The Traverse

Civil Engineering StudentsYear (1)Second semester Phase II Dr. Kamal M. Ahmed

Surveying Measurements Surveyors, regardless of how complicated the

technology, measure two quantities: angle and distances.

They do two things: map or set-out Angles are measured in horizontal or vertical planes

only to produce horizontal angles and vertical angles. Distances are measured in the horizontal, the vertical,

or sloped directions. Our calculations are usually in a horizontal or a

vertical plane for simplicity, sloped values can be calculated if needed.

For example: maps are horizontal projections of data, distances are horizontal on a map and so are the angles.

Assume that you are given the horizontal coordinates X (E), and Y (N) of two points A and B: (20,20) and (30, 40). If you measure the horizontal angle CBA and the horizontal distance AC, found them to be: 110 and 15m, then the coordinates of C can easily be computed, here is how: Calculate the azimuth of AB, then BC Calculate (E, N) for BC Calculate (E, N) for C

A

B

C

But, if you were given a slope distance or a slope angle, you wont be able to compute the location (Coordinates) of C.

What we did was to map point C, we found out its coordinates, now you plot it on a piece of paper, a map is a large number of points such as C, a building is four points, and so on.

Now, if point C was a column and we wanted to set it out , then we know the coordinates of C from the map:

Calculate the angle ABC and the length of BC Setup the instrument ,such as a theodolite, on B,

aim at A Rotate the instrument the angle ABC, measure a distance

BC, mark the point. You set out a point, then you can set out a project. In both cases, you need two known points such as A

and B to map or set out point C We call precisely known points such as A and B

control points In horizontal, we do a traverse to construct

new control points based on given points. You need at least two points given in horizontal ( or

one and direction) and one in vertical to begin your project

Traversing Definition:

A traverse is a series of consecutive lines whose lengths and directions have been measured.

Why? The purpose of establishing a traverse is to extend

the horizontal control. A survey usually begins with one given vertical control and two ( or one and direction) given in horizontal

You need more than two points to control the project, have enough known points to map any point, and set-out any object any where in a large project.

Grass

A

C

D

E

B

Procedure

F

You need to construct new control points points of known precise coordinates such as C, D, and E to measure from.

You do that with a traverse

Assume that you wanted to map calculate coordinates of the building, trees, and the fence in the drawing, you are given points A and B only, cannot measure angle and distance to corner F or the trees!!

1. Walk around and decide which are the best locations to have new control points

2. Construct the points, nails on asphalt, concrete and bolts, etc.

3. Measure all the angles and all the lengths of the traverse

4. Check if the angles and lengths are accepted

5. If rejected, re-do the work6. If accepted, adjust the errors

and compute coordinates.

Grass

A

C

D

E

B

Procedure

Coordinate Computations Assume that we were given a site to map, and

the coordinates of one point (A), and the azimuth of the line (AB), we need more known (control stations)

We marked three more points around the site, the four points make rectangle (or a square).

We then measured all the internal angles and the length of all the sides (lines).

Using the given azimuth of AB and all angles, we computed the azimuth of all the sides, we get the following table:

0.100.30399.80Sum

350.10200.30A

0.00- 99.7027000'0099.70DA

350.10300.00D

-100.000.0018000'00"100.00CD

450.10300.00C

0.00100.0090 00' 00"100.00BC

450.10200.00 B

100.100.000 00' 00''100.10AB

350.00200.00A

NEN =

d cos( )E =d sin( )

Azimuth()LengthLinePoint

Coordinate Computations

Note that the coordinates of A when computed at the bottom of the table, are not the same as given coordinates.

Also note the relationship between that error and the sum of Northingsand Eastings.

How do you explain that?

Assume that the traverse was a perfect square of 100 m side length and oriented towards the north, what you notice in our measurements, and how can you relate that to the error in A?

Questions

AB c

D

E=0.3

N=0.1

Closing error = 0.32mNotice that if the corrections are ignored, the value of the errors willAppear when you re-compute the coordinates of the first point (A).

