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Bowditch Rule (Compass Rule) Traverse Adjustment

Updated · GeoCAD Bridge engineering team · 8 min read

Summary

The Bowditch rule, also called the compass rule, adjusts a closed or link traverse by distributing the misclosure in easting and northing to each leg in proportion to that leg’s length. It assumes angles and distances were measured with comparable precision, and it is the default adjustment in most survey software. In the worked example below a 0.100 m misclosure over 2406.58 m (1:24,100) is removed so the loop closes exactly.

Definition

Bowditch rule: a traverse adjustment in which the correction to each departure and latitude equals the total misclosure in that component multiplied by the leg length divided by the total traverse length, with opposite sign.

  • Correction ∝ leg length
  • Assumes equal angular and linear precision
  • Also called the compass rule
  • Proposed by Nathaniel Bowditch (1807)

When to use the Bowditch rule

Use the Bowditch rule for closed loop and link traverses observed with a total station, where angular and distance precision are roughly balanced. It is the standard choice for boundary, topographic and construction control traverses and is the method most regulators and clients expect to see in a traverse report.

It should not be used to hide blunders. If the relative precision is worse than the project specification, find and remeasure the faulty observation before adjusting.

Closed traverse A-B-C-DA four-sided closed loop traverse. The computed end point A-prime falls slightly away from the start A; the gap is the linear misclosure. ABCDA′ (computed) Leg 1Leg 2Leg 3Leg 4
Figure 1. In a closed loop the computed coordinates of the last point should equal the start. The small gap A–A′ is the linear misclosure that an adjustment distributes back through the legs.

Formula

For each leg i of length Li, with total traverse length ΣL and misclosures fE and fN (computed minus known):

Correction ΔE_i = −f_E × L_i / ΣL Correction ΔN_i = −f_N × L_i / ΣL

The corrected departures and latitudes are then summed from the starting control point to give adjusted coordinates. Because the corrections sum to −fE and −fN, the adjusted traverse closes exactly on the known end point.

Angles should be balanced before computing departures and latitudes; see angular misclosure.

Bowditch versus Transit rule

The Transit rule distributes corrections in proportion to the absolute departure or latitude of each leg instead of its length. It assumes angles were measured more precisely than distances, which was true for transit-and-tape surveys but is rarely true for modern total stations. The Transit rule also changes bearings more on legs that run nearly north–south or east–west.

Comparison of the two classical traverse adjustment rules.
Bowditch (compass)Transit
Weight of correctionLeg length L|ΔE| or |ΔN|
Precision assumptionAngles ≈ distancesAngles ≫ distances
Changes bearingsYes, moderatelyYes, more on cardinal legs
Typical useTotal station traversesLegacy transit and tape

For networks with redundant observations or mixed precision, a least squares adjustment is the rigorous choice.

Common mistakes

  • Adjusting before balancing the angles, so the angular error is pushed into the coordinates.
  • Using slope distances instead of reduced horizontal distances (see distance reduction).
  • Mixing ground and grid distances when the control coordinates are on a projected grid such as UTM.
  • Adjusting an open traverse, which has no independent check and therefore no misclosure to distribute.
  • Reporting the adjusted traverse without the unadjusted misclosure and precision ratio.

Worked example

The loop A–B–C–D–A below has already had its angles balanced. Start coordinates are A = (5000.000 E, 3000.000 N).

Closed loop A–B–C–D–A. Departure = L·sin(Az), latitude = L·cos(Az).
LegAzimuthDistance (m)Departure ΔE (m)Latitude ΔN (m)
A–B26°10′00″285.10+125.724+255.882
B–C104°35′00″610.45+590.783-153.704
C–D195°30′00″720.48-192.540-694.276
D–A318°29′30″790.55-523.920+592.011
Sum2406.58+0.047-0.088
fE = ΣΔE = +0.047 m fN = ΣΔN = -0.088 m Linear misclosure = √(fE² + fN²) = 0.100 m Perimeter ΣL = 2406.58 m Relative precision = ΣL / misclosure = 1:24,100

Each correction is the misclosure times Li/ΣL with opposite sign. For leg A–B: −(+0.0470) × 285.10 / 2406.58 = -0.0056 m in easting.

Bowditch (compass) rule corrections and adjusted coordinates. Start A = (5000.000, 3000.000).
LegCorr. ΔECorr. ΔNAdj. ΔEAdj. ΔNPointEasting (m)Northing (m)
A–B-0.006+0.010+125.719+255.892B5125.7193255.892
B–C-0.012+0.022+590.771-153.682C5716.4903102.210
C–D-0.014+0.026-192.554-694.250D5523.9362407.960
D–A-0.015+0.029-523.936+592.040A5000.0003000.000

The adjusted coordinates return exactly to A, and every leg absorbed a share of the 0.1 m misclosure proportional to its length. You can reproduce this table in the Traverse Adjustment tool by entering the four azimuths and distances.

Step by step

  1. Reduce observations. Reduce slope distances to horizontal and, if working on a projected grid, apply the combined scale factor.
  2. Balance angles. Compare the sum of measured angles with the theoretical sum and distribute the angular misclosure.
  3. Compute azimuths. Propagate azimuths from the reference direction using the balanced angles.
  4. Compute latitudes and departures. ΔN = L·cos(Az), ΔE = L·sin(Az) for every leg.
  5. Compute misclosure. Sum ΔE and ΔN (loop) or compare with the closing control (link), then compute the linear misclosure and precision ratio.
  6. Check against specification. Stop and investigate if the precision ratio is worse than the project tolerance.
  7. Apply Bowditch corrections. Correct each ΔE and ΔN by −f × L_i/ΣL.
  8. Compute adjusted coordinates. Accumulate corrected ΔE and ΔN from the start control and confirm closure on the end control.

FAQ

Why is it called the compass rule?

It was originally developed for compass traverses, in which bearings and chained distances had similar precision, so corrections in proportion to distance were appropriate.

Does the Bowditch rule change the angles?

Yes. Because easting and northing corrections differ leg by leg, adjusted bearings and distances differ slightly from the observed ones. The changes are small when the misclosure is small.

What precision ratio is acceptable before adjusting?

It depends on the specification. Typical values are 1:5,000 for low-order work, 1:10,000 for property and construction control and 1:20,000 or better for higher-order control. Always use the project or regulatory standard.

Can I use the Bowditch rule on an open traverse?

No. An open traverse has no closing control, so there is no misclosure to distribute and no check on blunders.

Is the Bowditch rule the same as least squares?

No. Least squares minimizes the weighted sum of squared residuals and can use redundant observations and individual weights. The Bowditch rule is a simple proportional distribution that gives similar results only for a single well-balanced traverse.

Tools

Articles

References

  1. Ghilani, C. D. & Wolf, P. R. Elementary Surveying: An Introduction to Geomatics, 15th ed., Pearson, 2018 (traverse computations and adjustments).
  2. Ghilani, C. D. Adjustment Computations: Spatial Data Analysis, 6th ed., Wiley, 2017.
  3. Uren, J. & Price, W. F. Surveying for Engineers, 5th ed., Palgrave Macmillan, 2010.
  4. Bowditch, N. Letter on the correction of traverse surveys, The Analyst, 1808.

Cite this page

GeoCAD Bridge. “Bowditch Rule (Compass Rule) Traverse Adjustment.” GeoCAD Bridge Knowledge Center, updated 2026-10-05. https://www.geocadbridge.com/knowledge/survey-bowditch-rule/