8. THEORY OF ERRORS (SUR) 3140601 GTU

1
PREPARED BY : ASST. PROF. VATSAL D. PATEL
MAHATMA GANDHI INSTITUTE OF
TECHNICAL EDUCATION &
RESEARCH CENTRE, NAVSARI.

 Measurements of distances and angles are made in various
surveying operations. These survey measurements may
involve several elementary operations, such as setting up,
centering, levelling, bisecting and reading.
 In performing these operations, it is impossible to determine
the true values of these quantities (distance or angles) because
some errors always creep in all measurements.
2

 The errors can be broadly classified into three types :
1. Gross errors or mistakes
2. Systematic or cumulative errors
3. Accidental or random errors
3

 REASONS :
 Inattention
 Inexperience
 Carelessness
 Poor judgment/confusion in mind
 SOLUTION :
 Cross checking
4

 Under same condition always be the same size and sign.
 Always follow some mathematical /physical law.
 Correction can be determined and applied.
 Effect is +ve/-ve, Cumulative effect.
 Constant in character +ve OR –ve ,Results too great/too small.
 If undetected systematic errors are very serious.
5

 SOLUTION OF SYSTEMATIC ERRORS :
 Instrument design- errors should auto elimination.
 Find out the relationship of error.
6

 Remains after mistakes and systematic errors have been
eliminated.
 Beyond the ability of observer to control.
 Represent the limitation of precision.
 Obey the laws of chance.
 Must handle as per laws of probability.
7

 Lack of perfection in human eye.
 Observation in cm. tape 10mm,9mm,11mm.
 May compensate each other- compensating error.
 Smaller error – better precision.
8

 The following definitions of some of the terms should be
clearly understood in adjustment of the measurement :
9
Observation
Observed
value of a
quantity
True value
of a quantity
True error
Most
probable
value (MPV)
Most
probable
error
Residual
error
Observation
equation
Conditioned
equation
Normal
equation

 Observation :
 The measured numerical value of a quantity is known as
observation. For example, 15.36m, 45˚20'15'', etc.
10
Direct observation Indirect observation
If the value of a quantity
is measured directly.
Value of the quantity calculated
indirectly from direct observation
i.e. Angle A= 45˚20'15'' i.e. Angle computed at true station
from satellite station

 Observed value of a quantity :
 The observed value of a quantity is the value obtained from
the observation after applying corrections for systematic errors
and eliminating mistakes.
11

 Observed value of a quantity :
 The observed value of a quantity may be classified as :
12
Independent quantity Dependent or Conditioned
quantity
Whose value is independent
of the value of other quantity
( i.e. R.L. of B.M.)
Whose value is dependent upon
the value of one or more quantity
(i.e. R. L. of other points other
than B.M.)

 True value of a quantity :
 The value which is absolutely free from all the errors
(hypothetical quantity)
 True error :
 Difference between the true value and observed value.
 True error = True value - Observed value
13

 Most probable value (MPV) :
 Is the one which has more chance of being true than has any
other
 Most probable error:
 Which is added to and subtracted from most probable value
fixes the limits of true value.
 Residual error :
 Residual error = Observed value – Most probable value
14

 Observation equation :
 Relation between observed quantity and its numerical value
 For example, A + B = 78˚15' 20''
 Conditioned equation :
 Which is expressing the relation between several dependent
quantities.
 For example, In triangle ABC (A+ B+ C = 180˚)
15

 Normal equation :
 Multiplying each equation with unknown quantity
 No of normal equation= No of unknown
 Using normal equation MPV evaluated
16

 Accidental error follows law of probability.
 Occurrence of errors can be expressed by equation.
 Equation used to find probable error/precision.
 e.g. 45˚22' 30'' ± 2.10''
17

 Most important features of accidental error :
 +ve and –ve errors are equal in size and frequency, as the
curve is symmetrical; that is, they are equally probable.
 Small errors are more frequent than large errors; that is, they
are the most probable.
 Very large error seldom occur and are impossible.
19

