Measurements And Errors

1

• Chapter (5)
• Measurements And Errors
• Statistics of Measurement
• An understanding of the measurement process is
  essential in the geomatic sciences, anybody can
  be taught to use an instrument to measure a
  quantity by only a few will ever truly understand
  the significance of those measurements.
• Luckily geomatic practitioners are some of those
  privileged few.

2

• We will start with a few statements
• no measurement is exact
• every measurement contains errors
• the true value is never known
• the true error is never known
• If we conduct surveying measurements
  understanding the relevance of these statements
  then our measurement project will be considerable
  more successful than if we merely took the
  readings on face value.
• This set of lecture notes will give a little more
  detail on how measurements are treated in
  geomatic science, however the material really
  only deals with the simple cases.
• Considerably more detail on the adjustment of
  survey measurements will be given in later year
  surveying subjects.

3

• Sources of Errors
• Natural errors
• There are caused by natural conditions such as
  temperature, wind, humidity and cloud amount.
• Personal errors
• These includes an error in graduation of leveling
  staff when reading.
• Instrumental errors
• Result from an imperfect design or in adequate
  adjustment of the instrument used.

4

• Types of Errors
• There are three types of things that can go
  wrong, only two of which are really errors.
• Gross errors (blunders)
• Systematic errors
• Random errors
• Elimination of Gross and Systematic Errors
• Mistakes are simple to eliminate, don't make
  them in the first place. Systematic errors can
  usually be eliminated by applying a correcting
  algorithm.
• Gross Errors - eliminate by checking,
  re-measuring, using suitable field procedures,
  and above all taking care.
• Systematic Errors - eliminate by calibration,
  checking instruments against a standard, applying
  corrections.

5

• Once gross and systematic errors have been
  eliminated then only random errors remain.
• These are the errors that will be dealt with in
  the following sections, it is assumed that the
  data contain no mistakes, and has had all
  systematic error removed.
• Random Errors
• It has been found in practice that these errors
  are generally
• small in nature
• positive or negative in equal proportions, that
  is with a large sample the algebraic sum will
  tend towards zero. (This is the simple case,
  studies in later years will show that there are
  other situations where this does not apply).

6

• Methods of Survey to Minimize affect of Errors
• One of the most effective methods of controlling
  the effect of random errors is to "work from the
  whole to the part".
• The initial control over the whole area is
  established carefully, and then the rest of
  survey work is carried out within the control
  network, like the field exercise involving the
  mapping of the football oval on campus.
• Other field procedures we use to reduce the
  effect of errors include
• developing observing procedures to eliminate
  gross and systematic errors (for example checking
  the difference between face left and face right
  angle readings

7

• taking additional or redundant observations to
  check for mistakes
• and specifically with respect to random errors
• take observations many times.
• If we assume the simple case, that is that given
  a large sample the spread of values would conform
  to the normal distribution, we can estimate the
  most probable value of a certain quantity
• measured several times as the mean.

n number of readings taken The Standard
Deviation 's' gives a measurement of 'dispersion'
or spread about the mean, that is it will give an
estimate of precision of the observations.
8
where the top line represents the sum of the
squares of the deviations of the observations
from the mean, also known as the residuals. The
square of the standard variation is the
variance. This can also be expressed as
which is more convenient for calculations.
9
Example consider the following table showing
readings for an angle measured 12 times
10
This is the standard deviation/error of one
observation from the set. The standard deviation
of the mean is given by another formula The
standard deviation of the mean In the example
above, the standard deviation of the mean
11

• We would naturally hope that the precision of the
  average value was better than the precision of a
  single pointing, which is the case.
• Hence the best estimate of the angle 2 21'
  52" 0. 84" It is conventional to show the
  significant figures of the mean to one place more
  than that of the observations.
• How Many Times?
• The formula for the precision of the mean can
  also show us how many observations are required
  to attain a given precision.
• For example, suppose an angle can be measured to
  5" Typically this is measured by experience or
  experiment (as above), or is a function of the
  level of precision of the equipment being used.
  The specifications for the project call for a
  mean angle with a precision of 2". How many
  observations are required to achieve this
  precision?
• s 5? precision of single reading, sm 2?
  precision for mean

12
From sm s/ vn , 2 5/ vn , n 25/4 6.25
Therefore 7 observations are required with this
instrument to give an angle with a precision of
2 seconds.
13

• The Normal Distribution
• If we recall the example before where we observed
  an angle with a theodolite, and then using that
  data we plotted a frequency count of the angles,
  we would get the figure shown.
• If we were to take many more observations, and
  reduce the class intervals in size the
  distribution would probably tend to look like the
  one below.

