geomatics engineering credit hrs: 2 · principle of compass surveying • the principle of compass...

GEOMATICS ENGINEERING

CHAPTER 3

Compass Surveying and Traversing

Different types of Horizontal Angles

• Horizontal angles can be

• Interior angles

• Can be clockwise or anticlockwise considering the turning direction

• Deflection angles

• Can be right or left considering the turning direction

left

right

Direction of a Line

• Direction of a line is the horizontal angle from a reference line called

the meridian.

• There are four basic types of meridians:

• Astronomic meridian

• It is an imaginary line on the earth’s surface passing through the north-

south geographical poles.

• Magnetic meridian

• It is direction of the vertical plane shown by a freely suspended magnetic

needle.

• Grid meridian

• A line through a point parallel to the central meridian or y-axis of a

rectangular coordinate system

• Arbitrary meridian

• An arbitrary chosen line with a direction value assigned by the observer.

Quadrantal Bearing or Bearing

• Quadrantal Bearing: The magnetic bearing of a line measured clockwise or anticlockwise from NP or SP (whichever is nearer to the line) towards the east or west is known as QB. This system consists of 4-quadrants NE, SE, NW, SW. The values lie between 0-90°

• QB of OA = N a E

• Since bearing is with reference to N-S pole, angles are measured

clockwise in the first (NE) and 3rd (SW) quadrants. It is measured

anticlockwise in 2nd and 4th quadrants (NW and SE).

• When bearings are measured with reference to astronomical or true

meridian it is called as true bearing.

• If the bearing is from magnetic meridian, it is called as magnetic

bearing.

Azimuths or Whole Circle Bearing

• Azimuths are angles measured clockwise from any

reference meridian. They are measured from the north

and vary from 00 to 3600 and do not need letters to

identify their quadrant.

• Calculate azimuths of all lines?

• Reduced Bearing: When the whole circle bearing of a line is converted to quadrantal bearing it is termed as reduced bearing.

• Fore and Back Bearing:

• If the instrument is set up at O and bearing OA is taken it is termed as forward bearing of OA line, but if the instrument is set up at A and bearing AO is taken, it is termed as forward bearing of AO and back bearing of OA.

• Back bearings have the same numerical value but opposite letters. E.g. forward bearing of line OA is N300E and back bearing of OA is S300W.

•

• In WCB the difference between FB and BB should be exactly 180°

• BB=FB+/-180°

• Use the +ve sign when FB<180°

• Use the –ve sign when FB> 180°

Reduced Bearing, Fore and Back Bearing

Summary: Bearings to Azimuths

• When a line is in the 1st quadrant, the azimuth varies from 00 to 900

and azimuth and bearing of a line are the same (line OA).

• When a line is in the 2nd quadrant, the azimuth lies between 900 to

1800 and it can be obtained from bearing by subtracting it from 1800

(line OC).

• When a line is in the 3rd quadrant, the azimuth lies between 1800 and

2700 and it can be obtained from bearing by adding 1800 (line OD).

• When a line is in the 3rd quadrant, the azimuth is obtained by

subtracting the bearing from 3600 and it lies between 2700-3600. (line

OB)

Azimuth (WCB) to Bearing

• Convert the following azimuths to bearings.

• OA = 50020’

• OB = 154025’

• OC = 261025’

• OD =312038’

Angles, Bearings, and Azimuths

Prismatic compass

Compass traversing: Important Definition

• True meridian: Line or plane passing

through geographical north pole and

geographical south pole.

• Magnetic meridian: When the

magnetic needle is suspended freely

and balanced properly, unaffected by

magnetic substances, it indicates a

direction. This direction is known as

magnetic meridian. The angle between

the magnetic meridian and a line is

known as magnetic bearing or simple

bearing of the line.

North

pole

True meridian

Magnetic meridian

True bearing

Magnetic bearing

• Magnetic declination: The horizontal angle between the magnetic meridian and true meridian is known as magnetic declination.

• Local Attraction

• Method of correction for traverse:

• First method: Sum of the interior angle should be equal to (2n-4) x 90. if not than distribute the total error equally to all interior angles of the traverse. Then starting from unaffected line the bearings of all the lines are corrected using corrected interior angles.

• Second method: Unaffected line is first detected. Then, commencing from the unaffected line, the bearing of other affected lines are corrected by finding the amount of correction at each station.


Adding Angles Numerically

Definition of Traverse • A traverse is a series of consecutive lines whose lengths and directions have

been determined from field measurements.

