Scale factor and unit conversion

A distance is never just a number. It belongs to a surface — the ground you taped it on, or the grid a projection defined — and it belongs to a unit, which in the United States means one of two feet that differ by two parts in a million. This page handles both problems, because they are the two ways a correct measurement turns into a wrong coordinate.

What a grid scale factor is, and why a projection needs one

The earth is curved and a plan sheet is flat, and no mapping between the two preserves every distance. A projection chooses what to preserve and where to give ground. The State Plane systems use conformal projections — a Lambert conformal conic for zones that run east and west, a transverse Mercator for zones that run north and south — which preserve angles exactly and let distances stretch and shrink instead.

Both projections are made secant rather than tangent: the projection surface cuts through the ellipsoid rather than touching it. Between the two lines of intersection the grid is slightly smaller than the ellipsoid and the scale factor is below 1; outside them the grid is slightly larger and the scale factor is above 1. Zone widths were chosen so that the departure from unity stays inside roughly 1 part in 10,000, which is why a scale factor is typically something like 0.9999 or 1.0001 and why it matters at all: 1 part in 10,000 is 0.1 ft in 1,000 ft, well inside what an ordinary boundary survey has to hold.

The elevation factor

The grid scale factor takes you between the grid and the ellipsoid. It says nothing about the fact that you measured on the ground, which is somewhere above the ellipsoid. Two monuments a mile apart at 5,000 ft are further apart than their projections onto the ellipsoid, for the same reason that two radii of a circle diverge as you move outward, and the elevation factor is that ratio of radii.

H is the elevation you know — height above the vertical datum, the number on a benchmark. N is the geoid height, the separation between geoid and ellipsoid, and it is what turns an orthometric height into the ellipsoid height the geometry wants. Across the conterminous United States N is negative, commonly between about −10 m and −35 m. Omitting it is a small error, but a systematic one, and it is free to avoid.

The combined factor, and which way to go

The two factors multiply into one number that takes a distance all the way from the ground to the grid.

Getting the direction backwards is the classic error in this arithmetic, and it is nastier than it looks. It does not produce a small mistake — it produces the correction applied twice with the sign reversed. At a combined factor of 0.99971 a 1,000 ft distance should become 999.71 ft going to grid; dividing instead gives 1,000.29 ft, which is 0.58 ft away from the right answer and still looks like a perfectly ordinary distance. Nothing downstream will flag it.

The check that survives a bad day: on a combined factor below 1, the ground distance is the larger of the two, because the ground is further from the center of the earth than the grid surface. If your conversion made the ground distance shorter, you divided where you should have multiplied.

The two feet

Two feet have been in legal use in the United States, and both are exact by definition rather than measured.

The ratio between them is exact in a pleasing way: an international foot is 0.999998 US survey feet, exactly, so a US survey foot is 1/0.999998 international feet, which expands to 1.000002000004000008… The difference is therefore quoted as 2 parts per million, and that is correct to seven significant figures — but not exactly. The exact figure is 2.000004 ppm, and the trailing digits are recurring, not rounding error.

Two ppm is nothing on a tape length: 0.0002 ft over 100 ft. It is 0.02 ft over a 10,000 ft traverse, still inside the noise. It is 2 ft on a 1,000,000 ft State Plane easting and 4 ft on a 2,000,000 ft northing — not a measurement problem but a blunder, and one that lands the project a car length from where it belongs.

Why State Plane data in feet is usually US survey feet

The original State Plane Coordinate System was defined in the 1930s, before the international foot existed, in feet later formalised as the US survey foot. When the system was readjusted onto NAD83 the defining constants became metric, but the state legislatures that had written coordinates into statute mostly specified the US survey foot for the conversion back. So a NAD83 State Plane coordinate labelled only as feet is, in most states, in US survey feet — and guessing is not a method. The defining statute or the NGS zone documentation says which.

The failure mode is quiet. The wrong foot shifts a State Plane coordinate by 2 ppm of its own magnitude — a few feet — and every distance inversed from the shifted file is still right to 2 ppm, so the internal geometry looks perfect. Only the tie to published control fails, and only if you check it.

Areas square the error

An area is a length times a length, so a 2 ppm ambiguity in the foot becomes a 4 ppm ambiguity in the acreage: both sides of the figure carry it. Precisely, the area ratio is 1.000002000004…², or 4.000008 ppm. On a 100 acre parcel that is 0.0004 acres, and on a quarter section it is about 0.0006 acres — small enough never to matter to the description, large enough to explain why a recomputed acreage disagrees with the deed in the fourth decimal when every distance checks.

The deprecation, and what it does and does not change

NIST and NOAA announced the deprecation of the US survey foot for new work in a Federal Register notice at 84 FR 55510, dated 17 October 2019, with the unit to be retired after 31 December 2022. Since then the international foot is the foot for new applications, and the modernised National Spatial Reference System is defined in meters. The 1959 notice at 24 FR 5348 is the other half of the story: it is where the international foot was defined as exactly 0.3048 m and the US survey foot was retained, for surveying purposes only, at 1200/3937 m.

• Deprecation is not retroactive. Coordinates published in US survey feet before the retirement are still in US survey feet, and re-interpreting them as international feet corrupts them.
• New work uses meters or international feet, and says which, in the metadata and on the face of the drawing.
• Any file that carries feet without naming the foot should be treated as unknown until the source is checked, however inconvenient that is.
• Chains, links and rods remain defined on the US survey foot, because the rectangular system was run in them and its record is what they are used to retrace.

Chains, links and rods

Gunter's chain is 66 ft — 4 rods, 100 links — the unit the public lands were surveyed in. It survives because it makes areas come out whole: a square chain is 4,356 ft², ten square chains are an acre, and 80 chains are a mile, so a nominal section is 80 by 80 chains and 640 acres. A description in chains and links is the record, and converting it to feet is the first step in retracing it. All three are converted here on the US survey foot, the foot the original notes were run in.