Traverse Survey 7525u

Topic



4

Traverse  Survey 

LEARNING OUTCOMES By the end of this topic, you should be able to: 1.

Outline the basic components of a theodolite;

2.

Carry out temporary adjustments of a theodolite;

3.

Compute and adjust a theodolite traverse;

4.

Compute coordinates for traversing; and

5.

Determine the nature of errors affecting the traverse work.

 INTRODUCTION Traversing is a form of a control survey that requires the establishment of a series of stations that are linked together by angles and distances. The angles are measured by theodolites, and the distances are measured conventionally by tapes or electronic distance measuring equipment. The use of theodolite in traversing surveys is very fundamental and has become one of the most common methods in geomatic engineering work such as: 

general purpose angle measurement;



provision of control surveys;



contour and detail mapping; and



setting out and construction work.

This topic will describe the construction and use of the theodolite in traversing. It will explain the traverse design and the procedures of computing and adjusting a

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traverse. The application of coordinates for point location will also be covered in this topic which will be very useful in civil engineering project.

4.1

THEODOLITE PRINCIPLES AND APPLICATIONS

A theodolite is an instrument which is capable of measuring angles to the nearest whole second [Figure 4.1]. This can be done for both vertical and horizontal angles. Vertical angles are required for the calculation of elevation of points for example the reduction of slope distance to the horizontal.

Figure 4.1: Atheodolite Source: SouthGeosystems

Horizontal angles are required to obtain the relative direction to a survey control station or points of detail. Basically there are two types of modern theodolite which are in use today. These are the: (i)

Optical theodolite; and

(ii)

Electronic Digital theodolite.

Both types of instrument can be made to read to the nearest whole 1” which is considered accurate enough for most engineering purposes. With the advancement of modern electronics, most of the theodolites made today are of the electronic digital type. But the older optical types are still being used except that it will take longer time to read the angles than with an electronic one. The value of the angle observed however will be the same. Electronic theodolites are more versatile than

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the optical type. Useful features in the form of software can be added to an electronic theodolite. Thus modern instruments can be used in a variety of surveying situations. EXERCISE 4.1 What is the major difference between the optical and the electronic theodolite? Differentiate between each type in of its construction.

4.1.1

Construction of a Theodolite

All theodolites have the same common features [Figure 4.2] which can be described as follows: 1.

A Tribrach Allows the instrument to be connected to the top of a tripod and also allows the instrument to be levelled with respect to a plate bubble.

2.

The Horizontal Circle Compartment This compartment is comprised of: (i)

The lower plate that carries the horizontal circle. In most instruments it is made of glass with the graduations from 0ºto 360ºphotographically etched around the edge.

(ii) The upper plate that carries the horizontal circle indexing device and fits concentric with the lower plate. Attached to the upper plate is the plate bubble. When centered, the plate bubble ensures that the instrument axis is vertical. In modern electronic theodolites, the spirit bubble has been replaced with an electronic one. This electronic means of levelling has made initial levelling of the instrument a less time consuming task. 3.

The Vertical Circle Compartment The vertical circle is similar to the horizontal circle but is fixed to the telescope. Thus it revolves with the rotation of the telescope. This compartment has the vertical circle index. Most modern theodolites employ an automatic compensator but some of the more elderly instruments use an altitude bubble.

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4.

The Horizontal Clamp This clamp ensures that when engaged the horizontal circle is fixed. The slow motion screws allow for the movement of the instrument around the vertical axis. There is a similar slow motion screw and plate clamp for the vertical circle.

5.

The Optical Plummet Built in the base of the instrument to allow the instrument to be precisely centered over the station. The line of sight through the optical plummet is exactly the same as the vertical axis of the theodolite.

Figure 4.2: The theodolite construction

4.1.2

Axis System of a Theodolite

The axis system of a theodolite is shown in Figure 4.3 and is explained as follows: 1.

The horizontal circle [HC] is attached at the lower part of theodolite tribrach. It is perpendicular and centric to the vertical axis [V]. The horizontal circle reading microscope is attached to the upper body of the theodolite.

2.

The vertical circle [VC] is attached to the telescope centric and it is perpendicular to the horizontal axis H. The reading microscope for the vertical circle is attached to the adjustable (Vertical Bubble) pointer.

3.

