Surveying Theory and Practice PDF

Surveying Theory and Practice I. Concepts on Surveying II. Basic Survey Measurements III. Survey Operations IV. Mapping V. Types of Surveys I. Concepts on Surveying Surveying Uses of Surveying Types of Surveying Survey Measurements and...

Surveying Theory and Practice I. Concepts on Surveying II. Basic Survey Measurements III. Survey Operations IV. Mapping V. Types of Surveys I. Concepts on Surveying Surveying Uses of Surveying Types of Surveying Survey Measurements and Adjustments II. Basic Survey Measurements Distance Measurement Vertical Distance Measurement Angle and Direction Measurement III. Survey Operations Traverse Intersection and Resection Triangulation and Trilateration Astronomic Observation Modern Positioning Systems IV. Mapping Mapping and Map Drafting Map Projections V. Types of Surveys Land Surveys Mining Surveys Hydrographic Surveys Route and Construction Surveys  Survey Operations Traverse Traverse – is a series of lines connecting successive points, whose lengths and directions are determined from field observations. Traversing – is the process of measuring the lengths and directions of the lines of a traverse for the purpose of locating the position of certain points Traverse Station – any temporary or permanent point of reference over which the instrument is set-up Traverse Lines – lines connecting traverse stations whose length and direction is determined Types of Traverse 1. Circuit Traverse – a traverse that closes to the point of origin 2. Loop Traverse – a traverse starting from a station and closing into another station of the same or another traverse 3. Connection Traverse – a traverse which does not close into another station of the same or any other traverse Methods of Running a Traverse 1. Interior Angle Traverse – used principally in land surveying Ex. The interior angles of a 5 sided closed traverse were measured as follows: A = 55-50 C = 68-10 E=? B = 104-20 D = 196-12 If all angles are assumed to be correct, determine the angle E and find the bearing of each line if AB is N 20-15 E. 2. Deflection Angle Traverse – is used frequently for the location survey of roads, railroads, pipelines, etc. and other similar types of survey Ex. Following are the observed deflection angles of a closed traverse: A = 28-25-00 (L) E = 108-13-30 (L) B = 68-03-30 (L) F = 16-50-00 ( R ) C = 120-34-00 (L) G = 110-00-30 (L) D = 58-30-00 (R) Compute the error of closure and adjust the angular values by assuming that the error is the same for each angle. Note: For any closed traverse that the sides do not cross one another, the difference of deflection angles should be equal to 360 degrees. If the lines cross once or any odd number of time, the sum of the Left and Right deflection angles should be equal. 3. Traverse by Angle to the Right – is employed for city, tunnel, and mine surveys, and in locating details for a topographic map. Ex. A five sided closed traverse proceeds in a clockwise direction and the angle to the right of each station were observed as follows: A = 240-40 D= 220-04 B = 238-15 E – 271-13 C = 289-53 n= 5 Determine the error of closure and adjust the observed values on the assumption that the error is the same for each angle. Note: If clockwise = (n + 2) 180deg If counterclockwise = (n-2) 180 deg 4. Azimuth Traverse Ex. Given the tabulation of data of a closed traverse, determine the bearing and the angle to the right of each station,. Sta Occ. Sta Obs. Distance Azimuth from South A E 210.10 90-28 B 170-30 B A 155.34 350-30 C 123-05 C B 206.85 303-05 D 56-13 D C 174.50 236-13 E 357-58 E D 330.00 177-58 A 270-28 Traverse Computations For a closed traverse, some computations include: 1. Determining latitudes and departures 2. Calculating the total error of closure 3. Balancing the survey 4. Determining the new coordinates 5. Computing the Area 6. Subdivision of tracts of land Latitude – the projection of a line in the reference meridian or north-south line - is either North (+) or South (-) Departure - the projection in the reference parallel or east-wast line - is either E (+) or West (-)  N Lat (-) = d Cos θ Dep (-) = d Sin θ W E d θ S Error of Closure - the algebraic sum of the north and south latitudes should be zero and the algebraic sum of east and west departure should zero, else there exist an error. Linear error of closure - a short line of unknown length and direction connecting the initial and final stations of the traverse LEC = wCL2 + CD2 and Tan θ = - CD /- CL where: LEC = linear error of