Combined scale factor (grid and ground)

The elevation factor and combined factor from a grid scale factor and height, and conversion of a distance between grid and ground.

A published result changed on 2026-09-19: Traverse closure and the combined grid-to-ground factor are now stable, so their results no longer carry the EXPERIMENTAL_TOOL warning; the perfect-closure and elevation-as-ellipsoid-height warnings moved to the first position. Both match published worked examples (University of Memphis traverse notes and NOAA Manual NOS NGS 5) to every printed digit. No numbers changed. Changelog

Computing your own values needs JavaScript and WebAssembly. The worked example below was computed when this page was built, and the documentation under "How we got this" applies.

0.99983826

The combined factor is 0.99983826. A ground distance of 1,000 ft is 999.838 ft on the grid.

Elevation factor
Grid distance
Ground distance

Provenance

Computed by: survey.reduction.combined-factor 1.0.0, core 0.1.0
Model: Elevation factor = R/(R + h) with h the ellipsoid height; combined factor = grid factor × elevation factor (Stem 1990)
Accuracy: About 1 ppm with the mean radius; better with a local radius of curvature
Notes: None
Cites: Stem, J. E., National Geodetic Survey, State Plane Coordinate System of 1983, NOAA Manual NOS NGS 5

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Grid scale factor Point scale factor from the projection, like 0.99991 Ellipsoid heightoptional Height above the ellipsoid, like 1,500 ft (or give elevation and geoid height) · plain numbers mean ft Ground distanceoptional To convert to grid, like 1000 ft · plain numbers mean ft

More options 4

Orthometric elevationoptional Like 1,530 ft · plain numbers mean ft Geoid height Noptional Like -30 ft (negative in the conterminous US) · plain numbers mean ft Grid distanceoptional To convert to ground, like 1000 ft · plain numbers mean ft Earth radiusoptional Default 20,906,000 ft (6,372,000 m), as in NOAA Manual NOS NGS 5 · plain numbers mean ft

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How we got thisFormula, worked example, sources, and proof

Model: Elevation factor = R/(R + h) with h the ellipsoid height; combined factor = grid factor × elevation factor (Stem 1990)

Show your work

Elevation factor
$EF = R / (R + h)$
6,372,000 m / (6,372,000 m + 457.2 m) = 0.99992825
Combined factor
CF = grid scale factor × elevation factor
0.99991 × 0.99992825 = 0.99983826

The same steps an agent gets from the MCP server with explain: true.

Accuracy: About 1 ppm with the mean radius; better with a local radius of curvature

When to use this: Use this before a grid distance becomes a ground distance or the other way round: the elevation factor from the height, the combined factor with the grid scale factor, and the converted distance. It is the reduction every project on State Plane or UTM coordinates needs.

Limitations: The elevation factor uses the mean radius unless you give a local radius of curvature, which is about a part per million either way, and heights must be above the ellipsoid rather than above sea level. The grid scale factor belongs to the point or the line you are reducing; using one factor across a long project is where the error usually comes from.

Worked example: Grid factor 0.99991 at 1,500 ft ellipsoid height. Source: NOAA Manual NOS NGS 5 formula with R = 20,906,000 ft. It is golden test vector v001, and every build checks the tool still gives its answer within its tolerance.

You enter

Ellipsoid height: 1500 ft
Grid scale factor: 0.99991
Ground distance: 1000 ft

You get

Combined factor: 0.99983826
Elevation factor: 0.99992825
Grid distance: 999.838 ft
Ground distance: 1,000 ft

Review: Not yet independently reviewed by a licensed surveyor.

Last verified: 2026-09-18, when a maintainer last confirmed this tool's sources at the issuer. See the sources ledger.

Status: version 1.0.0, core 0.1.0. See this tool in the verification report.

Changes

2026-09-19, result change: Traverse closure and the combined grid-to-ground factor are now stable, so their results no longer carry the EXPERIMENTAL_TOOL warning; the perfect-closure and elevation-as-ellipsoid-height warnings moved to the first position. Both match published worked examples (University of Memphis traverse notes and NOAA Manual NOS NGS 5) to every printed digit. No numbers changed. Changelog

Checked against: 22 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.

Sources

State Plane Coordinate System of 1983, NOAA Manual NOS NGS 5, Stem, J. E., National Geodetic Survey, Reprint with corrections. Section 4 (grid scale factor, elevation factor, combined factor; R = 20,906,000 ft).

Terms

NGS: National Geodetic Survey: The NOAA office that defines the US National Spatial Reference System: datums, geoid models, and state plane coordinates. Source: State Plane Coordinate System of 1983 (NOAA Manual NOS NGS 5)
NOAA: National Oceanic and Atmospheric Administration: The US agency that includes the National Geodetic Survey and the National Weather Service. Source: NOAA Solar Calculator
NOS: National Ocean Service: The part of NOAA that includes the National Geodetic Survey. Source: State Plane Coordinate System of 1983 (NOAA Manual NOS NGS 5)

Learn the concept: Grid vs ground distance explained