Module 3

Module 3

• Traverse Surveying - Methods of traversing, Checks in closed traverse,

Traverse computations, Balancing the traverse- Bowditch’s rule, Transit
rule, graphical method based on Bowditch’s rule, omitted measurements (a
line and an angle only)
• Theory of Errors – Types, theory of least squares, Weighting of
observations, Most probable value, Computation of indirectly observed
quantities - method of normal equations.
 Outline

• Introduction
• Types of traversing
• Method of traversing
• Check in traversing
• Latitude, Departure
• Balancing the traverse

3
 Outline
• Introduction • Check in traversing
• Types of traversing • Closed traverse
• Closed traverse Linear error
• Open traverse Angular error
• Method of traversing • Open Traverse
• Chain traversing • Plotting traverse
• Chain and compass traversing • Angle and distance method
• Transit type traversing • Protractor method
• By measuring direct • Tangent method
bearings • Chord method
• By Direct observation of • Coordinate method
angles
• Plane table traversing • Latitude departure
• Balancing the traverse
• Bowditch's rule
• Transit Rule
• Third rule
• Graphical method- Axis method 4
 Introduction
• Working from the “whole to the part”. First provide control
using methods with higher accuracy followed by detail mapping
using lower accuracy and cheaper methods.
• Horizontal control
• Vertical control

Topographic mapping,
dimensional control of construction,
extension or densification of existing network
5
Introduction
Techniques used in the provision of horizontal control surveys include:
Classical methods
(1) Traversing
(2) Triangulation
(3) Trilateration
(4) Triangulateration
Modern approaches
(5) Satellite position fixing (Global Navigation Satellite Systems)
(6) Inertial position fixing
(7) Continuously Operating Reference Stations (CORS).

Whilst the above systems establish a network of points, single points

may be fixed by intersection and/or resection.

6
Advantages of traversing
• Since the advent of EDM equipment, traversing has
emerged as the most popular method of establishing
control networks not only in engineering surveying but
also in geodetic work.
• Traverse networks are, to a large extent, free of the
limitations imposed on the other systems and have the
following advantages
(1) Much less reconnaissance and organization required in
establishing a single line of easily accessible stations
compared with the laying out of well-conditioned
geometric figures.
(2) The limitations imposed on the other systems by
topographic conditions do not apply to traversing.
7
Advantages of traversing
(3)The extent of observations to only two stations at a time
is relatively small and flexible compared with the
extensive angular and/or linear observations at stations in
the other systems. It is thus much easier to organize.
(4) Traverse networks are free of the strength of figure
considerations so characteristic of triangular systems.
Thus once again the organizational requirements are
reduced.
(5) Scale error does not accumulate as in triangulation,
whilst the use of longer sides, easily measured with EDM
equipment, reduces azimuth swing errors.

8
Advantages of traversing
(6) Traverse stations can usually be chosen so as to be easily
accessible, as well as convenient for the subsequent
densification of lower order control.
(7) Traversing permits the control to closely follow the route
of a highway, pipeline or tunnel, etc., with the minimum
number of stations.
From the logistical point of view, traversing is far superior to
the other classical horizontal control methods and offers
at least equivalent accuracy.

9
Introduction
• The fundamental network of points whose horizontal positions are
accurately known are called horizontal control.
• Traversing is most frequently employed method for establishment of
horizontal control points for surveys of limited extent or where the
desired points lie along a devious route.

• To fix alignment of roads, canals, boundaries etc

• To locate details
• Layout of engineering works

10
Traverse
Traverse consists of a series of straight lines connected
successively at established points, along the route of a
survey

Points defining the ends of the traverse line are called

traverse stations or traverse points.

Distances between traverse stations are known as traverse

side or traverse legs.

11
At Sides are measured either by direct measurement
using a Tape or Electronic Distance Measuring
(EDM) equipment, or by indirect measurement
using tacheometer.

At stations where a traverse side changes its

direction, relative direction are measured with a
transit or theodolite, compass etc

12
Uses of Traversing
• Used as control surveys to locate topographical detail for the
preparation of plans
• To locate engineering works
• For the processing and ordering of earthwork
• Provide ground control for aerial mapping
• To establish the positions of boundary lines.
• To establish control for locating railroads, highways, and other
construction work.

13
Types of traversing
• Closed traverse; the lines ends at the starting point or runs between
stations of known coordinates
• link

• Open traverse; the lines which are connected but ends elsewhere
except starting point

14
Closed Traverse

15
Principle of Traversing
• If the coordinates of one station, true bearing and length of traverse
leg connected to it are known, the coordinates of other traverse
station can be calculated.

16
Method of traversing
There are several methods of traversing, depending on the
instruments used in determining the relative directions of
the traverse lines. The following are the principal methods:

1. Chain traversing
2. Chain and compass traversing
3. Transit type traversing
4. Plane table traversing

17
18
Methods of traversing
1. By chain angle method/ chain traversing
Entire work is done with chain/tape .
Angles are measured using chain.
Angles fixed by measurements are chain angles
2. Chain and compass traversing (Free or loose needle method)
Magnetic bearing of survey lines by compass.
Length measured by chain/tape
Magnetic meridian at each station established independently.

