1

Chapter 3 – Distance measurement (Tape and offset

surveying)

The method of tape surveying is often referred to as chain surveying, deriving its

name from the fact that the principal item of equipment traditionally used was a

measuring chain. owadays, the chain has been replaced by the more accurate steel

band.

Distance measuring techniques

Pacing

i. Pacing is a very useful (although imprecise) technique of distance measurement.

ii. An individual’s length of pace can be determined by repeatedly pacing between

two marks a set distance apart (e.g. 100ft or 30m).

iii. Pacing is particularly useful when looking for previously set survey markers

(property and construction layout markers).

Odometer

i. A measuring wheel (12-in to 24-in diameter) equipped with an odometer.

ii. It is used by assessors and other real-estate personnel to recorded property

frontages and areas.

2

Distance obtained from positioning techniques

Satellite positioning – once the position coordinates are known, it is simple enough to

compute the distance between those positions.

EDM (Electronic Distance Measurement)

It is an instruments function by sending a light wave or microwave along the path to be

measured and then measuring the phase differences between the transmitted and received

signals.

Gunter’s chain

The chain was robust, easily read, and easily repaired in the field if broken. It was liable

to vary somewhat in length, however, because of wear on the metal-to-metal surfaces,

bending of the links, mud between the bearing surfaces etc.

i. normally either 20 m or 30 m long

ii. made of tempered steel wire

iii. made up of links that measured 200 mm from center to center of each middle

connecting ring

iv. swiveling brass handles were fitted at each end, the total length was measured

over the handles

3

v. Tally markers, made of plastic, were attached at every whole metre position, and

those giving 5 m positions were of a different colour.

To be precise, it was 66 ft long and was composed of 100 links. The length of 66 ft was

apparently chosen because of its relationships to other units in the Imperial system:

80 chains = 1 mile

10 square chains = 1 acre (10 x 662 = 43560 ft

2)

4 rods (1 rod = 16.5 ft) = 1 chain

Taping

Taping is used for short distances and in many construction applications.

i. Fiber-glass tapes

a. Fiberglass tapes are used for applications where lower precision is

acceptable.

b. Fiberglass tapes can give accuracies in the centimeter range.

c. Typical uses of fiberglass tapes involve topographic, fencing

measurements and etc.

4

ii. Steel tapes

a. Steel tapes are used for precise measuring.

b. Typical engineering surveying accuracy ratios in the range of 1:3000 to

1:5000 can be readily attained when measuring with a steel tape if the

proper techniques are employed.

c. Steel tapes can measure to the nearest millimeter (or nearest hundredth of

a foot).

d. The most commonly used tape length in metric units is 30m (lengths of

20m, 50m, and 100m can also be obtained).

e. Graduated every 10mm and figured every 100mm. Whole meter figures

are shown in red at every meter.

f. Steel tapes come in two common cross-sections:

i. heavy duty is 8mm x 0.45mm – generally used in route surveys

(e.g. high-ways, railroads)

ii. normal usage is 6mm x 0.30mm – generally used in structural

surveys.

5

6

Taping accessories

i. Plumb bob

a) Plumb bobs are normally made of brass and weight from 8oz to 18oz.

10oz and 12oz plumb bobs most widely used.

b) Plumb bobs come with about 6 ft of string.

c) Plumb bobs are used in taping to permit the surveyor to hold the tape

horizontal when the ground is sloping.

7

ii. Hand level

a) Hand levels are small rectangular or cylindrical sighting tubes equipped

with tubular bubbles and horizontal crosshairs that permit the surveyor to

make low-precision horizontal sightings.

b) The bubble location and the crosshair can be viewed together via a 45°

mounted mirror.

8

c) Hand levels can be used to assist the surveyor in keeping the tape

horizontal while the tape is held off the ground.

d) The hand level is held by the surveyor at the lower elevation and a sight is

taken on the uphill surveyor.

iii. Ranging rods (or range poles)

a) Range poles are 2 m, 2.5 m or 3m wood or steel poles with pointed steel

shoes (or steel points).

b) These poles are usually painted alternately red and white in 500mm (or

1ft) sections.

c) The range pole can be used to provide theodolite and total station sightings

for angle and line work.

d) These poles were also used in taping to help with alignment for distances

longer than one tape length.

e) The pole was set behind the measurement terminal point and the rear tape

person could keep the forward tape person on line by simply sighting on

the pole and then waving the forward tape person left or right until they

are on line.