The horizontal and vertical circles are made of optical quality glass of a diameter of about 10 cm with degrees, minutes and high-accuracy units of

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20” or 10” markings ached into its surface. Both circles are read through the microscopes. The grading of the horizontal circle increases from 0 degree clockwise to 360 degrees. 4.

If the theodolite is viewed with the eyepiece towards you and the vertical circle [VC] is to the left of the telescope, the theodolite is defined as being in the ‘Face Left’. If the vertical circle [VC] is to the right of the telescope then it is in the ‘Face Right’ position.

5.

The grading of the vertical circle is such that when in Face Left with the telescope being horizontal, the zero (0) degree marking is vertically above the horizontal axis H. The grading increases from the zero mark in the direction of the eyepiece (90º) with 180º vertically below H and 270º at the object-lens.

Figure 4.3: The theodolite axis system

EXERCISE 4.2 Explain using a simple outline sketch the essential parts of a theodolite. Name these parts and describe their purpose.

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Temporary Adjustments of Theodolite

The temporary adjustments are steps that must be carried out every time a theodolite is used. It is a procedure of setting up a theodolite that involves the following process: 

centering;



leveling; and



removing parallax.

(a)

Centering the Theodolite The instrument must be vertically above the survey station to ensure that horizontal angle observations are correct. The steps are as follows: 1.

Start with a plumbob to get it approximately right above the survey station [Figure 4.4].

2.

Using the foot screws, move the optical plummet cross hairs on to the survey station.

3.

Roughly level the instrument using the legs of the tripod – the theodolite should stay almost on target.

4.

Level with foot screws. Move instrument above target; repeat level and move until done.

Figure 4.4: Centering the theodolite

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Levelling the Theodolite 1.

Turn bubble parallel to two foot screws A and B [Figure 4.5], to bring the horizontal bubble to the centre of its run by moving the foot screws in opposite directions (the bubble moves in the direction of your left thumb).

2.

Turn the instrument through 90º and bring the bubble to the centre of its run by adjusting the third foot screw C only.

3.

Turn the instrument through a further 90º to check the adjustment of the plate bubble.

4.

If the bubble remains in centre, then it is adjusted.

5.

If not, move it back one-half of the movement from the centre and readjust for a further 90º turn.

6.

Repeat the whole procedure; assuming this is the correct, the bubble will stay in a stationary position.

7.

The bubble must remain in the same place in the tube during a 360º rotation of the instrument.

8.

If the stationary position of the bubble is still off the centre, then a permanent adjustment should be made.

Figure 4.5: Levelling the theodolite

(c)

Removing the Parallax Point instrument to infinity (the sky), adjust eyepiece so cross-hairs are fine and dark. Then focus on the target using the focusing knob or collar and check for parallax by moving your eye slightly, and check for a stationary image with respect to the cross-hairs

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Permanent Adjustments of Theodolite

These adjustments are carried out once and will not alter unless it is being roughly handled or tampered with. There are certain basic requirements for a theodolite that must be established particularly when using it. The basic requirements are as follows: (a)

The vertical axis of a theodolite should be truly vertical.

(b)

The line of sight should be perpendicular to the horizontal axis.

(c)

The horizontal axis should be truly horizontal.

(d)

The cross hairs should be truly vertical and horizontal.

(e)

The vertical circle should be at zero when the line of sight is horizontal.

For this study it is appropriate to know only the basic requirements for permanent adjustments. The steps in carrying out the adjustments should be handled by the qualified person at the laboratory. EXERCISE 4.3

4.2

1.

Explain in detail how to set up a theodolite over a station mark.

2.

Why is it necessary to remove parallax before an angle observation is made?

3.

Describe in detail the five permanent theodolite adjustments that should be tested from time to time.

HORIZONTAL AND VERTICAL ANGLE MEASUREMENT

There are three main systems of reading the horizontal or vertical angles in a theodolite; 

optical scale reading;



optical micrometer reading; and



electronic digital display.

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(a)

Optical Scale Reading This system is usually employed on theodolite which reads to 20” or less. Both horizontal and vertical circles are displayed and read through the use of an auxillary telescope.

(b)

Optical Micrometer Reading This system is employed in a non-electronic instruments reading up to 1”. Generally only one circle reading can be displayed and read. A switch is required to read and display the other circle.

