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- Traversing
Module Notes 6. Traversing
Control observation techniques The commonly used methods for establishing (observing and computing) the plan co-ordinates of new survey control stations are: traversing intersection resection network All can be observed with the minimum amount of data required to give a unique determination of the coordinates of the new points, but this is very undesirable as there is no check on the quality of the observations or the correctness of the calculations. All should have redundant observations (extra observations), beyond those necessary for a unique solution, and networks in particular are generally observed in such a fashion, with a computational procedure known as the principle of least squares being used to generate the most probable values for the new co-ordinates, together with statistical data to show their reliability. Resections, intersections and traverses can be similarly treated, although traversing, which is still possibly the most commonly used method of establishing survey control, rarely tends to include redundant observations and is inherently the weakest of the methods mentioned. Observation by all methods using total stations is rapidly being superseded by the use of GPS, and most (if not all) GPS processing software uses the principle of least squares adjustment in its computations. Introduction to Traversing Traversing is a method by which survey control for either detail surveying or setting-out purposes can be provided utilising the methodology of ‘working from the whole to the part’ in terms of relating a series of baselines to one another. Traversing introduces angular measurements to the surveying observables of plan lengths and height differences used in basic linear surveying techniques. This has the advantage of increasing the potential accuracy and size of the survey that was somewhat limited when using basic linear surveying techniques. Using the three fundamental survey observations of slope lengths, height differences, (hence plan lengths) together with angular observations the positions, in terms of relative co-ordinates, of a series of points can be determined in relation to one another such that they can serve as the control for subsequent surveys.
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- Traversing Traverse types A traverse starts and finishes at control points whose co-ordinates are known. If the start and end points are the same, it is called a loop traverse; if they are different, it is a link traverse (Figure 1). a. Link b. Loop Figure 1. Traverse types The start/end station of a loop traverse may have co-ordinates known from previous control observations, or may have arbitrary co-ordinate values assigned. A link traverse always runs between control stations whose co-ordinates are already known in some specified co-ordinate system. The object of both loop and link traverses is the same – to allow determination of the co-ordinates of new control stations between the start and end of the traverse which can then be used for any subsequent survey operations. Traverse orientation At the start and end of any traverse it is necessary to be able to sight along a line of known bearing. This is often to another control point of known co-ordinates – a ‘Reference Object’, or RO. From the known co- ordinates of the starting station and its RO, the bearing from the starting station to the RO can be calculated; similarly with the closing station and its RO (Figure 2). If there is no external RO for a loop traverse, one of the traverse legs from the starting station may be given an arbitrary bearing (Figure 3a) or if an external RO exists orientation is similar to that of the link traverse, see Figure 3b. Figure 2. Link traverse opening & closing bearings Station with known co-ordinates Station with unknown co- ordinates T The co-ordinates of P1, P2 and P3 are to be determined from the traverse. A, Q, T & V have known co-ordinates; hence, opening bearing AQ and closing bearing TV are also known. A V (RO) P P P Q (RO) AQ TV
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- Traversing Traverse observation procedure Observations consist of setting-up at the starting (or opening) station, each new station in turn, and then the end or closing station (Figure 4). Figure 4. Traverse horizontal angles At each station; multiple rounds (FL & FR) of horizontal angles are observed between the back station and forward station; vertical angles are observed on both faces to both the back and forward station; distances are observed to both the back station and forward station. If it is intended to calculate height differences between the stations; the height of instrument and target(s) must also be measured. An example completed booking sheet and notes regarding best practice are shown on the following pages. A P V (RO) T P P Q (RO) βAP
QAP
βAQ θAP1P 2
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- Traversing Example traverse booking sheets EDM TRAVERSE OBSERVATION SHEET – Type A CLASS/GROUP CHFC Group 4^ DATE INSTRUMENT Make Nikon^ OBSERVER J Wren^ 02/07/ Model NPL632^ BOOKER C Robin Serial No (^105858) REDUCTIONS BY A Sparrow WEATHER Fair, clear skies^ CHECKED BY J Wren JOB/EXERCISE Castle Head Topo Control^ Page^2 of^6 HORIZONTAL ANGLES At Station: C To Station Face Circle reading Reduced angle Mean angle per round Mean angle B L 000 15 20 125 07 06
D L 125 22 25 125 07 05 125 07
D R 305 22 29 125 07
B R 180 15 22
B L 000 26 56
D L 125 34 04 125 07
D R 305 34 08 125 07 08 125 07
B R 180 27 00
B L 001 45 18
D L 126 52 22 125 07 04 125 07
D R 306 52 24 125 07
B R 181 45 20
VERTICAL ANGLES Ht Inst = 1.635 m Observed Reduced + / - Mean To Station: B Ht Tgt = 1.565 m
FL 090 13 20 00 13
- 00 13 25 FR (^269 46 30 00 13 ) To Station: D Ht Tgt = 1.356 m
FL 088 45 24 01 14
- 01 14 35 FR (^271 14 34 01 14 ) DISTANCES Slope / Horizontal To Station: B (m) To Station: D (m) 56.225, 56.227, 56.226 74.159, 74.156, 74. Mean 56.226 74. Reduced Horizontal ∆H Horizontal ∆H 56.226 - 0.149 74.141 1.