• Traversing is the act of establishing traverse stations.

There are two basic types of traverses:

• A closed traverse returns to its starting point or to another known point.

• An open traverse consists of a series of lines which are connected but do

not return to either the original starting point or another known point.

• A hub (typically a wooden stake with a nail on top) is set at each traverse

station.

Principle of Compass Surveying • The principle of compass surveying is traversing, which involves a

series of connected lines. The magnetic bearings of the lines are measured by prismatic compass and the distances of the lines are measured by tape or chain. Such surveys does not require the formation of a network of triangles.

• Interior details are located by taking offsets from the main survey lines. Compass surveying is recommend when • A large area to be surveyed

• The course of a river or coast line is to be surveyed

• The area is crowded with many details and triangulation is not possible.

• Compass surveying is not recommended for areas where local attraction is suspected due to the presence of magnetic substances like steel structures, iron ore deposits, electric cables, etc.

Methods of Traversing

• Compass traversing: Fore bearings and back bearings between the

traverse leg are measured

• Theodolite traversing: Horizontal angles between the traverse legs

are measured. The length of the traverse legs are measured by

chain/tape or by stadia method

Checks on Traverse: Closed Traverse

• Check on closed traverse:

• Sum of the measured interior angles (2n-4) x 90°

• Sum of the measured exterior angles (2n+4) x 90 °

• The algebraic sum of the deflection angles should be equal to 360°.

• Right hand deflection is considered +ve, left hand deflection –ve

• Check on linear measurement

• The lines should be measured once each on two different days (along

opposite directions). Both measurement should tally.

• Linear measurement should also be taken by the stadia method. The

measurement by chaining and stadia method should tally.

Problems:

• Convert the following WCBs to QBs

• (a) WCB of AB = 45°30’

(Ans N 45°30’ E)

• (b) WCB of BC = 125°45’

(Ans 180- 125°45’ = S 54° 15’ E)

• Fore bearing of the following lines are given. Find back bearing

• AB=S 30°30’ E

• BC=N 40°30’ W

Problem

Contd…

Problem

Included angle at A= 280-180-40=60

=FB of DA-180-FB of AB

Included angle at B= 40+180-70= 150

=FB of AB+180-FB of BC

Included angle at C= 70+180-210

=FB of BC+180-FB of CD

Formula: BB of previous line-FB of

next line

Problem

• The whole circle bearings of the sides of a travers

ABCDEF are given below. Compute the internal angles.

• Bearing of AB = 290045’

• Bearing of BC = 250048’

• Bearing of CD = 196012’

• Bearing of DE = 175024’

• Bearing of EF = 112018’

• Bearing of FA = 30000’

Total internal angles of a closed

traverse = (2n – 4)*90

n = number of sides of a traverse

Methods of Angel Measurement

Introduction

• Measurement of angles is basic to any survey operation

• When an angle is measured in a horizontal plane, it is

horizontal angle, when measured in vertical plane it is

vertical angle.

• Angle measurements involve three steps:

• Reference or starting point

• Direction of turning

• Angular value (value of the angle)

MEASUREMENT OF HORIZONTAL ANGLES

There are three methods of measuring horizontal

angles:-

i) Ordinary Method.

ii) Repetition Method.

iii) Reiteration Method.

MEASUREMENT OF HORIZONTAL ANGLES:

i) Ordinary Method. To measure horizontal angle POQ:-

i) Set up the theodolite at station point O and level it

accurately.

ii) Set the vernier A to the zero or 3600 of the

horizontal circle. Tighten the upper clamp.

iii) Loosen the lower clamp. Turn the instrument and

direct the telescope towards P to bisect it

accurately with the use of tangent screw. After

bisecting accurately check the reading which

must still read zero. Read the vernier B and

record both the readings.

iv) Loosen the upper clamp and turn the telescope

clockwise until line of sight bisects point Q on

the right hand side. Then tighten the upper clamp

and bisect it accurately by turning its tangent

screw.

o

P Q

HORIZONTAL ANGLE POQ

Tangent Screw: for minor

or macro movements of

telescope

i) Ordinary Method. To measure horizontal angle POQ :-

v) Read both verniers. The reading of the vernier a

which was initially set at zero gives the value of

the angle POQ directly and that of the other

vernier B by deducting 1800. The mean of the

two vernier readings gives the value of the

required angle POQ.

vi) Change the face of the instrument and repeat the

whole process. The mean of the two vernier

readings gives the second value of the angle

POQ which should be approximately or exactly

equal to the previous value.

vii) The mean of the two values of the angle POQ,

one with face left and the other with face right,

gives the required angle free from all

instrumental errors.

o

P Q


i) Ordinary Method. To measure horizontal angle POQ:-


ii) Repetition Method. This method is used for very accurate work. In

this method, the same angle is added several

times mechanically and the correct value of the

angle is obtained by dividing the accumulated

reading by the no. of repetitions.

