Advanced Survey

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NOTES OF LESSON SURVEYING II

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NOTES OF LESSON

CE2254 SURVEYING II

Prepared by

M.UMAMAGUESVARI M.Tech

Sr. Lecturer

Department of Civil Engineering

RAJALAKSHMI ENGINEERING COLLEGE




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Tacheometry

It is a method of surveying in which horizontal distances and (relative) vertical elevations

are determined from subtended intervals and vertical angles observed with an instrument

Uses of Tacheometry

Tacheometry is used for

1. preparation of topographic map where both horizontal and vertical distances are required

to be measured;

2. survey work in difficult terrain where direct methods of measurements are inconvenient;3. reconnaissance survey for highways and railways etc;

4. establishment of secondary control points.

Instrument

The instruments employed in tacheometry are the engineer's transit and the leveling rod or stadia

rod, the theodolite and the subtense bar, the self-reducing theodolite and the leveling rod, the

distance wedge and the horizontal distance rod, and the reduction tacheometer and the horizontal

distance rod

Systems of Tachometric Measurement

Depending on the type of instrument and methods/types of observations, tacheometric

measurement systems can be divided into two basic types:

(i) Stadia systems and

(ii) Non-stadia systems

Stadia Systems

In there system's, staff intercepts at a pair of stadia hairs present at diaphragm, are considered.

The stadia system consists of two methods:

• Fixed-hair method and

• Movable-hair method

Fixed - Hair method




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In this method, stadia hairs are kept at fixed interval and the staff interval or intercept

(corresponding to the stadia hairs) on the leveling staff varies. Staff intercept depends upon the

distance between the instrument station and the staff.

Movable - Hair method

In this method, the staff interval is kept constant by changing the distance between the stadia

hairs. Targets on the staff are fixed at a known interval and the stadia hairs are adjusted to bisect

the upper target at the upper hair and the lower target at the lower hair. Instruments used in this

method are required to have provision for the measurement of the variable interval between the

stadia hairs. As it is inconvenient to measure the stadia interval accurately, the movable hair

method is rarely used

Non - stadia systems

This method of surveying is primarily based on principles of trigonometry and thus telescopes

without stadia diaphragm are used. This system comprises of two methods:

(i) Tangential method and

(ii) Subtense bar method.

Tangential Method

In this method, readings at two different points on a staff are taken against the horizontal

cross hair and corresponding vertical angles are noted.

Subtense bar method

In this method, a bar of fixed length, called a subtense bar is placed in horizontal position. The

angle subtended by two target points, corresponding to a fixed distance on the subtense bar, at the

instrument station is measured. The horizontal distance between the subtense bar and the

instrument is computed from the known distance between the targets and the measured horizontal

angle.

Fixed-hair method or Stadia method

It is the most prevalent method for tacheometric surveying. In this method, the telescope of the

theodolite is equipped with two additional cross hairs, one above and the other below the main

horizontal hair at equal distance. These additional cross hairs are known as stadia hairs. This is

also known as tacheometer.

Principle of Stadia method




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A tacheometer is temporarily adjusted on the station P with horizontal line of sight. Let a and b be

the lower and the upper stadia hairs of the instrument and their actual vertical separation be

designated as i. Let f be the focal length of the objective lens of the tacheometer and c be

horizontal distance between the optical centre of the objective lens and the vertical axis of the

instrument. Let the objective lens is focused to a staff held vertically at Q, say at horizontal

distance D from the instrument station.

.

By the laws of optics, the images of readings at A and B of the staff will appear along the stadia

hairs at a and b respectively. Let the staff interval i.e., the difference between the readings at A

and B be designated by s. Similar triangle between the object and image will form with vertex at

the focus of the objective lens (F). Let the horizontal distance of the staff from F be d. Then, from

the similar s ABF and a' b' F,




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as a' b' = ab = i. The ratio ( f / i) is a constant for a particular instrument and is known as stadia

interval factor, also instrument constant. It is denoted by K and thus

d = K.s --------------------- Equation (23.1)

The horizontal distance (D) between the center of the instrument and the station point (Q) at

which the staff is held is d + f + c. If C is substituted for (f + c), then the horizontal distance D

from the center of the instrument to the staff is given by the equation

D = Ks + C ---------------------- Equation (23.2)

The distance C is called the stadia constant. Equation (23.2) is known as the stadia equation for a

line of sight perpendicular to the staff intercept.

Determination of Tacheometric Constants

The stadia interval factor (K) and the stadia constant (C) are known as tacheometric constants.

Before using a tacheometer for surveying work, it is reqired to determine these constants. These

can be computed from field observation by adopting following procedure.

Step 1 : Set up the tacheometer at any station say P on a flat ground.

Step 2 : Select another point say Q about 200 m away. Measure the distance between P and Q

accurately with a precise tape. Then, drive pegs at a uniform interval, say 50 m, along PQ. Mark

the peg points as 1, 2, 3 and last peg -4 at station Q.

Step 3 : Keep the staff on the peg-1, and obtain the staff intercept say s 1 .

Step 4 : Likewise, obtain the staff intercepts say s 2, when the staff is kept at the peg-2,

Step 5 : Form the simultaneous equations, using Equation (23-2)

D1 = K. s 1 + C --------------(i)

and D 2 = K. s 2+ C -------------(ii)

Solving Equations (i) and (ii), determine the values of K and C say K 1 and C 1 .

Step 6 : Form another set of observations to the pegs 3 & 4, Simultaneous equations can be

obtained from the staff intercepts s3 and s 4 at the peg-3 and point Q respectively. Solving those

equations, determine the values of K and C again say K 2 and C 2.




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Step 7 : The average of the values obtained in steps (5) and (6), provide the tacheometric

constants K and C of the instrument.

Anallactic Lens

It is a special convex lens, fitted in between the object glass and eyepiece, at a fixed distance from

the object glass, inside the telescope of a tacheometer. The function of the anallactic lens is to

reduce the stadia constant to zero. Thus, when tacheometer is fitted with anallactic lens, the

distance measured between instrument station and staff position (for line of sight perpendicular to

the staff intercept) becomes directly proportional to the staff intercept. Anallactic lens is provided

in external focusing type telescopes only.

Inclined Stadia Measurements

It is usual that the line of sight of the tacheometer is inclined to the horizontal. Thus, it is

frequently required to reduce the inclined observations into horizontal distance and difference in

elevation.

Let us consider a tacheometer (having constants K and C) is temporarily adjusted on a station, say

P (Figure 23.2). The instrument is sighted to a staff held vertically, say at Q. Thus, it is required

to find the horizontal distance PP1 (= H) and the difference in elevation P 1Q. Let A, R and B bethe staff points whose images are formed respectively at the upper, middle and lower cross hairs

of the tacheometer. The line of sight, corresponding to the middle cross hair, is inclined at

an angle of elevation θ and thus, the staff with a line perpendicular to the line of sight.

Therefore A'B' = AB cos θ = s cos θ where s is the staff intercept AB. The distance D (=




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OR) is C + K. scos θ (from Equation 23.2). But the distance OO1 is the horizontal

distance H, which equals OR cos θ. Therefore the horizontal distance H is given by theequation.

H = (Ks cos θ + C) cos θ

Or H = Ks cos2 θ + C cos θ ----------------- Equation (23.3)

in which K is the stadia interval factor (f / i), s is the stadia interval, C is the stadia

constant (f + c), and θ is the vertical angle of the line of sight read on the vertical circle ofthe transit.

The distance RO1, which equals OR sin θ, is the vertical distance between the telescopeaxis and the middle cross-hair reading. Thus V is given by the equation

V = (K s cos θ + c) sin θ

V = Ks sin θ cos θ + C sin θ ----------------- Equation (23.4)

----------------- Equation (23.5)

Thus, the difference in elevation between P and Q is (h + V - r), where h is the height ofthe instrument at P and r is the staff reading corresponding to the middle hair.

Ex23-1 In order to carry out tacheometric surveying, following observations were takenthrough a tacheometer set up at station P at a height 1.235m.

Staffheld

Verticalat

Horizontaldistancefrom P

(m)

StaffReading

(m)

Angle of

Elevation

Q 100 1.01 0°

R 200 2.03 0°

S ? 3.465,

2.275, 1.2805° 24' 40"

Compute the horizontal distance of S from P and reduced level of station at S if R.L. ofstation P is 262.575m




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Solution :

Since the staff station P and Q are at known distances and observations are taken athorizontal line of sight, from equation 23.2

i.e. from D = K.s + C, we get

100 = K. 1.01 + C --------------- Equation 1

200 = K. 2.03 + C --------------- Equation 2

where K and C are the stadia interval factor and stadia constant of the instrument.

Therefore Solving equation 1 and 2 ,

Substituting, value of K in Equation 1, we get

C = 100 - 1.01 x 98.04 = 0.98




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Now, for the observation at staff station S, the staff intercept

s = 3.465 - 1.280 = 2.185 m;

Given, the angle of elevation (of a observation at S), θ = 5° 24' 40"

Using equation 23.3 i.e., D = K s cos2 θ + C.cos θ, the horizontal distance of S from P is

D = 98.04 x 2.185 x cos2 5° 24' 40" + 0.98 cos 5° 24' 40"

= 212.312 + 0.9756 = 213.288 m

= (20.11 + 0.0924)m = 20.203 m

Thus R.L. of station S = R.L. of P + h + V - r

= 262.575 + 1.235 + 20.203 - 2.275

= 281.738 m

Uses of Stadia Method

The stadia method of surveying is particularly useful for following cases:

1. In differential leveling, the backsight and foresight distances are balanced conveniently if

the level is equipped with stadia hairs.