Resume 19
Methods of traversing
3. Transit Traversing
A. Traversing by direct observation of angles
1. Measuring included angles
2. Measuring deflection angles

B. Traversing by direct observation of bearings of traverse

legs
1. Direct method with transiting
2. Direct method without transiting
3. Back bearing method

20
Methods of traversing
TRANSIT TRAVERSING

A. Traversing by direct observation of angles

• In this method, the bearing of initial line is observed from astronomical
observation or arbitarily assumed.
• The angles between the lines are directly measured by a theodolite
• The magnetic bearing of other lines can be calculated in this method.
• Done for long traverse when great accuracy is required

• The angles measured at different stations may be either

1. Included angles
2. Deflection angles

➢This method is adopted during the theodolite survey of a big region.

21
1
Methods of traversing
1. By chain angle method/ chain traversing
Entire work is done with chain/tape .
Angles are measured using chain.
Angles fixed by measurements are chain angles
2. Chain and compass traversing (Free or loose needle method)
Magnetic bearing of survey lines by compass.
Length measured by chain/tape
Magnetic meridian at each station established independently.

Resume 2
Methods of traversing
3. Transit Traversing
A. Traversing by direct observation of angles
1. Measuring included angles
2. Measuring deflection angles

B. Traversing by direct observation of bearings of traverse

legs
1. Direct method with transiting
2. Direct method without transiting
3. Back bearing method

3
Methods of traversing
TRANSIT TRAVERSING

A. Traversing by direct observation of angles

• In this method, the bearing of initial line is observed from astronomical
observation or arbitarily assumed.
• The angles between the lines are directly measured by a theodolite
• The magnetic bearing of other lines can be calculated in this method.
• Done for long traverse when great accuracy is required

• The angles measured at different stations may be either

1. Included angles
2. Deflection angles

➢This method is adopted during the theodolite survey of a big region.

4
Traversing by included angles
• This method is used for the area whose survey is to be done by
making closed traverse .

• Angle is read accurately by repetition method.

• Only the bearing of the first line is read while the included
angles of the consecutive lines are read.

• If the arrangement of the lines of traverse is clock wise then

the external angles of the traverse are found out while if it is
counter - clock wise , the included angles are found out (angles
measured in clock wise direction)

• Generally the survey of a closed traverse is carried out in

counter-clockwise direction.
5
Traversing by included angle
• Theodolite is set at the first point A of the traverse ABCDEFGHA.
• By setting of the theodolite is meant its centering leveling focusing etc
• Then the bearing of the first line AB is read with the help of compass fitted in
the theodolite

6
Traversing by included angles
• To measure the angle HAB, the telescope is faced toward the point H and back
sight is taken. After this the theodolite is turned clockwise and faced toward
the point B and fore-sight is taken .
• The difference between these two readings gives the measurement of the
angle H A B.

7
Traversing by deflection angles
• This method is used especially for an open traverse
Eg:canal, railway track and pipe lie etc .

• Deflection angle is the angle between a line and the extended adjacent
line. These angles are read in clock-wise as well as couter-clock-wise
direction

• If these angle are read in clockwise direction then the method is called
the angle to the right method. While if these angle are read in counter-
clockwise direction then the method is known as the angle to the left
method .

8
Traversing by deflection angles
• Deflection angle is the angle which a survey line makes with the
prolongation of the preceding line.
• For reading the deflection angle at B , theodolite is set at the point B
and the face of theodolite is turned back to point A , set vernier A to 0
and the telescope is transited.

9
Traversing by deflection angles
• At this the face of telescope is diverted to the line AB. Then the
reading is taken .
• After this the theololite is turned clockwise to the next point C and
reading is taken. This is the required angle at B.
• The process is repeated by changing the face of the theodolite. The
average of these is taken . All angles of the traverse are read in this
manner.

10
Methods of traversing
TRANSIT TRAVERSING
A. Traversing by direct observation of angles
A. Measuring included angles
B. Measuring deflection angles
B. Traversing by direct observation of bearings of traverse
legs
A. Direct method with transiting
B. Direct method without transiting
C. Back bearing method

11
Methods of traversing
TRANSIT TRAVERSING
B. By measuring direct bearings of traverse leg (Traversing by fast
needle method)
• The method in which the magnetic bearings of traverse lines are
measured by a theodolite fitted with a compass is called traversing
by fast needle method.
• Used for short surveys , when better accuracy is not required.
• Used for open traverse where no checks or closing error cannot be
ascertained
• Three methods of observing the bearings of lines
• Direct method with transiting
• Direct method without transiting
• Back bearing method

12
Direct method with transiting
• Set the theodolite to zero at A
• Loosen the magnetic needle
• Point telescope to magnetic meridian.
• Loose upper clamp and bisect the next station B
• Both plates clamped, shift the instrument to B and back sight to earlier point A.
• Transit the telescope
• Loose upper clamp and bisect the next station C

13
 Direct method without transiting
• Set the theodolite to zero at A
• Loosen the magnetic needle
• Point telescope to magnetic meridian.
• Loose upper clamp and bisect the next station B
• Both plates clamped, shift the instrument to B and back sight to earlier point A.
• Loosen upper clamp and rotate clockwise to take a foresight on next station C
• Apply correction of 180.