 It is calculated from the probability curve of errors.
 In any large series of observations the probable error is an
error of such a value that the number of errors numerically
greater than it is the same as the number of errors numerically
less than it.
 The probable error is indicated along with the value of the
quantity with a ± sign.
20

 Purposes :
 Measure of precision of any series of observation.
 Means of assigning weights to two or more quantities.
 From the probability curve following six aspects are derived :
21
1
Probable
error of
Single
measurement
2
Probable
error of an
average
3
Probable
error of Sum
of
observations
4
Mean
Square
Error
5
Average
Error
6
Probable error
of Single
measurement

24
5
Average Error
It is arithmatic mean of seperate
errors of equal weight

 Several standards exist for assessing the precision of a set of
observation. The most popular is the standard deviation (σ).
 It is a numerical value which indicates the amount of variation
about a central value.
 The standard deviation establishes the limits of the error
bound within which 68.3% of the values of the set should lie.
 The smaller the value of the standard deviation, the greater the
precision.
25

 In carrying a line of level across a river, the following eight
readings were taken with level under identical condition.
2.322,2.346,2.352,2.306,2.312,2.300,2.306,2.326. Calculate :
1. The probable error of single observation
2. The probable error of mean
3. Most probable value of staff reading
4. Standard deviation
26

Sr.No. Staff readings Mean
v
(2)-(3)
v2
(1) (2) (3) (4) (5)
1 2.322 2.321 0.001 0.000001
2 2.346 2.321 0.025 0.000625
3 2.352 2.321 0.031 0.000961
4 2.306 2.321 -0.015 0.000225
5 2.312 2.321 -0.009 0.000081
6 2.300 2.321 -0.021 0.000441
7 2.306 2.321 -0.015 0.000225
8 2.326 2.321 0.005 0.000025
Total ∑ 18.57 ∑V2 0.0025831
27

29
Case-1
Direct observation of
equal weight
Probable error of
single observation
of unit weight
Probable error of
single observation
of weight w
Probable error of
single arithmetic
mean
a
b
c
Case-2
Un-Equal weights
Probable error of
single observation
of unit weight
Probable error of
single observation
of weight w
Probable error of
weighted
arithmetic mean
a
b
c
Case-3
Indirect observation
of Independent
quantity
Case-4
involving
conditional
Equations
Case-5
Computed
Quantities

30
Case-1
Direct observation of equal
weight
Case-2
Direct observation of Un-
Equal weights

31
Case-3
Indirect observation of
independent quantities
Case-4 Indirect
observations involving
conditional Equations
Case-5
Computed
Quantities
The
probable
error of
computed
quantities
may be
calculated
form the
laws…..

 The observed values of an angle are given below :
 Find :
 Probable error of single observation value of unit weight.
 Probable error of weighted arithmetic mean
 Probable error of single observation of weight 3
32
Angle Weight
85° 40' 20'' 2
85° 40' 18'' 2
85° 40' 19'' 3

Value Weight Value * Weight v v2 wv2
(1) (2) (3) = (1)*(2) (4) (5) = (4)*(4) (6)=(2)*(5)
20'' 2 40'' +1 1 2
18'' 2 36'' -1 1 2
19'' 3 57'' 0 0 0
n=3 ∑w=7 ∑= 133''/7 =19'' ∑v = 0 ∑V2 = 0 ∑wv2 = 4
33

 Weight :
 The weight of a quantity is trust worthiness of a quantity.
 The relative precision and trustworthiness of an observation as
compared to the precision of other quantities is known as
weight of the observation.
 The weights are always expressed in numbers.
35

 Weight :
 Higher number indicate higher precision and trust as compared
to lesser numbers.
36
Law of weight
1
Law of weight
2
Law of weight
3
Law of weight
4
Law of weight
5
Law of weight
6
Law of weight
7

 Law of weight 1 :
 The weight of the arithmetic mean of a number of
observations of unit weight, is equal to the number of the
observations.
37

