Engineering Surveying Lecture notes

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A Course Material on Surveying IICE6404 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 controlMethodsspecificationstriangulation- baselineinstruments and accessoriescorrectionssatellite stationsreduction to centre- trigonometrical levellingsingle and reciprocal observationstraversingGale s table. UNIT II SURVEY ADJUSTMENT 9 Errors Sources- precautions and correctionsclassification of errorstrue and most probable Values- weighed observations method of equal shifts  principle of least squares -0 normal Equationcorrelates- 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 systemsTraversing 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 segmentHand 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 surveyingTides - MSL - Sounding methods - Three-point problem - Strength of fix - Sextants and station pointer- Astronomical Surveyingfield observations and determination of Azimuth by altitude and hour angle methods  fundamentals of Photogrammetry and Remote Sensing TOTAL: 45 PERIODS OUTCOMES: On completion of this course students shall be able to 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", th 7 Edition, McGraw Hill, 2001. th 2. Bannister and S. Raymond, "Surveying", 7 Edition, Longman 2004. 3. Laurila, S.H. "Electronic Surveying in Practice", John Wiley and Sons Inc, 1993 REFERENCES: rd 1. Alfred Leick, "GPS satellite surveying", John Wiley & Sons Inc., 3 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 , 2007TABLE OF CONTENTS S.NO TOPICS PAGE NO 1 CHAPTER 1( CO NTROL SURVEYING) 1.1 HORIZONTAL CONTROLS & ITS METHODS 1 1.2 VERTICAL CONTROL & ITS METHODS 1 1.3 CLASSIFICATION OF TRIANGULATION SYSTEM 5 1.3.1 FIRST ORDER OR PRIMARY TRIANGULATION 5 1.3.2 SECONDARY ORDER OR SECONDARY TRIANGULATION 6 1.3.3 THIRD ORDER OR TERTIARY TRIANGULATION 7 2 CHAPTER 2 SURVEY ADJUSTMENTS 2.1 9 TYPES OF ERROR 2.2 THE LAW OF ACCIDENTAL ERRORS 10 2.3 PRINCIPLES OF LEAST SQUARES 10 2.4 LAW OF WEIGHTS 10 2.5 DISTRIBUTION OF ERROR OF THE FIELD MEASUREMENT 12 3 CHAPTER 3 TOTAL STATION BASIC PRINCIPLE 13 3.1 3.2 CLASSIFICATION OF TOTAL STATIONS 14 3.2.1 LASER DISTANCE MEASUREMENT 16 3.2.2 ELECTRO- OPTICAL SYSTEM 3.3 FEATURES OF TOTAL STATIONS 22 3.5 SOFTWARE APPLICATIONS 27 3.6 SOURCES OF ERROR FOR TOTAL STATIONS 31 CHAPTER 4 GPS SURVEYING INTRODUCTION 4.1 40 4.2 40 SEGMENTS OF GPS 4.3 SPACE SEGMENT 41 4.4 OBSERVATION PRINCIPLE AND SIGNAL 43 STRUCTURE4.5 STRUCTURE OF THE GPS NAVIGATION DATA 45 4.6 CONTROL SEGMENT 46 GROUND CONTROL SEGMENT 4.7 48 USER SEGMENT 4.8 48 4.9 BASIC CONCEPT OF GPS RECEIVER AND ITS 48 COMPONENTS 4.10 CLASSIFICATION OF GPS RECEIVERS 52 4.11 ACCURACY 60 4.12 DIFFERENTIAL THEORY 61 4.13 62 DIFFERENTIAL GPS CHAPTER 5 ADVANCED TOPICS IN SURVEYING 5.1 INTRODUCTION 63 5.2 PHOTOGRAMMETRIC SURVEYING 63 5.3 PRINCIPLES BEHIND TERRESTRIAL PHOTOGRAMMETRY 64 5.4 MEAN SEA LEVEL? EXPLAIN WHY IT IS USED AS 83 DATUM. 5.5 5.5 ASTRONOMICAL SURVEYING 83 5.5.1 CO-ALTITUDE OR ZENITH DISTANCE (Z ) AND A ZIMUTH (A ). 84CE6404 SURVEYING II CHAPTER 1 CONTROL SURVEYING 1.1 HORIZONTAL CONTROLS & ITS METHODS The horizontal control consists of reference marks of known plan position, from which salient points of designed structures may be set out. For large structures primary and secondary control points are used. The primary control points are triangulation stations. The secondary control points are reference to the primary control stations. Reference Grid Reference grids are used for accurate setting out of works of large magnitude. The following types of reference grids are used: 1. Survey Grid 2. Site Grid 3. tructural Grid 4. Secondary Grid Survey grid is one which is drawn on a survey plan, from the original traverse. Original traverse stations form the control points of the grid. The site grid used by the designer is the one with the help of which actual setting out is done. As far as possible the site grid should be actually the survey grid. All the design points are related in terms of site grid coordinates. The structural grid is used when the structural components of the building are large in numbers and are so positioned that these components cannot be set out from the site grid with sufficient accuracy. The structural grid is set out from the site grid points. The secondary grid is established inside the structure, to establish internal details of the building, which are otherwise not