The concept of Linear Closing Error

Now let us assume the same perfect square of 90angles and 100 m sides. When measured all lengths were correct, while the angle at B was in error by 10when measured, the surveyor reported 100. What happens?

The concept of Angular Closing Error

Here is the perfect traverse that we are trying to measure:

A

B c

D

Error in angles OR error in distances will result in a closing error: last point will not be at the first point.

The problem is that we do not know where the errors are and how much each error?

Measurements are never exact, we always assume that we have errors in angles and distances.

Before we learn how to compute the errors and how to adjust for them, let us learn some issues with traverse

The concept of Closing Error

Closed and Open Traverses

A closed traverse is the one that starts and ends at known points and directions, whether the shape is closed or not

A closed traverse can be a polygon {closed shape} or Link {closed geometry-open shape

Closed (polygon or link) traverses

Polygon

Link

Open TRAVERSE

A

B

X=XB-XA

Y

=

Y

B

-

Y

A

XObserved

Y

O

b

s

e

r

v

e

d

MX

M

Y

M XY

True Location Observed Location

Open Traverses are not used in engineering control applications, why?

The problem: there is no way to check the for the errors; you will have to accept whatever coordinates computed.

L1 L2 L3 L4 L5

Traverse Notations We will only cover the closed Traverse with interior

angles measured.

Traverse Stations Successive stations should be inter visible. Stations are chosen in safe, easy to access

places. Lines should be as long as possible

To reduce the number of lines Short lines will produce less accurate angles,

the traverse gets distorted as shown below.

A

BT1

T2

T3

T4

Angles should be as equal as possible and better be 30 to 150, why????

Lines should be and as equal as possible, Why? Stations must be referenced to retrieve them if lost.

We produce a descriptive card for each point

Traverse Stations

Descriptive card for a traverse point

. 3

(52.011)

( 1.8)

42.11

76.8

37.21

( ) ( 2.8)

:

.

Traversing by Interior Angles

All internal angles and all horizontal distances are measured

Each angle is measured in direct and reverse, Each angle is observed at least three times. A line of known direction should either be given or

assumed, what is a line with known direction? If the line of known direction is not a member of the

traverse, the angle to a traverse member should be measured. Why?

Closing Error In Traverses

We measure two values: angles and distances

Because of errors in both measurements, we get angle misclosure and linear misclosure ( closing error).

Both types of errors result in error in closure, we need a way to separate the error of angles from the error in distances to check and adjust them separately

Here is how the measured traverse will look:

The concept of Angle Misclosure

A

B c

D

Line AB was correct

Line BC was correct, but angle A was wrongThe rest of the lines and angles are correct A

Computations and Adjustments of Angle Misclosure

The sum of internal angles of a polygon of (n) points = (n - 2) * 180o

Angle misclosure = difference between the sum of the measured angles and the geometrically correct total for the polygon.

The misclosure is divided equally among the readings keeping in mind the measuring accuracy, and should be done at the beginning of the adjustment.

Judging The Angle Misclosure Usually the standards give an equation of the form: Max allowed angle misclosure c = k * n where (n)

is the number of points and K is a constant defined according to which standards used

If angles are accepted, correct by dividing the error equally among the angles

For example: The Federal Geodetic Control Subcommittee: 1.7, 3, 4.5, 10, and 12 for first-order, second-order class I, second-order class II, third-order class I, third-order class II

If no standards were given, then assume that:Allowed angle misclosure = 3n =

3 * theodolite accuracy * No. of anglesExample (1)For 5 points traverse, the sum of internal angles is 538 40 and the accuracy of the measured angles is

10. Do you accept the observations?.Answer Ang. Clos. Err. = 538 40 180 x (5 2)

= 538 40 540 = - 1 20 = - 80 Allowable Angle Closing Error = 3 x 10 5 = 67 Reject the observations. You have to re-observe

Example (2)

If the angles were observed to the nearest 30 in 5 points traverse. If the angular closing error was 2, correct the angles.

AnswerAllowable angular closing error = 3 x 30 5 = 201As = 120 ( 201 ) , accepted.