9

iv. Clamp handle

The clamp handle helps the surveyor to grip the tape at any intermediate point

without kinking the tape.

v. Chaining pin (or Marking arrows)

When measuring the length of a long line, the tape has to be laid down a number

of times and the positions of the ends marked with arrows, which are steel

skewers about 40 mm long and 3-4 mm diameter. A piece of red ribbon at the top

enables them to be seen more clearly.

a) Chaining pin come in a set of eleven.

10

b) They are painted alternately red and white and are 350mm to 450mm long.

c) Chaining pins are used to mark intermediate points on the ground while

making long measurements.

vi. Tension handle

The tension handle used in precise work to ensure that the correct tension is being

applied.

vii. Pegs

a) Points that need to be more permanently marked, such as the intersection

points of survey lines, are marked by nails set in the tops of wooden pegs

driven into ground by a mallet.

b) A typical size is 40 mm x 40 mm x 0.4 m long.

c) In very hard ground, steel dowels are used instead, while in asphalt roads

small 5 or 6 mm square brads are used.

11

viii. Abney hand level

The Abney level consists essentially of a sighting tube, to which is attached a graduated

arc. An index arm, pivoted at the centre of the arc, carries a small bubble tube, whose

axis is normal to the axis of the arm, so that as the tube is tilted the index moves over the

graduated arc.

By means of an inclined mirror mounted in one-half of the sighting tube, the bubble is

observed on the right-hand side of the field of view when looking through the eyepiece of

the sighting tube.

In using the instrument for the measurement of vertical angles, when measuring mean

ground slope, if the sight is taken onto a point whose height above the ground is the same

as that of the observer’s eye, then the line of sight will be parallel to the mean ground

surface.

A ranging rod with a mark on it at the required height makes a suitable target. To

measure the angle, a sight is taken onto the mark, and the bubble is brought into the field

of view by means of the milled head to be bisected by the sighting wire at the same time

as the wire is on the target. The angle is now read on the vernier.

12

Taping techniques

(i) Measuring distance with tape

The operation is carried out by two assistants, known as chainmen because the

measurement was traditionally made by chain, one acting as leading chainman and the

other as follower. The chainmen take one end each of the steel band and the band is

13

pulled out full length and examined for defects. The leader is equipped with ten arrows

and a ranging rod, and the follower also takes a ranging rod. Then, to measure line AB

having previously positioned ranging rods at both A and B:

(1) The leader drags his end of the band forward to A1 and holds his ranging rod

about 0.3 m short of the end.

(2) The follower has to holds end of the band firmly against station at A, and the

surveyor lines in the leader’s pole between A and B by closing one eye, sighting

poles A and B, and signaling the leader till he brings his pole into line AB. The

system of signaling usually adopted is to swing the left arm out to the left as an

instruction to the leader to move his pole in that direction: the right arm is

similarly used to indicate movement to the right, while both arms extended above

the head, then brought down, indicates that the pole is one line.

(3) The leader straightens the band past the rod by sending gentle ‘snakes’ down the

band.

(4) The follower indicates that the band is straight, and the leader puts an arrow at the

end, A1. (At this stage offsets or ties may be taken from known positions to

required detail.)

(5) The leader then drags his end to A2, taking nine arrows and his pole.

(6) The follower moves to A1, and puts his pole behind the arrow; the surveyor again

lines in from here or from A.

The above procedure is repeated, the follower picking up the first arrow before he moves

from A1. The leader moves to A3, carrying eight arrows. The follower moves to A2,

carrying the arrow from A1.

If the line measured is longer than ten times the band length, the leader will exhaust his

supply of arrows, so that when the eleventh band length is stretched out, the follower will

have to hand back the ten arrows to the leader. This fact is pointed out to the surveyor,

who notes it in his field book. The number of arrows held by the follower serves as a

check on the number of full band lengths measured in the line.

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(ii) Measurements along slopes

The terrain that has to be surveyed in general will not be a horizontal plane, whereas the

distance required for the preparation of a plan or map is the horizontal distance.

When distance is measured along a slope, it has to be converted into horizontal distance

for the purpose of plotting.

In general, there are two methods of chaining along a slope:

a) To measure distances horizontally in steps and transfer the points to the

ground, and

b) To measure along the slope and convert the distances into horizontal

distances.

