Chapter 4 Traverse Computations

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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).

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• 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).

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• 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.

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

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• 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.

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• Closed Traverse
• This traverse starts and ends at stations of
  known control.
• There are two types of closed traverseclosed 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.

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• 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
  (n2)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

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• 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

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• 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.

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• 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

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• 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.

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• 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.

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• 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.

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• Choice of points
• 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.

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• 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
• S ? (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.

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• A larger misclosure could be expected when using
  a magnetic compass, but in any case it must be
  calculated and removed.
• The Angular Misclosure
  S Measured Angles - S Internal Angles
• Maximum Angular Misclosure 2Accuracy of
  Theodolite v (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 (a).

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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
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Solution Calculation of Angular misclosure S
(Internal Angles) 360º 00? 12" S (Internal
Angles should be (N-2)180º
360º 00? 00" The Angular
Misclosure(?) S Measured Angles - S Internal
Angles 360º 00? 12" - 360º
00? 00" 12" Allowable 2 6" v4
24" OK Therefore distribute
error The correction / angle -12/4 3"
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Calculation of Whole circle Bearing
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• 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.
• ? Easting D . Sin ?
• ? Northing D . Cos ?
• 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.

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• 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).

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linear misclosure In above example can be
calculated as follow
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• e v (? E2 ? N2 )
• v (0.0942 0.6542) 0.661m
• e is the LINEAR MISCLOSURE
• Fractional Linear Misclosure (FLM) 1 in (S 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.

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Example Consider the following traverse and
traverse table
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Solution Calculation of Angular misclosure S
(Internal Angles) 539º 59? 10" S
(Internal Angles) should be (n -2)180
(5-2)180 540º 00? 00" The Angular
Misclosure(?) S Measured Angles
- S Internal Angles 539º
59? 10" - 540º 00? 00" - 50" Allowable 2
20" v5 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.
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Linear closure
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• From the table, S ?E -0.029, and S ?N -0.026
  This is then converted to a vector, expressing
  the misclosure in terms of a bearing and
  distance.
• Distance ( S ?E 2 S ?N2 )1/2 0.039 meters,
• Bearing tan-1 (S ?E/ S ?N) 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 113,795 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,

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If the coordinates of point A ( 2000,5000 ) Now
we will go to correct the coordinates of the
points of the traverse
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Final corrected coordinates
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• 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

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• 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.

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Example In a theodolite survey the following
details were noted and some of the observations
were found to be missing.
Calculate the missing data ?
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Solution
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In Departure 415.69 1069.44 0.826 L 1205 Sin
(?) 00.00 1205 Sin (?) 0.826 L
1485.13 1452025Sin2(?)0.682L22453.43L2205611.12
(1) In Latitude 240 498.69 0.564 L 1205
Cos (?) 00.00 1205 Cos (?) 0.564 L
258.69 1452025Cos2(?)0.318 L2291.8 L 6692 0.52
(2) For length CD Add Eq.(1) and
Eq.(2) 1452025 Sin2(?)Cos2(?)L22161.63 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 (?) 0.826 x
491.455 1485.13 Sin (?)
-0.8955
R.B. of CD N 63º 34?22"
W W.C.B. of CD 296º 25? 38"
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Example (Mansoura 4/1/2006) C and D are two
stations whose coordinates are given below
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?
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Solution
N
D (-680,1350.50)
130º
C (380,835)
220 m
A
B
640 m
o
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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?
L sin ? - 652.983 m , L cos ? 377.011 m ? (
W.C.B of AD ) 325º 59' 57? , Length of AD
754.005 m