14

• This should be quite familiar, and is known as
  the normal distribution. It is a theoretical
  distribution with the characteristics N (m, s).
• The Greek letter m is the population mean, and s
  the population variance.
• The formula is as follows

• The probability of a measurement x having a value
  between a and b is given by the area under the
  curve as follows

15
We estimate the mean of the normal distribution
by x and the standard deviation by s. With a
suitable large sample these approximate m and
s. The Standard Normal Distribution Normally
distributed data has a particular mean and
variance, it is a function of the data being
sampled. The standard normal distribution has
been re-scaled so that it has a mean value of
zero, and a variance of one. t ( x - m ) / s
Values of t are found in normal probability
tables The standard normal distribution will give
the probability of finding a given quantity
within certain limits. It gives probability by
calculating the area under the curve from
negative infinity to the value being tested.
16
When Should an Observation Be Rejected? We can
use the standard normal distribution to test
whether a value is probable or not, that is
whether to accept or reject certain measurements.
For example, lets look at a set of distance
measurements
17
From tables of the standard normal distribution,
we can determine the probability of an
observation lying within three standard
deviations of the mean as follows
In other words there is a 99.74 chance of a
measurement lying between ( -3 s lt x lt 3 s) of
the mean. If we are happy with odds of 99.74
then in the table above mean 93.827, s
0.024, so the value should be between 93.899
and 93.755 which it is (but only just).
18

• In general, around 68 of the measurements will
  fall within one sigma (one standard deviation),
  95 of the measurements will fall within 2
  standard deviations of the mean, and around 99
  within three.
• In the example above, if the criterion was that
  we would reject observations outside of two
  standard deviations, then by rejecting the value
  we get a mean of 93.820 and a standard deviation
  of 0.004.
• The standard deviation of the mean would then be
  0.001, a considerable improvement.

19
Weighted measurements It may have occurred to
the reader that not all measurements are made
with equal care or effort. If one measurement has
been made with greater precision or has been
repeated more times than another measurement ,
the more precise measurement is said to have
greater weight. Relative weight is designated w
. For a series of measurement M , each having
relative weighted mean M may be obtained
from Mw (w1M1 w2M2 wnMn)/ (w1w2.wn)
Swi Mi / Swi
20
Example Distance AB was measured with a
reinforced cloth tape , a steel tape and EDM ,
and found to be 566.93 , 566.72 and 566.74 ft.
respectively . If the cloth tape measurement is
assigned a relative weight of 1 , the steel tape
measurement a relative weight of 6 and the EDM
measurement a relative weight of 8 , what is the
weight mean of the three measurements? Solution M
w (1x566.936x566.728x566.74)/(168)
566.75 ft.
21
Example A line was measured four times with a
steel tape on a day when the wind condition
varied from claim strong gusts. Based on wide
conditions at the time of measurements, weights
were assigned to each measurements as
follows 426.47 m , weight 4 426.43
m , weight 3 426.64 m , weight 1
426.56 m , weight 2 What is the MPV of
line? Solution Mw (4x426.473x426.431x426.642x
426.56)/(4312) 426.493 m
22
sm vS(wivivi /Swi(n-1) v0.0445/10(4-1)
0.038 m , MPV of line 426.493 0.038 m
23

• Standard Errors of Derived Quantities
• If we have a known precision of reading of a
  level staff, and we perform a 5km level traverse,
  what is the likely precision of the Reduced
  Levels? This question is common to many surveying
  and measurement tasks, what will be the derived
  precision of a quantity based on the measurements
  we take.
• One way this is determined is to use the rule for
  the propagation of variances.
• If X is a function of n independent random
  variables xi with an expected mean of m and
  standard deviation of s.
• The data needs to be free of correlations, that
  is that a change in one observation does not
  affect any other observation.