2. In profile leveling and cross sectioning, stadia is a convenient means of finding distances

from level to points on which rod readings are taken.

3. In rough trigonometric, or indirect, leveling with the transit, the stadia method is more

rapid than any other method.

4. For traverse surveying of low relative accuracy, where only horizontal angles and

distances are required, the stadia method is a useful rapid method.

5. On surveys of low relative accuracy - particularly topographic surveys-where both the

relative location of points in a horizontal plane and the elevation of these points are

desired, stadia is useful. The horizontal angles, vertical angles, and the stadia interval are

observed, as each point is sighted; these three observations define the location of the

point sighted.




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Errors in Stadia Measurement

Most of the errors associated with stadia measurement are those that occur during

observations for horizontal angles and differences in elevation .Specific sources of errorsin horizontal and vertical distances computed from observed stadia intervals are as

follows:

1. Error in Stadia Interval factor

This produces a systematic error in distances proportional to the amount of error in the

stadia interval factor.

2. Error in staff graduations

If the spaces on the rod are uniformly too long or too short, a systematic error

proportional to the stadia interval is produced in each distance.

3. Incorrect stadia Interval

The stadia interval varies randomly owing to the inability of the instrument operator toobserve the stadia interval exactly. In a series of connected observations (as a traverse)

the error may be expected to vary as the square root of the number of sights. This is the principal error affecting the precision of distances. It can be kept to a minimum by properfocusing to eliminate parallax, by taking observations at favorable times, and by care in

observing.

4. Error in verticality of staff

This condition produces a perceptible error in measurement of large vertical angles thanfor small angles. It also produces an appreciable error in the observed stadia interval and

hence in computed distances. It can be eliminated by using a staff level.

5. Error due to refraction

This causes random error in staff reading.

6. Error in vertical angle

Error in vertical angle is relatively unimportant in their effect upon horizontal distance if

the angle is small but it is perceptible if the vertical angle is large.




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non-stadia systems of tacheometric surveying

Tangential Method

The tangential method of tacheometry is being used when stadia hairs are not present in thediaphragm of the instrument or when the staff is too far to read.

In this method, the staff sighted is fitted with two big targets (or vanes) spaced at a fixed vertical

distances. Vertical angles corresponding to the vanes, say θ1 and θ2 are measured. Thehorizontal distance, say D and vertical intercept, say V are computed from the values s (pre-

defined / known) θ1 and θ2 . This method is less accurate than the stadia method.

Depending on the nature of vertical angles i.e, elevation or depression, three cases of tangentialmethods are thereWhen Both vertical Angles are Angles of Elevation

From Figure 24.1,

V = D tan θ1

and V+s = D tan θ2

Thus, s = D ( tan θ2 - tan θ1 )




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-------------------- Equation (24.1)

-------------------- Equation (24.2)

Therefore R.L. of Q = (R.L. of P + h) + V – r -------------------- Equation (24.3)

where, h is the height of the instrument, r is the staff reading corresponding to lower vane

When Both Vertical Angles are Depression Angles

From Figure 24.2,

V = D tan θ1

and V-s = D tan θ2

Thus, s = D ( tan θ1 - tan θ2 )




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-------------------- Equation (24.4)

-------------------- Equation (24.5)

Therefore R.L. of Q = (R.L. of P + h) - V – r -------------------- Equation (24.6)

where, h is the height of the instrument, r is the staff reading corresponding to lower vane.

When one of the Vertical Angles is Elevation Angle and the other is Depression Ang le

From Figure 24.3,

V = D tan θ1

and s - V = D tan θ2

Thus, s = D ( tan θ2 + tan θ1 )

-------------------- Equation (24.7)

-------------------- Equation (24.8)




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Therefore R.L. of Q = (R.L. of P + h) - V – r -------------------- Equation (24.9)

where, h is the height of the instrument, r is the staff reading corresponding to lower vane

Ex24-1 In a tangential method of tacheometry two vanes were fixed 2 m apart, the lower vane being 0.5 mabove the foot of the staff held vertical at station A. The vertical angles measured are +1° 12' and -1° 30'.find the horizontal distance of A and reduced level of A, if the R.L. of the observation station is 101.365 mand height of instrument is 1.230 m.

Figure Ex24-1

Solution :

Let D be the horizontal distance between the observation station P and staff point A. Then, from FigureEx24.1,

V = D tanα1

s - V = D tanα1

Or, s = D tanα2 + D tanα1

Given, s = 2 m; α1= 1° 30' & α2 = 1° 12'




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Therefore R.L. of A = 101.365 + 1.230 -1.11 - 0.5 = 100.985 m

Subtense Bar Method

Instrument

Subtense bar is a graduated bar of fixed length mounted horizontally on a tripod stand(Figure 24.4). The bar is centrally supported on a leveling head and is fitted with a

sighting device at the centre. At its ends, there are targets and these are at fixed distance

apart. The bar can rotate about the vertical axis in a horizontal plane. It can be fixed atany position using a clamping and its tangent screw. It is used to measure horizontal

distance and difference in elevation indirectly where the terrain is rough and requirement

of accuracy is not high.

Method




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shows a schematic diagram of a subtense bar having centre at C fitted with targets at A and B.Let the separation between targets be s. A theodolite is set up at O. The bar is kept perpendicularto the line of sight OC by means of a sighting device at the centre of the bar. The horizontal angle

between the two targets at the ends of the bar is measured, let it be θ. The horizontal distance

----------------- Equation (24.10)

If the vertical angle of the centre of the subtence bar is β, then the difference in elevation (V)between the centre of the bar and the centre of the telescope is

V = D tan β ---------------- Equation (24.11)

Effect of Angular Error

The effect of an error in the measurement of the angle θ on the computed length D is as follows(Figure 24.5):

---------------- Equation (24.12)




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where δD is t

he error in the distance D; δθ is the error in the angle θ. The angle θ is in radians.

It may be noted that the nature of error in computation of distance is opposite to the nature of

error in measurement of horizontal angle ie., a positive error in δθ produces a negative error in Dand vice versa.

Ex24-2 The horizontal angle observed at a theodolite station by a subtense bar with vanes 2.0 mapart, is 0° 30'. Find the horizontal distance between the theodolite station and subtense bar.Ifthe bar is 1° out from the normal direction to the line of sight, determine the error in themeasurement of horizontal distance.




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Figure Ex24-2

Solution :

Referring from Figure 24.5, given s = 2.0 m; θ = 30', Then from

The horizontal distance between the subtense bar and the theodolite,

Referring figure Ex 24.2,




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Therefore Error in horizontal distance = D - D'

= 229 .18 - 229.145 = 0.035 m

Introduce the basic definitions and fundamental relations used in field astronomy.

Introduction

In a clear night, the sky appears as a vast hollow hemisphere with its interior studded

with innumerable stars. On observing the sky for some duration it appears that the

celestial bodies are revolving around the earth with its centre at the position of theobserver. The stars move in a regular manner and maintain same position relative to each

other. Consequently, terrestrial position or direction defined with reference to celestial

body remains absolute for all practical purposes in plane surveying. Thus, the absolutedirection of a line can be determined from the position / direction of a celestial body

Ast ronomical Terms and Defini tions

To observe the positions / direction and movement of the celestial bodies, an imaginarysphere of infinite radius is conceptualized having its centre at the centre of the earth. Thestars are studded over the inner surface of the sphere and the earth is represented as a

point at the centre. Figure 25.1 shows a celestial sphere and some principal parameters

necessary to understand astronomical observation and calculations for determination ofabsolute direction of a line. The important terms and definitions related to field

astronomy are as follows:




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Celestial sphere : An imaginary sphere of infinite radius with the earth at its centre and

other celestial bodies studded on its inside surface is known as celestial sphere.

Great Circle (G.C) : The imaginary line of intersection of an infinite plane, passing

through the centre of the earth and the circumference of the celestial sphere is known as

great circle.

Zenith (Z) : If a plumb line through an observer is extended upward, the imaginary pointat which it appears to intersect the celestial sphere is known as Zenith. The imaginary




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point at which it appears to intersect downward in the celestial sphere is known as Nadir(N).

Vertical circle : Great circle passing through zenith and nadir is known as vertical circle.

Horizon: Great circle perpendicular to the line joining the Zenith and Nadir is known as horizon.

Poles : If the axis of rotation of the earth is imagined to be extended infinitely in both directions,

the points at which it meets the celestial sphere are known as poles. The point of intersection in

the northern hemisphere is known as north celestial pole and that in the southern hemisphere as

south celestial pole.

Equator : The line of intersection of an infinite plane passing through the centre of the earth and

perpendicular to the line joining celestial poles with the celestial sphere.

Hour circle : Great circle passing through celestial poles is known as hour circle, also known as

declination circle.

Meridian : The hour circle passing through observer's zenith and nadir is known as (observer's)

meridian. It represents the North-South direction at observer station.

Altitude (h) : The altitude of a celestial body is the angular distance measured along a vertical

circle passing through the body. It is considered positive if the angle measured is above horizon

and below horizon, it is considered as negative.

Azimuth (A) : The azimuth of a celestial body is the angular distance measured along the horizon

from the observer's meridian to the foot of the vertical circle passing through the celestial body

(Figure 25.2).




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Declination ( ) : The declination of a celestial body is the angular distance measured from theequator to the celestial body along the arc of an hour circle. It is considered positive in Northdirection and negative in South.