1800

14
 Back bearing method
• Set the theodolite to zero at A
• Loosen the magnetic needle
• Point telescope to magnetic merdian.
• Loose upper clamp and bisect the next station B
• Both plates clamped, shift the instrument to B.
• Set the vernier to read back bearing of first line.
• Using lower clamp take backsight to A
• Loose upper clamp and rotate clockwise to next station
• The reading obtained is direct bearing of line BC

75 +180 = 255

15
Sources of error in Traversing
• The sources of error in traversing include:
(1) Errors in the observation of horizontal and vertical angles (angular
error).
(2) Errors in the measurement of distance (linear error).
(3) Errors in the accurate centring of the instrument and targets,
directly over the survey point (centring error).

16
Checks in closed traverse

• Check in traversing
• Closed traverse
Linear error
Angular error
• Open Traverse

17
Checks in closed traverse

The errors involved in closed traversing are two kinds:

1. Linear
2. Angular

The most satisfactory method of checking the linear measurements

consists in chaining each survey line a second time, preferably in the
reverse direction on different dates and by different parties.

18
Checks for the angular work

1.Traverse by included angles:

A) The sum of measured interior angles should be equal to (2N-4)x90, where

N=number of sides of the traverse.
B) If the exterior angles are measured, their sum should be equal to (2N+4)x90

2. Traverse by deflection angles: The algebraic sum of the deflection angles should
be equal to 360°, taking the right hand and deflection angles as a positive and left
hand angles as negative.

3.Traversing by direct observation of bearings: The fore bearing of the last line
should be equal to its back bearing ±180° measured from the initial station.
19
Checks in open traverse

No direct checks of angular measurement are available.

As illustrated in the Fig(a) the addition to the observation of bearing of

AB at station A, bearing of AD can also be measured., if possible.
Similarly, at D, bearing of DA can be measured and check applied. If the
two bearings differ by 180°, the work may be accepted as correct.
D
E

C
A
B
Checks in open traverse

• Another method, which furnishes a check when work

is plotted is shown as in Fig (b) and consists reading
the bearing to any prominent point P from each of
the consecutive stations. The check in plotting
consists in laying off the lines AP, BP, CP, etc and
noting whether the lines pass through one point.

Q
A C

B 21
Plotting a traverse survey
• Angle and distance method
• Protractor method
• Tangent method
• Chord method
• Coordinate method

22
Field work of Theodolite traverse survey
• Reconnaissance
• Selection and marking stations
• Measurement of traverse legs
• Measurement of traverse angles
• Booking of field notes

23
Traverse computations: Latitude(L) and departure(D)
• Latitude (L) – of a survey line is defined as the co-ordinate length
measured parallel to an assumed meridian direction (true north or
magnetic north or any reference meridian)
ie, L = l cos θ (+ in North direction, -ve in South direction)
• Departure (D) - of a survey line is defined as the co-ordinate length
measured at right angles to the meridian direction.
ie, D = l sin θ (+ in East direction and
–ve in West direction)
Traverse computations: Latitude(L) and departure(D)

•∑L=0
• ∑D = 0

25
Co-ordinates
• Consecutive or dependent co-ordinates: L and D of a line with
reference to the preceding point
Consecutive co-ordinate of any point = latitude and departure of
the line joining the preceding point to the point under
consideration.
• Independent or total co-ordinates: total L and D of a point with
reference to common origin
Total latitude or departure of line = total latitude (or departure)
of first point of traverse + algebraic sum of all preceding latitudes
(or departures)
26
Problem 1
Consecutive or dependent Independent or total

