Chapter (4) Traverse Computations Introduction The survey procedure known as traversing is fundamental to much survey measurement. The procedure consists of using a variety of instrument combinations to create polar vector in space, that is 'lines' with a magnitude (distance) and direction (bearing). These vectors are generally contiguous and create a polygon which conforms to various mathematical and geometrical rules (which can be used to check the fieldwork and computations).

The equipment used generally consists of something to determine direction like a compass or theodolite, and something to determine distance like a tape or Electromagnetic Distance Meter (EDM). There are orderly field methods and standardized booking procedures to minimize the likelihood of mistakes, and routine methods of data reduction again to reduce the possible occurrence of errors. The most fundamental of these checks is to perform a closed traverse that is a traverse that starts and finishes on either the same point or known points, (similar in concept to a level run).

The Function of Traverses Traverses are normally performed around a parcel of land so that features on the surface or the boundary dimensions can be determined. Often the traverse stations will be revisited so that perhaps three-dimensional topographic data can be obtained, so that construction data can be established on the ground. A traverse provides a simple network of 'known' points that can be used to derive other information.

Types of traverse There are two types of traverse used in survey. These are open traverse, and closed traverse.

Open Traverse An open traverse begins at a point of known control and ends at a station whose relative position is known only by computations. The open traverse is considered to be the least desirable type of traverse, because it provides no check on the accuracy of the starting control or the accuracy of the fieldwork. For this reason, traverse is never deliberately left open. Open traverse is used only open projects like roads, cannels, railway lines, shoreline protection.

Closed Traverse Closed loop traverse This traverse starts and ends at stations of known control. There are two types of closed traverse–closed on the starting point and closed on a second known point. Closed loop traverse This type of closed traverse begins at a point of known control, moves through the various required unknown points, and returns to the same point. This type of closed traverse is considered to be the second best and is used when both time for survey and limited survey control are considerations. It provides checks on fieldwork and computations and provides a basis for comparison to determine the accuracy of the work performed.

The first step in checking a closed traverse is the addition of all angles. Interior angles are added and compared to (n-2)*180o. Exterior angles are added and compared to (n+2)*180o. Deflection angle traverses are algebraically added and compared to 360o. The allowable misclosure depends on the instrument, the number of traverse stations, and the intention for the control survey. c =K * n 0.5

where c = allowable misclosure. K = fraction of the least count of the instrument, dependent on the number of repetitions and accuracy desired (typically 30" for third-order and 60" for fourth-order) n = number of angles. Exceeding this value, given the parameters, may indicate some other errors are present, of angular type, in addition to the random error. The angular error is distributed in a manner suited to the party chief before adjustment of latitude and departures. Adjustment of latitudes and departures is the accepted method. The relative point closure is obtained by dividing the error of closure (EC ) by the line lengths. Relative point closure = EC / S of the distances

Closed traverse between two known control points This type of closed traverse begins from a point of known control, moves through the various required unknown points, and then ends at a second point of known control. The point on which the survey is closed must be a point established to an equal or higher order of accuracy than that of the starting point. This is the preferred type of traverse. It provides checks on fieldwork, computations, and starting control. It also provides a basis for comparison to determine the accuracy of the work performed.

The procedure for adjusting this type of traverse begins with angular error just as in a loop traverse. To determine the angular error a formula is used to generalized the conversion of angles into azimuth. The formula takes out the reciprocal azimuth used in the back sight as (n-1) stations used the back-azimuth as a back sight in recording the angles. A1 + a1 + a2 + a3+ ...+ an -(n-1)(180o )=A2

If the misclosure is exceeded, the angular error may have been exceeded or the beginning and ending azimuths are in error or oriented in different meridian alignments. If beginning and ending azimuths were taken from two traverses, and the angle repetitions were found to be at least an order of magnitude better than the tabulated angular error, the ending azimuths may contain a constant error which may be removed to improve the allowable error. GPS or astronomic observations may be used to find the discrepancy if the benefit of this.

Traverse fieldwork The easiest way of visualizing the traversing process is to consider it to be the formation of a polygon on the ground using standard survey procedures. If the traverse is being measured using a theodolite (which is the normal case) then angles are observed to survey stations on both faces for a given number of rounds, and booked and reduced accordingly. The stations being observed are pre marked and targeted with range poles or traversing targets, or simply by a plumb-bob string for the duration of the angle measurement.