Ecliptic : The great circle along which the sun appears to move round the earth in ayear is called the ecliptic.

Equinoctial points : The points of intersection of the ecliptic circle with the equatorialcircle are known as equinoctial points. The point at which the sun transits from Southern

to Northern hemisphere is known as First point of Aeries (γ) and from Northern to

Southern hemisphere as First point of Libra (Ω).

Right ascension : The right ascension of a celestial body is the angular distance along

the arc of celestial equator measured from the First point of Aeries (γ) to the foot of the

hour circle. It is measured from East to West direction i.e., anti-clockwise in Northernhemisphere.

Prime meridian : Reference meridian that passes through the Royal Naval Observatoryin Greenwich, England is known as prime meridian; it is also known as Greenwichmeridian.

Longitude (λ) : The longitude of an observer's station is the angular distance measuredalong the equator from the prime meridian to the observer's meridian. It varies from zerodegrees to 180° E and 0° to 180° W.




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Latitude ( ): The latitude of an observer's station is the angular distance measuredalong the observer's meridian from the equator to the zenith point. It varies from zerodegree to 90° N and 0° to 90° S.

Figure 25.3 Hour angles




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Hour angle (HA) : The hour angle of a celestial body is the angle at the equatorial planemeasured westward from meridian to the hour circle passing through the celestial body(Figure 25.3).

Local hour angle (LHA): The angular distance of a celestial body measured westwardfrom the point of intersection of the equator and the meridian of the observer to the footof the hour circle passing through the celestial body.

Greenwich hour angle (GHA) : Angle at the equatorial plane measured westward fromthe prime (Greenwich) meridian to the hour circle through the celestial body.

Spherical triangle: Triangle formed by the intersection of three arcs of great circles (onthe surface of the celestial sphere) is known as spherical triangle.

Note :

The dimension of the celestial sphere is so large that the position of the observer andthe centre of the earth appears to be the same point

Astronomical Triangle

The spherical triangle formed by arcs of observer's meridian, vertical circle as well as

hour circle through the same celestial body is known as an astronomical triangle. Thevertices of an astronomical triangle are Zenith point (Z), celestial pole (P) and thecelestial body (S) and thus termed as ZPS triangle (Figure 25.4a). In each astronomicaltriangle, there are six important elements. Three of them are the three sides and otherthree are the three angles of the triangle. It is important to know these elements as someof these will be required to be observed in the field and others are to be computed to findthe position / direction of celestial body.




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Polar distance (PS or p) : The angular distance from the celestial pole (P) to the celestial body (S) alongthe hour circle is known as polar distance. It is also known as co-declination and is designated by p (= 90°-

δ), where δ is the declination of the celestial body, S.

Zenith distance (ZS or z) : The angular distance from observer's zenith (Z) to the celestial body (S) alongthe vertical circle is known as zenith distance. It is also known as co-altitude and is designated by z (= 90°-h), where h is the altitude of the celestial body, S.

Co-latitude, ZP : The angular distance from observer's zenith (Z) to the celestial pole (P) along the

observer's meridian is known as co-latitude and is given by (90°- φ), where φ is the latitude of the observer .

Ang le Z : The angle at the zenith (A) is measured from the observer's meridian to the vertical circle passingthrough the celestial body in a plane parallel to the observer's horizon. It is nothing but the azimuth of thecelestial body (Figure 25.2). It is measured clockwise from the observer's meridian and its value ranges fromzero to 360°.

Ang le P : The angle at the pole (P) is measured from the observer's meridian to the hour circle passingthrough the celestial body in a plane parallel to the equatorial plane. It is nothing but (360°– H, hour angle ofthe celestial body) (Figure 25.4b). Hour angle is measured clockwise from the upper branch of theobserver's meridian.

Ang le S : angle at a celestial body between the hour circle and the vertical circle passing through thecelestial body. It is known as the parallactic angle.




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If any of the three elements are known, the remaining three can be computed from

formulae of spherical trigonometry

Figure 25A shows a spherical triangle ZSP. In all there are six quantities in a spherical

triangle, namely, three angles Z, S and P and three sides z, s and p. If any three

quantities are known, the remaining three can be computed from different formulae of

the spherical trignometry given below.

Figure 25A




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1. Sine formula

--------------- Eq 25A.1

2. Cosine formula

(i) cos Z = sin S sin P - cos S cos P --------------- Eq 25A.2

(ii) cos z = cos s cos p + sin s sin p cos Z --------------- Eq 25A.3

or --------------- Eq 25A.3a

(iii) cos z cos s = sin z cot P - sin S sin P cos z --------------- Eq 25A.4

3. Half the angles fo rmula

(i ) --------------- Eq 25A.5

(ii ) --------------- Eq 25A.6

(iii) --------------- Eq 25A.7

Where

Equation 25A.5 to 25A.7 should be used when there is some confusion about the signs of the angles.Because the value of the angle A / 2 is always less than 90°, there is no ambiguity. As far as possible, Eq.25A.5 should be used when A < 90°. Eq. 25A.7 can be used for any value of the angle.

4. Half the sides Formula




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(i) --------------- Eq 25A.8

(ii ) --------------- Eq 25A.9

(iii ) --------------- Eq 25A.10

where

5. Tangent of ang les formulae

(i) --------------- Eq 25A.11a

(ii) --------------- Eq 25A.11b

6. Tangent of sides for mula

(i) --------------- Eq 25A.12a




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(ii) --------------- Eq 25A.12b

7. Combined formula

(i) sin z cos s = cos s sin p - sin s cos p cos Z --------------- Eq 25A.13

(ii) sin z sin s = sin s sin z --------------- Eq 25A.14

Ex25-1 Find the shortest distance between a station (29° 52' N, 77° 54' E) at Roorkee and to a station (28°34' N, 77° 06' E) at Delhi. Determine the azimuth of the line along which the direction of the shortestdistance to be set out starting from Roorkee.

Figure Example 25-1

Solution :

The shortest distance between two stations on the surface of the earth lies along the circumfrence of thegreat circle passing through the stations.




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Refering Figure Example 25-1, let us consider a great arc RD passing through the Roorkee and Delhirespectively. Thus, arc RD is the shortest distance between the stations. Let P be the pole of the earth. andPD and PR are arcs of meredians passing through Delhi and Roorkee stations respectively.

Then, PDR is a astronomical triangle, where

P = 77° 54' - 77° 06' = 48'

Distance PD, r = 90° - 28 ° 34' = 61 ° 26

Distance PR, d = 90° - 29 ° 52' = 60 ° 08'

Using equation (25A.3a),

Or, cos p = cos d . cos r + cos P . sin d . sin r

= cos 60° 08' . cos 61 ° 26' + cos 48' . sin 60 ° 08' . sin 61 ° 26'

= 0.23826 + 0.76154 = 0.99966837

Therefore p = 1° 28' 33".48

Assuming, rdius of the earth, R = 6370 km.

Arc distance, RD =

For the determination of the direction from R to D, the angle R of the spherical triangle is

required to be determined.

Using equation 25A.11a,




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Therefore R + D = 179° 36' 34".12 ------------------------Equation 1

Again, using equation 25A.11b

= 3° 43' 24".07

Or, R - D = 123° 31' 05".20 ------------------------Equation 2

Solving Equation 1 & 2,

R = 151° 33' 49".66

Thus, the azimuth of the line to be set out at station R to proceed along shortest path to

the station at Delhi is = 360° - R = 208 ° 26' 10".34

Ex25-2 Determine the azimuth of the sun having declination 8° 24' S during sunset at Roorkee (latitude 29°52' N).




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Figure Example 25-2 (a)

Solution :

Given, latitude of observer, φ = 29° 52'

declinationof the sun, δ = 8° 24' S

The Sun at sunset, thus altitude α = 0°

Therefore φ = 90°

z = 90° + δ = 98° 24'

s = 90° - φ = 60° 08'

Figure Example 25-2 (b)

The triangle ZPS is a right angled astronomical triangle, where RZ (in ZPS) is in westernhemisphere.

Now, using equation 25A.3a, we get




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= - 0.168

Z = 99° 41' 53".07 (W)

Thus, azimuth of the sun, A = 360° - Z = 261° 59' 32".26

To define azimuth and to explain hour angle method for determination of azimuth

Azimuth of a Li ne

Azimuth of a line is its horizontal angle measured clockwise from geographic or true

meridian.

For field observation, the most stable and retraceable reference is geographic north.

Geographic north is based on the direction of gravity (vertical) and axis of rotation of the

earth. A direction determined from celestial observations results in astronomic

(Geographic) north reference meridian and is known as geographic or true meridian.

The azimuth of a line is determined from the azimuth of a celestial body. For this, the

horizontal angle between the line and the line of sight to the celestial body is required to

be observed during astronomic observation along with other celestial observation




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Let AB be the line whose azimuth (A AB) is required to be determined (Figure 26.1). Let O

be the station for celestial observations. Let S be the celestial body whose azimuth (As)

is determined from the astronomical observation taken at O. The horizontal angle from

the line AB to the line of sight to celestial body (at the station O) is observed to be θ°

clockwise. The azimuth of the line, AB can be computed from

A AB = AS - θ° (clockwise).

If A AB computes to be negative, 360° is added to normalize the azimuth.




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In order to compute the azimuth of a line with proper sign, it is better to draw the known

parameters. The diagram itself provides the azimuth of the line with proper sign. For

example, in Figure 26.2, first a line of sight to celestial body, OS is drawn. Then, the

azimuth of the celestial body, AS is considered in counter-clockwise from the line OS and

obtained the true north direction i.e, the line ON. Similarly, the horizontal angle θ° is

represented in counter clockwise (since the angle from the line to the celestial body is

measured clockwise) direction from OS to obtain the relative position of the line. The

angle NOB represents the azimuth of the line AB.