 Calculate the weight of the arithmetic mean of the following
observations of an angle of unit weight.
38
Angle Weight
A= 65° 30' 10'' 1
A= 65° 30' 15'' 1
A= 65° 30' 20'' 1

 The number of observations of unit weight, n=3.
 Arithmetic mean
= 65° 30' 15''
 From law (1), the weight of the arithmetic mean = n
= 3
 Hence, the weight of the arithmetic mean 65° 30' 15'' is 3.
39

 The weight of the weighted arithmetic mean of a number of
observations is equal to the sum of the individual weights of
observations.
40

 An angle A was observed three times as given below with their
respective weights. What is the weight of the weighted
arithmetic mean of the angle ?
41
Angle Weight
A= 40° 15' 10'' 1
A= 40° 15' 14'' 2
A= 40° 15' 12'' 3

 Weighted arithmetic mean of A
= 40° 15' 12.33''
 Sum of the individual weights = 1 + 2 + 3 = 6
 From law (2), the weight of the weighted arithmetic mean of
the angle = sum of the individual weights
 The weighted arithmetic mean 40° 15' 12.33'‘ has the weight
of 6.
42

 The weight of the arithmetic sum of two or more quantities is
equal to the reciprocal of the sum of reciprocals of individual
weights.
43

 Calculate the weights of (A + B) and (A - B) if the measured
values and the weights of A and B, respectively are :
A = 40° 50' 30'' wt. 3
B = 30° 40' 20'' wt. 4
44

 A + B = 40° 50' 30'' + 30° 40' 20'' = 71° 30' 50''
 A - B = 40° 50' 30'' - 30° 40' 20'' = 10° 10' 10''
 From law (3), the weights of (A + B) and (A - B)
= = =
Hence, A + B = 71° 30' 50'' wt.
and A - B = 10° 10' 10'' wt.
45
w1 w2

 The weight of the product of any quantity multiplied by a
constant, is equal to the weight of that quantity divided by the
square of that constant.
46

 What is the weight of 3A if A = 30° 25' 40'' and its weight is
3?
 Solution :
 From law (4) the weight of (3A) is
 Weight of (3A) = 91° 17' 00''
= = =
47
C2
32

 The weight of the quotient of any quantity divided by a
constant, is equal to the weight of that quantity multiplied by
the square of that.
48

 Compute the weight of if A = 36° 20' 40'' of weight 3.
 Solution :
 The constant C of division is 4 and weight w of the
observation is 3.
 From law (5), the weight of is wC2
 weight of = = 3 x 42
= 9° 5' 10'' wt. 48
49

 The weight of the quotient remains unchanged if all the signs
of the equation are changed or if the equation is added to or
subtracted from a constant.
50

 If weights of A + B = 76° 20' 30'' is 3, what is the weight of
- (A + B) or 180° - (A + B) ?
 Solution :
 From law (6), the weights of - (A + B) will remain the same.
 Weight of - (A + B) = - 76° 20' 30'' wt. 3 or
 Weight of 180° - (A + B) or 103° 39' 30'' is equal to 3.
51

 If an equation is multiplied by its own weight, the weight of
the resulting equation is equal to the reciprocal of the weight
of that equation.
52

 Calculate the weight of the equation (A + B) if weight of
(A + B) is . The observed value of (A + B) is 120° 20' 40''
 Solution :
 As the equation is being multiplied by its own weight, from
rule (7), the weight of { w (A + B) } will be if the w is the
weight of (A + B).
 Weight of [ (A + B) = 90° 15' 30'']= =
53

 The principle of distributing errors by the method of least
squares is of great help to find the most probable value of a
quantity which has been measured for several times.
 It is found from the probability equation that the most
probable values of a series of errors arising from observations
of equal weight are those for which the sum of the squares is a
minimum. The fundamental law of least squares is derived
from this.
54

 The fundamental principle of the method of least squares may
be stated as follows :
 “In observation of equal precision, the most probable value of
the observed quantities are those that render the sum of the
square of the residual error a minimum”
55