If bearings are being observed with a magnetic compass then care must be taken to reduce the effect of variation in declination over the period of the survey, and especially to avoid the effects of local attraction. This is done by avoiding nearby metallic objects, and by observing both forward and reverse bearings for each traverse line. Whatever method is used for the measurement of distance then all appropriate corrections should be made, and the distances reduced to horizontal.

Choice of points Station marking Observations Planning - establish requirements for accuracy, density and location of control points. Reconnaissance - nature of terrain, access, location of points. Station marking Station marking - type of mark, reference. Protection. Description Card. Observations Angular and Distance Measurements. Angular Measurement – Targets, Reading and booking procedure. Linear Distance - Standard, slope, temp. Booking procedure.

Traverse Computations 1 Angular Closure of Closed loop traverse Using a theodolite we can measure all the internal angles. The sum of the internal angles of a polygon (traverse) is given by the rule: Σ θ = (n -2)* 180O Where n is the number of sides of the traverse, and each internal angle. Any variation from this sum is known as the misclosure and must be accounted for, either through compensation (if it is an acceptable amount) or elimination by repetition of the observations. An angular closure is computed for traverses performed with either Theodolites or magnetic compasses.

The Angular Misclosure = Σ Measured Angles - Σ Internal Angles A larger misclosure could be expected when using a magnetic compass, but in any case it must be calculated and removed. The Angular Misclosure = Σ Measured Angles - Σ Internal Angles Maximum Angular Misclosure =2*Accuracy of Theodolite *√ (No. of Angles) Calculation of Whole Circle Bearing When the angles is adjusted, then a bearing is adopted for one of the lines (or a known bearing is used) and bearings for all the lines are computed. The bearing of a line is computed by adding 180° to the bearing of the line before, and then subtracting the included angle (α).

Example: Observations, using a 6" Theodolite, were taken in the field for an anti - clockwise polygon traverse, A, B, C, D. The bearing of line AB is to be assumed to be 0o and the co-ordinates of station A are (3000.00 m E ; 4000.00 m N). N C B A D

observed clockwise horizontal angle horizontal distance ( m ) line observed clockwise horizontal angle traverse station 638.57 AB 132º 15΄ 30" A 1576.20 BC 126º 12΄ 54" B 3824.10 CD 069º 41΄ 18" C 3133.72 DA 031º 50΄ 30" D 360º 00΄ 12" Σ

Σ (Internal Angles should be (N-2)*180º = 360º 00΄ 00" Solution Calculation of Angular misclosure Σ (Internal Angles)= 360º 00΄ 12" Σ (Internal Angles should be (N-2)*180º = 360º 00΄ 00" The Angular Misclosure(Δ)= Σ Measured Angles - Σ Internal Angles = 360º 00΄ 12" - 360º 00΄ 00" = 12" Allowable = 2 * 6" * √4 = 24" OK Therefore distribute error The correction / angle = -12/4= 3"

corrected horizontal angle correction observed clockwise horizontal angle traverse station 132º 15΄ 27" -3" 132º 15΄ 30" A 126º 12΄ 51" 126º 12΄ 54" B 069º 41΄ 15" 069º 41΄ 18" C 031º 50΄ 27" 031º 50΄ 30" D 360º 00΄ 00" -12" 360º 00΄ 12" Σ

Calculation of Whole – circle Bearing back line (W.C.B.) forward station angle corrected 00 AB 180 BA 51 12 126 B 306 BC CB 15 41 69 C 06 54 195 CD DC 27 50 31 D 33 44 47 DA 227 AD 132 A Check

Linear closure The method of checking the distance component of the closed traverse is known as performing a linear closure. In its simplest form this consists of converting the corrected angles into bearing and then computing the partial Easting and Northing for each line. Δ Easting = D . Sin θ Δ Northing = D . Cos θ These values are then summed, and any deviation from the expected value is assessed. In a traverse that starts and finishes on the same point the total change in position should be zero, and in a traverse that starts and finishes on points that have a known position the sum should equal the known displacement.