(c)

Electronic Digital Display This system provides reading to both circles and is displayed on a screen as numbers. Therefore it reduces the likelihood of error in reading the circles. There are two systems which can be used: 

incremental; and



code method.

In both systems a pattern is etched into the glass circles and these are read and scanned electronically. The final reading is converted to a series of figures which are then displayed. Thus because the display is free from misreading errors, it is in a form which is suitable for automatic recording and data processing.

4.2.1

Horizontal Angle Measurement

The concept of measuring the horizontal and vertical circles is simple in either the traditional theodolite or the modern electronic theodolite. The following procedures should be used to measure the horizontal angles between three stations A, B and C [Refer Figure 4.6]. 1.

Setup the theodolite and center the instrument on station B. The theodolite instrument has two faces; “Face left” or “Face right”.

2.

Starting from the face left, the telescope is pointed at station A. The horizontal reading is then noted. E.g. 25º 30’ 00’.

3.

The instrument is then turned in a clockwise direction to point at station C. Again the horizontal reading is noted. E.g. 145º 50’ 00’.

4.

The horizontal angle α can be calculated, by finding the difference between the two horizontal readings,

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i.e., C – A = 145º5000 – 25º3000 α = 120º2000 5.

Change the face of the theodolite instrument. Whilst pointing at station C the horizontal reading is again recorded. E.g. 325º5000.

6.

Turn the instrument in a clockwise manner and point at station A. Record the horizontal reading. E.g. 205º3000. This time the readings must be subtracted in the correct order, i.e., C – A = 325º5000 – 205º 3000 α = 120º2000

Figure 4.6: Horizontal angle measurements

Note that changing the face will change the readings by 180º. This gives a check on the observations and ensures that reading errors can be eliminated. If there is a great difference in the two readings, the observations are repeated until readings agree.

4.2.2

Vertical Angle Measurement

A vertical angle is the angle measured vertically from a horizontal plane of reference [Refer Figure 4.7]. 1.

When the telescope is pointed in the horizontal plane (level), the reading of the vertical angle is zero (0º).

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2.

When the telescope is pointed up [elevated], then the vertical angle increases from zero and the reading is a positive (+ve) vertical angle [known as angle of elevation]. The reading increase from 0º to +90º when the telescope is pointed straight up.

3.

If the telescope is depressed [pointed down], then the angle reading will increase in numerical value. The depressed telescope reading indicates that it is below the horizontal plane and the reading is a negative (–ve) vertical angle or [known as angle of depression]. These numerical values increase from 0º to –90º when the telescope is pointed straight down.

Figure 4.7: Vertical angle reading

EXERCISE 4.4 1.

Name the three main reading systems for angular measurement in theodolite.

2.

The horizontal angles read at station A and station C are 75º3000 and 145º2000 respectively. If the theodolite is set up at station B. What will be the internal angle ABC?

3.

If the face left reading of the horizontal angle is 225º2000, what is the most probable reading that you will obtain if the theodolite is transit and read in the face right?

4.

The vertical angle reading of a theodolite is 105º30 indicating that it is above the horizontal plane. What will be the true vertical angle?

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TRAVERSING AND COORDINATE SYSTEM

This section describes briefly the meaning of traversing in of its applications, positioning by coordinates, the equipment and field procedures and methods of reduction and adjustment of data.

4.3.1

Traverse Definition

A traverse is a series of straight lines that are used to connect a series of selected points. This selected points are called traverse stations where distance and angle measurements are made. The relative positions of the traverse stations are then computed using some coordinate systems. For a better understanding, the definition of traverse survey can be summarized as follows [Figure 4.8]: 

A measurement of straight lines using measuring tapes or other electronic distance measuring instruments from one point to another and the horizontal angle between them is observed using theodolite.



The sides can be expressed as either polar coordinates or as rectangular coordinates.



A traverse framework comprising of a series of connected lines where the lengths and directions are observed and measured. The traverse framework can be an open or closed traverse i.e, starts at a known point and ends at another known point or the same start point.

Figure 4.8: Travers survey

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Coordinates System

The following information enables you to understand the concept of coordinates referring to the location of points either on the surface of the earth or on a plan. Points on the surface of the earth can be accurately positioned by taking measurements to a known, stable point of reference. Assume that the axes of a graph are referred to as North, South, East and West as shown in Figure 4.9.