visible directly from the structural grid. 1.2 VERTICAL CONTROL & ITS METHODS: The vertical control consists of establishment of reference marks of known height relative to some special datum. All levels at the site are normally reduced to the nearby bench mark, usually known as master bench mark. The setting of points in the vertical direction is usually done with the help of following rods: 1. Boning rods and travelers 2. Sight Rails 3. Slope rails or batter boards 4. Profile boards SCE 1 Department of Civil EngineeringCE6404 SURVEYING II A boning rod consist of an upright pole having a horizontal board at its top, forming a ‘T ‘shaped rod. Boning rods are made in set of three, and many consist of three ‘T’ shaped rods, each of equal size and shape, or two rods identical to each other and a third one consisting of longer rod with a detachable or movable ‘T’ piece. The third one is called traveling rod or traveler. Sight Rails: A sight rail consist of horizontal cross piece nailed to a single upright or pair of uprights driven into the ground. The upper edge of the cross piece is set to a convenient height above the required plane of the structure, and should be above the ground to enable a man to conveniently align his eyes with the upper edge. A stepped sight rail or double sight rail is used in highly undulating or falling ground. Slope rails or Batter boards: hese are used for controlling the side slopes in embankment and in cuttings. These consist of two vertical poles with a sloping board nailed near their top. The slope rails define a plane parallel to the proposed slope of the embankment, but at suitable vertical distance above it. Travelers are used to control the slope during filling operation. Profile boards: These are similar to sight rails, but are used to define the corners, or sides of a building. A profile board is erected near each corner peg. Each unit of profile board consists of two verticals, one horizontal board and two cross boards. Nails or saw cuts are placed at the top of the profile boards to define the width of foundation and the line of the outside of the wall An instrument was set up at P and the angle of elevation to a vane 4 m above the foot of the staff held at Q was 9° 30′. The horizontal distance between P and Q was known to be 2000 metres. Determine the R.L. of the staff station Q given that the R.L. of the instrument axis was 2650.38. Solution: Height of vane above the instrument axis = D tan α = 2000 tan 9° 30′ = 334.68 m Correction for curvature and refraction C = 0.06735 D² m, when D is in km = 0.2694 ≈ 0.27 m ( + ve) SCE 2 Department of Civil EngineeringCE6404 SURVEYING II Height of vane above the instrument axis = 334.68 + 0.27 = 334.95 R.L. fo vane = 334.95 + 2650.38 = 2985.33 m R.L. of Q = 2985.33 – 4 = 2981.33 m An instrument was set up at P and the angle of depression to a vane 2 m above the foot of the staff held at Q was 5° 36′. The horizontal distance between P and Q was known to be 3000 metres. Determine the R.L. of the staff station Q given that staff reading on a B.M. of elevation 436.050 was 2.865 metres. Solution: The difference in elevation between the vane and the instrument axis = D tan α = 3000 tan 5° 36′ = 294.153 Combined correction due to cuvature and refraction C = 0.06735 D² metres , when D is in km = 0.606 m. Since the observed angle is negative, the combined correction due to curvature and refraction is subtractive. Difference in elevation between the vane and the instrument axis = 294.153 – 0.606 = 293.547 = h. R.L. of instrument axis = 436.050 + 2.865 = 438.915 \ R.L. of the vane = R.L. of instrument aixs – h = 438.915 – 293.547 = 145.368 R.L. of Q = 145.368 – 2 = 143.368 m. In order to ascertain the elevation of the top (Q ) of the signal on a hill, observations were made from two instrument stations P and R at a horizontal distance 100 metres apart, the SCE 3 Department of Civil EngineeringCE6404 SURVEYING II station P and R being in the line with Q. The angles of elevation of Q at P and R were 28° 42′ and 18° 6′ respectively. The staff reading upon the bench mark of elevation 287.28 were respectively 2.870 and 3.750 when the instrument was at P and at R, the telescope being horizontal. Determine the elevation of the foot of the signal if the height of the signal above its base is 3 metres. Solution: Elevation of instrument axis at P = R.L. of B.M. + Staff reading = 287.28 + 2.870 = 290.15 m Elevation of instrument