1. Adjusting a Traverse

Adjusting a traverse (also known as balancing a traverse) is used to distributed the closure error back into the angle and distance measurements.

Summing the latitudes and departures for the raw field traverse:

img19
Eqn (E-1)
img10
  Figure E-1
Loop Traverse Misclosure

On an adjusted (balanced) traverse:

img18

Eqn (E-2)
img11
  Figure E-2
Adjusted (Balanced) Loop Traverse

 

The condition for an adjusted traverse is that the adjusted Lats and Deps sum to 0.00. As with other survey adjustments, the method used to balance a traverse should reflect the expected error behavior and be repeatable. Table E-1 lists primary adjustment methods with their respective advantages and disadvantages.

Table E-1

Method Premise Advantage Disadvantage
Ignore Don't adjust anything. Simple; repeatable Ignores error
Arbitrary Place error in one or more measurements Simple Not repeatable; ignores error behavior
Compass Rule Assumes angles and distances are measured with equal accuracy so error is applied to each. Simple; repeatable; compatible with contemporary measurement methods. Treats random errors systematically
Transit Rule Assumes angles are measured more accurately than distances; distances receive greater adjustment. Simple; repeatable; compatible with older transit-tape surveys. Treats random errors systematically; not compatible with contemporary measurement methods.
Crandall Method Quasi-statistical approach. Angles are held and errors are statistically distributed into the distances. Allows some random error modeling; repeatable. Models only distance errors, not angle errors.
Least squares Full statistical approach. Allows full random error modeling; repeatable; can mix different accuracy and precision measurements; provides measurement uncertainties. Most complicated method

 

The Compass Rule works sufficiently well for simple surveying projects and is the one we will apply.


2. Compass Rule

The Compass Rule (also known as the Bowditch Rule) applies a proportion of the closure error to each line.

For any line IJ, Figure E-3,

img12
Figure E-3
Adjusted Latitude and Departure

 

img20      Eqn (E-3)

 

img21       Eqn (E-4)

 

The Compass Rule distributes closure error based on the proportion of a line's length to the entire distance surveyed.


3. Adjusted Length and Direction

Regardless of the adjustment method applied, changing a line's Lat and Dep will in turn change the length and direction of the line.

 img13
Figure E-4
Adjusted Length and Direction

 

The adjusted length can be computed from the Pythagorean theorem:

img22        

Eqn (E-5)

 

Computing direction is a two-step process: (1) Determine β, the angle from the meridian to the line (2) Convert β into a direction based on the line's quadrant.

To determine β:

img23       Eqn (E-6)

 

β falls in the range of -90° to +90°.

The sign on β indicates the direction of turning from the meridian: (+) is clockwise, (-) is counter-clockwise. The combined signs on the adjusted Lat and Dep will identify the line's quadrant.

Figure E-5 shows the quadrant and direction computation for the various mathematic combinations of the adjusted Lat and Dep:

img14 1
Figure E-5
Converting ß to a Direction

4. Examples

These examples are a continuation of those from the Latitudes and Departures chapter.

a. Traverse with Bearings

img15
Figure E-6
Bearing Traverse Example

 

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
too far N
Dep err
too far E

 

(1) Adjust the Lats and Deps

Setup Equation (E-3):

img24

 

Line AB

img25

Line BC

img26

Line CD

 img27

Line DA

 img28

Check the closure condition

  Adjusted
Line Lat (ft) Dep (ft)
AB -176.386 -438.574
BC +203.382 -73.105
CD +192.340 +198.635
DE -219.336 +313.044
sums: 0.000 0.000
  check check

 

A common mistake is to forget to negate Lat err and Dep err in the correction equations. If that happens, the closure condition will be twice what it originally was as the corrections were applied in the wrong direction.

(2) Compute adjusted lengths and directions

 Use Equations (E-5) and (E-6) along with Figure E-5 to compute the new length and direction for each line.

Line AB

Adj Lat = -176.386 <- South
Adj Dep = -438.574 <- West

img29

 img30

Because it's the SW quadrant, Brng =S 68°05'27.4" W.

