Traverse Surveying

Introduction

The fundamental network of points whose horizontal positions are accurately known are called horizontal control. Traversing is most frequently employed method for establishment of horizontal control points for surveys of limited extent or where the desired points lie along a devious route.

Traverse

A traverse consists of a series of straight lines connected successively at established points, along the route of a survey. The points defining the ends of the traverse line are called traverse stations or traverse points. Distances between traverse stations are known as traverse side and are measured either by direct measurement using a Tape or Electronic Distance Measuring (EDM) equipment, or by indirect measurement using tacheometer. At stations where a traverse side changes its direction, relative direction are measured with a transit or theodolite.

Types of Traverse

Fundamentally, there are two types of traverses:

Open Traverse Closed Traverse.

Open Traverse

An open traverse originates from a point whose position may be known or unknown but terminates to a point whose position is not known. In this type of traverse, computational check is not possible to detect error or blunder in distances or directions. To minimize error, repeated observations for measurements need to be taken. In Figure 28.1 consider a traverse ABCDE that originates from the point A which may be unknown or may be defined with reference to known point P lying on the line PQ of known azimuth, but it terminates to an unknown point E. Thus, traverse ABCDE is an open traverse. An open traverse is generally used for exploratory purpose such as mine surveying. It should generally not be used in civil engineering works unless situation dictates. So no further discussion on this will be done

Closed Traverse

When a traverse originates from a known position and also terminates to known position then it is called a closed traverse. (If the origin and terminating points are the same then it is called closed-loop traverse (Figure 28.2a). This type of traverse permits an internal check on the accuracy of angular measurements, provides an indication of the consistency of measuring distances as well as angles. But detection of systematic errors in linear measurement or errors in the orientation of the traverse, is not possible. This type of traverse is recommended for minor projects). A closed traverse that originates from a known point and terminates to another known point (Figure 28.2b) is the most reliable. This type of traverse henceforth called as open looped close traverse provides computational checks allowing detection of systematic errors in both distance and direction and, therefore, preferred to all other types of traverse.

Traverse Surveying

Surveying for establishment of control points of a traverse consists of different steps of operation. These are

1. Reconnaissance 2. Selection of stations sites 3. Marking of stations 4. Field measurements 5. Computations.

Reconnaissance

During reconnaissance, extensive inspection and examination of the entire area to be surveyed is carried out in order to identify suitable location for traverse stations; to ensure inter visibility among traverse stations; to decide the method of traversing and type of instruments suitable to field condition; to collect miscellaneous information related to traverse surveying such as: (i) Transport facilities; (ii) Availability of food and

water for the survey party; (iii) Camping ground or stay arrangements; (iv) Availability of labour etc

Selection of the station sites

During selection of sites for traverse stations, following considerations are required to be taken into account:

1. Location should be such that the basic principle of surveying, i.e., working from the whole to the part gets implemented.

2. Number of stations should be as minimum as possible but the possible figures joining the stations should be well conditioned.

3. Length of the traverse lines better be as long as possible to reduce the effect of centering error.

4. The intervisibility among stations and at least between adjoining stations is required.

5. Stations may be positioned on firm and level ground.

Marking of Stations

After finalizing the sites for locations of traverse stations, their positions are marked on the ground. The station mark should be of permanent nature, so that same stations can be used in future also, if required. For that necessary monument is constructed.

Field Measurements

The field measurements in traverse surveying consist of measurement of length and direction of traverse sides and thus involve

Linear measurements and Angular measurements.

Linear Measurements

The length of traverse sides are measured by direct method using tape or EDM (Module 3) and by indirect method using tacheometer (Module 7). However, the choice of the method depends on the method of traversing employed. The checking of the linear measurements consists in taping each traverse line a second time, preferably in the reverse direction on different dates and by different parties.

Angular Measurements

The direction of traverse sides are measured by deflection angles, interior angles, azimuth etc (Module 6). Immediately after completion of a set of observation, an arithmetical check on the angular error of closure is required to be performed in the field itself to detect any blunder or excessively large error in angular measurement. For this, higher accuracy azimuth of one line or two lines, depending on the method adopted, are

required to be known or determined before starting of the traverse surveying. The method of traversing depends on the type of parameter used to define the direction of sides; such as

Deflection Angle Traverse Interior Angle Traverse Azimuth Traverse

Computations

The details of computation leading to determination of coordinates of stations is discussed in Lesson 29.

Deflection Angle Traverse

In this method of traverse surveying, successive stations are occupied and deflection angles are measured by a transit or theodolite. At each station, a backsight is taken to the previous station with vernier A set at zero. The telescope is then reversed by plunging it. A foresight to the next station is then taken by turning the instrument about the vertical axis on its upper motion, and the deflection angle is observed. The angle is recorded as right R or left L, according to whether the upper motion is turned clockwise or counter clockwise. Usually, deflection angles are observed twice, once with the telescope in face left and the other in face right condition. Immediately after completion of traverse observation, an arithmetical check on the angular error of closure should be performed to detect any blunder or excessively large error in angular measurement.

To check the angular closure, azimuths are calculated from previously known azimuth of a line. The unknown azimuth of a line is computed by adding the right deflection angle and subtracting the left deflection angle with the forward azimuth of the previous line. Thus,

W.C.B. of a traverse line = Forward W.C.B. of the preceding line ± deflection angle.

360° needs to be subtracted from computed W.C.B., if it is more than 360° and to be added, if the computed value is negative. Figure 28.3 illustrates the computations of azimuth using deflection angles to the left and right.

The error of closure in a traverse observation can be determined by two ways. These are either based on the computed azimuth or on the observed deflection angles.

In the azimuth adjustment method, the computed azimuth of the closing line is compared with its previously observed value. If both the values agrees each other, there is no error of closure otherwise it exist. To remove the error, a correction equal in magnitude but opposite in nature to the error of closure is to be distributed among the observed deflection angles.

