traversing. required readings: 9-1,9-2.1,9-2.2, 9-3 to 9-8 &9-10 10-1 to 10-7.1 &10-8,...

Traver

sin

g

Required readings: 9-1,9-2.1,9-2.2, 9-3 to 9-8 &9-10

10-1 to 10-7.1 &10-8, 10-10, 10-11, 10-15, and 10-17

Required solved examples: 10-1 to 10-4

Required figures: 10-1, tables 10-1 to 10-5

Traversing• Definition: A traverse is a series of

consecutive lines whose lengths and directions have been measured.

• Traversing: The act of establishing traverse stations and making the necessary measurements.

• Why?• Closed (polygon or link) and opened

traverses

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!!

Traverse Notations• We will only discuss closed Traverse with interior

angles measured.• The polygon corners will be numbered or lettered

in anti-clockwise direction.• All angles are measured in a clockwise direction,

and the average of direct and reverse readings is computed at all the angles.

• Angles are designated with three letters, the backsight station will be given first, the occupied station second, and the forsight station third.

Traverse Stations• Successive stations should be inter visible.• Stations are chosen in a safe, easy to access places. • Lines should be as long as possible, and as equal as

possible, Why?• Stations must be referenced to retrieve them if lost

Traversing by Interior Angles• A polygon is established around the site• All internal angles and all horizontal distances are

measured• Each angle is measured in direct and reverse, the average

is a single observation of the angle, how many readings?• Each angle is observed at least three times, how many

readings?• 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? (SITES 1 AND 2 PROJECT 1)

• 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 B was wrong

The rest of the lines and angles are correct

A’

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.

• Accuracy Standards: c = k * n where (n) is the number of points.

• K: a constant defined according to which standards used, 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

XAB XBC

XDA XCD

+ ve+ ve

- ve- ve

If the traverse is closed, then

ΔX = 0 and

ΔY = 0

A’

ΔY

ΔX

If the traverse is not closed,

Then ΔX = Xw and ΔN = Ycw

?

?

Computations of Linear Closing Error

• If he closing error is (W) then

Xw = ΔX and

Yw = ΔY,

W = length of closing error = Xw2 + Yw

2

Fractional Closing error = traverse precision = W / L

Traverse Adjustment• Two kinds of misclosures.• 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 X direction = sum of the

departures.– Compute the Misclosure in Y direction = sum of the latitudes.– Compute the linear misclosure– Use the Compass (Bowditch) rule to adjust:

Correction in dep or lat for AB = -(total dep or lat misclosure)traverse perimeter

x AB length

Correction in departure for AB = -(total departure misclosure)traverse perimeter

x length AB

Correction in latitude for AB = -(total latitude misclosure)traverse perimeter

x length AB

- Use Bowdich (Compass) rule to compute the adjustments for departures and latitudes of all sides, for a line such as AB:

And,

• Add the corrections to the departure or the latitude of each line. Get the adjusted departure latitude

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

Xi = X i-1 + D X Yi = X i-1 + D Y

• Check that the misclosure is zero.

• To solve a problem, it is easier to use a table such as table 10-4• Review equations in section 10-10.• Three checks:

• Compare adjustments to errors• After corrections are added, check that the sum of longitudes

is zero, same for longitudes• Compare coordinates of last and first points after adjustment

• It is important to practice how to compute length and azimuth from departure and latitude, or from coordinates:

tan(azimuth) = departure

latitude

length =departure

sin (azimuth)latitude

= cos (azimuth)

departure = D X = d (sin azimuth)latitude = D Y= d (cos azimuth)

D sin (Az) D cos (Az) Correction Balanced

Station Length (ft) L

Azimuth AZ

Departure Latitude Departure-(Wx/P)* L

Latitude-(Wy/P)* L

Departure Latitude X Y

A 10,000 10,000

285.10 26 10.0’ 125.72 255.88 -0.06 +0.08 +125.66 +255.96

B 10,125.66 10,255.96

610.45 104 35.2’ 590.77 -153.74 -0.13 +0.18 +590.64 -153.56

C 10,716.3 10,102.4

720.48 195 30.1’ -192.56 -694.27 -0.15 +0.21 -192.72 -694.04

D 10,523.58 9408.34

203.00 358 18.5’ -5.99 202.91 -0.05 +0.06 -6.04 +202.97

E 10,517.54 9611.31

747.02 306 54.1 -517.40 388.5 -0.14 +0.19 -517.54 +388.69

A 10,000 10,000

Sum P=2466.05

Wx =+0.54 Wy =-0.72 -0.54 +0.72 0.00 0.00

Linear Misclosure = 0.90 ftRelative precision = 0.90 / 2466 = 1: 2700

Traverse Area

A

B

CD

E

Traverse area = 1 S { Xi (Yi+1 - Yi-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• Formula page 27-4 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 X and the Y.