1. General

In this chapter we will cover computational differences between closed loop and link traverses.Unlike a loop traverse, a link traverse does not close back on itself. In order for a link traverse to be closed, the positions of its endpoints must be known.

The endpoints must be known either relative to each other, Figure H-1(a) and (b):

img1

(a)

img2

(b)

Figure H-1
Link Traverse Closed by Known Latitude and Departure

 

or,

by coordinates in the same coordinate system, Figure H-2(a) and (b):

img3

(a)

img4

(b)

Figure H-2
Link Traverse Closed byKnown Coordinates


2. Angles; Misclosure & Adjustment

a. Concept

Because a link traverse doesn't close back on itself there are is no interior angles sum against which to check.

Typically, angles on a link traverse are either consistently turned in the same direction,  Figure H-3, or deflection angles are measured, Fugure H-4.

img5

Figure H-3
No Angle Condition

Figure H-4
No Deflection Angle Condtion

 

In order to check angle closure, the traverse must start and end with known directions, Figure H-5.

Figure H-5
Directions at Both Ends

 

To determine and distribute the angular misclosure is a two step process

1. Using the start direction and measured angles, compute the raw direction of each line. The difference between the computed and known end directions is the angular misclosure.

2. Using whatever desired correction method, correct each raw direction for the misclosure.

The known directions can be explicit (bearing or azimuth) or be computed from coordinates.

If one or other direction at the traverse ends is missing then angle misclosure and adjustment cannot be done. This step would be skipped and the process would continue with latitude and departure computations.

b. Examples

(1) Link Traverse 1

The crossing traverse in Figure H-6 begins and ends on known bearings and uses angles measured to the left at each point.

Figure H-6
Angle left Link Traverse Example

 

Starting with BrngQB, the remaining bearings are computed using the measured angles. The angular misclosure is the difference between computed and known bearings of line QB.

Compute raw bearings

QR

116°48'53" - 60°25'19"
BrngQR = S 56°23'34" E

RS

180°00'00" - (56°23'34" + 48°18'36")
BrngRS = S 75°17'50" W

ST

75°17'50" - 32°11'51"
BrngST = N 43°05'59" E

TY

91°05'25" - 73°05'59"
BrngTY = S 43°59'26" E

 

Compare computed and known bearings of line TY, Figure H-7.

img27

Figure H-7
Angular Misclosure

 

Angular misclosure = 47°59'42" - 47°59'26" = +0°00'16"

Snce there are four angles, each would be corrected by 0°00'04". We could either:

Apply the correction to each angle and then then recompute the directions, or,

Apply the correction to each direction.

We'll demonstrate the latter method.

From Figure H-7, it can be seen that each direction must be rotated counter-clockwise. The first direction would be rotated 0°00'04", the second 0°00'08", and so on, Figure H-8.

img29

Figure H-8
Effect of Applying Correction

 

Because bearings are used, we need to examine which quadrant they fall in to determine if the bearing angle increases or decreases. Based on Figure H-7, bearing angles in the SE and NW quadrants will be increased, those in the NE and SW decreased.

Line

Raw bearing

Correction

Adjusted Bearing

QR

S 56°23'34"E

+0°00'04"

S 56°23'38"E

RS

S 75°17'50"W

-0°00'08"

S 75°17'42"W

ST

N 43°05'59"E

-0°00'12"

N 43°05'47"E

TY

S 43°59'26"E

+0°00'16"

S 43°59'42"E

 

If you compute the new angles between the adjusted bearings, you would find that each angle is increased by 0°00'04".

 

(2) Link Traverse 2

The link traverse in Figure H-9 starts and ends on known azimutns and uses deflection angles measured at each point.

img30

Figure H-9
Deflection Angle Link Traverse Example

 

As with the bearing example, the ancglar misclosure is the difference between the computed and knwn direction of the last line.