In the deflection angle adjustment method, the algebraic sum of the deflection angles is being computed taking the right deflection angles as positive and left deflection angles as negative. The general relation to compute the error of closure by this method is as follows:

Forward azimuth of the first side of traverse (at first station) + R - L = Forward

azimuth of the last side of the traverse (at terminating station) + 360° ---- (Equation 28-1)

where R is the sum of the right deflection angles and L is the sum of the left

deflection angles. If the computed value differs from the previously observed values, an error of closure exists in the measurements of traverse. To find adjusted azimuth of the sides of the traverse, first the deflection angles get adjusted by applying correction equal to the amount of error of closure and then adjusted azimuths are computed.

A deflection angle should never be measured without double centering the instrument, because the error caused by the line of sight not being normal to the horizontal axis of the instrument may be too large to be tolerated. Moreover, the angular error of closure should not exceed the estimated standard deviation for observing an angle from a single setup times the square root of the number of instrument stations. In practice, this estimated standard deviation in angular measurement is usually taken equal to 0.5 to 1.0 times the least count of the instrument used in measuring the angles. If the error exceeds the permissible limit, then observation should be repeated till permissible limit has been achieved.

Deflection angle method of traversing is being generally used for open looped closed traverse. It is most useful for the location survey of linear engineering works such as highways, railways, canals and pipelines etc.

Ex28-1: Determine the azimuth of the sides of the open looped close traverse (Figure 28.4) by azimuth correction method.

Solution :

Figure 28.4 shows a deflection-angled closed traverse. It originates at point A lying on the line PQ having azimuth of 47° 53' 20" (determined from a previous survey). The traverse closes on point F lying on a line XY whose azimuth is 319° 55' 20" (also determined from a previous survey). The deflection angle observed at different stations are shown in columm (2) of Table 28.1 of a azimuths of the sides of traverse (computed using Eq 28-1) are as shown in column (3) of Table 28.1. It is found that the calculated azimuth of XY (319° 52' 20") fails to agree with its previously observed value (319° 55' 20"). Thus, there is an error of angular closure by - 3' 00" for the traverse and thus needs a corrction of + 3' 00". Since six deflection angles are measured, the correction to each angle is 30". The azimuth of AB receives a 30" correction [column (4) of Table 28.1], since this azimuth has been obtained by considering only one measured angle; azimuth of BC receives a 1' 00" correction, since this azimuth has been obtained by considering two measured angles and so on. The correction to the last azimuth is 6 x 30" = 3' 00", since this azimuth has been obtained by using all six measured deflection

angles. Finally, the adjusted value of azimuth has been found (column 5) by making correction (columm 4) to computed value (columm 3).

Table 28.1 Calculation of Adjusted of Azimuths of a Closed Traverse from Deflection Angles (Figure 28.4)

Line (1)

Deflection Angle (2) Computed Azimuth from

observed angles (3)

Correction (4)

Adjusted

Azimuth (5) At (a) Nature

(b) Value ©

PQ 47° 53' 20" a priori

observed 47° 53' 20"

A R (+) 70° 21' 40"

AB 118° 14' 00" +0' 30" 118° 14' 30"

B L(-) 67° 48' 40"

BC 50° 25' 20" +1' 00" 50° 26' 20"

C R (+) 40° 15' 20"

CD 90° 40' 40" +1' 30" 90° 42' 10"

D R (+) 114°46' 00"

DE 205° 26' 40" +2' 00" 205° 28' 40"

E L(-) 131°58' 00"

EF 73° 28' 40" +2' 30" 73° 31' 10"

F L(-) 113°36 20"

XY - 40° 07' 40" +3' 00" - 40° 04' 40"

XY Computed 319° 52' 20" +3' 00" 319° 55' 20"

XY a priori

observed 319° 55' 20"

Angular error of closure = - 3' 00"

Example 28-2: Determine the azimuth of the sides of a open looped closed traverse (Figure 28.5) by deflection angle correction method.

Solution :

Figure 28.5 illustrates a closed loop traverse with deflection angles. This traverse originates and closes on traverse station A. Since, the initial and terminating station is the same, if there is no error of closure, the algebraic sum of the deflection angles should be equal to 360°. Column (1c) of Table 28.2 shows the algebraic sum of the observed values of the deflection angles as 360° 01' 40". Thus, there is an error of closure error of +1' 40". To adjust the observed angles, this error is distributed equally (20") among all the five deflection angles, assuming all angles are of equal weight. Since the sum of right deflection angles is more than the sum of left deflection angles and the error of closure is positive, so 20" is subtracted from each right deflection angle and 20" is added to the left deflection angle. Thus, the adjusted value of deflection angles are determined [Column (3) of Table 28.2].

To find the direction of the traverse lines, adjusted value of azimuth of lines are computed using the adjusted deflection angle starting from the previously observed value of azimuth of the line AB [Column (3) of Table 28.3]. Note that the observed azimuth of the starting side AB agrees with its computed value, since the angles had been adjusted before azimuth computation.

