Scale Factors

Overview#

Every distance a surveyor measures in the field is a ground distance -- a measurement taken at the physical elevation of the terrain, across the curved surface of the Earth. But when that measurement enters a coordinate system like State Plane or UTM, it must be expressed as a grid distance -- a distance on the flat projection surface defined by the coordinate system. These two distances are never exactly equal.

The difference arises from two independent geometric realities. First, map projections distort distances as they flatten the curved ellipsoid onto a plane. Second, the Earth's surface sits above the ellipsoid, and distances measured at elevation are systematically longer than their ellipsoidal equivalents. Scale factors quantify both of these effects and provide the mathematical bridge between what the surveyor measures on the ground and what the coordinate system reports.

"Measured distances must be reduced to their equivalents on the mapping surface before state plane or UTM coordinates can be computed from them." -- Ghilani & Wolf, Elementary Surveying: An Introduction to Geomatics (13th Ed.), Ch. 20, p. 595

For boundary surveys over small areas at low elevations, the difference between ground and grid distances may be negligible -- a few thousandths of a foot over a typical lot. But for control surveys, route surveys, large subdivisions, and any work tied to geodetic coordinates, ignoring scale factors introduces systematic error that accumulates with distance. A surveyor who understands scale factors can move confidently between field measurements and coordinate computations, and can explain to clients why a measured distance does not match the distance computed from coordinates.

Grid Scale Factor ($k$)#

What It Represents

The grid scale factor, commonly denoted $k$, is the ratio of a distance on the projection surface (the grid) to the corresponding distance on the reference ellipsoid. It quantifies the distortion introduced by the map projection at a specific location:

$k = \frac{\text{distance on the projection (grid)}}{\text{distance on the ellipsoid}}$

No map projection preserves distances everywhere. Every projection introduces scale distortion that varies by position within the zone. The grid scale factor captures exactly how much distortion exists at any given point.

Behavior by Projection Type

The two projection types used in State Plane Coordinate Systems behave differently:

Lambert Conformal Conic (used for states wider east-west): Scale factor varies with latitude. At the two standard parallels -- the latitudes where the cone intersects the ellipsoid -- $k = 1.0$ exactly. Between the standard parallels, $k < 1$ (grid distances are shorter than ellipsoidal distances). Outside the standard parallels, $k > 1$ (grid distances are longer).

Transverse Mercator (used for states longer north-south): Scale factor varies with distance from the central meridian. Along the two standard lines (lines parallel to and on either side of the central meridian where the cylinder intersects the ellipsoid), $k = 1.0$ exactly. Between the standard lines (near the central meridian), $k < 1$. Beyond the standard lines (farther from the central meridian), $k > 1$.

In both cases, the projection is designed so that $k$ never deviates far from 1.0 within the zone. For SPCS 83 zones, the grid scale factor typically ranges from about 0.9999 to 1.0001 -- a maximum distortion of roughly 1 part in 10,000, or about 1 foot per 2 miles.

"In the Lambert conformal conic projection, the scale is exact along the two standard parallels and changes in a north-south direction only. In the transverse Mercator projection, the scale is exact along two lines roughly parallel to the central meridian." -- Ghilani & Wolf, Elementary Surveying: An Introduction to Geomatics (13th Ed.), Ch. 20, p. 586

Determining the Grid Scale Factor

The grid scale factor for a specific location can be obtained from:

• NGS software and tools -- Programs like the State Plane Coordinate conversion tools compute $k$ for any given latitude/longitude.
• Published tables -- NGS provides tabulated scale factors by latitude or distance from the central meridian for each SPCS zone.
• Direct computation -- For Lambert zones, $k$ is a function of latitude. For Transverse Mercator zones, it can be approximated as:

$k \approx k_0 + \frac{E'^2}{2R^2}$

where $k_0$ is the scale factor at the central meridian (typically 0.9999 or similar for State Plane zones), $E'$ is the distance from the central meridian, and $R$ is the mean radius of curvature of the ellipsoid.

For UTM zones, $k_0 = 0.9996$ at the central meridian, producing larger distortions than State Plane zones but covering wider areas.

When a line spans a significant distance within a zone, the grid scale factor varies along its length. In such cases, the scale factor for the line can be computed as the average of the scale factors at the endpoints and the midpoint, weighted by Simpson's Rule:

$k_{\text{line}} = \frac{k_1 + 4k_m + k_2}{6}$

where $k_1$ and $k_2$ are the scale factors at the endpoints and $k_m$ is the scale factor at the midpoint.

Elevation Factor#

The Geometry

The elevation factor -- also called the sea level factor or ellipsoid factor -- accounts for the geometric difference between a distance measured at ground elevation and the corresponding distance on the ellipsoid surface. It has nothing to do with the map projection; it is purely a consequence of measuring at a height above the reference surface.

