SKP Engineering College - civil.skpec.edu.incivil.skpec.edu.in/wp-content/uploads/sites/4/2017/11/IV-Surveying... · CE6404 SURVEYING II L T P C 3 0 0 3 OBJECTIVES: • This subject

SKP Engineering College

Tiruvannamalai – 606611

A Course Material

on

Surveying-II

By

R.Muralidaran

Assistant Professor

Civil Department

Quality Certificate

This is to Certify that the Electronic Study Material

Subject Code:CE6404

Subject Name:Surveying II

Year/Sem:II/IV

Being prepared by me and it meets the knowledge requirement of the University

curriculum.

Signature of the Author

Name: R.Muralidaran

Designation: Assistant Professor

This is to certify that the course material being prepared by Mr.D.Murali is of the

adequate quality. He has referred more than five books and one among them is from

abroad author.

Signature of HD Signature of the Principal

Name: A.Saravanan Name: Dr.V.Subramania Bharathi

Seal: Seal:

S.K.P.Engineering College,Tiruvannamalai IV Sem

Civil Department Surveying-II

CE6404 SURVEYING II L T P C

3 0 0 3

OBJECTIVES:

• This subject deals with geodetic measurements and Control Survey methodology and

its adjustments. The student is also exposed to the Modern Surveying.

UNIT I CONTROL SURVEYING 9

Horizontal and vertical control - Methods - specifications - triangulation- baseline - instruments and

accessories - corrections - satellite stations - reduction to centre- trigonometrical levelling - single

and reciprocal observations - traversing - Gale's table.

UNIT II SURVEY ADJUSTMENT 9

Errors Sources- precautions and corrections - classification of errors - true and most probable

values- weighed observations - method of equal shifts -principle of least squares -0 normal

equation - correlates- level nets- adjustment of simple triangulation networks.

UNIT III TOTAL STATION SURVEYING 9

Basic Principle - Classifications -Electro-optical system: Measuring principle, Working

principle, Sources of Error, Infrared and Laser Total Station instruments. Microwave system:

Measuring principle, working principle, Sources of Error, Microwave Total Station instruments.

Comparis on between Electro-optical and Microwave system. Care and maintenance of Total

Station instruments. Modern positioning systems - Traversing and Trilateration.

UNIT IV GPS SURVEYING 9

Basic Concepts - Different segments - space, control and user segments - satellite configuration -

signal structure - Orbit determination and representation - Anti Spoofing and Selective Availability

- Task of control segment - Hand Held and Geodetic receivers -data processing - Traversing and

triangulation.

UNIT V ADVANCED TOPICS IN SURVEYING 9

Route Surveying - Reconnaissance - Route surveys for highways, railways and waterways -

Simple curves - Compound and reverse curves - Setting out Methods - Transition curves -



Functions and requirements - Setting out by offsets and angles - Vertical curves - Sight distances-

hydrographic surveying - Tides - MSL - Sounding methods - Three-point problem - Strength of fix

- Sextants and station pointer- Astronomical Surveying - field observations and determination of

Azimuth by altitude and hour angle methods - fundamentals of Photogrammetry and

Remote Sensing

OUTCOMES:

On completion of this course students shall be able to

TOTAL: 45 PERIODS

• Understand the advantages of electronic surveying over conventional surveying methods

• Understand the working principle of GPS, its components, signal structure, and error

sources

• Understand various GPS surveying methods and processing techniques used in GPS

• observations

TEXT BOOKS:

1. James M. Anderson and Edward M. Mikhail, "Surveying, Theory and Practice",

7th Edition, McGraw Hill, 2001.

2. Bannister and S. Raymond, "Surveying", 7th Edition, Longman 2004.

3. Laurila, S.H. "Electronic Surveying in Practice", John Wiley and Sons Inc, 1993

REFERENCES:

1. Alfred Leick, "GPS satellite surveying", John Wiley & Sons Inc., 3rd

Edition, 2004.

2. Guocheng Xu, " GPS Theory, Algorithms and Applications", Springer - Berlin, 2003.

3. Satheesh Gopi, rasathishkumar, N. madhu, "Advanced Surveying, Total Station GPS and

Remote Sensing" Pearson education , 2007



CONTENTS

S.No Particulars Page

2 Unit – II 31

5 Unit – V 96

4 Unit – IV 71

3 Unit – III 58

1 Unit – I 01

S.K.P.Engineering College IV Sem

1


UNIT 1 - CONTROL SURVEYING

PART A

1. What is the main principle involved in triangulation? (CO1-L1)(AUC Nov/Dec 2010)

The principle of triangulation is the accurate measurement of one side and two adjacent

angles of a triangle, the length of other two sides can be calculated accurately.

2. Briefly write on the Effect of curvature of earth. (CO1-L1) (AUC Nov/Dec 2010)

The effect of curvature of the earth is determining the altitude of a target. Distant targets

which are close to the ground cannot be seen by radar because they will be below the horizon.

The height of a distant target that is above the horizon will be underestimated if the curvature of

the earth is not taken into account.

3. What is meant by phase of a signal? (CO1-L1) (AUC Apr/May 2010)

When a cylindrical signal is partly illuminated and partly in shade, the observer sees

only the illuminated portion and bisects it. The error of bisection thus introduced is called phase.

It is apparent displacement of the centre of the signal. 4. What do you understand by eccentricity of signal? (CO1-L1) (AUC Apr/May 2010)

Sometimes it is impossible to set up the instrument exactly over or under the signal

which has been observed from the station points. Such a situation arises in traverse surveying

and in triangulation surveying. In triangulation survey this happens when a leaning beacon or

eccentric signal is observed. In both cases the instrument is set up near the signal at a satellite

station or eccentric station or false station and the angles are observed. The observed angles at

each point are reduced to the centre.

5. What is the object of geodetic surveying? (CO1-L1) (AUC Apr/May 2011)

) To provide the most accurate system of horizontal control points on which the less

precise triangles may be based.

) To form a frame work to which cadastral, topographical, hydrographical, engineering and

other surveys may be referred to. ) To assist in the determination of the size and shape of the earth by making observations

for latitude, longitude and gravity.

6. What do you mean by a well-conditioned triangle? (CO1-L1) (AUC Apr/May 2011)

The shape of the triangle formed by the selected triangulation stations should be such

that any error in the measurement of the angle shall have a minimum effect upon the length of

the calculated sides. Such a triangle is called Well - conditioned triangle.

7. What is a base net? (CO1-L1) (AUC May/June 2009)

Some site conditions may not be favourable to get the required length of a base line. In

such a situation a short base line is selected and the same is then extended. Such group of

triangles which are meant for extending the base is known as base net.


2


8. Give the specification of first order triangulation. (CO1-L1) (AUC May/June 2009)

Average triangle closure : less than 1 second

Maximum triangle closure : not more than 3 seconds

Length of the base line : 5 to 15 km

Length of the sides of triangles : 30 to 150 km

Actual error of base : 1 in 300000

Probable error of base : 1 in 1000000

Discrepancy between two measures of section: 10 mm km

Probable error of computed distance : 1 in 60000 to 1 in 250000

Probable error in astronomic azimuth : 0.5 seconds

9. Name the different corrections to be applied to the length of a base line.

(CO1-L1) (AUC May/June 2013) (AUC May/June 2012) (AUC May/June 2009)

) Correction for absolute length

) Correction for temperature

) Correction for pull or tension

) Correction for sag

) Correction for slope

) Correction for alignment

) Reduction to sea level

10. Triangulation networks for covering a large area are composed of any one or a

combination of basic figures arranged as a series of chains or a connected centralized

network. Enumerate any two such arrangements. (CO1-L1)(AUC May/June 2012)

) Single chain of triangles

) Double chain of triangles

) Central point figures

) Quadrilateral or interlacing triangles

i) Single chain of triangles:

This type of figure is used where a narrowstrip of terrain is to be covered. This system is

economical and rapid. As the number of conditions to be fulfilled in the figure adjustment is

relatively small, it is not accurate for primary work.

ii) Double chain of triangles:

This arrangement is similar to the single chain as shown in figure. This is also

economical and rapid. It is used to cover greater area.


3


11. What is meant by control surveying? (CO1-L1) (AUC Nov/Dec 2012)

A control survey is a survey that provides positions horizontal or vertical points to which

supplementary surveys are adjusted.

12. What is satellite station and reduction to center? (CO1-L1) (AUC Nov/Dec 2012)

Satellite station:

It is selected as near as possible to the true station. From this station observations are

taken to the other triangulation stations with the same precision. Reduction to centre:

Angles taken from satellite station are corrected and reduced to what they would have

been if the true station was occupied. This operation of applying corrections to the observed

angles due to the eccentricity of the station is termed as Reduction to centre.

13. Describe signals. (CO1-L1) (AUC May/June 2013 )

A signal is any device erected to define the exact position of an observed station.

Requirements:

) It should be clearly visible against any background.

) It should be feasible to centre accurately over the station mark.

) It should be suitable for accurate bisection.

) It should be free from phase.

14. What is meant by third order or tertiary triangulation? (CO1-L1)

Third order or tertiary triangulation consists of a number of points fixed within the

framework of secondary triangulation. These are the points which form the immediate control for

engineering and other surveys.

15. Name two groups of people involved in the measuring the base line. (CO1-L1)

The field work for the base-line measurement is carried out by two parties, viz, setting

out party and measuring party.

Setting out party consists of two surveyors and a number of porters. The duty of the

porters is to place the measuring tripods at correct intervals, in alignment in advance.

Measuring party consists of two observers, recorder, leveler and staffman for actual

measuremet.

16. Enlist the types of signals used in triangulation. (CO1-L2)

) Day light or Non - luminous signals

) Sun or Luminous signals

) Night signals

17. Give the classification of triangulation system. (CO1-L2)

) First order or primary triangulation

) Second order or secondary triangulation

) Third order or tertiary triangulation

18. List the equipments used for measurement of base line. (CO1-L2)

Standardized tape, Straining device, Spring balance, Thermometers, Steel tape and

tripods.


4


1

1

1

PART B

1. The following observations were made in a trigonometric leveling :

Angle of depression to G at S = 10 45' 2''

Height of Instrument at S = 1.180 m

Height of signal at G = 4.220 m

Horizontal distance between G and S = 6945 m

Co-efficient of refraction = 0.07

R sin 1" = 30.88 m. If RL of S is 345.32 m. Calculate RL of G.

(CO1-H2) (AUC Nov/Dec 2010)

Solution:

d = 6945 m; β = 10 45' 32''; h = 1.180 m; s = 4.220 m; R sin 1" = 30.88 m; m = 0.07;

RL of S = 345.32 m

d sin m

H 2

Here

cos ( 1 m

1

)

s h

4.22 1.18

90.28" ( ve )d sin 1" 6945 X sin 1"

1O

45' 32" 90.28"

= 10 45' 2.28"

d

6945

224.90" 3' 44.9"R sin 1" 30.88

Curvature correction,

2

3' 44.9"

1' 52.45" 2

Refraction correction, r m 0.07 X 3' 44.9" 15.74"

d sin m

H 2

cos ( 1 mO

)

6945 sin ( 1 45' 2.28" ( 0.07 X 3' 44.9" ) 1' 52.45" )

cos

H = 209 m

( 1O 45' 2.28" ( 0.07 X

3' 44.9" ) 3' 44.9" )

RL of G = RL of S + H = 345.32 + 209

RL of G = 554.32 m

2. The following reciprocal observations were made at two points M and N.

Angle of depression of N at M = 00 7' 5''

Angle of depression of M at N = 00 9' 5''

Height of signal at M = 4.820 m

Height of signal at N = 3.950 m Height

of instrument at M = 1.150 m Height of

instrument at N = 1.280 m Distance

between M and N = 36320 m.


5

Civil Department

Surveying-II

Calculate:

i) The R. L. of N if that of M is 395.460 m

ii) The average Co-efficient of refraction at the time of observation.

Take R sin 1'' = 30.880 m. (CO1-H2) (AUC Nov/Dec 2010)

Solution:

d = 36320 m; β = - 9' 05''; α = - 7' 35''; h1 = 1.150 m; h2 = 1.280 m; 81 = 4.820 m;

82 = 3.950 m; R sin 1" = 30.88 m; m = 0.07; RL of M = 395.46 m

The difference in elevation (H) is given by,

d sin 1 1

H 2

cos

1 1 2 2

Axis signal correction at M = 1 s1 h1

d sin 1"

4.82 1.15 20.84" ( ve )

36320 X sin 1"

Axis signal correction at N =

s2 h2

3.95 1.28 15.16" ( ve )2

d sin 1" 36320 X sin 1"

1 1 7' 35" 20.84"

7' 55.84"

1 2 9' 05" 15.16" 9' 20.16"

1 1

2

1 1

2

9' 20.16" 7' 55.84"

42.16" 2

9' 20.16" 7' 55.84"

8' 38" 2

d

36320

1176.17" 19' 36.1"R sin 1" 30.88

19' 36.1" 9' 48"

2 2

r 2 1 1

2

9' 48"

8' 38" 1' 10"

m r

1' 10" 0.0595

19' 36.1"

d sin 1 1

H 2 36320 X sin ( 42.16" )

cos 1 1

cos ( 42.16" 9' 48" )


6


2 2

= 7.42 m

RL of N = RL of M + H = 395.46 + 7.42

RL of N = 402.88 m 3. What is meant by a satellite station and reduction to centre? Derive the expression for reducing the angles measured at the satellite station to centre. (CO1-H2) (AUC Apr/May 2010)

Satellite station:

Sometimes in order to form well-conditioned triangles of triangulation and also to have

better visibility objects such as church spirals, towers of temples, flag poles, etc are selected.

But the instrument cannot be set up over these true stations for the measurement of angles. In

such cases, a subsidiary station called as satellite station or eccentric station or false station is

selected as near as possible to the true station. From this station observations are taken to the

other triangulation stations with the same precision.

Reduction to centre:





7

Civil Department

Surveying-II


8

Civil Department

Surveying-II

Positions of satellite stations:

Case 1: Position 81 to the left of B (fig.1)

True angle, 1 2

Case 2: Position 82 to the right of B (fig.2)

True angle, 1 2

Case 3: Position 83 between AC and B (fig.3)

True angle, 1 2

Case 4: Position 84 below B (fig.4)

True angle, 1 2

The observed angles are reduced to the meridian and the corrections are computed as

( in sec onds ) d sin D sin 1"

Where θ = observed angle reduced to the assumed meridian.

D = distance from the true station to the observed station.


9


4. The following observations were made on a satellite station S to determine angle BAC.

Calculate the angle BAC. (CO1-H2) (AUC Apr/May 2010)

Line Length Line Bearing

SA 9.500 m SA 0° 00' 00"

AB 2950 m SB 78° 46' 00"

AC 3525 m SC 100° 15' 00"

Solution:

The correction to any direction is given by,

d sin D sin 1"

sec onds

a) For the line AB:

780

46' 00" ; d = A8 = 9.5 m; D = AB = 2950 m;

d sin

9.5 X sin ( 780 46' 00" )

D sin 1" 2950 X sin 1"

= 651.52" = 1O' 51.5"

Direction of AB = direction of SB + β =

= 780 56' 51.5"

780

46' 00" + 1O' 51.5"

b) For the line AC:

1000 15' 00" ; d = A8 = 9.5 m; D = AC = 3525 m;

d sin

9.5 X sin ( 1000 15' 00" )

D sin 1" 3525 X sin 1"

= 547.02" = 9' 7"


10


Direction of AC = direction of BC + β =

= 1000 24' 7"

1000 15' 00" + 9' 7"

Angle BAC = 1000 24' 7" - 780 56' 51.5"

BAC = 210 27' 15.5"

5. How do you determine the intervisibility of triangulation station? (CO1-H2) (AUC Apr/May 2010)

If the distance between stations is more or difference in elevation is less, the intervisibility

has to be checked by calculations. It may be necessary to raise both the instrument and the

signal in order to overcome the curvature of the earth and the intervening obstructions. The

following three conditions may decide the height of the instrument and the signal.

Distance between stations

Elevation of stations

Intervening ground

i) Distance between stations:

Considering the condition of no intervening ground the distance of visible horizon from a

station of known elevation above datum is given by 2

h (1 2m ) D 2R

Where h = height of the station above datum

D = distance to the visible horizon

R = mean radius of the earth

m = mean coefficient of refraction

m = 0.07 for sights over land

= 0.08 for sights over sea

Taking D and R in kilometers with m = 0.07

h = 0.06728 D2, in which h is in metres.

ii) Elevations of stations:

If there is no obstruction due to intervening ground, the elevation of a station at a

distance may be calculated, when it may be visible from another station of known elevation.

Then


11


D

D

1

1

h (1 2m ) D 2

2R

Substituting m = 0.07 and R in km.

h = 0.06728 D2

h1 = 0.06728 2 (metres)

where h1 = known elevation of station A above datum

D1 = distance from A to the point of tangency

D = known distance between A and B

D1

h1

0.06728

D1 3.853 h1

Knowing D1, D2 = D - D1

Where D2 = distance from B to the point of tangency.

Knowing D2, h2 the required elevation of B above datum may be calculated from

h2 = 0.06728 2 (metres)

If the actual ground level of B is known, it can be known whether it is necessary to

elevate the station B above the ground. If found to be necessary the required height of tower

can be calculated.

It is a point to be noted that the line of sight should not graze the surface at the point of

tangency but should be atleast 2 to 3 m above. iii) Intervening ground:

In general during the reconnaissance itself the elevations and positions of peaks in the

intervening ground between the proposed stations should be determined.

A comparison of the elevations of stations should be made to the elevation of the

proposed line of sight. If the line of sight is clear off the obstruction then the work is proceeded.

If not the problem can be solved based on the principle discussed in the previous sections.

6. The elevation of two triangulation stations A and B 150 km apart are 250 m and 1050 m

above MSL. The elevation of two peaks C and D on the profile between satellite stations

are 300 m and 550 m respectively. The distance AC = 50 km and AD = 85 km. design a

suitable signal required at B, so that it is visible from the ground station A.

(CO1-H2) (AUC Apr/May 2010)

Solution:

Let acedb be the visible horizon and a horizontal sight Ab1, through A meet the

horizon tangentially at e. let AO, CO, DO and BO be the vertical lines through A, C, D and

B respectively and O being the centre of the earth. Distance Ae to the visible horizon from

station A of an elevation 250 m is given by

D Ae 3.8553

h 3.8553

250 60.96 km

Let a, c, d and b be the points in which the vertical lines through A, C, D and B cuts

the level line. Here AC = 50 km; AD = 85 km; AB = 150 km

ce = Ae - AC = 60.96 - 50 = 10.96 km


12

Civil Department Surveying-

II

ed = AD - Ae = 85 - 60.96 = 24.04 km

eb = AB - Ae = 150 - 60.96 = 89.04 km

Let c1, d1 and b1 be the points in which a horizontal line through A cut the vertical

lines through C, D and B respectively. The corresponding heights cc 1, dd1 and bb1 are given as

cc1

0.06728 ( ce ) 2

0.06728 (10.96 ) 2

8.08 m

dd1

bb

0.06728 ( ed ) 2

0.06728 ( eb ) 2

0.06728 ( 24.04 ) 2

0.06728 ( 89.04 ) 2

38.88 m

533.4 m1

Now, Bb = elevation of B = 1050 m

Bb1 = Bb - bb1 = 1050 - 533.4 = 516.6 m

Let AB be the line of sight. Now from Ac1c2 . Ad1d2

and

Ab1 B

c1c2

d1d 2

Bb1 X

Bb1 X

Ac1

Ab1

Ad1

Ab1

516.6 X

516.6 X

50 172.2 m

150

85 292.74 m

150

Elevation of line of sight at C = elevation of c2


13


b S

b S

= cc1 c1c2 8.08 172.2 180.28 m

Elevation of line of sight at D = elevation of d2

= dd1 d1d2

38.88 292.74

331.62 m

Elevation of C = 300 m and elevation of D = 550 m

Thus the line of sight clears the peak D, but fails to clear the peak at C by

c2C = 300 - 180.28 = 119.72 m

Let Ac3 be the new line of sight, such that Cc3 = 3 m (minimum)

c2c3 = Cc3 + c2C = 3 + 119.72 = 122.72 m

Hence Bb3 = c2 c3 X

AB

AC

2

122.72 X

150

85

Bb3 = 216.56 m 217 m (say)

Hence minimum height of scaffold at B required is 217 m.

7. After measuring the length of a baseline, the correct length of the line is computed by

applying various applicable corrections. Discuss the following corrections and provide

expressions for

i) Correction for temperature.

ii) Correction for pull.

iii) Correction for sag. (CO1-H2) (AUC Apr/May 2011)

i) Correction for temperature:

If the temperature in the field is more than the temperature at which the tape was

standardized, the length of the tape increases measured distance becomes less and the

correction is additive. Similarly if the temperature is less, the length of the tape decreases

measured distance becomes more and the correction is negative. The temperature correction is

given by

C (

Tm

T0 ) L

Where α = coefficient of thermal expansion

Tm = mean temperature of tape

TO = standardized temperature of tape

L = measured length of tape

If however steel and brass wires are used simultaneously as in Jaderin's method, the

corrections are given by

( L L ) C ( brass ) b S b

t

( L L ) C ( steel ) S S b

t

ii) Correction for pull or tension:

If the pull applied during measurement is more than the pull at which the tape was

standardized, the length of the tape increases, measured distance becomes less and the


14


correction is positive. Similarly, if the pull is less, the length of the tape decreases, measured

distance becomes more and the correction is negative.

If Cp is the correction for pull, we have

c ( P P0 ) L

P AE

Where, P = pull applied during measurement (N)

PO = standard pull (N)

L = measured length (m)

A = cross sectional area of the tape (cm2)

E = young's modulus of elasticity (N I cm2)

The pull is applied in the field should be less than 20 times the weight of the tape.

iii) Correction for sag:

When the tape is stretched on supports between two points, it takes the form of a

horizontal catenary. The horizontal distance will be less than the distance along the curve. The

difference between horizontal distance and the measured length along catenary is called the

sag correction. For the purpose of determining the correction, the curve may be assumed to be

a parabola.

cs l ( wl ) 2

24 n 2

P 2

l W 2

24 n 2 P 2

Where CS = tape correction per tape length

l = total length of the tape

W = total weight of the tape

n = number of equal spans

P = pull applied

If L = total length measured

N = number of whole length tape

Total sag correction = Ncs sag correction for any fractional tape length

8. From an eccentric station S, 12.25 m to the west of the main station B, the following

angles were measured.

Angle of BSC = 76° 25' 32"

Angle of CSA = 54° 32' 20"

The stations S and C are to the oppose sides of the line AB. Calculate the correct angle

ABC if the length AB and BC are 5286.5 m and 4932.2 m respectively.

(CO1-H2) (AUC Apr/May 2011)

Solution:

BS = d = 12.25 m; AB = c = 5286.5 m; BC = a = 4932.2 m;

540

32' 20" ;


15


760

25' 32"

Correct angle, 1 2

d sin ( )

1 c

x 206265

12.25 x sin ( 54o 32' 20" 76o 25' 32") x 206265

5286.5

β1 = 360.92 sec = 6' 0.92"

2 d sin

b

12.25 x

x 206265

sin ( 76o

25' 32" ) x 206265

4932.2 β2 = 497.98 sec = 8' 17.98"

1 2

= 54o 32' 20" 6' 0.92" 8' 17.98"

α = 540 30' 2.94"

9. A steel tape 20 m long standardized at 55° F with a pull of 98.1 N was used for measuring

a baseline. Find the correction per tape length, if the temperature at the time of

measurement was 80° F and the pull exerted was 156.96 N. Weight of 1 cubic metre of

steel = 77107 N. weight of tape = 7.85 N and E = 2.05 x 105 N/mm2. Coefficient of linear

expansion of tape per degree F = 6.2 x 10-6. (CO1-H2) (AUC Apr/May 2011)

Solution:

L = 20 m; T0 = 55oC; Tm = 80oC; Po = 98.1 N; P = 156.96 N; α = 6.2 x 10-

6; Weight of steel = 77107 N; Weight of tape = 7.85 N; E = 2.05 x 105 N I

mm2

i) Correction for Temperature:

Ct = α (Tm - T0) L

= 6.2 x 10-6 (80 - 55) x 20

Ct = 0.0031 m

ii) Correction for Pull:

P Po CP =

L

AE


16


Here, weight of tape = (Area x 1 x weight of steel) x length

7.85 = (A x 1 x 77107) x 20 7.85 6 2

2

A = 5.1 x 10 m 5.1 mm 77107 x 20

CP = 156.96 98.1

x 205.1 x 2.05 x 10

5

iii) Sag Correction:

CP = 0.00112 m

Cs =

LW 2

24n 2 P

2

20 x =

24 x 12

7.852

x 156.96 2

Cs = 0.00208 m

Total correction = Ct + CP - Cs

= 0.0031 + 0.00112 - 0.00208

Total correction = 0.00214 m

True length = Length + correction

= 20 + 0.00214

True length = 20.00214 m

10. Explain the criterion of strength of a figure with reference to a well conditioned triangle.

(CO1-H2) (AUC May/June 2009)

The shape of the triangle formed by the selected triangulation stations should be such

that any error in the measurement of the angle shall have a minimum effect upon the length of

the calculated sides. Such a triangle is called a well-conditioned triangle.