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(a) Stepping (or measuring horizontally)

Suppose one has to measure the distance between the stations A and B, the procedure is

the following.

1) At least two persons are required for the chaining. One end of the chain is held at

A and a convenient length is selected and the chain is held horizontally. The chain

tends to sag due to its own weight and has to be counteracted by applying

sufficient pull to it so that it remains horizontal.

2) The follower holds the end of the chain at A. the leader goes along the line with a

selected length of the chain and a ranging rod and faces the follower. The

follower directs the leader to be in line and both pull the chain to eliminate the sag

of the chain.

3) The length of the chain is selected such that it can be held truly horizontal and

pull applied by hand. Once the chain has been stretched to be horizontal, the point

C’ of the end of the chain is transferred to the ground using a plumb bob. The

point C is thus obtained. The distance between A and C is the length of the chain

held horizontally.

4) The process can be repeated starting at C to get points D, E, F, etc. till the end B

is reached.

5) The total horizontal distance is the sum of the lengths of the number of steps

taken to reach B from A.

6) In general, measurement using this method should be done downhill. If the

distance has to be measured uphill, the second method is preferable, as the

follower, while going uphill, has to direct the leader for ranging, hold the chain,

and transfer the point to the ground, which is very difficult.

7) The horizontal distance (selected as the step distance) also depends upon the

slope. For steep slopes, the distance has to be small.

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(b) Measuring along the slope

Length can be measured along the slope and the slope distance converted into horizontal

distance.

To measure the slope either as an angle or gradient and calculate the horizontal distance;

- The slope of the terrain may be determined as an angle θ with an angle-measuring

instrument. The simplest instrument is a clinometers.

- When the slope of the ground has been measured, it is easy to find the horizontal

distance:

Horizontal distance = Ɩcosθ

Where, Ɩ is the length along the slope

θ is the angle made by the ground with the

horizontal.

- If the slope is gentle and uniform for a long distance, this method can be used.

- For undulating ground, this method is not suitable.

17

Standard conditions for the use of steel tapes

Because steel tapes can give different measurements when used under various tension,

support, and temperature conditions, it is necessary to provide standards for their use.

Standard taping conditions are shown below:

Metric system, 30 m steel tape:

1. Temperature = 20°C

2. Tape fully supported

3. Tape under a tension of 50N (because 1 lb force = 4.448N, 50N = 11.24 lbs)

In the real world of field surveying, the above-noted standard conditions seldom occur at

the same time. The temperature is usually something other than standard, and in many

instances the tape cannot be fully supported (one end of the tape is often held off the

ground to keep it horizontal). If the tape is not fully supported, the tension of 50N does

not apply. When standard conditions are not present, systematic errors will be introduced

into the tape measurements.

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Taping corrections

Systematic Taping Errors Random Taping Errors

1. Slope 1. Slope

2. Erroneous length 2. Temperature

3. Temperature 3. Tension and sag

4. Tension and sag 4. Alignment

5. Marking and plumbing

19

Offset Surveying

Objective: To understand how to set a perpendicular line from a point using tapes

only

Equipment for measuring right angles

The cross staff

The optical square

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Setting out right angles

Because this operation is often required in connection with the measurement of offsets,

this is a convenient point at which it may be discussed. There are two cases to consider:

(1) dropping a perpendicular from a point to a line; and

(2) setting out a line at right angles to the survey line from a given point on the steel

band.

Dropping a perpendicular from a point to a line

(a) For short offsets, the end of the tape is held at the point to be located, and the

right angle is estimated by eye. Although this is a usual method in practice, it

is not so accurate as the following methods.

(b) Swing the tape with its zero as center about the point, and the minimum

reading at which it crosses the band is noted. This occurs when the tape is

perpendicular to the band.

(c) Swing the tape with the free end of the tape at center P (the point), strike an

arc to cut the band at A and B. Bisect AB at Q. Then angle PQA = 90°. P

A Q B

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(d) Run the tape from P to any point A on the band. Bisect PA at B, and with the

center B and radius BA strike an arc to cut the band at Q. Then angle AQP =

90°, being the angle in a semicircle.

P

B

Survey line

A Q

Setting out a line at right angles to the survey line from a given point on the steel band

(a) Cross staff: this is mounted on a short ranging rod, which is stuck in the ground at

the point at which the right angle is to be set out. The cross staff is turned until a

sight is obtained along the survey line, and the normal is then set out by sighting

through the slits at right angles to this.