Consider two points on the ground separated by a horizontal distance $D_{\text{ground}}$. Both points sit at an elevation $h$ above the ellipsoid. The corresponding distance on the ellipsoid surface, $D_{\text{ellipsoid}}$, is shorter because the ellipsoid has a smaller radius of curvature than the surface at elevation $h$. The relationship is:

$\text{Elevation Factor} = \frac{R}{R + h}$

where:

• $R$ is the mean radius of the Earth (approximately $6{,}371{,}000$ m, or $20{,}906{,}000$ ft)
• $h$ is the elevation of the survey above the ellipsoid (the ellipsoid height, not the orthometric elevation)

Practical Considerations

The elevation factor is always less than 1.0, meaning that the ellipsoidal distance is always shorter than the ground distance. The higher the elevation, the larger the correction:

A few important notes:

• Strictly speaking, the height $h$ in the elevation factor formula should be the ellipsoid height, not the orthometric elevation (elevation above the geoid). The difference between the two is the geoid undulation $N$, so $h = H + N$, where $H$ is the orthometric height. In the conterminous United States, geoid undulations range from about $-8$ m to $-53$ m (the geoid is below the ellipsoid). For most practical applications, the difference between using $H$ and $h$ is negligible, but for high-precision geodetic work the distinction matters.

• When a line spans a significant elevation range, the average elevation of the two endpoints should be used.

"The scale of the projection surface is transferred to the Earth's surface at sea level, and then it is necessary to further reduce measured distances to sea level before computing coordinates." -- Ghilani & Wolf, Elementary Surveying: An Introduction to Geomatics (13th Ed.), Ch. 20, p. 595

Combined Factor (CF)#

The combined factor (sometimes called the combined scale factor) is the single multiplier that directly converts a ground distance to a grid distance. It is the product of the grid scale factor and the elevation factor:

$CF = k \times \frac{R}{R + h}$

And the fundamental conversion equation is:

$D_{\text{grid}} = D_{\text{ground}} \times CF$

Conversely, to go from grid distance back to ground distance:

$D_{\text{ground}} = \frac{D_{\text{grid}}}{CF}$

The combined factor accounts for both sources of distortion simultaneously -- the map projection distortion and the elevation effect. In some locations and elevations, the two effects partially cancel each other. For example, in a region near the center of a State Plane zone (where $k < 1$) at a moderate elevation (where the elevation factor is also $< 1$), both factors reduce the ground distance, and the combined effect is additive. In a region outside the standard parallels/lines (where $k > 1$) at elevation, the grid scale factor increases the distance while the elevation factor decreases it, producing partial cancellation.

"The combined factor, which is the product of the grid scale factor and the elevation factor, is applied to convert ground-level distances to grid distances, or vice versa." -- Ghilani & Wolf, Elementary Surveying: An Introduction to Geomatics (13th Ed.), Ch. 20, p. 596

Practical Application#

Worked Example

Problem: A surveyor measures a horizontal distance of $1{,}000.000$ ft on the ground. The survey is at an average elevation of $2{,}500$ ft above the ellipsoid, in a State Plane zone where the grid scale factor is $k = 0.99990$. What is the grid distance?

Step 1: Compute the Elevation Factor

$\text{Elevation Factor} = \frac{R}{R + h} = \frac{20{,}906{,}000}{20{,}906{,}000 + 2{,}500}$

$\text{Elevation Factor} = \frac{20{,}906{,}000}{20{,}908{,}500} = 0.999880$

Step 2: Compute the Combined Factor

$CF = k \times \text{Elevation Factor} = 0.99990 \times 0.999880$

$CF = 0.999780$

Step 3: Compute the Grid Distance

$D_{\text{grid}} = D_{\text{ground}} \times CF = 1{,}000.000 \times 0.999780$

$D_{\text{grid}} = 999.780 \text{ ft}$

Interpretation: The grid distance is $0.220$ ft (about $2\frac{5}{8}$ inches) shorter than the ground distance. Over a 1,000-foot line, this is roughly 1 part in 4,500 -- well within the threshold where the correction matters for control-quality work but might be ignored for a residential lot survey.

Reverse Conversion

If a surveyor needs to stake a point at a grid distance of $500.000$ ft from a known control point, the ground distance to set on the instrument is:

$D_{\text{ground}} = \frac{D_{\text{grid}}}{CF} = \frac{500.000}{0.999780} = 500.110 \text{ ft}$

The surveyor must measure $500.110$ ft on the ground to place the point at the correct grid position.

When Scale Factors Matter#

Not every survey requires rigorous application of scale factors. The decision depends on the size of the project, the required accuracy, and the coordinate system in use.

When They Can Be Ignored

• Small boundary surveys at low elevation where coordinates are not tied to a geodetic system.
• Relative measurements within a project where all distances are measured on the ground and no grid coordinates are involved.
• Topographic surveys where the mapping precision does not warrant the correction.

When They Cannot Be Ignored

• Control surveys tied to State Plane or UTM coordinates.
• Route surveys (highways, pipelines, railways) spanning miles, where the systematic error accumulates.
• Large subdivisions where lot dimensions must agree with both field measurements and computed coordinate geometry.
• Geodetic work of any kind.
• ALTA/NSPS Land Title Surveys when coordinates are reported on a geodetic datum.
• Construction staking from grid coordinates -- failing to apply the combined factor means every point will be set in the wrong position.

