Free Gale's Table Generator for Major Closed Traverse
| Line (From–To) |
Length (m) |
Bearing (WCB) Degrees — Minutes — Seconds |
|||
|---|---|---|---|---|---|
What Is a Traverse Survey?
A traverse is a series of connected lines (called legs or courses) whose lengths and directions are measured in the field. It is one of the most common methods of control surveying — used to establish the positions of ground points that serve as reference stations for detailed surveys, setting out, and mapping.
Types of traverse
| Type | Description | Check condition |
|---|---|---|
| Closed traverse | Starts and ends at the same point (or at two known points). Allows error detection and correction. | \(\sum L = \sum D = 0\) (or known closing difference) |
| Open traverse | Starts at one point and ends at a different, unconnected point. No closure check possible. | None — errors cannot be detected |
| Link traverse | Connects two known control points. Provides closure check without returning to start. | Check bearings and coordinates at end point |
Why use a closed traverse? The closure condition (\(\sum\text{Latitude} = 0\) and \(\sum\text{Departure} = 0\)) provides a built-in mathematical check on the fieldwork. Any discrepancy (closing error) can be detected and distributed proportionally among the legs using correction rules such as Bowditch or Transit.
Bearings
A bearing is the horizontal angle between a reference direction (usually north) and the survey line, measured in a specified direction. Two bearing systems are used:
- Whole Circle Bearing (WCB): measured clockwise from north, ranging from 0° to 360°. Also called azimuth. Easiest for calculations.
- Quadrant Bearing (QB) / Reduced Bearing: measured from north or south toward east or west, maximum 90°. Written as N45°30'E or S60°W.
Latitude and Departure
For each traverse leg of length \(L\) and quadrant bearing \(\theta\), the latitude is the north-south component and the departure is the east-west component:
Closing error and precision
For a perfect closed traverse, the algebraic sum of all latitudes and all departures should both be zero. In practice, measurement errors cause small discrepancies:
Typical precision standards: Rough survey: 1:1000. Ordinary survey: 1:3000 to 1:5000. Good survey: 1:10,000. First-order control: better than 1:50,000.
Gale's Table — Structure and Purpose
The Gale's Traverse Table (named after its systematic tabular format) is the standard computation sheet used in engineering surveying for processing closed traverse data. It organises all intermediate and final results in a logical sequence that allows errors to be spotted easily and corrections to be applied systematically.
Column-by-column explanation
| Column | Content | How computed |
|---|---|---|
| Line | Survey leg identifier (e.g. AB, BC) | Field data |
| Length (L) | Measured distance of each leg (m) | Field measurement (tape or EDM) |
| WCB | Whole circle bearing of the leg | Field measurement (theodolite or compass) |
| Reduced Bearing | Equivalent quadrant bearing | Converted from WCB using quadrant rules |
| Latitude (+N / -S) | North/south component of the leg | \(L\cos\theta_{QB}\), sign from quadrant |
| Departure (+E / -W) | East/west component of the leg | \(L\sin\theta_{QB}\), sign from quadrant |
| Correction to Latitude | Bowditch/Transit correction for that leg | See correction method formulas |
| Correction to Departure | Bowditch/Transit correction for that leg | See correction method formulas |
| Corrected Latitude | Latitude after applying correction | Latitude ± correction |
| Corrected Departure | Departure after applying correction | Departure ± correction |
| Consecutive Northing | Running northing coordinate from start | \(N_i = N_{i-1} + \text{Corrected Latitude}_i\) |
| Consecutive Easting | Running easting coordinate from start | \(E_i = E_{i-1} + \text{Corrected Departure}_i\) |
| Adjusted Length | Recomputed leg length from corrected components | \(L_{adj} = \sqrt{L_{corr}^2 + D_{corr}^2}\) |
| Adjusted Bearing | Recomputed bearing from corrected components | \(\theta_{adj} = \arctan(D_{corr}/L_{corr})\), then convert to WCB |
Check rows in the Gale's Table
The bottom rows of the table provide three important checks:
- Sum of uncorrected latitudes and sum of uncorrected departures — these reveal the closing errors \(\delta L\) and \(\delta D\).
- Sum of corrections — must equal \(-\delta L\) (for latitudes) and \(-\delta D\) (for departures) to confirm corrections cancel the error exactly.
- Sum of corrected latitudes and sum of corrected departures — must be exactly zero (or the computed closing difference for link traverses).
- Closing check — final consecutive northing and easting must return to the starting values.
Correction Methods
Bowditch Rule (Compass Rule)
The Bowditch rule, also called the compass rule, assumes that errors in the traverse are proportional to the length of each leg. It is the most widely used correction method for engineering surveys and is appropriate when linear measurements and angular measurements have equal reliability (i.e. the expected error is proportional to distance).
The corrections for all legs must sum to exactly \(-\delta L\) (latitudes) and \(-\delta D\) (departures). Rounding adjustments (typically ±1 mm) are applied to the largest legs to ensure the sums balance.
Transit Rule
The transit rule assumes that angular measurements are more accurate than linear measurements. Corrections are proportional to the magnitude of the latitude or departure of each leg, not its length. Used less frequently than Bowditch.