Figure 4.9: Measurement of positions

In the above Figure, Point A is at (1,2) and Point B is at (9,5). The referencing used is that the x-axis is known as EASTINGS and the y-axis is known as NORTHINGS. There are two methods of referring the point: 

Rectangular coordinates; or



Polar coordinates.

(a)

Rectangular Coordinates (Grid) Rectangular coordinates are a system of locating points by means of the measurement of two perpendicular distances from the principal axes to that point. These two perpendicular distances are the easting and northing. The origin of these points is usually located at the extreme south and west of the area so that all the coordinates have positive values.

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Polar Coordinates The polar coordinate is explained as follows [Refer Figure 4.10]: If R is the origin and P is the reference object, then point P can be located by its polar coordinates of angle and distance i.e. Ө and D, where D is the distance from the origin and Ө is a clockwise angle between R and P. Coordinates of this type are often used in survey calculations when fixing and plotting detail, especially when setting out data on the ground. Northing P [ Reference Object ]

D

R

Figure 4.10: Polar coordinates

EXERCISE 4.5 1.

What are the characteristics of a traverse?

2.

Explain the significant difference between the two methods of coordinate referencing in traverse?

4.3.3

Northing

There are three (3) reference directions [or datum meridian] that are used as traverse reference we should be associated with. They are: 

Magnetic North;



Grid North; and



True North.

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Magnetic North The earth has its magnetic field with its North and South poles in the vicinity of the true positions of the North and South poles of the planet. This magnetic field will orientate a free swinging magnetic needle in a north/south direction. Based on this principle, it is therefore possible to orientate angular measurements to magnetic north. The magnetic north moves about the central axis of the earth. This variation is known as ‘Magnetic Declination.’ Magnetic Declination is the angular difference between the true Whole Circle Bearing (WCB) (measured relative to true north) and magnetic WCB (measured relative to magnetic north). Magnetic declination varies according to the observer’s position relative to its distance to the pole.

(b)

Grid North It is a grid of lines parallel to the true meridian of one point on the grid, usually the origin of the grid. Since the central meridian points to true north, therefore as we move east or west away from the central meridian, the difference between grid north and true north increases.

(c)

True North This is the point at which all the lines of longitude converge (the axis of rotation of the planet).

4.3.4

Bearing Calculation

The two main types of bearing that are commonly used in geomatic engineering are: 

Whole Circle Bearings (WCB); and



Quadrant Bearings (QB) or Reduced Bearing (RB).

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Whole Circle Bearing

Figure 4.11: Whole Circle Bearing (WCB)

The Whole Circle Bearing (WCB) of a line AB is defined as the clockwise angle from 0º to 360º at A between the direction to North and the direction to B. This is the standard way of defining a bearing in surveying. In Figure 4.11; Bearing of AB =  and Bearing BA = α Bearing of BA = Bearing AB ±180º and Bearing AB = Bearing of BA ±180º (b)

Quadrant Bearing A quadrant bearing can be defined as the angle lying between 0º and 90º, between the direction to the north or south and the direction of the line [Refer Figure 4.12]. East and West directions are never used as reference lines, but they are included since they indicate direction either east or west of the line from grid north.

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Figure 4.12: Quadrant Bearing

4.3.5

Angular Observation and Booking

Two most common methods in making angular observation in theodolite traversing are: 

Internal angle method; and



Bearing method.

(a)

The Internal Angle Method The procedure and booking to be followed are as follows [Refer to Figure 4.13]: (i)

Theodolite is set up at station B and station A is sighted in face left with circle reading of 0o 00’ 00”.

(ii)

With the zero reading fixed, the upper clamp of the theodolite is released and station C is sighted. The circle reading is observed and recorded [Refer to Table 4.1].

(iii) The telescope is then transitted to change to the face right. (iv) Station A is again sighted and the angle reading in the face right is recorded. (v)

Station C is sighted and the angle is read.

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Figure 4.13: Internal angle method Table 4.1: Booking for Internal Angle Method From Station

To Station

Face Left

Face Right

Angle

A

0º0000

180º0000

30º 2620

B

30º 2630 C

(b)

Mean

30º2620

210º2640

30º 2640

The Bearing Method The procedure and bookings to be followed are as follows [Refer to Figure 4.14]: (i)

Instrument is set up at station B and station A is sighted in face left. A known bearing e.g. 45º2020 is set on station A using the lower clamp.