Determination of Azimuth of Celestial body

In field astronomy, a celestial body provides the reference direction. So, from the

geographic location (latitude and longitude) of the station, ephemeris data of celestial

body and either time or altitude of the same celestial body, the azimuth of the celestial

body is computed by solving astronomical triangle. If time is used, the procedure is

known as the hour-angle method. Likewise, if altitude is measured, the procedure is

termed as the altitude method. The basic difference between these two methods is that




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the altitude method requires observation of approximate time and an accurate vertical

angle of the celestial body, whereas the hour angle method requires observation of

accurate time. Recent developments of time receivers and accurate timepieces,

particularly digital watches with split-time features, and time modules for calculators, the

hour-angle method is more accurate, faster. It requires shorter training for proficiency. It

has fewer restrictions on time of day and geographic location and thus is more versatile.

The method is applicable to the sun, Polaris, and other stars. Consequently, the hour-

angle method is emphasized, and its use by surveyors is encouraged.

Hour Angle Method

In this method, precise time is being noted when the considered celestial body is being

bisected. The observed time is used to derive the hour angle and declination of the

celestial body at the instant of observation.The geographic position (latitude and

longitude) of the observation station is required to be known a priori for the hour angle

method. Usually, these values are readily obtained from available maps. However, to

achieve better accuracy, latitude and longitude must be more accurately determined

specially during observations for celestial bodies close to the equator--e.g., the sun-than

for bodies near the pole--e.g., Polaris.

Figure 26.3 Astronomical Triangle




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The declination of the celestial bodies at the instant of observation is required to beknown for computation of azimuth of the celestial body. It is available in star almanac atthe 0, 6, 12 and 18 hours of UTI of each day ( Greenwich date). Thus, the declination atthe instant of observation (of celestial body) is determined by linear interpolation forcorresponding the UT1 time of observation. However, since the declination of the sunvaries rapidly, its interpolation is done using the relation:

Declination, d = Decl 0h + (Decl 24h - Decl 0h) ( ) + (0.0000395) (Decl 0h) sin (7.5UT1) -------------(Equation 26.2)

The hour-angle of the celestial body is being derived using the GHA (available in staralmanac with reference to Greenwich date) and the longitude of the observation station.For observations in the Western Hemisphere, if UTI is greater than local time, theGreenwich date is the same as local date and if UTI is less than local time, Greenwichdate is the local date plus one day. For the Eastern Hemisphere, if UTI is less than localtime (24-hr basis), Greenwich date is the same as local date and if UTI is greater thanlocal time, Greenwich date is local date minus one day. The hour angle of the celestialbody at the observation station is the LHA. Thus, it is the LHA at UTI time of observationwhich is necessary to compute the azimuth of a celestial body. Hence, as can be seenfrom Figure 26.3, the equation for the LHA is

LHA = GHA - W λ (west longitude) ----------------- (Equation 26.3)

Or LHA = GHA + E λ (east longitude) -------------------(Equation 26.4)

LHA should be normalized to between 0° and 360° by adding or subtracting 360°, ifnecessary.

The Greenwich hour angle (GHA) of celestial bodies-the sun, Polaris, and selectedstars-is tabulated in star almanac from 0 hr to 24 hr at an interval of 6 hours of UTI timeof each day (Greenwich date). Thus, to find GHA at the time of observation linearinterpolation is required to be performed.

The GHA can also be derived by making use of the equation of time E (apparent timeminus mean time) by using the relation:

GHA = 180° + 15 E ---------------------------Equation (26.5)

where E is in decimal hours. In those cases where E is listed as mean time minusapparent time, the algebraic sign of E should be reversed.

Once the parameters (declination and Hour angle of the celestial body, latitude of theobservation station) required to compute the azimuth of the celestial body are available,




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the computation of azimuth of the celestial body is carried out using the relations ofastronomical triangle ( Appendix- 25A).

Ex26-1 A celestial body is observed at a station (latitude 36° 04' 00" N and longitude 94°10' 08" N). The UTI at the instant of observation was (15h 16m 41s.57).The GHA at 0hr(UT1) was found to be 177° 04 ' 44".1 on the day of observation and 177° 08' 06".3 onthe next day. The declination of the selected body at 0hr (UT1) was + 6° 15' 05".9 S and+ 5° 5' 54".7 S on the day of observation and next day respectively. Determine theazimuth of the celestial body.

Figure Example 26.1

Soluton : Given, GHA at 0hr

= 177° 04 ' 44".1

GHA at 24hr

= 177° 08' 06".3

Instant of observation of the celestial body = 15h 16

m 41

s.57

Using linear interpolation method,

the GHA at the instant of observaton =

= 177° 04' 44".1 + (177° 08' 06".3 - 177° 04' 44".1 + 360°) x

= 406.28788° - 360° = 46.28788°

Thus,




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Local Hour angle at the instant of observation, LHA = GHA - Wλ (The station is at west to Greenwich)

= (46.28788° - 94° 10' 08") = - 47.88101° = -47.88101° + 360°

= 312.11899°

Declination (δ)of the celestial body at the instant of observation (Implementing linear interpolation)

δ = - 6° 15' 05".9 + ( - 5° 51' 54".7 + 6° 15' 05".9) x = - 6.00563°

Now, using the relation (Equation 26.3), azimuth of the celestial body

Z =

=

= - 57.10487745°

Since, LHA is between 180° and 360° and A is negative (Figure Ex 26.1), thus

Azimuth of the body, A = Z + 180° = - 57.10487745° + 180°

= 122.8951226° = 122° 53' 42 ".44

To explain the field method for sun observation and subsequent computation of

azimuth

Sun Observations (Hour- Angle Method)

Among all the celestial bodies, the sun is the most prominent body that can be observed easily

and accurately. Thus, the sun observation provides surveyors a convenient method for

determination of astronomic azimuth. In the hour-angle method, a horizontal angle from a line to

the sun is measured. Knowing accurate time of the observation and geographic position of the

observation station, the sun's azimuth is computed using the relations of astronomical triangle.

This azimuth and horizontal angle are combined to yield the line's azimuth.

The sun is observed through the telescope fitted with either an eyepiece sun glass or an objective

lens filter. For total stations, an objective lens filter is mandatory (to protect EDMI components).

It is to be noted that the sun's image is large in diameter-approximately 32 min of arc-making




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accurate pointing on the center impractical. Thus, in lieu of pointing the centre, observation to the

sun are usually taken with the edges tangential to both the horizontal and vertical cross hairs. It is

usually achieved by allowing sun's trailing edge to move onto the vertical cross hair or the

leading edge is pointed by moving the vertical cross hair forward, until it becomes tangent to the

sun's image.

The field work involved in the determination of azimuth of a line from sun observation consists

of following steps:

1. Carry out temporary adjustment of a theodolite at the observation station with face left

condition.




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2. Open the lower plate main screw and swing the telescope to bisect the reference object. Fix the

lower plate main screw and bisect accurately using the lower plate tangent screw. Note down the

horizontal circle reading.

3. Swing the telescope by opening the upper plate main screw. Bring the image of the sun into the

upper left quadrant of the diaphragm (Figure 27.1a). Close the upper plate main screw and then

bring the vertical hair tangent to right limb of the sun using the upper plate tangent screw. As

soon as the lower limb of the sun makes contact with the horizontal hair, the chronometer time is

recorded. Note down the horizontal (as well as vertical, if altitude of the sun is required to be

observed) circle readings.

4. Next, bring the image of the sun into lower right quadrant using the upper and vertical platetangent screws. The vertical hair is kept in tangent to left limb of the sun using by the upper plate

tangent screw. As soon as the upper limb of the sun makes contact with the horizontal hair, the

chronometer time is recorded (Figure 27.1b). Note down the horizontal (as well as vertical, if

altitude of the sun is required to be observed) circle readings.

5. Open the upper plate main screw and swing back the telescope to the reference object. Close

the upper plate main screw and bisect the reference object. Note down the horizontal circle

reading.

6. Change the face left of the instrument into the face right condition and repeat step 2 with the

instrument in face right condition.

7. Swing the telescope by opening the upper plate main screw. Bring the image of the sun into the

upper right quadrant of the diaphragm (Figure 27.1c). Close the upper plate main screw and then

bring the vertical hair tangent to left limb of the sun using the upper plate tangent screw. As soonas the lower limb of the sun makes contact with the horizontal hair, the chronometer time is

recorded. Note down the horizontal (as well as vertical, if altitude of the sun is required to be

observed) circle readings.

Figure 27.1 (c) Position of the sun during celestial observation




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8. Next, bring the image of the sun into lower left quadrant using the upper and vertical plate

tangent screws. The vertical hair is kept in tangent to right limb of the sun using the upper plate

tangent screw. As soon as the upper limb of the sun makes contact with the horizontal hair, the

chronometer time is recorded (Figure 27.1d ). Note down the horizontal (as well as vertical, if

altitude of the sun is required to be observed) circle readings.

9. Repeat step 5.

Thus, a set of observation is to be taken. It consists of four instants of time and four horizontal

circle readings (and four vertical circle readings). An azimuth of the line AZL should be

computed for each pointing on the sun.

For any field observation, it may consist of one or more sets, but a minimum of three sets is

recommended.