Rules of Thumb

These approximations help surveyors quickly assess whether scale factors are significant for a given project:

• Elevation factor: At 1,000 ft elevation, the correction is approximately 1 ft per 20,000 ft (1 part in 20,906). At 5,000 ft elevation, it is roughly 1 ft per 4,000 ft.
• Grid scale factor: In a typical State Plane zone, the maximum deviation from 1.0 is about 1 part in 10,000, or approximately 0.5 ft per mile.
• Combined factor: As a rough test, if the combined correction would amount to less than the survey's measurement uncertainty over the distances involved, it can reasonably be omitted. For a 1:10,000 accuracy survey over a 500-foot boundary line, the correction at low elevation (a few hundredths of a foot) is generally below the noise.

Project-Specific Scale Factors#

Ground Coordinate Systems

Many surveyors, particularly in construction, find it impractical to work with grid distances that do not match field measurements. A common solution is to create a ground coordinate system -- a modified version of the State Plane system that has been scaled to read true ground distances at the project site.

The process is straightforward:

1. Determine the combined factor ($CF$) for the project area, using the average grid scale factor and the project's average elevation.
2. Divide all State Plane coordinates by $CF$ to produce ground-level coordinates. Alternatively, apply the reciprocal of $CF$ as a scale factor to the coordinate system.
3. Optionally apply a translation (false northing/easting adjustment) so that ground coordinates are clearly distinguishable from standard State Plane values.

The result is a coordinate system where distances computed from coordinates match distances measured on the ground -- at least within the project area where the combined factor is reasonably uniform.

Example

If the combined factor for a project site is $CF = 0.999780$, the ground-level scale factor applied to State Plane coordinates is:

$\text{Ground Factor} = \frac{1}{CF} = \frac{1}{0.999780} = 1.000220$

State Plane northing and easting values are multiplied by this factor to produce ground coordinates. A point with State Plane coordinates of $N = 2{,}000{,}000.00$ ft, $E = 6{,}000{,}000.00$ ft would become approximately $N = 2{,}000{,}440.00$ ft, $E = 6{,}001{,}320.00$ ft in the ground system.

Cautions

Ground coordinate systems are powerful tools, but they carry risks if not properly documented:

• The scale factor must be recorded on every deliverable that uses the modified system. A future surveyor who treats ground coordinates as State Plane coordinates will introduce the very systematic error the original surveyor was trying to avoid.
• The system is only valid within the project area. The combined factor changes with location and elevation; a ground system designed for one site should not be extended to distant areas.
• Mixing systems is dangerous. If some project data is on State Plane and other data is on a ground system, combining them without accounting for the scale factor will produce errors.
• Clearly label the datum and projection basis. A note such as "Ground coordinates based on [State Plane Zone], combined scale factor 0.999780 applied" should appear on plans, base maps, and digital files.

"When coordinates are computed on a ground coordinate system, the basis of the coordinate system, including the scale factor used, must be clearly stated on all documents." -- Ghilani & Wolf, Elementary Surveying: An Introduction to Geomatics (13th Ed.), Ch. 20, p. 601

Key Takeaways#

• Ground distances and grid distances are never exactly equal. Two independent factors -- the map projection distortion and the elevation above the ellipsoid -- cause the difference.
• Grid scale factor ($k$) is the ratio of grid distance to ellipsoidal distance. It varies by position within the projection zone and equals 1.0 only along the standard parallels (Lambert) or standard lines (Transverse Mercator).
• Elevation factor ($R / (R + h)$) reduces ground distances to the ellipsoid. It is always less than 1.0 and becomes more significant at higher elevations.
• Combined factor ($CF = k \times R/(R+h)$) is the single multiplier that converts ground distance to grid distance. Grid Distance = Ground Distance $\times$ CF.
• The correction matters for control surveys, route surveys, large subdivisions, geodetic work, and construction staking from grid coordinates. It can typically be ignored for small boundary surveys at low elevation.
• Ground coordinate systems apply a project-specific combined factor to make coordinates read true ground distances -- but the factor used must be documented on every deliverable.
• Always know which distance you are working with. Confusing ground and grid distances is a systematic error that affects every measurement on the project.

References#

1. Ghilani, C.D. & Wolf, P.R. Elementary Surveying: An Introduction to Geomatics (13th Ed.). Pearson, 2012. Chapters 20--21.
2. National Geodetic Survey. "State Plane Coordinate System." NOAA/NGS. https://geodesy.noaa.gov/SPCS/
3. National Geodetic Survey. "SPCS2022 -- State Plane Coordinate System of 2022." NOAA/NGS. https://geodesy.noaa.gov/INFO/SPCS2022/
4. Meyer, T.H. Introduction to Geometrical and Physical Geodesy: Foundations of Geomatics. ESRI Press, 2010. Chapters 8--9.
5. Stem, J.E. State Plane Coordinate System of 1983. NOAA Manual NOS NGS 5, National Geodetic Survey, 1989.
6. Burkholder, E.F. "The 3-D Global Spatial Data Model." CRC Press, 2008.