Table 28.2 Adjustment of Deflection Angles (Figure 28.5)

Deflection Angle (1) Correction for deflection angle (2)

Adjusted

Deflection Angle (3) At (a) Nature (b) Value ©

A R (+) 98° 31' 20" - 20" 98° 31' 00"

B R(+) 73° 48' 40" - 20" 73° 48' 20"

C R (+) 120° 15' 20" - 20" 120° 15' 00"

D L (-) 74° 16' 20" + 20" 74° 16' 40"

E R(+) 141° 42' 40" - 20" 141° 42' 20"

R = 434° 18' 00" R = 434° 16' 40"

L = 74° 16' 20" L = 74° 16' 40"

Difference = 360° 01' 40" R - L = 360° 00' 00"

(Check)

Error in Closure = +1' 40"

Table 28.3 Computation of Azimuths Using Adjusted Deflection Angles (Figure 28.5)

Traverse side (1)

Adjusted Deflection Angle (2)

Computed Azimuth (3)

At station

(a) Nature (b) Value ©

AB 62° 00' 00"

(a priori azimuth)

B R(+) 73° 48' 20"

BC 135° 48' 20"

C R (+) 120° 15' 00"

CD 256° 03' 20"

D L (-) 74° 16' 40"

DE 181° 46' 40"

E R(+) 141° 42' 20"

EA 323° 29' 00"

A R (+) 98° 31' 00"

AB 422° 00' 00"

AB 62° 00' 00"

Interior Angle Traverse

The field operations in the interior angle method of traversing consist of occupation of the successive stations and a transit or theodolite is being used to measure horizontal angle. At each station the vernier is set at zero, and a backsight to the preceding station is taken. The instrument is then turned on its upper motion until the advance station is sighted and the interior angle is observed. All interior angles is generally observed twice, once with telescope direct and other with the telescope reversed. Immediately after completion of observation, an arithmetical check on the angular error of closure should be performed to detect any blunder or excessively large error in angular measurement

Figure 28.6 shows an interior-angle closed traverse that originates at point A lying on the line PQ having a known azimuth of 42° 00' 00" as determined from a previous survey. In this traverse, point A was occupied first and an angle was observed between line PQ and traverse line AB. The angle is observed to be 18° 00' 00". After the interior angle at A was observed, points B, C, D and E were occupied and interior angles was observed by method of repetition (at least twice) at each traverse station. Directions for the traverse sides are reckoned using the angle QAB turned from the line of known azimuth PQ to line AB. Thus, the azimuth of line AB is 42° 00'00" + 18° 00' 00"= 62° 00' 00".

To check the angular closure, azimuths are calculated from previously known azimuth of a line. The azimuth of each succeeding traverse line is computed by adding the

clockwise interior angle or subtracting the anti-clockwise interior angle to the back azimuth of the preceding line. Thus,

W.C.B. of a traverse line = Back W.C.B. of the preceding line ± included angle ------------

--------- (Equation 28.2)

where + for clockwise interior angle and - for anti-clockwise interior angle. Further, 360° needs to be subtracted from computed W.C.B., if it is more than 360° and to be added, if the computed value is negative.

The azimuth of each succeeding traverse line is then calculated by using Eq 28-2 and are as shown in column (3) of Table 28.4.

In this method also there are two ways for finding the error of closure in a traverse observation. These are

Azimuth adjustment method Interior angle adjustment method

In azimuth adjustment method, the computed azimuth of the starting side is checked with its priori observed value. If both the values agrees, there is no error in the measurements for traverse. Otherwise there is error in measurement. To remove the error, a correction equal in magnitude but opposite in nature to the error of closure is distributed to the computed azimuth to find corrected azimuth.

Figure 28.7 illustrates the computation of azimuths using interior angles. It can be found that the calculated azimuth of AB (62° 01' 40") fails to agree with its previously observed value (62° 00' 00"). Thus, there is an error of angular closure by 1' 40" for the traverse. Since five interior angles are measured, the correction to each angle is - 20". The azimuth of BC receives a - 20" correction [column (4) ofTable 28.4], since this azimuth has been obtained by considering only one measured angle; azimuth of CD receives a -40" correction, since this azimuth has been obtained by considering two measured angles and so on. The correction to the last azimuth is 5 X (-20") = -1' 40", since this azimuth has been obtained by using all five measured included angles. Finally, the adjusted value of azimuth is then found [column (5) of Table 28.4].

Table 28.4 Calculation of Adjusted of Azimuths of a Closed Traverse fromInterior Angle

Line (1) Interior Angle (2)

Azimuth (3) Correction

(4)

Adjusted

At (a) Value (b) Azimuth (5)

AB 62° 00 ¢ 00" Fixed

BA 242° 00 ¢ 00"

B 106°11 ¢ 20"

BC 135° 48 ¢ 40" -20" 135° 48 ¢ 20"

CB 315° 48 ¢ 40"

C 59° 44 ¢ 40"

CD 256° 04 ¢ 00" -40" 256° 03 ¢ 20"

DC 76° 04 ¢ 00"

D 254°16 ¢ 20"

DE 181°47 ¢ 40" -1 ¢ 00" 181°46 ¢ 40"

ED 1°47 ¢ 40"

E 38°17 ¢ 20"

EA 323°30 ¢ 20" -1 ¢ 20" 323°29 ¢ 00"

AE 143°30 ¢ 20"

A 81° 28 ¢ 40"

AB 62°01 ¢ 40" -1 ¢ 40" 62° 00 ¢ 00"

AB 62° 00 ¢ 00" Fixed

Angular error in closure = 1 ¢ 40"

In interior angle adjustment method, the algebraic sum of the interior angles is being computed and needs to be (2n - 4) X 90° where n is the number of sides in the traverse. If there is no difference, no error is associated with the observation. Otherwise there is an error in observation of interior angles. The amount of error is distributed equally among all the interior angles to find their corrected values. Then, azimuth of the line are computed using the adjusted interior angles.

In any case, the sum of the interior angles of a traverse should not deviate from (2n - 4) X 90° by more than the square root of the number of instrument setups times the estimated standard deviation in observing the angles. In practice, this estimated standard deviation in angular measurement is usually taken equal to 0.5 to 1.0 times the least count of the instrument used in measuring the angles. If the misclosure is within the permissible limit, it is to be adjusted. However, if it is large, the error should be located and corrected before leaving the field. If necessary the whole work should be repeated.

In the closed loop traverse, although the calculated azimuths are internally consistent, the absolute orientation is based entirely on one angle observed between the known

azimuth line and the side of a traverse. This is a weakness in close loop method of traversing. To eliminate the weakness, another angle should be observed from some other traverse point to another independent line of known azimuth.