axis at R = R.L. of B.M. + staff reading = 287.28 + 3.750 = 291.03 m Difference in level of the instrument axes at the two stations S =291.03 – 290.15 = 0.88 m. α = 28° 42 and α = 18° 6′ s cot α- = 0.88 cot 18° 6′ = 2.69 m = 152.1 m. h = D tan α = 152.1 tan 28° 42′ = 83.272 m R.L. of foot of signal = R.L. of inst. aixs at P + h - ht. of signal = 290.15 + 83.272 – 3 = 370.422 m. Check : (b + D) = 100 + 152.1 m = 252.1 m h = ( b + D) t an α = 252.1 x tan 18° 6′ = 82.399 m R.L. of foot of signal = R.L. of inst. axis at R + h+ ht. of signal = 291.03 + 82.399 – 3 = 370.429 m. SCE 4 Department of Civil EngineeringCE6404 SURVEYING II 1.3 CLASSIFICATION OF TRIANGULATION SYSTEM: The basis of the classification of triangulation figures is the accuracy with which the length and azimuth of a line of the triangulation are determined. Triangulation systems of different accuracies depend on the extent and the purpose of the survey. The accepted grades of triangulation are: 1. First order or Primary Triangulation 2. Second order or Secondary Triangulation 3. Third order or Tertiary Triangulation 1.3.1 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 ( us ually the whole of the country) . 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: 1. Average triangle closure : Less than 1 second 2. Maximum triangle closure : Not more than 3 seconds 3. Length of base line : 5 to 15 kilometers 4. Length of the sides of triangles : 30 to 150 kilometers 5. Actual error of base : 1 in 300,000 6. Probable error of base : 1 in 1,000,000 7. Discrepancy between two measures of a section : 10 mm kilometers 8. Probable error or computed distance : 1 in 60,000 to 1 in 250,000 9. Probable error in astronomic azimuth : 0.5 seconds 1.3.2 SECONDARY 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 SCE 5 Department of Civil EngineeringCE6404 SURVEYING II 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: 1. Average triangle closure : 3 sec 2. Maximum triangle closure : 8 sec 3. Length of base line : 1.5 to 5 km 4. Length of sides of triangles : 8 to 65 km 5. Actual error of base : 1 in 150,000 6. Probable error of base : 1 in 500,000 7. Discrepancy between two measures of a section : 20 mm kilometers 8. Probable error or computed distance : 1 in 20,000 to 1 in 50,000 9. Probable error in astronomic azimuth : 2.0 sec SCE 6 Department of Civil EngineeringCE6404 SURVEYING II 1.3.3 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 specifications for a third-order triangulation are as follows: 1. Average triangle closure : 6 sec 2. Maximum triangle closure : 12 sec 3. Length of base line : 0.5 to 3 km 4. Length of sides of triangles : 1.5 to 10 km 5. Actual error of base : 1 in 75, 0000 6. Probable error of base : 1 in 250,000 7. Discrepancy between two Measures of a section : 25 mm kilometers 8. Probable error or computed distance : 1 in 5,000 to 1 in 20,000 9. Probable error in astronomic Azimuth: 5 sec. Explain the factors to be considered while selecting base line. The measurement of base line forms the most important part of the triangulation operations. The base line is laid down with great accuracy of measurement and alignment as it forms the basis for the computations of triangulation system. The length of the base line depends upon the grades of the triangulation. Apart from main base line, several other check bases are also measured at some suitable intervals. In India, ten bases were used, the lengths of the nine bases vary from 6.4 to 7.8 miles and that of the tenth base is 1.7 miles. Selection of Site for Base Line. Since the accuracy in the measurement of the base line depends upon the site conditions, the following points should be taken into consideration while selecting the site: 1. The site should be fairly level. If, however, the ground is sloping, the slope should be uniform and gentle. Undulating ground should, if possible be avoided. 2. The site should be free from obstructions throughout the whole of the length. The line clearing should be cheap in both labour and compensation. SCE 7 Department of Civil EngineeringCE6404 SURVEYING II 3. The extremities of the base should be intervisible at ground level. 