(b) Optical square: this is used as already described, being either held in the hand or

else propped on a short ranging rod.

(c) Pythagoras’ theorem (3, 4, 5 rule or any multiple thereof, say 9, 12, 15, or

11.62=8.4

2+8

2): with the zero end of the tape at P take the 24 m mark of the tape

to A, where AP = 12 m on the band. Take the 9 m mark on the tape in the hand

and, ensuring that the tape is securely held at A and P, pull both parts of the tape

taut to Q. Then angle APQ = 90°. Q

15 m 9 m

survey line

A P

(d) Take A and B on the band so that PA = PB. Strike arcs from A and B with equal

radii to intersect at Q. Then angle APQ = 90°.

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Errors in linear measurement and their correction

In all surveying operations, as indeed in any operation involving measurement, errors are

likely to occur, and so far as is possible they must be guarded against or their effects

corrected for. The types of error that can occur have been classified as follows:

1) Mistakes

2) Systematic error

3) Random error

Mistakes

Mistakes occur due to carelessness of human.

Examples of mistake include:

1.) Miscounting the number of tape length when measuring a long distance,

2.) Misreading the graduation on measuring tape,

3.) Erroneous booking

The possibility of occurrence of these mistakes can be minimized by taking suitable

check measurement.

- Omitting an entire band length in booking - This is prevented by noting down

each band length, and by the leader keeping careful count of the arrows.

- Misreading the steel band - It is best if two people make important readings.

- Erroneous booking sometimes occurs - It is prevented by the chainman carefully

calling out the result and the surveyor repeating it, paying attention when calling 5

or 9, 7 or 11.

Systematic or cumulative error

Systematic errors are defined as the errors whose magnitude and algebraic sign can be

determined, allowing the surveyor to eliminate them from the measurements, and hence

improve the accuracy. In other words, under the same measurement conditions, these

errors will have the same magnitude and direction. An example of systematic errors is the

error resulting from the effects of temperature on a steel tape. If the temperature is

known, the lengthening effects on the steel tape can be precisely determined.

If appropriate corrections are not made, these errors can accumulate and cause significant

discrepancies between measured values. By keeping equipment in proper working order

and following established surveying procedures, many of the systematic errors can be

eliminated.

These arise from sources that may be taken to act in a similar manner on successive

observations, although their magnitude can vary. Their effects, when known, may be

eliminated.

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1) Slope

All measurements in surveying must either be in the horizontal plane, or be corrected to

give the projection on this plane. Lines measured on slopping land must be longer than

lines measured on the flat, and if the slope is excessive, then a correction must be applied.

There are two methods.

i) Stepping

On ground that is of variable slope this are the best method, and no need calculation. The

measurement is done in short lengths of 5-10 m, the leader holding the length horizontal.

The point on the ground below the free end of the band is best located by plumb bob, as

shown in the figure below.

Plumb bob

5 – 10 m held horizontally

Stepping

ii) Measuring along the slope

This method is applicable where the ground runs in long regular slopes. The slope is

measured either by an instrument such as the Abney level, or by leveling, a procedure

that gives the surface height at points along the slope.

(a) Measurement of slope angle, a

Correct length = measured length × cos a where a = angle of slope

Correction = - L(1 – cos a) where L = measured length

L

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(b) slope can be expressed also as 1 in n, which means a rise of 1 unit vertically for n

units horizontally: for small angles a = 1/n radians.

(c) Slope can also be expresses in terms of the difference in level, h, between two

points.

Correction = - [L – (L2 – h2)1/2]

L

h

2

2

−≈

(d) Finally Pythagoras’ theorem may be used.

L

h

a

correct length = √(L2 – h

2)

2) Erroneous length (or Incorrect length of chain)

The most careful measurements will not produce an accurate survey if, for example, the

band has been damaged and is therefore of incorrect length, because every time the band

is stretched out it will measure not 30 m but 30 m ± (some constant or systematic error).

If uncorrected, such an error could have serious effects. By checking the band against a

standard, such as two marks measured for the purpose, the exact error per band length is

known. If this error cannot be eliminated, a correction can be applied which will enable

the effect of error to be removed.

dardsoflength

usedbandoflengthxlengthmeasuredlengthcorrect

tan=

Example:

A measurement was recorded as 171.278m with a 30m tape that was only 29.996m under

standard conditions. What is the corrected measurement?