 If the measurement are of equal weight, the most probable
value is that which makes the sum of the square of the residue
(v) a minimum.
 Thus, ∑v2 = a minimum
 If the measurement are of unequal weight, the most probable
value is that which makes the sum of the products of the
weight (w) and the square of the residuals a minimum.
 Thus, ∑w(v)2 = a minimum
56

 When a quantity is being deduced from a series of
observations, the residual error will be the difference between
the adopted value and the several observed values.
 Let X1, X2, X3, X4, …. Be the observed values
 Z = most probable value
57

 Then, Z - X1 = e1
Z – X2 = e2
Z – X3 = e3
... ... ....
Z – Xn = en
 Where e’s are the respective residual errors of the observed
values.
58
...................... (1)

 If M = arithmetic mean, then
=
 Where n = number of observed values
59
........(2)

 From equation (1),
nZ - ∑X = ∑e
or Z = + , but = M from equation (2)
Z = M +
 If the number of observations is large, n is very large and e is
kept small by making precise measurement, and the second
term becomes nearly equal to zero.
60
........(3)

 Thus Z = M
 Thus, when the number of observations is large, the arithmetic
is the true value or most probable value.
Z = M
61

 Now, we calculated the residual errors from the mean value. If
v1, v2, v3 etc. are the residual errors, then
M - X1 = v1
M – X2 = v2
M – X3 = v3
... ... ....
M – Xn = vn
62
...................... (4)

 Adding the above,
nM - ∑X = ∑v
or M = +
As M = , under the preceding conditions and by preceding
equation and hence = 0
 Hence the sum of the residuals equals zero and the sum of plus
residual equals the sum of the minus residuals.
63

 Rules for giving weights to the field observations :
 The weights of an angle varies directly to the number of
observations made on the angle.
 The weight of the level lines vary inversely as the lengths of
their routes.
 The weights of any angle measured a large number of times, is
inversely proportional to the square of the probable error.
64

 Rules for giving weights to the field observations :
 The correction to be applied to various observed quantities, are
in inverse proportion to their weights.
65

 Distribution of errors to the field observations :
 Whenever observations are made in the field, there is always
some error (accidental error).
 It is always necessary to check the observations made in the
field for the closing error, if any.
 The sign of correction is opposite to that of the closing error.
The closing error should be distributed to the observed
quantities.
66

 The correction to be applied to an observation is inversely
proportion to the weight of the observation. In other words,
the greater the weight, the smaller the correction.
 The correction to be applied to an observation is directly
proportional to the length of the probable error.
67

 In case of line of levels, the correction to be applied is
proportional to the length of the route.
 If all the observations are of the same weight, the error is
distributed to observed quantities equally, and therefore the
corrections are equal.
68

 The following are the three angles A,B and C observed at a
station O closing the horizon, along with their probable errors
of measurement. Determine their corrected values.
A = 82° 15' 20'' ± 2''
B = 128° 26' 10'' ± 4''
C = 149° 18' 15'' ± 3''
69

 Sum of three angles = 359° 59' 45''
 Discrepancy = 360° 00' 00'' - 359° 59' 45'' =15''
 Hence each angle is to be increased, and the error of 15'' is to
be distributed in proportionate to the square of the probable
error.
70

71
Hence Corrected angles
are
A 82° 15' 20'' + 2.07'' 82° 15' 22.07''
B 128° 26' 10'' + 8.28'' 128° 26' 18.28''
C 149° 18' 15'' + 4.65'' 149° 18' 19.65''
TOTAL 360° 00' 00'' OK

 Adjust the following angles closing the horizon
72
Angle Weight
A = 112° 20' 47'' 2
B = 90° 30' 15'' 3
C = 58° 12' 05'' 1
D = 98° 57' 00'' 4

 Sum of four angles = 360° 00' 08''
 Discrepancy = 360° 00' 00'' - 360° 00' 08'' = - 08''
 Hence each angle is to be decreased, and the error of 08'' is to
be distributed in to the angles in an inverse proportionate to
their weights.
73