Example 28-3 : Determine the azimuth of a closed traverse from (Figure 28.6) Interior angles by making correction using by azimuth adjustment method.

Solution :

Table 28.4 Calculation of Adjusted of Azimuths of a Closed Traverse from Interior Angle (Figure 28.6)

Line (1) Interior Angle (2)

Azimuth (3) Correction

(4)

Adjusted

At (a) Value (b) Azimuth (5)

AB 62° 00' 00" Fixed

BA 242° 00' 00"

B 106° 11' 20"

BC 135° 48' 40" -20" 135° 48' 20"

CB 315° 48' 40"

C 59° 44' 40"

CD 256° 04' 00" -40" 256° 03' 20"

DC 76° 04' 00"

D 254° 16' 20"

DE 181° 47' 40" -1' 00" 181° 46' 40"

ED 1° 47' 40"

E 38° 17' 20"

EA 323° 30' 20" -1' 20" 323° 29' 00"

AE 143° 30' 20"

A 81° 28' 40"

AB 62° 01' 40" -1' 40" 62° 00' 00"

AB 62° 00' 00" Fixed

Angular error in closure = 1' 40"

Example 28-4: Determine the azimuth of the sides of a closed traverse (Figure 28.6) from Interior Angles by making correction using interior angle adjustment method.

Solution :

Figure 28.6 shows an interior-angle closed traverse that originates at point A lying on the line PQ having a known azimuth of 42° 00' 00" as determined from a previous survey. In this traverse, point A was occupied first and an angle was observed between line PQ and traverse line AB. The angle is observed to be 18° 00' 00". After the interior angle at A was observed, points B,C,D and E were occupied and interior angles was observed by method of repetition (at least twice) at each traverse station. The algebraic sum of the interior angles (539° 58' 20") is then computed and found to be less than (2n - 4) X 90° i.e., 540° 00' 00" [column (1b) of Table 28.5]. Thus, there is an error of closure error of - 1' 40". To adjust the observed angles, this error is distributed equally (20") among all the five angles, assuming all angles are of equal weight. Since the sum

of observed angles is less than (2n - 4) X 90°, 20" is added to each of the angles. Thus, the adjusted values of interior angles are determined [Column (3) of Table 28.5]

Table 28.5 Adjustment of Interior Angles (Figure 28.6)

Interior Angle (1) Correction (2)

Adjusted

At (a) Value (b) Interior Angles(3)

A 81° 28' 40" + 20" 81° 29' 00"

B 106° 11' 20" + 20" 106° 11' 40"

C 59° 44' 40" + 20" 59° 45' 00"

D 254° 16' 20" + 20" 254° 16 ' 40"

E 38° 17' 20" + 20" 38° 17' 40"

S 539° 58' 20" 540° 00' 00"

(2.n-4)X90° 540° 00' 00"

Error in Closure = -1' 40"

Directions for the traverse sides are reckoned using the angle QAB turned from the line of known azimuth PQ to line AB Thus, the azimuth of line AB is 42° 00' 00" + 18° 00' 00"= 62° 00' 00". The azimuths of each succeeding traverse line are then calculated by using the adjusted interior angle in Eq 28-2 and are as shown in column (3) of Table 28.6. It has been found that the observed azimuth of the starting side AB agrees with its computed value.

Table 28.6 Calculation of Azimuths from Adjusted Interior Angle (Figure 28.6)

Line (1) Interior Angle (2)

Azimuth (3) At (a) Value (b)

AB 62° 00' 00"

BA 242° 00' 00"

B 106° 11' 40"

BC 135° 48' 20"

CB 315° 48' 20"

C 59° 45' 00"

CD 256° 03' 20"

DC 76° 03' 20"

D 254° 16' 40"

DE 181° 46' 40"

ED 1° 46' 40"

E 38° 17' 40"

EA 323° 29' 00"

AE 143° 29' 00"

A 81° 29' 00"

AB 62° 00' 00"

Traverse Computations

Traverse surveying in the field yields observed angles or directions and length of the traverse sides. Thus, these parameters are used in traverse computations which are performed in a plane rectangular coordinate system. The traverse computations involve calculation of consecutive coordinates of traverse stations, checking in error of closure, determination of the amount of closing error, adjustment of traverse by balancing of consecutive coordinates, calculation of independent coordinates and determination of corrected distances and azimuth of sides.

Consecutive Coordinates

Consecutive coordinates of a station is designated by its departure and latitude from its previous station as origin. Departure of a traverse side is defined as its component perpendicular to the reference meridian and the component of the traverse side along

or parallel to the reference meridian is known as latitude. Thus, if l and are the length and azimuth of a traverse side, then departure and latitude of the side are given by l

sin and l cosine respectively. The algebraic sign of the departure and latitude of a traverse side depends on its azimuth value thus on the sign of the trigonometric parameters associated with these. In Figure 29.1, the sides AB has length and bearing as 442.56m and 34° 52' respectively. Thus, its departure and latitude are 253.0m (442.56m sine 34° 52') and 363.11m (442.56 cosine 34° 52') respectively. Similarly, the departure and latitude of the sides BC (478.23m, 112°) is 441.93m and 182.75m respectively

Ex29-1 : A closed-loop traverse was run among stations A, B, C and D having following observation

Sides Length ( li ) m Azimuth ( i )

AB 372.222 0° 42'

BC 164.988 94° 42'

CD 242.438 183° 04'

DA 197.145 232° 51'

Find the consecutive coordinates of the stations.

Solution : The consecutive coordinates of the stations A, B, C and D are as calculated in Table 29.1.