In a triangle one side may be computed from the computation of adjacent triangle. The

error in other two sides may affect the rest of the figure. If the two sides are to be equally

accurate, then they should be of equal length, which could be possible only by making the

triangle isosceles. In order to find the magnitude of the angle of triangle, let ABC be an

isosceles triangle with AB of known length.

The sides BC and CA are to be computed. As the triangle is isosceles A B.

sin ABy sine rule, a = c

sin c …………………. (1)

Let δA be the error in the measurement of angle.

δa1 be the corresponding error in the side a partially.

Differentiating eqn.(1) with respect to A we get


17


a1

a1

a

c cos A .A

sin c

cos A

A A . cot A ………………….. (2) sin A

Similarly δC be the error in the measurement of C and

δa2 be the corresponding error in the side a.

Differentiating eqn.(2) again partially with respect to C, then

a c sin A cos c

c 2

sin 2 c

a2 cos c

c c . cot c …………….. (3) a sin c

If A and c are the probable errors in angles, then they are equal to .

Then the probable fraction error in the side a

cot 2

A cot 2 c

This is minimum when cot 2 A cot 2 c is minimum.

But C = 1800 - A - B = 180

0 - 2A (since A B )

cot 2 A cot 2 2 A should be minimum.

Differentiating the above equation with respect to A and equating to zero, we get after reduction

4 cos 2 A 2 cos 2 A 1 0

From which A is got 560 14' approximately.

Hence the best shape of a triangle is an isosceles triangle with base angles 56 0 14'.

However for practical consideration 560 14' = 600.

For all practical purposes, an equilateral triangle is the most suitable. In general,

however, triangles having angles smaller than 300 or greater than 1200 should be avoided.

11. A tape 20 m long of standard length at 290C was used to measure a line, the mean

temperature during measurement being 190C. the measured distance was 882.10 m, the

following being the slopes: 2o 20' for 100 m; 4o 12' for 150 m; 1o 06' for 50 m; 7o 48' for

200 m; 3o

00' for 300 m; 5o

10' for 82.10 m. find the true length of the line if the coefficient

of expansion is 6.5 x 10-6 per degree F. (CO1-H2) (AUC May/June 2009)

Solution:

L = 882.10 m; T0 = 29oC = 860 F; Tm = 19oC = 680 F; α = 6.5 x 10-6;


Ct = α (Tm - T0) L

= 6.5 x 10-6

(68 - 86) x 882.10

Ct = - 0.103 m

ii) Correction for Slope:

Csl l ( 1 cos )

= 100 ( 1 cos 20 10' )

+ 150 ( 1 cos 40 12' ) +

50 ( 1 cos 10 06' ) +

200 ( 1 cos 70 48' ) +

Cs = 3.078 m

300 ( 1 cos 30 00' ) + 82.1 ( 1 cos 50 10' )


18


Total correction = Ct - Csl

= - 0.103 - 3.078

Total correction = - 3.181 m


= 882.1 - 3.181


12. Write short notes on Selection of site for base line. (CO1-H1) (AUC May/June 2009)

The length of the base line to be adopted depends on the magnitude of triangulation

work ie., the grade of the triangulation. Apart from main base line additional check bases are

also provided at some suitable intervals.

The location of the base line should be such that the site affords accurate measurement . The

following factors should be considered in the selection of the location.

) The ground selected should be as plain as possible. However, gentle slope may also be

adopted.

) All the main stations of triangulation should be visible from both the ends of the base

line.

) It should be possible to build up a network of well-proportioned triangles on the base.

) The site should be free from obstructions throughout the length of the base line. The

expenses involved in clearing the site should be minimum.

) The ground should be reasonably firm and water gaps, if any, should not be wider than

the length of a tape.

) The site should be possible for extension to primary triangulation. This is an important

aspect, as the error in extension may exceed the error in measurement.

13. A steel tape of nominal length 30 m was suspended between two supports to measure

the length on a slope of 04o 25' is 29.861 m. the mean temperature during measurement

was 15oC and pull applied was 120 N. if standard length of the tape was 30.008 m at 27oC

and the standard pull of 50 N, calculate the correct horizontal length. Take the weight of

the tape as 0.16 N/m, its cross sectional area equal to 2.75 mm2 coefficient of linear

thermal expansion = 1.2 x10-5 per degree Celsius and E = 2.05 x 105 N / mm-2.


Solution:

Lt = 30 m; Lsl = 29.861 m; Ls = 30.008 m; T0 = 270

C; Tm = 150

C; Po = 50 N; P = 120 N;

α = 1.2 x 10-5; Area = 2.75 mm2; Weight of tape = 0.16 Nim; E = 2.05 x 105 N i mm2

i) Correction for slope:

h 2

c 2L

Here h = Lsl sin = 29.861 x sin (40 25') = 2.3 m

2.32

c = 0.0886 m 2 x 29.861

ii) Correction for absolute length:

c Lc

29.861 x ( 30.008 29.861) a

/ 30.008


19


2

Ca = 0.146 m iii) Correction for Temperature:

Ct = α (Tm - T0) Lsl

= 1.2 x 10 - 5 (15 - 27) x 29.861

Ct = · 0.0043 m

iv) Correction for Pull:

P Po 120 50CP =

L =

AE

2.75 x

2.05 x 105

x 29.861

CP = 0.0037 m

v) Sag Correction:

LW 2

29.861 x

0.16

2

Cs =

24n 2 P

2

= 24 x 12

x 120 2

Cs = 0.0000022 m

Total correction = - C + Ca + Ct + CP - Cs

= - 0.0886 + 0.146 - 0.0043 + 0.0037 - 0.0000022



= 29.861 + 0.0568


14. Two stations P and Q are 81 km apart. They are situated on either side of a sea. The

instrument axis at P is 39 m above MSL. The elevation of Q is 207 m above MSL.

Calculate the minimum height of the signal at Q. The coefficient of refraction is 0.08 and

the mean radius of earth is 6370km. (CO1-H2) (AUC May/June 2012)

Solution:

There is no intervening ground.

Hence the height of the signal at Q, h = ( 1 2 m ) D 2 R

81 2= ( 1 ( 2 x 0.08 ) ) x

Minimum height of signal, h = 0.43 m

15. Briefly explain the following:

i) Satellite stations

2 x 6370

ii) Phase of a signal. (CO1-H1) (AUC May/June 2012)

Satellite stations:

Sometimes in order to form well-conditioned triangles of triangulation and also to have

better visibility objects such as church spirals, towers of temples, flag poles, etc are selected.

But the instrument cannot be set up over these true stations for the measurement of angles. In

such cases, a subsidiary station called as satellite station or eccentric station or false station is

selected as near as possible to the true station. From this station observations are taken to the

other triangulation stations with the same precision.





20


The distance between the true station and the false station may be obtained either by

method of trigonometrical levelling or by triangulation. It is to be noted that in primary

triangulations satellite stations should be avoided.

Phase of a signal:

Phase is the error of bisection of some type of signals when they are partly in light and

partly in shade. This is commonly the case with cylindrical signals as the observer sees only its

illuminated portion and bisects it. This is the apparent displacement of the centre of the signal.

Thus the phase correction is necessary.

The correction may have to be applied under two conditions when

i) The observation is made on the bright portion.

ii) The observation is made on the bright line. i) The observation is made on the bright portion:

When the observation is made on the bright portion FD is shown in figure (a).

Let A be the position of the observer and

B be the centre of the signal

FD be the visible portion of the illuminated surface.

AE be the line of sight.

E be the mid-point of FD

β be the phase correction

1 and 2 be the angles which the extremities of the visible portion make with AB.

α be the angle which the direction of the sun makes with AB.

r be the radius of the signal.

D be the distance AB.

The phase correction,

1 1

(2

2

1 )


21

Civil Department

Surveying-II

D

D

1

( )

But 2

2 1 2

r

radians and D

r sin ( 90 0 ) r cos

radians

1 2

D

r cos

D

r

r (1 cos )

2D

(0r)

r cos 2 r cos 2

2 radians 2 secD

206265

r cos 2

D sin 1"

2 D

sec onds

ii) Observation is made on the bright line:

0bservation is made on the bright line formed by the reflected rays as indicated by the

path SE is shown in figure (b). AE is the observed line of sight. 90O

1 ( )

2

Let β be equal to EAB

As S E and S1 A are parallel.

sEA 180O

( )

Δ BEA = 1800 1 sEA 180O

2

1

[ 180O

2

( ) ]

= 900 + 1

( ) 2

EBA 180O

( BEA )

= 1800 - β - 900 1

( ) 2

= 900

1 ( )

2

= 900 2

since β is small in comparison to α.

r sin 90O

2

D

radians

r cos

= 2 D

r cos

= 2 D sin 1"

radians

sec onds


22


206265 r cos

2 D

sec onds

16. Explain about the curvature and refraction correction in trigonometrical leveling.


Correction for refraction:

Figure represents two stations A and B located at very far distance whose difference in

elevation is to be found. Let 0 be the point represent the centre of the earth.

Let AA' be the tangent to the level line at A.

AA1 be the horizontal line at A.

BB1 be the horizontal line at B.

A' AO' 1 be the observed angle of elevation from A to B.

B' BB2 1 be the observed angle of depression from B to A.

r be the angle of refraction = A' AB B' BA

AA' be the tangent at A to the curve line of sight AB.

BB' be the tangent at B to the curve line of sight BA.

d be the horizontal distance between A and A1.

R be the mean radius of the earth (6370 km).

m be the coefficient of refraction.

be the angle subtended at the centre by the distance AA1 over which the

observations are made.

The actual line of sight between A and B should have been along straight line AB.

Because of the effect of terrestrial refraction the actual line of sight is curved concave towards


23


the ground surface. AA' is therefore the apparent sight from A to B and BB' is the apparent sight

from B to A.

The angle measured at A towards B is the angle between the apparent sight A'A and the

horizontal line O'A. Hence the A' AO' is the observed angle α1. Without the effect of refraction

the true angle of elevation is BAO' . Hence the correction for refraction is

the angle r). Then the correction is subtractive.

Correct angle BAO' = A' AO' O' AB

= α1 - r

A' AB (Say

Similarly the angle measured at B towards A is B1 BB2

the absence of refraction is ABB2 .

1 . The true angle of depression in

Correct angle ABB2 = B' BB2

= β1 + r

B' BA

Thus the correction for refraction is subtractive to the angle of elevation and additive to the

angle of depression.

Coefficient of refraction:

It is defined as the ratio of angle of refraction and the angle subtended at the centre

of the earth by the distance over which observations are taken, thus

m r

r m

The coefficient of refraction may be determined for the following two cases:

i) Distance d small and H large:

In this case one angle α1 is the angle of elevation and the other β1 is the angle of

depression.

In Δ ABO'

ABB BAO' BO'

A

1 r 1 r

r

2

1 1 2

Substituting r = m

1 1 (1 2 m )

ii) Distance d large and H small:

In this case both α1 and β1 are angles of depression.

Changing the sign of α1 we get,

r 2

1 1 2

Which reduces to 1 1 (1 2 m )

Correction for curvature:


24


The angle α1 is measured with reference to the horizontal lines AB' but it should be

measured with the chord AA1 where A1 is the vertical projection of B on a level line passing

through A.

Hence the correction = O' AA1 2

and is additive.

Similarly the angle 1 was measured with reference to the horizontal line BO' while it should be

measured with the chord BB1.

Hence the correction, B BB

and is subtractive.2 1 2

Thus the correction for curvature is

depression.

Combined correction:

for angles of elevation and 2

for angles of 2

Combined angular correction = ( 1 2 m ) d

2R sin 1"

sec

The combined correction is positive for angles of elevation and negative for angles of

depression.

17. From a satellite station S, 5.8 m from main triangulation station A, the following

directions were measured.

A = 0o

0' 0"; B = 132o

18' 30"; C = 232o

24' 06"; D = 296o

06' 11"; AB = 3265.5 m;

AC = 4022.2 m; AD = 3086.4 m. determine the directions of AB, AC and AD.


Solution:

The correction to any direction is given by,

d sin D sin 1"

sec onds

a) For the line AB:


25


1320 18' 30" ; d = AS = 5.8 m; D = AB = 3265.5 m;

d sin

5.8 x

sin ( 1320 18' 30" )

D sin 1" 3265.5 x sin 1"

= 270.9" = 4' 30.9"

Direction of AB = direction of SB + β =

= 1320

23' 0.9"

1320 18' 30" + 4' 30.9"

b) For the line AC:

2320 24' 6" ; d = AS = 5.8 m; D = AC = 4022.2 m;

d sin

5.8 x

sin (

2320

24' 6" )

D sin 1" 4022.2 x sin 1"

= - 235.7" = - 3' 55.7"

Direction of AB = direction of SC + β =

= 2320 20' 10.3"

2320

24' 6" - 3' 55.7"

c) For the line AD:

2960 6' 11" ; d = AS = 5.8 m; D = AD = 3086.4 m;

d sin

5.8 x

sin (

2960

6' 11" )

D sin 1" 3086.4 x sin 1"

= - 348.1" = - 5' 48.1"

Direction of AB = direction of SD + β =

= 2960 0' 22.9"

2960

6' 11" - 5' 48.1"

18. How are the triangulation system classified and how triangulation survey work carried

out? (CO1-H1) (AUC May/June 2013) (AUC Nov/Dec 2012)

Classification of a triangulation system is based on the accuracy with which the length

and angle of a line of the triangulation are determined. The following are the classification based

on the order of grades:

i) First order or primary triangulation

ii) Second order or secondary triangulation

iii) Third order or tertiary triangulation

i) First order or primary triangulation:

The first order triangulation is of the highest order and is employed either to determine

the earth's figure or to furnish the most precise control points to which secondary triangulation

may be connected. The primary triangulation system embraces the vast area. Every precaution

is taken in making linear and angular measurements and in performing the reductions. The

following are the general specifications of the primary triangulation:

Average triangle closure : less than 1 second

Maximum triangle closure : not more than 3 seconds

Length of the base line : 5 to 15 km


Actual error of base : 1 in 300000

Probable error of base : 1 in 1000000


Probable error of computed distance : 1 in 60000 to 1 in 250000


26


Probable error in astronomic azimuth : 0.5 seconds

ii) Second order or secondary triangulation:

The secondary triangulation consists of a number of points fixed within the framework of

primary triangulation. The stations are fixed at close intervals so that the sizes of the triangles

formed are smaller than the primary triangulation. The instruments and methods used are not

of the same utmost refinement. The general specifications of the secondary triangulation are:

Average triangle closure : 3 seconds

Maximum triangle closure : 8 seconds

Length of the base line : 1.5 to 5 km


Actual error of base : 1 in 150,000

Probable error of base : 1 in 500,000


Probable error of computed distance : 1 in 20,000 to 1 in 50,000

Probable error in astronomic azimuth : 2 seconds

iii) Third order or Tertiary triangulation:

The third order triangulation consists of a number of points fixed within the framework of

secondary triangulation and forms the immediate control for detailed engineering and other

surveys. The sizes of the triangles are small and instrument with moderate precision may be

used. The general specifications of the third order triangulation are:

Average triangle closure : 6 seconds

Maximum triangle closure : 12 seconds

Length of the base line : 0.5 to 3 km

Length of the sides of triangles : 1.5 to 10 km

Actual error of base : 1 in 750,000

Probable error of base : 1 in 250,000


Probable error of computed distance : 1 in 5,000 to 1 in 20,000

Probable error in astronomic azimuth : 5 seconds

19. Find the sag correction for 30 m steel tape under a pull of 80 N in three equal spans of 10

m each. Mass of one cubic cm of steel = 7.86 g/cm3. Area of cross section of the tape

= 0.10 sq.cm. (CO1-H2) (AUC May/June 2013 )

Solution:

Sag Correction:

Cs = LW 2

24n 2 P

2

Here each span = 30

10 m 3

Weight of tape, W = (Area x 1 x weight of steel) x length

= (0.10 x 1 x 7.86 x 10 - 3) x 30 x 100

= 2.358 kg

30 x 100 x 2.3582

Cs =

24 x 32

x ( 80 9.81 ) 2


27


2

Cs = 1.16 cm

20. A 30 m steel tape was standardized on the flat and was found to be exactly 30 m under

no pull at 66o F. it was used in catenary to measure a base of 5 bays. The temperature

during the measurement was 92o F and the pull exerted during measurement was 100N.

The area of cross section of the tape was 8 mm2. The specific weight of steel is

78.6 kN/m2, α = 0.63 x 10·5 Fo and E = 2.1 x 105 N/mm2. Find the true length of the tape.


Solution:

L = 30 m' T0 = 660 F' Tm = 920 F' P0 = 0' P = 100 N' α = 0.63 x 10

Weight of steel = 78.6 kNim2' E = 2.1 x 10

5 N i mm

2


Ct = α (Tm - T0) L

= 0.63 x 10 - 5

(92 - 66) x 30

Ct = 0.00491 m

ii) Correction for Pull:

- 5

' A = 8 mm '

P Po

100 0CP =

L =

AE 8 x

2.1 x 105

x 30

CP = 0.00178 m

iii) Sag Correction:

Cs =

LW 2

24n 2 P

2

Here each span = 30

6 m 5

Weight of tape, W = (Area x 1 x weight of steel) x length

= (8 x 10 - 6

x 1 x 78.6 x 103) x 6

= 3.773 N

6 x 3.7732

Cs =

24 x 12

x 100 2

Cs = 0.000356 m

Total sag correction, Cs = 5 x 0.000356 = 0.00178 m

Total correction = Ct + CP - Cs

= 0.00491 + 0.00178 - 0.00178



= 30 + 0.00491


21. Explain with reference to signals, Non·luminous, luminous and night signals.


Signal:


28


A signal is any device erected to define the exact position of an observed station.

Requirements:

) It should be clearly visible against any background.

) It should be feasible to centre accurately over the station mark.

) It should be suitable for accurate bisection.

) It should be free from phase.

Non·luminous or Opaque signal:

Day light or non-luminous signals consist of the various forms of mast, target or tin

cone types. They are generally used for direct signals less than 30 kilometers. For sights under

6 kilometers, pole signals consisting of round pole painted black and white in alternate section

and supported on a tripod may be used. A target signal consists of a pole carrying two square or

rectangular targets placed at right angles to each other. The targets are made of cloth stretched

on wooden frames. The signals should be of dark colour for visibility against the sky and should

be painted white or black strips against a dark background. The top of the mast should carry a

flag. Its height above the station should be roughly proportional to the length of the longest sight

upon it. A height in the vertical plane corresponding to at least 30" is necessary. The following

rules may serve as a guide:

Diameter of signal in cm = 1.3D to 1.9D, where D is in kilometers

Height of signal in cm = 13.3D, where D is in kilometers

Luminous or Sun signals:

Luminous or sun signals reflect the sun's rays directly or indirectly in the direction of the

observer. The heliotrope and heliograph fall into this category. The heliotrope consists of a

mirror to reflect the sun's rays and a line of sight to direct the reflected rays towards the

observer. The latter may be simply a sight vane with an aperture and cross hairs or it may be

telescopic. Flashes are sent from the observing station to establish the line if sight. The mirror

should be adjusted every few minutes since the direction of the sun goes on changing. A form of

heliotrope is the Galton sun signal used for lines of sight exceeding 30 km.

Night signals:

Night signals are used in observing the angles of a triangulation system at night. Various

forms of night signals used are:


29


Various forms of oil lamps with reflectors or optical collimators for lines of sight less than

80 kilometers.

Acetylene lamp designed by captain G.T.Mccaw for lines of sight up to 80 kilometers.

22. The altitude of two proposed stations A and B, 100 km apart, are respectively 420 m and

700 m. The intervening obstruction situated at C, 70 km from A as an elevation of 478 m.

Ascertain if A and B are intervisible, and if necessary find how much B should be raised

so that the line of sight must be less than 3 m above the surface of the ground. (CO1-H2)

Solution:

Let aceb be the visible horizon and a horizontal sight Ab1, through A meet the horizon

tangentially at e. Distance Ae to the visible horizon from station A of an altitude 420 m is given

by

D Ae 3.8553 h 3.8553 420 79.01km

Let a, c and b be the points in which the vertical lines through A, C and B cuts the

level line. Here AC = 70 km' AB = 100 km

ce = Ae - AC = 79.01 - 70 = 9.01 km

eb = AB - Ae = 100 - 79.01 = 20.99 km

The corresponding heights cc1 and bb1 are given as

cc1

0.06728 ( ce ) 2

0.06728 ( 9.01) 2

5.46 m

bb1

0.06728 ( eb ) 2

0.06728 ( 20.99 ) 2

29.64 m

Now, Bb = elevation of B = 700 m


30

Civil Department

Surveying-II

Bb1 = Bb - bb1 = 700 - 29.64 = 670.36 m

Let AB be the line of sight. Now from Ac1c2

and

Ab1 B

c1c2

Bb1

x

Ac1

Ab1

670.36 x

70

100

469.25 m

Elevation of line of sight at C = elevation of c2

Elevation of C = 478 m

= cc1 c1c2

5.46 469.25

474.71 m

Thus the line of sight fails to clear the peak at C by

c2C = 478 - 474.71 = 3.29 m

Let Ac3 be the new line of sight, such that Cc3 = 3 m (minimum)

c2c3 = Cc3 + c2C = 3 + 3.29 = 6.29 m

Hence Bb3 = c2 c3 x

AB

Ac2

6.29 x

100

70

Bb3 = 8.99 m 9 m (say)

Hence height of scaffold at B required is 9 m.


31


SURVEYING II

UNIT 2 - SURVEY ADJUSTMENTS

PART A

1. Write a note on Accidental Errors. (CO2-L1) (AUC Nov/Dec 2010)

Accidental errors occur by a combination of reasons beyond the ability of the observer to

control. They sometimes occur in one direction and sometimes in the other side. Thus they are

likely to make the apparent result too large or too small. These errors represent the limit of

precision in the determination of a value.

2. Give any four random errors occur in linear measurements. (CO2-L1)(AUC Nov/Dec 2010 & 11)

(i) Mistakes

(ii) Systematic error

(iii) Accidental error

(iv) True error

(v) Residual error

(vi) Most probable error.

3. Define conditioned quantity. (CO2-L1) (AUC Apr/May 2010)

A quantity is said to be conditioned when its value is dependent upon the values of one or

more quantities. It is also called as dependent quantity.

4. What is meant by weight of an observation? (CO2-L1) (AUC Apr/May 2010,11, Nov/Dec 2012)

Weight of an observation is a measure of its relative worth which may be indicated by a

number. Thus if a certain observation is said to have weightage 5, it is meant to say that it is five

times as much as an observation of weight 1.

5. Differentiate 'most probable error' from 'residual error'. (CO2-L1) (AUC Apr/May 2011)

Most probable error:

Most probable error is defined as that quantity which added to and subtracted from the

most probable value which fixes the limits. By this limits there is an even chance the true value of

the measured quantity may lie.

Residual error:

A residual error is the difference between the most probable value of a quantity and its

observed value.

6. Distinguish between true error and residual error. (CO2-L1) (AUC May/June 2009)

True error:

A true error is the difference between the true value of a quantity and its observed value.

Residual error:


observed value.


32


7. What do you mean by figure adjustment in triangulation? (CO2-L1) (AUC May/June 2009)

Figure adjustment is the determination of the most probable values of the angles

involved in any geometrical figure so as to fulfill the geometric requirements.

It invariably involves one or more conditional equations. Conditional equations may

be framed by the method of normal equation or by the method of correlates. In case of

more condition equations, the solution may be applied easily by the method of correlates.

8. Distinguish between the observed value and the most probable value of a quantity. (CO2-L2)(AUC May/June 2012 & 13)

Observed value:

The observed value of a quantity is the value obtained as a result of an observation

which is corrected for all errors.

Most probable value:

Most probable value of a quantity is the value which is more likely to be the true value

than any other value.

9. What are normal equations? (CO2-L1) (AUC May/June 2012)

A normal equation is an equation of condition by means of which the most probable

value of any unknown quantity may be determined corresponding to a set of values assigned to

other unknown quantities.

10. What is method of correlates? (CO2-L1) (AUC May/June 2013)

Correlates are the unknown multiples or independent constants employed for finding the

most probable values of unknowns.

In this method of correlates all the condition equations are collected. One more

equation of condition, i.e., the sum of the squares of the residual errors should be minimum is

added.

11. How do you determine the most probable values? (CO2-L1)

Direct observation of quantities of equal weights.

Direct observation of quantities of unequal

weights.

Indirect observations involving unknowns of equal weights.

Indirect observations involving unknowns of unequal weights.

Observation equations are accompanied by condition

equations.

12. Define principles of least squares. (CO2-L1)

The principle of least squares may be defined as "In observation of equal precision

the most probable values of the observed quantities are those that render the sum of the squares

of the residual errors a minimum". This is also known as Method of least squares. 13. What are the laws of accidental errors? (CO2-L1)

Small errors occur often compared to large errors and such errors are denoted as

most probable.

Additive and subtractive errors occur frequently which may be of same size, such

errors are called as equally probable.

Large errors occur rarely and of impossible category.

14. What are the conditions to be satisfied when correcting the measured angles? (CO2-L1)

Correction to be applied to an observation is inversely proportional to the weight of the observation.


33


v w

Correction to be applied to an observation is directly proportional to the square of

the probable error.

Correction to be applied to an observation is proportional to the length in case of line of

levels.