74
?
* 4
Hence
Corrected
angles are
A 112° 20' 47'' - 0.96'' 112° 20' 45.04''
B 90° 30' 15'' - 0.64'' 90° 30' 13.36''
C 58° 12' 05'' - 1.92'' 58° 12' 2.08''
D 98° 57' 00'' - 0.48'' 98° 56' 59.92''
TOTAL 360⁰ 00’ 00”

 MPV : The most probable value of a quantity is the value
which has more chances of being true than any other value.
(close to the true value)
 It can be determined form principle of least square.
 If systematic errors are eliminated from the observations, the
arithmetic mean will be the most probable value of the
quantity being observed.
75

 Methods of determination of MPV :
76
Determination of MPV
Direct observations
Direct observations of
equal weights
unequal weights
Indirect observations
involving unknown of
equal weights
involving unknown of
unequal weights
Observation equations
accompanied by
condition equations
The Normal equation
The method of differences
The method of correlates

 CASE-1 Direct observation of equal weight :
 The MPV of directly observed quantity of observation of
equal weight is the arithmetic mean of observations.
 Thus, if X1, X2, X3, X4, …… Xn , X is Most probable value.
 n = No. of observation
77

 Following direct measurements of a base line were taken :
2523.32 m; 2523.25 m; 2523.17 m; 2523.38 m; 2523.47 m;
2523.68 m. Calculate the most probable value of the length of
the base line.
 Solution : MPV = Arithmetic mean M
78
= 2523.378 m. MPV of base line length X = 2523.378 m

 CASE-2 Direct observation of unequal weight :
 The MPV of directly observed quantity of observation of
unequal weigh is weighted arithmetic mean of observations.
 Thus, if X1, X2, X3, X4, …… Xn , X is Most probable value.
 W = weight of observation w₁, w₂, w₃….
 n = No. of observation
79

 Find the most probable value of the angle from the following
observations :
Angle A = 76° 35' 00'' wt. 1
Angle B = 76° 33' 40'' wt. 2
 Solution : MPV = The weighted arithmetic mean of the
observed quantities
80
= 76° 33' 6.67''

 CASE-3 Indirectly observed quantities involving
unknowns of equal weights :
 Most probable value found by method of normal equations.
 To form normal equations, multiply equations by their
coefficients of unknowns and add the result.
81

 Find the most probable value of the angle A from the
following observation equations :
Angle A = 30° 28' 40'' weight 2.
Angle 3A = 91° 25' 55'' weight 3.
82

 MPV = The normal equation of A
 Multiply first eq. by 2 = (1 * 2) and second eq. by 9 = (3 *3)
 2*A = 2 * 30° 28' 40''  2A = 60° 57' 20'' ____(1)
 9*3A = 9* 91° 25' 55'' 27A= 822° 53' 15''____(2)
 Sum (1) + (2) = 29A = 883° 50' 35''
 So Angle A = 883° 50' 35'' / 29 = 30° 28' 38.45''
 Most probable value of angle A = 30° 28' 38.45''
83

 CASE-4 Indirectly observed quantities involving
unknowns of unequal weights :
 Most probable value found by method of normal equations.
 To form normal equations, “multiply each observation
equation by the product of the algebraic coefficient of the
unknown quantity in the equation and the weight of the that
observation and add the equations thus formed”
84

 CASE-5 Observation equations accompanied by condition
equations, conditioned quantities :
 One or more conditional equations are available.
 Methods to compute MPV
1. The Normal equations
2. The Methods of differences
3. The Method of correlates
85

 The Normal equations :
 By avoiding the conditioned equations and forming the normal
equations of the unknowns.
 The Methods of differences :
 Direct method of forming the normal equations for the
observed quantities is suitable for simple cases.
86

 Method of correlates :
 The method of correlates is also called the method of
condition equation or the method of Lagrange multiplier.
87

8. THEORY OF ERRORS (SUR) 3140601 GTU

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8. THEORY OF ERRORS (SUR) 3140601 GTU