Table 29.1 Consecutive Coordinates of stations of a closed-loop traverse

Sides

Length ( li )

m

Azimuth

( i )

Stations

Consecutive coordinates of stations, (m)

Departure ( Xi ),

li sin i

Latitude ( Yi ),

li cos i

East (+) West (-) North

(+) South(-)

AB 372.222 0° 42'

B 4.547 372.194

BC 164.988 94° 42'

C 164.576 -11.653

CD 242.438 183° 04'

D -12.970 -242.091

DA 197.145 232° 51'

A -157.136 -119.056

Thus, consecutive coordinates of station are A (-157.136m, -119.056m), B (4.547m, 372.194m), C (164.576m, -11.653m) and D (-12.970m, -242.091m)

Independent Coordinates

The departure and latitude of a station with reference to an origin are known as independent coordinates. The independent coorinate of at least one of the stations with reference to the considered origin is required to be known a priori. Thus, if the independent coordinates of any station, say i, is known to be (Xi, Yi), the independent coordinates of another station say j, (Xj , Yj) can be determined by using the following relations:

Xj = Xi + xij ------------ (Equation 29.1)

Yj = Yi + yij ------------ (Equation 29.2)

Where ( xij, yij) are the departure and latitude of the side ij.

Ex29-2 : Calculate the Independent Coordinates for the traverse defined in Example 29.1 . Given that the independent Coordinates of the stations A as ( 7200.054 , 7640.842).

Solution :

Table 29.2 Computation of Independent Coordinates of a closed-loop traverse

Stations

(1)

Sides

(2)

Length ( li ) m

(3)

Azimuth

(i)

(4)

Consecutive Coordinates, (m)

(5)

Independent Coordinates (m)

(6)

Departure

(a)

Latitude

(b)

X

(a)

Y

(b)

A 7200.054 7640.842

AB 372.222 0° 42' 4.547 372.194

B 7204.601 8013.036

BC 164.988 94° 42' 164.576 -11.653

C 7369.177 8001.383

CD 242.438 183° 04' -12.970 -242.091

D 7356.207 7759.292

DA 197.145 232° 51' -157.136 -119.056

A 7199.071 7640.236

Error of Closure

A check in the accuracy of traverse observation with length and azimuth considered together can be done by checking the error of closure of the traverse. The error of closure considering both length and azimuth together can be done by using consecutive coordinates of the station points of the traverse. Since, in any closed traverse, the algebraic sum of the departure should be equal to the difference between the X coordinates of the initial and terminating stations of the traverse. Similarly, the algebraic sum of the latitudes should be equal to the difference between the Y coordinates of the initial and terminating stations. Thus, for a n-stations traverse,

Xn - X1 = departures -------------- (Equation 29.3)

Yn - Y1 = latitudes -------------- (Equation 29.4)

( X1 ,Y1 ) and ( Xn , Yn ) are the independent coordinates of the initial and terminating stations of the traverse.

In a closed-loop traverse, since the initial and terminating station of the traverse is the same, Eq (29.3) and Eq (29.4) reduces to

departures = 0 -------------- (Equation 29.5)

latitudes = 0 -------------- (Equation 2963)

Ex29-3 : Check the error of closure for the traverse defined in Example 29.1

Solution: Referring Table 29.2, from Columm 6

Thus, the observations for traverse sides are associated with errors.

Closing Error

The amounts by which Eq 29.3 and Eq 29.4 fail to be satisfied are called the errors in closure in position or simply the closures of a traverse (Figure 29.2). The closing correction in terms of departure and latitude dD and dL are thus can be found as

dD= (Xn - X1) + departures --------------- Eq (29.7)

dL= (Yn - Y1) + latitudes --------------- Eq (29.8)

and the direction of the error is given by

--------------- Eq (29.9)

The sign of dD and dL finally defines the value of azimuth of error of clo

Ex29-4 : Compute the error of closure for the traverse defined in Example 29.1

Solution : Refering table 29.2 and using equations 29.7 and 29.8 respectively, we get

Error in departure, dD= departures = - 0.983

Error in latitude, dL= latitudes = - 0.606

and the direction of the error is given by

= 58° 20' 49".16

Since the both the errors in departure and latitude are with negative sign thus, it indicates the error is in third quadrant and 180° is to be added to find the azimuth of the error of closure. Thus, the azimuth of the error of closure is (180° + 58° 20' 49".16) = 238° 20' 49".16

Adjustment of Traverse

Traverse adjustment is required to provide a mathematically closed figure by making closure in latitudes as well as closed departures. Methods for adjustment of traverse are classified into two types: approximate methods and rigorous methods.

The approximate methods for traverse adjustment are based on the conditions prevailing in the combinations of linear and angular precision in the observations. On these basis, conditions are divided into three types. (1) Precision in angular measurement is higher than in linear measurement; (2) Precision in angular as well as in linear measurements are same; (3) Precision in linear measurement is more than that in angular measurement.

The method of least square provides the most rigorous method of traverse adjustment, which allows variation in precision in the observations, minimizes random variations in the observations, provides the best estimates for positions of all traverse stations, and yields statistics relative to the accuracies of adjusted observations and positions. This method does require more of a computational effort than the approximate adjustment. But, the results are well worth effort. However, the method is beyond the scope of discussion of this course and further discussion on adjustment of traverse are based on approximate methods.

Approximate Methods for Adjustment of Traverse

In these methods, corrections are applied to the individual departures and latitudes to satisfy equations 29.3 and 29.4 for the considered traverse. The operations involved are known as adjustment or balancing a traverse. The approximate methods usually adopted for balancing a traverse are the

Transit method Bowditch's method.