15. Define direct and indirect observation. (CO2-L1)

Direct observation:

An observation is the numerical value of a measured quantity, and may be either direct or

indirect. A direct observation is the one made directly on the quantity being determined, e.g., the

measurement of a base, the single measurement of an angle etc.

Indirect observation:

An indirect observation is one in which the observed value is deduced from the

measurement of some related quantities, e.g., the measurement of angle by repetition (a

multiple of the angle being measured.)

16. Explain level net. (CO2-L1)

A level net is an interconnecting net work of level circuits formed by level lines

interconnecting three or more bench marks.

In adjusting a level net, the method of least squares may be adopted.

17. State Gauss's rule. (CO2-L1)

Gauss's rule is applied when the weights of the observation are not directly known. If the residual error of each observation is known the weights can be calculated by gauss's rule given by the following expression:

n 2 2 2

Where w is the weight to be assigned to a quantity.

n is the number of observations made for the quantity.

Lv2 is the sum of squares of residuals.

18. Why figure adjustment is made? (CO2-L1)

Figure adjustment is needed so as to fulfill the geometric conditions of any geometrical

figure. The figure adjustment involves one or more condition equations.

19. What is method of equal shift? (CO2-L1) Method of equal shift indicates that any shift which is necessary to satisfy the local equilibrium should be the same for each triangle of a polygon. Similarly any shift necessary to satisfy the side equation should be the same for each triangle. 20. Differentiate between conditioned quantity and conditional equation. (CO2-L1)

Conditioned quantity:

A quantity is said to be conditioned when its value is dependent upon the values of one or

more quantities. It is also called as dependent quantity.

Conditional equation:

A conditional equation is an equation expressing the relation existing between the several

dependent quantities.


34


2 2

2 3

( (

0 2 2 0 2 2 1 2 3

wv2

0.6745

1

1

0

2 2

2 3

( (

0 1

wv 2

0.6745 w (n 1)

PART B

1. The following are the observed values of the angle A with the corresponding

weights. (i) 510 20' 30'' Weight 2

(ii) 510 20' 28'' Weight 3

(iii) 510 20' 29'' Weight 2.

Determine:

(1) the standard deviation

(2) the standard error of the weighted mean

(3) the probable error of single observation of weight 3

(4) the probable error of the weighted mean. (CO2-H2) (AUC Nov/Dec 2010)

Solution:

As the error is in seconds only, the needed details are arranged as under.

Value Weight (value x weight) v v2 wv

2

30" 2 60" 1.14 1.2996 2.5992

28" 3 84" - 0.86 0.7396 2.2188

29" 2 58" 0.14 0.0196 0.0392

Lw = 7 Weighted mean = 28.86" L wv2 = 4.86

Weighted arithmetic mean = ( 30 x 2 ) 28 x 3 )

2

29 x 2 ) 02 8.86"

7

v1 0" 8.86" .14"; v2 8" 8.86" .86"; v3 9" 8.86" .14"

Probable error of single observation of unit weight = ES

=

= 0.6745

n

4.86

3

i) Standard deviation:

Standard deviation =

w v 2

w

4.86

7

= 1.05

.69

ii) Standard error of weighted mean:

Weighted arithmetic mean = ( 30 x 2 ) 28 x 3 )

2

29 x 2 ) 02 8.86"

7

iii) Probable error of single observation of weight 3 :

= ES

w

.05

3

.61

iv) Probable error of weighted arithmetic mean:

= =

0.6745

4.86

7 x 2

= 0.4


35


1 : 4 : 9

2. Find the most probable values of the following angles closing the horizontal at a station.

P = 450 23' 37'' Weight = 1

Q = 750 37' 1 '' Weight = 2

R = 1250 21' 21'' Weight = 3

S = 1130 37' 9'' Weight = 3. (CO2-H2) (AUC Nov/Dec 2010)

Solution:

Sum of observed angles = 450

23' 37'' + 750

37' 15'' + 1250

21' 21'' + 1130

37' 59''

= 3600 O' 12''

Error = + 12"

Total correction = - 12" Let C1, C2, C3 & C4 be the corrections to the observed angles P, Q, R and S. the

error will be distributed to the angles in an inverse proportion to their weights.

P = 450 23' 37'' + C1

Q = 750 37' 15'' + C2

R = 1250 21' 21'' + C3

S = 1130 37' 59'' + C4

C1 : C2 : C3 : C4 = (1)2 : (2)2 : (3)2 :

(3)2

: 9 ……………..……………. (1)


36


4 Cl

9 Cl

9 Cl

12 0.521"

4 Cl

4 X 0.522" 2.08"

9 Cl

9 X 0.522" 4.70"

9 Cl

9 X 0.522" 4.70"

Also, C1 + C2 + C3 + C4 = 12" ……………..……………. (2)

From (1) C2

C3

C4

Substituting these values of C2, C3 & C4 in (2), we get

C1 + 4 C1 + 9 C1 + 9 C1 = 12"

C1 23

C2

C3

C4

Hence the corrected angles are

P = 450 23' 37'' - 0.521" = 450 23' 3 .

8'' Q = 750 37' 15'' - 2.08" = 750 37'

12.92'' R = 1250 21' 21'' - 4.70" =

1250

21' 1 .3'' S = 1130

37' 59'' -

4.70" = 1130 37' .3''

Sum = 3600 00' 00''

3. What do you understand by the terms station adjustment and figure adjustment and

also explain the method of adjustment by least squares. (CO2-H1) (AUC Apr/May 2010)

Station adjustment:

It is the process of obtaining the most probable values of two or more angles

measured at a station so as to satisfy the condition of being geometrically consistent. Figure adjustment:

It is the determination of the most probable values of the angles involved in

any geometrical figure so as to fulfill the geometric requirements.

Method of adjustment of least squares:


37



38

Civil Department

Surveying-II


39


16 : 9 :

9 C

1

4 c

1 c

1 1

12 X16 6.62"

9 x 6.62" 3.72"

1 2 3

Solving equation (4a), (4b), (4c) and (4d), the values of 1 ,

2 ,

3 and

4 can be known. substituting

the values of 1 ,

2 ,

3 and

4

in equations (3a), (3b) …… (3h), the correction e1, e2, e3……. en can

be calculated and hence the corrected angles can be found.

4. The angles of a triangle ABC recorded were as follows:

Inst station Angle Weight

A 77° 14' 20" 4

B 49° 40' 35" 3

C

53° 04' 53"

2

Give the corrected values of the angles. (CO2-H2) (AUC Apr/May 2010) (AUC May/June 2009 & 13)

Solution:

Sum of observed angles = 77° 14' 20" + 49° 40' 35" + 53° 04' 53"

= 1790 59' ''

Error = - 12"

Total correction = 12" Let C1, C2 & C3 be the corrections to the observed angles A, B and C. The error will

be distributed to the angles in an inverse proportion to their weights.

A = 77° 14' 20" + C1

B = 49° 40' 35" + C2

C = 53° 04' 53" + C3

C : C : C = (4)2 : (3)2 : (2)2 4 ……………..……………. (1)

Also, C1 + C2 + C3 + C4 = 12" ……………..……………. (2)

From (1) C2 16

C3 16 4

Substituting these values of C2 & C3 in (2), we get

9 C1 + C1

+ 16

1 C = 12"

4 1

C1 29

C2 16


40


1 x 6.62" 1.66"

1 3

A Weight A Weight

30° 20′ 8" 2 30° 20′ 10" 3

30° 20′ 10" 3 30° 20′ 9" 4

30° 20′ 6" 2 30° 20′ 10" 2

Sum of weights = 2 + 3 + 2 + 3 + 4 + 2 =16

C3 4


A = 77° 14' 20" + 6.62" = 77° 14' 26.62"

B = 49° 40' 35" + 3.72" = 49° 40' 38.72"

c = 53° 04' 53" + 1.66" = 53° 4' 54.66"

Sum = 1800 00' 00''

5. What is meant by weight of an observation and enumerate laws of weights giving

examples. (CO2-H2) (AUC Apr/May 2010) (AUC Nov/Dec 2012)

Weight of an observation:

Weight of an observation is a measure of its relative worth which may be indicated by a

number. Thus if a certain observation is said to have weightage 5, it is meant to say that it is five

times as much as an observation of weight 1.

Laws of weights: (1) The weight of the arithmetic mean of the measurements of unit weight is equal to

the number of observations.

For example, let an angle A be measured six times, the following being the values:

A Weight A Weight

30° 20′ 8" 1 30° 20′ 10" 1

30° 20′ 10" 1 30° 20′ 9" 1

30° 20′ 7" 1 30° 20′ 10" 1

1 Arithmetic mean = 30° 20′ +

6

(8" + 10" + 7" + 10" + 9" + 10")

= 30° 20′ 9".

Weight of arithmetic mean = number of observations = 6.

(2) The weight of the weighted arithmetic mean is equal to the sum of the individual

weights.

For example, let an angle A be measured six times, the following being the values:

Arithmetic mean = 30° 20′ + 1/16 (8"X2 + 10" X3+ 7"X2 + 10"X3 + 9" X4+ 10"X2)

= 30° 20′ 9".

Weight of arithmetic mean = 16.

(3) The weight of algebric sum of two or more quantities is equal to the reciprocals of

the sum of individual weights.

For Example angle, A = 30° 20′ 8", Weight 2

B = 15° 20′ 8", Weight 3

1Sum of reciprocals of individual weights =

4 2 4


41


1 4

1 1 3 3

4 2 4

1 4

1 1 3 3

4 2 4

2

B)

1 Weight of A + B = 45° 40′ 16" =

1 Weight of A - B = 15° 00′ 00" =

(4) If a quantity of given weight is multiplied by a factor, the weight of the result is

obtained by dividing its given weight by the square of the factor.

For example, let A = 42° 10′ 20", weight 6.

Then weight of 3A = 126o 31' = 6

32

3

(5) If a quantity of given weight is divided by a factor, the weight of the result is

obtained by multiplying its given weight by the square of the factor.

For example, let A = 42° 10′ 30", weight 4.

AThen weight of 3

= 14o 3' 30" = 4 (3)2

= 36

(6) If a equation is multiplied by its own weight, the weight of the resulting equation is

equal to the reciprocal of the weight of the equation.

3 For example, let A + B = 98° 20′ 30", weight .

5 3 5Then weight of (A 5

= 59o 0' 18" is equal to . 3

(7) The weight of the equation remains unchanged, if all the signs of the equation are

changed or if the equation is added or subtracted from a constant.

For example, let A + B = 80° 20′ 00", weight 3.

Then weight of 180o - (A + B) or 99o 40' 00" is equal to 3.

6. The following are the observed values of an angle

Angle

Weight

18° 09' 18"

2

18° 09' 19" 3

18° 09' 20"

2

Determine probable error of observation of weight 3 and that of the weighted arithmetic

mean. (CO2-H2) (AUC Apr/May 2010)

Solution:

As the error is in seconds only, the needed details are arranged as under.


42


(19 x 3) (20 x 2) 133 19"

3 2

1 1 2 1 1 1 1 1

wv2

0.6745

1

0.6745 1

wv 2

0.6745 w (n 1)

0.6745 4

0.36

0.95 0.55

Value Weight (value x weight) v v2 wv2

18" 2 36" -1 1 2

19" 3 57" 0 0 0

20" 2 40" 1 1 2

Lw = 7 Weighted mean = 19" L wv2 = 4

Weighted arithmetic mean = (18 x 2)

2 7

v1 8" 9" "; v2 9" 9" 0 ; v3 0" 9" "

Probable error of single observation of unit weight = ES

=

n

4 =

= 0.95

i) Probable error of weighted arithmetic mean:

= =

ii) Probable error of single observation of weight 3 :

3

7 X 2

= ES

w 3

7. Find the most probable values of the angles A, B, C from the following observations at a

station P. (CO2-H2) (AUC Apr/May 2011)

A = 38° 25' 20" Weight 1

B = 32° 36' 12" Weight 1

A+B = 71° 01' 29" Weight 2

A+B+C = 119° 10' 43" Weight 1

B+C = 80° 45' 28" Weight 2

Solution:

Normal equation of A:

A = 38° 25' 20"

2A + 28 = 142° 2' 58"

A + 8 + c = 119° 10' 43"

4A + 38 + c = 299° 39' 1" ……………. (1)


43


Normal equation of B:

8 = 32° 36' 12"

2A + 28 = 142° 2' 58"

28 + 2c = 161° 30' 56"

A + 8 + c = 119° 10' 43"

3A + 68 + 3c = 455° 20' 49" ……………. (2)

Normal equation of C:

28 + 2c = 161° 30' 56"

A + 8 + c = 119° 10' 43"

A + 38 + 3c = 280° 41' 39" ……………. (2)

The three normal equations are

4A + 38 + c = 299° 39' 1"

3A + 68 + 3c = 455° 20' 49"

A + 38 + 3c = 280° 41' 39"

8y solving above equations we get,

A = 38° 25' 17.71"

B = 32° 36' 11.52"

C = 48° 09' 15.57"

8. i) Form the normal equations for x, y and z in the following equation of equal weight:

3x + 3y + z - 4 = 0

x + 2y + 2z - 6 = 0

5x + y + 4z - 21 = 0

ii) If the weights of the above equation are 2, 3 and 1 respectively form the normal

equations for x, y and z. (CO2-H2) (AUC Apr/May 2011)

Solution:

i) Form the normal equations for x, y and z with equal weight:

a) Forming the normal equation for x:

Coefficients of x are 3, 1, 5.

9x + 9y + 3z - 12 = 0

x + 2y + 2z - 6 = 0

25x + 5y + 20z - 105 = 0

35x + 16y + 25z - 123 = 0 …………….. (1)

b) Forming the normal equation for y:

Coefficients of x are 3, 2, 1.

9x + 9y + 3z - 12 = 0

2x + 4y + 4z - 12 = 0

5x + y + 4z - 21 = 0

16x + 14y + 11z - 45 = 0 …………….. (2)


44


\

c) Forming the normal equation for z: Coefficients of x are 1, 2, 4.

3x + 3y + z - 4 = 0

2x + 4y + 4z - 12 = 0

20x + 4y + 16z - 84= 0

25x + 11y + 21z - 100 = 0 …………….. (3)

Hence the normal equations for x, y and z are

35x + 16y + 25z - 123 = 0

16x + 14y + 11z - 45 = 0

25x + 11y + 21z - 100 = 0

ii) Form the normal equations for x, y and z with weights 2, 3 & 1:

a) Forming the normal equation for x:

Coefficients of x are (3x2), (1x3), (5x1).

18x + 18y + 6z - 24 = 0

3x + 6y + 6z - 18 = 0

25x + 5y + 20z - 105 = 0

46x + 29y + 32z - 147 = 0 …………….. (1)

b) Forming the normal equation for y:


18x + 18y + 6z - 24 = 0

6x + 12y + 12z - 36 = 0

5x + y + 4z - 21 = 0

29x + 31y + 22z - 81 = 0 …………….. (2)

c) Forming the normal equation for z:


6x + 6y + 2z - 8 = 0

6x + 12y + 12z - 36 = 0

20x + 4y + 16z - 84 = 0

32x + 22y + 30z - 128 = 0 …………….. (3)

Hence the normal equations for x, y and z are

46x + 29y + 32z - 147 = 0

29x + 31y + 22z - 81 = 0

32x + 22y + 30z - 128 = 0

9. Explain the laws of accidental errors. (8) (CO2-H2) (AUC May/June 2009)

Investigations of observations of various types show that accidental errors follow a

definite law, the law of probability. This law defines the occurrence of errors and can be

expressed in the form of equation which is used to compute the probable value or the probable

precision of a quantity. The most important features of accidental errors which usually occur are:


45


v 2

0.6745 1

v 2

l) n

Es

2 2 2 2

El E2 E3 E4

(i) Small errors tend to be more frequent than the large ones; that is they are the most

probable.

(ii) Positive and negative errors of the same size happen with equal frequency ; that is, they

are equally probable.

(iii) Large errors occur infrequently and are impossible.

Some of the errors can be defined under laws of accidental errors as follows:

i) Probable error of a single measurement is given by

Es

n

Where Es = probable error of single observation

v = difference between any single observation and the mean

n = number of observations.

ii) Probable error of an average is given by

E m

( n

Em

n

iii)Probable error of a sum

Probable error of measurement (sums and differences)

Where E1, E2, E3 ……….. En are probable errors.

10. What is meant by triangulation adjustment? Explain the different conditions and cases

with sketches. (8) (CO2-L3) (AUC May/June 2009)

Triangulation adjustment:


46

Civil Department

Surveying-II


47


11. Give the general rules for the adjustments of a geodetic triangle. (8)

(CO2-L1) (AUC May/June 2009)


48

Civil Department

Surveying-II


49


e

0 e e

) )

2

12. Some leveling was carried out with the following results.

Rise or Fall Weight

P to Q +4.32m 1

Q to R +3.17m 1

R to S +2.59m 1

S to P -10.04m 1

Q to S +5.68 m 2

The R.L of P is known to be 131.31 m above datum. Determine the probable levels of other

points. (CO2-H2) (AUC May/June 2012)

Solution:

PQRSP is closed circuit.

Error of closure = 4.32 + 3.17 + 2.59 - 10.04 = 0.04 Let e1, e2, e3 and e4 be the corrections to be observed quantities taken in order.

Hence the condition equation is

e1 + e2 + e3 + e4 = - 0.04

As the weightage is 1.

e1 = e2 = e3 = e4 = - 0.01

RL of P = 131.31 m

And rise = 4.32 - 0.01 = 4.31 m

RL of Q = 131.31 + 4.31 = 135.62 m

And rise = 3.17 - 0.01 = 3.16 m

RL of R = 135.62 + 3.16 = 138.78 m

And rise = 2.59 - 0.01 = 2.58 m

RL of S = 138.78 + 2.58 = 141.36 m

Subtract fall = - 10.04 - 0.01 = - 10.05 m

RL of P = 141.36 - 10.05 = 131.31 m

Hence satisfied. However the measurement from Q to S gave a rise of + 5.68 m with

weightage of 2.

considering the points Q and S again

Difference in RL from Q to S = 141.36 - 135.62 = 5.74 m

Error = 5.74 - 5.68 = 0.06 Let e2 & e4 be the corrections considered, then

e2 + e4 = - 0.06 ………………. (i)

From least square condition

2 e 2 2 4

= 0 ………………… (ii)

Differentiating eqn. (i) & (ii)

2

2 e2

δ e2

4

e4

δ e4

0

Multiplying eqn. (i) by - λ and adding with eqn. (ii) we have,

e ( 2 e 2 2

e ( e 0 4 4


50


3 e

i.e., 2 e2

e4

0 ( or ) e2 2

0 ( or ) e4

i.e., e2 4

0 0.06

i.e., 0.04

Hence probable levels of points are

RL of P = 131.31 m

RL of Q = 135.62 - 0.02 = 135.60 m

RL of R = 138.78 m

RL of S = 141.36 - 0.04 = 141.32 m

13. The following are the mean values observed in the measurement of three angles A, B and

C at a station.

A 76o 42' 46.2" Weight 4

A+B 134o 36' 32.6" Weight 3

B+C 185o 35' 24.8" Weight 2

A+B+C 262o 18' 10.4" Weight 1

Calculate the most probable value of each angle using normal equation.


Solution:

Normal equation of A: 4A = 306o 51' 4.8"

3A + 38 = 403° 49' 37.8"

A + 8 + c = 262° 18' 10.4"

8A + 48 + c = 972° 58' 53" ……………. (1)


3A + 38 = 403° 49' 37.8"

28 + 2c = 371° 10' 49.6"

A + 8 + c = 262° 18' 10.4"

4A + 68 + 3c = 1037° 18' 37.8" ……………. (2)

Normal equation of C:

28 + 2c = 371° 10' 49.6"

A + 8 + c = 262° 18' 10.4"

A + 38 + 3c = 633° 29' 00" ……………. (2)


8A + 48 + c = 972° 58' 53"


51


4A + 68 + 3c = 1037° 18' 37.8"

2A + 38 + 3c = 633° 29' 00"


A = 76° 42' 46.11"

B = 57° 53' 46.49"

c = 127° 41' 38.14"

14. Find the most probable value of the following.

A = 28o 24' 27.4"

B = 32o

14' 16.3"

c = 51o 18' 18.8"

A+B = 60o 38' 45.6"

B+c = 83o 32' 28.2". (CO2-H2) (AUc Nov/Dec 2012)

Solution:

8y using normal equation method we find the most probable values of A, 8 and c.

Normal equation of A: A = 28

o 24' 27.4"

A + 8 = 60o 38' 45.6"

2A + 8 = 89° 03' 13" ……………. (1)


8 = 32o 14' 16.3"

A + 8 = 60o 38' 45.6"

8 + c = 83o 32' 28.2"

A + 38 + c = 176° 25' 30.1" ……………. (2)

Normal equation of c:

c = 51o

18' 18.8"

8 + c = 83o 32' 28.2"

8 + 2c = 134° 50' 47" ……………. (1)


2A + 8 = 89° 03' 13"

A + 38 + c = 176° 25' 30.1"

8 + 2c = 134° 50' 47"


A = 28° 24' 28.98"

B = 32° 14' 15.05"

c = 51° 18' 15.98"


52


VI

eI

V2

e2

Vn

en

..........

V2

....... Vn v v

l ............(2)

v e

v e

v

e M .......

e

M

vn

rn

.........

vl

rl

v2

r2

15. Explain the general principles of least squares. (CO2-H2) (AUc Nov/Dec 2012)

According to the principle of least squares, the most probable value of an observed

quantity available from a given set of observations is the one for which the sum of the squares of

the residual errors is a minimum. When a quantity is being deduced from a series of observations,

the residual errors will be the difference between the adopted value and the several observed

values,

Let V1, V2, V3 etc. be the observed values

x = most probable value

then, x

x

x .. (l)

Where e's are the respective errors of the observed value.

If M = arithmetic mean, then

M n n

Where, n = number of observed values.

From equation (1)

n x

x n n

but n

x

= M from (2)

.......(3) n

If n is large and e is kept small by making precise measurement,

infinitesimal with respect to M. Hence x

becomes practically n

Thus the arithmetic mean is the true value where the number of observed value is very large.

Let rl , r2 , r3 ,........rn , be the residuals,

M

M

M ..(4)

Adding the above,


53


v r

v r

v

r 0 ........

vl

rl

'

v2

r2

'

vn

rn

' ........

r 2

nM 2

v 2

2M v ........

r '2

nN 2

v 2

2N v ........

v in e

r 2 M v 2M v v 2 v 2 M v

v 2 v r 2 v 2 where, M

v 2 v 2 r 2 ....

2

2 v r '2 nN 2 r 2N v

n

2

2 v v r

2 n N 2N

2 n n

2

v r '2 r2 n N

n 2

v N

n

nM

M n n

M n

here, ............(5) n

The sum of the residuals equals zero and sum of plus residual equals the sum of the minus

residuals.

Let N be any other value of the unknown other than arithmetic mean.

N

N

N ...(6)

Squaring eqn (4) and adding we get,

......(7)

Similarly squaring eqn (6) and adding we get,

......(8)

nM qn (7)

n n

or

...............(9) n

Substituting L92

of eqn (9) in eqn (8),

Where is always positive, Lr2 is less than Lr'2.

"The sum of the squares of the residuals found by the use of the arithmetic mean is a

minimum".


54


l 2

16. Adjust the following angles closing the horizon at a station.

A = 122o 05' 58.9" weight 1

B = 86o 45' 16.4" weight 1

c = 72o 50' 31.2" weight 3

D = 78o 18' 16.6" weight 1. (CO2-H2) (AUc Nov/Dec 2012)

Solution:

Sum of observed angles = 122o 05' 58.9" + 86o 45' 16.4" + 72o 50' 31.2" + 78o 18' 16.6"

= 3600 0' 3.1''

Error = + 3.1"

Total correction = - 3.1"

Let c1, c2, c3 & c4 be the corrections to the observed angles A, 8, c and D. The error

will be distributed to the angles in an inverse proportion to their weights.

A = 122o 05' 58.9" + c1

8 = 86o 45' 16.4" + c2

c = 72o 50' 31.2" + c3

D = 78o 18' 16.6" + c4

c1 : c2 : c3 : c4 = (1 )

: (1 )2

: ( 3 )2

: (1 )2

: 1 : 9 : 1 ……………..……………. (1)

Also, c1 + c2 + c3 + c4 = 3.1" ……………..……………. (2)

From (1) c c 2 1

c 9 c 3 1

c c 4 1

Substituting these values of c2, c3 & c4 in (2), we get

c1 + c1 + 9 c1 + c1 = 3.1"

c 0.258" 1

c 0.258" 2

c 9 X 0.258" 3

c 0.258" 4

2.322"


A = 122o 05' 58.9" - 0.258" = 1220 05' 58.64''

8 = 86o 45' 16.4" - 0.258" = 860 45' 16.14''

c = 72o 50' 31.2" - 2.322" = 720 50' 28.88''

D = 78o 18' 16.6" - 0.258" = 780 18' 16.34''

Sum = 3600 00' 00''


55


2 2 2 w2 r2 w3r3 ....... wn nr min imum

N vl

N v2

N v3

N vn

2 2 2 2

Vl ) w2 ( N V2 ) w3 ( N V3 ) ....... wn ( N Vn ) 0 ............. w2v

2 w

3v

3 ......... w

nv

n wlvl

w2

w3

......... wn

wr 2

0.6745

l

probable error of sin gle observation of unit weight ES ................(5)

w r

17. Define the following terms (i) True error, (ii) Residual error, (iii) Most probable error.

(CO2-L2) (AUc May/June 2013)

i) True error:

A true error is the difference between the true value of a quantity and its observed value.

ii) Residual error:


observed value.

iii) Most probable error:

Most probable error is defined as that quantity which added to and subtracted from the

most probable value which fixes the limits. 8y this limits there is an even chance the true

value of the measured quantity may lie.

