvature
β’ AβB = H = D tan Ξ¦
β’ Ξ¦ = Ξ± +(C - πΎ)
β’ H = D tan [Ξ± +(C - πΎ) ]
β’ H = D tan [Ξ± +(
α
2
- m α)]
β’ H = D tan [Ξ± +( 1-2m)
π·
2π πππ1"
]
Expression for height difference
β’ π = Angle of depression
β’ πΎ = Angle of refraction
β’ C = Angle of Curvature
β’ BβA = H = D tan Ξ¦
β’ Ξ¦ = π - C + πΎ = π - (C - πΎ)
β’ H = D tan [π - (C - πΎ)]
β’ H = D tan [π - (
α
2
- m α)]
β’ H = D tan [π - ( 1-2m)
π·
2π πππ1"
]
β’ Expression for height difference