Comparison
| Criterion | Bowditch Rule | Transit Rule |
|---|---|---|
| Assumption | Linear and angular errors equally proportional to distance | Angular measurements more precise than linear |
| Correction proportional to | Length of each leg | Absolute latitude or departure of each leg |
| Best for | Tape/EDM traverses, most engineering surveys | Traverses where angles are measured very precisely |
| Common use | Standard engineering traverses, land surveying | City surveys, triangulation-connected traverses |
| Formula memory | Proportion = leg length / perimeter | Proportion = leg lat (or dep) / sum of all lats (or deps) |
Which to use? For most engineering traverses measured with EDM or tape and a theodolite, the Bowditch rule is standard and recommended. The Transit rule is appropriate only when the angular measurement is known to be significantly more precise than the linear measurement.
Simple worked example — Bowditch correction
Consider a 4-leg closed traverse with total perimeter ΣL = 500 m, latitude closing error δL = +0.050 m, departure closing error δD = −0.030 m:
| Leg | Length (m) | Proportion (l/ΣL) | cL = −δL × proportion | cD = −δD × proportion |
|---|---|---|---|---|
| AB | 120 | 120/500 = 0.240 | −0.050 × 0.240 = −0.012 m | +0.030 × 0.240 = +0.007 m |
| BC | 150 | 150/500 = 0.300 | −0.050 × 0.300 = −0.015 m | +0.030 × 0.300 = +0.009 m |
| CD | 110 | 110/500 = 0.220 | −0.050 × 0.220 = −0.011 m | +0.030 × 0.220 = +0.007 m |
| DA | 120 | 120/500 = 0.240 | −0.050 × 0.240 = −0.012 m | +0.030 × 0.240 = +0.007 m |
| Σ | 500 | 1.000 | −0.050 m ✓ | +0.030 m ✓ |
The sum of latitude corrections = −0.050 m = −δL ✓. The sum of departure corrections = +0.030 m = −δD ✓. The corrected ΣLat = +0.050 + (−0.050) = 0 ✓. The corrected ΣDep = −0.030 + (+0.030) = 0 ✓.
Simple worked example — Transit correction
Using the same closing errors (δL = +0.050 m, δD = −0.030 m), suppose the four legs have absolute latitudes |L| of 80, 60, 90, 70 m (total Σ|L| = 300 m) and absolute departures |D| of 95, 105, 70, 80 m (total Σ|D| = 350 m):
| Leg | |Lat| (m) | cL = −δL × |Li|/Σ|L| | |Dep| (m) | cD = −δD × |Di|/Σ|D| |
|---|---|---|---|---|
| AB | 80 | −0.050 × 80/300 = −0.013 m | 95 | +0.030 × 95/350 = +0.008 m |
| BC | 60 | −0.050 × 60/300 = −0.010 m | 105 | +0.030 × 105/350 = +0.009 m |
| CD | 90 | −0.050 × 90/300 = −0.015 m | 70 | +0.030 × 70/350 = +0.006 m |
| DA | 70 | −0.050 × 70/300 = −0.012 m | 80 | +0.030 × 80/350 = +0.007 m |
Key observation: Notice that the Bowditch corrections are the same for legs AB and DA (both 120 m) because their lengths are equal. Under Transit rule, the corrections differ between those legs because their latitudes and departures differ. This is the fundamental difference — Bowditch weights by geometry (length), Transit weights by magnitude of the computed components.
Adjusted bearing and length
After applying corrections, the adjusted bearing and length of each leg are recomputed from the corrected latitude and departure:
Precision, Accuracy and Angular Check
Linear precision ratio
The precision of a closed traverse is expressed as the ratio of the linear closing error to the total perimeter:
| Survey type | Precision ratio | Typical application |
|---|---|---|
| Rough stadia survey | 1:500 to 1:1000 | Reconnaissance, topographic sketch |
| Ordinary engineering | 1:1000 to 1:5000 | Road alignment, earthworks, pipe laying |
| Good engineering | 1:5000 to 1:10,000 | Building setting out, control surveys |
| First-order geodetic | 1:50,000 or better | Primary control, national grid |
Angular closure check
For a closed polygon traverse with \(n\) sides, the theoretical sum of interior angles is:
Sources of error
- Instrumental errors: poorly calibrated theodolite, collimation error, incorrect levelling of instrument.
- Natural errors: temperature variations affecting tape length, wind affecting the EDM signal, atmospheric refraction.
- Personal errors: incorrect reading of the circle, poor centring over the station, booking mistakes.
- Systematic errors: tape that is not exactly the nominal length, sag in suspended tape, incorrect temperature/tension corrections.
Omitted measurements
If one leg of a closed traverse has unknown length or bearing (but not both), it can be computed from the closure conditions. For a missing bearing \(\theta\) and length \(l\):
Frequently Asked Questions
1. What is a Free Gale's Table Generator and what does this tool do?
This free Gale's Table generator is an online tool for computing a complete closed traverse table. Enter your survey leg lengths and WCB bearings (manually or via CSV), select Bowditch or Transit correction, and the tool instantly generates: reduced bearings, latitudes, departures, closing errors, Bowditch corrections, corrected latitudes and departures, consecutive northing/easting coordinates, adjusted bearings and adjusted lengths. A full step-by-step solution and SVG traverse plot are also shown.
