A distance measured with a total station or a chain is a ground distance. A distance computed by inversing between two State Plane coordinates is a grid distance. They are not the same number, the difference is systematic rather than random, and getting the direction of the correction backwards doubles the error instead of removing it.
This is not a subtle topic. It is a topic that is simple, well documented, and still responsible for a steady supply of stakeout disputes, because the numbers look close enough to pass a casual glance.
The two rules
grid distance = ground distance × combined factorground distance = grid distance ÷ combined factorIn the United States the combined factor is almost always less than 1, because both of its components usually are. That gives a useful sanity check that requires no arithmetic: grid distances are normally shorter than ground distances. If your correction made the grid distance longer, look again.
Where the two factors come from
The combined factor bundles two entirely separate effects and it helps to keep them separate in your head, because they respond to different things.
| Factor | What it accounts for | Depends on | Typical size in CONUS |
|---|---|---|---|
| Grid scale factor | The projection flattening the ellipsoid | Position within the zone | ±100 ppm in State Plane, −400 to +970 ppm in UTM |
| Elevation factor | Your height above the ellipsoid | Orthometric height plus geoid height | −15.7 ppm per 100 m of height |
| Combined factor | Both together | Both | Usually between about 0.9995 and 1.0000 |
A full worked example
A crew is staking a building corner from published State Plane coordinates on a site at 1250 US survey feet elevation with a geoid height of −29.5 m. The grid scale factor at the project is 0.99992360.
- Ellipsoid height h = 1250 × 1200/3937 + (−29.5) = 381.001 − 29.5 = 351.501 m.
- Elevation factor = 6 372 000 ÷ 6 372 351.5 = 0.99994484.
- Combined factor = 0.99992360 × 0.99994484 = 0.99986844.
- The inverse between the two published coordinates gives a grid distance of 2468.225 ft.
- Ground distance = 2468.225 ÷ 0.99986844 = 2468.550 ft.
- The crew should expect to measure 2468.55 ft in the field, not 2468.23 ft.
The difference is 0.325 ft over 2468 ft — about 132 ppm. On a building layout that is enough to matter, and on a long boundary line it is enough to create a discrepancy against a record distance that will need explaining.
A larger example, at elevation
The same arithmetic at 5280 US survey feet with a geoid height of −17.2 m and a grid scale factor of 0.99990120 gives a combined factor of 0.99965142. A published grid distance of 14 500.000 ft corresponds to a ground distance of 14 500.000 ÷ 0.99965142 = 14 505.056 ft.
That is 5.06 ft. A crew that ignores the correction entirely will be 5.06 ft out. A crew that applies it in the wrong direction will be about 10.1 ft out. The second crew will also have a traverse that closes beautifully, because the error is a uniform scale change and closure ratios are blind to uniform scale.
Why a good closure does not protect you
This deserves its own heading because it is genuinely counterintuitive. A traverse closure test compares the sum of latitudes and departures against the known misclosure. If every distance in the traverse is scaled by the same factor, the shape of the traverse is unchanged and the misclosure scales with it, so the closure ratio comes out the same.
In other words, a scale error is invisible to the check most crews rely on. The errors that closure catches are blunders and random errors. Systematic scale errors pass straight through. The only way to catch them is to check against known, published coordinates — inverse between two control points and compare the computed distance with the measured one, corrected.
Ground coordinate systems, and the labelling problem they create
Many projects avoid the arithmetic altogether by working in a modified ground coordinate system: take the State Plane coordinates, divide by the combined factor about a project origin, and work in a grid on which measured distances match computed ones directly. It is a sensible practice and it is very widely used.
It is also a labelling hazard. Modified ground coordinates look exactly like State Plane coordinates, especially if the origin was left unshifted, and they will be mistaken for them by the next person unless the drawing says otherwise in plain language. Many practices deliberately apply a large offset to the coordinates so that nobody can confuse the two.
- Say on the drawing whether coordinates are grid or ground.
- If ground, state the combined factor used and the point or line it was computed at.
- If a shift was applied, state the shift.
- Never publish a coordinate labelled with a State Plane zone name if it has been scaled to ground.
- Check the whole thing by inversing between two published control points before you trust the setup.
The habit that prevents nearly all of this: at the start of every project, occupy two published control points, inverse between them, apply the combined factor, and compare with a measured distance. Five minutes of work that turns an assumption into an observation.