An angular closure must be performed first, as these formulae contain two measured variables (direction and distance) the bearings must have their error eliminated so we can attribute the remaining error to the distances. If the linear misclosure is acceptable, then this can be adjusted out of the network, but if the misclosure is too large then the fieldwork should be repeated (unless the source of the problem can be isolated).

linear misclosure In above example can be calculated as follow D cosӨ ( m ) D sinӨ bearing circle whole Distance line +638.57 0.00 00" 00΄ 00º 638.57 AB +931.227 -1271.701 51" 12΄ 306º 1576.20 BC -3677.764 -1047.754 06" 54΄ 195º 3824.10 CD +2107.313 +2319.361 33" 44΄ 47º 3133.72 DA -0.654 -0.094 9172.59 Σ

Acceptable FLM values :- e = √ (Δ E2 + Δ N2 ) = √ (0.0942 + 0.6542) = 0.661m e is the LINEAR MISCLOSURE Fractional Linear Misclosure (FLM) = 1 in (Σ D / e ) 1 in (9172.59 / 0.661) = 1 in 13900 Acceptable FLM values :- 1 in 5000 for most engineering surveys 1 in 10000 for control for large projects 1 in 20000 for major works and monitoring for structural deformation etc.

Observed clockwise horizontal angle Example Consider the following traverse and traverse table: horizontal Distance (m) line Observed clockwise horizontal angle traverse station 127.54 AB 096º 54΄ 10" A 086.32 BC 107º 32΄ 30" B 078.45 CD 141º 27΄ 10" C 149.68 DE 087º 15΄ 40" D 096.02 EA 106º 49΄ 40" E 538.11 ΣL 539º 59΄ 10" ΣӨ

96o 54’ 10” 127.54 m. 107o 32’ 30” 96.02 m. 78.45 m. 86.32 m. 149.68 m. 141o 27’ 10” 106o 49’ 40” 87o 15’ 40” A B C D E N

Solution Calculation of Angular misclosure: The Angular Misclosure(Δ) Σ (Internal Angles) = 539º 59΄ 10" Σ (Internal Angles) should be (n -2)*180 = (5-2)*180 =540º 00΄ 00" The Angular Misclosure(Δ) = Σ Measured Angles - Σ Internal Angles = 539º 59΄ 10" - 540º 00΄ 00" =- 50" Allowable = 2 * 20" * √5= 89.44" OK Therefore distribute error. The correction/angle = 50"/5= 10“ The angles area adjusted for this misclosure amount, this case 10 seconds would be added to each angle to distribute the misclosure evenly throughout the traverse.

W.C.B. corrected angle line 000º 00΄ 00" (A) 096º 54΄ 20" AB 072º 27΄ 20" (B) 107º 32΄ 40" BC 111º 00΄ 00" (C) 141º 27΄ 20" CD 203º 44΄ 10" (D) 087º 15΄ 50" DE 276º 54΄ 20" (E) 106º 49΄ 50" EA

هنا بيستخدم زاوية انحراف الخط وليست الزوايا الداخلية للترافيرس Linear closure Latitu. ( m ) E Depar. N bearing distance line 127.54 0.000 000º 00΄ 00" AB 26.021 82.305 072º 27΄ 20" 086.32 BC -28.114 73.239 111º 00΄ 00" 078.45 CD -137.018 -60.250 203º 44΄ 10" 149.68 DE 11.545 -95.323 276º 54΄ 20" 096.02 EA -0.026 -0.029 538.11 Σ

Distance =( Σ ΔE 2 + Σ ΔN2 )1/2 = 0.039 meters, From the table, Σ ΔE = -0.029, and Σ ΔN = -0.026 This is then converted to a vector, expressing the misclosure in terms of a bearing and distance. Distance =( Σ ΔE 2 + Σ ΔN2 )1/2 = 0.039 meters, Bearing = tan-1 (Σ ΔE/ Σ ΔN) = 227° 30‘ Then the work is repeated. Conventionally the misclosure is expressed as a ratio of the total perimeter of the traverse and referred to as the 'accuracy'. In this case this is 1:13,795 which satisfies requirements under the Survey Coordination Act. If the misclosure was found to be large then it is likely that a mistake had occurred during the field process. The bearing of the misclosure vector can be used as an indication of the line in which the mistake occurred, however this is a guide only. Naturally if the misclosure was close to one physical length of the measuring device (say 50m) then it is likely that a chain length was omitted somewhere. If the source of the mistake cannot be isolated,