Figure 27.1 (d) Position of the sun during celestial observation

Errors and Correction

The errors involved in determining a line's azimuth can be divided into two categories: (I)

measuring horizontal angles from a line to the sun and (2) errors in determining the sun's azimuth.Except when pointing on the sun, errors in horizontal angles are similar to any other field angle.

Since, the observations are taken at the edges of the sun, a correction for semi-diameter is

required. The correction (C) to be applied to the measured horizontal angles is:

C = ± sec α




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where α is the (correct) altitude of the sun and is the semi-diameter of the sun (can beobtained from star almanac).

The width of a theodolite cross hair is approximately 2 or 3 arc-sec. With practice, the sun's edge,

particularly the trailing one, can be pointed to within this width. In many instances, pointing the

sun may introduce a smaller error than pointing the backsight mark.

Total error in the sun's azimuth is a function of errors in obtaining UTI time, and scaling latitude

and longitude. The magnitude these errors contribute to total error which is a function of the

observer's latitude, declination of the sun, and time from local noon .

Since errors in scaling latitude and longitude are constant for all data sets of an observation, each

computed azimuth of the sun contains a constant error. Errors in time affect azimuth in a similar

manner. Consequently, increasing the number of data sets does not appreciably reduce the sun's

azimuth error. The increase can, however, improve horizontal angle accuracy and therefore have

a desirable effect.

After azimuths of the line have been computed, they are compared and, if found within

acceptable limits, averaged. Azimuths computed with telescope direct and telescope reversed

should be compared independently. Systematic instrument errors and use of an objective lens

filter can cause a significant difference between the two. An equal number of direct and reverse

azimuths should be averaged.

An alternate calculation procedure averages times and angles, or points on opposite edges of the

sun and averages to eliminate semi-diameter corrections. Due to the sun traveling on an apparent

curved path and its changing semi-diameter correction with altitude, this procedure usually

introduces a significant error in azimuth. Also, it does not provide a good check on the final

azimuth.

Determination of the altitude of the sun

The altitude of the sun can be computed from astronomical triangle (Figure 26.3) using the

relation




Page 44 of 74

h = sin-1 (sin φ sin δ + cos φ cos δ cos LHA) --------------------- Equation (27.1)

The altitude of the sun can also be obtained from the observation of the sun by takingthe vertical angle measurement along with horizontal angle. In this case, the

observations need to be updated with the following corrections:

(a) Correction for refraction: Mean value of refraction (r 0) based on observed altitude isavailable from star almanac. A table for modification factor (f), depending on thetemperature and pressure during observation, is also available in star almanac. Thus,

Correction for refraction from star almanac is C r = r o φ. Other wise, α can be computed

using Cr = -58" cos α′. It is always subtractive from the observed altitude.

(b) Correction for parallax: A correction strictly needs to be applied for the Sun's parallax,

which will suffice to be considered as 0.1′ for all altitudes less than 70°. Otherwise, itmay be computed using

Cp = +8.82 cos α′

The correction for parallax is to be added always to the observed altitude.

(c) Correction for semi-diameter

α = α′ ±




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The correction is + for the lower limb and - for the upper limb.

Ex27-1 On September 29, 2003 the sun was observed on the rooftop of Geomatics Engineering section ofI.I.T. Roorkee having geographical location (Latitude 29° 50' 44'.83 and Longitude 77° 54' 00”.39). Theobservation was recorded as follows: Temp 14° C and pressure = 969 millibars.

Horizontal Angle Time

Vernier A Vernier B

Vernier C and DObservationto

Face

Hr ' " ° ' " ° ' " ' "R.O L 99 17 40 17 40

L 10 04 36 59 05 40 05 20 22 37 00 37 20

L 10 05 58 58 50 40 50 40 22 33 20 33 20

R 10 07 49 238 10 20 10 20 22 30 20 30 40

R 10 08 32 237 10 40 10 20 22 25 20 25 40

R.O R 279 18 20 18 20

Find out the azimuth of the line joining the station and reference object.

Figure Example 27.1




Page 46 of 74

Solution : Refer figure 26.3

The latitude (φ) of the station = 29° 51' 44”. 83

The longitude (λ) of the station =77° 54' 00”. 39

The date of observation September 29, 2003.

To determine the azimuth of the line joining the observation station and the reference object, say AZ.

Let as consider observation with face left condition of the instrument.

The time of observation, UTI = 10h 04

m 36

s

Then, GHA = UTI+ R (Exess)

From star almanac, R on September 29 (2003) at 10h 04

m 36

s

= 0h 30

m 21.7

s + 41.3

s = 0

h 31

m 3.01

s

GHA = 0h 31

m 3.01

s + 10

h 04

m 3.01

s

=10h 35

m 39.01

s

Thus, LHA = GHA + λ

= 10h 35

m 39.01

s +77° 54' 00”.39 = 236° 48' 45".54

From star almanac, apparent declination of the sum, δ (at the instant of observation) = S 2° 15' 30" + 3' 54"= S 2° 19' 24"

Now, from equation 26.3

Since the sun was observed in the afternoon, it was observed in western hemisphere.

Thus, azimuth of the sun = 360° - 74° 09' 54”.34




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= 285° 50' 05”.65

From, the observation of vertical angle, the apparent altitude (α' )of the sun = 22° 37' 10”

Parallax corection = + 8.8" cos α'

= + 8.8" cos 22° 37' 10" = + 8".123

Semi diameter = + = + 16'.0 (from star almanac)

The mean refraction error (r o) = 139"

As temperature and pressure during observation are 14° C and 969 millibar, the factor for refractioncorrection, f = 0.95 (From star almanac).

Thus refraction correction = - 139" x 0.95

= - 132".05 = -2' 12"

Therefore, The altitude of the sun

= 22° 37' 10" + 0° 16'.0 + 0° 0' 8".123 - 0° 2' 12"

= 22° 51' 06".123

Thus, horizontal angle of the sun = observed horizontal angle of the sun - Semi diameter correction

= 59° 05' 40" - sec α

= 59° 05' 40" - 17' 21".4589 = 58° 48' 18".54

Therefore, The horizontal angle between the sun and the R.O.

HA = 99° 17' 40" - 58° 48' 10".54 = 40° 29' 29".46 (Left)

The azimuth of the line joining the station and the reference object = Azimuth of the sun + HA

= 285° 50' 05".65 + 40° 29' 29".46 = 326° 19' 35".11

Ex.27-1 At a place having geographic location (Latitude (θ) = 37° 57' 23" N and Longitude (λ) = 91° 46' 35"

W) the polaris has been observed to find the azimuth of a R.O (Reference object) as follows:




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Time Horizonatal AnglePoints Face

Hr ' " ° ' " Set

R.O. L 0 0 08.2

Remarks

Polaris L 2 16 21.5 94 35 02.3 GHA = 163° 05' 46".1

Polaris R 2 19 36.1 274 35 18.5 24h = 164° 04' 58".6I

R.O R 180 00 03.4

R.O L 60 03 20.7 Declination at

P L 2 28 10.0 154 39 11.9 0h = 89° 12' 14".98

P R 2 30 14.4 334 39 24.7 24h = 89° 12' 14".63

II

R.O R 240 03 16.4R.O L 120 06 44.5

P L 2 41 57.6 214 43 52.2

P R 2 44 18.9 394 44 08.1III

R.O R 300 06 39.6

Compute the azimuth of the R.O

ANS

Direct Reverse

Set ° ' " ° ' " AverageI 264 26 55 264 26 48.7

II 264 26 54.7 264 26 48.3

III 264 26 55 264 26 48.7

264° 26' 51".4

Introduction

• The photogrammetry has been derived from three Greek words:

o Photos: means light

o Gramma: means something drawn or written

o Metron: means to measure

• This definition, over the years, has been enhanced to include interpretation as well as

measurement with photographs

Definition

The art, science, and technology of obtaining reliable information about physical objects and the

environment through process of recording, measuring, and interpreting photographic images and

patterns of recorded radiant electromagnetic energy and phenomenon (American Society of

Photogrammetry, Slama).




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• Originally photogrammetry was considered as the science of analysing only photographs.

• But now it also includes analysis of other records as well, such as radiated acoustical

energy patterns and magnetic phenomenon.

Definition of photogrammetry includes two areas:

1) Metric

It involves making precise measurements from photos and other information source to determine,

in general, relative location of points. Most common application: preparation of plannimetric and

topographic maps.

Interpretative:

It involves recognition and identification of objects and judging their significance through careful

and systematic analysis. It includes photographic interpretation which is the study of

photographic images. It also includes interpretation of images acquired in Remote Sensing using

photographic images, MSS, Infrared, TIR, SLAR etc.

Definitions

Aerial Photogrammetry

Photographs of terrain in an area are taken by a precision photogrammetric camera mounted in an

aircraft flying over an area

Terrestrial PhotogrammetryPhotographs of terrain in an area are taken from fixed and usually known position or near the

ground and with the camera axis horizontal or nearly so.

Photo-interpretation

Aerial/terrestrial photographs are used to evaluate, analyse, and classify and interpret images of

objects which can be seen on the photographs

Applications of photogrammetry

Photogrammetry has been used in several areas. The following description give an overview of

various applications areas of photogrammetry (Rampal, 1982)

(1) Geology:

Structural geology, investigation of water resources, analysis of thermal patterns on earth'ssurface, geomorphological studies including investigations of shore features

• engineering geology

• stratigraphics studies

• general geologic applications

• study of luminescence phenomenon

• recording and analysis of catastrophic events

• earthquakes, floods, and eruption.






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Following names are popularly used to indicate type of sensor system used in recording imagery.