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Traversing Notes for good practice in traverse observations:
Observe each item (HAs, VAs and Distances) as a separate operation, i.e. do not observe, for example, HA and VA to a station at the same time. Doing so can easily lead to mis-pointings and thus observing errors, and/or booking errors, and actually can take more time than observing the different items separately.
Observe each round of horizontal angles in the shortest possible time; similarly, observe face left and face right vertical angle observations to any station as rapidly as possible. This is to minimise the effect of changing atmospheric conditions between pointings.
Complete the sheet from top (header information) to bottom in the logical order shown, ensuring that you record all required data, especially height of instrument and target which can otherwise easily be forgotten.
When calculating the mean and accepted angles, round either to the odd or to the even (consistently to either one or the other, not to a mixture), rather than rounding up or down, to avoid incurring a systematic error.
All reductions must be completed before the instrument is moved to the next station, to avoid time-wasting re-setting of the instrument if a problem is found with the observations.
All reductions should be independently checked and signed off. Traverse computations Calculation is by successive application of the equations for calculating co-ordinates from distance and bearing, i.e. with reference to Figure 4 where the starting fixed coordinates for station ‘A’ are known EP1 = EA + DAP1 sin βAP and NP1 = NA + DAP1 cos βAP with the bearing βAP1 being determined from the known fixed bearing AQ plus the observed clockwise angle QAP1. The co-ordinates of station P2 can subsequently be determined by extending this principle, i.e. since the co-ordinates of station P1 are now known, as is the bearing of the line P1A (180 degrees different from bearing AP1), then EP2 = EP1 + DP1P2 sin βP1P and NP2 = NP1 + DP1P2 cos βP1P where βP1P2 = βP1A + AP1P2 = βAP1 180 o^ + (^) AP1P
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- Traversing When computing in a co-ordinate system other than a plane grid, e.g. when working with Ordnance Survey National Grid co-ordinates, items such as reduction of distances to mean sea level (MSL) and application of local scale factor (LSF) have to be included in the traverse computation. These issues are not dealt with in this module and further details may be found in most of the standard surveying textbooks if required. Regardless of the traverse type or the co-ordinate system used, in practice the co-ordinates calculated for the final station (T in the above example, Figure 4) are not usually equal to its known values, nor is
the observed final bearing ( TV in Figure 4) equal to its fixed value. This is because there will usually
be small errors built up due to the accuracy of the instrument set-up (centring), small observational errors when sighting targets and possible minor errors in the instrument itself. These differences between the observed position/bearing and the given position/bearing are referred to as positional and angular or bearing misclosures. If these misclosures come within pre-set tolerances they are usually distributed through the traverse by an adjustment process. It should be noted that an adjustment does not eliminate the misclosures; it only spreads them throughout the traverse observations such that the final co-ordinates will become the ‘Most Probable Values’ based upon the data available. The pre-set tolerance for angular or bearing misclosure is deemed to be sufficiently small to exclude gross errors, i.e. blunders. Therefore, if the misclosure is within tolerance, it can be assumed that it is caused by the cumulative effect of a small error in each observation. The misclosure is therefore distributed evenly through the angular observations. This process will become apparent during the worked examples. For positional misclosures, two of the methods of adjustment commonly used are: Bowditch Equal shifts with a third method, Transit, also being mentioned in some textbooks. The principle of the Bowditch adjustment is that the adjustment to the co-ordinates at the end of each leg of the traverse is proportional to the length of the leg. This stems from the days when traverse distances were taped; the longer the distance, the more times the tape had to be used, and consequently the larger the error in the distance measurement, i.e. a long leg would contain more error in the distance measurement than a short leg. That is not the case when the traverse distances are measured by EDM, and Bowditch is not an appropriate adjustment technique for EDM traverses. The standard error of EDM distance measurements is of the form ±(x mm + y ppm) where ppm stands for ‘parts per million’ i.e. mm per kilometre. The
