E. Traverse Adjustment
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:


Figure E1 Loop Traverse Misclosure 

On an adjusted (balanced) traverse:


Figure E2 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 E1 lists primary adjustment methods with their respective advantages and disadvantages.
Table E1  
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 transittape surveys.  Treats random errors systematically; not compatible with contemporary measurement methods. 
Crandall Method  Quasistatistical 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 
Most simple surveying projects use total stations which measure distances and angles with comparable quality. The Compass Rule is an appropriate adjustment method for these traverses. Although least squares would provide a "better" adjustment solution, it is generally overkill for a basic traverse. For more information on traverse adjustment by least squares see XVIII. Least Squares Lite.
We will concnetrate on the Compass Rule.
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 E3,
Figure E3 Adjusted Latitude and Departure 
Equations E1 and E2 
Equations E3 and E4 
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.
Figure E4 Adjusted Length and Direction 
The adjusted length can be computed from the Pythagorean theorem:
Equation E5 
Computing direction is a twostep 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:
Equation E6 
β falls in the range of 90° to +90°.
The sign on β indicates the direction of turning from the meridian: (+) is clockwise, () is counterclockwise. The combined signs on the adjusted Lat and Dep will identify the line's quadrant.
Figure E5 shows the quadrant and direction computation for the various mathematic combinations of the adjusted Lat and Dep:
Figure E5 Converting ß to a Direction 
4. Examples
These examples are a continuation of those from the Latitudes and Departures chapter.
a. Traverse with Bearings
Figure E6 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 Equations E1 and E2:
Now solve Equations E3 and E4 for each line:
Line AB
Line BC
Line CD
Line DA
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 E5 and E6 along with Figure E5 to compute the new length and direction for each line.
Line AB
Adj Lat = 176.386 < South
Adj Dep = 438.574 < West
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
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
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
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
Figure E7 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 Equations E1 and E2:
Solve Equations E3 and E4 for each line:
Line ST
Line TU
Line UV
Line VS
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 E5 and E6 along with Figure E5 to compute the new length and direction for each line.
Line ST
Adj Lat = +218.836 < North
Adj Dep = 269.332 < West
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
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
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
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.
Figure E8 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 Equations E1 and E2:
Solve Equations E3 and E4 for each line:
Line EF
Because it's in the SE quadrant: Az = 180°00'00"+(46°56'57.1") = 133°03'02.9"
Line FG
Because it's in the NE quadrant: Az = 24°33'19.7"
Line GH
Because it's in the SW quadrant: Az = 180°00'00"+(61°05'18.8") =241°05'18.8"
Line HE
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 Compass Rule demonstrated here works well for simple traverses having minimal redundant measurements. The examples thus far are closed loop traverses. Chapter H. Closed Link Traverse shows how to perform traverse computations, including adjustments, on closed link traverses.
The Compass and Transit Rules and Crandall Method are compared using a numeric example in Chapter K. Comparison of Adjustment Methods. Traverse Adjustment, an Excel workbook for the three methods, is in the Software area. As traverses become more complex with additional measurements added, particularly with mixed quality, a least squares adjustment is the best strategy to employ. This concept is discussed in greater detail in XV. Least Squares Lite topic.