line length WCB R.B L= D= L Total STATIO

Lcos θ sinθ Coordinates N

AB 232 32 12

BC 148 138 36

CD 417 202 24

DE 372 292 0

27
line length WCB R.B L = Lcos θ D= L sinθ Total Coordinates STATION

Y X

Y X

AB 232 32° 12’ N 32° 12’ E + 196.32 + 123.63 400 400 A

BC 148 138° 36’ S 41° 24’ E - 111.02 + 97.88 596.32 523.63 B

CD 417 202° 24’ S 22° 24’ W - 385.54 - 158.90 485.30 621.51 C

DE 372 292° 0 N 68° 0 W + 139.36 - 344.9 99.76 462.61 D

239.12 117.68 E
28
Omitted measurements

• Often it becomes impossible to measure all the lengths and bearings

of a closed traverse. (due to obstacles)
• The values of these missing quantities can be determined, provided
they do not exceed two in number.
• The omitted measurements or missing quantities can be calculated by
latitudes and departures.
• It is based on the principle that for a closed traverse, the algebraic
sum of the latitudes = 0 and algebraic sum of departures = 0.
ie, ΣL = 0 and ΣD = 0
Types of omitted measurements
The following are some of the more common types of omitted measurements:
I. Omitted Measurements are in One Side
a. Length and bearing of one side unknown
II. Omitted Measurements Involving Adjacent Sides
a. Lengths of two sides unknown
b. Bearings of two sides unknown
c. Length of one side and bearing of another side unknown
III. Omitted Measurements Involving Non-Adjacent Sides
a. Lengths of two sides unknown
b. Bearings of two sides unknown
c. Length of one side and bearing of another side unknown
Omitted measurement-Length and bearing of side omitted
 Problem
1. The consecutive co-ordinates of a line PQ are +352.5 and -126.3 with
reference to the magnetic meridian. If the magnetic declination is 6°35’
west, find out the co-ordinates of line PQ with reference to the true
meridian.
2. The co-ordinates of two points A and B are as follows. Find the
length and bearing of AB.
Points Co-ordinates

Northing Easting
A 500.25 640.75
B 840.78 315.60
3. It was impossible to observe the length and bearings of line PQ
directly. The following observations were taken from the two stations A
and B. Compute the length and bearing of PQ.
Line Length in m Bearing
AP 126.00 S 65°36’ W
AB 314.40 N 24°12’ E
BQ 115.50 N 76°48’ W

4. Following are the lengths and bearings of a traverse ABCD. Calculate

the length and bearing of line DA.
Line Length in m Bearing
AB 248 30°
BC 320 140°
CD 180 210°
Co-ordinates
• Consecutive or dependent co-ordinates: L and D of a line with
reference to the preceding point
Consecutive co-ordinate of any point = latitude and departure of
the line joining the preceding point to the point under
consideration.
• Independent or total co-ordinates: total L and D of a point with
reference to common origin
Total latitude or departure of line = total latitude (or departure)
of first point of traverse + algebraic sum of all preceding latitudes
(or departures)
1
Problem 1
Consecutive or dependent Independent or total

line length WCB R.B L= D= L Total STATIO

Lcos θ sinθ Coordinates N

AB 232 32 12

BC 148 138 36

CD 417 202 24

DE 372 292 0

2
line length WCB R.B L = Lcos θ D= L sinθ Total Coordinates STATION

Y X

Y X

AB 232 32° 12’ N 32° 12’ E + 196.32 + 123.63 400 400 A

BC 148 138° 36’ S 41° 24’ E - 111.02 + 97.88 596.32 523.63 B

CD 417 202° 24’ S 22° 24’ W - 385.54 - 158.90 485.30 621.51 C

DE 372 292° 0 N 68° 0 W + 139.36 - 344.9 99.76 462.61 D

239.12 117.68 E
3
Problem 2

line length R.B L = Lcos θ D= L sinθ

AB 305 N 30 30 E

BC 550 S 42 42 E

CD 830 S 48 48 W

DE 530 N60 0 W

4
Balancing the traverse
• Operation of applying the corrections to latitudes and departures so
that ΣL = 0 and ΣD = 0. (for a closed polygon)
• Methods of adjusting traverse
• Bowditch’s method
• Transit method
• Graphical method
• Axis method
Balancing the traverse- Error of closure
• In a closed traverse, ΣL = 0 and ΣD = 0
• Relative error of closure = error of closure e’ /perimeter of traverse

6
Problem :2
The length and magnetic bearings of legs of a traverse ABCDA is
given . Later it is found that there is a magnetic declination of 50
30’W. Find the error of closure.

LINE LENGTH MAG TRUE REDUCE L =l D =l SIN

BEARING BEARING D COSθ θ
BEARING

AB 470 343 52 338 22 N 21 38 +436.90 - 173.30

W
BC 635 87 50 82 20 N 82 20 +84.72 +629.40
E
CD 430 172 40 167 10 S 12 50 E -419.30 +95.52

DA 563 265 12 259 42 S 79 42 -100.50 -552.90

W
2098 ∑L= ∑D = -
+1.67 2.37
7
Balancing the traverse- Error of closure
• ∑L= 1.67, ∑D= -2.37 θ

• Tan θ = 2.37/1.67
• Error of closure e= √(1.67)2 + (-2.37)2 =
• Precision or Relative error of closure
= e/perimeter
= /2098
= 1 in 741

8
(1) BOWDITCHS RULE
• Also Known as Compass rule, used when angular measure and linear
measure are equally precise
• The “Bowditch rule” as devised by Nathaniel Bowditch, surveyor,
navigator and mathematician, as a proposed solution to the problem
of compass traverse adjustment, which was posed in the American
journal The Analyst in 1807.