(ii)

By releasing the upper clamp, station C is sighted and the reading is recorded [Refer to Table 4.2].

(iii) Next, station D is sighted and the bearing is read. (iv) The telescope is then transit to change to the face right. (v)

With the face right setting, bearings to station A, C and D are recorded.

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Figure 4.14: Bearing method Table 4.2: Booking for Bearing Method From Station

To Station

Face Left

Face Right

Mean

A

45º2020

225º2025

45º2022.5

C

105º2540

285º2547

105º2543.5

60º0521

D

140º3052

320º3058

140º3055.0

35º0511.5

Angle

B

For improved precision the angle measurement can be repeated any number of times. The number of face left observations must equal the number of face right observations. EXERCISE 4.6 1.

Name the FOUR quadrant bearings by referring to the North and South directions.

2.

The whole circle bearing (WCB) of a traverse line is 115º 30. What will be the value if it is described in quadrant bearing?

3.

What is the relation between magnetic declination and magnetic north?

4.

Explain in detail the procedure and booking method of angular observation in theodolite traverse.

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LINEAR MEASUREMENT IN TRAVERSE

Most theodolite traverses would now use an electronic distance measuring instrument [EDM] to measure distance. The EDMs are usually built with theodolites e.g. total station instrument. The procedure of linear measurement is as follows [Refer to Figure 4.15]

Figure 4.15: Linear measurements in traverse

Tripod with targets is set up at Stations A and C. Theodolite or total station is set up on station B. Linear measurement or distance to BA and BC are taken and recorded. The tripod is then moved from station A to station D. The theodolite is removed from station B and set up at station C. Targets will be set up at station B and D simultaneously. Finally, distance CD is measured and recorded. If the process of linear measurement requires the use of steel tape, spring balance is required for applying tension, vertical angle must be taken for slope correction, and thermometer required for temperature correction.

4.5

TRAVERSE COMPUTATION

The following steps describe the traverse adjustment based on a closed loop traverse. Step 1 – Angular Misclosure The general limits for the angular misclosure of a traverse is 20 n , where n is the number of instrument setups. This equates to a second order engineering survey. If the traverse is a closed loop type, then the internal angles should sum to: (2n – 4)  90º; where n is the number of stations.

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In the following example [Figure 4.16], the traverse have five sides with five internal angles. The sum of internal angles was found to be 539º5900, whereas the sum should be (2  5 – 4)  90º = 540º0000. The angular misclosure is +01 00 or 60. Therefore the correction to be applied is 60/5 = +12 per angle.

Figure 4.16: Internal angles of closed traverse

The value of the misclosure lies within acceptable limits such as those for a second order traverse i.e.  20 5   45 . Since all the angles were measured with the same degree of accuracy, then each angle can be adjusted equally because they were all equally liable to error. The distribution of error equally to all angles to adjust out error is shown in Table 4.3. Table 4.3: Adjustment to Internal Angles Observed Angles

Adjusted Angles

A

101º2400

+12

101º2412

B

149º1300

+12

149º1312

C

80º5830

+12

80º5842

D

116º1900

+12

116º1912

E

92º0430

+12

92º0442

Sum

539º5900

540º0000

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Step 2 – Calculations of Bearing Once the starting bearing has been obtained, the whole circle bearings [WCB] of all the other lines can be calculated. The general rule to determine the bearing of a line at a given station is: 

Add the included (Direct Angle) at the station to the WCB of the previous line.



If the sum obtained is <180º, then add 180º to it.



If the sum obtained is >180º, then subtract 180º from it.



If the sum obtained is >540º, then subtract 540º from it.

The calculated bearings of sides from the internal angles are described in Table 4.4 and Figure 4.17. Table 4.4: Bearing Calculations Bearing Bearing Angle Bearing Bearing Angle Bearing Bearing Angle Bearing Bearing Angle Bearing Bearing Angle Bearing

AB BA B BC CB C CD DC D DE ED E EA AE A AB

51º2200(given) [180 + 51º2200] 149º1312 [231º2200 – 149º312] [180 + 82º0848] 80º5842 [262º0848 – 80º5842] [181º1006 – 180] 116º 1912 360 + [1º1006] – 116º1912 [244º5054 – 180] 92º0442 360 + [64º5054] – 92º0442 332º4612 – 180 101º2412 152º4612 – 101º2412

231º2200 82º0848 262º0848 181º006 1º006 244º5054 64º5054 332º4612 152º4612 51º2200

On the last line the calculation [AB] comes back to the starting bearing, providing a check on the work.