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- Traversing Example of a Loop Traverse Computation and Adjustment A loop traverse incorporating four stations has been observed using a total station to provide control for a detail survey of a green field site. An abstract of the field survey data together with the opening control station co-ordinates and orientation is given below. Co-ordinates of station Barn 500.00 mE, 2000.00 mN Bearing of line Barn - Oak 180 o^ 00' 00" The following clockwise horizontal angles were observed at each station: - Oak - Barn - Gate 272 o^ 36' 26" Barn - Gate - Pond 271 o^ 13' 44" Gate - Pond - Oak 252 o^ 52' 40" Pond - Oak - Barn 283 o^ 17' 18" Together with the plan distances: - Barn - Gate 264.672 m Gate - Pond 189.111 m Pond - Oak 258.705 m Oak - Barn 260.156 m Compute the above traverse to provide the control stations with final accepted co-ordinates, distributing any positional misclosure using the Equal Shifts method of adjustment. Check that the accuracy specifications given below are satisfied Angular misclosure < 5" x No. of stations Positional misclosure < 1/ 10 ,000 of traverse length Solution (in addition refer to traverse computation sheet on page 15 ) The first stage is to draw a figure that approximately represents the data given in the survey abstract showing the line lengths, the observed horizontal clockwise angles, the known control station co-ordinates and opening bearing for the traverse.
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- Traversing Determination of forward bearings Utilising the above sketch and the principles covered earlier determine the observed forward bearings for each of the survey lines, i.e. utilising βBC = βBA + B => βAB 180 o^ + B Considering each line in turn starting with the known bearing of line Barn Oak βBarn-Gate = βBarn-Oak + clockwise angle (Oak - Barn - Gate) βBarn-Gate = 180o^ 00’ 00” + 272o^ 36’ 26” = 452o^ 36’ 26” => 92 o^ 36’ 26” (since there are only 360o^ in one full circle) βGate - Pond = βBarn-Gate ± 180o^ + clockwise angle (Barn - Gate - Pond) βGate - Pond = 92o^ 36’ 26” ± 180o^ 00’ 00” + 271o^ 13' 44" = 183 o^ 50’ 10” βPond - Oak = βGate - Pond ± 180o^ + clockwise angle (Gate - Pond - Oak) βPond - Oak = 183 o^50 ’ 10 ” ± 180o^ 00’ 00” + 252o^ 52' 40" = 256 o^42 ’ 5 0” βOak - Barn = βPond - Oak ± 180o^ + clockwise angle (Pond - Oak - Barn) βOak - Barn = 256 o^42 ’ 50 ” ± 180o^ 00’ 00” + 283o^ 17' 18" = 000 o^0 0’ 08 ” The angular/bearing misclosure is the difference between the fixed opening value and the closing observed value for the bearing of the line Barn Oak and is equal to 180 o^ 00’ 00” - …………………… = ………………….. The angular/bearing misclosure, if falling within the pre-set tolerance, is then distributed evenly
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- Traversing For a closed loop traverse the sum of these partial co-ordinate differences should be equal to zero since the traverse starts and finishes at the same point. As stated in the introduction, the likelihood of this happening is minimal due to the errors inherent in the observations of the angles and distances which were used to determine the adjusted bearings and the horizontal distances used to calculate the E and N values. In this example the sum of the partial co-ordinate differences are ΣΔE = - 0.029 m and ΣΔN = - 0.030 m. These differences from zero for a closed loop traverse are termed the Easting and Northing misclosures , with the total positional vector misclosure for the traverse being given by:
2 2
vector _ misclosure Emisc Nmisc