• Radargrammetry: Radar sensor

• X-ray photogrammetry: X-ray sensor

• Hologrammetry: Holographs

• Cine photogrammetry: motion pictures

• Infrared or colour photogrammetry: infrared or colour photographs

On the basis of principle of recreating geometry

When single photographs are used with the stereoscopic effect, if any, it is called monoscopic

photogrammetry. If two overlapping photographs are used to generate three dimensional view tocreate relief model, it is called stereophotogrammetry. It is the most popular and widely used

form of photogrammetry

On the basis of procedure involved for reducing the data from photographs

Three types of photogrammetry are possible under this classification

Instrumental or analogue photogrammetry

It involves photogrammetric instruments to carry out tasks.

Semi-analytical or analyticalAnalytical photogrammetry solves problems by establishing mathematical relationship between

coordinates on photographic image and real world objects. Semi-analytical approach is hybrid

approach using instrumental as well analytical principles

Digital Photogrammetry or softcopy photogrammetry

It uses digital image processing principle and analytical photogrammetry tools to carry out

photogrammetric operation on digital imagery

On the basis of platforms on which the sensor is mounted

If the sensing system is spaceborne, it is called space photogrammetry, satellite photogrammetry

or extra-terrestrial photogrammetry

Out of various types of the photogrammetry, the most commonly used forms are

stereophotogrammetry utilizing a pair of vertical aerial photographs (stereopair) or terrestrial

photogrammetry using a terrestrial stereopair.

Classification of Photographs

The following paragraphs give details of classification of photographs used in different

applications

) On the basis of the alignment of optical axis

A) Vertical : If optical axis of the camera is held in a vertical or nearly vertical position.




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(b) Tilted : An unintentional and unavoidable inclination of the optical axis from vertical

produces a tilted photograph.

(c) Oblique : Photograph taken with the optical axis intentionally inclined to the vertical.

Following are different types of oblique photographs

(i) High oblique: Oblique which contains the apparent horizon of the earth.

(ii) Low oblique: Apparent horizon does not appear.

(iii) Trimetrogon: Combination of a vertical and two oblique photographs in which the central

photo is vertical and side ones are oblique. Mainly used for reconnaissance.

(iv) Convergent: A pair of low obliques taken in sequence along a flight line in such a manner

that both the photographs cover essentially the same area with their axes tilted at a fixedinclination from the vertical in opposite directions in the direction of flight line so that the

forward exposure of the first station forms a stereo-pair with the backward exposure of the next

station.

1.Comparison of photographs

2). On the basis of the scale

(a) Small scale - 1 : 30000 to 1 : 250000, used for rigorous mapping of undeveloped terrain and

reconnaissance of vast areas.

(b) Medium scale - 1 : 5000 to 1 : 30000, used for reconnaissance, preliminary survey and

intelligence purpose.

(c) Large scale - 1 : 1000 to 1 : 5000, used for engineering survey, exploring mines.

3). On the basis of angle of coverageThe angle of coverage is defined as the angle, the diagonal of the negative format subtends at the

real node of the lens of the apex angle of the cone of rays passing through the front nodal point of

the lens

Name Coverage angle Format size(cm)

Focal length(cm)

Standard or normal angle 60o (i) 18

(ii) 23(i) 21(ii) 30

Wide angle 90o (i) 18

(ii) 23(i) 11.5(ii) 15

Super wide or ultra wide angle 120o (i) 18

(ii) 23(i) 7

(ii) 8.8

Type of photo Vertical Low oblique High oblique

Characteristics Tilt < 3o Horizon does not appear Horizon appears

Coverage Least Less Greatest

Area Rectangular Trapezoidal Trapezoidal

Scale Uniform if flat Decreases from foreground

to background

Decreases from foreground

to backgroundDifference with map Least Less Greatest

Advantage Easiest to map - Economical and illustrative




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Narrow angle < 60o

Information recorded on photographs

The following information is recorded on a typical aerial photograph

1. Fiducial marks for determination of principal points.

2. Altimeter recording to find flying height at the moment of exposure.

3. Watch recording giving the time of exposure.

4. Level bubble recording indicating tilt of camera axis.5. Principal distance for determining the scale of photograph.

6. Number of photograph, the strip and specification no. for easy handling and indexing.

7. Number of camera to obtain camera calibration report.

Date of photograph

Figure 1: Typical aerial photograph

Types of projections




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1. Parallel : The projecting rays are parallel.

2. Orthogonal : Projecting rays are perpendicular to plane of projection. This is a special

case of parallel projection. Maps are orthogonal projection. The advantage of this

projection is that the distances, angles, and areas in plane are independent of elevation

differences of objects.

Central : Central projection is the starting point for all photogrammetry. In this projection rays

pass through a point called the projection center or perspective center. The image projected by a

lens system is treated as central projection although in strictest senses it is not so.

Introductory definitions for photographs

Vertical photograph A photograph taken with the optical axis coinciding with direction of gravity.

Tilted or near vertical

Photograph taken with optical axis unintentionally tilted from vertical by a small amount (usually

< 3°)

Focal length (f)

Distance from front nodal point to the plane of the photograph (from near nodal point to image

plane).

Exposure station (point L)

Position of frontal nodal point at the instant of exposure (L)

Flying height (H)

Elevation of exposure station above sea level or above selected datum

Figure 3: Aerial photographs showing various elements as defined(a) Elements of vertical photograph (b) Section of imaging geometry showing various elements




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X-axis of photo

Line on photo between opposite collimation marks, which most nearly parallels the flight

direction

Y-axis

Line normal to x-axis and join opposite collimation marks

Principal point (o)

The point where the perpendicular dropped from the front nodal point strikes the photograph or

the point in which camera axis pierces the image plane.

Camera axis

It is a ray of light incident at front nodal point in the object space and at right angles to the image

plane.

Fiducial marks or collimation marks

Index marks usually four in number, rigidly connected with the camera lens through the camera

body and forming images on the photographs to which the position on the photograph can be

referred

Photographs center

The geometrical center of the photograph as defined by the intersection of the lines joining the

fiducial marks.

Format

It is the planar dimension of photograph (9" x 9", 7" x 7", 23 cm x 23 cm, 18 cm x 18 cm, 15 cm

x 15 cm).

Photogram

Photograph taken with a photogrammetric camera having fixed distance between negative plane

and lens and equipped with fiducial or collimating marks. For photograms the bundle of rays on

the object side at the moment of exposure can be reproduced. To achieve this the following data

known as the elements of interior orientation must be known:

• Calibrated focal length

• Lens distortion data

Location of the principal point with reference to the photograph center (normally these two

coincide)

Hence, a photogram is a photograph with known interior orientation

Difference between near vertical photographs and map

1. Production : Quickest possible and most economical method of obtaining informationabout areas of interest. Boon for difficult areas. Enlarging and reducing easier in case of

photographs than maps.




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2. Content : Map gives an abstract representation of surface with a selection from nearly

infinite number of features on ground. Photograph shows images of surface itself. Maps

often represent non-visible phenomenon (like text) This may make interpretation difficult

for photograph. Special films like color and infrared films can bring about special

features of terrain.

3. Metric accuracy: Map is geometrically correct representation, photos are generally not.

Maps are orthogonal projections, photo is central projection. Map has same scale

throughout photo has variable scale. Bearing on photographs may not be true.

Training requirement: A little training and familiarity with the particular legend used in the map

enables proper use of map. Photo-interpretation requires special training although initially it may

appear quite simple as it gives a faithful representation of ground.

Stereoscopic vision

• Method of judging depth may be classified as stereoscopic or monoscopic.

• Persons with normal vision are said to have binocular vision. They can view with both

eyes simultaneously. These persons are said to have stereoscopic viewing

The term monocular vision is applied to viewing with one eye only.

Depth perception

• Monoscopic viewing provides only rough depth impression which is based on the

following clues:

o Relative size of objects

o Hidden objects

o Shadows

o Differences in focussing of eye required for viewing objects at varying distances.

• For stereoscopic depth perception usually two clues are involved

o Double image phenomenon.

o Relative convergence of optical axis of two eyes.

• The human eye functions in a similar manner to a camera. The lens of the eye is biconvex

in shape and is composed of refractive transparent medium. The separation between eyes

is fixed (called eye base). Therefore, in order to satisfy the lens formula for varyingobject distance, the focal length of eye lens changes. For example, when a distant object

is seen, the lens muscle relax, causing the spherical surface of the lens to become flatter.

This increases the focal length to satisfy the lens formula and accommodate the long

object distance. When close objects are viewed, a reverse phenomenon happens. The

eye's ability to focus varying object distance is called accommodation.

• The figure shows two situations where eyes separated by eye base (b) are focussing on

two objects A and B located at two distances d A and d B respectively.

• In binocular vision, there is convergence of axes of eyes when focusing at point A and B.

The corresponding angles are φ1 and φ2. Angle φ1 tells the mind that object A is distance

d A. Similarly for point B. These angle are called parallactic angles for points. The nearer




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the object, the greater the parallactic angle and vice versa. Difference ( φ1 - φ2 ) tells the

mind that the distance (depth) between two points is e = d B - d A.

For average separation of eye and distinct vision of about 10 inch, the limiting upper value of Φ ≈ 16°. Lower limiting value of φ ranges from 10 to 20 seconds and represents a distance ofabout 1700 to 1500 ft for average eye separation. It is called stereoscopic acuity of the person.