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Figure 4.17: Traverse bearings

Step 3 – Reduction of Length Reduce length for slope, temperature, sag, tension and standard if conventional measurement of distance is being applied. This part has being discussed earlier, please refer to Topic 2 – Linear distance meaurement. Step 4 – Linear Misclosure and Partial Coordinates The determination of linear misclosure, calculated by summing the positive (+ve) and negative (–ve) partial coordinates for both eastings and northings (departure ∆E and latitude ∆N), is known as partial coordinates. The algebraic sum should be zero in both cases, i.e. the traverse should finish at the same point at which it started. The following steps should be followed: 1.

Take algebraic sum of columns distance, ∆E and ∆N.

2.

For a closed traverse Σ∆E = Σ∆N = 0.

3.

Calculate linear misclosure:









 E 2  N 2  / L      0.0562  / 2081.19      0.483 / 2081.19  1: 4325.

4.

Linear misclosure is acceptable for second order engineering traverse [Acceptable misclosure is greater than 1:4000 for 2nd class traverse].

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Once the linear misclosure is accepted, the partial coordinates of the traverse stations can be calculated. The coordinates for the traverse stations are derived from easting and northing differences. Thus the next step is to calculate the partial coordinates for each line in the traverse. The partial coordinates can be obtained from the two equations below: Departure ∆E = Horizontal Length L * sin WCB [ β ] Latitude ∆N = Horizontal Length L * cos WCB [ β ] The tabulation of bearings and distances, depature ∆E and latitude ∆N is described in Table 4.5 and Figure 4.18: Table 4.5: Partial Coordinates

AB BC CD DE EA

WCB [β] Deg Min Sec 51 22 00 82 08 48 181 10 06 244 50 54 332 46 12 Sum [Σ]

Length [L] 401.58 382.20 368.28 579.03 350.10 2081.19

Departure ∆E

Latitude ∆N

L  sin [β] 313.697 378.615 –7.509 –524.130 –160.193

L  cos [β] 250.720 52.223 –368.203 –246.097 311.301

0.48

Figure 4.18: Northing and easting

–0.056

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4.5.1

TRAVERSE SURVEY

Bowditch Rule of Traverse Adjustment

The Bowditch rule of traverse adjustment is the most commonly used method, since it is easy to implement. It assumes that the error in the bearing of a line caused by inaccurate angular measurement produces a displacement at one end of a line relative to the other end which is equal and perpendicular to the displacement along that line due to an error in linear measurement, which is taken to be the square root of the length of the line. This leads to the following relationships: 1.

Adjustment for partial eastings [δ∆N] for a given line is:

E   2.

 E  d d

Adjustment for partial northings [δ∆N] for a given line is:

N  

 N  d d

Adjust traverse by Bowditch Rule. The corrections to the delta eastings and delta northings are proportional to the lengths of the traverse sides. ∆E' = ∆E + δ∆E and ∆N' = ∆N + δ∆N Calculate the adjusted bearings and distances as shown in Table 4.6. Table 4.6: Bowditch Adjustment Distance

Easting

Northing

d

∆E

∆N

Corrections

Corrected

δ∆E

δ∆N

∆E

∆N

A

401.58

313.697

250.720

–0.093

0.011

313.604

250.731

B

382.20

378.615

52.223

–0.088

0.010

378.527

52.233

C

368.28

–7.509

–368.203

–0.084

0.010

–7.593

–368.193

D

579.03

–524.130

–246.097

–0.134

0.016

–524.264

–246.081

E

350.10

–160.193

311.301

–0.081

0.009

–160.274

311.310

2081.19

∑0.48

∑-0.056

0.00

0.00

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For a closed line type traverse, the traverse will finish on a point of known coordinates. By summing the partial coordinates for both eastings and northings, a determination of the coordinates for the closing station can be obtained. In this case, the coordinates should be the same as those given for this closing coordinates [Table 4.7]. Table 4.7: Coordinates of Traverse Coordinates Corrected ∆E