24
This is estimated by the use of 's' in the
formula instead of the unknown s. This is an
important formula for geomatic engineers. Now,
getting back to our leveling question
25
DH bs1 - fs1 bs2 - fs2 . . . .bsn fsn
(bs1 bs2 . . . .bsn) - (fs1 fs2
fsn) So, in order to determine the final
precision of a level traverse, we use the rule
for the propagation of variances
and if all staff readings have the same
precision,
26
root of the number of set-ups, which is the
criterion used in the practical exercises. As the
number of set-ups is proportional to the
distance, the precision of the RL is also
proportional to the distance, which is how the
Survey Coordination Regulations express
accuracies for Leveling traverses.
27
For another example, consider a change in height
determined by measuring a radiation with a total
station
From the diagram ?H HI D Sin ? - HT,
so.... S2?H (s?H/ sH1)2 sH12 (s?H/ sD)2 sD2 (
s?H/ s?)2s?2 (s?H/ sHT)2sHT2 (s?H/ sH1) 1 ,
(s?H/ sD) Sin ? , ( s?H/ s?) ? Cos ? , (s?H/
sHT) -1 s 2?H s 2HI Sin2? s 2? D2 Cos2? s
2? s 2HT
28
If we considered the precision of say the angle
reading only, then the precision of the change in
height will be s 2 ? H D2Cos2?s2? s? H
D Cos? s? So, for a distance of 500m, a slope
of 8 00 and a precision of angle measurement of
10", the precision will be s?H 500 x Cos(8
00') x (0.0028 x p/180º) 0.024 m Taking
another example, what is the derived precision of
the coordinates of a point obtained from the
measurement of a radiation with a total station?
29
Data coordinates of A and B Measurements length
AC (l), variance s2I angle CAB, a, variance s
2a The quantities that are to be derived are the
bearing between A and C, and then the coordinates
of C. EC EA l x Sin ? AC EA l x Sin (?
AB - a) NC NA l x Cos ? AC NA l x Cos (?
AB - a) So then the variance of the computed
coordinates is s2E Sin2( ?AB-a)2sI2 l2Cos2(?
AB - a) s2a Similarly s2N Cos2( ?AB-a)2sI2
l2Sin2(?AB - a) s2a
30
For example, let's compute the precisions of some
coordinates given a distance and a series of
bearings Distance 100m, precision of distance
measurement 0.01m Bearing 0, 30, 60, 90,
precision of angle measurement 05"
It can be seen that the precision of the
coordinate varies depending on the bearing of the
line, whereas the precisions of the measurements
are constant.
31
The rule of propagation of variances can be
applied to most measurement applications,
enabling the precision of even complex
derivations to be determined or predicted.
Propagation of Small Systematic Errors A similar
concept to the rule of propagation of variances
is used to determine how a small error in a
measurement can be propagated through a
calculation. If X f (x1, x2, ......x n) then d
X (df/ dx1) dx1 (df/ dx2) dx2 .. (df/ dxn)
dxn For example, if the required precision of a
linear distance is to be 110,000, and we have a
clinometers that can measure slope to 10?, can
we use this clinometers to measure a line 200m
long on a slope of 10?
32
So a precision of 0.1010 over 200m equals
12000, so we would need to use something else to
measure the slope angle.
33
Example El-Mansoura2006 What is the most
probable value in horizontal distance measured by
the tangential system in tacheometeric survey,
where the distance on staff measured 16 times
with average 3.44 m and probable error 4 cm for
one observation, the two elevation angles
measured 9 times with average 49º , 51º with
probable error 4', 12' for one observation
respectively? Solution D S / (tan a1- tan a2)
3.44 / ( tan 51º- tan 49º) D 40.70 m
dD / dS 1/ (tan a1- tan a2) 1/ ( tan 51º-
tan 49º) dD / dS 11.83 m
34
dD / da1 -S sec2a1/ (tan a1- tan a2)2 -
1215.64 m dD / da2 S sec2a1/ (tan a1- tan a2)2
1118.57 m ss s /vn 4 /v16 1
cm sa1 4 /v9 1' 20" sa2
12 /v9 4' 00" sD v(11.83)2X(1/100)2(
1215.64X80p/180X60X60)2
(1118.57X240p/180X60X60)2
1.93 m Then the most
probable value of the distance40.701.93 m
35
Example ( El-Mansoura 2002 ) A staff was
observed at a point B from an instrument (
having constants of 100 and 0 ) at point A . Find
the horizontal distance AB , the reduced level of
point B and its standard errors (s) ? if Stadia
readings ( 2.08,2.60,3.12 ) and (s) for each
0.002 , vertical ( depression ) angle
4º 40' and its (s) 10' , the reduced level of
the horizontal axis of the instrument ( 12.00 )
and its (s) 0.01m. Solution DAB 100 S cos 2
? 100x1.04xcos2 4º 40' 103.312 m dD / dS 100
cos2 ?99.338 m dD / d? - 100 S x cos?x sin? -
16.867 m
36
sD v (99.338)2x(0.002)2 (16.867)2x(10/60xp/180
)2 0.205 m RL of point B 12 0.5x100
S sin2? 3.567 m dRL / dS - 50sin 9º 20'
-8.109 m dRL / d? - 50 S cos 9º 20' x2
-102.623 m s R v (8.109)2x(0.002)2
(102.623)2x(10/60xp/180)2 0.299 m