📐 Combined Scale Factor Formula
E = R / (R + H)
(R ≁E6,371,000 m; H = orthometric height in metres)
Step 2 —Grid Scale Factor:
k = from State Plane tables or NGS NCAT (typically 0.9999—.0001)
Step 3 —Combined Scale Factor:
C = E ±k
Step 4 —Convert ground to grid:
D_grid = D_ground ±C
Step 5 —Convert grid to ground:
D_ground = D_grid ÷ C
📊 Reference Table
| Elevation H (m) | Elevation Factor E | Typical k (SPCS) | Combined Factor C | 1 km E(mm) |
|---|---|---|---|---|
| 0 m (sea level) | 1.000000 | 0.99996 | 0.999960 | -40 mm |
| 200 m | 0.999969 | 0.99996 | 0.999929 | -71 mm |
| 500 m | 0.999921 | 0.99999 | 0.999911 | -89 mm |
| 1,000 m | 0.999843 | 1.00001 | 0.999853 | -147 mm |
| 2,000 m | 0.999686 | 1.00001 | 0.999696 | -304 mm |
⚠️ngineering Consequences
A documented real project showed that neglecting the combined scale factor on a 1-mile survey traverse produced a misclosure of 0.351 feet (107 mm) —yielding a ratio of 1:8,400. This fails the minimum closure requirement for:
- Cadastral boundary surveys (1:20,000 required)
- DOT ROW surveys (1:50,000 required)
- Structural engineering control (1:25,000 required)
Frequently Asked Questions
Do I need a grid-to-ground correction for State Plane coordinates?
Yes, always. Even within a single State Plane zone, the two effects (elevation factor and grid scale factor) both apply. At high elevations (1,000+ m), the elevation factor alone can introduce errors of 150+ mm per kilometer if ignored.
When is the combined scale factor exactly 1.000000?
The combined factor C = 1.000 only when the elevation factor and grid scale factor perfectly cancel —this happens at specific elevations and distances from the central meridian. In practice, no real project can assume C = 1 without computing it.
Using the wrong datum or applying coordinates without grid-to-ground correction can cause 1-400 metre positional errors —a leading cause of surveying negligence claims and contract disputes.