The No. of repetitions made usually in this

method is six, three with the face left and three

with the face right. In this way, angles can be

measured to a finer degree of accuracy than that

obtainable with the least count of the vernier.

o

P Q




Station Object Face No. of

Reading

Initial angle on vernier Final reading on

vernier

Angle on vernier Mean

angle of

vernier

Mean

Angle

A B A B A B

O

P

Q

Left 1

2

3

0-0-00

30-40-0

61-40-0

180-0-0

210-40-20

240-20-20

30-40-0

61-40-0

91-20-20

210-40-20

240-20-20

271-0-20

30-26-46 30-20-6 30-23-26

30-23-24

O

P

Q

Right 1

2

3

0-0-00

30-40-20

61-40-20

180-0-0

210-40-0

240-20-40

30-40-20

61-40-20

91-20-20

210-40-0

240-20-40

271-0-0

30-26-46 30-20-0 30-23-23

Angel POQ


To measure horizontal angle by

repetitions:-

i) Set up the theodolite at starting point O

and level it accurately.

ii) Measure The horizontal angle POQ.

iii) Loosen the lower clamp and turn the

telescope clock – wise until the object

(P) is sighted again. Bisect Q accurately

by using the upper tangent screw. The

verniers will now read the twice the

value of the angle now.

o

P Q


ii) Repetition Method contd...

iv) Repeat the process until the angle is repeated the

required number of times (usually 3). Read again

both verniers . The final reading after n repetitions

should be approximately n X (angle). Divide the

sum by the number of repetitions and the result thus

obtained gives the correct value of the angle POQ.

v) Change the face of the instrument. Repeat exactly in

the same manner and find another value of the angle

POQ. The average of two readings gives the

required precise value of the angle POQ.


o

P Q



o

S P

Reiteration Method

R

Q

This method is another precise and

comparatively less tedious method of

measuring the horizontal angles.

It is generally preferred when several

angles are to be measured at a

particular station.

This method consists in measuring

several angles successively and

finally closing the horizon at the

starting point. The final reading of

the vernier A should be same as its

initial reading.

MEASUREMENT OF HORIZONTAL ANGLES:


If not ,the discrepancy is equally

distributed among all the measured

angles.

Procedure

Suppose it is required to measure the

angles POQ, QOR, ROS and SOP.

Then to measure these angles by

repetition method :

i) Set up the instrument over station

point O and level it accurately.

o

S P

Reiteration Method

R

Q


Procedure

ii) Direct the telescope towards point

P which is known as referring object.

Bisect it accurately and check the

reading of vernier as 0 or 3600 .

Loosen the lower clamp and turn the

telescope clockwise to sight point Q

exactly. Read the verniers again and

The mean reading will give the value

of angle POQ.

iii) Similarly bisect R & S

successively, read both verniers at-

each bisection, find the value of the

angle QOR and ROS.

o

S P

Reiteration Method

R

Q

iii) Reiteration Method (contd.).

iv) Finally close the horizon by sighting

towards the referring object (point P).

v) The vernier A should now read 3600. If

not note down the error .This error occurs

due to slip etc.

vi) If the error is small, it is equally

distributed among the several angles .If

large the readings should be discarded

and a new set of readings be taken.

o

S P

Reiteration Method

R

Q

iii) Reiteration Method (contd.).

MEASUREMENT OF VERTICAL ANGLES

Vertical Angle : A vertical angle is an angle between the inclined line

of sight and the horizontal. It may be an angle of elevation or

depression according as the object is above or below the horizontal

plane.

P

Q

O O

P

Q

P

Q

O HORI. LINE

HORI. LINE

β

HORI. LINE

VERTICAL ANGLE

Fig.a

Fig. b Fig. c

POQ= α + β

POQ= α - β

β

β

α α

α

Traversing

(Balancing of a Traverse and Area Calculation)

Closure of Error in a Traverse or Balancing of a Traverse

• After we have corrected our angles, we now adjust

(correct or balance) our traverse in order to create a

perfect or an exact closure.

• How do we do that?