4.5.2

∆N

East

North

10,000.000

20,000.000

A

313.604

250.731

10,313.604

20,250.731

B

378.527

52.233

10,692.131

20,302.964

C

–7.593

–368.193

10,684.538

19,934.771

D

–524.264

–246.081

10,160.274

19,688.690

E

–160.274

311.310

10,000.000

20,000.000

A

Transit Rule of Traverse Adjustment

In the Bowditch method of traverse adjustment, all the lines will have some corrections made in both eastings and northings, as called for in the totals of ∆E and ∆N. In the Transit Rule, the lengths of the individual lines do not enter into the calculations and the errors in easting and nothings are distributed uniformly, throughout the traverse. If the misclosure in eastings and northings are given as δ∆E and δ∆N and there are n legs in the traverse, then for each leg of the traverse, the adjustment to both partial eastings and northings is given by: ∆E = ∆E + δ∆E /n and ∆N = ∆N + δ∆N/n where ∆E and ∆N are the differences in eastings and northings as obtained in the preliminary calculations.

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Work Example 1 Bearing

Distan ce

1–2

106°08'50"

50.502

2–3

142°22'40"

16.341

3–4

186°14'30''

26.236

4–5

267°27'50''

36.454

5–6

279°47'40"

25.034

6–7

349°21'10"

30.974

7–1

29°10'50"

22.892

The table above shows a simple traverse booking by whole circle bearing [WCB] method. Your task is to carry out the necessary traverse adjustment to determine the correct coordinates of each station. Coordinate of station is given 1 as E.5000 N.5000 Solution Step1 – Determine the linear misclosure and partial coordinates 1. Find sum of distance, Σd. 2. Calculate ∆E = d * sin WCB 3. Calculate ∆N = d * cos WCB 2

2

4. Calculate linear misclosure = [√(Σ∆E + Σ∆N ) ]/ ΣL Step 2 – Adjust the traverse using Bowditch’s Rule  E  d and N    N  d E   d d ∆E' = ∆E + δ∆E

and

∆N' = ∆N + δ∆N

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Step 3 – Calculate the final coordinates of Easting and Northing for all the traverse stations Present all your calculation in a tabular form as shown: (i)

Linear misclosure and partial coordinates

Station

Bearing

d (Meters)

Easting ∆E

Northing ∆N

1–2

106°08'50"

50.502

48.510

–14.045

+0.004

–0.001

2–3

142°22'40"

16.341

9.975

–12.943

+0.001

0

3–4

186°14'30''

26.236

–2.852

–26.080

+0.002

–0.001

4–5

267°27'50''

36.454

–36.418

–1.613

+0.003

–0.001

5–6

279°47'40"

25.034

–24.669

4.259

+0.002

+0.001

6–7

349°21'10"

30.974

–5.723

30.441

+0.002

+0.001

7–1

29°10'50"

22.892

11.161

19.987

+0.002

+0.001

208.433

Σ0.016

Σ–0.006

Σ



  E 2    N 2

d



 0.016 2    0.006 2 208.433



Corrections δ∆E δ∆N

0.01709 208.433

Linear Misclosure ≈ 1 : 12000 (ii)

Bowditch Adjustment and Final coordinates Corrected ∆E

Corrected ∆N

1–2

48.514

2–3

Station

Final Coordinate

–14.046

E 5000.000 5040.514

N 5000.000 4985.954

9.976

–12.943

5058.490

4973.011

3–4

–2.850

–26.081

5055.640

4946.930

4–5

–36.415

–1.614

5019.225

4945.316

5–6

–24.667

4.258

4994.558

4949.574

6–7

–5.721

30.440

4980.837

4980.014

7–1

11.163

19.986

5000.000

5000.000

Σ0.000

Σ0.000

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TRAVERSE SURVEY

TRAVERSE FIELD WORK

Traversing is carried out with a minimum of three tripods. One is for the instrument and the other two are for the back and front stations. A minimum of three people is required in a traversing team. The leader of the team, reads the instrument, while the 2nd person has the important job of recording the readings on the booking sheet. The 3rd person has the task of moving and setting up the tripods as the traverse progresses. There are several steps which should be followed that will lead to a smooth traverse [refer Figure 4.19]. 1.