This amount of misclosure has occurred over the total traverse length, i.e. linear accuracy = vector misclosure / traverse length This can be more understandably expressed as a proportional misclosure of 1 in …. Proportional misclosure = 1 in (Total Traverse Length / Vector Misclosure) Distribution of proportional misclosure Subject to the proportional misclosure lying within the pre-set tolerance it is adjusted through the traverse to allow the final accepted co-ordinates of the points to be established. The positional misclosure can be adjusted utilising one of three commonly used techniques: the Bowditch Method, the Transit Method or the Equal Shifts Method – although as stated earlier only Bowditch (for taped distances) or Equal Shifts (for EDM distances) are recommended. In this module, the Equal Shifts method is used for traverse computations as total stations using EDM are most often used for distance measurement. Distribution of co-ordinate misclosures Equal Shifts (This is the method used in this module) In this method the misclosure is spread as evenly as possible throughout the traverse as in the angular misclosure adjustment. Hence in this example as there are 4 lines the adjustment is E = - Emisc / 4 - (-0.029) / 4 = +0.00725 m say +0.007, +0.00 8 , +0.00 7 & +0. N = - Nmisc /4 - (-0.030)/4 = +0.0075 m say +0.00 8 , +0.00 7 , +0.00 8 & +0.00 7 Note not working to fractions of a mm on adjustment values as it can become meaningless. It is useful for you to be aware that there are other methods, such as Bowditch and Transit. Bowditch Method - (not used for traverse computations in this module) The adjustment of each traverse line is directly proportional to the length of the line. E = - Emisc x (length of traverse line/total traverse length)
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- Traversing N = - Nmisc x (length of traverse line/total traverse length) For example for the line Barn Gate EBarn - Gate = - (-0. 0 29) x (264.672/972.644) = +0.008 m NBarn - Gate = - (-0.030) x (264.672/972.644) = +0.008 m Transit Method – (not used for traverse computations in this module) The adjustment for each traverse line is proportional to the E or N value for that particular line. E = - Emisc x absolute E value for particular traverse line sum of absolute E values for whole traverse N = - Nmisc x absolute N value for particular traverse line sum of absolute N values for whole traverse Again using the line Barn Gate as an example EBarn - Gate = - (-0. 0 29) x (264.398/528.825) = +0.014 m NBarn - Gate = - (-0.030) x (12.037/520.342) = +0.001 m Each of the above adjustment processes will result in slight changes to the previously determined adjusted bearings for the traverse lines, less so in the case of the Transit adjustment. Determination of final co-ordinates The final co-ordinate values for the new stations are equal to the cumulative sum of the partial co-ordinate differences together with the positional misclosure adjustments added to the initial fixed co-ordinates of the opening traverse control station. The following page shows a type of tabulation that can be used to help layout the traverse calculation process, it is not obligatory and many people prefer to calculate long-hand, see link example, or develop spread-sheets. When the bearing between two control stations is required for subsequent setting-out operations, or as an opening bearing for further control observations, or as a reference bearing for detail surveying, it must be calculated from the final adjusted co-ordinates determined from the traverse calculation. The initial observed and adjusted bearings used in the traverse computation are never used for these purposes.
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- Traversing Example of a Link Traverse Computation and Adjustment In order to find the plan distance between two points, R and E, that are not inter-visible, a link traverse is observed incorporating fixed control stations together with two unknown points. From the data below compute the traverse, using equal shifts adjustment, to obtain co-ordinates for the unknown stations and hence the horizontal distance RE. Given Control Station Co-ordinates Station Easting (m) Northing (m) A 1033.132 2813. B 1136.452 2743. E 1521.081 3109. F 1712.450 3033. Observed Horizontal Clockwise Angles and Horizontal Distances Clockwise Angle Observed Value (o^ ' ") Horizontal Distance (m) ABR 84 36 27 BR = 237. BRS 253 21 48 RS = 188. RSE 101 11 20 SE = 215. SEF 268 05 00 Check that the accuracy specifications given below are satisfied Angular misclosure < 10" x No. of stations Positional misclosure < 1/10,000 of traverse length Solution The first observed angle (at a point with known co-ordinates) is at B and the last is at E: these are therefore the opening and closing stations of the traverse respectively. The reference object (RO) from B is A, and from E is F, these providing the opening and closing bearings of the traverse. As with the loop traverse the first stage is to draw a figure that approximately represents the data given in the survey abstract, showing the known control stations and the line lengths and observed horizontal clockwise angles for the traverse.
