a well explained topic about errors in surveying

1. 1
ERROR THEORY
• Scope
– Depending on their respective purposes,
measurement have to be performed to a
certain accuracy and reliability.
– Due to shortcoming of measuring
instruments and human senses it is
impossible to obtain truly error free
measurements.

2. 2
• As a rule, measurements are repeated
several times, and if possible supported
by additional measurements.
• Example
– Measuring hypotenuse in addition to the
sides of right angled triangle, or
– Observing the third angle of a triangle in
addition to the two that are needed.
• The evaluation of the measurements
leads to the following tasks:

3. 3
– To derive the most probable mean value of
the desired unknown quantity.
– To provide a figure for the accuracy or the
dispersion of a single measurement.
– To estimate the accuracy or dispersion of
the mean and its confidence region.

4. 4
• Surveying measurements are subjected
to three types of errors
– Gross Errors/mistakes
– Systematic Errors
– Random Errors

5. 5
• Gross Errors
– Often called mistakes or blunders
– Usually much larger than other categories
– Usually result due to inexperience of the
observer who is not familiar with equipment
and methods used
– Gross errors are due carelessness or
incompetence of the observer
– Can be spotted by check measurements
and then eliminated.

6. 6
• Examples
– displacement of arrows or station marks
– miscounting tape lengths
– misreading the tape
– wrong booking

7. 7
• Systematic Errors
– Systematic Errors are those which follow
some mathematical law and they will have
the same magnitude and sign in a series of
measurements when repeated under the
same condition.
– They are cumulative in nature
– Can be eliminated by applying some
mathematical corrections.
– Can also be removed by calibrating the
observing equipment and quantifying the
the errors

8. 8
– Proper selection of measuring procedure
• Examples
– Wrong length of tape
– Poor ranging
– poor straightening
– slope
– sag
– temperature variation
– wrong tensioning

9. 9
• Random Errors
– Random errors are errors that remain after
all gross and systematic errors are removed
– Random errors cannot be removed from
measurement but methods can be adopted
to ensure that they are kept within
acceptable limits.
– Random errors are inherent in all types of
measurement, the magnitude and the sign of
which are not constant.
– Random errors can either be positive or
negative and hence they tend to
compensate each other.

10. 10
• In surveying the true value of a quantity is
usually never known
• The exact error in a measurement or
observation can never be known.
• Random errors follow general laws of
probability and these are:
– Small errors occur more frequently and therefore
are more frequent than large ones
– Large errors happen infrequently and are
therefore less probable, very large errors may be
mistakes and not random errors
– Positive and negative errors of the same size are
equally probable and happen with equal
frequency.

11. 11
• Examples
– holding and marking
– variation in tension
0
-10 10
Magnitude of Error
Error
frequency

12. 12
In general, the distance measurement obtained in the field will be
in error. Errors in the distance measurement can arise from a
number of sources:
1) Instrument errors. A tape may be faulty due to a defect in
its manufacturing or from kinking.
2) Natural errors. The actual horizontal distance between
the ends of the tape can vary due to the effects of:
temperature,
elongation due to tension, and
sagging.
3) Personal errors. Errors will arise from carelessness by
the survey crew:
poor alignment
tape not horizontal

13. 13
• Redundancy
– Redundant observations are additional
measurements taken for a quantity in order
to evaluate standard errors and establish
probabilities.
– Redundant observation are also used to
detect mistakes in the field work
• Precision
– Pertain to the closeness to one another of a
set of repeated observations in a random
variable

14. 14
– If such observation are clustered together,
then they are said to have been obtained
with a high precision
Accuracy
– Refers to the closeness between
measurements and their true values.
– The further the measurement is from the
true value, the less accurate it is.
– Quite often words like the most probable
value, expected value and mean are used
as the true value can never be achieved

15. 15
• Reliability
– In all surveying measurements, attempts
are made to detect and eliminate mistakes
in fieldwork and computations and the
degree to which a survey is able to do this is
a measure of its reliability.
– Unreliable observations are those which
may contain gross errors without the
observer knowing
– Reliable observations are unlikely to contain
undetected mistakes.

16. 16
• Mean
– The single li, which would be obtained if a
quantity were to be determined by an
infinite number of equal precise
independent measurements (n ) with
random errors, would oscillate around a
mean value .
– Mean is called the mathematical
expectation or the “true” value of the
quantity.

17. 17
– Usually only a limited number of
measurements is available (a random
sample of size n) the sample mean
(arithmetic mean is used to estimate the
true value.
ẋ =1/n(l1+l2+…+ ln) = 1/nli
– The residuals or correction which remain
after the formation of arithmetic mean
v1 = ẋ -l1; v2 = ẋ-l2; vn = ẋ-ln
are such that the sum of the square (v) is
minimum

18. 18
– This the basic requirement for the method
called least square
• Measure of spread/dispersion
– To obtain a measure of spread of a single
measurement, the deviations of the
observation li from the true value  are
considered.
– For n  this leads to the so called “true
errors”.
1= 1-l1; 2= 1-l2; …; n= 1-ln

19. 19
– The measure of accuracy of a single
observation li is defined by the “theoretical
standard deviation”.
 = ±(2/n)1/2
for n 
– Usually  and i are unknown, sample mean
x and residuals vi are used.

20. 20
– Based on the estimation the sample
standard error of the observation is
obtained as an approximation or
estimation of as:
s = ± (v2/n-1)1/2
– The sample standard error of the mean of
n observations becomes
sẋ =±s/(n)1/2

21. 21
x
f(x)
measurements
High precision
small 
Low precision
high 
2 2
1 1

22. 22
• A small deviation (1) indicates a small spread
amongst results
• A large deviation (2) indicates a large spread
• In other words the set of measurements with
small standard deviation (1) has a higher
precision than the other set with larger
standard deviation (2)
• For statistical reasons standard deviation
should be derived from a large number of
observation.