TOPIC 5 TRAVERSING MS SITI KAMARIAH MD SA’AT LECTURER SCHOOL OF BIOPROCESS ENGINEERING sitikamariah@unimap.edu.my

Stationing Stations are dimensions measured along a baseline. The beginning point is described as 0+00. A point 100 ft(m) from the beginning is 1+00. A point 565.98 ft(m) from the beginning is 5+65.98. Points measured before the beginning station are 0-50, -1+00, etc.

Baseline Stations and Offset Distances

Overview In this lecture we will cover : Rectangular and polar coordinates Definition of a traverse Applications of traversing Equipment and field procedures Reduction and adjustment of data

Rectangular coordinates Point A Point B North East EB NB (EB,NB) N=NB-NA EA NA (EA,NA) E=EB-EA

Polar coordinates North Point B d  Point A  ~ whole-circle bearing East Point A Point B d   ~ whole-circle bearing d ~ distance

Whole circle bearings North Bearing are measured 0o clockwise from NORTH and must lie in the range 0o    360o 4th quadrant 1st quadrant West 270o East 90o 3rd quadrant 2nd quadrant South 180o

Coordinate conversions Rectangular to polar Polar to rectangular E N d  d  E N

What is a traverse? Control survey A series of established stations tied together by angle and distance. The angles are measured using theodolites/total station, while distances can be measured using total stations, steel tapes or EDM.

What is a traverse? A polygon of 2D (or 3D) vectors Sides are expressed as either polar coordinates (,d) or as rectangular coordinate differences (E,N) A traverse must either close on itself Or be measured between points with known rectangular coordinates A closed traverse A traverse between known points

Types of Traverses Open Traverse using deflection angles. Closed traverse using interior angles.

Open Traverse

Closed Traverse

Applications of traversing Establishing coordinates for new points (E,N)known (,d) (,d) (,d) (E,N)new (E,N)new

Applications of traversing These new points can then be used as a framework for mapping existing features (E,N)new (E,N)known (E,N)new (E,N)new (E,N)new (,d) (,d)

Applications of traversing They can also be used as a basis for setting out new work (E,N)new (E,N)known

Equipment Traversing requires : An instrument to measure angles (theodolite) or bearings (magnetic compass) An instrument to measure distances (EDM or tape)

Computation of Latitudes and Departures Latitude-north/south rectangular component of line (North +;South -) Latitude (ΔY) = distance(H) cos α Departure-east/west rectangular component of line (East +;West -) Departure (ΔX) = distance(H) sin α Where: α = bearing or azimuth of the traverse course H = the horizontal distance of the traverse course

Location of a Point

Closure of Latitudes and Departures

Latitude / Departure Computations

Measurement sequence C B D A E 232o 168o 60.63 99.92 56o 352o 205o 77.19 129.76 21o A 118o 32.20 303o 48o E

Computation sequence Calculate angular (bearing/azimuth) misclose Adjust angular (bearing/azimuth) misclose Calculate adjusted bearings Reduce distances for slope etc… Compute (E, N) for each traverse line Calculate linear misclose Calculate accuracy Adjust linear misclose.

Calculate internal angles Point Foresight Bearing Backsight Internal Angle Adjusted A 21o 118o 97o B 56o 205o 149o C 168o 232o 64o D 352o 120o E 303o 48o 105o  =(n-2)*180 Misclose Adjustment At each point : Measure foresight bearing Measure backsight bearing Calculate internal angle (back-fore) For example, at B : Bearing to C = 56o Bearing to A = 205o Angle at B = 205o - 56o = 149o

Calculate angular misclose Point Foresight Bearing Backsight Internal Angle Adjusted A 21o 118o 97o B 56o 205o 149o C 168o 232o 64o D 352o 120o E 303o 48o 105o  =(n-2)*180 535o Misclose -5o Adjustment -1o

Calculate adjusted angles Point Foresight Bearing Backsight Internal Angle Adjusted A 21o 118o 97o 98o B 56o 205o 149o 150o C 168o 232o 64o 65o D 352o 120o 121o E 303o 48o 105o 106o  =(n-2)*180 535o 540o Misclose -5o Adjustment -1o

Compute adjusted bearings Adopt a starting bearing Then, working clockwise around the traverse : Calculate reverse bearing to backsight (forward bearing 180o) Subtract (clockwise) internal adjusted angle Gives bearing of foresight For example (bearing of line BC) Adopt bearing of AB 23o Reverse bearing BA (=23o+180o) 203o Internal adjusted angle at B 150o Forward bearing BC (=203o-150o) 53o