9
Problem 3
• The following are lengths and consecutive coordinates of traverse
ABCDA. Balance using Bowditch’s rule

Line Length Consecutive coordinates

(m) Latitude Departure

AB 235.10 +218.50 -86.80
BC 317.40 +42.16 +314.59
CD 215.0 -209.49 +48.36
DA 281.6 -50.51 -277.03

10
 Solution:
Total error in latitudes=+218.50+42.16-209.49-50.51
=+0.66
Total error in departures=-86.80+314.59+48.36-277.03
=-0.88
Perimeter of the traverse
=235.10+317.40+215+281.60
=1049.10m.

11
(a) Correction to latitude of
AB=Length of AB/Perimeter of traverse*total error in latitudes
=235.10/1049*0.66=0.148
BC=317.40/1049.10*0.66=0.200
CD=215.00/1049.10*0.66=0.135
DA=281.60/1049.10*0.66=0.177
CHECK Total correction =0.66 m

12
As the error is positive, the sign of the correction is
negative.
corrected Latitudes of,
AB=+218.50-0.148 = + 218.352
BC=+42.16-0.200 = +41.960
CD=-209.49-0.135 = -209.625
DA=-50.51-0.177 = -50.687
Total = 0.000

13
(b) Correction to departure of
AB = Length of AB/perimeter of traverse*Total error in departures
=235.10/1049.10*0.88=0.197
BC =317.40/1049.10*0.88=0.266
CD =215.00/1049.10*0.88=0.180
DA =281.60/1049.10*0.88=0.236
Total correction=0.88

14
As the error is negative, the sign of the correction is
positive corrected departure of
AB=- 86.80+ 0.197=- 86.603
BC=+314.59+0.266=+314.856
CD=+ 48.36+0.180=+ 48.540
DA=-277.03+0.236=-276.794
Total = 0.001 = 0.000
15
Line Length Consecutive Corrected Consecutive Corrected
coordinates latitude coordinates departure

(m) Latitude Departure

AB 235.10 +218.50 +218.352 -86.80 -86.603
BC 317.40 +42.16 +41.960 +314.59 +314.856
CD 215.0 -209.49 _209.625 +48.36 +48.540
DA 281.6 -50.51 -50.687 -277.03 -276.794

16
(2) TRANSIT RULE

• Used when angular measure is more precise than linear measure

17
Problem 4 (Balancing the traverse – Transit rule)
The following are the length and consecutive coordinates of
closed traverse ABCDA.

Line Length(m) Consecuti

ve
coordina
tes
Latitude Departure
AB 235.10 +218.50 -86.80
BC 317.40 +42.16 +314.59
CD 215.00 -209.49 +48.36
DA 281.60 -50.51 -277.03

18
Solution:
Total error in latitude = +0.66
Total error in departures =-0.88
Total perimeter of the traverse = 1049.10m
Sum of latitudes = 218.50+42.16+209.49+50.51
=520.66
Sum of departures =86.60+314.59+48.36+277.03
=726.78

19
Correction to latitude of
AB = Latitude of AB/Total sum of latitudes*total errorin latitude
= 218.50/520.66*0.66=0.277
BC = 42.16/520.66*0.66=0.053
CD = 209.49/520.66*0.66=0.266
DA = 50.51/520.66*0.66=0.064
Total correction = -0.66

20
As the error is positive, the sign of the correction is negative.
Calculation of corrected latitudes:
Line Latitude Correction Corrected
latitude
AB +218.50 -0.277 +218.223
BC +42.16 -0.053 +42.107
CD -209.49 -0.266 -209.756
DA -50.51 -0.064 -50.574
Sum= +0.66 -0.66 0.000

21
Correction to departure of
AB = departure of AB/sum of departures*total error in departure
= 86.80/726.78*0.88=0.105
BC = 314.59/726.78*0.88=0.381
CD = 48.36 /726.78*0.88=0.059
DA = 277.03/726.78*0.88=0.335
Total correction = 0.88
As the error is negative, the sign of the correction is positive.

22
Calculation of corrected departures:

Line Departure Correction Corrected

departure
AB -86.80 +0.105 -86.695
BC +314.59 +0.381 +314.971
CD +48.36 +0.059 +48.419
DA -277.03 +0.335 -276.695
Sum= -0.88 +0.88 0.000

23
(3) Third Rule:
If the correction are to be applied separately, then the third rule may
be used:
(¡) Correction to northing of anyline
=northing of that line/sum of northings*1/2(total error in latitude)
(¡¡) Correction to southing of any line
=southing of that line/sumof southings*1/2(total error in latitude)
(¡¡¡) Correction to easting of any line
=easting of that line/sum of eastings*1/2(total errorin departure)
(¡v) Correction to westing of any line
=westing of that line/sum of westings*1/2(totalerror in departure)

NOTE: If the error is positive, correction will be negative, and vice versa. 24
(4) Graphical method

