GEOMATICS ENGINEERING

CHAPTER 2

Direct and Indirect Distance Measurement Methods

Distance Measurement

• Methods

• Equipment

• Classification

• Usage

• Fieldwork procedure

• Booking system

• Adjustment and plotting

Introduction

• Linear measurement is the basis of all surveying.

• There are two main methods to measure distance:

• Direct method

• Measurements are made by tape or chain

• Indirect method

• EDM, Transit, stadia or theodolite are used

Types of Distances

• One of the fundamentals of surveying is the need to measure

distance. Distances are not necessarily linear, especially if they occur

on the spherical earth.

• We will deal with distances in geometric space, which we can

consider a straight line from one point or feature to another.

Methods of Measuring Horizontal Distance

• Pacing

• A surveyor must walk a known distance a number of times in his

natural way: accuracy = 1:50

• Odometer

• An odometer converts the number of revolutions of a wheel of

known circumference to a distance. Can be used for preliminary

surveys. Odometer distance must be converted to horizontal

distance when the slope of the ground is steep.

• Tachometry

• Distance is measured indirectly with the help of an optical

instrument called tachometer.

• Theodolite can also be used with leveling staff.

Methods of Measuring Horizontal Distance

• Chaining

• Used where great precision is not required.

• Chains of 30 m lengths are frequently used.

• Taping

• Used for accurate work and may be o iron, or cloth

• Usually are 30 m in lengths

• Accuracy: 1:10,000

• Electronic distance measurement (EDM)

• Waves are utilized to measure the distance.

• Accuracy: 1:10,000 to 1:100,000

Pacing

Tapping or Distance Measurement by Tape

• Equipment's

• Tension handles allow the user to apply a specified tensile force

on the tape.

• Steel tapes are manufactured under fixed conditions of

temperature and tensile force (30 meter tape).

• Tape grips: These allow the user to firmly grasp the steel tape and

resist the pull of the tape from the person located on the other end

of the tape.

• Plumb bobs: These are used to locate the tape precisely over a

specified point.

• Chaining pins: These are used to mark tape lengths.

• Hand level: These are used to establish an approximate horizontal

line of sight.

Tapping Equipment

General Procedure of Tapping

• Person A: Holds the tension handle located at the “zero” end of the tape.

• Person B: Holds the tape reel and uses a tape grip to pull the tape.

• Person B: Pinches the plumb bob string at a convenient point on the tape.

• Person A: Holds the plumb bob string along the edge of the tape above his/her intended mark.

• Person B: Holds the plumb bob over his/her mark.

• Calls out “mark….mark…..mark” to indicate that the plumb bob is being held over the intended mark.

• Braces for the tension applied by person A.

• Person A: Does the actual pulling (10 lbf or N).

• Reports the tension value and the corresponding tape measurement.

• Person B: Reports the tape reading at his/her end of the tape.

Using a hand level to establish a horizontal

line of sight

Person ‘B’ using the tape grip to hold the

plumb bob over her mark (a chaining pin)

Person ‘A’ applying 10 lb or N

of tension using the tension

handle while maintaining the

plumb bob over the mark

Measured Distance • The measured length is determined by subtracting tape reading

A from tape reading B.

• For example: Person B calls out a tape reading of 27.900 Person A reports a tape reading of 0.031. The measured length is computed as 27.900 - 0.031 or 27.869 meters.

• The measurement procedure is repeated with one important difference.

• Person B holds the plumb bob string at a different point on the tape.

• The two measured lengths are then compared.

• If the difference between the measurements is less than the allowable tolerance, then the average of the measured lengths between the two points can be considered accurate.

• If they are not, the length measurements are be repeated until two successive measurements are obtained within the allowable tolerance.

General Procedure of Tapping

Distance Measurement for Slope Surface

Distance Measurement for Slope Surface

Distance Measurement for Slope Surface

Tapping Corrections

• In general, the distance measurement obtained in the field will be in error.

• Errors in a distance measurement can arise from a number of sources:

• Instrument errors. • A tape may be faulty due to a defect in its manufacturing or from kinking.

• Natural errors. • The actual horizontal distance between the ends of the tape can vary due to

the effects of:

• 1. temperature,

• 2. elongation due to tension, and

• 3. sagging.

• Personal errors. • Errors will arise from carelessness by the survey crew:

• 1. poor alignment

• 2. tape not horizontal

• 3. improper plumbing

• 4. faulty reading of the tape

• When tapes are manufactured, a standardized temperature and a standardized tension value are specified by the manufacturer.