closure CL = Closure in latitude or the algebraic sum of the north and south latitudes CD = closure in departure or the algebraic sum of the east and west departures θ = bearing angle of the side error Relative precision - is defined by the ratio of the linear error of closure to the perimeter or total length of the traverse RP = LEC / D where: LEC = linear error of closure D = total length of the perimeter RP = Relative precision Ex. 1 In a given closed traverse the sum of the north latitudes exceeds the sum of the south latitudes by 2.74 m and the sum of the west departures exceeds the sum of the east departures by 3.66 m. Determine the linear error of closure and the bearing of the side of error. Ex. 2 Given the observations data of a closed traverse, determine: a) latitude and departure b) LEC c) bearing of the side error d) Relative precision Line Length Azimuth Latitude Departure from N N (+) S(-) E(+) W(-) AB 233.10 122-30 BC 242.05 85-15 CD 191.50 20-00 DE 234.46 333-35 EF 270.65 254-08 FA 252.38 213-00 Sum Traverse Adjustment (Balancing a traverse)– the procedure of computing the linear error of closure and applying corrections to the individual latitudes and departures for the purpose of providing a mathematically closed figure. Methods of Traverse Adjustment 1. Arbitrary Method – adjustment by the discretion of the surveyor’s assessment of the conditions surrounding the survey. 2. Compass Rule or Bowditch Rule – is based on the assumption that all lengths were measured with equal care and all angles taken with approximately the same precision. It is also assumed that the errors in the measurement are accidental and that the total error in any side of the traverse is directly proportional to the total length of the traverse.  cl = CL(d/D) and cd = CD(d/D) where: cl = corr. to be applied to the lat. of any course cd = corr. to be applied to any dep. of any course CL = Closure in latitude or the algebraic sum of the north and south latitudes CD = closure in departure or the algebraic sum of the east and west departures d = length of any course D = total length or the perimeter of the course 3. Transit Rule – is based on the assumption that angular measurements are more precise than the linear measurements and that the errors in traversing are accidental. cl = Lat(CL)/(NL - SL) and cd = Dep(CD)/(ED - WD) where: cl = corr. to be applied to the lat. of any course cd = corr. to be applied to any dep. of any course CL = Closure in latitude or the algebraic sum of the north and south latitudes CD = closure in departure or the algebraic sum of the east and west departures NL = summation of north latitudes SL = summation of south latitudes ED = summation of east departures WD = summation of west departures 4. Least Squares Method – is based on the theory of probability where it is employed to simultaneously adjust the angular and linear measurements to make the sum of the squares of the residual a minimum. 5. Crandall Method – is an application of the theory of least squares and is suitable for use if the linear measurements made are less precise than the angular measurements. 6. Graphical Method – is basically an application of the compass rule by graphically moving each traverse point parallel to the error of closure by an amount proportional to the distance along the traverse from the initial point to the given point. 7. Coordinate Method – is also an application of compass rule since the corrections applied are still proportionate to the lengths being adjusted. Note: In the board exam, you only need to know 1 or 2 methods of adjustment and the easiest, in my experience are the transit rule and compass rule. Adjusted Lengths and Directions Because there was an adjustment in the latitudes and departures, it follows that the lengths and directions of each corrected course are changed. To get the adjusted length and directions: L = eLatcorr2 + Depcorr2 and Tan θ = Depcorr /Latcorr Ex. 1 Adjust the tabulated data using the compass rule and transit rule. Also determine the linear error of closure, bearing of the side of error, and the relative error of closure. Course Distance Bearing Latitude Departure N (+) S(-) E(+) W(-) AB 495.85 N 05 O 30’ E BC 850.62 N 46 O 02’ E CD 855.45 S 67 O 38’ E DE 1020.87 S 12 O 25’ E EF 1117.26 S 83 O 44’ W FA 660.08 S 55 O 09’ W Sum Ex. 2 Given in the accompanying tabulation are the known and computed coordinates of stations along a traverse. The traverse originates on station Baguio whose known coordinates are X = 6,208.67 and Y = 8,601.44, and closes on station Acupan whose known coordinates are X = 5,226.10 and Y=5,782.62. Adjust the coordinates of the traverse stations and tabulate values accordingly. Sta Computed Coordinates X Y Baguio 6,208.67 8,601.44 A 7,030.45 8,299.54 B 6,984.53 7,698.69 C 7,001.14 7,260.00 D 7,112.99 6,774.08 E 6,586.70 5,941.82 F 6,147.28 6,058.24 G 5,467.06 6,066.64 Acupan 5,226.18 5,782.98 Angular Error of Closure Allowable Angular Error of Closure: Primary Traverse- 2.5”SP (Least Reading of 1”) Secondary Traverse– 10”SS (Least Reading of 15”) Tertiary Traverse– 30”ST (Least Reading of 1’) Ex. 1 In a secondary traverse of 60 stations has an azimuth error of 30”. How many groups should the traverse be divided in order to distribute the error? Ex. 2 Find the correct azimuth of line T3-T4 Sta. Occ. Sta Obs Azimuth T1 28-00 T4 T4 313-55 T3 T3 215-00 T2 T2 143-20 T4 208-01 T1 Ex. 3 A 180 primary station traverse has an angular error of closure of +4 sec. What is the correct azimuth of P109-P110 if the azimuth is 121-32-12? Methods of Determining Area 1. Area by Triangles (MATH) 1. Known Base and Altitude A = (1/2) bh 2. Two Sides and Included Angle Measured A = ( ½ ) a b sin θ 3. Three Sides Measured A = s(s-a)(s-b)(s-c) where: s = (1/2)(a + b + c) 2. Area by Coordinates (MATH) A = (1/2) X1 X2 X3 …… XN X1 Y1 Y2 Y3 ….… YN Y1 Ex. 1 A Surveyor sets-up a transit at P which is located in the middle portion of a four sided tract of lnad and reads the direction and measures distances, as given below. Find the Area. Line Bearing Distance PA N 41-30 W 410.52 PB N 39-10 E 532.18 PC S 70-20 E 450.75 PD S 60-15 W 590.08 Ex. 2 Assuming that the origin of the coordinate system is at station 1, determine the area using coordinate method. Line Adjusted Latitude Adjusted Departure (+) N (-) S (+) N (-) S 1–2 490.71 47.27 2- 3 587.12 608.89 3–4 327.41 786.78 4–5 1002.76 218.32 5- 6 122.67 1116.62 6 –1 375.01 544.64 Sum 1452.84 - 1452.84 1661.26 - 1661.26 Tabulated Solution Sta Coordinates Double Areas Total Lat Total Dep (+) AREAS (-) AREAS 1 2 3 4 5 6 7 SUMS 3. Area by Double Meridian Distance (DMD) Method or Double Parallel Distance (DPD) Method Meridian Distance of a Line – is the shortest distance from the midpoint of the line to the reference meridian. Double Meridian Distance (DMD) of a Line - meridian distance of a line times 2 Parallel Distance of a Line – the distance from the midpoint of the line to the reference parallel or east-west line. Double Parallel Distance of a line – parallel distance of a line times 2 Rules: 1. The DMD(DPD) of the first course is equal to the departure(latitude) of the course. 2. The DMD(DPD) of any other course is equal to the DMD(DPD) of the preceding course, plus departure(latitude) of the preceding course, plus the departure(latitude) of the course itself. 3. The DMD(DPD) of the last course is numerically equal to the departure(latitude) of the initial course, but with an opposite sign. Double Area = DMD(Latitude) or DPD(Departure) A = (1/2) (NDA + SDA) or (1/2) (EDA + WDA) Note: Make sure that the enclosed traverse data is already balance before computing for areas. Ex. Find the Area by DMD and DPD method. Line Adjusted Latitude Adjusted Departure (+) N (-) S (+) N (-) S 1–2 490.71 47.27 2- 3 587.12 608.89 3–4 327.41 786.78 4–5 1002.76 218.32 5- 6 122.67 1116.62 6 –1 375.01 544.64 Sum 1452.84 - 1452.84 1661.26 - 1661.26 Tabulated Solution for DMD Sta DMD Double Areas (+) NDA (-) SDA 1–2 2-3 3-4 4-5 5-6 6-1 SUMS Tabulated Solution for DPD Sta DPD Double Areas (+) EDA (-) WDA 1–2 2-3 3-4 4-5 5-6 6-1 SUMS 4. Area by offsets from a Straight Line 1. Trapezoidal Rule – the assumption in using the trapezoidal rule is that the ends of the offsets in the boundary line are assumed to be connected by straight lines, thereby forming a series of trapezoids. fig. A = d (h1 + hn + h2 + h3 + h4 … +hn-1) 2 where: A = Summation of areas of the trapezoids d= common spacing between offsets  n= number of offsets h1= first offset hn = last offset h2 ,h3, etc = intermediate offset hn-1 = last intermediate offset Ex. A series of perpendicular offsets were taken 2.5 meters apart and were measured in the following order; 0.0, 2.6, 4.2, 4.4, 3.8, 2.5,4.5,5.2,1.6 and 5.0 meters. Obtain the area included between the transit line, curve boundary, and the end offsets using the trapeziodal rule. 2. Simpson’s One-Third Rule – is based on the assumption that the curved boundary consists of a series of parabolic arcs, where each arc is continuos over three adjacent offsets