18. How will you obtain error from direct observations of unequal weights on a single

quantity? (CO2-L1)

In case of observations made with unequal weights, the most probable value of the

observed quantity is equal to the weighted arithmetic mean of the observed quantities.

8ased on the principle of least squares the most probable values of the observed quantities with

unequal weights are those which make the sum of the weighted squares of the residual errors a

minimum.

Then by the above concept 2

1 1

where,

r1

r2

r3

rn …………………………….. (1)

N = the most probable value of quantity.

V1, V2, V3, Vn = the observed quantities with weights w1, w2, w3, etc.

w1 ( N ... (2)

Rearranging, N w1

.........................(3)

With the knowledge of N, the residual errors r1, r1, r1, etc can be found from eqn (1)

i) Probable error of a single observation of unit weight (ES)

ES n

.....................(4)

ii) Probable error of single observation of weight (w)

w weight w


56


wr 2 0.6745

1)

wr 2 0.6745

w (n 1)

2

0

0 2

0 e e e e

0 e e e e

0

0

0

0

1 3 e

w w ( n

........................(6)

iii) Probable error of weighted arithmetic mean (Em)

Em ............................(7) 19. The following angles were measured at a station 'O' so as to close the horizontal angles:

Adjust the angles by method of correlates. (CO2-H2)

Inst station Angle Weight

AOB 83 ° 42' 28.75" 3

BOc 102 ° 15' 43.26" 2

cOD 94 ° 38' 27.2" 4

DOA 79 ° 23' 23.77" 2

Solution:

Sum of observed angles = 83 ° 42' 28.75" + 102 ° 15' 43.26" + 94 ° 38' 27.2" + 79 ° 23' 23.77"

= 360° 00' 2.98"

Error = + 2.98"

Total correction = - 2.98"

Let e1,e

2,e

3 & e

4 be the corrections.

e1

e2

e3 e4 .98" ………………. (1)

From least square condition, w e2

3 e 2

2 e 2 2

4 e 2

4 2

………………. (2)

Differentiate (1) & (2) we get,

1 2 3 4 ……………….. (3)

3e1 1

2 e2 2 4 e

3 3 2 e4 4 ……… (4)

Multiply (3) by - & add with (4), we get

3e1

2 e

2

4 e

3

2 e4

; e1 3

; e2 2

; e3 4

; e4 2

Substitute the above value in (1)

3 +

2 +

4 +

2 = 2.98"


57


= 1.88"

e1

0.63"

e2

0.94"

e3

0.47"

e4 0.94"

The most probable values are

A = 83 ° 42' 28.75"- 0.63" = 830 42' 28.12''

8 = 102 ° 15' 43.26" - 0.94" = 1020 15' 42.32''

c = 94 ° 38' 27.2" - 0.47" = 940 38' 26.73''

D = 79 ° 23' 23.77" - 0.94" = 790 23' 22.83''

Sum = 3600 00' 00''

S.K.P. Engineering College,Tiruvannamalai IV Sem

58


UNIT-III PART –A 1. Define Total Station. [CO3 –L2] EDMs are now being incorporated with an Electronic theodolite and a microprocessor with memory unit, so called Total Station. They can simultaneously and automatically measure both distances and angles. They record field notes electronically and transmit them to computers, plotters and other office equipment for processing. These instruments can record horizontal and vertical angles together with slope distances. 2. What are the basic principles of EDM? [CO3 –L2] The basic principle of EDM instrument is the determination of time required for electro-magnetic waves to travel between two stations. Here the velocity of electro-magnetic wave is the basis for computations of the distance. Electromagnetic energy propagates through to atmosphere in accordance with the following equation: Where V is the velocity of electromagnetic energy, in meters per seconds; f the modulated frequency of the energy, in hertz; and the wavelength, in meters. 3. What are the errors in EDM. [CO3 –L1] Zero error, Cyclic error, Scale error. 4. What is electromagnetic wave? [CO3 –L1] Electromagnetic waves are energy transported through space in the form of periodic disturbances of electric and magnetic fields. All electromagnetic waves travel through space at the same speed, c=2.99792458x10^8 mms commonly known as the speed of light. An electromagnetic wave is characterized by a frequency and a wavelength. These two quantities are related to the speed of light by the equation, Speed of light = frequency x wavelength. 5. What are types of EDM based on range? [CO3 –L1]

EDMs based on range: Sort Range EDM: This type EDM is used up to the range of 5km with the use of infra-red light as signal. Medium Range EDM: This type EDM is used up to the range of 100km with the use of microwave light as signal. Long Range EDM: This type EDM is used up to the range of 100km with the use of radio-waves light as signal. 6. List out the total station Instruments. [CO3 –L2] Distomat: Geodimeter: Tellurometer: 7. what are the corrections in EDM? [CO3 –L1] Zero corrections, Prism Intger. 8.What are the components of total station? [CO3 –L1] Total station is an instrument which consists of the following

(i) Distance measuring instrument (EDM) (ii) An angle measuring instrument (Theodolite)

(iii) A simple microprocessor.

9. Write the advantages of Total Station Survey? [CO3 –L3]


59


(i) quick setting of the instrument on the tripod using laser plummet (ii) On-board are computation programme to compute the area of the field

3. Greater accuracy in area computation because of the possibility of taking acrs in area computation. 4. Graphical view of plots and land for quick visualization. 5. Coding to do automated mapping. As soon as the field jobs are finished, the map of the area with dimensions is ready after data transfer.

10. Write the demerits of Total Station Surveying? [CO3 –L3] (i) Their use does not provide hard copies of field notes. Hence, it may be difficult for

the surveyor to look over and check the work while surveying. (ii) For an overall check of the surveying, it will be necessary to return to the office and

prepare the drawings using appropriate software. (iii) They should not be used for observations of the sun, unless special filters, such as

the Troelof’s prism are used. If not, the EDM part of the instrument will be damaged.

(iv) The instrument is costly, and for conducting surveys using total station skilled

personnel are required.

11. What are the applications of Total Station surveying? [CO3 –L3] Total stations are mainly used by land surveyors and civil engineers, either to record features as in topographic surveying or to set out features (such as roads, houses or boundaries). They are also used by archaeologists to record excavations and by police, crime scene investigators, private accident reconstructions and insurance companies to take measurements of sense. Meterologists also use total stations to track weather balloons for determining upper-level winds. 12. What are the types of accuracy of Total Station? [CO3–L2] Accuracy is highly dependent on leveling the instrument. Thus two leveling bubbles are provided on the instrument and are referred to the circular level is on horizontal axis of instrument just below scope of the total station. The accuracy of a total station is dependent on instrument type. Angle Accuracy (Horizontal of Vertical) can range from2”5”. Distance Accuracy can range from: +/- (0.8 +1 ppm x D ) mm to+/- (3 +3 ppm x D)mm where D =distance measured Sensitivity of Circular Level = 10’/2 m Sensitivity of plate Level = 30”/2 mm.

PART-B

1.Explain the basic principle of a total station (CO3-H1)

Although taping and theodolites are used regularly on site - total stations are also used extensively in surveying, civil engineering and construction because they can measure both

distances and angles.

A typical total station is shown in the figure below


60


Fig 3.1 Total Station

Because the instrument combines both angle and distance measurement in the same unit, it is known as an integrated total station which can measure horizontal and vertical angles as wellasslopedistances.

Using the vertical angle, the total station can calculate the horizontal and vertical distance components of the measured slope distance.

As well as basic functions, total stations are able to perform a number of different survey tasks and associated calculations and can store large amounts of data.

As with the electronic theodolite, all the functions of a total station are controlled by its microprocessor, which is accessed thought a keyboard and display. To use the total station, it is set over one end of the line to be measured and some reflector is positioned at the other end such that the line of sight between the instrument and the reflector is unobstructed (as seen in the figure below).

-The reflector is a prism attached to a detail pole

-The telescope is aligned and pointed at the prism

-The measuring sequence is initiated and a signal is sent to the reflector and a part of this signal is returned to the total station

-This signal is then analysed to calculate the slope distance together with the horizontal and vertical angles.


61


-Total stations can also be used without reflectors and the telescope is pointed at the point that needs to be measured

-Some instruments have motorised drivers and can be use automatic target recognition to search and lock into a prism - this is a fully automated process and does not require an operator.

-Some total stations can be controlled from the detail pole, enabling surveys to be conducted by one person

Fig 3.2 Measuring with a Total Station

Most total stations have a distance measuring range of up to a few kilometres,

when using a prism, and a range of at least lOOm in reflector less mode and an accuracy of 2-3mm at short ranges, which will decrease to about 4-5mm at lkm.

Although angles and distances can be measured and used separately, the most common applications for total stations occur when these are combined to define position in control surveys.

As well as the total station, site surveying is increasingly being carried out using GPS equipment. Some predictions have been made that this trend will continue, and in the long run GPS methods may replace other methods.


62


Although the use of GPS is increasing, total stations are one of the predominant instruments used on site for surveying and will be for some time.

Developments in both technologies will find a point where devices can be made both methods.

2.Explain in deatail about classification of total stations (CO3-H1) ELECTRO- OPTICAL SYSTEM

DISTANCE MEASUREMENT

When a distance is measured with a total station, am electromagnetic wave or pulse is used for the measurement - this is propagated through the atmosphere from the instrument to reflector or target and back during the measurement.

Distances are measured using two methods: the phase shift method, and the pulsed laser method.

This technique uses continuous electromagnetic waves for distance measurement although these are complex in nature, electromagnetic waves can be represented in their simplest from as periodic waves.

Fig 3.3 Sinusoidal wave motion

The wave completes a cycle when moving between identical points on the wave and the number of times in one second the wave completes the cycle is called the frequency of the wave. The speed of the wave is then used to estimate the distance.

LASER DISTANCE MEASUREMENT

In many total stations, distances are obtained by measuring the time taken for a pulse of


63


laser radiation to travel from the instrument to a prism (or target) and back. As in the

phase shift method, the pulses are derived an infrared or visible laser diode and they are transmitted through the telescope towards the remote end of the distance being measured, where they are reflected and returned to the instrument.

Since the velocity v of the pulses can be accurately determined, the distance D can be obtained using 2D = vt, where t is the time taken for a single pulse to travel from instrument - target - instrument.

This is also known as the timed-pulse or time-of-flight measurement technique.

The transit time t is measured using electronic signal processing techniques. Although only a single pulse is necessary to obtain a distance, the accuracy obtained would be poor. To improve this, a large number of pulses (typically 20,000 every second) are analysed during each measurement to give a more accurate distance.

The pulse laser method is a much simpler approach to distance measurement than the phase shift method, which was originally developed about 50 years ago.

SLOPE AND HORIZONTAL DISTANCES

Both the phase shift and pulsed laser methods will measure a slope distance L from the total station along the line of sight to a reflector or target. For most surveys the horizontal distance D is required as well as the vertical component V of the slope distance.

Horizontal distance D = L cosa = L sin z

Vertical distance = V = L sina = L cos z

Where a is the vertical angle and z is the is the zenith angle. As far as the user is concerned, these calculations are seldom done because the total station will either display D and V automatically or will dislplay L first and then D and V after pressing buttons

Fig Slope and Distance Measured


64


How accuracy of distance measurement is specified All total stations have a linear accuracy quoted in the form

±(a mm + b ppm)

The constant a is independent of the length being measured and is made up of internal

sources within the instrument that are normally beyond the control of the user. It is an

estimate of the individual errors caused by such phenomena as unwanted phase shifts in

electronic components, errors in phase and transit time measurements.

The systematic error b is proportional to the distance being measured, where 1 ppm (part

per million) is equivalent to an additional error of 1mm for every kilometre measured.

Typical specifications for a total station vary from ±(2mm + 2ppm) to ±(5mm + 5 pmm).

For example: ±(2mm + 2ppm), at 1OOm the error in distance measurement will be

±2mm but at 1.5km, the error will be ±(2mm + [2mm/km * 1.5km]) = ±5mm Reflectors used in distance measurement

Since the waves or pulses transmitted by a total station are either visible or infrared, a plane

mirror could be used to reflect them. This would require a very accurate alignment of the mirror,

because the transmitted wave or pulses have a narrow spread. To get around this problem special mirror prisms are used as shown below.

Fig 3.5 Reflector used in total station 3. What are the Features Of Total Stations (CO3-L2)

Total stations are capable of measuring angles and distances simultaneously and combine an

electronic theodolite with a distance measuring system and a microprocessor. ANGLE MEASUREMENT


65


All the components of the electronic theodolite described in the previous lectures are found total

stations. The axis configuration is identical and comprises the vertical axis, the tilting axis and line of

sight (or collimation). The other components include the tribatch with levelling footscrews, the

keyboard with display and the telescope which is mounted on the standards and which rotates

around the tilting axis. Levelling is carried out in the same way as for a theodolite by adjusting to centralise a plate level

or electronic bubble. The telescope can be transited and used in the face left (or face I) and

face right (or face II) positions. Horizontal rotation of the total station about the vertical axis is

controlled by a horizontal clamp and tangent screw and rotation of the telescope about the tilting

axis. The total station is used to measure angles in the same way as the electronic theodolite.

Distance measurement

All total stations will measure a slope distance which the onboard computer uses, together with

the zenith angle recorded by the line of sight to calculate the horizontal distance. For distances taken to a prism or reflecting foil, the most accurate is precise measurement.

For phase shift system, a typical specification for this is a measurement time of about 1-

2s, an accuracy of (2mm + 2ppm) and a range of 3-5km to a single prism.

Although all manufacturers quote ranges of several kilometres to a single prism.

For those construction projects where long distances are required to be measured, GPS methods

are used in preference to total stations. There is no standard difference at which the change

from one to the other occurs, as this will depend on a number of factors, including the accuracy

required and the site topography. Rapid measurement reduces the measurement time to a prism to between 0.5 and 1' s for both

phase shift and pulsed systems, but the accuracy for both may degrade slightly. Tracking measurements are taken extensively when setting out or for machine control, since

readings are updated very quickly and vary in response to movements of the prism which is

usually pole-mounted. In this mode, the distance measurement is repeated automatically at

intervals of less than O.5s. For reflector less measurements taken with a phase shift system, the range that can be

obtained is about 1OOm, with a similar accuracy to that obtained when using a prism or foil. KEYBOARD AND DISPLAY

A total station is activated through its control panel, which consists of a keyboard and multiple

line LCD. A number of instruments have two control panels, one on each face, which

makes them easier to use. In addition to controlling the total station, the keyboard is often used to code data generated by

the instrument - this code will be used to identify the object being measured. On some total stations it is possible to detach the keyboard and interchange them with other total stations and with GPS receivers.This is called integrated surveying


66


Fig 3.6 Key Board and Display SOFTWARE APPLICATIONS

The microprocessor built into the total station is a small computer and its main function is

controlling the measurement of angles and distances. The LCD screen guides the operator

while taking these measurements. The built in computer can be used for the operator to carry out calibration checks on

the instrument. The software applications available on many total stations include the

following: Slope corrections and reduced levels

Horizontal circle orientation

Coordinate measurement

Traverse measurements

Resection (or free

stationing) Missing line

measurement Remote

elevation measurement

areas

Setting out. 4.What are sources of error for total stations (CO3-H1)

1 CALIBRATION OF TOTAL STATIONS

To maintain the high level of accuracy offered by modern total stations, there is now much more

emphasis on monitoring instrumental errors, and with this in mind, some construction sites


67


require all instruments to be checked on a regular basis using procedures outlined in the quality

manuals. Some instrumental errors are eliminated by observing on two faces of the total station and

averaging, but because one face measurements are the preferred method on site, it is

important to determine the magnitude of instrumental errors and correct for them. For total stations, instrumental errors are measured and corrected using electronic calibration

procedures that are carried out at any time and can be applied to the instrument on site. These

are preferred to the mechanical adjustments that used to be done in labs by technician. Since calibration parameters can change because of mechanical shock, temperature changes

and rough handling of what is a high-precision instrument, an electronic calibration should

be carried our on a total station as follows: Before using the instrument for the first time

After long storage periods

After rough or long transportation

After long periods of work

Following big changes in temperature

Regularly for precision surveys

Before each calibration, it is essential to allow the total station enough to reach the ambient

temperature. 2 HORIZONTAL COLLIMATION (OR LINE OF SIGHT ERROR)

This axial error is caused when the line of sight is not perpendicular to the tilting axis. It affects

all horizontal circle readings and increases with steep sightings, but this is eliminated by

observing on two faces. For single face measurements,an on-board calibration function is

used to determine c, the deviation between the actual line of sight and a line perpendicular to the

tilting axis. A correction is then applied automatically for this to all horizontal circle readings.

Fig 3.7 Line of Sight error


68


.3 TILTING AXIS ERROR

This axial errors occur when the titling axis of the total station is not perpendicular to its

vertical axis. This has no effect on sightings taken when the telescope is horizontal, but

introduces errors into horizontal circle readings when the

telescope is tilted, especially for steep sightings. As with horizontal collimation error,

this error is eliminated by two face measurements, or the tilting axis error a is measured in

a calibration procedure and a correction applied for this to all horizontal circle readings -

as before if a is too big, the instrument should be returned to the manufacture.

Fig tilting axis error

4 COMPENSATOR INDEX ERROR

Errors caused by not levellinga theodolite or total station carefully cannot be eliminated by

taking face left and face right readings. If the total station is fitted with a compensator it

will measure residual tilts of the instrument and will apply corrections to the horizontal and

vertical angles for these. However all compensators will have a longitudinal error l and traverse error t known as

zero point errors. These are averaged using face left and face right readings but for single

face readings must be determined by the calibration function of the total station.


69


Fig 3.8 Compensator Index Error

A vertical collimation error exists on a total station if the Oo to 18Oo line in the

vertical circle does not coincide with its vertical axis. This zero point error is present in all

vertical circle readings and like the horizontal collimation error, it is eliminated by taking FL

and FR readings or by determining i

For all of the above total station errors (horizontal and vertical collimation, tilting axis

and compensator) the total station is calibrated using an in built function. Here the function

is activated and a measurement to a target is taken as shown below.

Following the first measurement the total station and the telescope are each rotated

through 180o and the reading is repeated.

Any difference between the measured horizontal and vertical angles is then

quantified as an instrumental error and applied to all subsequent readings automatically.

The total station is thus calibrated and the procedure is the same for all of the above error

type.


70


Fig 3.9 Compensator Index Error

71 Civil Department Surveying-II

UNIT—IV PART—A 1.What is meant by satellite constellation? [CO4 –L1] Space segment consist 21 GPS satellites with an addition of 3 active spares. These satellites are placed in almost six circular orbits with an inclination of 55 degree. Orbital height of these satellites is about 2,200 km corresponding to about 26,600 km provides repeated satellite configuration every day advanced by four minutes with respect to universal time. 2.What are the types of segments? [CO4 –L1] The GPS system of the system consists of three segments: The Space Segment, The Control Segment, and The User Segment. The Space and Control segments are directly operated, maintained, and developed by the US Air Force (USAF) on behalf of the Department of Defense . 3. Write short notes on space segments? [CO4 –L2] The Space Segment of the system consists of the system consists of the GPS Satellites. These space vehicles send radio signals from space. The nominal GPS Operational Constellation consists of 24 satellites that orbit the earth at approximately 20,200 km every 12 hours. There are often more than24 operational satellites as new ones are launched to replace older satellites. The satellite orbits repeat almost the same ground track (as the earth turns beneath them) once each day. 4.What short notes on control segments? [CO4 –L2] The Control Segment is a world-wide network of monitor and control stations that maintains the satellites in their proper orbits. It also tracks the GPS satellites, uploads data and software updates, and maintains the health and status of the entire constellation. The operational hub of this network is at Schriever Air Force Base east of Colorado Springs. 5.Write short notes on user segments? [CO4 –L2] The user segment of the system is your GPS receiver, which receives GPS signals and uses the received information to calculation its position and the time. The user Segment includes the equipment of the military personal and civilians who receive GPS signals. Military GPS user equipment has been integrated into fighters, bombers, tankers, helicopters, ship, submarines, tanks, jeeps, and soldiers’ equipment. 6.What is satellite? [CO4 –L2] A satellite is an object in space that orbits or circles around a bigger object. There are two kinds of satellites: natural (such as the moon orbiting the Earth) or an artificial body placed in orbit round the earth or another planet in order to collect information or for communication. 7.What is GPS? [CO4 –L2] The Global Positioning System (GPS) is a satellite-based system that can be used to locate positions anywhere on the earth. Operated by the U.S. Department of Defense (D o D), NAVSTAR (NAVigation Satellite Timing and Ranging) GPS provides continuous (24 hours/day), real-time, 3-dimensional positioning, navigation and timing worldwide. 8.What are the sources of error in GPS. [CO4 –L3]

1. Ionospheric and atmosphere delays. 2. Satellite and Receiver Clock Errors. 3. Multipath. 4. Dilution of Precision. 5. Selective Availability (S/A)


6. Anti Spoofing (A-S)

9.Mention any four applications using GPS. [CO4 –L3] a. GPS Applications in Agriculture. b. GPS Navigation: Land, Sea and Air c. GPS Applications: Mapping and Surveying

10.What are the GPS surveying techniques? [CO4 –L1] Following are the techniques that are commonly used:

i. Static. ii. Fast Static (Rapid Static). iii. Kinematic. iv. Pseudo-Kinematic (pseudo-static).

v. Real Time Kinematic.

PART-B 1. Briefly explain the Characteristics of GPS Navigation and Satellite navigation?(CO4-H1)

Traditional methods of surveying and navigation resort to tedious field an d a st ro n o mi ca l o bs er va tio n f or d eri v i n g p o s it ion a l an d di re cti o na l

information. Diverse field conditions, seasonal variation and many unavoidable

circumstances always bias the traditional field approach. However, due to rapid

advancement in electronic systems, every aspect of human life is affected to a great

deal. Field of surveying and navigation is tremendously benefited through electronic

devices. Many of the critical situations in surveying/navigation are now easily and

precisely solved in short time.

Astronomical observation of celestial bodies was one of the standard methods of

obtaining coordinates of a position. This method is prone to visibility and

weather condition and demands expert handling. Attempts have been made by USA

since early 1960's to use space based artificial satellites. System TRANSIT was widely

used for establishing a network of control points over large regions. Establishment of

modern geocentric datum and its relation t o l o c a l d at u m wa s s u c ce s s fu l l y a ch ie

ved through TRANSIT. Ra pi d improvements in higher frequently transmission and

precise clock signals along with advanced stable satellite technology have been

instrumental for the development of global positioning system.

The NAVSTAR GPS (Navigation System with Time and Ranging Global Positioning System) is a satellite based radio navigation system providing precise three-

dimensional position, course and time information to suitably equipped user.

GPS has been under development in the USA s ince 1973. The US

department of Defence as a worldwide navigation and positioning resource for

military as well as civilian use for 24 hours and all weather conditions primarily

developed it.

In its final configuration, NAVSTAR GPS consists of 21 satellites (plus 3 active

spares) at an altitude of 20200 km above the earth's surface (Fig. 1). These

satellites are so arranged in orbits to have atleast four satellites visible above the

horizon anywhere on the earth, at any time of the day. GPS Satellites transmit at


frequencies L1=1575.42 MHz and L2=1227.6 MHz modulated with two types of code

viz. P-code and CIA code and with navigation message. Mainly two types of observable

are of interest to the user. In pseudo ranging the distance between the satellite and the

GPS receiver plus a small corrective

GPS Nominal Constellation

24 Satellites in 6 Orbital Planes

4 Satellites in each Plane

20,200 km Altitudes, 55 Degree Inclination

Fig 4.1 The Global Positioning System (GPS), 21-satellite configuration

term for receiver clock error is observed for positioning whereas in carrier phase

techniques, the difference between the phase of the carrier signal transmitted by the

satellite and the phase of the receiver oscillator at the epoch is observed to derive the

precise information.

The GPS satellites act as reference points from which receivers on the ground

detect their position. The fundamental navigation principle is based on the

measurement of pseudoranges between the user and four satellites (Fig.2). Ground

stations pre c ise ly monitor the orbit of every satellite and by measuring the travel

time of the signals transmitted from the satellite four distances between receiver and

satellites will yield accurate position, direction and speed. Though three-range

measurements are suffic ient, the fourth observation is essential for solving clock synchronization error between receiver and satellite. Thus, the term "pseudoranges" is

derived. The secret of GPS measurement is due to the ability of measuring carrier

phases to about 1/100 of a cycle equaling to 2 to 3 mm in linear distance.

Moreover the high frequency L1 and L2 carrier signal can easily penetrate the ionosphere

to reduce its effect. Dual frequency observations are important for large station separation

and for eliminating most of the error parameters.


R1

Figure 4.2: Basic principle of positioning with GPS

There has been significant progress in the design and miniaturization of stable

clock. GPS satellite orbits are stable because of the high altitudes and no

atmosphere drag. However, the impact of the sun and moon on GPS orbit though significant, can be computed completely and effect of solar radiation pressure on the

orbit and tropospheric delay of the signal have been now modeled to a great

extent from past experience to obtain precise information for various applications.

