Traver

sin

Traver

sin

gg

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.

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

N (mag)

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

N (mag)

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:

Point Line LengthAzimuth )

E = d sin( )

N = d cos( ) E N

A           200.00 350.00

  AB 100.10 0° 00' 00''  0.00  100.10  

B           200.00 450.10

  BC 100.00 90° 00' 00" 100.00  0.00   

C           300.00 450.10

  CD 100.00 180°00'00" 0.00  -100.00   

D           300.00 350.10

  DA 99.70 270°00'00” - 99.70  0.00    

A           200.30 350.10

Sum   399.80    0.30  0.10    

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 Northings and 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

A

B c

D

ΔE=0.3

ΔN=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 90° angles and 100 m sides. When measured all lengths were correct, while the angle at B was in error by 10° when 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

• 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 wrong

The rest of the lines and angles are correct

A’

• 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

∑∆

YO

bse

rved

MX

MY

MX

Y

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

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 wrong

The 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

A

B c

D

ΔE

ΔN

The concept of Linear Closing Error

Assume that the traverse in reallity was 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

N

E

D

C

B

A

EAB EBC

EDA ECD

+ve +ve

-ve -ve

If the traverse is closed, then

ΔE = 0 and

ΔN = 0

N

E

D

C

B

A

EAB EBC

EDA ECD

+ve +ve

-ve -ve

If the traverse is closed, then

ΔE = 0 and

ΔN = 0

A’

ΔN

ΔE

If the traverse is not closed ,

Then ΔE = Ec and ΔN = Nc

Computations of Linear Closing Error

• If he closing error is (W) then

Ew = ΔE and

Nw = ΔN,

W = length of closing error = Ew2 + Nw

2

Fractional Closing error = traverse precision = W / L

Direction of the error = Azimuth = tan-1 (Ew / Nw) = tan-1 (ΔE / ΔN)

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) Rule Correction in departure for AB = -

( ΔE L ) ( LAB)

Correction in latitude for AB = - ( ΔN L ) ( LAB)

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

• -Advantages and disadvantages of this method

Computations of Coordinates

• Add the corrections to the departure or the latitude of each line to get the adjusted departure or latitude (maintain signs)

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

Ei = E i-1 + ΔE

Ni = N i-1 + ΔN

• Check that the misclosure is zero.

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

Order1st 2 nd 3 rd

Class I II I II

Angular Closure

1.7”√n 3.0”√n 4.5”√n 10.0”√n 12.0”√n

Linear Closure(after angul. adj.)

0.04√∑Lor,

1/100,000

0.08√∑Lor,

1/50,000

0.20√∑Lor,

1/20,000

0.40√∑Lor,

1/10,000

0.80√∑Lor,

1/5,000

Example of Standards

point Length L Azimuth

AZDeparture L sin (Az)

LatitudeL cos (Az)

Correction Balanced E N

Departure(WN/L)* L

Latitude(WE/ L)* L

DepartureE

LatitudeN

A

285.10 26 10.0’ 125.72 255.88

B

610.45 104 35.2’

590.77 -153.74

C

720.48 195 30.1’

-192.56 -694.27

D

203.00 358 18.5’

-5.99 202.91

E

747.02 306 54.1’

-517.40 388.5

A

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

 Pnt.

 Length

 Azimuth

Departure= L sin (AZ)

Latitude= L cos (AZ)

Correction Balanced  E

 N

Departure(WE/ L)* L

Latitude Dep. Lat.

L AZ (WN/ L)* L

A                 10000.00 10000.00

  285.1 26 10’ 125.72 255.88 0.06- 0.08 125.66 255.96    

B                 10125.66 10255.96

  610.45 104 35’ 590.77 -153.74 0.13- 0.18 590.64 153.56-    

C                 10716.29 10102.40

  720.48 195 30.1’

- 192.56 - 694.27 0.16- 0.21 192.72- 694.06-    

D                 10523.58 9408.34

  203 358 18.5’

- 5.99 202.91 0.04- 0.06 -6.03 202.97    

E                 10517.54 9611.31

  647.02 306 54.1 -517.4 388.5 0.14- 0.19 -517.54 388.69    

A                check 10000.00 10000.00

Sum =L2466.05

  W =+0.54E WN=0.72 -0.54 0.72 0.00 0.00    

=(0.54/2466.06)x285.1=(0.72/2466.06)x285.1

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 area•The 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.

Example

• Calculate the area of a traverse whose corners are (100,100), (300,100), (300,300), (100,300).