that are equally spaced. Note: This rule can only be used when there is an odd number of offsets and if they are equally spaced. If it is even, the area can be obtained up to the 2nd to last offset, in which trapezoidal rule must be used. fig. A = (d/3) [(h1 + hn) + 2(Hodd) + 4(Heven)] where: A = area of the tract d= common spacing between offsets n= number of offsets h1= first offset hn = last offset Hodd = summation of odd offsets Heven = summation of even offsets Ex. Using Simpson’s One-third rule, compute the area of defined by the edge of the river, transit line and a perpendicular offsets of 0.5, 1.4, 2.5, 5.6, 8.5, 7.4, 3.8, 5.1, and 2.3 meters separated equally by 4 meters. 5. By Planimeteric computations – areas are being measured from a amp drawn to scale by an instrument called planimeter. Practice Problem Determine the Area by DMD, DPD and Coordinate Method. Also get the LEC, bearing of the side error and relative precision. Adjust by using transit and compass rule. Sta. Occ. Sta Obs Azimuth Distance Bearings T1 28-00 88.12 T4 T4 313-55 15.65 T3 T3 215-00 81.35 T2 T2 143-20 25.64 T4 208-01 T1 Omitted Measurements – where sometimes its is not practical or possible to determine by field observation the length and direction of lines in a closed traverse, missing quantities can be determined by analytical method as long as they do not exceed two in number. Limitations: provides no check, throws all possible errors in the computed length and direction, mistakes can not be detected Advantages: useful for partition of land, able to solve problems like rugged terrain, lack of time, unfriendly landowners, etc. Common Types of Omitted Measurements 1. Omitted Measurements are in One Side 1st Case: Length and Bearing of One Side Unknown  Ex. Given the following data in the closed traverse, calculate the bearing of line 2-3. Line Bearing Distance Latitude Departure 1-2 S 35-49 W 44.37 -35.98 -25.97 2-3 3-4 N 11-39 E 12.82 12.56 2.59 4-5 N 73-57 E 63.83 17.65 61.34 5-1 S 49-18 E 105.80 -68.99 80.21 2. Omitted Measurements Involving Adjoining Sides 1st Case: Length of One Side and Bearing of Another side Unknown Ex. Find the missing quantities Line Length Bearing Latitudes Departures AB 1,084.32 S 75-48 E -265.99 1,051.19 BC 1,590.51 S 15-18 W -1,534.14 -419.69 CD 1,294.74 S 68-06 W -482.92 -1,201.31 DE N 28-39 W EA 1,738.96 2nd Case: Bearing of Two Sides Unknown Ex. From the technical description of a closed traverse, det. the bearing of line AB and line BC Line Distance Bearing Latitude Departure AB 64.86 BC 107.72 CD 44.37 S 35-30 W DE 137.84 N 57-15 W EA 12.83 N 1-45 E  3rd Case: Lengths of Two Sides Unknown Ex. Determine the unknown Quantities of a closed traverse. Line Distance Bearing Latitudes Departure AB 639.32 N 09-30 W 630.55 -105.52 BC N 56-55 W CD S 56-13 W DE 570.53 S 02-02 E -570.17 20.24 EA 1,082.71 S 89-31 E -9.13 1,082.67 3. Omitted Measurements Involving Non-Adjoining Sides 1st Case: Length of One Side and Bearing of Another Side Unknown 2nd Case: Bearing of Two Sides Unknown 3rd Case: Lengths of Two Sides Unknown Subdivision of Land Procedures: 1. Resurvey 2. Determine Lat. and Dep. 3. Balance the Traverse 4. Compute the Area 5. Using the Adjusted Lat. and Dep., Subdivide the Area Methods of Subdividing Parcel of lands 1. Dividing an Area into Two Parts by a Line between Two Points Ex. Give the following data, determine the areas of the two lots produced if a line from F to C was used to divide it. Also determine the length and bearing of FC. 2. Dividing A tract of Land by a Line Running in a Given Direction Ex. Given the following data of land, find the area of each of the two parts into which the tract of land is divided by a line through A with a bearing of N 75-30 E. Course ADJ Lat ADJ Dep DMD Double Area AB 490.71 47.27 BC 587.12 608.89 CD -327.41 786.78 DE -1002.76 218.32 EF -122.67 -1,116.62 FA 375.01 -544.64 Sum 0.00 0.00 C Fig. D B A E F 3. To Cut Off a Required Area by a Line Through a Given Point. Ex Subdivide into two equal parts, with the dividing line passing through corner B. 4. To Cut Off a Required Area by a Line in a Given Direction Ex. Divide the area into two equal parts using a east-west line, compute the length of the dividing line, and determine the distances from the line to the adjacent traverse station. Problems 50 25-00-00 1. 20 100 Find the length of dividing line if the lot is divided into two equal parts. Note: L = S(mb12 + nb22) / (n + m) 100 2. 50 Find the area appropriated for the road with a right of way of 10m.