1. Definition; Equations

Latitude is the north-south component of a line; departure the east-west. North latitudes are positive, South are negative; similarly East departures are positive, West are negative.

img13

(a)
The latitude of line AB is North (+),
its departure is East (+).

(b)
The latitude of line CD is South (-),
its departure is West (-).

Figure D-1
Latitudes and Departures

Latitude (Lat) and Departure (Dep) are computed from:

img4         Eqn (D-1) 

 

Dir can be either a bearing angle, Figure D-2(a), or azimuth angle, Figure D-2(b).

 

img15 img16

(a)
LatAB is North
DepAB is East

(b)
LatBA is South
DepBA is West

Figure D-2
Bearings or Azimuths

 

Because a bearing angle never exceeds 90°, the Lat and Dep equations will always return positive values.

img17

Sin(0°) to Sin(90°) ranges from 0 to +1.0

Cos(0°) to Cos(90°) ranges from +1.0 to 0

The correct mathematical sign for the Lat and Dep come from the bearing quadrant.

A bearing of S 47°35' E has a negative Lat (South) and a positive Dep (East).

Figure D-3
Quadrants
 

 

An azimuth angle ranges from 0° to 360°, so the sine and cosine return the correct signs on the Lat and Dep.

Examples

img10

d 05

  Figure D-4
NE Azimuth

img11

img19

  Figure D-5
SW Azimuth

 

Reversing a line direction results in the same magnitude Lat and Dep but reversed signs:

Line A to B Line B to A
img22 img21
 (a) (b) 
Figure D-6
Reverse Latitude and Departure

 


2. Closure

On a closed loop traverse, the sum of the Lats should equal zero as should the sum of the Deps.

Closure condition:

 

img29
Eqn (D-2)

 

 

 

 

  Figure D-7
Perfect Closed Loop

Most traverses won’t close perfectly due to the presence of errors. If the field crew followed careful procedures and compensated distance measurements for atmospheric conditions, only random errors prevent achieving perfect closure:event achieving perfect closure:

Actual closure:

 

img30
Eqn (D-3 )

 

 

 

 

  Figure D-8
Cumulative Random Errors

 

Figure D-9
Linear Closure

 

The Linear Closure, LC, is the total misclosure distance. It is computed from the Lat and Dep errors:

 

img31       Eqn (D-4)

 

Traverse precision is a function of the linear closure and the total survey distance. The precision is stated as a relative value, usually as 1 unit of error per D units measured: 

img32       Eqn (D-5)

 

What is allowable minimum precision? It depends on the survey purpose. Traditional first order surveys have a required precision of 1/100,000; in Wisconsin a property survey must be at least 1/3000. In order to achieve a formal minimum precision, specific equipment and field procedures must be used which match that precision requirement. Achieving a minimum precision should be by design not by accident.


3. Examples

In the following examples shown, all calculations are shown with an additional significant figure. Because these are generally intermediate computations, carrying an additional digit minimizes roundoff error in subsequent calculations.

When reporting results of an intermediate calculation, those should be stated to the correct number of significant figures so as not to imply an accuracy beyond that of the measurements.

a. Traverse with bearings

img26

 

 

 

img33
Eqn (D-6)

 

Lat and Dep will always compute as positive; must assign correct mathematical sign based on the bearing quadrant.

 

Figure D-10
Bearing Traverse
 

Line AB

img43

Because the bearing is South and West, the Lat and Dep are -176.357' and -438.548' respectively.

Line BC

img44

Because the bearing is North and West, the Lat and Dep are +203.395' and -73.093' respectively.

Line CD

img45

Because the bearing is North and East, the Lat and Dep are +192.357' and +198.651' respectively.

Line DA

img46

Because the bearing is South and East , the Lat and Dep are -219.312' and +313.065' respectively

In tabular form:

Line

Bearing

Length (ft)

Lat (ft)

Dep (ft)

AB

S 68°05'35"W

472.68  

-176.357

-438.548

BC

N 19°46'00"W

216.13  

+203.395

-73.093

CD

N 45°55'20"E

276.52  

+192.357

+198.651

DA

S 54°59'15"E

382.24  

-219.312

+313.065

 

sums: 

1347.57  

+0.083

+0.075

   

Distance  

Lat err

Dep err

   

 

too far N

too far E

 

 img36

img37

 

b. Traverse with azimuths

 

img34
Eqn (D-7)

 

Lat and Dep will always compute directly with the correct sign when using azimuths.

Figure D-11
Azimuth Traverse
 

 

Line ST

img38

Line TU

img39

Line UV

img41

Line VS

img42

Line

Azimuth  

Length (ft)

Lat (ft)

Dep (ft)

ST

309°05'38"

347.00  

+218.816

-269.311

TU

258°34'22"

364.55  

-72.226 

-357.324

UV

128°04'44"

472.74  

-291.560

+372.123

VS

60°21'26"

292.94  

+144.885

+254.602

 

sums:

1477.23  

-0.085

+0.090

   

Distance  

Lat err

Dep err

     

too far S

too far E 

 

img47

img48

 

c. Crossing Traverse

A four-sided parcel has two obstructed lines.

 
Figure D-12
Parcel Boundaries

 

In order to create a closed traverse, the survey crew measures a crossing traverse which connects all four points.

 

As long as a traverse closes back on its beginning point, the closer condition is still:

img29

regardless of how many times it may cross itself.

 

Given this traverse data, determine its closure and precision.

 

Figure D-13
Closed Crossing Traverse

 

 

Rather than write out each Lat and Dep computation separately, we can simply set up the table and record the computations in it. 

Line

Azimuth  

Length (ft)

Lat (ft)

Dep (ft)

EF

133°02'45"

455.03  

-310.780

+332.737

FG

24°33'35"

228.35  

+207.691

+94.912

GH

241°05'15"

422.78  

-204.403

-370.084

HE

349°25'20"

213.85  

+307.534

-57.430

 

sums:

1419.28  

+0.042

+0.135

   

Distance  

Lat err

Dep err

     

too far N

Too far E  

 

 img49

img50