Solution:

Correction per tape length = - 0.004

Number of times the tape was used = 171.278/30

Total correction = -0.004 x 171.278/30 = - 0.023m

Corrected distance = 171.278 – 0.023 = 171.255m

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Or

dardsoflength

usedbandoflengthxlengthmeasuredlengthcorrect

tan=

= 171.278 x 29.996/30

= 171.255m

Example:

You must lay out the side of a building, a distance of 210.08ft. The tape to be used is

known to be 100.02ft under standard conditions.

Solution:

Correction per tape length = 0.02ft

Number of times that the tape is to be used = 2.1008

Total correction = 0.02 x 2.1008 = +0.04ft

(When involve laying out a distance, the sign of the correction must be reversed before

being applied to the layout measurement. We must find the distance that, when corrected

by +0.04, will give 210.08ft, that is 210.08 – 0.04 = 210.04ft. This is the distance to be

laid out with the tape (100.02ft) so that the corner points will be exactly 210.08ft apart.)

3) Tape standardization

For very accurate work a spring balance should be attached to one end of the steel band.

The purpose of this is to ensure that the band is tensioned up to the value at which it was

standardized: i.e. if the band is 30 m long at 20° C under a 5 kg pull on the flat, then a

tension of 5 kg should be applied to eliminate any correction for pull. The balance is

usually attached to a short cord. The far end of the band will then be attached to a second

rod, and if these rods are set firmly on or in the ground and levered backwards, the

tension applied to the tape can be regulated to any value. All good-quality bands should

have a standardization certificate, which, for example, might say that the band, nominally

30 m long, is in fact only 29.999 m on the flat at 20° C with a tension of 5 kg applied.

This data is used to make corrections to the length as taped to refine the procedure.

In addition to the standardization and slope corrections mentioned above the following

factors might have to be considered:

- elasticity and thermal changes in those cases where the field conditions differ

from those at which the tape was standardized;

- deviation from the straight;

- height above mean sea level;

- sag, if the tape has been standardized on the flat, not in catenary.

(a) Temperature

Similarly a correction is required if the tape temperature, T, is not equal to the standard

temperature, Ts :

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Correction for Temperature = α (T – Ts) L

where α = coefficient of linear expansion.

L = length

If T is less than Ts, the tape has contracted, and if T is more than Ts, the tape has

expanded.

The correction to the length measured is positive (correction has to be added) when (T –

Ts) is positive.

The correction to the length measure is negative (correction has to be subtracted) when (T

– Ts) is negative.

Example:

A distance was recorded as being 471.37ft at a temperature of 38°F. What is the distance

when corrected for temperature? If the standardized temperature for the tape is 68° F and

α = 0.00000645 per unit length per degree Fahrenheit (°F).

Solution

Ct = 0.00000645(38 – 68)471.37 = -0.09

Correct distance = 471.37 – 0.09 = 471.28ft

Example:

You must lay out two points in the field that will be exactly 100.000m apart. Field

conditions indicate that the temperature of the tape is 27°C. What distance will be laid

out? If the standardized temperature for the tape is 20°C and α = 0.0000116 per unit

length per degree Celsius (°C).

Solution

Ct = 0.0000116(27 – 20)100.000 = +0.008m

(when corrected by +0.008, will give 100.000m: Layout distance is 100.000 – 0.008 =

99.992m)

(b) Tension

As mentioned, the correct tension can be applied to the band by attaching a spring

balance to the handle at one end. If the standard tension is not applied a correction should

be made, because the length of the tape will have changed:

Pull correction = AE

LPP sa )( −

27

where Pa, Ps = field and standard tension respectively,

A = cross-sectional area of band,

E = Young’s modulus of elasticity for the band (generally taken as

21x105kg/cm

2), and

L = length measured.

Example:

A 30m tape is used with a 100N force, instead of the standard tension of 50N. If the

cross-sectional area of the tape is 0.02cm2, what is the tension error per tape length?

Solution

Cp = (100 – 50)30 / [0.02 x 21 x 105

x 9.807] = + 0.0036m

If a distance of 182.716m had been measured under these conditions, the total correction

would be:

Total Cp = 182.716/30 x 0.0036 = +0.022m

The corrected distance would be 182.738m.