Figure 1: Binocu lar vision (Wolf and Dewitt, 2000)

To understand the stereoscopic vision by viewing photographs, let us replace the eyes by two

cameras and take photographs as shown in figure 2 assuming that the photographic plates are

between the objects and lenses. If these photographs could be placed in the same position and

seen with both eyes, one would get the same depth. Since it is not possible to keep eyes at such a

wide separation (called air base B), a scaled version is shown in figure (ii) where camera position

are replaced by a scaled down geometrical arrangement (eyes separated by eye base-b e ), the two

images are fused and the brain perceives the scaled down depth between objects.




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Figure 2: Stereoscopic vision (i) object imaged with cameras (ii) scaled down version with eyes

Stereoscope

• If two photographs taken from two exposure stations L1 and L2 are laid on table so thatleft photograph is seen by left eye and the right photo is seen with right eye, a 3D-model

is obtained. However, viewing in this arrangement is quite difficult due to following

reasons:

o Eye strain and focusing difficulty due to close range.

o There is disparity in viewing. The eyes are focused at short range on photos lying

on table where as the brain perceives parallactic angles which tends to form the

stereoscopic model at some depth below the table.

By using stereoscope, these problems can be alleviated. Different types of stereoscopes are

available for different purposes - from pocket (inexpensive) for viewing small area to mirror

(expensive) for viewing larger area.

Figure 3: Photograph showing mirror stereoscope (Wolf and Dewitt, 2000)




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Figure 4: Princip le of stereoscope (Wolf and Dewitt, 2000) (a) pocket (b) mirror

Conditions of good stereo viewing

1. The two images must be obtained from two different positions of the camera.

2. The left photo and right photo should be presented to the left and right eye in the same

order.

3. Photographs of the object should be obtained from nearly the same distance, i.e. scales of

the two adjacent photographs should be nearly same. Normal eye can accommodate 5 to

15% variations in scale of photograph.4. During stereovision, keep the area under vision in near vertical epipolar planes as far as

possible. The plane containing optical centers for eyes (S1 S 2), object point A and

corresponding images a and a' on left and right photographs respectively is called the

epipolar plane. Thus camera axes should be approximately in one plane, though the eyes

can accommodate the differences to a limited degree. The term epipolar planes is used if

it refers to eyes; the term basal planes is used if it refers to cameras.

5. For a good stereovision, the pair of photos should be in the same orientation with eachother as they were at the time of photography.

6. The brightness of both the photographs should be similar.




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The ratio B/H should be appropriate. If this ratio is too small say smaller than 0.02, we can obtain

the fusion of two pictures, but the depth impression will not be stronger than if only one

photograph was used. The ideal value of B/H is not known, but is probably not far from 0.25. In

photogrammetry values up to 2.0 are used.

Geometry of overlapping vertical photos

• Figure 5 shows two photographs taken from two exposure stations L 1 (left) and L 2 (right). Ground point A appears in these two photos at a and a' in left and right photos.

The image coordinates of these photo points a and a' are (xa, ya ) and (x' a, y'a )respectively.

A pair of vertical photographs which have some common area imaged is known as a stereopair.

This stereopair can be used for estimation of depth by measurement of parallax which will

explained next.

Figure 5: Geometry of overlapping vertical photos

Figure 6: A stereopair

Parallax




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• It is the central concept to the geometry of overlapping photographs and is defined as the

apparent shift in the position of a body with respect to a reference point caused by a shift

in the point of observation

In photogrammetry, it refers to the relative difference in position of an image point that appears in

each of the overlapping photo

Absolute Parallax: Absolute X parallax (P X ) is given by (x l - x r ), where x l and x r are x coordinates on left and

right photographs respectively

P x = x l - x r

Other names:

X-parallax, horizontal parallax, linear parallax, absolute stereo parallax, or just parallax

For the following photograph pair, the parallax is given as follows.

P x = x b - (-x b' )

It can be noted that the value of parallax has been used with sign

Figure 7: Parallax in photograph pair

Parallax and its relation to height

From figure 5, the absolute parallax PA at point A is given as follows where B is air base.

Similarly for another point B, we can write




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Height differences between two points ( dh BA = h B - h A ) is given by:

dP BA = P B - P A is the difference in absolute parallaxes between two points. Above equation can bemodified to give

• where dp BA = p B - p A is the difference in parallax bar readings between two points. Thus theabove equation says that dPBA = dpBA under the assumption that flying height for both photographsis same (i.e. H1 = H2 = H) and photograph is truly vertical.

• It may, however, be noted that for a given point, there may be parallax in both directions

X and Y. The Y-parallax is caused due to the following reasons:

o Unequal flying height

o Photographic tilt

o Misalignment of flight line

o Misalignment of stereoscope

Great difference in parallax between adjacent images (in highly mountainous/rugged terrain)

Other forms of parallax equation

If the principal point are assumed to have same elevation as known point A, then the parallax of

principal point is the same as the parallax of point A i.e. P A = b m , the mean principal base. Using

this assumption

dp = absolute parallax of unknown point - absolute parallax of known point

If dp is small then

Some other forms of parallax equation are given in adjacent formats




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Sources of errors in using parallax equation

• Locating and marking the flight line on photo.

• Orientation of stereopair for parallax measurement.

• Parallax and photo co-ordinate measurements.

• Shrinkage and expansion of photographs.

• Unequal flying height.

• Tilted photo.

• Error in ground control.

Camera lens distortion, atmospheric refraction etc

Measurement with Parallax bar and floating marks

• From the parallax equation, we note that

o parallax of any point is directly related to its elevation

o

parallax is greater for point which are higher in elevation• If we could measure the parallax of the point, its height could be measured easily.

Unfortunately there is no device for this. However, difference in absolute parallaxes

which can be related to height differences. Therefore, if the elevation of one point is

known, the elevation of other point can be measured.

Parallax bar (figure 8) or stereometer is used to measure the difference in parallaxes between

corresponding points on stereopair which have been oriented as they were at the time of

photography






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• If the mark floats too far above the surface then stereofusion will be broken because the

eyes can not accommodate the range in convergence, and a double image of the terrain

will result

• If the dots are moved slightly apart the floating mark will split into two separate dots as

the brain of the observer can not imagine seeing the floating mark beneath the surface

Again if the dots are moved too far apart a double image of the terrain will result.

Figure 9: Floating mark

Base lining

• In order to get fused 3D images, the orientation relationship between the two photographs

and stereoscopic lenses should be same as that between the photographed ground and the

camera lenses at the time of photography. Base lining is the process to achieve such

orientation. The step by step procedure to achieve base lining is as follows:

o Find out the photographic center of the first photograph by joining the opposite

fiducial marks and getting the intersecting lines. Let it be A. Mark a

corresponding point A' on the other photo of the stereopair.o Find the photograph center of the second photo B and locate corresponding point

B' on the first photograph of the stereopair.

o Join A, B' to get AB' and B and A' to get BA'.

o Fix a white sheet (called the base sheet) on table and draw a straight line on this

sheet in the middle.

o Set up stereoscope on this sheet such that the line is midway between stereoscope

legs.

o Put the first photograph in such a way that the line AB' coincides with the line on

the base sheet.




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o Now by trial and error, and viewing through the stereoscope, put the second

photograph (right photograph) in such a way that the image of point A coincides

with A' and B coincides with B' i.e line AB' coincides with A'B.

o This arrangement will give 3D view of stereopair.

o Now with the help of stereometer, the floating mark of left plate is put over a

well defined point on the left photograph. Now by giving lateral and longitudinal

motion to the other plate (with the help of stereometer drum), we try to bring the

floating mark of other glass plate on the corresponding point of second

photograph. When the images of left and right floating marks are fused, the

reading on the parallax bar can be recorded as the parallax bar reading for that

point.

• Operations with EDMI Measurement with EDMI involves four basic steps:(a)Setup

(b)Aim

(c)Measure

(d) Record

• Setting up: The instrument is centered over a station by means of tribrach or by mounting

over a compatible theodolite. Reflector prisms are set over the remote station either on

tribrach or on a prism pole. Observations related to height or instrument and prism arerecorded. These are usually kept the same to avoid any additional corrections.

• Aiming: The instrument is aimed at prisms by using sighting devices or theodolite

telescope. Slow motion screws are used to intersect the prism centre. Some kind of

electronic sound or beeping signal helps the user to indicate the status of centering.

• Measurement: The operator presses the measure button to record the slope distance whichis displayed on LCD panel.

Recording: The information on LCD panel can be recored manually or automatically. All

meteorological parameters are also recorded.

Error sources in EDMI

The fundamental distance measured by EDMI can be put into the following generalized form:

M integerambiguity

λ wavelengthofmodulationwave

Φ measuredphasedifference

n g grouprefractiveindex

V 0 velocityofEMRinvacuum

K 1 scaleerror

K 2 zeroerror

K 3 cyclic error




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Measurement with EDMI has the following error sources which have to be accounted for while

reporting the distance (Kennie and Petrie, 1990):

i)Instrumentoperationerrors

One has to be careful for

• precise centering at the master and slave station

• pointing/sighting of reflector

• entry of correct values of prevailing atmospheric conditions

(ii)AtmosphericerrorsMeteorological conditions (temperature, pressure, humidity, etc.) have to betaken into account to correct for the systematic error arising due to this. These errors can be

removed by applying an appropriate atmospheric correction model that takes care of different

meteorological parameters from the standard (nominal) one.

(iii)Instrumenterror:

Consists of three components - scale error, zero error and cyclic error. These are systematic in

nature.