• Although the standard values vary from tape to tape, typical values are • Ts = 20 °C

• Ps = 50 N.

• Corrections

• In the field, all your measured lengths are wrong!!

• A series of corrections are typically required to account for the effects of: • Temperature

• Sagging

• Tension

• Scale (Erroneous tape length)

• Slope correction

Tapping Corrections

1. Erroneous Tape Length

• Tape has a nominal length under certain conditions, a tape stretches

with time.

• Standardization needs to be carried out frequently by using reference

tape or baseline.

2. Slope Correction

• All plan distances are always quoted as horizontal distances L,

therefore any distance not measured on the horizontal will need to be

corrected for slope. Slope correction must ALWAYS be considered,

and either eliminated in the field or mathematically compensated.

3. Tension or Pull Correction

AxE

LPPe ms

pull

)(

OR

4. Temperature Correction

• Most materials expand and contract with temperature change, and this effects taped distances. If a tape has stretched due to heat it will read shorter than it would at its normal (or standard) temperature.

)( smtemp TTCxLe

5. Sag Correction

• If the tape cannot be supported for its

length then it will hang freely under the

influence of gravity. The shape of the tape

will take is known as (sag) and can be

determined mathematically.

Corrected Measured Length

• Actual length is:

orPtensionpullslopeSsagLstTtempma CCCCCLL )()()()(

OR

Problem 1

• A steel tape of nominal length 30 m was used to measure

a line AB by suspending it between supports. The

following measurements were recorded.

• The standardization length of the tape against a reference

tape was known to be 30.014 m at 20 oC and 50 N.

• If the tape weighs 0.17 N/m and has a cross sectional

area of 2 mm2, calculate the horizontal length of AB.

• Temp. correction factor = 0.0000112 m/oC

Solution to Problem 1

Problem 2

Solution to Problem 2

Solution to Problem 2

Theodolite Surveying

Distance Measurement by Stadia Method

Introduction to Theodolite

Theodolite Surveying

• The system of surveying in which the angles are

measured with the help of a theodolite, is called

Theodolite surveying.

• In combination with Leveling Staff theodolite can

also be used to measure the distance i.e. called as

stadia or tachometric method of distance

measurement.

THEODOLITE

The Theodolite is a most accurate surveying instrument mainly

used for :

• Measuring horizontal and vertical angles

• Locating points on a line

• Prolonging survey lines

• Finding difference of level

• Setting out grades

• Ranging curves

• Tachometric Survey (distance measurement)

TRANSIT VERNIER THEODOLITE

TRANSIT VERNIER THEODOLITE

Details of Upper & Lower Plates

TRANSIT VERNIER THEODOLITE

CLASSIFICATION OF THEODOLITES

Theodolites may be classified as ;

A.

i) Transit Theodolite.

ii) Non Transit Theodolite.

B.

i) Vernier Theodolites.

ii) Micrometer Theodolites.

CLASSIFICATION OF THEODOLITES

A. Transit Theodolite: A theodolite is called a transit theodolite

when its telescope can be transited i.e revolved through a

complete revolution about its horizontal axis in the vertical

plane, whereas in a-

Non-Transit type, the telescope cannot be transited. They

are inferior in utility and have now become obsolete.

CLASSIFICATION OF THEODOLITES

B. Vernier Theodolite: For reading the graduated circle if

verniers are used, the theodolite is called as a Vernier

Theodolite.

Whereas, if a micrometer is provided to read the graduated

circle the same is called as a Micrometer Theodolite.

Description of a Transit Vernier Theodolite

A Transit vernier theodolite essentially consist of the

following :

1. Leveling Head. 6. T- Frame.

2. Lower Circular Plate. 7. Plumb –bob.

3. Upper Plate. 8. Tripod Stand.

4. Telescope.

5. Vernier Scale.

Terms Used in Manipulating a Transit Vernier Theodolite 1. Centering

Centering means setting the theodolite exactly over an instrument- station

so that its vertical axis lies immediately above the station- mark. It can be

done by means of plumb bob suspended from a small hook attached to

the vertical axis of the theodolite. The center shifting arrangement if

provided with the instrument helps in easy and rapid performance of the

centering.

2. Transiting

Transiting is also known as plunging or reversing. It is the process of

turning the telescope about its horizontal axis through 1800 in the vertical

plane thus bringing it upside down and making it point, exactly in

opposite direction.