Line BC

Adj Lat = +203.382 <- North
Adj Dep = -73.105 <- West

img31

img32

Because it's the NW quadrant, Brng = N 19°46'14.9" W

Line CD

Adj Lat = +192.340 <- North
Adj Dep = +198.635 <- East

img33

 img34

Because it's the NE quadrant, Brng = N 45°55'20.7" E

Line DA

Adj Lat = -219.336 <- South
Adj Dep = +313.044 <- East

img35

img36

Because it's the SE quadrant, Brng = S 54°58'58.0" E


(3) Adjustment summary

  Adjusted Adjusted
Line Lat (ft) Dep (ft) Length Bearing
AB -176.386 -438.574 472.715 S 68°05'27.4" W
BC +203.382 -73.105 216.122 N 19°46'14.9" W
CD +192.340 +198.635 276.479 N 45°55'20.7" E
DE -219.336 +313.044 382.237 S 54°58'58.0" E

 

b. Traverse with Azimuths

img16 
Figure E-7
Azimuth Traverse Example

 

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
too far S
Dep err
too far E

 

(1) Adjust the Lats and Deps

Setup Equation (E-3):

img37

Line ST

img38

Line TU

img39

Line UV

img40

Line VS

img41

Check the closure condition

  Adjusted
Line Lat (ft) Dep (ft)
ST +218.836 -269.332
TU -72.205 -357.346
UV -291.533 +372.094
VS +144.902 +254.584
sums: 0.000 0.000
  check check

 

(2) Compute adjusted lengths and directions

Use Equations (E-5) and (E-6) along with Figure E-5 to compute the new length and direction for each line.

Line ST

Adj Lat = +218.836 <- North
Adj Dep = -269.332 <- West

img42

 img43

Because it's in the NW quadrant: Az = 360°00'00"+(-50°54'20.4") =309°05'39.6"

Line TU

Adj Lat = -72.205 <- South
Adj Dep = -357.346 <- West

img44

img45

Because it's in the SW quadrant: Az = 180°00'00"+(78°34'36.0") = 258°34'36.0"

Line UV

Adj Lat = -291.533 <- South
Adj Dep = +372.094 <- East

img47

img48

Because it's in the SE quadrant: Az = 180°00'00"+(-51°55'17.6") = 128°04'42.4"

Line VS

Adj Lat = +144.902 <- North
Adj Dep = +254.584 <- East

img49 

 img50

Because it's in the NE quadrant: Az = 60°21'09.7"

(3) Adjustment summary

  Adjusted Adjusted
Line Lat (ft) Dep (ft) Length (ft) Azimuth
ST +218.836 -269.332 347.029 309°05'39.6"
TU -72.205 -357.346 364.568 258°34'36.0"
UV -291.533 +372.094 472.700 128°04'42.4"
VS +144.902 +254.584 292.933 60°21'09.7"

 

c. Crossing Loop Traverse

As long as a traverse closes back on its beginning point, it can be adjusted the same as any other loop traverse.

img17 
Figure E-8
Crossing Loop Traverse Example

 

Line Azimuth Length (ft) Lat (ft) Dep (ft)
EF 133°02'45" 455.30 -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" 312.85 +307.534 -57.430
  sums: 1419.28 +0.042 +0.135
    Dist Lat err
too far N
Dep err
too far E

 

(1) Adjust and recompute each line.

Setup Equation (E-3):

img51

Line EF

img52

img53

img54

Because it's in the SE quadrant: Az = 180°00'00"+(-46°56'57.1") = 133°03'02.9"

Line FG

img55

img56

img57

Because it's in the NE quadrant: Az = 24°33'19.7"

Line GH

img58

img59

img61

Because it's in the SW quadrant: Az = 180°00'00"+(61°05'18.8") =241°05'18.8"

Line HE

img63

 img64a

img65

Because it's in the NW quadrant: Az = 360°00'00"+(-10°35'00.5") = 349°24'59.5"

(2) Adjustment summary

  Adjusted Adjusted
Line Lat (ft) Dep (ft) Length (ft) Azimuth
EF -310.794 +332.694 455.278 133°03'02.9"
FG +207.684 +94.890 228.335 24°33'19.7"
GH -204.416 -370.124 422.821 241°05'18.8"
HE +307.525 -57.460 312.847 349°24'59.5'
sums: -0.001 0.000
  check (rounding) check

5. Summary

A traverse is adjusted, or balanced, to distribute remaining random errors back into the measurements. There are a number of ways to accomplish this differing in how the errors are modeled and computation complexity. The Comapss Rule demonstrated here works well for simple traverses having minimal redunant measurements. As traverses become more complex with additional measurements added, particularly with mixed quality, a least sqaures adjustment is the best to employ. 

The examples used in this chapter are all closed loop traverses. A later chapter will show how to perform traverse computations, including adjustments, on closed link traverses.