25
Axis Rule
• Adopted when angles are measured very accurately
• Only length of sides are changed, direction remains unchanged, the general shape
of diagram is preserved
• Correction to any length = that length × 1/2 length of closing error/Length of axis
of adjustment

26
Gales Traverse Table
• Traverse computations can be done in a tabular form, called Gales Traverse
table

Steps:
• Adjust the interior angles to satisfy the geometric condition (check (2n-4) 90ᵒ)
• In case of compass bearings are adjusted for Local Attraction
• Starting from observed bearings , calculate bearings of all lines.
• Reduce WCB to QB.
• Calculate consecutive coordinates (latitudes and departures).
• Calculate ∑ L and ∑ D
• Apply necessary corrections to latitude and departure using any rule.
• Using corrected consecutive coordinates, calculate independent coordinates
27
Gales Traverse table
Sources of errors in traversing

1. Poor selection of stations, resulting in bad sighting conditions

caused by
(a) alternate sun and shadow
(b) visibility of only the rod's top
(c) line of sight passing too close to the ground
(d) lines that are too short
(e) sighting into the sun.
2. Errors in observations of angles and distances.
3. Failure to observe angles an equal number of times direct and
reversed. 30
Mistakes in traversing
1. Occupying or sighting on the wrong station.
2. Incorrect orientation.
3. Confusing angles to the right and left.
4. Mistakes in note taking.
5. Misidentification of the sighted station.

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Theory of errors
Errors of measurement are of three kinds:
(i) Mistakes
(ii) systematic errors
(iii) accidental errors
(i) Mistakes - Mistakes are errors that arise from inattention, inexperience, carelessness and poor

judgment or confusion in the mind of the observer. If a mistake is undetected, it produces a serious

effect on the final result. Hence every value to be recorded in the field must be checked by some

independent field observation.

(ii) Systematic Error. A systematic error is an error that under the same conditions will always be of

the same size and sign.

• A systematic error always follows some definite mathematical or physical law, and a

correction can be determined and applied.

• Such errors are of constant character and are regarded as positive or negative according as

they make the result too great or too small. Their effect is therefore, cumulative.
(iii) Accidental Error- Accidental errors are those which remain after
mistakes and systematic errors have been eliminated and are caused by
a combination of reasons beyond the ability of the observer to control.

• They tend sometimes in one direction and some times in the other,
i.e., they are equally likely to make the apparent result too large or
too small.

• They tend to compensate each other and are called compensating

errors.
The law of accidental errors
• Investigations of observations of various types show that accidental errors follow a definite law,
the law of probability.

• This law defines the occurrence of errors and can be expressed in the form of equation which is
used to compute the probable value or the probable precision of a quantity.

• The most important features of accidental errors which usually occur are:
(i) Small errors tend to be more frequent than the large ones ; that is they are the most
probable.
(ii) Positive and negative errors of the same size happen with equal frequency ; that is,
they are equally probable.
(iii) Large errors occur infrequently and are impossible.
Probability curve
• Probable Error of a single observation:

• Probable error of a single observation in a set of observation is derived using the formula,

Es = +-0.6745√(∑v2/ (n-1))

where, Es = probable error of single observation

v = difference between the observation and the mean value of the set of observation.

n = numbers of observations in the given set.

• Probable Error of an average (mean):

• Probable error of an average or mean is given by,

Em = 0.6745√(∑v2/n (n-1)) = Es/ √n

where, Em = probable error of mean.

Most probable value
• Most probable value of a quantity is the one which has more chances of being
true than any other.
a) Most probable value of a quantity is equal to the arithmetic mean if
the observations are of equal weight.
b) Most probable value of a quantity is equal to the weighted
arithmetic mean if the observations are of unequal weight.

• Weight of an observation: Weight of an observation is its relative importance to

the other observations taken under the identical conditions.

• Mean value: Mean value of a set of observation is the arithmetic or the weighted
arithmetic mean of the observations.
Problem
• In carrying a line of levels across a river, the following eight readings
were taken with a level under identical conditions:
2.322, 2.346, 2.352, 2.306, 2.312, 2.300, 2.306, 2.326.
Calculate (i) the probable error of single observation
(ii) the probable error of mean
Answer :
Rod reading v v2
2.322 0.001 0.000001
2.346 0.025 0.000625
2.352 0.031 0.000961
2.306 0.015 0.000225
2.312 0.009 0.000081
2.300 0.021 0.000441
2.306 0.015 0.000225
2.326 0.005 0.000025
Mean = 2.321 Σ v2 = 0.002584

Es = +-0.6745√(∑v2/ (n-1))
= +-0.6745√(0.002584/(8-1)) = +- 0.01295 m

Em = +- Es/ √n
= +- 0.00458 m
• Principle of least squares:

• Most probable value of a given quantity from the given available set of
observation is the one for which the sum of the squares of the residual errors is a
minimum.

• Alternatively, the most probable values of the errors in the given set of
observations of equal weight are those for which the sum of their squares is a
minimum.