Correction in each angle = - 120 / 5 = - 24But the angles were observed to the nearest 30, the corrections will be : - 30, - 30, - 30 , - 30 & 0 .

AB c

D

E

N

The concept of Linear Closing Error

Assume that the traverse in reallitywas a perfect square.

Assume that there was an error in measuring the length AB only, all other lengths and angles were correct

A

- A will close at A,

- AA is the linear closing error

NE

D

C

B

A

EAB EBC

EDA ECD

+ ve+ ve

- ve- ve

If the traverse is closed, then

E = 0 and

N = 0

NE

D

C

B

A

EAB EBC

EDA ECD

+ ve+ ve

- ve- ve

If the traverse is closed, then

E = 0 and

N = 0

A

N

E

If the traverse is not closed,

Then E = Ec and N = Nc

Computations of Linear Closing Error

If he closing error is (W) thenEw = E and Nw = N,

W = length of closing error = Ew2 + Nw2Fractional Closing error = traverse precision =

W / LDirection of the error = Azimuth =

tan-1 (Ew / Nw) = tan-1 (E / N)

Adjustment of Linear Misclosure Compute and adjust the angle misclosure Compute the linear misclosure:

Compute the azimuth of a traverse side Compute the azimuth of all the sides Compute the departure and latitude of all the sides Compute the Misclosure in (E) direction =

sum of the departures. Compute the Misclosure in (N) direction =

sum of the latitudes. Compute the linear misclosure If accepted, use the Compass (Bowditch) rule to

adjust:

Compass (Bowditch) RuleCorrection in departure for AB = - ( E L ) ( LAB)Correction in latitude for AB = - ( N L ) ( LAB)Where:

L is the length of a line, and ( L) is the perimeter

Computations of Coordinates

Add the corrections to the departure or the latitude of each line to get the adjusted departure or latitude

Compute the adjusted point coordinates using the corrected departure or latitude:

Ei = E i-1 + E Ni = N i-1 + N

Check that the misclosure is zero.

Horizontal Control Accuracy Standards For Traverse (By The Federal Geodetic Control Subcommittee (FGCS))

0.80Lor,

1/5,000

0.40Lor,

1/10,000

0.20Lor,

1/20,000

0.08Lor,

1/50,000

0.04Lor,

1/100,000

Linear Closure

(after angul. adj.)

12.0n10.0n4.5n3.0n1.7nAngular Closure

IIIIIIClass

3 rd2 nd1stOrder

Example of Standards

Acceptable Relative precision 1 in 5000 & 30 n for most engineering surveys1 in 10000 & 10 n for control, for large projects1 in 20000 & 2 n for major works and monitoring for

structural deformation etc.Ex.: The angles were observed to the nearest 30 in 5

points traverse. If the angular closing error was 2, correct the angles.

AnswerAllowable angular closing error = 3 x 30 5 = 201As = 120 ( 201 ) , accepted.Correction in each angle = - 120 / 5 = - 24But the angles were observed to the nearest 30, the

corrections will be : - 30, - 30, - 30 , - 30 & 0 .

WN =- 0.72WE =+0.54P=2466.05Sum

A

388.5- 517.40306 54.1

747.02

E

202.91- 5.99358 18.5

203.00

D

- 694.27- 192.56195 30.1

720.48

C

- 153.74590.77104 35.2

610.45

B

255.88125.7226 10.0285.10

A

NE

LatitudeN

DepartureE

Latitude(WE/ L)* L

Departure(WN/L)* L

LatitudeL cos (Az)

Departure L sin (Az)

AzimuthAZ

Length Lpoint BalancedCorrection

Other Methods

There are several methods that are used to adjust or balance traverses;

1. Arbitrary method

2. Transit rule

3. Least-Squares method

Traverse Area

A

B

CD

E

Traverse area = 1 { Ei (Ni+1 - Ni-1)}2

Multiply the X coordinate of each point by the difference in Y between the following and the preceding points, half the sum is the areaThe formula will work for traverses lettered in a clockwise direction, but it will give a correct area with a negative sign.The formula should work if you switch the N and the E.