Comparison of main ch a rac te rist ic s o f TRANS IT and GPS re veal technological advancement in the field of space based positioning system

(Table1).

Table 1. TRANSIT vs GPS

Details TRANSIT GPS

Orbit Altitude 1000 Km 20,200 Km

Orbital Period 105 Min 12 Hours

Frequencies 150 MHz

400 MHz 1575 MHz

1228 MHz

Navigation data 2D : X, Y 4D : X,Y,Z, t velocity

Availability 15-20 minute per pass Continuously

Accuracy fi 30-40 meters

(Depending on velocity

fi15m (Pcode/No. SA

0.1 Knots

Repeatability — fi1.3 meters relative

Satellite 4-6 21-24

Geometry Variable Repeating

Satellite Clock Quartz Rubidium, Cesium

GPS has been designed to provide navigational accuracy of ±10

m to ±15 m. However, sub meter accuracy in differential mode has


Segmen

Input Function Output

Space Navigation Generate and P-Code message Transmit code CIA Code

and carrier Ll,L2

Control P-Code Produce

GPS Navigation

Observations time predict

message

Time

User Code

observation

Navigation Position

velocity Carrier phase

solution time

observation Surveying

been achieved and it has been proved that broad varieties of problems in geodesy and geodynamics can be tackled through GPS.

Versatile use of GPS for a civilian need in following fields have been

successfully practiced viz. navigation on land, sea, air, space, high precision

kinematics survey on the ground, cadastral surveying, geodetic control network

densification, high precision aircraft positioning, photogrammetry without ground

control, monitoring deformations, hydrographic surveys, active control survey and

many other similar jobs related to navigation and positioning,. The outcome of a

typical GPS survey includes geocentric position accurate to 10 m and relative positions

between receiver locations to centimeter level or better.

2.What are the space, control and user segments of GPS and their

Functions?(CO4-H1) For better understanding of GPS, we normally consider three major segments

viz. space segment, Control segment and User segment. Space segment deals

withGPS satellites systems, Control segment describes ground based time and

orbit control prediction and in User segment various types of existing GPS

receiver and its application is dealt .

Table 2 gives a brief account of the function and of various

segments along with input and output information.

Table 2. Functions of various segments of GPS

ephemeris

Space Segment


User Segment

Control Segment

Figure 4.3: The Space, Control and User segments of GPS GLONASS (Global Navigation & Surveying System) a similar system to GPS is being

developed by former Soviet Union and it is considered to be a valuable complementary

sys te m to GPS for future application.

SPACE SEGMENT

Space segment will consist 2l GPS satellites with an addition of 3 active spares. These

satellites are placed in almost six circular orbits with an inclination of 55 deg re e. Orbital

height of thes e satellite s i s about 20,200 km corresponding to about 26,600

km from the semi major axis. Orbital period is exactly 12 hours of s idereal time

and this provides repeated satellite configuration every day advanced by four

minutes with respect to universal time.

Final arrangement of 21 satellites constellation known as "Primary

satellite constellation" is given in Fig. 4. There are six orbital planes A to F

with a separation of 60 degrees at right ascension (crossing at equator). The

position of a satellite within a particular orbit plane can be identified by argument of latitude or mean anomaly M for a given epoch.

Figure 4. 4: Arrangement of satellites in full constellation


GPS satellite s are broadly divided into three blocks: Block-I satellite pertains to

development stage, Block II represents production satellite and Block IIR are

replenishment/spare satellite. Under Block-I, NAVSTAR 1 to 11 satellites were launched before 1978 to 1985 in two

orbital planes of 63-degree inclination. Design life of these prototype test satellites was

only five years but the operational period has been exceeded in most of the cases.

The first Block-II production satellite was launched in February 1989 using channel

Douglas Delta 2 booster rocket. A total of 28 Block-II satellites are planned to support

21+3 satellite configuration. Block-II satellite s have a designed lifetime of 5-7 years.

To sustain the GPS facility, the development of follow-up satellites under Block-II R has

started. Twenty replenishment satellites will replace the current block-II satellite as and

when necessary. These GPS satellites under Block-IR have additional ability to

measure distances between satellites and will also compute ephemeris on board

for real timeinformation gives a schematic view of Block-II satellite. Electrical power

is generated through two solar panels covering a surface area of 7.2 square meter

each. However, additional battery backup is provided to provide energy when the

satellite moves into ear th's shadow region. Each satellite weighs 845kg and has a

propulsion system for positional stabilization and orbit maneuvers.

Fig 4.5 Schematic view of a Block II GPS satellite

GPS sate llite s have a very high performance frequency standard with an accuracy

of between lXl0-l2 to lXl0-l3 and are thus capable of creating precise time base. Block-I

satellites were partly equipped with only quartz oscillators but Block-II satellites have

two cesium frequency standards and two rubidium frequency standards. Using

fundamental frequency of l0.23 MHz, two carrier frequencies are generated to transmit

signal codes.

4.4 OBSERVATION PRINCIPLE AND SIGNAL STRUCTURE


NAVSTAR GPS is a o ne-wa y ran g in g s ystem i.e. s ignals are o n l y

transmitted by the satellite. Signal travel time between the satellite and the receiver

is observed and the range distance is calculated through the knowledge of

signal propagation velocity. One way ranging means that a clock reading at the

transmitted antenna is compared with a clock reading at the receiver antenna. But

since the two clocks are not strictly synchronized, the observed signal travel time

is biased with systematic synchronization error. Biased ranges are known as

pseudoranges. Simultaneous observations of four pseudoranges are necessary to

determine X, Y, Z coordinates of user antenna and clock bias.

Real time positioning through GPS signals is possible by modulating carrier

frequency with Pseudorandom Noise (PRN) codes. These are sequence of binary

values (zeros and ones or +1 and -1) having random character but identifiable

distinctly. Thus pseudoranges are derived from travel time of an identified PRN

signal code. Two different codes viz. P-code and CIA code are in use. P means

precision or protected and CIA means clearIacquisition or coarse acquisition.

P- code has a frequency of 10.23 MHz. This refers to a sequence of 10.23 million binary

digits or chips per second. This frequency is also referred to as the chipping rate of

P- code. Wavelength corresponding to one chip is 29.30m. The P-code sequence

is extremely long and repeats only after 266 days. Portions of seven days each

are assigned to the various satellites. As a consequence, all satellite can transmit

on the same frequency and can be identified by their unique one-week segment. This

technique is also called as Code Division Multiple Acces s (CDMA). P-code is the

primary code for navigation and is available on carrier frequencies L1 and L2.

The CIA code has a length of only one millisecond; its chipping rate is 1.023 MHz with corresponding wavelength of 300 meters. CIA code is only transmitted on L1 carrier.

GPS receiver normally has a copy of the code sequence for determining the signal propagation time. This code sequence is phase-shifted in time step- by-

step and correlated with the received code signal until maximum correlation is achieved. The necessary phase-shift in the two sequences of codes is a measure of the

signal travel time between the satellite and the receiver antennas. This technique can be explained as code phase observation.

For precise geodetic applications, the pseudoranges should be derived from

phase measurements on the carrier signals because of much higher resolution. Problems of ambiguity determination are vital for such observations.

The third type of signal transmitted from a GPS satellite is the broadcast message sent at a rather slow rate of 50 bits per second (50 bps) and repeated every

30 seconds. Chip sequence of P-code and C IA code are se p arately combined with the stream of message bit by binary addition ie the same value for code and message

chip gives 0 and different values result in 1.

The main features of all three signal types used in GPS observation viz carrier, code and data signals are given in Table 3.


GPS Satellite Signals

Atomic Clock (G, Rb) fundamental 10.23. MHz

L1 Carrier Signal 154 X 10.23 MHz

L1 Frequency 1575.42 MHz

L1 Wave length 19.05 Cm

L2 Carrier Signal 120 X 10.23 MHz

L2 Frequency 1227.60 MHz

L2 Wave Length 24.45 Cm

P-Code Frequency (Chipping Rate) 10.23 MHz (Mbps)

P-Code Wavelength 29.31 M

P-Code Period 267 days : 7

CIA-Code Frequency (Chipping Rate) 1.023 MHz (Mbps)

CIA-Code Wavelength 293.1 M

CIA-Code Cycle Length 1 Milisecond

Data Signal Frequency 50 bps

Data Signal Cycle Length 30 Seconds

The signal structure permits both the phase and the phase shift (Doppler effect) to be

measured along with the direct signal propagation. The necessary bandwidth is

achieved by phase modulation of the PRN code as illustrated in Fig. 6.

Tim e Carrie r

PRN +1 Code -1

Fig 4.6 Generation of GPS Signals STRUCTURE OF THE GPS NAVIGATION DATA


Structure of GPS navigation data (message) is shown in Fig. 7. The user has to

decode the data signal to get access to the navigation data. For on line navigation

purposes, the internal processor within the receiver does the decoding. Most of the

manufacturers of GPS receiver provide decoding software for post processing purposes.

With a bit rate of

50 bps and a cycle time of 30 seconds, the total information content of a navigation data set is 1500 bits. The complete data frame is subdivided into five subframes of six- second duration comprising 300 bits of information. Each subframe contains the data words of 30 bits each. Six of these are control bits. The first two words of each subframe are the Telemetry Work (TLM) and the C/A-P-Code Hand over Work (HOW). The TLM work contains a synchronization pattern, which facilitates the access to the navigation data. Since GPS is a military navigation system of US, a limited access to the total system accuracy is made availab le to the c ivilian users. The service available to the civilians is called Standard Positioning System (SPS) while the service available to the authorized users is called the Precise Positioning Service (PPS). Under current policy the accuracy available to SPS users is 100m, 2D- RMS and for PPS users it is 10 to 20 meters in 3D. Additional limitation viz. Anti-Spoofing (AS), and Selective Availability (SA) was further imposed for civilian users. Under AS, only authorized users will have the means to get access to the P-code. By imposing SA condition, positional accuracy from Block-II satellite was randomly offset for SPS users. Since May 1, 2000 according to declaration of US President, SA is switched off for all users.

Fig 4.7 Data block

The navigation data record is divided into three data blocks:

Data Block I appears in the firs t subframe and contains

the c lock coefficient/bias.

Data Block II appears in the second and third subframe

and contains all necessary parameters for the

computation of the satellite coordinates.

Data Block III appears in the fourth and fifth subframes and

contains the almanac data with clock and

ephemeris parameter for all available satellite of

GPS


system. This data block includes also

ionospheric correction parameters and particular

alphanumeric information for authorized users.

Unlike the first two blocks, the subframe four and five are not

repeated every 30 seconds.

International Limitation of the System Accuracy

The GPS system time is defined by the cesium oscillator at a selected monitor

station. However, no clock parameter are derived for this station. GPS time is

indicated by a week number and the number of seconds since the beginning of

the current week. GPS time thus varies between 0 at the beginning of a week to

6,04,800 at the end of the week. The initial GPS epoch is January 5, 1980 at 0 hours Universal Time. Hence, GPS week starts at Midnight (U T ) between

Satu rda y and Su n da y. Th e GPS time is a continuous time scale and is defined

by the main clock at the Master Control Station (MCS). The leap seconds is UTC

time scale and the drift in the MCS clock indicate that GPS time and UTC

are not identical. The difference is continuously monitored by the control

segment and is broadcast to the users in the navigation message. Difference of about

7 seconds was observed in July,

1992.

Figure 4.8 Data Flow in the determination of the broadcast ephemeris GPS satellite is identified by two different numbering schemes. Based on launch

sequence, SVN (Space Vehicle Number) or NAVSTAR number is allocated. PRN

(Pseudo Random Noise) or SVID (Space Vehicle Identification) number is related to

orbit arrangement and the particular PRN segment allocated to the individual

satellite. Usually the GPS receiver displays PRN number.

4.6 CONTROL SEGMENT

Control segment is the vital link in GPS technology. Main functions of

82


the control segment.

- Monitoring and controlling the satellite system continuously

- Determine GPS system time

- Predict the satellite ephemeris and the behavior of each satellite clock.

- Update periodically the navigation message for each particular satellite.

For continuos monitoring and controlling GPS satellites a master

control stations (MCS), several monitor stations (MS) and ground

antennas (GA) are located around the world (Fig. 9). The operational control segment (OCS) consists of MCS near Colorado springs (USA),

three MS and GA in Kwajaleian Ascension and Diego Garcia and two

more MS at Colorado Spring and Hawai. 4.7 GROUND CONTROL SEGMENT

The monitor station receives all visible satellite signals and determines their

pseudorages and then tran sm it s the ran g e da t a alo n g w it h the local

meteorological data via data link to the master control stations. MCS then

precomputes satellite ephemeris and the behaviour of the satellite clocks and

formulates the navigation data. The navigation message data are transmitted to the

ground antennas and via S-band it links to the satellites in view. Fig. 9 shows this

process schematically. Due to systematic global distribution of upload antennas, it is

possible to have atleast three contacts per day between the control segment and each

satellite.

4.8 USER SEGMENT

Appropriate GPS receivers are required to receive signal from GPS satellites for

the purpose of navigation or positioning. Since, GPS is still in its development

phase, many rapid advancements have completely eliminated bulky first

generation user equipments and now miniature powerful models are frequently

appearing in the market.

3.Explain in detail about receivers (CO4-H1) The main components of a GPS receiver are shown in Fig. 10. These are:

- Antenna with pre-amplifier

- RF section with signal identification and signal processing

- Micro-processor for receiver control, data sampling and data processing

- Precision oscillator

83


- Power supply

- User interface, command and display panel

- Memory, data storage

An t en n a a

nd p r

e amplifie

r

Signal processor Code

tracking loop Carrier

tracking loop

Precision

oscillator Micro proces

sor Memory

External

power supply Command &

display unit External data

logger


Fig 4.9 Major components of a GPS receiver

ANTENNA Sensitive antenna of the GPS receiver detects the electromagnetic wave

signal transmitted by GPS satellites and converts the wave energy to electric current]

amplifies the signal strength and sends them to receiver electronics.

Several types of GPS antennas in use are mostly of following types (Fig.).

Mono pole Helix Spiral helix Microstrip Choke ring

Types of GPS Antenna

- Mono pole or dipole

- Quadrifilar helix (Volute)

- Spiral helix

- Microstrip (patch)

- Choke ring

Microstrip antennas are most f req uen tl y used because of its added advantage for airborne application, materialization of GPS receiver and easy construction.

However, for geodetic needs, antennas are designed to receive both carrier frequencies Ll and L2. Also they are protected against multipath by extra ground planes or by using

choke rings. A choke ring consists of strips of conductor which are concentric with the vertical axis of the antenna and connected to the ground plate which in turns reduces

the multipath effect.

RF Section with Signal Identification and Processing

The incoming GPS signals are down converted to a lower frequency in the RS

section and processed within one or more channels. Receiver channel is the primary electronic unit of a GPS receiver. A receiver may have one or more channels. In

the parallel channel concept each channel is continuouslymfranking one particular

satellite. A minimum of four parallel channels is required to determine position and time. Modern receivers contain upto l2 channels for each frequency.


In the sequencing channel concept the channel switches from satellite to

satellite at regular interval. A single channel receiver takes atleast four times of 30 seconds to establish first position fix, though some receiver types have a dedicated

channel for reading the data signal. Now days in most of the cases fast sequencing

channels with a switching rate of about one-second per satellite are used.

In multiplexing channel, sequencing at a very high speed between different satellite s is

achieved using one or both fre q ue n cie s. The switching rate is synchronous with

the navigation message of 50 bps or 20 milliseconds per bit. A complete sequence

with four satellites is completed by 20 millisecond or after 40 millisecond for dual

frequency receivers. The navigation message is continuous, hence first fix is achieved

after about 30 seconds.

Though continuous tracking parallel channels are cheap and give good overall

performance, GPS receivers based on multiplexing technology will soon be available at

a cheaper price due to electronic boom. Microprocessor

` To control the operation of a GPS receiver, a microprocessor is essential

for acquiring the signals, processing of the signal and the decoding of the

broadcast message. Additional capabilities of computation of on-line position and

velocity, conversion into a given local datum or the determination of waypoint

information are also required. In future more and more user relevant software will

be resident on miniaturized memory chips.

Precision Oscillator

A reference frequency in the receiver is generated by the precision

oscillator. Normally, less expensive, low performance quartz oscillator is used in receivers

since the precise clock information is obtained from the GPS satellites and the user

clock error can be eliminated through double differencing technique when all

partic ipating receivers ob ser ve at exactl y the same epoch. For navigation with two or

three satellites only an external high precision oscillator is used.

Power Supply

First generation GPS receivers consumed very high power, but modern

receivers are designed to consume as little energy as possible. Most receivers have

an internal rechargeable. Nickel-Cadmium battery in addition to an external power

input. Caution of low battery signal prompts the user to ensure adequate arrangement of

power supply.

Memory Capacity


For port processing purposes all data have to be stored on internal or

external memory devices. Post processing is essential for multi station techniques

ap p licab l e t o geo dati c and su rve yin g p ro b lem s . G P S o b servatio n fo r

pseudoranges, phase data, time and navigation message data have to be

recorded. Based on sampling rate, it amount to about 1.5 Mbytes of data per hour for

six satellites and 1 second data for dual frequency receivers. Modern receivers have

internal memories of 5 Mbytes or more. Some receivers store the data on magnetic

tape or on a floppy disk or hard-disk using external microcomputer connected through

RS-232 port.

Most modern receivers have a keypad and a display for communication between

the user and the receivers. The keypad is used to enter commands, external

data like station number or antenna height or to select a menu operation.

The display indicates computed coordinates, visible satellites, data quality indices and

other suitable information. Current operation software packages are menu driven

and very user friendly.

5.What are the classification of gps receivers (CO4-H1)

GPS receivers can be divided into various groups according to different criteria. In the

early stages two basic technologies were used as the classification criteria viz.

Code correlation receiver technology and sequencing receiver technology, which

were equivalent to code dependent receivers and code free receivers. However, this

kind of division is no longer justifiable since both techniques are implemented in

present receivers. e.g.

Another classification of GPS receivers is based on acquisition of data types

- ClA code receiver

- ClA code + Ll Carrier phase

- ClA code + Ll Carrier phase + L2 Carrier phase

- ClA code + p_code + Ll, L2 Carrier phase

- Ll Carrier phase (not very common)

- Ll, L2 Carrier phase (rarely used)

as:


Based on technical realization of channel, the GPS receivers can be classified

- Multi-channel receiver

- Sequential receiver

- Multiplexing receiver

GPS receivers are even classified on the purpose as:

- Military receiver

- Civilian receiver

- Navigation receiver

- Timing receiver

- Geodetic receiver

For geodetic application it is essential to use the carrier phase data as observable. Use of Ll and L2 frequency is also essential along with P-code.

Examples of GPS Receiver

GPS receiver market is developing and expanding at a very high speed. Receivers are becoming powerful, cheap and smaller in size. It is not possible

to give details of every make but description of some typical receivers given may be regarded as a basis for the evaluation of future search and study of

GPS receivers.

Classical Receivers

Detailed description of code dependent Tl 4lOO GPS Navigator and code free

Macrometer VlOOO is given here:

T1 4100 GPS Navigator was manufactured by Texas Instrument in 1984. It was the first GPS receiver to provide CIA and P code and L1 and L2 carrier phase observations. It is

a dual frequency multiplexing receiver and suitable for geodesist, surveyor and navigators. The observables through it are:

- P-Code pseudo ranges on L1 and L2

- CIA-Code pseudo ranges on L1

- Carrier phase on L1 and L2

The data are recorded by an external tape recorder on digital cassettes or are

downloaded directly to an external microprocessor. A hand held control display unit (CDU) is used for communication between observer and the receiver. For

navigational purposes the built in microprocessor provides position and velocity in real

time every three seconds. T1


4100 is a bulky instrument weighing about 33 kg and can be packed in

two transportation cases. It consumes 90 watts energy in operating mode of 22V - 32V. Generator use is recommended. The observation noise in P-Code is between 0.6

to 1 m, in CI A code it ranges between 6 to 10 m and for carrier phase it is between 2 to

3 m.

T1 4100 has been widely used in numerous scientific

and applied GPS projects and is still in use. The main disadvantages

of the T1 4100 compared to more modern GPS equipment's are

- Bulky size of the equipment

- High power consumption

- Difficult operation procedure

- Limitation of tracking four satellites simultaneously

- High noise level in phase measurements

Sensitivity of its antenna for multipath and phase centre variation if two receivers are

connected to one antenna and tracking of seven satellites simultaneously is possible. For long distances and in scientific projects, T1 4100 is still regarded

useful. However, due to imposition of restriction on P- code for civilian, T1 4100 during Anti Spoofing (AS) activation can only be used as a single frequency CIA code

receiver.

The MACROMETER V 1000, a code free GPS receiver was introduced in 1982 and

was the first receiver for geodetic applications. Precise results obtained through it

has demonstrated the potential of highly accurate GPS phase observations. It is a

single frequency receiver and tracks 6 satellites on 6 parallel channels. The complete

system consists of three units viz.

- Receiver and recorder with power supply

- Antenna with large ground plane

- P 1000 processor

The processor is essential for providing the almanac data because the Macrometer V

1000 cannot decode the satellite messages and process the data. At pre determined epoches the phase differences between the received carrier signal and a

reference signal from receiver oscillator is measured. A typical baseline accuracy

reported for upto 100 km distance is about 1 to 2 ppm (Parts per million).

Macrometer II, a dual frequency version was introduced in 1985. Though it is

comparable to Macrometer V 1000, its power consumption and weight are much less. Both systems require external ephemeredes. Hence specialized operators

of few companies are capable of using it and it is required to synchronize the clock

of all the instruments proposed to be used for a particular observation session. To


overcome above disadvantages, the dual frequency Macrometer II was further

miniaturized and combined with a single frequency CIA code receiver with a brand

name MINIMAC in

1986, thus becoming a code dependent receiver.

Examples of present Geodetic GPS Receivers

Few of the currently available GPS receivers that are used in

geodesy surveying and precise navigation are described. Nearly all models

started as single frequency CIA-Code receivers with four channels. Later L2 carrier

phase was added and tracking capability was increased. Now a days all le a d in g

manufacturers have gone for code-less, non- sequencing L2 technique. WILDI LEITZ

(Heerbrugg, Switzerland) and MAGNAVOX (Torrance, California) have jointly

developed WM 101 geodetic receiver in 1986. It is a four channel L1 CIA code receiver.

Three of the channels sequentially track upto six satellites and the fourth channel, a

house keeping channels, collects the satellite message and periodically calibrates the inter

channel biases. CIA-code and reconstructed L1 carrier phase data are observed once per

second.

The dual frequency WM 102 was marketed in 1988 with following key features:

- L1 reception with seven CIA code channel tracking upto six

satellites simultaneously.

- L2 reception of up to six satellites with one sequencing P- code channel

- Modified sequencing technique for receiving L2 when P-code signals are encrypted. The observations can be recorded on built in data cassettes or can be

transferred on line to an external data logger in RS 232 or RS 422 interface.

Communication between operator and receiver is established by alpha

numerical control panel and display WM 101/102 has a large variety of receiver resident

menu driven options and it is accompanied by comprehensive post processing

software.

In 1991, WILD GPS system 200 was introduced. Its hardware comprises the Magnavox SR 299 dual frequency GPS sensor, the hand held CR 233 GPS controller and a

Nicd battery. Plug in memory cards provide the recording medium. It can track 9

satellites simultaneously on L1 and L2. Reconstruction of carrier phase on L1 is

through C/A code and on L2 through P-code. The receiver automatically switches to

codeless L2 when P-code is encrypted. It consumes 8.5 watt through 12-volt power

supply.

TRIMBLE NAVIGATION (Sunny vale, California) has been producing TRIMBLE

4000 series since 1985. The first generation receiver was a L1 C/ A code receiver with five parallel channels providing tracking of 5 satellites s imu ltaneo u s l y. Further

upgradation included increasing the number of channels upto tweleve, L2

sequencing capab ilit y and P-code capability. TRIMBLE Geodatic Surveyor 4000


SSE is the most advanced model. When P-Code is available, it can perform

following types of observations, viz.,

- Full cycle L1 and L2 phase measurements

- L1 and L2, P-Code measurements when AS is on and

P- code is encrypted

- Full cycle L1 and L2 phase measurement

- Low noise L1, C/A code

- Cross-correlated Y-Code data

Observation noise of the carrier phase measurement when P-code is available

is about fi 0-2mm and of the P-code pseudoranges as low as fi 2cm. Therefore, it is

very suitable for fast ambiguity solution techniques with code/ carrier combinations.

ASHTECH (Sunnyvale , California) developed a GPS receiver with 12 parallel

channels and pioneered current multi-channel technology. ASHTECH X II GPS receiver

was introduced in 1988. It is capable of me a su r ing pseudoranges, carrier

phase and integrated dopler of up to 12 satellite s on L1. The pseudorange s

measurement are smoothed with integrated Doppler. Postion velociy, time and

navigation informations are displayed on a keyboard with a 40-characters display. L2

option adds 12 physical L2 squaring type channels.

ASHTECH XII GPS receiver is a most advanced system, easy to handle and does

not require initialization procedures. Measurements of all satellites in view are carried

out automatically. Data can be stored in the internal solid plate memory of 5

Mbytes capacity. The minimum sampling interval is 0.5 seconds. Like many other

receivers it has following additional options viz.

