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How long is a foot? Does it surprise you that in many surveying and mapping applications the true answer is, “it depends”? There are two definitions for the foot as used in surveying and mapping, namely the U.S. Survey Foot and the International Foot, and the difference between the two can be both significant and troublesome if care isn’t exercised in their application.
I have been personally involved in the analysis and problem resolution of more than 100 projects in the last 12 years of my professional career that directly related to the use of the wrong definition for the foot. If you don’t have a perfect understanding of the U.S. Survey and International Foot and how the differences between the two can result in problems in your work, then make sure to read on.
By DefinitionThe history and application of the relationship between metric and English measurements of length is interesting. The U.S. Metric Law of 1866 provided the relationship that one meter is equal to 39.37 inches, exactly. This relationship forms the basis of the U.S. Survey Foot, which continues to have significant use in surveying and mapping today. Between 1893 and 1959 the U.S. further defined the yard as 3,600/3,937 meters. This math is fairly simple to work out given the constants of 36 inches per yard and 39.37 inches per meter. With a little further math reduction it should be simple to follow that one foot is equal to 1200/3937 meters, or basically 0.304800609601 meters. In my work, I have memorized the inverse of that relationship, namely that 3.28083333333 U.S. Survey Feet are equivalent to one meter.
A fundamental change in the relationship between metric and English or Imperial units, however, was created on June 25, 1959, when the definition of the U.S. and British yard were altered to reflect the relationship used in other countries. That relationship is based on one yard equal to 0.9144 meters, exactly. From that, it should follow that one foot is equal to one-third of that constant, or 0.3048 meters. This is also equivalent to 2.54 centimeters equal to 1 inch, which is the conversion I learned in college physics. This forms the basis of the International Foot. The inverse of this constant provides us with the relationship that 3.28083989501 International Feet are equal to one meter. This change in 1959 resulted in two possible methods for conversion between the units of meters and feet.
Does It Really Matter?
3.28083333333 feet per meter U.S. Survey Foot
3.28083989501 feet per meter International Foot
The two constants only differ beginning with the sixth place right of the decimal. Another way to analyze these two relationships is to divide one by the other. If we do so, the ratio is 0.999998, or two parts per million.
A difference of two parts per million is inconsequential for many things we do, like running differential levels or extending a traverse two miles with a total station (the difference would be two-hundredths of a foot in two miles). However, it can have a profound effect when we work with rectangular plane coordinates with values that are typically very large. And most professionals I know work with State Plane, UTM or similar coordinate systems on a frequent basis.
In a rectangular coordinate system, starting values are assigned to the grid origin to ensure that positive values are found within the limits of the typical boundaries of the projection. These starting values are known as the false northing and false easting. The northing of any point on that projection then becomes the sum of the false northing and the distance from the grid origin to the point measured in the direction of grid north. Similarly, the easting is the sum of the false easting and the distance from the grid origin to the point measured perpendicular to grid north. These values are normally very large, in the millions if not tens of millions, for most State Plane Coordinate or UTM projections, and the difference between the two conversion methods produce starkly different results for values in this range.
Grizzly Flat lies in the mountains east of Sacramento, Calif. Just east of Grizzly Flat is a location with the following metric coordinates in the California 2 Zone of California’s State Plane Coordinate System:
N 609,601.219 m
E 2,133,604.267 m
When this position is converted to English units, the possible values using the conversions for the U.S. Survey and International Foot are as follows:
U.S. Survey Foot
N 609,601.219 * (3937/1200) = 2,000,000.00 sFT
E 2,133,604.267 * (3937/1200) = 7,000,000.00 sFT
N 609,601.219/0.3048 = 2,000,004.00 iFT
E 2,133,604.267/0.3048 = 7,000,014.00 iFT
Is the difference between the two enough to be concerned about? The answer should now be clear. Using the wrong conversion factor for this point would result in an error of 4 feet in the northing and 14 feet in the easting. Two parts per million is very significant when working with the large coordinate values typical for plane coordinate systems!
NAD 83 Legislation
The U.S. Survey Foot has been the unit of choice for most surveying and mapping projects performed in the United States throughout time. The U.S. Coast and Geodetic Survey (USC&GS), now the National Geodetic Survey (NGS), originally computed and published values using the U.S. Survey Foot for State Plane Coordinates referenced to the North American Datum of 1927 (NAD27). In fact, metric coordinates based on NAD27 were never published by USC&GS/NGS.
However, at the time of publication of the North American Datum of 1983 (NAD83) in 1986, the federal government was moving toward consistency with the international community and therefore the NGS adopted conversion formulas from geographic coordinates (latitude/longitude) to State Plane Coordinates that produced values in meters, not feet. Today NGS only works in metric units with NAD83.
But much of the work required in surveying and mapping today is carried out in the units of feet. We must therefore make the conversion from meters to feet before using any control points in the National Spatial Reference System (NSRS), and we just proved in our example that the selection of the wrong conversion can result in significant errors.
Most states, at the request of the NGS, have adopted legislation that defines the appropriate conversion method. In a February 2006 document, the NGS states “that policy was drafted, in part to encourage states and territories to enact legislation to adopt NAD83 as their official reference system and to legitimize a specific foot conversion for all spatial data users in their communities.”1 The current status of the legislation and defined conversion on a state-by-state basis is shown in Table 1.
As you can see in Table 1, California has enacted NAD83 legislation2 that specifically defines the conversion method as the U.S. Survey Foot, and therefore the first set of coordinates in the Grizzly Flat example would be appropriate for use in California.
The datasheets for existing monuments found on the NGS Web site at www.ngs.noaa.gov provide additional insight into the proper coordinate conversion method for the states. Excerpts from actual datasheets are shown in Table 2. Notice the control point located in Maine only has metric units for the Maine West zone listed. Maine is one of the few states that has not defined the appropriate conversion method in state law. Also note the control point in Arizona has published coordinates for the Arizona Central zone in both meters and International Feet (the “iFT” designation) while the control point in Florida has published coordinates for the Florida West zone in both meters and U.S. Survey Feet (the “sFT” designation). Also notice that only metric values are listed for the UTM projection. NGS does not publish UTM values in either International or U.S. Survey Feet.
You must be particularly careful when processing GPS data, working in some CAD systems, developing spreadsheets, or using services such as OPUS or OPUS-RS. Many software packages and services require you to specify the conversion method, and the example provided earlier should convince you that the difference is considerable. MicroStation by Bentley Systems (Exton, Pa.) requires a definition of the working units for all drawing files. This must be specified during the setup prior to beginning any actual drawing work. Be very careful with the selection here as the primary choices for English units are “Survey Feet” and “Feet.” In this case, “Feet” signifies the International Foot definition; confusion in this particular area has been the root cause of about 90 percent of the problems I have consulted on in my professional career.