4. The ground should be reasonably firm and smooth. Water gaps should be few, and if possible not wider than the length of the long wire or tape. 5. The site should suit extension to primary triangulation. This is an important factor since the error in extension is likely to exceed the error in measurement. In a flat and open country, there is ample choice in the selection of the site and the base may be so selected that it suits the triangulation stations. In rough country, however, the choice is limited and it may sometimes be necessary to select some of the triangulation stations that at suitable for the base line site. Standards of Length. The ultimate standard to which all modern national standards are referred is the international meter established by the Bureau International der Poids at Measures and kept at the Pavilion de Breteuil, Sevres, with copies allotted to various national surveys. The meter is marked on three platinum- iridium bars kept under standard conditions. One great disadvantage of the standard of length that are made of metal are that they are subject to very small secular change in their dimensions. Accordingly, the meter has now been standardized in terms of wavelength of cadmium light. SCE 8 Department of Civil EngineeringCE6404 SURVEYING II CHAPTER 2 SURVEY ADJUSTMENTS 2.1 TYPES OF ERROR Errors of measurement are of three kinds: (i ) mistakes, (i i) systematic errors, and ( i ii) accidental errors. (i ) Mistakes. Mistakes are errors that arise from inattention, inexperience, carelessness and poor judgment or confusion in the mind of the observer. If mistake is undetected, it produces a serious effect on the final result. Hence every value to be recorded in the field must be checked by some independent field observation. (i i) Systematic Error. A systematic error is an error that under the same conditions will always be of the same size and sign. A systematic error always follows some definite mathematical or physical law, and a correction can be determined and applied. Such errors are of constant character and are regarded as positive or negative according as they make the result too great or too small. Their effect is therefore, cumulative. If undetected, systematic errors are very serious. Therefore: (1 ) All the surveying equipments must be designed and used so that whenever possible systematic errors will be automatically eliminated and ( 2 ) all systematic errors that cannot be surely eliminated by this means must be evaluated and their relationship to the conditions that cause them must be determined. For example, in ordinary levelling, the levelling instrument must first be adjusted so that the line of sight is as nearly horizontal as possible when bubble is centered. Also the horizontal lengths for back sight and foresight from each instrument position should be kept as nearly equal as possible. In precise levelling, every day, the actual error of the instrument must be determined by careful peg test, the length of each sight is measured by stadia and a correction to the result is applied. ( i ii) Accidental Error. Accidental errors are those which remain after mistakes and systematic errors have been eliminated and are caused by a combination of reasons beyond the ability of the observer to control. They tend sometimes in one direction and some times in the other, i.e., they are equally likely to make the apparent result too large or too small. An accidental error of a single determination is the difference between ( 1 ) the true value of the quantity and ( 2 ) a determination that is free from mistakes and systematic errors. Accidental error represents limit of precision in the determination of a value. They obey the laws of chance and therefore, must be handled according to the mathematical laws of probability. The theory of errors that is discussed in this chapter deals only with the accidental errors after all the known errors are eliminated and accounted for. SCE 9 Department of Civil EngineeringCE6404 SURVEYING II 2.2 THE LAW OF ACCIDENTAL ERRORS 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: (i ) Small errors tend to be more frequent than the large ones; that is they are the most probable. (i i) Positive and negative errors of the same size happen with equal frequency ; that is, they are equally probable. (i ii) Large errors occur infrequently and are impossible. 2.3 PRINCIPLES OF LEAST SQUARES It is found from the probability equation that the most probable values of a series of errors arising from observations of equal weight are those for which the sum of the squares is a minimum. The fundamental law of least squares is derived from this. 