- 1 ppm timing signal output

- Photogrammetric camera input - Way point navigation

- Real time differential navigation and provision of port

processing and vision planning software

In 1991, ASHTECH P-12 GPS receiver was marketed. It has 12 dedicated channels of

L1, P-code and carrier and 12 dedicated channels of L2, P-code and carrier. It also has

12 L1, CIA code and carrier channels and 12 code less squaring L2 channels. Thus

the receiver contains 48 channels and provides all possibilities of observations to all

visible satellites. The signal to noise level for phase measurement on L2 is only slightly

less than on L1 and significantly better than with code-less techniques. In cases of

activated P- code encryption, the code less L2 option can be used.

TURBO ROGUE SNR-8000 is a portable receiver weighing around 4 kg, consumes


15-watt energy and is suitable for field use. It has 8 parallel channels on L1 and L2.

It provides code and phase data on both frequencies and has a codeless option. Full

P-code tracking provides highest precision phase and pseudo rages measurements,

codeless tracking is automatic "full back" mode. The code less mode uses the

fact that each carrier has identical modulation of P-code/Y- code and hence the L1

signal can be cross-correlated with the L2 signal. Results are the differential phase

measurement (L1-L2) and the group delay measurement (P1-P2)

Accuracy specifications are :

P-Code pseudo range 1cm (5 minutes integration) Codeless

pseudo range 10cm (5 minutes integration)

Carrier phase

0.2 - 0.3 mm Codeless phase 0.2 - 0.7 mm

One of the important features is that less than 1 cycle slip is expected for

100 satellite hours.

Navigation Receivers

Navigation receivers are rapidly picking up the market. In most cases a single C/A code sequencing or multiplexing channel is used. However, modules with four or five parallel

channels are becoming increasingly popular. Position and velocity are derived from C/A

code pseudoranges measurement and are displayed or downloaded to a

personal computer. Usually neither raw data nor carrier phase information is

available. Differential navigation is possible with some advanced models.

MAGELLAN NAV 1000 is a handheld GPS receiver and weighs only 850 grams. It was

introduced in 1989 and later in 1990, NAV 1000 PRO model was launched. It is a

single channel receiver and tracks 3 to 4 satellites with a

2.5 seconds update rate and has a RS 232 data

port.

The follow up model in 1991 was NAV 5000 PRO. It is a 5-channel receiver

tracking all visible satellites with a 1-second update rate. Differential navigation is

possible. Carrier phase data can be used with an optional carrier phase module.

The quadrifilar antenna is integrated to the receiver. Post processing of data is also

possible using surveying receiver like ASHTECH XII located at a reference station.


Relative accuracy is about 3 to 5 metres. This is in many cases sufficient for thematic

purposes.

Many hand held navigation receivers are available with added features.

The latest market situation can be obtained through journals like GPS world etc.

For most navigation purpose a single frequency C IA code receiver is suffic ient. For

accu rac y requirements better than 50 to 100 meters, a differential option is

essential. For requirement below 5 meters, the inclusion of carrier phase data is

necessary. In high precision navigation the use of a pair of receivers with full

geodetic capability is advisable. The main characteristics of multipurpose geodetic

receiver are summarized in Table 4.

Table 4. Overview of geodetic dual-frequency GPS satellite receiver (1992)

Receiver Channel Code Wavelen Anti-

L1 L2 L1 L2 L1 L2 spoofing

TI 4100 4 4 P P Single

MACROMET 6 6 - - I2 No influence

ASHTECH 12 12 CIA - I2 No influence

ASHTECH P 12 12 CIA, P Squaring

TRIMBLE 8-12 8-12 CIA - I2 No influence

TRIMBLE 9-12 9-12 CIA, P Codeless

SSE WM 102 7 1 CIA P Squaring

WILD GPS 9 9 CIA p Codeless

TURBO 8 8 CIA, P Codeless

Some of the important features for selecting a geodetic receiver are :

- Tracking of all satellites

- Both frequencies

- Full wavelength on L2

- Low phase noise-low code noise

- High sampling rate for L1 and L2

- High memory capacity

- Low power consumption

- Full operational capability under anti spoofing condition

Further, it is recommended to use dual frequency receiver to minimize ion-

spherical influences and take advantages in ambiguity solution.


ACCURACY

In general, an SPS receiver can provide position information with an error of less than 25

meter and velocity information with an error less than 5 meters per second. Upto

2

May 2000 U.S Government has activated Se le ct ive Availability (SA ) to maintain

optimum mil i ta r y e f fec tivene ss . Se le c t ive Availability inserts random errors into the

ephemeris information broadcast by the satellites, which reduces the SPS accuracy

to around 100 meters.

For many applications, 100-meter accuracy is more than acceptable. For

applications that require much greater accuracy, the effects of SA and

environmentally produced errors can be overcome by using a technique called

Differential GPS (DGPS), which increases overall accuracy.

DIFFERENTIAL THEORY

Differential positioning is technique that allows overcoming the effects of environmental

errors and SA on the GPS signals to produce a highly accurate position fix. This is done by determining the amount of the positioning error and applying it to position fixes

that were computed from collected data.

Typically, the horizontal accuracy of a single position fix from a GPS receiver is 15

meter RMS (root-mean Square) or better. If the distribution of fixes about the true position is circular normal with zero mean, an accuracy of 15 meters RMS implies

that about 63% of the fixes obtained during a session are within 15 meters of the

true position.

6.Describe briefly about sources of errors in GPS (CO4-H2)

There are two types of positioning errors: correctable and non-correctable. Correctable

errors are the errors that are essentially the same for two GPS receivers in the

same area. Non-correctable errors cannot be correlated between two GPS receivers in

thesamearea.

CORRECTABLE ERRORS

Sources of correctable errors include satellite clock, ephemeris data and

ionosphere and tropospheric delay. If implemented, SA may also cause a

correctable positioning error. Clock errors and ephemeris errors originate with the GPS

satellite. A clock error is a slowly changing error that appears as a bias on the

pseudorange measurement made by a receiver. An ephemeris error is a residual error in

the data used by a receiver to locate a satellite in space.

Ionosphere delay errors and tropospheric delay errors are caused by atmospheric

conditions. Ionospheric delay is caused by the density of electrons in the ionosphere

along the signal path. A tropspheric delay is related to hu midit y, te mperature,


and altitude along the s i gnal path. Usuall y, a tropospheric error is smaller than an

ionospheric error.

An o th er co rr ect a bl e e rro r i s ca us e d b y S A wh ic h i s u se d b y U. S Department of

Defence to introduce errors into Standard Positioning Service (SPS) GPS signals to

degrade fix accuracy.

The amount of error and direction of the error at any given time does not change

rapidly. Therefore, two GPS receivers that are sufficiently close together will

observe the same fix error, and the size of the fix error can be determined.

NON-CORRECTABLE ERRORS

Non-correctable errors cannot be correlated between two GPS receivers that

are located in the same general area. Sources of non-correctable errors include

receiver noise, which is unavoidably inherent in any receive r, and multipath errors,

which are environmental. Multi-path errors are caused by the receiver "seeing"

reflections of signals that have bounced off of surrounding objects. The sub-meter

antenna is multipath- resistant; its use is required when logging carrier phase data.

Neither error can be eliminated with differential, but they can be reduced

substantially with position fix averaging. The error sources and the approximate RMS

error range are given in the Table.

Error Sources

Error Source Approx. Equivalent Range

Error (RMS) in meters

Correctable with Differential

Clock (Space Segment) 3.0

Ephemeris (Control Segment) 2.7

Ionospheric Delay (Atmosphere) 8.2

Tropospheric Delay (Atmosphere) 1.8

Selective Availability (if implemented) 27.4 Total 28.9

Non-Correctable with Differential

Receiver Noise (Unit) 9.1

Multipath (Environmental) 3.0

Total 9.6

Total user Equivalent range error (all sources)

30.5

Navigational Accuracy (HDOP = 1.5) 45.8

7.What is meant by differential gps (CO4-H2)


Most DGPS techniques use a GPS receiver at a geodetic control site whose position is

known. The receiver collects positioning information and calculates a position fix , which is

then compared to the known co-ordinates. The difference between the known position

and the acquired position of the control location is the positioning error.

Because the other GPS receivers in the area are assumed to be operating under similar conditions, it is assumed that the position fixes acquired by other receivers in the area (remote units) are subject to the same error, and that the correction computed for the control position should therefore be accurate for those receive rs. The correction is communicated to the remote units by an operator at the control site with radio or cellular equipment. In post-processed differential, all units collect data for off-site processing; no corrections are determined in the field. The process of correcting the position error with differential mode is shown in the Figure .

The difference between the known position and acquired position at the control point is the

DELTA correction. DELTA, which is always expressed in meters, is parallel to the surface of

the earth. When expressed in local coordinate system, DELTA uses North-South axis (y)

and an East-West axis (x) in2D operation; an additional vertical axis (z) that is

perpendicular to the y and x is used in 3D operation for altitude. 8.What are the applications of gps (CO4-H1)

z Providing Geodetic control.

z Survey control for Photogrammetric control surveys and mapping.

z Finding out location of offshore drilling.

z Pipeline and Power line survey. z Navigation of civilian ships and planes.

z Crustal movement studies.

z Geophysical positioning, mineral exploration and mining.

z Determination of a precise geoid using GPS data.

z Estimating gravity anomalies using GPS.

z Offshore positioning: shiping, offshore platforms, fishing boats etc.


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UNIT 4

PART A

1. What do you understand by parallax? [CO5 –L2] (AUC Nov/Dec 2012/AUC Apr/May 2010)

Parallax is defined as the apparent displacement of an object due to the real displacement

of the observer. For example the apparent movement of the stars is due to the real

displacement of the observer from one position to another upon the earth's orbit.

2. Distinguish between crab and drift. [CO5 –L2] (AUC Apr/May 2010) Crab:

The angle formed between the flight line and the edges of the photograph in the direction

of flight is designated by a term called crab. The crab is caused in the photograph when the

focal plane of the camera is not square with the direction of flight.

Drift:

Drift is caused by the failure of the photograph to stay on the predetermined flight line. If

an aircraft is allowed to go on its course without allowance for wind velocity, it will drift.

3. What do you mean by sounding? [CO5–L1] (AUC Nov/Dec 2010/ Apr/May 2011)

The measurements of depths below the water surface are called soundings. It is to find the

depth measurement in land with reference to a datum.

4. Distinguish between 'terrestrial photogrammetry' and 'aerial photogrammetry'.

[CO5 –L2] (AUC Apr/May 2011)

Terrestrial photogrammetry:

Photographs taken from a fixed position on or near the ground and the branch deals on

such aspects are called terrestrial photogrammetry. Aerial photogrammetry:

Aerial photogrammetry is the other branch wherein the photograph are taken by cameras

mounted on an aircraft flying over the area. 5. What is meant by scale of a photograph? [CO5 –L1] (AUC May/June 2009)

Scale of photograph is obtained from the ratio of the distance of any two points on the

photograph and the distance between the corresponding points on the ground. The two points

chosen for scaling should lie nearly equidistant on either side of the principal point.

6. Write the concept of map - marking in cartography? [CO5–L2] (AUC May/June 2009)

While there are many steps involved in the map making process, they can be grouped

into three main stages: data collection, organization, and manipulation; map design and artwork

preparation; and map reproduction.

7. What is a fathometer? [CO5 –L1] (AUC May/June 2012)

A fathometer is used for measuring depth of large rivers and seas with depth more than

10 m. by this instrument the depth of water is obtained by sending a sound impulse from the

surface of water towards the bottom of the river or sea bed.


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8. Differentiate between 'tilted photograph' and 'oblique photograph'. (AUC May/June 2012)

Tilted photograph: [CO5 –L2]

A tilted photograph is an aerial photograph made with the camera axis unintentionally.

The tilt from the vertical axis is usually less than 30. Oblique photograph:

An oblique photograph is the one made in an aerial photograph intentionally between the

horizontal and the vertical. 9. Define hydrographic surveying. [CO5 –L1] (AUC Nov/Dec 2010) (AUC May/June 2013)

Hydrographic surveying is that branch of surveying connected with all the observations

and measurements concerned with bodies of water. These observations and measurements are

needed for the design of marine structures, hydraulic structures and other cross-drainage

works.

10. Define EDM. [CO5 –L2] (AUC May/June 2013)

Electro-magnetic distance measurement is a general term used collectively in the

measurement of distances applying electronic methods. Basically the EDM method is based on

generation, propagation, reflection and subsequent reception of electromagnetic waves.

11. What are the equipments used for sounding? [CO5 –L2] (AUC Nov/Dec 2012)

i) Sounding rods or poles.

ii) Lead lines or sounding cables.

iii) Fathometer.

12. What is meant by three point problem in hydrographic surveying? [CO5 –L1]

If a sounding is located by two angles from the boat by observations to three known points

on the shore, the plotting can be done adopting three-point problem. The three point problem

may be solved by mechanical, graphical or analytical methods.

13. Explain the term 'Cartography'. [CO5 –L1]

Cartography:

Cartography has always been closely associated with Geography and Surveying. Its

recognition as a distinct discipline is relatively recent. Scientific journals dealing with

Cartography began to appear in the middle of the twentieth century. Numerous definitions of

Cartography have appeared in the literature. Earlier definitions tend to emphasize map making

while more recent definitions also include map use within the scope of Cartography.

14. What are lunar and solar tides? [CO5 –L1]

Lunar tides:

The periodical variations in natural water level are called as tides. The resultant force

between the earth and moon causes lunar tides. Solar tides:

The production of solar tides is due to force of attraction between earth and sun which is

similar to the lunar tides. 15. What is meant by photo-theodolite? [CO4 –L1]

Photo-theodolite is a combination of a camera and a theodolite. It is used to take

photographs and measuring angles. 16. Define tilt displacement. [CO5–L1]

Tilt displacement:

Tilt distortion or tilt displacement is defined as the difference between the distance of the

image of a point on the tilted photograph from the isocentre and the distance of the image of the

same point on the photograph from the isocentre if there had been no tilt.


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17. Name the different methods for soundings. [CO5–L1]

) By cross rope

) By range and time intervals

) By range and one angle from the shore

) By range and one angle from boat

) By two angles from the shore

) By two angles from the boat

) By one angle from shore and one angle from boat

) By intersecting ranges

) By tacheometry

18. Give the significance of trilateration. [CO5–L1]

There is no angular measurement is made. The three sides of triangles are measured

precisely using the EDM equipment. This technique is useful when angular measurement is

difficult or impossible due to any reason.

19. Define cadastral surveying. [CO5–L1]

Cadastral surveying is the one which is conducted in order to determine the boundaries

of fields, estates, houses, etc.

20. What are the uses of photogrammetry? [CO5–L1]

) Construction of planimetric and topographic maps.

) Mountainous and hilly areas with less number of trees can be very satisfactorily

surveyed.

) Aerial surveying is most suitable for reconnaissance.

) Acquisition of military intelligence.

) Interpretation of geology and soil details.

) Largely used for the surveys of buildings.

21. Define Celestial Horizon. [CO5–L1] (AUC Nov/Dec 2010)

Celestial horizon is the great circle traced upon the celestial sphere by that plane which

is perpendicular to the Zenith-Nadir line, and which passes through the center of the earth.

22. What is meant by solar Apparent Time? [CO5–L1] (AUC Nov/Dec 2010)

The apparent solar time is the time calculated on the basis of the daily motion of the sun.

As the sun does not move uniformly along the ecliptic, the apparent solar time or the solar day

is not uniform. Thus it cannot be recorded by a clock which moves with a uniform rate.

Apparent solar time = hour angle of the sun + 12 hours

23. What is equation of time? [CO5–L1] (AUC May/June 2013) (AUC Apr/May 2010)

At any instant the difference between apparent solar time and mean solar time is known

as the equation of time.

Values of equation of time are sometimes prefixed with the plus sign (sun after clock) or minus

sign (sun before clock).

Equation of time = R.A. of the mean sun - R.A. of the sun 24. Distinguish between latitude and co-latitude. [CO5–L2] (AUC Apr/May 2010)

Latitude ( ):

It is angular distance of any place on the earth's surface north or south of the equator,

and is measured on the meridian of the place. It is marked + or - (N or S) according as the


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place is north or south of the equator. The latitude may also be defined as the angle between

the zenith and the celestial equator. Co-latitude (c):

The Co-latitude of a place is the angular distance from the zenith to the pole. It is the

complement of the latitude and equal to (90°- ).

25. Distinguish between the 'Zenith' and 'Nadir'. [CO5–L2] (AUC Apr/May 2011)

Zenith:

The Zenith (Z) is the point on the upper portion of the celestial sphere marked by plumb

line above the observer. It is thus the point on the celestial sphere immediately above the

observer's station.

Nadir:

The Nadir (Z') is the point on the lower portion of the celestial sphere marked by the plum

line below the observer. It is thus the point on the celestial sphere vertically below the

observer'station.

26. Differentiate 'Tropic of cancer' from 'Tropic of Capricorn'. [CO5–L2] (AUC Apr/May2011)

Tropicofcancer:

The parallel of latitude 23o 27 l' 2

Tropic of capricorn:

The parallel of latitude 23o 27 l' 2

north of equator is known as tropic of cancer.

south of equator is known as tropic of capricorn.


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27. Explain the term "sidereal time". [CO5–L1] (AUC May/June 2009)

The sidereal time at any instant is the hour angle of the first point of aries reckoned

westward from Oh to 24h. The right ascension of the meridian of a place is known as Local

sidereal time (L.S.T).

Local sidereal time (L.S.T) = Right ascension of a star + westerly hour angle of a star

28. What is the correction for parallax when the altitude of celestial body is observed?

[CO5–L1] (AUC May/June 2009)

When the sun or star is viewed from different points, change in the direction of the body

is observed due to parallax. The parallax in altitude is called diurnal parallax.

Correction for parallax 8.8" cos '

Where α' is the observed altitude. This correction is always additive.

29. Define the right ascension (R.A). [CO5–L1] (AUC May/June 2012)

Right ascension is the equatorial angular distance measured eastward from the First

Point of Aries to the hour circle through the heavenly body.

30. Enumerate the properties of a spherical triangle. [CO5–L1] (AUC May/June 2012)

) Any angle is less than two right angles or .

) Sum of the three angles is less than six right angles or 3 and greater than two right

angles or .

) Sum of any two sides is greater than the third.

) If the sum of any two angles is equal to two right angles or , the sum of the angles

opposite them is equal to two right angles.

) The smaller angle is opposite to the smaller side and vice - versa.

31. Define celestial sphere and azimuth axis. [CO5–L1] (AUC May/June 2013) (AUC

Nov/Dec 2012)

Celestial sphere:

The millions of stars that we see in the sky on a clear cloudless night are all at varying

distances from us. Since we are concerned with their relative distance rather than their actual

distance from the observer. It is exceedingly convenient to picture the stars as distributed over

the surface of an imaginary spherical sky having its center at the position of the observer. This

imaginary sphere on which the star appears to lie or to be studded is known as the celestial

sphere.

Azimuth axis (A):

The azimuth of a heavenly body is the angle between the observer's meridian and the

vertical circle passing through the body.

32. What is Latitude and Longitude? [CO5–L1] (AUC Nov/Dec 2012)

Latitude ( ):

It is angular distance of any place on the earth's surface north or south of the equator,

and is measured on the meridian of the place. It is marked (+ or -) (N or S) place is north or south of the equator. The latitude may also be defined as the angle between the zenith and the celestial equator.

Longitude ():

The longitude of a place is the angle between a fixed reference meridian called the prime

of first meridian and the meridian of the place. The prime meridian universally adopted is that of

Greenwich. The longitude of any place varies between 0° and 180°, and is reckoned as Φ° east

or west of Greenwich.

33. Give the relationship for conversion of sidereal time to mean time. [CO5–L2]

Local sidereal time = R.A. of mean sun 12 h + mean time at that place


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34. Describe nautical almanac. [CO5–L1]

Nautical almanac is an official publication of a country wherein astronomical data are

provided. It provides data which are useful for surveyors and for practice of astronomical,

navigation, etc.

It provides the following data:

) Greenwich hour angle of the sun.

) Equation of time.

) Semi-diameter of the sun.

) Sidereal hour angle.

) Polar star tables.

35. What is meant by mean solar time? [CO5–L1]

The mean sun is an imaginary body and is assumed to move at a uniform rate along the

equator in order to make a solar day of uniform period. It starts from the vertical equinox at the

same time as the true sun and to return to the vertical equinox with the true sun. The time when

measured by the diurnal motion of the mean sun is called the mean solar time or mean time.

This is the time kept by our clocks and watches.

36. What is the relation between the Right ascension and Hour Angle? [CO5–L2] Right ascension of the sun = local sidereal time - Hour angle of the sun

37. Define standard time. [CO5–L1]

The standard meridian usually lies at an exact number of hours from Greenwich. The

mean time associated with this meridian is called standard time.

Standard time = L.M.T (difference of longitude in time between the given place and the

standard meridian)

38. What is meant by declination? [CO5–L1]

The declination of a celestial body is angular distance from the plane of the equator,

measured along the star's meridian generally called the declination circle, (i.e., great circle

passing through the heavenly body and the celestial pole). Declination varies from 0° to 90°,

and is marked (+ or -) according as the body is north or south of the equator.

39. What are the relationships between hour angle, right ascension and time? [CO5–L1]Apparent solar time = hour angle of the sun + 12 hours

Mean solar time = hour angle of the mean sun + 12 hours

Local sidereal time (L.S.T) = R.A. of the mean sun + hour angle of the mean sun

Sidereal time of apparent noon = R.A. of the sun

Sidereal time of mean noon = R.A. of the mean sun

40. What are the corrections to be applied to the observed altitude of sun? [CO5–L1]

The observed altitude has to be corrected to obtain the true altitude. The corrections to

be applied are1. Instrumental corrections i) index error

ii) bubble error

2. 0bservational corrections

i) Correction for parallax

ii) Correction for refraction

iii) Correction for dip of the horizon

iv) Correction for semi-diameter


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PART B

1. At a point in latitude 550 46' 12'' N, the altitude of sun's centre was found to be 230 17'

32'' at 5h 17m, P.M. (G.M.T.) The horizontal angle at the R.M. and Sun's centre was 680

24' 30''. Find the azimuth of the sun. [CO5-H2]

Data:

i) Sun's declination of G.A.N. on day of observation = 170 46' 52'' N

ii) Variation of declination per hour = - 37''

iii) Refraction of altitude 230

20' 00'' = 00

2' 12''

iv) Parallax for altitude = 00 0' 8''

v) Equation of time (App. - Mean) = 6m 0s (IRSE). (AUC Nov/Dec 2010)

Solution:

i) Calculation of declination:

G.M.T. of observation = 5h 17m P.M.

Add equation of time = 0h 6m 0s

G.A.T. of observation = 5h 23m 0s P.M.

Now declination at G.A.T. = 170 46' 52'' N

Apparent time interval,

G.A.N. = 5h 23m 0s

Variation in the declination in this time interval at the rate of 37" per hour = 3' 39"

(decrease).

Declination at G.A.T. of observation = 170 46' 52'' - 3' 39" = 170 43' 13''

ii) Calculation of altitude:

Observed altitude of sun's centre = 230 17' 32''

subtract refraction correction = 00 2' 12''

= 230 15' 2O''

Add parallax correction = O' 8"

Correct altitude = 230

15' 28''

Now, co-latitude = c = 900 - = 900 - 550 46' 12'' = 340 13' 48''

co-declination = p = 900 - = 900 - 170 43' 13'' = 720 16' 47''

co-altitude = z = 900 - = 900 - 230 15' 28'' = 660 44' 32''

2s = 1730 15' 7''

s = 860 37' 33.5''

s - c = 520 23' 45.5'' ; s - p = 140 2O' 46.5" ; s - z = 190 53' 1.5"

Azimuth of sun is given by,

0 0

tan A

sin ( s z) sin ( s c)

sin (19 53' 1.5" ) sin ( 52 23' 45.5" )

1.04372 sin s sin ( s p) sin ( 860 37' 33.5") sin (140 20' 46.5")


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A = 460 13' 29.84"

2 A = 230 6' 44.92'

2. What is the equation of time? Show that it vanishes four times a year. [CO5-H1]

(AUC Apr/May 2010)

At any instant the difference between apparent solar time and mean solar time is known

as the equation of time. The values of equation of time at 0 hour (midnight) at Greenwich are

tabulated in the nautical almanac for everyday of the year.

Values of equation of time are sometimes prefixed with the plus sign (sun after clock) or minus

sign (sun before clock).

Equation of time = R.A. of the mean sun - R.A. of the sun

At different seasons of the year the value of the equation of time varies from 0 to

16 minutes. During the following dates on a year viz., April 15, June 14, september 1 and

December 25 the true sun and the mean sun are on the same meridian, ie., the apparent time

and mean time are the same resulting in the equation time.

The difference between mean time and apparent time is due to obliquity of real sun

and the mean sun along different orbits as shown in figure.

The following relationships are of interest. We know

L.s.T. = R.A. of the mean sun + hour angle of the mean sun

L.s.T. = R.A. of the sun + hour angle of the sun

Equating the above both equations

R.A. of the mean sun - R.A. of the sun = hour angle of the sun - hour angle of the mean sun

Equation of time = hour angle of the sun - hour angle of the mean sun

Equation of time = apparent time - mean time

Apparent time = mean time + equation time


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3. Determine the hour angle and declination of star from the following data:

Altitude of star = 22° 30'

Azimuth of the star = 145° E

Latitude of the observer = 49° N. [CO5-H2] (AUC Apr/May 2010)

Solution:

The azimuth of the star is 145° E, the star is in the eastern hemisphere.