Transit Method

This method is developed for balancing a traverse in which angles are measured with a higher degree of precision than the lengths of the sides. It is based on the assumption that the error in departure (or latitude) of a traverse side is proportional to its departure (or latitude). Thus, according to the transit rule, the corrections to the departure (or

latitude) of a traverse side can be calculated by using

where

dij = Correction in departure of a traverse side ij

lij = Correction in latitude of a traverse side ij

dD = total error in departure (or Algebric sum of the departures of all sides of the traverse)

dL = total error in latitude (or Algebric sum of the latitudes of all sides of the traverse)

dij = departure of the traverse side ij

lij = latitude of the traverse side ij

D = Arithmetic sum of the departures of all the sides of the traverse

L = Arithmetic sum of the latitudes of all the sides of the traverse

The corrections in transit rule do not take into consideration of the algebraic nature of the departure (or latitude) of traverse sides. This made the transit rule valid when the traverse lines are parallel with the grid system used for the traverse computations. So, further discussion regarding transit rule is being restricted in this course work.

Bowditch's Method

The Bowditch's method is used when both the linear and angular measurements are compatible to each other, i.e., they are of equal precision. The corrections may be applied either analytically or may be carried out graphically. This method of balancing of traverse is widely prevalent and most commonly used

Bowditch Analytical Method

Analytical method of correction by Bowditch's rule can be applied either to its coordinates directly or to the departure (or latitude) of a traverse side.

(a) The corrections to the coordinates can be calculated by using

where

Xi = Correction to Xi coordinates of a station i;

Yi = Correction to Yi coordinates of a station i;

dX = total closure correction of the traverse in departure;

dY = total closure correction of the traverse in latitude;

Li = distance from the initial station to the station i, measured along the sides of the traverse;

L = perimeter of the traverse

Ex29-5 : Compute the adjusted Independent Coordinates for the traverse defined in Example 29.1 by applying Bowditch's method of direct correction to coordinates having error of closure.

Solution :

Table 29.3 Computation of Independent Coordinates by direct correction to coordinates traverse

Stations

(1)

Sides

(2)

Distance (m)

(3)

Uncorrected Independent

Coordinates (m)

(4)

Bowditch's Corrections

(m)

(5)

Adjusted Independent

Coordinates (m)

(6)

X Y Xi Yi X Y

A 7200.054 7640.842 7200.054 7640.842

B 372.222 7204.601 8013.036 0.137 0.085 7204.738 8013.121

C 537.210 7369.177 8001.383 0.335 0.206 7369.527 8001.588

D 779.648 7356.207 7759.292 0.623 0.348 7356.830 7759.640

A 976.793 7199.071 7640.236 0.983 .606 7200.054 7640.842

2665.873 dX = - 0.983

dY = -0.606

Bowditch Analytical Method (contd)

(b) The corrections to the departure (or latitude) of a traverse side, starting from station I and

terminating to station j, can be calculated by using

Where

dij = Correction in departure of a traverse side ij;

lij = Correction in latitude of a traverse side ij;

dD = total error in departure of the traverse;

dL = total error in latitude of the traverse;

dij = length of the traverse side ij;

L = perimeter of the traverse ;

Ex29-6 : Compute the Coordinates for the traverse defined in Example 29.1 by applying correction to consecutive coordinates by Bowditch's method.

Solution :

Table 29.5 Adjustment of Coordinates of a closed-loop traverse using Bowditch's Rule

Sides

Length

( dij )m

Azimuth

( i )

Consecutive coordinates, (m)

Bowditch's Correction (m) Adjusted

Consecutive coordinates, m

Departure ( Di)

Latitude ( Li )

Departure, dij

Latitude, l

ij

Departure

Latitude

AB 372.22

2 0° 42' 4.547

372.194

0.375 0.231 4.922 372.425

BC 164.98

8 94° 42' 164.576

-11.653

0.166 0.102 164.742 -11.551

CD 242.43

8 183° 04'

-12.970 -

242.091

0.244 0.151 -12.726 -

241.94

DA 197.14

5 232° 51'

-157.136

-119.056

0.198 0.122 -

156.938

-118.934

L = 976.79

3

dD = -0.983

dL = -0.606

= 0.983 = 0.606 =

0.000

=

0.000

Bowditch Graphical Method

For rough surveys or traverse of small area, adjustment can also be carried out graphically. In this method of balancing, the locations and thus the coordinates of the stations are adjusted directly. Thus, the amount of correction at any station is proportional to its distance from the initial station .

Let Po Qo Ro So To P' is the graphical plot of a closed-loop traverse PQRSTP. The observed length and direction of traverse sides are such that it fails to get balanced and is depicted in its graphical presentation by an amount Po P'. Thus, the closing error of the traverse is Po P' (Figure 29.2). The error Po P' is to be distributed to all the sides of the traverse in such a way that the traverse gets closed i.e., P' gets coincides with Po in its plot. This is carried out by shifting the positions of the station graphically. In order to obtain the length and direction of shifting of the plotted position of stations, first a straight line is required to be drawn, at some scale, representing the perimeter of the plotted traverse. In this case, a horizontal line Po P' is drawn [Figure 29.3 (ii)]. Mark the traverse stations on this line such as Qo, Ro, So and To in such a way that distance between them represent the length of the traverse sides at the chosen scale. At the terminating end of the line i.e., at P', a line P' P a is drawn parallel to the correction for closure and length equal to the amount of error as depicted in the plot of traverse. Now, join Po to Pa and draw lines parallel to P' Pa at points Qo, Ro, So and To. The length and direction of Qo Qa, Ro Ra, So Sa and To Ta represent the length and direction of errors at Qo, Ro, So and To respectively. So, shifting equal to Qo Qa , Ro Ra, So Sa and To Ta and in the same direction are applied as correction to the positions of stations Qo, Ro, So and To respectively. These shifting provide the corrected positions of the stations as to Qa, Ra, Sa,Ta and Pa. Joining these corrected positions of the stations provide the adjusted traverse Pa Qa Ra, Sa Ta [Figure 29.3 (i)].

