Error due to use of wrong scale

𝑅.𝐹 𝑜𝑓 𝑤𝑟𝑜𝑛𝑔 𝑠𝑐𝑎𝑙𝑒.

● Correct length = 𝑅.𝐹. 𝑜𝑓 𝐶𝑎𝑟𝑟𝑒𝑐𝑡 𝑠𝑐𝑎𝑙𝑒
× Measured length.
2
● Correct Area = ( 𝑅.𝐹. 𝑜𝑓 𝑤𝑟𝑜𝑛𝑔 𝑠𝑐𝑎𝑙𝑒
𝑅.𝐹 𝑜𝑓 𝑐𝑜𝑟𝑟𝑒𝑐𝑡 𝑠𝑐𝑎𝑙𝑒 ) × Calculated Area.

Principale of Least Square.

∑𝑣
2 𝐸𝑠
𝐸= ± 0. 6745 𝑛(𝑛−1)
=
𝑛

Where

E = Probable errors of single observation.

𝑉𝑠 = Difference between any single observation and mean of the values.

𝑛 = Number of observations of the mean.

Tap Corrections

𝐿.𝐶
● Correction for Standardization 𝐶𝑎 = 𝑡

2
2 2 ℎ
● Correction for slope 𝐶𝑠 = 𝐿 − 𝐿 + ℎ 𝐴𝑙𝑠𝑜 𝐶𝑠 = 2𝐿
2
ħ
● Correction for Alignment or bad ranging 𝐶𝑎𝑙 = 2𝐿

● Corrrection for Temperature 𝐶𝑡 = α(𝑇𝑚 − 𝑇0)𝐿

(𝑃−𝑃0)𝐿
● Correction for Pull or Tension 𝐶𝑝 = 𝐴𝐸

(
𝐿1 𝑊𝐿1 )2 𝑙𝑤
2
● Correction for Sag 𝐶𝑠𝑙 = 2 = 2
24𝑃 24𝑃

Error due to Incorrect length of chain or Tape.

a) True length of the line, (ℓ).

Actual true length of the line =

𝐴𝑐𝑡𝑢𝑎𝑙 𝑏𝑢𝑡 𝑤𝑟𝑜𝑛𝑔 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑐ℎ𝑎𝑖𝑛/𝑇𝑎𝑝
𝐷𝑒𝑠𝑖𝑔𝑛𝑎𝑡𝑒𝑑 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑇𝑎𝑝𝑒/𝑐ℎ𝑎𝑖𝑛
× 𝑤𝑟𝑜𝑛𝑔 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒

( )× 𝑙
'
𝐿 '
𝑙= 𝐿

Case I

( )
'
𝐿
a) In case of Area, 𝐴 = 𝐿
× 𝐴'

Where, A = True Area and A’ = wrong measured area

Case II

( ) . 𝑉'
'
𝐿'
b) In case of Volume 𝑉 = 𝐿

Where V = True Volume and V’ = wrong measured Volume

Important Terms.

●​ Bearing → Direction of a line with respect to a fixed meridian is Called bearing.

●​ True Meridian/Bearing

→ True meridian is a line joining the true north pole true south pole end point of reference

→ Angle measured for any line w.r.t. True meridian is called True bearing

Magnetic Meridian/Bearing.

●​ The line joining the magnetic north pole and magnetic south pole end point of reference is called a
magnetic meridian.
●​ Bearing taken w.r.t. magnetic meridian is called magnetic bearing.

Magnetic Declination
At any place, a horizontal angle b/w True meridian and magnetic meridian is called magnetic Declination.

For Eastern Declination α = β + θE or T.B = M.B + θE

For western Declination α = β - θw or T.B = M.B - θw

Note + sign is used for declination is to the east, and the sign (-) is used it declination is to the west.