Control

A control network is the framework of survey stations whose coordinates have been precisely

determined and are often considered definitive. The stations are the reference monuments, to

which other survey work of a lesser quality is related. By its nature, a control survey needs to be

precise, complete and reliable and it must be possible to show that these qualities have been

achieved. This is done by using equipment of proven precision, with methods that satisfy the

principles and data processing that not only computes the correct values but gives numerical

measures of their precision and reliability.

Since care needs to be taken over the provision of control, then it must be planned to ensure that it

achieves the numerically stated objectives of precision and reliability. It must also be complete as

it will be needed for all related and dependent survey work. Other survey works that may use the

control will usually be less precise but of greater quantity. Examples are setting out for

earthworks on a construction site, detail surveys of a greenfield site or of an as-built development

and monitoring many points on a structure suspected of undergoing deformation.




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The practice of using a control framework as a basis for further survey operations is often called

‘working from the whole to the part’. If it becomes necessary to work outside the control

framework then it must be extended to cover the increased area of operations. Failure to do so

will degrade the accuracy of later survey work even if the quality of survey observations is

maintained.

For operations other than setting out, it is not strictly necessary to observe the control before other

survey work. The observations may be concurrent or even consecutive. However, the control

survey must be fully computed before any other work is made to depend upon it.

BASIC MEASUREMENTS

Surveying is concerned with the fixing of position whether it be control points or points of

topographic detail and, as such, requires some form of reference system.

The physical surface of the Earth, on which the actual survey measurements are carried out, is not

mathematically definable. It cannot therefore be used as a reference datum on which to compute

position. Alternatively, consider a level surface at all points normal to the direction of gravity.

Such a surface would be closed and could be formed to fit the mean position of the oceans,

assuming them to be free from all external forces, such as tides, currents, winds, etc. This surface

is called the geoid and is defined as the equipotential surface that most closely approximates to

mean sea level in the open oceans.An equipotential surface is one from which it would require the

same amount of work to move a given mass to infinity no matter from which point on the surface

one started. Equipotential surfaces are surfaces of equal potential; they are not surfaces of equal

gravity. The most significant aspect of an equipotential surface going through an observer is that

survey instruments are set up relative to it. That is, their vertical axes are in the direction of the

force of gravity at that point. A level or equipotential surface through a point is normal, i.e. at

right angles, to the direction of gravity. Indeed, the points surveyed on the physical surface of the

Earth are frequently reduced, initially, to their equivalent position on the geoid by projection

along their gravity vectors.

The reduced level or elevation of a point is its height above or below the geoid as measured in the

direction of its gravity vector, or plumb line, and is most commonly referred to as its height above

or below mean sea level (MSL). This assumes that the geoid passes through local MSL, which is

acceptable for most practical purposes. However, due to variations in the mass distribution within




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the Earth, the geoid, which although very smooth is still an irregular surface and so cannot be

used to locate position mathematically.

The simplest mathematically definable figure which fits the shape of the geoid best is an ellipsoid

formed by rotating an ellipse about its minor axis. Where this shape is used by a country as the

surface for its mapping system, it is termed the reference ellipsoid. Figure 1.1 illustrates the

relationship between these surfaces.

The majority of engineering surveys are carried out in areas of limited extent, in which case the

reference surface may be taken as a tangent plane to the geoid and the principles of plane

surveying applied. In other words, the curvature of the Earth is ignored and all points on the

physical surface are orthogonally projected onto a flat plane as illustrated in Figure 1.2. For areas

less than 10 km square the assumption of a flat Earth is perfectly acceptable when one considers

that in a triangle of approximately 200km2, the difference between the sum of the spherical

angles and the plane angles would be 1 second of arc, or that the difference in length

of an arc of approximately 20 km on the Earth’s surface and its equivalent chord length is a mere

8 mm.

The above assumptions of a flat Earth, while acceptable for some positional applications, are not

acceptable for finding elevations, as the geoid deviates from the tangent plane by about 80 mm at

1 km or 8 m at 10 km from the point of contact. Elevations are therefore referred to the geoid, at

least theoretically, but usually to MSL practically.

Fig. 1.3 Basic measurements

to several kilometres with millimetre precision, depending on the distance measured; lasers and

northseeking gyroscopes are virtually standard equipment for tunnel surveys; orbiting satellites

are being used for position fixing offshore as well as on; continued improvement in aerial and

terrestrial photogrammetric and scanning equipment makes mass data capture technology an

invaluable surveying tool; finally, data loggers and computers enable the most sophisticated

procedures to be adopted in the processing and automatic plotting of field data.




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CONTROL NETWORKS

The establishment of two- or three-dimensional control networks is the most fundamental

operation in the surveying of large or small areas of land. Control networks comprise a series of

points or positions which are spatially located for the purpose of topographic surveying, for the

control of supplementary points, or dimensional control on site.

The process involved in carrying out the surveying of an area, the capture and processing of the

field data, and the subsequent production of a plan or map, will now be outlined briefly.The first and obvious step is to know the purpose and nature of the project for which the surveys

are required in order to assess the accuracy specifications, the type of equipment required and the

surveying processes involved.

For example, a major construction project may require structures, etc., to be set out to

subcentimetre accuracy, in which case the control surveys will be required to an even greater

accuracy. Earthwork volumes may be estimated from the final plans, hence contours made need

to be plotted at 2 m intervals or less.

If a plan scale of 1/500 is adopted, then a plotting accuracy of 0.5 mm would represent 0.25 m onthe ground, thus indicating the accuracy of the final process of topographic surveying from the

supplementary control and implying major control to a greater accuracy. The location of

topographic data may be done using total stations, GPS satellites, or, depending on the extent of

the area, aerial photogrammetry. The cost of a photogrammetric survey is closely linked to the

contour interval required and the extent of the area. Thus, the accuracy of the control network

would define the quality of the equipment and the number of observations required.

The duration of the project will affect the design of survey stations required for the control points.

A project of limited duration may only require a long, stout wooden peg, driven well into solid,reliable ground and surrounded by a small amount of concrete. A fine nail in the top defines the

geometrical position to be located.

The next stage of the process is a detailed reconnaissance of the area in order to establish the best

positions for the control points.

Initially, data from all possible sources should be studied before venturing into the field. Such

data would comprise existing maps and plans, aerial photographs and any previous surveying data

of that area.






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Fig. 1.5 A network Figure 1.5 illustrates possible lines of sight. All the horizontal angles shown would be measured

to the required accuracy to give the shape of the network. At least one side would need to be

measured, say AB, to give the scale or size of the network. By measuring a check baseline, say

ED, and comparing it with its value, computed through the network, the scale error could be

assessed. This form of survey is classical triangulation and although forming the basis of the

national maps of many countries, is now regarded as obsolete because of the need for lines of

sight between adjacent points. Such control would now be done with GPS.

If the lengths of all the sides were measured in the same triangular configuration without the

angles, this technique would be called ‘trilateration’. Although giving excellent control over scale

error, swing errors may occur. For local precise control surveys the modern practice therefore is

to use a combination of angles and distance. Measuring every angle and every distance, including

check sights wherever possible, would give a very strong network indeed. Using sophisticated

least squares software it is now possible to optimize the observation process to achieve the

accuracies required.

Probably the most favoured simple method of locating the relative coordinate positions of control

points in engineering and construction is traversing. Figure 1.6 illustrates the method of

traversing to locate the same control points A to F . All the adjacent horizontal angles and

distances are measured to the accuracies required, resulting in much less data needed to obtain

coordinate accuracies comparable with the previous methods. Also illustrated are minor or

supplementary control points a, b, c, d , located with lesser accuracy by means of a link traverse.

The field data comprises all the angles as shown plus the horizontal distances Aa, ab, bc, cd , and

dB. The rectangular coordinates of all these points would, of course, be relative to the major

control.




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Whilst the methods illustrated above would largely supply a two-dimensional coordinate position,

GPS satellites could be used to provide a three-dimensional position.

All these methods, including the computational processes, are dealt with in later chapters of

this book.

Fig. 1.6 Major traverse A to F. Minor link traverse A to B

LOCATING POSITION

Establishing control networks and their subsequent computation leads to an implied rectangular

coordinate system over the area surveyed. The minor control points in particular can now be used

to position topographic data and control the setting out of a construction design.

The methods of topographic survey and dimensional control will most probably be:

(a) by polar coordinates (distance and bearing) using a total station; or

(b) by GPS using kinematic methods.

Considering method (a), a total station would be set up over a control point whose coordinates are

known as ‘a’, and back-sighted to another control point ‘b’ whose coordinates are also known.

Depending on the software on board, the coordinates may be keyed into the total station.

Alternatively, the bearing of the line ‘ab’, computed from the coordinates, may be keyed in.

Assuming the topographic position of a road is required, the total station would be sighted to a

corner cube prism fixed to a detail pole held vertically at ( P 1), the edge of the road as shown in




Figures 1.7 and 1.8. The field data would comprise the horizontal angle baP 1(á1) and the

horizontal distance D1 ( Figure 1.8). Depending on the software being used, the angle á1 would

be used to compute the bearing aP 1 relative to ‘ab’ and, with the horizontal distance D1, compute

the rectangular coordinates of P 1 in the established coordinate system. This process is repeated

for the remaining points defining the road edge and any other topographic data within range

of the total station. The whole area would be surveyed in this way by occupying pairs of control

points situated throughout the area.

A further development is the integration of a total station with GPS. This instrument, produced by

Leica Geo systems and called Smart Station, provides the advantages of both the systems (a) and

(b). If using existing control, the local coordinate data is copied into the Smart Station and used in

the usual manner on existing pairs of control points. Where the GPS cannot be used because of

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Advanced Survey