In the astronomical triangle ZPM, we have

Co - altitude, ZM = 90o - α = 90o - 22o 30' = 67o 30'

Co - latitude, ZP = 90o - = 90o - 49o = 41o

A = 145o

Using cosine formula,

Cos PM = cos ZM cos ZP + sin ZM sin ZP cos A

cos (670 30 ' ) cos ( 410 ) sin (670 30 ' ) sin ( 410 ) cos (1450 )

Cos PM 0.2077

PM = 1010 59' 15.36"

Declination of star, δ = 900 - PM = 900 - 1010 59' 15.36" = 110 59' 15.36"

δ = . 110 59' 15.36" s

Using cosine rule,

cos H cos ZM cos PZ cos PM

sin PZ sin PM

0 0 0

cos H cos ( 67 30' ) cos ( 41 ) cos (101 59' 15.36" )

sin ( 410 ) sin (1010 59' 15.36")

Cos H = 0.8406

cos ( 360 o - H ) = 0.8406

( 360 o - H ) = 32 o 47' 47.28"

H = 360 o - 32 o 47' 47.28"

Hour angle, H = 3270 12' 12.72"


105


4. What are parallax and refraction and how do they affect the measurements of vertical

angles in astronomical work? [CO5-H1] (AUC May/June 2012) (AUC Apr/May 2010)


106



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5. If the GST of GMN is 13h 29m 28s, what will be the HA of the star of RA 22h 19m 20s at a

place in longitude 120° 32' W at 2.10 AM, GMT the same day. [CO5-H2] (AUC Apr/May

2010) Solution:

LsT = RA of the star + Hour angle of the star

Calculate LsT corresponding to the given LMT knowing GsT of GMN.

First calculate LST of LMN:

Longitude = 120° 32' W = 8h 2.4m

As the place is to the west, the acceleration has to be added at the rate of 9.8565s per hour of

longitude to the GsT of GMN to get the LsT of LMN.

Now,

8h x 9.8565 = 78.85 seconds


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2.4m x 0.1642 = 0.39 seconds

Total acceleration = 79.24 seconds

GsT of GMN = 13h 29m 28s Add

acceleration = 79.24s L8T

of LMN = 13h 30m 47.248

Now GMT = 2h 10m 00s

subtract Longitude = 8h 2.4m 00s

LMT of the event = - 5h 52m 24s (previous day)

LMN (day of given G.s.T. of G.M.N.) = 12h 00m 00s

subtract LMT of event (previous day) = 5h 52m 24s

Mean time interval between the event = 17h 52m 24s and the L.M.N.

Let us convert this mean time interval to the sidereal time interval by adding acceleration at the

rate of 9.8565s per mean hour.

17h

x 9.8565 = 167.56 seconds

52h x 0.1642 = 8.54 seconds

24h x 0.0027 = 0.06 seconds

Total acceleration = 176.16 seconds = 2m 56.16s

sidereal time b/n event and L.M.N. = 17h 52m 24s + 2m 56.16s = 20h 48m 33.6s (before L.M.N.)

L.s.T. of L.M.N. = 13h 30m 47.24s

subtract s.l. = 20h 48m 33.6s

L.s.T. of event = - 7h 17m 46.36s = 16h 42m 13.64s

Now, Hour angle = L.s.T. - R.A. = 16h 42m 13.64s - 22h 19m 20s = - 5h 37m 6.36s

= 24h - 5h 37m 6.36s

Hour angle = 18h

22m

53.648

(here 24 hrs is added to make HA positive)

6. Describe the Napier's rules of circular parts in obtaining the solution of right angle

spherical triangle. [CO5-H2] (AUC Apr/May 2011)


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The relationship of right-angled spherical triangle may be obtained from 'Napier Rules

of Circular Parts'.

From the above figure the spherical triangle is right angled at C. a circle is drawn and

divided into five parts. In order to starting from the side a, the two sides containing the right

angle (a and b) and the components of three parts A, c and B are shown in figure. Then if any

part is considered as the middle part the two parts adjacent to it as adjacent parts, we have the

following rules as per Napier:

sine of middle part = product of tangents of the adjacent parts.

sine of middle part = product of cosines of opposite parts.

Thus,

sin b = tan a tan (900 - A) and

sin b = cos (900 - B) cos (900 - c)

By choosing different parts in turn as the middle parts, we can obtain all the possible

relationships between the sides and angles.

7. Find the shortest distance between two places A and B, given that the latitudes of A and

B are 15° 00' N and 12° 06' N and their longitudes are 50° 12' E and 54° 00' E respectively.

Find also the directions of B on the great circle route. Radius of the earth = 6370 km.

[CO5-H2] (AUC Apr/May 2011)

Solution:

i) Distance between AB:

In the triangle ABP,

b = AP = 900 - latitude of A = 900 - 150 = 750

a = BP = 900 - latitude of B = 900 - 120 6' = 770 54'

p = APE 540 - 500 12' = 30 48'

Using cosine rule,

cos P = cos p - cos a cos b

sin a sin b

cos p = cosP sin a sin b + cos a cos b

= cos (30 48') sin (770 54') sin (750 ) + cos (770 54') cos (750 )

Cos p = 0.9966


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0

p = 40 43' 33.85"

Arc distance = radius x central angle =

AB = 525.43 km

ii) Direction of A from B:

6370 x

40 43 ' 33.85" x

1800

The direction of A from B is the angle B and vice versa.

Angles A and B can be found by the tangent semi-sum and semi-difference formulae.

cos 1

( a - b )

tan 1

(A +B) = 2 cot 1

P and

2 cos 1

( a +b ) 2 2

sin 1

( a - b )

tan 1

(A -B) = 2 cot 1

P

2 sin 1

( a +b ) 2 2

Here

a - b 77 O 54' - 75 O

= =10 27' 2 2

a + b 77 O 54' + 75 O

= =760 27' 2 2

P 3 O 48' = = 10 54'

2 2

1 cos (10 27 ') tan (A +B) = cot (10 54 ') 128.62

2 cos (760 27 ')

A E 890 33 '16.36" ........................................(1)

2

tan 1

(A B) = sin (1 27 ')

cot (10

54 ') 0.7846 2 sin (760 27 ')

A E 380 7 ' 3.83" ........................................(2)

2 By subtracting above equations, we get

Direction of A from B = angle B = 890 33' 16.36" - 380 7' 3.83" = N 510 26' 12.53" W

8. Write a detailed note:

i) Sidereal time

ii) Solar apparent time. [CO5-H1] (AUC Apr/May 2011)

i) Sidereal time:

since the earth rotates on its axis from west to east, all heavenly bodies appear to

revolve from east to west around the earth. such motion of the heavenly bodies is known as

apparent motion. We may consider the earth to turn on it axis with absolute regular speed. The

star appears to complete one revolution round the celestial pole as centre in constant interval of

time and they cross the observer's meridian twice.

The time interval between two successive upper transits of the first point of aries over

the same meridian is known as the sidereal day and the instant of crossing is called as sidereal


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noon. The day is divided into 24 hours which is reckoned consecutively from zero at one noon

to 24 hours at the following noon. Each hour is divided into 60 minutes and each m inute is

divided into 60 seconds.

Thus the sidereal time at any instant is the hour angle of the first point of aries reckoned

westward from 0h to 24h.

The right ascension of the meridian of a place is known as the local sidereal time.

Local sidereal time (L.s.T.) = R.A. of star + westerly hour angle of a star.

If the sum of above equation is greater than 24 hours, 24 hours has to be deducted and

24 hours should be added if it is negative. It is also expressed as

Local sidereal time (L.s.T.) = R.A. of mean sun 12 hours + mean time at that place.

When the position of the star is at its upper transit or culmination, its hour angle is zero.

Then

sidereal time of transit of star = R.A. of a starii) Solar apparent time:

The time interval between two successive lower transits of the centre of the sun over

the same meridian is called the apparent solar day. The apparent solar day is not of constant

length throughout the year but changes. Hence our modern clocks and chronometers cannot be

used to give us the apparent solar time. The non-uniform length of the day is due to two

reasons.

) The orbit of the earth round the sun is not circular but elliptical with sun at one of its foci.

The distance of the earth from the sun is thus variable.

) The apparent diurnal path of the sun lies in the elliptic. The time elapsing between the

departure of a meridian from the sun and it returns would vary because of the obliquity of

the ecliptic.

It is divided into 24 hours with each hour divided into 60 minutes and in turn each minute

is divided into 60 seconds. Thus the apparent solar time is the time calculated on the basis of

the daily motion of the sun. As the sun does not move uniformly along the ecliptic, the apparent

solar time or the solar day is not uniform. Thus it cannot be recorded by a clock which moves

with a uniform rate.

Apparent solar time = hour angle of the sun + 12 hours

9. The following observations of the sun were taken for azimuth of a line in connection with a

survey.

Mean time = 16h

30m

Mean hour angle between sun and referring object = 18° 20' 30"

Mean corrected altitude = 33° 35' 10"

Declination of the Nautical Almanac = + 22° 05' 36"

Latitude of the place = 52° 30' 20"

Determine the azimuth of the line. [CO5-H2] (AUC Apr/May 2011)

Solution:

Considering astronomical triangle, the hour angle ZPM = H,

Zenith distance, ZM = z = 900 - α = 900 - 330 35' 10" = 560 24' 50"

Polar distance, PM = 900 - δ = 900 - 220 5' 36" = 670 54' 24"

Co-latitude, ZP = 900 - = 900 - 520 30' 20" = 370 29' 40"

Using cosine rule,



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cos A =

cos PM - cos ZP cos ZM

sin ZP sin ZM

cos (670 54 ' 24") - cos (370 29 ' 40") cos (560 24 ' 50")

sin (370

29 ' 40") sin (560

24 ' 50")

Cos A = - 0.1238

Azimuth of sun, A = 970 6' 41.27"

10. Explain the three systems of coordinates by which the position of a heavenly body can

be specifies. [CO5-H1] (AUC May/June 2009)

i) Altitude and Azimuth system:

This system is also called a horizon system which is dependent on the position of

the observer. Here, the horizon is the plane of reference and the co -ordinates of a heavenly

body are azimuth and altitude. The horizon is the primary reference great circle and the

secondary reference great circle is the observer's meridian. This system is necessitated as only

horizontal and vertical angles could be measured using engineer's theodolite. The heavenly

body may be on the eastern or western part of the celestial sphere.

Let the heavenly body, M be in the eastern part of the celestial sphere. Let Z be the

observer's zenith and P be the celestial pole. A vertical circle passing through M and Z is drawn

to cut the horizon plane at M'. The azimuth (A), the angle between the observer's meridian

(through P) and the vertical circle through the body is the first co-ordinate. Azimuth is also equal

to the angle at the zenith between the meridian and the vertical circle through M. the other co -

ordinate of M is the altitude (α). It is the angle measured above or below the horizon on the

vertical circle through the body.


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Let the heavenly body, M be on the western part of celestial sphere. The concerned

angles NOM is the azimuth and MOM' is the altitude.

It should be noted that in the northern hemisphere, the azimuth is always measured from the

north to the east or west. In the southern hemisphere the azimuth is measured from the south to

the east or to the west.

Zenith distance = ZM - MM'

ii) Declination and Right Ascension system:

This system is also referred to as Independent Equatorial system. In this system the

coordinates are independent of observer's position. Accordingly, the two great circles of

reference are:

i) The equatorial circle - primary circle

ii) The declination circle - secondary circle.

This system is used in the publication of star catalogues. The first coordinate of the

heavenly body, M is the right ascension. It is the angle along the arc of the celestial equator

measured from the first point of aries. It is also the angle between the hour circle through . The

other coordinates in this system is the declination ( ). It is the angle of the body measured from

equator along the arc of the declination circle.


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iii) Declination and Hour angle system:

This system is also called as Dependent equatorial system. Here one coordinate is

dependent on the observer's position and the other is independent. Accordingly, the two great

circles of reference are:

i) The horizon - primary circle

ii) The declination circle - secondary circle

The first coordinate of M is the hour angle. As discussed earlier, the hour angle is the

angle measured along the arc of the horizon from the observer's meridian to the declination

circle passing through the body. It is also the angle subtended at the pole, between the

observer's meridian and the declination circle of the body.

In the northern hemisphere the hour angle is measured always from the south towards

the west up to the declination circle. It varies from 00 to 3600. When it is between 00 to 1800,

the star is in western hemisphere, otherwise in the eastern hemisphere. The other coordinate is

the declination as discussed in the previous system. In the above figure star is the hour angle and M1 M is the declination of the celestial body (M). The projection of M are M' and M1 on the

horizon and equator respectively.

11. Write the procedure for determination of true meridian. [CO5-H1](AUC May/June 09 & 13)

It is the simplest method of determining the direction of the celestial pole is probably

that observing at star at equal altitudes. In this method the knowledge of the latitude (or) local

time is not necessary.

The field observations are taken in the following steps:

) set the instrument at 0 and level it accurately.

) sight the reference mark with the reading 00 O' O" on the horizontal circle.

) 0pen the upper clamp and turn the telescope clockwise to bisect accurately the star at

position M1. Clamp both horizontal as well as vertical circle.

) Read the horizontal angle 1 as well as the altitudes α of the star.


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) When the star reaches the other side of the meridian, follow it through the telescope,

by unclamping the upper clamp and bisect it when it attains the same altitude.

The telescope is turned in azimuth and vertical circle reading remains unchanged. Read the

angle 2.

Case 1: Both positions of the star to the same side:

A = azimuth = ROP'

1 ROM1'

2

ROM 2 '

A 1 2 1

2 1 2

2

Hence the azimuth of the line is equal to half of the sum of the two observed angles.

Case 2: Both positions of the star are on opposite sides of line:

A = azimuth = M1OP M1OR

1

2 M1OM 2 M1OR

A 1 2

2

1

A 2 1

2

Hence the azimuth of the line is equal to half of the difference of the two observed angles.

12. At a certain place in longitude 1380 45' east, the star is observed east of the meridian at

6h 45m 218 P.M. with a watch keeping local mean time. It was again observed at the same

altitude to the west of meridian at 8h 48m 438 P.M. Find the error of the watch given below.

G.8.T: at G.M.N on that day = 9h 26m 128; R.A of the star =17h 12m 488.

[CO5-H2] (AUC May/June 2013)

8olution:

L.s.T. of star = R.A. of star = 17h 12m 48s

Let sidereal time is converted into mean time.

Longitude = 1380 45' E = 9h 15m E


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since the place has east longitude.

L.s.T. of L.M.N. = (G.s.T at G.M.N) - retardation

9h x 9.8565 = 88.71 sec

15m x 0.1642 = 2.46 sec

Total retardation = 91.17 sec = 1m 31.17s

G.s.T. at G.M.N. = 9h 26m 12s

subtract retardation = 1m

31.17s

L.s.T. at L.M.N. = 9h 24m 40.83s

Now local sidereal time = 17h 12m 48s

subtract L.s.T. at L.M.N. = 9h 24m 40.83s

s.l. since L.M.N. = 7h 48m 7.17s

Let us convert this s.l. into mean time interval by subtracting the retardation at the rate of

9.8296s per sidereal hour.

7h x 9.8296 = 68.81 sec

48m x 0.1638 = 7.86 sec

7.17s x 0.0027 = 0.02 sec


s.l. = 7h

48m

7.17s

subtract retardation = 1m 16.69s

M.l since L.M.N. = 7h

46m

50.48s

Local mean time of transit star = 7h 46m 50.48s P.M

Now L.M.T. of watch for east observation = 6h 45m 21s P.M

L.M.T. of watch for west observation = 8h 48m 43s P.M

= 15h 34m 04s

L.M.T. of transit of the star as shown by the chronometer = 7h 47m 13.52s P.M Chronometer error = 53.65 sec

13. A star was observed at western elongation at a place in latitudes 52o 20' N and latitude

52o 20' E when its clockwise horizontal angle from a survey line was 105o 49' 55". Find

the azimuth of the survey line and the local mean time of elongation given that the stars

declination was 73o 27' 30" N and its right ascension 14h 50m 54s the G8T of GMN being

5h 16m 54s. [CO5-H2] (AUC May/June 2009)

8olution:

i) Azimuth of star:

sin A = cos

cos

cos ( 73 0 27' 30" )

cos ( 52o 20' )

0.4659

A = 270 46' 6.54"

ii) Hour angle of star: o

Cos H = tan

tan

tan ( 52 20' ) tan ( 73o 27' 30")

0.3847

H = 670 22' 29.58" = 4h 29m 29.978

iii) Azimuth of the line:

since the star was at western elongation, also lies on west of the meridian.

Azimuth of the line AB = azimuth of the star + horizontal angle between line and star

= 270 46' 6.54" + 1050 49' 55" = 1330 36' 1.54"


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iv) Local mean time of observation:

First calculate the L.s.T. of L.M.N. from the given value of G.s.T. of G.M.N.

Longitude = 520 20' E = 3h 29m 20s

3h x 9.8565 = 29.57

29m x 0.1642 = 4.76

20s x 0.0027 = 0.054

Total acceleration = 34.384 sec

G.s.T. of G.M.N. = 5h 16m 54s

Add acceleration = 34.384s

L.s.T. of L.M.N. = 5h 17m 28.38s

Now L.s.T. of observation = R.A. of star + H.A. of the star

= 14h 50m 54s + 4h 29m 29.97s

L.s.T. = 19h 20m 23.97s

subtract L.s.T. of L.M.N. = 5h 17m 28.38s

s.l. from L.M.N. = 14h 2m 55.59s

Let us convert the s.l. into mean time interval by subtracting the retardation at the rate of

9.8296 per sidereal hour.

14h

x 9.8296 = 137.61 sec

2m x 0.1638 = 0.33 sec

55.59s

x 0.0027 = 0.15 sec


Mean time interval from L.M.N.

= s.l. - retardation

= 14h 2m 55.59s - 2m 18.09s

L.M.T. of observation = 14h Om 37.58

14. i) With the help of a sketch, explain the construction of an astronomical triangle.

Obtain the relations existing amongst the spherical coordinates. [CO5-H1]

lt is one formed by joining the pole, the zenith and any star M on the sphere by arcs of

great circles.

Let α be the altitude of the celestial body (M)

δ be the declination of the celestial body (M)


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be the latitude of the observer.

Then ZP = co-latitude of the observer

= 900 - = c

MP = co-declination (or) polar distance of M

= 900 - δ = p

ZM = zenith distance (or) co-altitude of the body

= 900 - α = z

Then

Angle at Z = MZP = azimuth (A) of the body

Angle at P = ZPM = hour angle (H) of the body

Angle at M=ZMP = parallactic angle

If the three sides (MZ, ZP and PM) of the astronomical triangles are known, the angles

A and H can be computed using the formulae of spherical trigonometry.

Also,

cos A = sin δ cos α . cos

- tan α . tan

tan A

sin ( s ZM ) sin ( s ZP )

2 sin s . sin ( s PM )

sin ( s z ) sin ( s c )

sin s . sin ( s p )

sin A

sin ( s z ) sin ( s c )

2 sin z . sin c

cos A

sin s sin ( s p )

Where

2 sin z . sin c

z 1

( ZM ZP PM ) 1

2 2

( z c p )

similarly,

cos H

sin

tan

tan

Also,

cos cos

tan H

sin ( s ZP ) sin ( s PM )

2 sin s . sin ( s ZM )

sin ( s c ) sin ( s p )

sin s . sin ( s z )

sin H

sin ( s c ) sin ( s p )

2 sin c . sin p


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cos H

sin s sin ( s z )

2 sin c . sin p

ii) Find the GMT corresponding to the LMT 9h 40m 12s A.M. at a place in longitude 42o 36'

w. (AUC May/June 2012)

Solution:

420 36' Longitude of the place is 42o 36' W =

15

= 2h 50m 24s

As the place is west of Greenwich, the GMT will be more.

GMT = LMT + longitude

= 9h 40m 12s + 2h 50m 24s

GMT = 12h 30m 36S

15. The mean observed altitude of the sun, corrected for refraction, parallax and level was

36o 14' 16.8" at a place in latitude 36o 40' 30" N and longitude 56o 24' 12" E. the mean

watch time of observation was 15h 49m 12.6", the watch being known to be about 3m fast

on LMT. Find the watch error given the following:

Declination of sun at the instant of observation = +17o 26' 42.1"

GMT of GAN = 11h 56m 22.Ss. [CO5-H2] (AUC May/June 2012)

Solution:

The hour angle of the sun is given by

tan A

sin ( s zp ) sin ( s zs )

2 sin s sin ( s ps )

Where, zp = 900 - ; ps = 900 - ; zs = 900 - ;

2s = zp + zs + ps

zp = 900 - = 900 - 360 40' 30'' = 530 19' 30''

ps = 900

- = 900

- 170

26' 42.1'' = 720

33' 17.9''

zs = 900 - = 900 - 360 14' 16.8'' = 530 45' 43.2''

2s = 1790 38' 31.1''

s = 890 49' 15.55''

(s - zp) = 890 49' 15.55'' - 530 19' 30'' = 360 29' 45.55''

(s - zs) = 890 49' 15.55'' - 530 45' 43.2'' = 360 3' 32.35''

(s - ps) = 890 49' 15.55'' - 720 33' 17.9'' = 170 15' 57.65''

0 0

tan A

sin ( s zp ) sin ( s zs )

sin ( 36 29' 45.55" ) sin ( 36 3' 32.35" )

1.08612 sin s sin ( s ps ) sin ( 89

0 49' 15.55" ) sin (17

0 15' 57.65" )

A = 470 21' 48.38''

2 A = 940 43' 36.75'' = 6h 18m 54.45s


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L.A.T. = 18h 18m 54.45s

Now the longitude 56o 24' 12" is converted as 3h 45m 36.8s

L.A.T. = 18h 18m 54.45s

Longitude = 3h 45m 36.8s

G.A.T. = 12h 04m 03.8s

G.M.T. of G.A.N. = 11h 56m 22.8s

G.A.T. = G.M.T. + E.T

12h 04m 03.8s = 11h 56m 22.8s + E.T

E.T = 7m 41s

Then L.M.T. = G.M.T + Longitude = 11h 56m 22.8s + 3h 45m 36.8s = 15h 41m 59.6s

Error = 15h 49m 12.6" - 15h 41m 59.6s = 7m 13S (East)

16. Explain about Mean solar time and Standard time system.

[CO5-H2] (AUC Nov/Dec 2012)

Mean solar time:

In order to circumvent the non-uniformity of apparent solar time, a fictitious body called

the mean sun is introduced. Thus the man sun is an imaginary body and is assumed to move at

a uniform rate along the equator in order to make a solar day of uniform period. It is presumed

to start from the vertical equinox at the same time as the true sun and to return to the vernal

equinox with the true sun. Thus the line when measured by the diurnal motion of the mean sun

is called the mean solar time or simply mean time. This is the time kept by clocks and watches.

The time interval between two successive lower transits of the mean sun over the same

meridian is called a mean solar day or civil day. Civil time and astronomical time are the two

systems which are in use. Prior to 31st December 1924, the astronomical day was reckoned to

begin at noon.

From 1st

January, 1925 both the civil day and astronomical day begin at zero hour midnight.

Civil day is divided into two portions, viz.,

Midnight to noon = anti meridian (AM)

Noon to midnight = post meridian (PM)

But astronomical day is from midnight to midnight, i.e., zero hour to 24 hours. Standard time:

Instead of using different local mean times by the people in a country it will be

appropriate to adopt the mean time on a particular meridian as the standard time for the whole

of a country. such a meridian is called the standard meridian.

The meridian passing the Greenwich is called Greenwich meridian which is the

standard meridian for Great Britain. Greenwich mean time (G.M.T.) is measured from the lower

transit of the Greenwich meridian by the mean sun, i.e., from Greenwich mean midnight, 0 to 24

hours. It is identical with Universal time (U.T), a term which was recommended by the

international astronomical union and which has now been generally adopted.

The standard meridian usually lies at an exact number of hours from Greenwich. The

mean time associated with this meridian is called the standard time. This is the time which is

kept by all watches and clocks throughout the country. The longitude of the standard meridian

adopted in India is 820 30' E or 5 hours 30 min East.

The standard time may be converted to the local mean time and vice-versa by the

relation.

standard time = L.M.T (difference of longitude in time between the given place and the

standard meridian)


121


Thus plus sign has to be used if the place is to the west of the standard meridian and

minus sign if it is to the east.

17. Determine the hour angle and declination of a star from the following data.

Altitude of the star = 21o 30'

Azimuth of the star = 140o E

Latitude of the observer = 4So N. [CO5-H2] (AUC Nov/Dec 2012)

Solution:

The azimuth of the star is 140° E, the star is in the eastern hemisphere.