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- Traversing Determination of fixed opening and closing bearings To determine the observed bearings and the angular/bearing misclosure of the traverse, it is first necessary to calculate the fixed bearings at the start and end of the traverse (the opening and closing bearings) using the known co-ordinate values of the control stations A, B, E and F. To aid in the determination of these fixed bearings sketch the relevant positions of A relative to B and F relative to E. Solve by basic trigonometry or by using the RP function on your calculator. Fixed Opening Bearing B to A From the signs of ΔEBA and ΔNBA, bearing B to A is in the 4th^ quadrant, therefore BA = 360 + where = tan-^1 (ΔEBA /NBA) = tan-^1 (-103.320 / 70.356) = - 55 44' 49"
Hence bearing B A BA= 304 o^ 15' 11"
Fixed Closing Bearing E to F From the signs of ΔEEF and ΔNEF, bearing E to F is in the 2nd^ quadrant, therefore EF = 180 + where = tan-^1 (EEF/NEF) = tan-^1 (191.369 / - 75.380) = - 68 o^ 30' 02"
Hence bearing E F EF = 111 o^ 29' 58"
B
A ΔEBA
ΔNBA
BA
F
E
ΔEEF
ΔNEF
EF
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- Traversing Note that the angular/bearing misclosure can also be determined from the following check formula:
Obs'd. closing brg = Opening fixed brg + (observed clockwise angles) – ((n+1) x 180o)
where n equals the number of observed angles. Hence, in this example, Observed closing bearing = 304º 15' 11" + 707º 14' 35" - (5 x 180º) = 111º 29' 46" Fixed closing bearing = 111 29' 58", hence misclosure is - 12" and the adjustment is +3" per angle as before. Determination of partial co-ordinate differences Having now obtained a set of ‘adjusted forward bearings’ for each of the traverse lines, the partial co- ordinate differences for each traverse line can be determined from as in the previous loop example: EBR DBR sin BR and NBR DBR cos BR etc. For a link traverse, the sum of these partial co-ordinate differences should be equal to the difference between the given fixed co-ordinates of the opening and closing stations, B and E in this example. As stated in the introduction, and seen in the loop example, the likelihood of this happening is minimal due to the errors inherent in the observations of the angles and distances which were used to determine the adjusted bearings and horizontal distances used to calculate E and N values. As before the difference from the known value in each case is termed the Easting and Northing misclosure. Vector misclosure, linear accuracy and proportional misclosure are all calculated as previously shown in the loop traverse example. Distribution of proportional misclosure As in the loop traverse example subject to the proportional misclosure lying within the pre-set tolerance it is adjusted through the traverse to allow the final accepted co-ordinates of the points to be established. The complete traverse calculation (with the exception of the calculation of the fixed opening and closing bearings) is shown on the next page using the equal shifts method. The calculation can be set out long hand as per the examples overleaf or undertaken on the Traverse Computation form (see copies on NOW) used for the loop traverse example. It is important that however the calculation process is undertaken to ensure a clear and concise set of calculations is provided.
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- Traversing Adj. Adjusted bearings Horiz Distance ΔE ΔN E N Station Opening bearing B-A 304 15' 11" 1136.452 2743.626 B Observed angle ABR 84 36' 27" Forward bearing B-R 28 51' 38" +3" 28 51' 41" 237.642 114.708 208. Back bearing R-B 208 51' 38" - 0.009 +0. Observed angle BRS 253 21' 48" 114.699 2 08.134 1251.151 2951.760 R Forward bearing R-S 102 13' 26" +6" 102 13' 32" 188.736 184.456 - 39. Back bearing S-R 282 13' 26" - 0.008 +0. Observed angle RSE 101 11' 20" 184.44 8 - 39.958 1435.599 2911.802 S Forward bearing S-E 23 24' 46" +9" 23 24' 55" 215.131 85.491 197. Back bearing E-S 203 24' 46" - 0.009 +0. Observed angle SEF 268 05' 00" 85.482 197.42 5 1521.081 3109.227 E Forward bearing E-F 111 29' 46" +12" 111 29' 58 " ------------ ------------ Closing bearing E-F 111 29' 58" εL = 641.509 384.655 365. Misclosure - 12" Given ΔE, ΔN: 384.629 365. Misclosure: +0.026 - 0.029 Vector misclosure = 0.0389m Proportional misclosure = 1/16, Adjustment per station: - 0.026/3 +0.029/ = - 0.0087 = 0. Equal Shifts Adjustment To check if the computation has worked correctly, check that the calculated Easting and Northing coordinates for station E are the same as the information given at the start of the exercise. In this example, the given coordinates of station E were (1521.081 mE, 3109.227 mN), therefore the traverse computation has been carried out correctly. As the tolerance requirements of bearing misclosure and proportional misclosure have been met, then the new coordinates of R and S can be relied upon to be accurate. In this example, the distance RE was also required. This can be calculated by Pythagoras' theorem or using the Rec( or PR function on your calculator: Distance RE = [(EE - ER)^2 + (NE - NR)^2 ] = [(1521.081 - 1251.151)^2 + (3109.227 – 2951.760)^2 ] = 312.503 m Note that in the equal shifts adjustment, unadjusted co-ordinates could have been calculated from the initial partial co-ordinates, and the adjustments applied cumulatively directly to those unadjusted co-ordinates to obtain adjusted co-ordinates, in the same way that the bearings were adjusted cumulatively rather than the angles being adjusted individually.