2. What is a Gale's Table in traverse surveying?
Gale's Table (traverse computation sheet) is the standard tabular format used in engineering surveying to process closed traverse data. It computes column-by-column: WCB, reduced bearing, latitude (+N/−S), departure (+E/−W), Bowditch or Transit corrections, corrected latitudes and departures, running easting/northing coordinates, and adjusted bearings and lengths. The systematic layout makes all intermediate steps easy to check and verify.
3. How does Bowditch correction work in a closed traverse — with a numerical example?
The Bowditch correction (compass rule) distributes the closing error proportionally to each leg's length. Example: total perimeter ΣL = 500 m, latitude closing error δL = +0.050 m. For leg AB (length 120 m): Bowditch correction to latitude = −0.050 × (120/500) = −0.012 m. For leg BC (length 80 m): correction = −0.050 × (80/500) = −0.008 m. The corrections for all legs sum to −δL = −0.050 m, exactly cancelling the error. The same proportional formula applies to departure corrections using δD.
4. When should I use Bowditch rule versus Transit rule for traverse correction?
Use the Bowditch rule (compass rule) for most engineering traverses — road surveys, drainage, building setting-out — where both linear and angular measurements have roughly equal proportional accuracy. Bowditch correction is proportional to leg length: c_L = −δL × (l_i/ΣL). Use the Transit rule only when angular measurements are known to be significantly more precise than linear measurements, such as a theodolite traverse connected to a triangulation network. Transit corrections are proportional to the absolute latitude or departure of each leg, not its length.
5. What is a closed traverse and why must ΣLat = ΣDep = 0?
A closed traverse is a series of survey legs that forms a closed polygon, starting and ending at the same known point. Geometrically, the algebraic sum of all north-south components (latitudes) must be zero and the sum of all east-west components (departures) must be zero because the traverse returns to its origin. Measurement errors produce a small discrepancy called the closing error. Bowditch or Transit corrections are applied to each leg to force both sums back to zero.
6. What are latitude and departure in a Gale's Table calculation?
For a traverse leg of length L and quadrant bearing θ: Latitude = L × cos(θ) = the north-south component (positive = northward, negative = southward). Departure = L × sin(θ) = the east-west component (positive = eastward, negative = westward). Example: leg 120 m at WCB 64°20' (NE quadrant, θ=64.333°): Latitude = 120×cos(64.333°) = +51.87 m (north). Departure = 120×sin(64.333°) = +108.20 m (east). These are the coordinate differences ΔN and ΔE for that leg.
7. How do I convert WCB to reduced (quadrant) bearing for latitude/departure calculation?
WCB (Whole Circle Bearing) is 0°–360° clockwise from north. To find the quadrant bearing and sign rules: NE (WCB 0–90°): QB = N(WCB)E, Lat=+N, Dep=+E. SE (WCB 90–180°): QB = S(180−WCB)E, Lat=−S, Dep=+E. SW (WCB 180–270°): QB = S(WCB−180)W, Lat=−S, Dep=−W. NW (WCB 270–360°): QB = N(360−WCB)W, Lat=+N, Dep=−W. Example: WCB 134°50' is SE quadrant, QB = S(180−134°50')E = S45°10'E, latitude negative (southward), departure positive (eastward).
8. What is precision ratio in a closed traverse and what values are acceptable?
Precision ratio = linear closing error / total perimeter, expressed as 1:n. The linear closing error e = √(δL²+δD²). Example: δL=0.05 m, δD=0.08 m, perimeter=500 m → e=√(0.05²+0.08²)=0.094 m → precision = 1:(500/0.094) = 1:5319 (Good). Acceptable standards: Rough survey 1:500–1:1000. Ordinary engineering 1:1000–1:5000. Good engineering 1:5000–1:10,000. First-order geodetic 1:50,000+. This tool automatically classifies the result.
9. How are adjusted bearing and adjusted length calculated in Gale's Table?
After applying Bowditch or Transit corrections, the adjusted bearing and length are recomputed from the corrected components: Adjusted length = √(corrected_lat² + corrected_dep²). The adjusted bearing angle θ = arctan(|corrected_dep| / |corrected_lat|), then the quadrant is assigned from the signs of corrected_lat (+N or −S) and corrected_dep (+E or −W), and finally converted to WCB. These adjusted values are slightly different from the original measured values because the corrections redistribute the closing error.
10. Why does the sum of Bowditch corrections sometimes differ slightly from the closing error?
Individual Bowditch corrections are rounded to 4 decimal places (0.1 mm precision). The sum of all rounded corrections may differ from the exact closing error by ±1 in the last decimal place. This is normal and expected in any manual or digital traverse computation. The standard practice is to assign the small rounding discrepancy (typically 0.0001–0.001 m) to the leg with the largest rounding residual, usually the longest leg. This calculator reports ΣCorrLat and ΣCorrDep in the check row so you can verify the balance to 5 decimal places.