 

Compute raw azimuths

JK

154°52'30" - 108°00'18"
AzJK = 46°52'12"

KL

46°52'12" + 92°13'46"
AzKL = 139°05'58"

LM

139°05'58" - 65°55'46"
AzLM = 73°10'12"

MW

73°10'12" - 29°21'42"
AzMW = 102°31'54"

 

Compare the computed and known direction of line MW, Figure H-10.

img35

Figure H-10
Determine Anguar Misclosure 

 

Angular misclosure = 102°32'06" - 102°31'54" = +0°00'12"

Snce there are four deflection angles, each would be corrected by 0°00'03". We could either:

Apply the correction to each deflection angle and then then recompute the directions, or,

Apply the correction to each direction.

We will again apply the corrections directly to the azimuths. The raw direction must be rotated clockwise into the known direction. Unlike bearings, we don't need to worry about quadrants for azimuth corrections - each azimuth is corrected in the same direction, Figure H-11..

img37

Figure H-11
Balancing the Angles

 

Because the computed closing direction must be rotated clockwise into the known closing direction, each raw azimuth will have a clockwise correction applied.

Line

Raw azimuth

Correction

Adjusted azimuth

JK

46°52'12"

+0°00'03"

46°52'15"

KL

139°05'58"

+0°00'06"

139°06'04"

LM

73°10'12"

+0°00'09"

73°10'21"

MW

102°31'54"

+0°00'12"

102°32'06"


3. Traverse Closure; Adjustment

a. Similarities; Differences

Latitudes and departures are computed same as those for a loop traverse:

img38

Eqn (H-1)

Where the two differ is in how their closure is determined and adjustments made.

On a loop traverse, the closure condition is:

img40

Eqn (H-2)

 

But because a link traverse does not close back on itself, that condition does not apply. Instead, we need to know the location, relative or absolute, of the traverse's end points.

If we know the relative location, Figure H--12,

img2

Figure H-12
Relative Positions of Endpoints Are Known

 

the closure condition is

img41

Eqn (H-3)

 

If we have coordinates of the endpoints, Figure H-13,

img4

Figure H-13
Endpoint Coordinates are Known

 

the closure condition is

img42

Eqn (H-4)

 

The latitude and departure errors would be a result of how well the closure condition was met. Linear closure and precision would be determined just as for a loop traverse.

 

b. Example

Given the link traverse in Figure H-14 with adjusted directions and known end point coordinates:

img43

Figure H-14
Adjust Angles on a Crossing Traverse

 

(1) Compute latitudes and departures

img44

img45

img46

 

Line Direction Length Lat Dep
QR S 56°23'38"E 398.75' -220.700' +332.104'
RS S 75°17'42"W 422.89' -107.347' -409.038
ST N 43°05'47"E 604.49' +441.402' +413.004'
  sums: 1426.13' +113.355' +336.070

 

(2) Compute closure and precision

From the coordinates

img47

The closure and precision are

img48

img49

(3) Adjusting by the Compass Rule

img50

img51

img52

img53

img54

 

Line Direction Length Lat Dep Adj Lat Adj Dep
QR S 56°23'38"E 398.75' -220.700' +332.104' -220.715' +332.124'
RS S 75°17'42"W 422.89' -107.347' -409.038 -107.363' -409.017'
ST N 43°05'47"E 604.49' +441.402' +413.004' +441.379' +413.034'
  sums: 1426.13' +113.355' +336.070 +113.301' +336.141'
          check check

 

Adjusted lengths and directions would be computed the same as for a loop traverse, as would coordinates.


4. Summary

A link traverse requires a few variations on the computations we covered for a loop traverse:

  • The angular misclosure is dependent on available directions at both ends of the traverse - if either is not available, then misclosure can neither be determined nor compensated.
  • The relative or absolute position of the endpoints must be known - latitude and departure closure conditions are dependent on the end point relationships.
  • A link traverse does not have an area. Although the Area by Coordinates or DMDs equations can be applied and yield a numeric result, that result will not make sense as an area.

All other computations; lats and deps, traverse adjustment, adjusted lengths and directions, and coordinate determination; are performed in the same manners as for a loop traverse.