3. Swinging the telescope

It means turning the telescope about its vertical axis in the horizontal

plane. A swing is called right or left according as the telescope is rotated

clockwise or counter clockwise.

4. Face Left

If the vertical circle of the instrument is on the left side of the observer while

taking a reading, the position is called the face left and the observation taken

on the horizontal or vertical circle in this position, is known as the face left

observation.

5. Face Right

If the vertical circle of the instrument is on the right side of the observer while

taking a reading ,the position is called the face right and the observation taken

on the horizontal or vertical circle in this position, is known as the face right

observation.

6. Changing Face

It is the operation of bringing the vertical circle to the right of the observer, if

originally it is to the left, and vice – versa. It is done in two steps; Firstly

revolve the telescope through 1800 in a vertical plane and then rotate it

through 1800 in the horizontal plane i.e. first transit the telescope and then

swing it through 1800.

7. Line of Collimation

LINE OF

COLLIMATION

DIAPHRAGM

TELESCOPE

It is also known as the line of sight. It is an imaginary line joining the

intersection of the cross- hairs of the diaphragm to the optical centre of the

object- glass and its continuation.

8. Axis of the telescope

It is also known an imaginary line joining the optical centre of the object-

glass to the center of eye piece.

AXIS OF THE TELESCOPE

TELESCOPE

OBJECT GLASS

.

9. Axis of the Level Tube

It is also called the bubble line. It is a straight line tangential to

the longitudinal curve of the level tube at the center of the tube.

It is horizontal when the bubble is in the center.

10. Vertical Axis

It is the axis about which the telescope can be rotated in the

horizontal plane.

11. Horizontal Axis

It is the axis about which the telescope can be rotated in the

vertical plane. It is also called the trunion axis.

The adjustments of a theodolite are of two kinds :-

1. Permanent Adjustments.

2. Temporary Adjustments.

1) Permanent adjustments: The permanent

adjustments are made to establish the relationship

between the fundamental lines of the theodolite and ,

once made, they last for a long time. They are essential

for the accuracy of observations.

Adjustment of a Theodolite

Adjustment of a Theodolite

2. Temporary Adjustment

The temporary adjustments are made at each set up

of the instrument before we start taking

observations with the instrument. There are three

temporary adjustments of a theodolite:-

i) Centering

ii) Leveling

iii) Focusing

Centering of a Theodolite

Leveling of a Theodolite

Focusing of a Theodolite

Leveling Staff

Stadia

• Stadia is a rapid method of measuring distances.

• Along with the center cross hairs, transits are equipped

with two additional cross hairs known as stadia hairs.

Definition sketch

• we can define D as the horizontal distance from the transit to the rod

• the stadia interval (S) as the distance between the rod reading at the

top stadia hair and the rod reading at the bottom stadia hair

Horizontal distance

• When the line of sight is horizontal, the distance (D) from the instrument to the rod is given by:

• where K is the stadia interval factor (commonly 100)

• C is a constant (a property of the instrument)

• S is the difference in the stadia hair rod readings.

• Many modern transits have the following properties:

• K = 100

• C = 0 (internal-focusing)

• C = 0.3 (external-focusing)

• It is your responsibility to check the constant values for your instrument. They are provided on the instrument case and they are not all the same.

Example

• Find the distance from the transit given the following

information:

• K = 100

• C = 0

• upper stadia reading = 1.841 m

• middle reading = 1.536 m

• lower stadia reading = 1.231 m

Inclined Sightings (Distance)

• Due to topography, many stadia shots are inclined.

• In other words, the line of sight associated with the transit is not

horizontal.

Inclined Sightings (Distance)

• We need to modify the equation to account for the inclined line of sight

• The horizontal distance (H) between the transit and the rod is found

using the following expression

• where α is the vertical angle corresponding to the line of sight

• note: the angle of inclination can be either positive or negative (i.e. you

are shooting down a steep hill)

Example

• Let’s use the same numbers as the previous example with

the exception that the line of sight is inclined at an angle

of 4 degrees.

• Find the horizontal distance from the transit given the

following information:

• K = 100

• C = 0

• S = 0.610

• α = 4 °

Vertical Distance

• we can define the vertical distance (V) as the distance between the rod

reading and the instrument (HI).

• The vertical distance (V) between the rod reading and the instrument is given

by:

Elevation of the Point

• finally, we are interested in computing the elevation of the

location situated under the survey rod or…

Elevation of the Point • Let’s finish off our example by assuming that our HI has been established to be

100.00.