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 2.4 LAW OF WEIGHTS From the method of least squares the following laws of weights are established: ( i ) 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: ÐA Weight ÐA 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 \ Arithmetic mean = 30° 20′ + 1/6 (8 ” + 10” + 7” + 10” + 9” + 10”) SCE 10 Department of Civil EngineeringCE6404 SURVEYING II = 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 : ÐA Weight ÐA 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 SCE 11 Department of Civil EngineeringCE6404 SURVEYING II Sum of weights = 2 + 3 + 2 + 3 + 4 + 2 =16 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 individual weights. For Example angle A = 30° 20′ 8”, Weight 2 B = 15° 20′ 8”, Weight 3 Weight of A + B = (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. ( 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. ( 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. (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. 2.5 DISTRIBUTION OF ERROR OF THE FIELD MEASUREMENT. Whenever observations are made in the field, it is always necessary to check for the closing error, if any. The closing error should be distributed to the observed quantities. For examples, the sum of the angles measured at a central angle should be 360°, the error should be distributed to the observed angles after giving proper weight age to the observations. The following rules should be applied for the distribution of errors: (1 ) The correction to be applied to an observation is inversely proportional to the weight of the observation. SCE 12 Department of Civil EngineeringCE6404 SURVEYING II ( 2 ) The correction to be applied to an observation is directly proportional to the square of the probable error. ( 3 ) In case of line of levels, the correction to be applied is proportional to the length. The following are the three angles α, β and y observed at a station P closing the horizon, along with their probable errors of measurement. Determine their corrected values. Solution. α = 78° 12′ 12” ± 2” β = 136° 48′ 30” ± 4” y = 144° 59′ 08” ± 5” Sum of the three angles = 359° 59′ 50” Discrepancy = 10” Hence each angle is to be increased, and the error of 10” is to be distributed in proportion to the square of the probable error. Let c1, c2 and c3 be the correction to be applied to the angles α, β and y respectively. c1 : c2 : c3 = (2 )² : ( 4 )² : ( 5 )² = 4 : 16 : 25 … (1 ) Also, c1 + c2 + c3 = 10” … ( 2 ) From ( 1 ), c2 = 16 /4 c1 = 4c1 And c3 = 25/4 c1 Substituting these values of c2 and c3 in ( 2 ), we get c1 + 4c1 + 25/4 c1 = 10” or c1 ( 1 + 4 + 25/4 ) = 10” \ c1 = 10 x 4/45 = 0”.89 \ c2 = 4c1 = 3”.36 And c3 = 25 /4 c1 = 5”.55 SCE 13 Department of Civil EngineeringCE6404 SURVEYING II Check: c1 + c2 + c3 = 0.”89 + 3”.56 + 5”.55 = 10” Hence the corrected angles are SCE 14 Department of Civil EngineeringCE6404 SURVEYING II α = 78° 12′ 12” + 0”.89 = 78° 12′ 12”.89 β = 136° 48′ 30” + 3”.56 = 136° 48′ 33”.56 and y = 144° 59′ 08” + 5”.55 = 144° 59′ 13”.55 - Sum = 360° 00′ 00”+ 00 An angle A was measured by different persons and the following are the values : Angle Number of measurements 65° 30′ 10” … 2 65° 29′ 50” … 3 65° 30′ 00” … 3 65° 30′ 20” … 4 65° 30′ 10” … 3 Find the most probable value of the angle. Solution. As stated earlier, the most probable value of an angle is equal to its weighted arithmetic mean. 65° 30′ 10” x 2 = 131° 00′ 20” 65° 29′ 50” x 3 = 196° 29′ 30” 65° 30′ 00” x 3 = 196° 30′ 00” 65° 30′ 20” x 4 = 262° 01′ 20” 65° 30′ 10” x 3 = 196° 30′ 30” Sum = 982° 31′ 40” Σ weight = 2 + 3 + 3 + 4 + 3 = 15 SCE 15 Department of Civil EngineeringCE6404 SURVEYING II \ Weighted arithmetic mean = 982° 31′ 40” - = 65° 30′ 6”.67 Hence most probable value of the angle = 65° 30′ 6”.67 The telescope of a theodilite is fitted with stadia wires. It is required to find the most probable values of the constants C and K of tacheometer. The staff was kept vertical at three points in the field and with of sight horizontal the staff intercepts observed was as follows. Distance of staff Staff intercept S( m ) from tacheometer D( m) 150 1.495 200 2.000 250 2.505 Solution: The distance equation is D = KS + C The observation equations are 150 = 1.495 K + C 200 = 2.000 K + C 250 = 2.505 K + C If K and C are the most probable values, then the error of observations are: 150 - 1.495 K - C 200 - 2.000 K - C 250 - 2.505 K – C SCE 16 Department of Civil Engineering

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