In the astronomical triangle ZPM, we have

cO altitude, ZM 900 900 21

0 30' 68

0 30'

cO latitude, ZP 900

A = 1400

Using cosine formula,

900

480 420


cos ( 680

30' ) cos ( 420

) sin ( 680

30' ) sin ( 420

) cos (1400

)

Cos P = - 0.2046

PM = 1010

48' 21.9"

Declination of star, δ = 900 - PM = 900 - 1010 48' 21.9" = - 110 48' 21.9"

δ = 110

4S' 21.9" s

Using cosine rule,

cos H

cos ZM cos PZ cos PM

sin PZ sin PM

0 0 0

cos H cos ( 68 30' ) cos ( 42 ) cos (101 48' 21.9" )

sin ( 420 ) sin (1010 48' 21.9")

Cos H = 0.7917

cos ( 3600 H ) 0.7917

( 3600 H ) 37

0 39' 19.21"

Hour angle, H = 3220 20' 40.79"

18. Explain about instrumental correction to be observed altitude and azimuth.


Instrumental correction:

The instrumental corrections are Index error and Bubble error corrections.

i) Index error corrections:

The small vertical angle between the line of collimation and the horizontal bubble

line of the altitude or azimuthal bubble is the index error. This is obtained by adopting the

following procedure.

With the telescope normal in face left position any well-defined object such as a

church spire is bisected and left the angle be α1.

The face is changed (right face) and the telescope is reversed and the telescope is

reversed and the same object is bisected again and let the angle be α2. The mean value,

' 1 2 .

2

With reference to the mean value the observed values are corrected. The

index error is said to be + E or - E according as this amount is to be added to or subtracted


122


from the observed altitude. sometimes it may not be practicable to take observations on

both faces; in such cases the correction for index error is necessary. The index error is

eliminated by taking both face observations.

ii) Bubble error corrections:

At the time of observation, when the bubble tube is not at the centre of its run, then the

correction for bubble error is needed.

0 E Correction for bubble error X V

n In which LO = sum of the readings of the object glass end of the bubble.

LE = sum of the readings of the eye-piece end of the bubble. n =

number of bubble ends read.

v = angular value of one division of the bubble in seconds.

The sign of the correction is plus or minus according as LO is greater or lesser than

LE. The observed altitude when corrected for index error and bubble error (semi-diameter) is

called the apparent altitude

19. Explain Direction and velocity of current by floats using three methods.



123



124



125


20. Explain the location of floats with two theodolite method. [CO5-H1] (AUC Nov/Dec 2010)

In this method, a point is fixed independent of the range by angular observations from

two points on the shore. The method is generally used to locate some isolated points. If this

method is used on an extensive survey, the boat should be run on a series of approximate

ranges. Two instruments and two instrument men are required. The position of instrument is

selected in such a way that a strong fix is obtained. New instrument stations should be chosen

when the intersection angle ( ) falls below 30°.

Thus A and B are the two instrument stations. The distance d between them is very

accuarately measured. The instrument stations A and B are precisely connected to the ground

traverse or triangulation, and their positions on plan are known. With both the plates clamped to

zero, the instrument man at A bisects B ; similarly with both the plates clamped to zero, the

instrument man at B bisects A. Both the instrument men then direct the line of sight of the

telescope towards the leadsman and continuously follow it as the boat moves. The surveyor on

the boat holds a flag for a few seconds, and on the fall of the flag the sounding and the angles

are observed simultaneously. The co-ordinates of the position P of the sounding may be

computed from the relations:

d tan d tan α tan x = ; y =

tan α + tan tan α + tan

Advantages:

) The preliminary work of setting out and erecting range signals is eliminated.

) It is useful when there are strong currents due to which it is difficult to row the boat

along the range line.

21. Explain Tilt distortion with neat sketch in photographic method.

[CO5-H1] (AUC May/June 2013) (AUC Nov/Dec 2010)


126



127


22. Write in detail about the methods of locating soundings.

[CO5-H1] (AUC May/June 2009) (AUC Apr/May 2011) (AUC Apr/May 2010)

The methods of locating soundings:

i) By cross rope.

ii) By range and time intervals.

iii) By range and one angle from the shore.

iv) By range and one angle from the boat.

v) By two angles from the shore.

vi) By two angles from the boat.

vii) By one angle from shore and one from boat.

viii) By intersecting ranges.

ix) By tacheometry.

i) Location by Cross-Rope:

This is the most accurate method of locating the soundings and may be used for

rivers, narrow lakes and harbours. It is also used to determine the quantity of materials

removed by dredging the soundings being taken before and after the dredging work is don e.

A single wire or rope is stretched across the channel etc. and is marked by metal tags at

appropriate known distance along the wire from a reference point or zero station on shore.

The soundings are then taken by a weighted pole. The position of the pole during a

sounding is given by the graduated rope or line.

ii) By range and time intervals:

In this method, the boat is kept in range with the two signals on the shore and is

rowed along it at constant speed. Soundings are taken at different time intervals. Knowing

the constant speed and the total time elapsed at the instant of sounding, the distance of the

total point can be known along the range. The method is used when the width of channel is

small and when great degree of accuracy is not required. However, the method is used in


128


conjunction with other methods, in which case the first and the last soundings along a range

are located by angles from the shore and the intermediate soundings are located by

interpolation according to time intervals.

iii) By range and one angle from the shore:

In this method, the boat is ranged in line with the two shore signals and rowed

along the ranges. The point where sounding is taken is fixed on the range by observation of

the angle from the shore. As the boat proceeds along the shore, other soundings are also

fixed by the observations of angles from the shore. Thus B is the instrument station, A 1 A2 is

the range along which the boat is rowed and α1, α2, α3 etc., are the angles measured at B

from points 1, 2, 3 etc.

To fix a point by observations from the shore, the instrument man at B orients his

line of sight towards a shore signal or any other prominent point (known on the plan) when

the reading is zero. He then directs the telescope towards the leadsman or the bow of the

boat, and is kept continually pointing towards the boat as it moves. The surveyor on the boat

holds a flag for a few seconds and on the fall of the flag, the sounding and the angle are

observed simultaneously.

The angles are generally observed to the nearest 5 minutes. The time at which the

flag falls is also recorded both by the instrument man as well as on the boat. In order to

avoid acute intersections, the lines of soundings are previously drawn on the plan and

suitable instrument stations are selected.

iv) By range and one angle from the boat:

The method is exactly similar to the previous one except that the angular fix is made

by angular observation from the boat. The boat is kept in range with the two shore signals

and is rowed along it. At the instant the sounding is taken, the angle, subtended at the point

between the range and some prominent point B on the sore is measured with the help of

sextant. The telescope is directed on the range signals, and the side object is brought into

coincidence at the instant the sounding is taken. The accuracy and ease of plotting is the

same as obtained in the previous method. Generally, the first and the last soundings, and

some of the intermediate soundings are located by angular observations and the rest of the

soundings are located by time intervals.


129


As compared to the previous methods, this method has the following advantages:

) Since all the observations are taken from the boat, the surveyor has better control

over the operations.

) The mistakes in booking are reduced since the recorder books the readings directly

as they are measured.

) On important fixes, check may be obtained by measuring a second angle towards

some other signal on the shore.

) Obtain good intersections throughout; different shore objects may be used for

reference to measure the angles.

v) By two angles from the shore:

In this method, a point is fixed independent of the range by angular observations

from two points on the shore. The method is generally used to locate some isolated points. If

this method is used on an extensive survey, the boat should be run on a series of

approximate ranges. Two instruments and two instrument men are required. The position of

instrument is selected in such a way that a strong fix is obtained. New instrument stations

should be chosen when the intersection angle ( ) falls below 30°.

Thus A and B are the two instrument stations. The distance d between them is very

accuarately measured. The instrument stations A and B are precisely connected to the

ground traverse or triangulation, and their positions on plan are known. With both the plates


130


clamped to zero, the instrument man at A bisects B ; similarly with both the plates clamped

to zero, the instrument man at B bisects A. Both the instrument men then direct the line of

sight of the telescope towards the leadsman and continuously follow it as the boat moves.

The surveyor on the boat holds a flag for a few seconds, and on the fall of the flag the

sounding and the angles are observed simultaneously. The co-ordinates of the position P of

the sounding may be computed from the relations:

d tan d tan α tan x = ; y =

tan α + tan tan α + tan

Advantages:

) The preliminary work of setting out and erecting range signals is eliminated.

) It is useful when there are strong currents due to which it is difficult to row the boat

along the range line.

vi) By two angles from the boat:

In this method, the position of the boat can be located by the solution of the three-

point problem by observing the two angles subtended at the boat by three suitable shore

objects of known position. The three-shore points should be well-defined and clearly visible.

Prominent natural objects such as church spire, lighthouse, flagstaff, buoys etc., are

selected for this purpose. If such points are not available, range poles or shore signals may

be taken.

Thus A, B and C are the shore objects and P is the position of the boat from which

the angles α and β are measured. Both the angles should be observed simultaneously with

the help of two sextants; at the instant the sounding is taken. If both the angles are observed

by surveyor alone, very little time should be lost in taking the observation. The angles on the

circle are read afterwards. The method is used to take the soundings at isolated p oints. The

surveyor has better control on the operations since the survey party is concentrated in one

boat.

Advantages:

) Preliminary work setting out and erecting range signals is eliminated.

) The position of the boat is located by the solution of the three point problem either

analytically or graphically.


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vii) By one angle from shore and one from boat:

This method is the combination of methods 5 and 6 described above and is used

to locate the isolated points where soundings are taken. Two points A and B are chosen on

the shore, one of the points (say A) is the instrument station where a theodolite is set up,

and the other (say B) is a shore signal or any other prominent object. At the instant the

sounding is taken at P, the angle α at A is measured with the help of a sextant. Knowing the

distance d between the two points A and B by ground survey, the position of P can be

located by calculating the two co-ordinates x and y.

viii) By intersecting ranges:

This method is used when it is required to determine by periodical sounding at the

same points, the rate at which silting or scouring is taking place. This is very essential on the

harbors and reservoirs. The position of sounding is located by the intersection of two


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ranges, thus completely avoiding the angular observations. Suitable signals are erected at the

shore. The boat is rowed along a range perpendicular to the shore and soundings are taken at the

points in which inclined ranges intersect the range, as illustrated in figure. However, in order to

avoid the confusion, a definite system of flagging the range poles is necessary. The position of the

range poles is determined very accurately by ground survey.

ix) By tacheometry:

The method is very much useful in smooth waters. The position of the boat is located by

tacheometric observations from the shore on a staff kept vertically on the boat. Observing the staff

intercept s at the instant the sounding is taken, the horizontal distance between the instrument

stations and the boat is calculated.

The direction of the boat (P) is established by observing the angle (α) at the instrument

station B with reference to any prominent object A The transit station should be near the water level

so that there will be no need to read vertical angles. The method is unsuitable when soundings are

taken far from shore.

23. Explain the following:

i) Scale of a vertical photograph.

ii) Relief displacement on a vertical photograph. [CO5-H1] (AUC Apr/May 2011)

i) Scale of a vertical photograph:


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ii) Relief displacement on a vertical photograph:


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24. What is a three point problem in hydrographic surveying? What are the various

solutions for the problem? Explain in detail. [CO5-H1] (AUC May/June 2009)

Given the three shore signals A, B and C, and the angles α and β subtended by AP, BP

and CP at the boat P, it is required to plot the position of P. 1. Mechanical Solution

(i) By Tracing Paper

Protract angles α and β between three radiating lines from any point on a piece of

tracing paper. Plot the positions of signals A, B, C on the plan. Applying the tracing

paper to the plan, move it about until all the three rays simultaneously pass through A, B

and C. The apex of the angles is then the position of P which can be pricked through.

(ii) By Station Pointer:

The station pointer is a three-armed protractor and consists of a graduated circle

with fixed arm and two movable arms to the either side of the fixed arm. All the three

arms have beveled or fiducial edges. The fiducial edge of the central fixed arm


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corresponds to the zero of the circle. The fiducial edges of the two moving arms can

be set to any desired reading and can be clamped in position. They are also

provided with verniers and slow motion screws to set the angle very precisely. To plot

position of P, the movable arms are clamped to read the angles α and β very precisely.

The station pointer is then moved on the plan till the three fiducial edges simultaneously

touch A, B and C. The centre of the pointer then represents the position of P which can

be recorded by a prick mark.

2. Graphical Solutions

(a) First Method:

Let a, b and c be the plotted positions of the shore signals A, B and C respectively

and let α and β be the angles subtended at the boat. The point p of the boat position p can

be obtained as under:

) Join a and c.

) At a, draw ad making an angle β with ac. At c, draw cd making an angle α with ca.

Let both these lines meet at d.

) Draw a circle passing through the points a, d and c.

) Join d and b, and prolong it to meet the circle at the point p which is the require d

position of the boat.

Proof: From the properties of a circle,

apd = acd = α and cpd = cad = β

which is the required condition for the solution.

(b) Second Method:


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) Join ab and bc.

) From a and b, draw lines ao1 and bo1 each making an angle (90° - α) with ab on the

side towards p. Let them intersect at 01.

) Similarly, from b and c, draw lines each making an angle (90° - β) with ab on the side

towards p. Let them intersect.


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) With - as the centre, draw a circle to pass through a and b. Similarly, with - as the

centre draw a circle to pass through b and c. Let both the circles intersect each other at

a point p. p is then the required position of the boat.

Proof: ao1b = 180° - 2 (90° - α) = 2α

apb = ½ ao1b = α

Similarly, bo2c = 180° - 2 (90° - β) = 2β

and bpc = ½ bo2c = β.

The above method is sometimes known as the method of two intersecting circles.

(c) Third Method:

) Join ab and bc.

) At a and c, erect perpendiculars ad and ce.

) At b, draw a line bd subtending angle (90° - α) with ba, to meet the perpendicular

through a in d.

) Similarly, draw a line be subtending an angle (90° - β) with bc, to meet the

perpendicular through c in e.

) Join d and e.

) Drop a perpendicular on de from b. The foot of the perpendicular (i.e. p) is then the

required position of the boat. 25. Explain briefly the different methods of prediction of tides.

[CO5-H2] (AUC May/June 2009)

i) Age of tide

ii) Lunitidal interval

iii) Mean establishment

iv) Vulgar establishment

i) Age of tide:

This condition is fulfilled only in southern ocean extending southwards from about 40 0

S

latitude. It is the only portion of ocean where equilibrium figure may be developed. Primary

tide waves are generated and secondary waves are propagated into pacific, Atlantic and

Indian oceans. The velocity of wave travel may exceed 1000 km per hour, though it is less in

shallow water. The amplitude of vertical range from crest to trough is not more than 60 to 90

cm. due to direction of propagation of tide wave, high or low wa ter occurs at different times

at various places on the same meridian. "The time which elapse between the generation of

spring tide and its arrival at the place is called Age of tide".

ii) Lunitidal interval:


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"It is the time interval that elapses between the moon's transits and occurrence of next

high water". The value is found to vary because of existence of priming and lagging. The


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values can be observed and plotted for a fortnight against the times of moon's transits, a

curve is obtained. A curve has approximately same for each fortnight and used for rough

prediction of time of tide. The time of transit of moon at Greenwich is given in nautical

almanac. The time of transit can be derived by adding 2 m for every hour of west longitude

and subtracting 2 m for every hour of east longitude.

iii) Mean establishment:

The average value of Lunitidal at a place is known as mean establishment as shown

by dotted line. If the value is known and Lunitidal interval and the time of high water can be

estimated. The procedure of determination are

) Find from charts, the age of tide and mean establishment for the place.

) Knowing the hour of moon's transit, determine the time of moon's transit on the day

of generation of tide.

Day of generation = day in question - age of tide

) Corresponding to time of transit of moon on the day of generation of tide, find out the

amount of priming or lagging correction. ) Add the priming or lagging correction to mean establishment to get Lunitidal interval

for day in question.

) Add the Lunitidal interval to the time of moon's transit on the day in question to get

approximate time of high water.

Hour of moon's transit 0 1 2 3 4 5 6 7 8 9 10 11 12

Correction in minutes 0 -16 -31 -41 -44 -31 0 31 44 41 31 16 0

iv) Vulgar establishment:

"It is defined as the value of Lunitidal interval on the day of full moon or change of

moon".

Its value is always more than establishment since lagging correction in second or

fourth quadrant is positive. The difference between vulgar establishment and mean

establishment depends upon age of tide. The value of vulgar establishment is approximately

equal to clock time at which high water occurs on day of full moon or change of moon.

Mean establishment = vulgar establishment - lagging correction

Height of tide:

The approximate height of tide of known rise at any time between high and low water can

be expressed

1 H = h + r cos

2

H = required height of tide above datum

h = height of mean tide level above datum

r = range of tide

= interval from high water

x 180O

interval between high and low water


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26.Explain the principle underlying "Electronic Distance Measurement". Write a note

on errors in EDM. [CO5-H1] (AUC May/June 2013) (AUC May/June 2012)

Principle:

Electronic Distance Measurement (EDM) is a general term used collectively in the

measurement of distances applying electronic methods. Basically the EDM method is based

on generation, propagation, reflection and subsequent reception of electromagnetic wave s.

Every EDM equipment should perform the following functions:

) Generation of carrier wave and measuring wave frequencies.

) Modulation and demodulation of the carrier wave.

) Measurement of the phase difference between the transmitted and received

measuring waves.

) Display, in some form, the result of this measurement.

Scale Error:

Scale error is proportional to the length of the line measured and is caused by:

) internal frequency errors, including those caused by external temperature

and instrument "warm up" effects;

) errors of measured temperature, pressure and humidity which affect the velocity of

the signal; and

) non-homogeneous emissions/reception patterns from the emitting and

receiving diodes (phase in-homogeneities).

Cyclic Error

The precision of an EDM instrument is dependent on the precision of the internal

phase measurement. Unwanted interference either through electronic/optical cross talk or

multi-path effects of the transmitted signal onto the received signal causes cyclic error. The

major form of the cyclic error is sinusoidal with a wavelength equal to the unit length of the

instrument.

The unit length is the scale on which the EDM instrument measures the distance, and is

derived from the fine measuring frequency. Unit length is equal to one half of the

modulation wavelength. The magnitude of the cyclic error can be of the order of 5 - 10 mm,

but it will vary depending on the actual length measured.

27. Given the three shore signals A, B and C and the angles α and β subtended by AP, BP

and CP at the boat P, it is required to plot the position of P (refer figure below). How

will you obtain the position of P using a station pointer? [CO5-H2] (AUC May/June

2012)


141


1

2

Solution to three-point problem is also obtained using mechanical devices. Most

usual solution is obtained by use of one such machine named as station -pointer. This

instrument consists of three arms, the fiducial edges of which radiate to a common centre.

The middle arm is fixed, while the two outer arms are capable of rotation about the centre of

the instrument. These two arms are fitted with verniers reading to 1 minute and with clamp

and tangent screw arrangements for accurate adjustment.

In order to use the instrument the arms are so set by means of the verniers a to

include the observed angles and . The instrument is moved over the paper until the

fiducial edges pass simultaneously through the three points a, b, c on the chart. The centre

is then marked

with a hard pencil or pricker or rays drawn along the edges of the arms which are produced to

intersect

27. write short notes on i) Aerial photograph ii) Stereoscopy. [CO5-H1] (AUC Nov/Dec 2012)

i) Aerial photograph:


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ii) Stereoscopy:

28. How to measure angles with the sextant? [CO5-H1] (AUC Nov/Dec 2012)


143


29. Explain the procedure to use fathometer in ocean sounding. [CO5-H1]

A Fathometer is used in ocean sounding where the depth of water is too much, and to

make a continuous and accurate record of the depth of water below the boat or ship at which it

is installed. It is an echo-sounding instrument in which water depths are obtained be

determining the time required for the sound waves to travel from a point near the surface of the

water to the bottom and back. It is adjusted to read depth on accordance with the velocity of

sound in the type of water in which it is being used. A fathometer may indicate the depth

visually or indicate graphically on a roll which continuously goes on revolving and provide a

virtual profile of the lake or sea.


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30. Explain the different types of tides in detail. [CO5-H1]

Tides:

All celestial bodies exert a gravitational force on each other. These forces of attraction

between earth and other celestial bodies (mainly moon and sun) cause periodical variations in

the level of a water surface, commonly known as tides.

Types of tides:

i) Lunar tides

ii) Solar tides

iii) Spring and neap tide (combined effect)

iv) Other effects

i) Lunar tides:

The figure shows the earth and the moon, with their centres of masses 01 and 02

respectively. Since moon is very near to the earth, it is the major tide producing force. To start

with, we will ignore the daily rotation of the earth on its axis. Both earth and moon attract each

other, and the force of attraction would act along 01 02. Let 0 be the common centre of gravity

of earth and moon. The earth and moon revolve monthly about 0, and due to this revolution

their separate positions are maintained. The distribution of force is not uniform, but it is more

for the points facing the moon and less for remote points. Due to the revolution of earth about

the common centre of gravity 0, centrifugal force of uniform intensity is exerted on all the

particles of the earth. The direction of this centrifugal force is parallel to 0102 and acts outward.

Thus, the total force of attraction due to moon is counter-balanced by the total centrifugal force,

and the earth maintains its position relative to the moon. However, since the fore of attraction

is not uniform, the resultant force will very all along. The resultant forces are the tide producing

forces. Assuming that water has no inertia and viscosity, the ocean enveloping the earth's

surface will adjust itself to the unbalanced resultant forces, giving rise to the equilibrium. Thus,

there are two lunar tides at A and B, and two low water positions at C and D. The tide at A is

called the superior lunar tide or tide of moon's upper transit, while tide at B is called inferior or

antilunar tide.

Now let us consider the earth's rotation on its axis. Assuming the moon to remain

stationary, the major axis of lunar tidal equilibrium figure would maintain a constant position.

Due to rotation of earth about its axis from west to east, once in 24 hours, point A would

occupy successive position C, B and D at intervals of 6 h. Thus, point A would experience

regular variation in the level of water. It will experience high water (tide) at intervals of 12 h and


145


low water midway between. This interval of 6 h variation is true only if moon is assumed

stationary. However, in a lunation of 29.53 days the moon makes one revolution relative to sun

from the new moon to new moon. This revolution is in the same direction as the diurnal

rotation of earth, and hence there are 29.53 transits of moon across a meridian in 29.53 mean

solar days. This is on the assumption that the moon does this revolution in a plane passing

through the equator. Thus, the interval between successive transits of moon or any meridian

will be 24 h, 50.5 m. Thus, the average interval between successive high waters would be

about 12 h 25 m. The interval of 24 h 50.5 m between two successive transits of moon over a

meridian is called the tidal day.

ii) Solar tides: The phenomenon of production of tides due to force of attraction between earth and

sun is similar to the lunar tides. Thus, there will be superior solar tide and an inferior or anti -

solar tide. However, sun is at a large distance from the earth and hence the tide producing

force due to sun is much less.

Solar tide = 0.458 Lunar tide

iii) Spring and neap tides:

Solar tide = 0.458 Lunar tide.

Above equation shows that the solar tide force is less than half the lunar tide force.

However, their combined effect is important, especially at the new moon when both the sun

and moon have the same celestial longitude, they cross a meridian at the same instant.


146


Assuming that both the sun and moon lie in the same horizontal plane passing through the

equator, the effects of both the tides are added, giving rise to maximum or spring tide of new

moon. The term 'spring' does not refer to the season, but to the springing or waxing of the

moon. After the new moon, the moon falls behind the sun and crosses each meridian 50

minutes later each day. In after 7 ½ days, the difference between longitude of the moon and

that of sun becomes 90°, and the moon is in quadrature. The crest of moon tide coincides with

the trough of the solar tide, giving rise to the neap tide of the first qua rter. During the neap

tide, the high water level is below the average while the low water level is above the average.

After about 15 days of the start of lunation, when full moon occurs, the difference between

moon's longitude and of sun's longitude is 180°, and the moon is in opposition. However, the

crests of both the tides coincide, giving rise to spring tide of full moon. In about 22 days after

the start of lunation, the difference in longitudes of the moon and the sun becomes 270° and

neap tide of third quarter is formed. Finally, when the moon reaches to its new moon position,

after about 29 ½ days of the previous new moon, both of them have the same celestial

longitude and the spring tide of new moon is again formed making the beginning of another

cycle of spring and neap tides.

iv) Other effects:

The length of the tidal day, assumed to be 24 hours and 50.5 minutes is not constant

because of

(i) varying relative positions of the sun and moon,

(ii) Relative attraction of the sun and moon,

(iii) Ellipticity of the orbit of the moon (assumed circular earlier) and earth,

(iv) Declination (or deviation from the plane of equator) of the sun and the moon,

(v) Effects of the land masses and

(vi) Deviation of the shape of the earth from the spheroid.

Due to these, the high water at a place may not occur exactly at the moon's upper or

lower transit. The effect of varying relative positions of the sun and moon gives rise to what

are known as priming of tide and lagging of tide.

At the new moon position, the crest of the composite tide is under the moon and

normal tide is formed. For the positions of the moon between new moon and first quarter, the

high water at any place occurs before the moon's transit, the interval between successive high

water is less than the average of 12 hours 25 minutes and the tide is said to prime. For

positions of moon between the first quarter and the full moon, the high water at any place

occurs after the moon transits, the interval between successive high water is more than the

average, and tide is said to lag. Similarly, between full moon and 3rd quarter position, the tide

primes while between the 3rd quarter and full moon position, the tide lag s. At first quarter, full

moon and third quarter position of moon, normal tide occurs.

Due to the several assumptions made in the equilibrium theory, and due to several

other factors affecting the magnitude and period of tides, close agreement between the results

of the theory, and the actual field observations is not available. Due to obstruction of land

masses, tide may be heaped up at some places. Due to inertia and viscosity of sea water,

equilibrium figure is not achieved instantaneously. Hence prediction of the tides at a place

must be based largely on observations.


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