(c) Sag (or correction for Catenary)

If the highest accuracy is required, rather than lie the band along the ground, it can be

suspended between tripod heads, i.e. hung in catenary, and a correction for the sag in the

tape applied if the tape has been standardized on the flat. Fig. 2 shows a simple

arrangement that could be used.

Straining lever Straining post

spring balance

Marking pegs

Fig. 2

When measuring a line the pegs are aligned, preferably by theodolite, and after the pegs

have been driven, zinc strips are tacked on. The levels of the tops are found, a traverse

scratch mark is made on the first peg to serve as the beginning of the base line; a

longitudinal scratch may also be line in. Lever-type straining arms and spring balance

may conveniently used for tensioning and supporting the tape, which is adjusted so that

the first zero is aligned with the scratched reference mark. A traverse scratch is then made

on the second peghead against the second, and this serves as reference mark for the

second bay, the process being repeated. The tape itself is aligned by theodolite, and

temperatures are measured as before. Note that in this operation the tape should float just

28

clear of the stakes. Instead of aligning the zero of the tape with the scratch made when

taping the previous bay, it is also possible to make another scratch to mark the beginning

of the new bay, the necessary correction to the bay length then being measuring the

distance between the two scratches.

Sag correction = 2

32

24P

Lw−

where w = weight per unit length of the tape and L is the measured length of span. If the

tape is standardized in catenary no correction is required for sag so long as the field

tension P is the same as standard tension Ps.

Example

Calculate the length between two supports if the recorded length is 50.000m, the mass of

the tape is 1.63kg, and the applied tension is 100N.

Solution

Cs = -(1.63 x 9.807)2 x 50.000 / (24 x 1002) = -0.053m

Therefore, the length between supports = 50.000 – 0.053 = 49.947m

4) Alignment

The higher the accuracy required the more critical the alignment of the band becomes.

Corrections can be applied for misalignment, but because this would require actually

measuring the misalignment, as in the figure below, it is generally easier to take care and

line in the band with a theodolite.

Correction = -[AB(1 – cos α) + BC(1 –cos β)]

A B1 C

α β B

5) Sea level

The length of the line as measured can be reduced to its equivalent length at mean sea

level. In Fig. 1

29

Ɩ = (R + H) θ and Ɩ1 = Rθ

Hence HR

Rll

+=1

Correction = Ɩ – Ɩ1 = HR

Hl

HR

Rl

+=

+− )1(

≈R

Hl and is deducted.

Where R = radius of earth (approx. 6367km)

30

Example

Four bays of base line AB were measured under a tension of 120 N and the data was given

below. If the tape was standardized on the flat under a pull of 89 N and at temperature 20°C.

Calculate the true length of the line.

Bay Length (m) Difference in level (m)

1 29.478 + 0.294

2 29.208 - 0.384

3 29.396 + 0.923

4 29.916 - 0.726

Field temperature 31°C Cross-sectional area of tape 3.24 mm

2

Density 7700 kg/m3

Coefficient of linear expansion 0.000 001/°C

Young’s Modulus 15.3 x 104 MN/m

2

Mean radius of earth 6367 km

Mean level of tape 76.56 m AOD

Solution

`

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Random errors (or accidental error)

Random (or accidental) errors are associated with the skill and vigilance of the surveyor.

It is not directly related to the conditions or circumstances of the observation. For a single

measurement or a series of measurements, it is the error remaining after all possible

systematic errors and mistakes have been eliminated.

As the name implies, random errors are unpredictable and are often caused by factors

beyond the control of the surveyor. Their occurrence, magnitude and direction (positive

or negative) cannot be predicted. Some random errors, by their very nature, tend to cancel

themselves. Because of their random nature, correction factors cannot be computed and

applied as some systematic errors. It is assumed that the positive and negative random

error measurements would tend to cancel each other out.

Difference between Systematic Error and Random Errors

The diagram below illustrates the distinction between systematic and random errors.

Systematic errors tend to be consistent in magnitude and/or direction. If the magnitude

and direction of the error is known, accuracy can be improved by additive or proportional

corrections.

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Additive correction involves adding or subtracting a constant adjustment factor to each

measurement;

Proportional correction involves multiplying the measurement(s) by a constant.

Unlike systematic errors, random errors vary in magnitude and direction. It is possible to

calculate the average of a set of measured positions, however, and that average is likely

to be more accurate than most of the measurements.