Determination of Corrected Distances and Azimuth

After the adjustment of traverse, the last step of traverse computation is to calculate adjusted (corrected) length and direction of the traverse sides. These can be obtained from adjusted values of station coordinates. Either of consecutive or independent coordinates can be used for these computations.

The distance between stations i and j is obtained from

and the azimuth of the line ij from the North is

In these computations, due consideration to the algebraic sign is to be given. The

values thus found would be used in the plan or map for which survey is being conducted.

Ex 29-7 : Compute the adjusted length and azimuth of the traverse sides defined in Example 29.1

Solution :

Table 29.5 - Computation of Adjusted length and azimuth of the sides of a closed-loop traverse

Sides

Length

( li ) m

Azimut

( i )

Corrected Consecutive coordinates, (m)

Corrected

Departure Latitude Length ( li ) m Azimuth ( i )

AB 372.222 0° 42' 4.922 372.425 372.457 0° 45' 25".85

BC 164.988 94° 42' 164.742 - 11.551 165.146 94° 00' 38".77

CD 242.438 183° 04' - 12.726 - 241.94 242.274 183° 00' 39".5

DA 197.145 232° 51' - 156.938 - 118.934 196.913 232° 50' 37".2

Azimuth Traverse

In this method, at each traverse station the back azimuth of the preceding line and the azimuth of the forward line are measured using a transit. The reference meridian may be either true or assumed. In this method, successive stations are occupied, beginning with the line of known or assumed azimuth. At each station the transit is "oriented" by setting the A vernier or horizontal circle index to read the back azimuth (forward

azimuth ± 180°) of the preceding line and then back sighting to the preceding traverse

station. The instrument is then turned on the upper motion, and a foresight on the following traverse station is secured. The reading indicated by the horizontal circle index on the clockwise circle is the azimuth of the forward line. Any angular error of closure of a traverse becomes evident by the difference between initial and final observations taken along the first line. The method is used extensively for topographic and other surveys where a large number of details are located by angular and linear measurements from the traverse stations.

Figure 28.8 shows a closed-loop traverse run by the azimuth method. The traverse is begun at

station A with a backsight along line AB for which the azimuth is known to be 62°00'00 " by

astronomical observation or using GNSS receiver. With the instrument set over station A, (the A

vernier is set to 62° 00' 00" clockwise) and a backsight is made on station B using the lower

motion. The instrument is then turned on the upper motion and a foresight is taken on station E.

Since zero is oriented toward true north (Figure 28.8), the clockwise circle read at the A vernier

(143° 30' 20") is the azimuth of line AE. When the instrument is moved to point B, the back

azimuth of line AB is computed by adding 180° to the forward azimuth (62° 00' 00" + 180° =

242° 00' 00"), and this value is set on the clockwise circle before a backsight, using the lower

motion, is taken on station A. The circle is oriented with zero toward true north and when the

instrument is turned on the upper motion for a foresight on station C, the angle observed on the

circle at the A vernier (135° 48' 40") is the azimuth of line BC from true north. This operation is

repeated for each successive traverse station. When the final setup is made over station A, the

azimuth observed on the foresight along line AB should equal the azimuth of line AE plus180°.

Any deviation between this last observation and the fixed direction of the closing line is the

angular closure. The error of closure in azimuth may be adjusted as discussed in interior angle

method of traversing.

Omitted Measurments

If the length and/or bearing of any side of a closed traverse gets omitted, that can be computed analytically by applying.

departures = (Xn - X1) --------------- Eq (31.1)

latitudes = (Yn - Y1) ------------------- Eq (31.2)

where (X1 ,Y1) and (Xn ,Yn) are the independent coordinates of initial and terminating control stations. Thus, in any traverse, maximum two omitted parameters can be computed from two available equations. However, no check on the accuracy of the field work can be done nor can the traverse be balanced as errors, if any, present in the survey work get propagated into the computed values of the omitted quantities. So, computation of omitted measurement, if any, is done for during computation of traverses of lower order

Type of Omitted Measurments

The number of omitted measurement may be one i.e., the length or bearing of any side of a traverse. The omitted element may be computed by using either of the Eq 31.1 or 31.2.

If the number of omitted measurements is two, the unknown elements may be computed using both the Eq 31.1 and Eq 31.2. But, when either both length and azimuth of the same side or lengths of any two sides (adjacent or non-adjacent) are omitted, their computation can be done easily adopting this same method.

But, if bearing of two adjacent sides or the length of one side and bearing of adjacent side are omitted then the above method for solution gets complicated. In these cases, trigonometric methods for solution better be applied to compute the omitted measurements conveniently. Example 31-1 provides a method of computation for omitted bearings of two sides and Example 31-2 for omitted the length of one side and bearing of other side.

Also, the omitted measurements may occur on sides that are not adjacent. In these cases, the sides of the traverse are shifted in parallel directions (thus departure and latitude of the sides get maintained) in such a way the sides having unknown elements are placed adjacent and then compute the unknown elements as outlined above. In Example 31-2 the length of one side and bearing of non- adjacent side are omitted. Before applying the method of solution for adjacent sides, the original traverse get reoriented and detail may be found in the solution of the example.

Examples

Ex31-1 Calculate the missing data of the closed loop traverse ABCD:

Line Length

(m) W.C.B.

AB 550 60°

BC 1200 ?

CD 880 ?

DA 1050 310°

Ex31-2 In a traverse ABCDE, the length of the sides DE and EA could not be measured.

Line AB BC CD DE EA

Length (m) 480 626 470 ? ?

WCB (m) 99° 00' 31° 05' 301° 20' 235° 00' 153° 25'

Determine the missing data.

Solution :

Let us consider the closed traverse ABCD. Let the line DA is having length and bearing

(WCB) be l and respectively.