• Closure of a traverse is initiated by computing the latitude and

departure of each line or course defining the traverse.

• The latitude of a course is its orthographic projection upon the

north-south axis of the survey.

• The departure of a course is its orthographic projection upon the

east-west axis of the survey.


• In traverse calculations, latitudes and departures can be either

negative or positive.

• North latitudes and east departures are considered positive, similar to

the x and y directions in a Cartesian coordinate system.

• South latitudes and west departures are considered negative.

• After applying our trigonometry theory, we have to manually modify

our latitudes and departures according to their proper sign:


• Let’s put in some numbers.

• Assume that the interior angle at hub A1 is 70 degrees

and that the length of A1-A2 is 125.000 meters.

• From simple trigonometry, we can find

• y12 = (125.000 m) [cos(70)] = 42.753 m

• x12 = (125.000 m) [sin(70)] = 117.462 m


• If we incorporate the azimuths of the traverse course, the sign of the

latitudes and departures are accounted for automatically.

• this will be very important if you want to perform your calculations on a

computer (EXCEL).

• Let’s examine our course A1-A2 one more time.

• Measured counter-clockwise from the east, the azimuth of course A1-

A2 is 340 degrees.


• Further calculations will be based on following equations

• where ‘θ’ represents the counter-clockwise azimuth of the specified

course measured from the east and L is the length of the traverse leg.

• For example

Problem We need to compute the latitude

and departure for each leg of the

traverse.

Solution to Problem

Measured counter-

clockwise from the east, leg

A-B has an azimuth of:

Latitude (AB) = L sin θ

= 28.510 x sin (63050’)

Departure (AB) = L cosθ

= 28.510 x cos (63050’)

63.83

Solution to Problem • how accurate was our survey? Are there errors?

• In theory, for a closed traverse, our starting point and finishing point must be the same.

• In other words, the algebraic sum of the latitudes must equal zero and the algebraic

sum of the departures must equal zero.

• The linear error of closure (z) represents the distance from the actual location of point A

to the computed location of point A:

Solution to Problem (Correcting or Balancing the Traverse by Transit Rule)

• For closed traverse, we need to distribute the closure error throughout the traverse.

• we will use a method called the TRANSIT RULE for distributing our error.

• for any leg of the traverse, the correction in latitude or departure is in proportion to the ratio of the absolute value of the latitude or departure for the traverse leg and the absolute summation of all latitudes or departures associated with the traverse.

• For leg A-B, we can write:

Solution to Problem (Correcting or Balancing the Traverse by Transit Rule))


With our survey, we were long in departure by 53mm

Similarly, the corrected departure for leg A-B is found using:

Solution to Problem The corrected lengths can now be determined for each leg of the traverse

Example

Latitude (L*Cosθ) Departure (L*Sinθ)

Line Length (L) WCB from North Q.B. (θ) Northing (+) Southing (-) Easting (+) Westing (-)

AB

BC

CD

DE

102.5

108.7

92.5

125.0

261041’

9006’

282022’

71030’

S 81041’ W

N 9006’ E

N 77038’ W

N 71030’ E

107.331

19.815

39.663

14.796

17.190

118.54

101.426

90.352

Σ 166.809 14.796 135.730 191.778

Diff of Northing-southing = 166.809-14.796 = 152.013 m

Diff of Easting-Westing = 135.730-191.778 = - 56.048 m

Length of AE = (152.0132 + 56.0482)^1/2 = 162.016 m

Reduced bearing (Bearing) of AE = tan-1 (56.048/152.013) = 20013’12”

Bearing of EA is in South and EAST Quadrant. Hence bearing of AE is in North

and West Quadrant.

Hence Whole Circle Bearing o AE is 360 - 20013’12” = 339046’48”

An open traverse was run from A to E in order to obtain the length and bearings of

the line ‘AE’ which could not be measured directly in the field. Using the below

information, it is required to calculate length and bearing of line AE.

Example

• The following lengths, latitudes, and departures were

obtained for a closed traverse ABCDEFA.

• Adjust the traverse by the BOWDITCH METHOD.