Level and center the instrument.

2.

Set the theodolite to read zero or near zero.

3.

Record face left horizontal reading to back station.

4.

Record face left vertical reading [ VA1] to back station.

5.

Record slope distance to backsight station [SD1]. Obtain three readings for this distance and mean (Readings should be within ±3 mm).

6.

Turn instrument and sight front station.

7.

Record face left horizontal reading to front station.

8.

Record face left vertical reading [VA2] to front station.

9.

Record slope distance [SD2] to foresight station. Obtain three readings for this distance and mean (Readings should be within ±3 mm).

Transit the instrument to change to the face right setting. 1.

Record face right horizontal reading to front station.

2.

Record face right vertical reading to front station.

3.

Turn instrument to face back station.

4.

Record face right horizontal reading to back station.

5.

Record face right vertical reading to back station.

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Figure 4.19: Field traversing

Repeat the field procedure so that two complete sets of horizontal and vertical readings are obtained. This will allow check to be made in the field, in case of gross reading errors. Ensure that all the required information has been obtained before the instrument is moved. When the leader and the booker are satisfied that all the information has been recorded then only the instrument is moved to the next station.

4.7

TRAVERSING ERRORS

Traversing errors normally falls into three categories, i.e. centering, angular and distance. By taking precautions during the field work, it is possible to reduce their effect. 1.

Centering It is important to ensure that the theodolite instrument and targets are centered correctly over each survey station. that angles and distances may be required from or to a known station. This will not be the case if the theodolite or targets are not centered correctly.

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2.

Angles When clamping the instrument, apply light clamp to the vertical and horizontal locks. Hard clamping can affect the pointing of the instrument and is not necessary. Failure to eliminate parallax and poor focusing can affect accurate pointing. Always keep the target in the center of the field of view. All movement of the theodolite should be kept as smooth as possible and all movement around the instrument should be kept to a minimum.

3.

Distances When recording these, all distances should be obtained to 3 decimal places and three readings should be taken and the mean calculated.

There is a possibility that that some of the errors outlined below will occur from time to time, so be aware of them. Don’t rush and hopefully you will not forget to record any information which is required. 

Turning the wrong screw.



Sighting the wrong target.



Using the stadia instead of the cross-hairs.



Forgetting to set the micrometer reading before taking a reading.



Misreading the circles.



Transposing the figures when booking the data. EXERCISE 4.7 1.

What personnel and instrument would be required in order to undertake a traverse survey?

2.

Explain briefly the field procedures usually adopted in theodolite traversing.

3.

State the precautions to be observed against making errors in traversing.

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

Traversing is a form of a control survey that requires the establishment of a series of stations that are linked together by the angles and distances. The angles are measured by theodolites, and the distances are measured conventionally by tapes or electronic distance measuring equipment.



A theodolite is an instrument which is capable of measuring both vertical and horizontal angles to the nearest whole seconds.



Basically there are two types of modern theodolite i.e. the optical theodolite and electronic digital theodolite. But both have the same common features in of their construction.



The theodolite system is comprised of the horizontal circle where it is perpendicular to the vertical axis and the vertical circle where it is perpendicular to the horizontal axis. Theodolites in correct adjustment have their axes and line of sight of the telescope mutually perpendicular. All three should intersect at one point.



The temporary adjustment of a theodolite is a procedure of setting up a theodolite that involves centering it over a ground mark, levelling it and collimating it (adjusting the eyepiece focus).



The three main systems of reading the horizontal or vertical angles in a theodolite are: (i)

optical scale reading;

(ii)

optical micrometer reading; and

(iii) electronic digital display. 

Angles must be read in both the ‘face left’ and face right’ positions to eliminate most instrumental errors.



Horizontal angle readings on face left and face right will be 180 different.



Vertical angle reading (zenith angles), face left and face right should sum to 360.



A bearing is a direction (in degrees, etc.) measured clockwise from north.

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

All angles and distances of a traverse are measured to provide redundancy.



Adjustments to angles, and to bearings and distances are made only when the error is within the expected tolerance.



Adjustment is not used to disguise gross errors.

Figure 4.1 Optical theodolite by SouthGeosystems [http://www.southgeosystems.com/index.html]