• It can be proved that the mean value is the true value in case the numbers of
observations are very large in numbers.
• The sum of the squares of the residuals found by the arithmetic mean value is a
minimum. This is thus the fundamental law of least squares.
Definitions:
• Independent quantity - is one whose value is independent of the values of other
quantities.

• Conditioned quantity - is one whose value is dependent of the values of one or more
quantities. It is also called dependent quantity.
Eg: In triangle ABC, <A+<B+<C = 180°. Any two angles may be regarded as
independent and third angle as dependent.

• Direct observation –An observation is the numerical value of a measured quantity. It may
be direct or indirect.
A direct observation is the one made directly on the quantity being determined.
• Indirect observation - is one in which observed value is deduced from the
measurement of some related quantities.

• Weight of an observation – is a number giving an indication of its precision and

trustworthiness when making a comparison between several quantities of
different worth.
It indicates the relative precision of a quantity within a set of observations.
• If a certain observation is of weight 4, it means that it is 4 times as much reliable as an
observation of weight 1.

• Observed value of a quantity – is the value obtained when it is corrected for all
the known errors

• True value of a quantity – is the value which is absolutely free from all the errors.
• Most probable value – is the one which has more chances of being true than any other.

• True error – is the difference between true value of a quantity and its observed value.

• Most probable error – is defined as that quantity which added to or subtracted from the
most probable value fixes the limits within which it is an even chance the true value of
the measured quantity must lie.

• Residual error – is the difference between the most probable value of the quantity and
its numerical value

• Observation equation – is the relation between the observed quantity and its numerical
value eg: A+B = 50°
• Conditioned equation – is the equation expressing the relation
existing between the several dependent quantities.
Eg: In triangle ABC, <A+<B+<C = 180°

• Normal equation – is the one which is formed by multiplying each

equation by the coefficient of the unknown whose normal equation is
to be found and by adding the equations thus formed
Laws of weights
1. The weight of arithmetic mean of measurements of unit weight is
equal to the number of observations.
Eg: Let an angle A be measured 6 times, the values are 30°20’18’’ weight 1,
30°20’10’’ weight 1, 30°20’7’’ weight 1, 30°20’10’’ weight 1, 30°20’9’’ weight
1, 30°20’10’’ weight 1
Then, arithmetic mean = 30°20’ + 1/6 (18”+10”+7”+10”+9”+10”)
= 30°20’9’’
Weight of arithmetic mean 30°20’9’’ = number of observations = 6
2. The weight of the weighted arithmetic mean is equal to the sum of the
individual weights.

Eg: Let an angle A be measured 6 times, the values are 30°20’8’’ weight 2, 30°20’10’’
weight 3, 30°20’6’’ weight 2, 30°20’10’’ weight 3, 30°20’9’’ weight 4, 30°20’10’’ weight
2
Sum of individual weights = 2+3+2+3+4+2 =16
Weighted arithmetic mean = 30°20’ + 1/16 (8”x2+10”x3+6”x2+10”x3+9”x4+10”x2)
= 30°20’9’’
Weight of weighted arithmetic mean 30°20’9’’ = sum of individual weights = 16
3. The weight of the arithmetic sum of two or more quantities is equal to the
reciprocal of the sum of reciprocals of individual weights.
Eg: Let angle A = 42°10’20’’ weight 4, B = 30°40’10’’ weight 2
Sum of reciprocal of individual weights = 1/4 +1/2 =3/4
Weight of A + B (72°50’30”) = 1/(3/4) = 4/3
Weight of A - B (11°30’10”) = 1/(3/4) = 4/3

4. If a quantity of given weight is multiplied by a factor, the weight of the

result is obtained by dividing its given weight by the square of the factor.
Eg: Let angle A = 42°10’20’’ weight 6

Weight of 3A ( 126°31’) = 6/(3^2) = 2/3

5. If a quantity of given weight is divided by a factor, the weight of the
result is obtained by multiplying its given weight by the square of the
factor.
Eg: Let angle A = 42°10’20’’ weight 4

Weight of A/3 (= 14°3’30”) = 4 x 3^2 = 36

6. If an equation is multiplied by its own weight, the weight of the resulting

equation is equal to the reciprocal of the weight of the equation.

Eg: Let angle A+B = 98°20’30’’ weight 3/5

Weight of 3/5 (A+B) (= 59°0’18”) = 1/(3/5) = 5/3

7. The weight of an equation remains unchanged, if all the signs of
the equation are changed or if the equation is added to subtracted
from a constant.