• The elevation at the corresponding point of interest would then be found using:

Electromagnetic Distance Measurement (EDM)

62

ELECTROMAGNETIC DISTANCE

MEASUREMENT (EDM)

• First introduced by Swedish physicist Erik Bergstrand (Geodimeter) in 1948.

Used visible light at night to accurately measure distances of up to 40km.

• In 1957, the first Tellurometer, designed by South African, Dr. T.L. Wadley, was

launched. The Tellurometer used microwaves to measure distances up to 80km

day or night.

• First models bulky and power hungry, they revolutionized survey industry which, until

their arrival, relied on tape measurements for accurate distance determinations.

• The picture above shows the remote unit of the CA1000 Tellurometer, which was used

extensively in the 70’s and 80’s.

63

Introduction to EDM

• EDM instruments, as the name implies utilizes

electromagnetic energy for measuring distances between

two points.

• Electromagnetic waves can be represented in the form of

periodic sinusoidal waves.

• The time taken for an alternating current to go through

one complete cycle of values is called period of the wave.

• One cycle of the wave motion is completed when one

period has been completed and number of cycles per unit

time is called as frequency (Hertz-one cycle per sec).

• Wavelength (λ) = velocity of radiation (V) / f

64

Propagation of Electromagnetic Energy

Velocity of EM energy V = ƒ λ ƒ is the frequency in hertz (cycles/second)

λ is the wavelength

In vacuum the velocity of electromagnetic waves equals the speed of light.

V = c/n n >1, n is the refractive index of the

medium through which the wave

propagates

c is the speed of light = 299 792 458 m/sec

f λ = c/n or λ = c/fn

Note that n in any homogeneous medium varies with the wavelength λ.

White light consists of a combination of wavelengths and hence ‘n’ for

visible light is referred to as a group index of refraction.

For EDM purposes the medium through which electromagnetic energy is

propagated is the earths atmosphere along the line being measured. It is

therefore necessary to determine ‘n’ of the atmosphere at the time and

location at which the measurement is conducted.

65

Propagation of Electromagnetic Energy The refractive index of air varies with air density and is derived from

measurements of air temperature and atmospheric pressure at the time and

site of a distance measurement. For an average wavelength λ:

na= 1 + ( ng-1 ) x p - 5.5e x 10-8

1 + 0.003661T 760 1 + 0.003661T

Where ng is the group index of refraction in a standard atmosphere

(T=0°C, p=760mm of mercury, 0.03% carbon dioxide)

ng = 1+ ( 2876.04 + 48.864/λ2 +0.680/ λ4 ) x 10-7

p is the atmospheric pressure in mm of mercury (torr)

T is the dry bulb temperature in °C and

e is the vapor pressure

Where e= e’+de and e’=4.58 x 10a, a=(7.5T’)/(237.3+T’),

de=-(0.000660p (1+0.000115T’) (T-T’) and

T’ is the wet-bulb temperature

So measuring p, T and T’ will allow for the computation of n for a specific λ

66

THE FRACTION OF A WAVELENGTH AND THE PHASE ANGLE

Am

pli

tud

e

θ

+ r

- r

0°

90°

180°

270° ¼λ ¼λ ¼λ ¼λ

λ

½λ ½λ

θ λ

360

A fraction of a wavelength can be determined from a corresponding phase angle θ

Note:

For θ = 0° the fraction is 0

For θ = 90° the fraction is ¼

For θ = 180° the fraction is ½

For θ = 270° the fraction is ¾

For θ = 360° the fraction is 1

EDM INSTRUMENTS CAN MEASURE PHASE ANGLES

67

Principles of Electronic Distance Measurement If an object moves at a constant speed of V over a straight distance L in a

time interval ∆t, then

L= V∆t = (c/n)∆t

Knowing the speed of light c and being able to determine the refractive

index, we could measure the time interval it takes for an electromagnetic

wave to move from A to B to determine the distance L between A and B. But

since the speed of light (c) is very high, the time interval ∆t would need to

be measured extremely accurately. Instead, the principle of EDM is based on

the following relationship:

L = (m + p) λ

1 2 3 4 5 6 7 8 9 10 11 12

p λ λ λ λ λ λ λ λ λ λ λ λ A

B

L

m is an integer number of whole wavelengths, p is a fraction of a wavelength

So L can be determined from λ, m and p

68

69

Distance Measuring by EDM

• The instrument broadcasts a focused signal that is returned by a prism or reflection from the object.