Line Length (m) WCB Latitude (m) Departure (m)

AB 480 99° 00' - 75.088 474.090

BC 625 31° 05' 535.261 322.678

CD 470 301° 20' 244.408 -401.454

DA l l cos l sin

Thus in closed traverse ABCD,

Latitude = 0 or, l cos + 704.581 = 0 or, l cos = - 704.581

Departure = 0 or, l sin + 395.314 or l sin = - 395.314

Plotting a Traverse Survey

The accuracy of a survey depends upon the accuracy with which its control points are plotted. The traverse stations, which are the control points, can be plotted either by the angle and distance method or by the coordinate method.

Angle and Distance method

In this method, distances between the stations are laid off to scale and angles or bearings are plotted. Thus, the traverse stations are plotted with reference to the previous station. Therefore, the accuracy of the location of each station on the plan evidently depends upon the accuracy with which previous points are plotted. Even if there is a slight error in plotting, the error gets accumulated and the position of the last station may be displaced to a considerable distance from its true position. So, this method is used for plotting of small traverse or for traverse of low accuracy

Coordinate Method

The independent coordinates of horizontal control points are required in the state plane coordinate system for preparation of topographic map. So, data from field surveying are used to calculate and adjusted in the office, correct within specified limits, and made available for direct plotting by the coordinate method. This plotting can be performed with high precision, since all measurements are linear distances measured from orthogonal axes.

In order to plot by the coordinate method, a series of grid lines are drawn on the base sheet for the map using a graticule (Figure 32.1). These grid lines are sets of X and Y axes of orthogonal coordinate systems. In the state plane coordinate system (and in most other coordinate systems), the Y grid axis is aligned toward North. These grid lines are spaced at some regular, uniform interval say, 50, 100, 200, 500 or 1000 m suitable for the scale of the map being compiled. When drawing the grid lines, extreme care must be exercised to ensure that the lines are straight and of uniform weight and that the X grid lines are perpendicular to the Y grid lines. Each X and Y grid line is labeled at the edges of the map sheet with its coordinate value. Grid lines should be drawn as quickly as possible on stable-base material under uniform temperature conditions so that all measurements are consistent. The positions of the control points are then plotted by laying off the differences between the coordinates for the point and the coordinates of a pair of intersecting grid lines close to the control point (Figure 32.2).

The grid lines are retained on the finished drawing or map or at the very least, the grid tick intersections are left. These grid lines or grid intersection points are invaluable for scaling the coordinates of points on the map. They can also be useful to evaluate dimensional changes in the map-base material and allow more realistic scaling of distances and locations from a given map sheet.

The advantages of plotting horizontal control by rectangular coordinates are as follows:

(1) adjusted coordinates are used to plot the locations so it is known that the data are correct to within the closure of the control survey network;

(2) each point is plotted independently so that there is no accumulation of error;

(3) high precision and uniform accuracy can be maintained on large map sheets and when multiple map sheets are required; and

(4) the method is very adaptable to automatic plotting routines and is compatible with modern data-storage procedure.

Example

Ex.32-1 Figure 32.2 shows a set of coordinate grid lines for plotting the stations of a traverse. For a small traverse such as this one, it is desirable to locate the grid lines so as to center the traverse lies approximately at the centre on the map sheet. This is done by examining the adjusted coordinates, noting the maximum and minimum values for X and Y coordinates, and then calculating the maximum changes in X and Y coordinates. These distances reduced to map scale, subtracted from the dimensions of the map sheet and divided by 2, yield the distances that should be measured from the edges of the map sheet to the control points having maximum and minimum X and Y coordinates.

For the example, let us consider the following data are available:

Map scale = 1 : 10,000

Map sheet dimensions are 282 by 218 mm

Grid interval = 200 m on the ground or 20 mm at map scale

Maximum change in X = 1297.84 m on the ground or 129.8 mm at map scale

Maximum change in Y = 422.34 m on the ground or 42.2 mm at map scale

Westernmost station is 1, for which X = 4321.40 m

Northernmost station is 5, for which Y = 6405.27 m

With a grid interval of 200 m and in view of the values of coordinates for the westernmost and northernmost stations, it is necessary to determine the locations for the western most Y grid line at X = 4200 m and for the northernmost X grid line at Y = 6400 m. It is possible to work either at map scale or in metres on the ground. Assume that the best metric scale available for the work is. graduated only in millimetres, so that working at map scale is more convenient. Since the change in X exceeds that in Y, the X axis is aligned with the long dimension of the map sheet. The location of the Y grid line for X = 4200 m is determined from the left or western edge of the sheet. Since the maximum change in X coordinates is 129.8 mm at map scale, the distance from the left edge of the sheet to the grid line is (282 -129.8)/2 -(4321.4 -4200)/10 = 64.0 mm at

map scale. This distance is measured from the left edge of the map sheet and the Y grid line for X = 4200 m is drawn parallel to ,the edge of the sheet. The location of the X grid line for Y = 6400 m is determined from the top of the map sheet next. The maximum change in Y is 42.2 mm at map scale, so that the distance from the top of the sheet to the X grid line for Y = 6400 m is (218 -42.2)/2 -(6405.3 -6400)/10 = 88.4 mm at map scale. This distance is then scaled from the top of the map sheet and the X grid line for Y = 6400 m is drawn through the point and perpendicular to the Y grid line. The balance of the grid lines are then drawn parallel to these two controlling lines at intervals of 20 mm at map scale or 200 m on the ground. After the grid lines are labeled, the position of station 1 is plotted by subtracting X = 4200 and Y = 6200 from the X and Y coordinates for 1 to yield increments in the X and Y directions of 121.4 m and 40.6 m, respectively. These distances are scaled along the grid lines and perpendiculars are erected, the intersection of which defines the control point. The remaining stations are plotted in similar fashion. When all points have been plotted, the distances between points should be scaled as a check against the adjusted distances between the stations of the traverse. In this way the accuracy of the plotting is verified analytically Further, verification of the plotted positions are done during plotting of feild details.