' 2 2 2
Closing error In the Traverse = 𝐸 = 𝐴𝐴 = (∑ 𝐿) + ∑ 𝐷 ( )

Bowditch’s Method (Compas Rule)

Permissible error in linear measurement 𝑒 ∝ 𝑙

1
Permissible error in angular measurement 𝑒 ∝
𝑙

1
Correction to latitude 𝐶𝐿 = ∑ 𝐿 × ∑ 𝑙'

1
Correction due to departure 𝐶𝐷 = ∑𝐷 × ∑𝑙

𝐿' 𝐷
Transit Method: 𝐶𝑙 = ∑𝐿 × 𝐿𝑟
, 𝐶𝐷 = ∑ 𝐷 × 𝐷𝑟
 1
𝑐𝑙𝑜𝑠𝑖𝑛𝑔 𝑒𝑟𝑟𝑜𝑟
Axis Method: correction of any length = True length × 2
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑎𝑥𝑖𝑠

𝑆 𝑙
Sensitivity: Angle b/w the line of sights in radius α = 𝐷
=𝑛× 𝑅

1
𝑅
= ( ) × 206265
𝑆
𝑛𝐷

(D) = Distance of the instrument from the staff

(n) = Number of divisions. (ℓ) = length of one division 2 mm
(R) = Radius of curvature (S) = Steft intercept

Check in Height of Instrument N/d:

∑ BS - ∑ FS = ∑ Rise - ∑ Fall = Last RL – First RL

2
𝑑 2
Curvature 𝐶𝑐 = 2𝑅
= 0. 07857𝑑

( )
2
1 𝑑
Refraction 𝐶𝑟 = 7 2𝑅

Final combination correction

( )
2
6 𝑑
𝐶 = 𝐶𝑐 − 𝐶𝑟 = 7 2𝑅

C = 0.06735d2, d = 3.85√c d = in km and c = in meter.

Reciprocal Levelling:
The true difference elevation.

𝐻=
1
2
⎡⎢ ℎ − ℎ + ℎ' + ℎ' ⎤⎥
(
⎣ 𝑎 𝑏 𝑎) 𝑏 ⎦ ( )
Determining Areas:
Mid ordinate rule Δ = Area = Avg ordinate × length of base
𝑂1+ 𝑂2+ ....+𝑂𝑛
∆= 𝑛
× 𝐿

Average ordinate Rule

Area D = Average ordinate length of the base.

= ( 𝑂0+ 𝑂1+ ...+𝑂𝑛

𝑛+1 )× 𝐿 1
𝐷 =
𝐿
(𝑛+1)
×∑𝐷

∑D = D0 + ….+ On

Simpson’s one-third Rule

​
∆=
𝑑
3 ( ) (
[ 𝑂0 + 𝑂𝑛 + 4 𝑂1 + 𝑂3 + )
...+ 𝑂𝑛−1 + 2(𝑂2 + 𝑂4 + ... 𝑂𝑛−2)]

Volume: Prismoidal formula (simpson’s rule)

𝑉 =
𝑑
3 ( ) (
[ 𝐴1 + 𝐴𝑛 + 4 𝐴2 + 𝐴 4 )] )

Trapezoidal formula (Area and area method)

𝑑
𝑉 = 2
(𝐴1 + 𝐴2)

Fixed Hair Method

𝑓
D=k×s+c 𝑘 = 1
= 100 (Multiplying constant)

Constant C = (f + d)
S = Staff intercept, I = stadia interval, f = focal length of obj
Staff is vertical and sight is inclined (upward)
sin𝑠𝑖𝑛 2θ
D = k.s.cos2θ + c cos θ 𝑉 = 𝑘. 𝑠 2
+ 𝐶 sin 𝑠𝑖𝑛 θ

Staff is normal to the sight and sight is inclined.

D = (k.s + c) cos θ + r sin θ V = L sin θ = (k.s + c) sin θ

Substense method
𝑓
𝐷 = 𝑖
× 𝑠 + (𝑓 + 𝑑) 𝑖 = 𝑚. 𝑝

m = Number of revolutions
p = pitch

Curve Surveying
Length of the curve.
π𝑅∆
𝑙 = 180 , Δ = the angle of centre in degree

Tangent length
T = R tan Δ/2

Length of cord
L = 2 R sin Δ/2

Mid ordinate
M = R (1 – cos Δ/2)