Compute adjusted bearings Line Forward Bearing Reverse Bearing Internal Angle AB 23o 203o 150o BC 53o CD DE EA 53o B 150o D 203o A E

Compute adjusted bearings Line Forward Bearing Reverse Bearing Internal Angle AB 23o 203o 150o BC 53o 233o 65o CD 168o DE EA 233o 65o 168o B D 23o A E

Compute adjusted bearings Line Forward Bearing Reverse Bearing Internal Angle AB 23o 203o 150o BC 53o 233o 65o CD 168o 348o 121o DE 227o EA 53o 348o B 121o D 23o 227o A E

Compute adjusted bearings Line Forward Bearing Reverse Bearing Internal Angle AB 23o 203o 150o BC 53o 233o 65o CD 168o 348o 121o DE 227o 47o 106o EA -59o 301o 53o 168o B D 23o 47o A 106o 301o E

Compute adjusted bearings Line Forward Bearing Reverse Bearing Internal Angle AB 23o 203o 150o BC 53o 233o 65o CD 168o 348o 121o DE 227o 47o 106o EA 301o 98o 23o (check) 53o 168o B D 23o 227o 98o A 121o E

(E,N) for each line The rectangular components for each line are computed from the polar coordinates (,d) Note that these formulae apply regardless of the quadrant so long as whole circle bearings are used

Vector components Line Bearing Distance E N AB 23o 77.19 30.16 71.05 BC 53o 99.92 79.80 60.13 CD 168o 60.63 12.61 -59.31 DE 227o 129.76 -94.90 -88.50 EA 301o 32.20 -27.60 16.58  (399.70) (0.07) (-0.05)

Closure Error and Closure Correction

Compass Rule – distributes the errors in lat/dep. C lat AB= AB Σ lat P C dep AB = AB Σ dep P Where: C lat AB = correction in latitude AB ∑ lat = error of closure in latitude AB = distance AB P = perimeter of traverse Where: C dep AB = correction in departure AB ∑ lat = error of closure in departure AB = distance AB P = perimeter of traverse

Linear misclose & accuracy Convert the rectangular misclose components to polar coordinates Accuracy is given by Beware of quadrant when calculating  using tan-1

Quadrants and tan function E +  negative add 360o + positive okay  positive add 180o  + negative add 180o

For the example… Misclose (E, N) Convert to polar (,d) Accuracy (0.07, -0.05) Convert to polar (,d)  = -54.46o (2nd quadrant) = 125.53o d = 0.09 m Accuracy 1:(399.70 / 0.09) = 1:4441

Bowditch adjustment The adjustment to the easting component of any traverse side is given by : Eadj = Emisc * side length/total perimeter The adjustment to the northing component of any traverse side is given by : Nadj = Nmisc * side length/total perimeter

The example… East misclose 0.07 m North misclose –0.05 m Side AB 77.19 m Side BC 99.92 m Side CD 60.63 m Side DE 129.76 m Side EA 32.20 m Total perimeter 399.70 m

Vector components (pre-adjustment) Side E N dE dN Eadj Nadj 1A 30.16 71.05 AB 79.80 60.13 BC 12.61 -59.31 CD -94.90 -88.50 D1 -27.60 16.58 Misc (0.07) (-0.05)

The adjustment components Side E N dE dN Eadj Nadj 1A 30.16 71.05 0.014 -0.010 AB 79.80 60.13 0.016 -0.012 BC 12.61 -59.31 0.011 -0.008 CD -94.90 -88.50 0.023 -0.016 D1 -27.60 16.58 0.006 -0.004 Misc (0.07) (-0.05) (0.070) (-0.050)

Adjusted vector components Side E N dE dN Eadj Nadj 1A 30.16 71.05 0.014 -0.010 30.146 71.060 AB 79.80 60.13 0.016 -0.012 79.784 60.142 BC 12.61 -59.31 0.011 -0.008 12.599 -59.302 CD -94.90 -88.50 0.023 -0.016 -94.923 -88.484 D1 -27.60 16.58 0.006 -0.004 -27.606 16.584 Misc (0.07) (-0.05) 0.070 -0.050 (0.000)

Summary of initial traverse computation Balance the angle Compute the bearing or azimuth Compute the latitude and departure, the linear error of closure, and the precision ratio of the traverse

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