Elevation factor and the combined factor

You measure distances on the ground, at whatever elevation the ground happens to be. The projection is defined on the ellipsoid, which is somewhere below you. Before a measured distance can be compared with a grid distance it has to be reduced to the ellipsoid, and that reduction is the elevation factor.

It is often called the sea level factor, which is a historical name and a slightly misleading one — the reduction is to the ellipsoid, not to sea level, and those differ by the geoid height. In the conterminous United States the geoid sits tens of meters below the ellipsoid, so the distinction is worth several parts per million.

The geometry

Two points at the top of a tower are farther apart than the same two directions measured at the base, because the radial lines from the Earth's center diverge. Reducing a distance from height H down to the ellipsoid is therefore just similar triangles: multiply by the ratio of the radii.

elevation factor = R ÷ (R + H + N)

R ≈ 6 372 000 m is the mean radius of the Earth, H is the orthometric height, and N is the geoid height at the point.

The sum H + N is the ellipsoid height, h. So the formula is really R ÷ (R + h) — the geoid height appears only because your elevations are normally orthometric, and you have to add N to get to the ellipsoid. In CONUS N is negative, typically between about −8 m and −50 m, so including it makes the elevation factor slightly closer to 1 than it would otherwise be.

How big is it?

The elevation factor changes by about 15.7 ppm per 100 m of height, or about 47.8 ppm per 1000 ft. That gives you a quick mental estimate: a project at 5000 ft sits around 240 ppm below the ellipsoid in scale, which is a quarter of a foot per thousand feet.

Elevation factor at various heights, computed with R = 6 372 000 m and H + N as the ellipsoid height
Ellipsoid height h = H + N	Elevation factor	Departure in ppm	Effect on a 10 000 ft line
0 m	1.00000000	0.0	0.00 ft
100 m	0.99998431	−15.7	−0.16 ft
351.5 m	0.99994484	−55.2	−0.55 ft
500 m	0.99992154	−78.5	−0.78 ft
1000 m	0.99984309	−156.9	−1.57 ft
1592.1 m	0.99975020	−249.8	−2.50 ft
2000 m	0.99968623	−313.8	−3.14 ft

The combined factor

Combined factor: The product of the grid scale factor and the elevation factor. It converts a measured ground distance directly to a grid distance in a single multiplication, which is why it is the number a crew actually carries.

combined factor = grid scale factor × elevation factor

Sometimes called the grid factor or the combined grid factor.

grid distance = ground distance × combined factor ground distance = grid distance ÷ combined factor

Grid is smaller than ground in most of the United States, because both factors are usually less than 1.

Worked example: reducing a measured distance to grid

A crew measures a horizontal ground distance of 2468.55 ft on a site at an orthometric height of 1250 US survey feet, where the geoid height is −29.5 m. The published grid scale factor for the project center is 0.99992360.

Convert the height: 1250 US survey feet × 1200/3937 = 381.001 m.
Add the geoid height: h = H + N = 381.001 + (−29.5) = 351.501 m.
Elevation factor = 6 372 000 ÷ (6 372 000 + 351.5) = 0.99994484, or −55.2 ppm.
Combined factor = 0.99992360 × 0.99994484 = 0.99986844, or −131.6 ppm.
Grid distance = 2468.55 × 0.99986844 = 2468.225 ft.
The line is 0.325 ft shorter on the grid than it was on the ground.

Worked example: going the other way

Now a higher site. Two published control points in a mountain state have a grid distance between them of 14 500.000 ft. The project sits at an orthometric height of 5280 US survey feet with a geoid height of −17.2 m, and the grid scale factor is 0.99990120. The crew needs to know what distance to expect on the ground.

Convert the height: 5280 US survey feet × 1200/3937 = 1609.347 m.
h = 1609.347 + (−17.2) = 1592.147 m.
Elevation factor = 6 372 000 ÷ (6 372 000 + 1592.147) = 0.99975020, or −249.8 ppm.
Combined factor = 0.99990120 × 0.99975020 = 0.99965142, or −348.6 ppm.
Ground distance = 14 500.000 ÷ 0.99965142 = 14 505.056 ft.
The ground distance exceeds the grid distance by 5.056 ft.

Five feet on fourteen thousand is not a rounding difference. A crew that stakes from grid coordinates while taping check distances on the ground will find that discrepancy, and if they do not know where it comes from they will chase it through the instrument, the prism constant and the atmospheric correction before finding it.

Practical notes

Use ellipsoid height, not orthometric height, in the denominator. That means adding the geoid height, which is negative in CONUS.
One combined factor for a project is fine when the terrain relief and the scale variation are both small. Check both before assuming it.
On a project with 500 ft of relief, the elevation factor changes by about 24 ppm from bottom to top. Whether that matters depends on the tolerance.
Do not apply a combined factor twice. This happens when software is set to reduce automatically and the crew also carries a factor in the data collector.
Areas scale as the square of the combined factor. For a combined factor of 0.99986844 the area factor is 0.99973690, so a 40.000 acre grid area is 40.011 acres on the ground.
Record the combined factor and the point it was computed at on the deliverable.

Questions

What is the formula for the elevation factor?

Elevation factor equals R divided by (R + H + N), where R is about 6 372 000 m, H is the orthometric height and N is the geoid height. The sum H + N is simply the ellipsoid height, so the reduction is to the ellipsoid rather than to sea level despite the older name.

Do I really need the geoid height in the elevation factor?

It is worth including. At a geoid height of −30 m, omitting N shifts the factor by about 4.7 ppm — roughly 0.05 ft on a 10 000 ft line. That is small but systematic, and adding one number costs nothing.

How do I convert grid distance to ground distance?

Divide by the combined factor. Ground equals grid divided by the combined factor; grid equals ground multiplied by it. Since the combined factor is usually less than 1 in the United States, ground distances are usually the larger number.

How much does elevation change the factor?

About 15.7 ppm per 100 m of height, or about 47.8 ppm per 1000 ft. A project at 5000 ft therefore carries an elevation factor around 240 ppm below 1 — roughly a quarter of a foot per thousand feet.

Sources

NOAA / National Geodetic Survey — Guidance on reducing observed distances to the ellipsoid and to the grid, and on the use of geoid heights in that reduction.