Eg: Let angle A+B = 80°20’ weight 3

Weight of 180° - (A+B) (=99°40’) = 3
Most probable value

• Most probable value of a quantity is the one which has more chances of
being true than any other.
1. Direct observations of equal weight
The most probable value of directly observed quantity of equal
weights is equal to the arithmetic mean of the observed values.
If v1,v2,v3,….. ,vn = observed value of quantity of equal weight,
M = arithmetic mean of observations, then,
Most probable value = arithmetic mean = M = v1+v2+v3+…….+vn
n
2. Direct observations of unequal weight
The most probable value of an observed quantity of unequal
weights is equal to the weighted arithmetic mean of the
observed quantities.
If v1,v2,v3,….. ,vn = observed value of quantity weights w1,
w2,…..wn
N = most probable value of the quantity, then,

N = w1 v1+w2 v2+w3 v3+…….+wn vn

w1+w2+w3+…………+wn
3. Indirectly observed quantities involving unknowns of equal or
unequal weights

When the unknowns are independent of each other, the most

probable value can be found by forming the normal equations
for each of the unknown quantities and treating them as
simultaneous equations to get the values of the unknowns.
4. Observation equations accompanied by condition equation

When the observation equations are accompanied by one or

more condition equations, the latter may be reduced to an
observation equation which will eliminate one of the unknowns.
The normal equation can then be formed for the remaining
unknowns.
Normal equation

• A normal equation is the one which is formed by multiplying each

equation by the coefficient of the unknown whose normal equation is
to be found and by adding the equation thus formed.
Example problem 1 :
• Form the normal equations for x,y,z in the following equations of equal
weight.
3x+3y+z-4 = 0 ----------- (1)
x+2y+2z-6 = 0 ------------(2)
5x+y+4z-21 = 0 ------------(3)
Ans:
Normal equation for x
Coefficient of x in eqn (1), (2) and (3) are 3,1,5
(1) X 3 9x + 9y + 3z -12 =0
(2) X 1 x + 2y + 2z - 6 =0
(3) X 5 25x + 5y + 20z – 105 =0
35x + 16y + 25z -123 =0
Normal equation for y
Coefficient of y in eqn (1), (2) and (3) are 3,2,1
(1) X 3 : 9x + 9y + 3z -12 =0
(2) X 2 : 2x + 4y + 4z - 12 =0
(3) X 1 : 5x + y + 4z – 21 =0
16x + 14y + 11z -45 = 0

Normal equation for z

Coefficient of z in eqn (1), (2) and (3) are 1,2,4
(1) X 1: 3x + 3y + z -4 =0
(2) X 2: 2x + 4y + 4z - 12 =0
(3) X 4: 20x + 4y + 16z – 84 =0
25x + 11y + 21z -100 =0
Normal equations of x : 35x + 16y + 25z -123 =0
Normal equations of y : 16x + 14y + 11z -45 =0
Normal equations of z : 25x + 11y + 21z -100 =0
Problem 2:
• Form the normal equations for x,y,z in the following equations of
given weight.
3x+3y+z-4 = 0, weight 2 ----------- (1)
x+2y+2z-6 = 0, weight 3 ------------(2)
5x+y+4z-21 = 0, weight 1 ------------(3)
Normal equation for x
Coefficient of x in eqn (1), (2) and (3) - 3,1,5 to be multiplied with
weights 2,3,1
(1) X (3x2) - 18x + 18y + 6z -24 =0
(2) X (1x3) - 3x + 6y + 6z - 18 =0
(3) X (5x1) - 25x + 5y + 20z – 105 =0
46x + 29y + 32z -147 =0

Normal eqn for x: 46x + 29y + 32z -147 =0

Normal equation for y
Coefficient of y in eqn (1), (2) and (3) : 3,2,1 to be multiplied with
weights : 2,3,1
(1) X (3x2) - 18x + 18y + 6z -24 =0
(2) X (2x3) - 6x + 12y + 12z - 36 =0
(3) X (1x1) - 5x + y + 4z – 21 =0
29x + 31y + 22z -81 =0

Normal eqn for y: 29x + 31y + 22z -81 =0

Normal equation for z
Coefficient of z in eqn (1), (2) and (3) - 1,2,4 to be multiplied with
weights 2,3,1
(1) X (1x2) - 6x + 6y + 2z -8 =0
(2) X (2x3) - 6x + 12y + 12z - 36 =0
(3) X (4x1) - 20x + 4y + 16z – 84 =0
32x + 22y + 30z -128 =0

Normal eqn for z: 32x + 22y + 30z -128 =0

Problem 3
• Find the most probable value of angle A from the following
observation equation.
A = 30°28’40” ------------ (1)
3A = 91°25’55” ------------ (2)
4A = 121°54’30” ------------ (3)
Ans:
Only one unknown (A) with equal weight.
Coefficients of A are 1,3,4.
(1) X 1 : A = 30°28’40”
(2) X 3 : 9A = 274°17’45”
(3) X 4 : 16A = 487°38’00”
26 A = 792°24’25” – Normal eqn for A

Therefore A = 30°28’37.9”
Problem 4:
Problem:
• Find the most probable values of angles A,B,C of triangle ABC from
the following observation equations.
A = 77°14’20” , weight 4
B = 49°40’35” , weight 3
C = 53°04’52” , weight 2