Length Latitude Departure

AB

BC

CD

DE

EF

FA

183.79

160.02

226.77

172.52

177.09

53.95

0

+128.72

+177.76

-76.66

-177.09

-52.43

+183.79

+98.05

-140.85

-154.44

0.00

+13.08

Solution to Example

Length Latitude Departure Correction Corrected

+ - + - Latitude Departure Latitude Departure

AB

BC

CD

DE

EF

FA

183.79

160.02

226.77

172.52

177.09

53.95

0

+128.72

+177.76

-76.66

-177.09

-52.43

+183.79

+98.05

+13.08

-140.85

-154.44

0.00

-0.06

-0.05

-0.07

-0.05

-0.05

-0.02

+0.07

+0.06

+0.09

+0.06

+0.07

+0.02

-0.06

+128.67

+177.69

-76.71

-177.14

-52.45

+183.86

+98.11

-140.76

-154.38

+0.07

+13.10

Σ 974.14 306.48 306.18 294.92 295.29 -0.30 +0.37 0.00 0.00

Σ Latitude = +0.30 Σ Departure = -0.37

Adjustment in latitude AB = length of AB x latitude disclosure / perimeter of traverse

= 183.79*0.30/974.14 = -0.06

Adjustment in departure AB = length of AB x departure disclosure / perimeter of traverse

= 183.79*0.37/974.14 = +0.07

Methods of Traverse Area Calculation

• Calculation of Traverse Area

• The area of a closed traverse may be calculated from:

• The coordinates (x and y)

• The latitude and double meridian distance

• The departure and total latitudes

61

Example for first method: The coordinates (x and y)

62

Find the area of the closed traverse having the following data, by the

coordinate method.

Line Latitude Departure

AB

BC

CD

DA

225.5

-245.0

-150.5

+170

+120.5

+210.0

-110.5

-220.0


63

The consecutive coordinates are arranged in independent coordinate form as

follows: The independent coordinates of the most easterly station A are

assumed to be (+200, +100). Thus, the independent coordinates of all stations

become positive (i.e. they come to the first quadrant).

Station

Side

Consecutive coordinates Independent Coordinates

Latitude (y) Departure (x) Latitude (y) Departure (x)

A

B

C

D

A

AB

BC

CD

DA

+225.5

-245.0

-150.5

+170

+120.5

+210.0

-110.5

-220.0

+200.00

+425.50

+180.50

+30.00

+200.00

+100.00

+220.50

+430.50

+320.00

+100.00


64

The coordinates are now arranged in determinant form as follows.

200.00 425.50 180.50 30.00 200.00

100.00 220.50 430.50 320.00 100.00

Sum of products of coordinates joined by solid lines,

ΣP = (200.00 x 220.5 + 425.5 x 430.5 + 180.5 x 320.0 + 30.0 x 100.0)

= 288037.75

Sum of products of coordinates joined by solid lines,

ΣQ = (100.0 x 425.5 + 220.5 x 180.5 + 430.5 x 30.0 + 320 x 200.0)

= 159265.25

Required Area = 0.5 (ΣP – ΣQ) = 0.5 (288037.75 – 159265.25)

= 64386.25 m2

Example for second method: The latitude and double

meridian distance

65

Double Meridian distance of a line = DMD of previous line + Departure of Previous Line +

Departure of this line

Calculation of DMD (Double meridian distance)

DMD of AB = Departure of AB = +120.5

DMD of BC = +120.5+120.5+210.0 = 451.0

DMD of CD = +451.0 +210.0 -110.5 = +550.5

DMD of DA = +550.5 - 110.5 – 220.0 = +220.0

Line Latitude Departure DMD Double area = (col. 2 x col. 4)

1 2 3 4 5 (+) 6 (-)

AB

BC

CD

DA

225.5

-245.0

-150.5

+170

+120.5

+210.0

-110.5

-220.0

+120.5

+451.0

+550.5

+220.0

27172.75

37400.00

110495.00

82850.25

Total +64572.75 -193345.25

Algebraic Sum -128772.5

Required area of traverse = 0.5 x algebraic sum = 0.5 x 128772.5 = 64386.25

Example for third method:

• The departure and total latitudes

66

Line Latitude Departure Station Total

latitude

Algebraic

sum of

adjoining

departure

Double area (col. 5 x

col. 6)

+ -

1 2 3 4 5 6 7 8

AB

BC

CD

DA

225.5

-245.0

-150.5

+170

+120.5

+210.0

-110.5

-220.0

B

C

D

A

+225.5

-19.5

-170.0

0.00

+330.5

+99.5

-330.5

-99.5

74527.75

56185.00

1940.25

0

Total = 130715.75-1940.25=

128772.50

Required area = 0.5 x 128772.50 = 64386.25 m2

Total latitude of C = Total latitude of B + Latitude of BC

geomatics engineering credit hrs: 2 · principle of compass surveying • the principle of compass...

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