If the coordinates of point A ( 2000,5000 ) Now we will go to correct the coordinates of the points of the traverse ∆N corr. ∆E corr. for norting easting uncorr. Line 127.546 0.007 0.006 127.54 0.000 AB 26.025 82.310 0.004 0.005 26.021 82.305 BC -28.110 73.243 -28.114 73.239 CD -137.011 -60.242 0.008 -137.018 -60.250 DE 11.550 -95.318 11.545 -95.323 EA 0.026 0.029 -0.026 -0.029 Σ

Final corrected coordinates: northing (m) Easting (m) point 5000.000 2000.000 A 5127.546 2000.007 B 5153.571 2082.317 C 5125.461 2155.560 D 4988.450 2095.318 E

Traverses - Missing Data: As a rule traverses are always closed, either onto them selves or between known points so that an estimate of accuracy and precision can be obtained, as well as a check on our fieldwork. There are rare occasions where traverses cannot be closed, and more commonly there are situations where open traverses run off a rigorous network are used to determine the dimensions of features that are not readily accessible. The use of traversing procedures and calculation to determine these dimensions is based on the mathematics of a closed traverse. That is, the data that is missing from the traverse is presumed to be that which would close the traverse. If we adopt this procedure, then an additional condition applying to our measurements is known

The missing elements of a traverse polygon that can be solved for are as follows: Bearing and Distance of One Line . Bearing of One Line, Distance of Another. Distance of two Lines. Bearing of two Lines.

Calculate the missing data ? Example In a theodolite survey the following details were noted and some of the observations were found to be missing. Length ( m ) W.C.B. line 480 060º 00΄ 00" AB 1180 115º 00΄ 00" BC ? 235º 40΄ 00" CD 1205 DA Calculate the missing data ?

Solution latitude R.B. 480 AB 1180 BC L CD 1205 DA Σ Departure line Length( m ) line +240 415.69 N 60º E 480 AB -498.69 1069.44 S 65º E 1180 BC -0.564 L -0.826 L S 55º 40΄W L CD 1205 cos Ө 1205 sin Ө Ө 1205 DA Σ

W.C.B. of CD = 296º 25΄ 38" In Departure 415.69 + 1069.44 –0.826 L +1205 Sin (θ) = 00.00 1205 Sin (θ) = + 0.826 L – 1485.13 1452025Sin2(θ)=0.682L2–2453.43L+2205611.12 (1) In Latitude 240 – 498.69 – 0.564 L + 1205 Cos (θ) = 00.00 1205 Cos (θ) = + 0.564 L + 258.69 1452025Cos2(θ)=0.318 L2+291.8 L+ 6692 0.52 (2) For length CD Add Eq.(1) and Eq.(2) 1452025 [Sin2(θ)+Cos2(θ)]=L2–2161.63 L+ 2272531.64 L2 – 2161.63 L + 820506.64 = 00.00 (3) Solving Eq. (3) L = 491.455 m. For Bearing of line CD Substitute in Eq.(1) 1205 Sin (θ) = + 0.826 x 491.455 – 1485.13 Sin (θ) = -0.8955 R.B. of CD = N 63º 34΄22" W W.C.B. of CD = 296º 25΄ 38"

Example (Mansoura 4/1/2006) Latitude (m) station +835 +380 C -680 D C and D are two stations whose coordinates are given below: Latitude (m) Departure (m) station +835 +380 C +1350.50 -680 D From station C is run a line CB of 220 m length with a bearing of 130º. From B is run a line BA of length 640 m and parallel to CD . Find the length and bearing of AD?

Solution: N D (-680,1350.50) 130º C (380,835) 220 m A 640 m B o

DC = {(1060)2 + (515.5)2 }0.5 = 1178.703 m W.C.B of DC = tan-11060/515.5 = 115º 56΄ 05″ W.C.B of BA = 295º 56′ 05″ Latitude Departure W.C.B Length (m) Line -515.50 +1060 115º 56΄ 05″ 1178.703 DC -141.413 +168.530 130º 220 CB +279.902 -575.574 295º 56′ 05″ 640 BA L cos Ө L sin Ө Ө L AD L sin Ө= - 652.983 m , L cos Ө = + 377.011 m Ө ( W.C.B of AD ) = 325º 59′ 57″ , Length of AD = 754.005 m