• Therefore, if the speed of the signal is known (speed of light), and the time for the signal to travel to the target and back is known, the distance can be calculated.

Velocity =Distance

Time

Rearranging the equation for distance results in:

Distance = Velocity x Time

How the process works can be shown

using the velocity equation.

70

EDM Advantages and Disadvantages

Advantages of EDM’s

1. Precise measurement of

distance.

2. Line of sight instrument

3. Capable of measuring long

distances

4. Reflector less are single

person operation

Disadvantages of EDM’s

1. Electronic = batterers

2. Accuracy affected by

atmospheric conditions.

3. Can be expensive

Error ± (2 mm + 2 ppm x D)

Distance Measurement by Total Station

71

Distance Measurement by Total Station

• Point the instrument at a prism or reflector (which is

vertical at the point).

• Push the measure button and record the distance.

• We can measure horizontal or slope distance, It is

important to note which readings is being collected.

• If your are measuring the slope distance, the zenith angle must be

recorded to allow the computation of horizontal distance.

• If you are collecting topographic data with elevations, it is important

that the height of the instrument and the height of the prism be

recorded.

72

Sources of Error in EDM:

Personal:

• Careless centering of instrument and/or reflector

• Faulty temperature and pressure measurements

• Incorrect input of T and p

Remember:

L = (m + p) λ

Instrumental

• Instrument not calibrated

• Electrical center

• Prism Constant (see next

slide)

Natural

• Varying ‘met’ along line

• Turbulence in air

73

Sources of Error in EDM:

A B C

Determination of System Measuring Constant

1. Measure AB, BC and AC

2. AC + K = (AB + K) + (BC + K)

3. K = AC- (AB + BC)

4. If electrical center is calibrated, K rep-

resents the prism constant.

Blunders:

• Incorrect ‘met’ settings

• Incorrect scale settings

• Prism constants ignored

• Incorrect recording settings

(e.g. horizontal vs. slope)

Good Practice:

Never mix prism

types and brands on

same project!!!

Calibrate regularly !!!

74

75

Distance Measuring Methods--GPS

• GPS (global Positioning System) is a system of 21-24 satellites in orbit around the earth.

• Each satellite knows its position and uses a unique signal to continuously broadcasts this information.

• Along with the position information is a time signal.

• When a GPS receiver receives a signal from at least four (4) satellites it

can compute its position by trilateration.

• The receiver position can be expressed in degrees of latitude and

longitude, or distance (meters) using Universal Transverse Mercator

(UTM) coordinates.

76

Distance Measuring Methods-- GPS~ cont.

• Because UTM distances are based on a x-y coordinate

system, distances between points can be determined by

simple math.

• Example: Determine the distance between Stillwater and

Oklahoma City when the UTM coordinates for Stillwater

are 675087E & 3998345N and the UTM coordinates for

Oklahoma City are 639982E & 3925518N

77

Distance Measuring Methods– GPS ~ Example

• Subtracting the

coordinates gives the two

sides of a right triangle.

• The hypotenuse of the

triangle is the distance

between the two towns

(44.6 mi).

HD = 399962 72255

2

= 82586211 m

= 44.6 mi

Numerical Problems-Distance Measurement

78

Problem-1

• The readings given below were made with a tachometric theodolite

having a multiplying constant (K) of 100 and no additive constant (C).

The reduced level at station A was 100.0 m and the height of the

instrument axis is 1.35 m above the ground. Calculate the gradient

expressed as the horizontal distance one meter rise or fall vertically

between the stations B and C.

79

Station To Vertical angle Stadia readings

A B

C

+11030’

-17000’

2.048/1.524/1.000

2.112/1.356/0.600

80

Problem-2

• A tachometric readings were taken from an instrument

station A the reduced level of which was 15.05 m to a staff

station B. Instrument multiplying constant (K) is 100,

additive constant 360 mm, staff held vertical. The height

of instrument is 1.38 m and vertical angle is 30 deg. The

stadia readings are 0.714, 1.007, and 1.300 respectively.

Calculate the horizontal and vertical distances.

81

Problem-3

• In a line ABC, AB measures 354.384 m, BC measures

282.092 m and AC measures 636.318 m using a

particular EDM reflector combination,. A line measures

533.452m with this instrument-reflector combination.

What is the correct length of the line?

• Solution:

K = AC – (AB+BC) = 636.318 – (354.384+282.092)

= -0.158

Corrected length of line = 533.452 + (-0.158) = 533.294 m

82