Survey News

Using the State Plane Coordinate System
(and How to Correct for SPCs Inherent Distortions)

The State Plane Coordinate System (SPC)

Most surveys are based on an arbitrary “flat earth” coordinate system, which generally ignores earth curvature. However, surveys of large areas must take curvature into account. This is done by mathematically projecting map data from the curved surface of the earth onto a mathematical plane. The key to understanding any map projection, not only SPC, is to understand the inevitable distortion resulting from such a projection.

The widespread use of GPS in surveying means that although the desired end product may still be flat earth, the global nature of the GPS satellite system means that geodetic and cartographic principles indeed come into play, though they may be transparent to the end user.

The SPC system is legally defined from state to state. Each has one or more SPC zones, using some type of cartographic projection. The type of projection used depends on the zone’s shape and extents. In Michigan’s case, the system was designed to result in scale distortion of no more than 1 part in 10,000. This amount of error is about the same as was achievable in land surveying prior to the advent of electronic distance measurement and GPS.

While the SPC system is indeed a grid coordinate system, with mutually perpendicular X- and Y-axes, there are implications for the surveyor and engineer that make its use very different from that of an arbitrary rectangular XY grid system. Significant problems can occur in creating and maintaining SPC survey data if one has an incomplete understanding of the process.

The Michigan SPC system consists of three zones, legally defined in Act 9 of 1964, (revised by Act 154 of 1988), entitled Michigan Coordinate Systems. South Zone is used from the south state line to about Clare, Central Zone from Clare to the Straits of Mackinaw, and the North Zone covers the Upper Peninsula. Each is defined to result in obviously different X (east) values, in the range of tens of millions of feet.

Geo-Referencing

The key to understanding the use of SPC lies in definition of the term “geo-reference.” This means that coordinate values represent, and are derived from, the latitude and longitude of an actual point on the ground. The XY values can be translated to and from latitude and longitude; however, because the XY value is a result of cartographic projection, some inherent distortion will be present. For example, two points on the earth’s surface that are 2,640 feet (a half mile) apart will scale about 2,639.5 feet apart when projected to SPC.

This amount of distortion may or may not be significant, depending on the intended use of the map. For example, a user who is interested in the distance from Grand Rapids to Marquette can ignore that amount of scale distortion; but, a surveyor who needs to relate an old survey distance to a geo-referenced data set, this distortion must be taken into account.

Scaling SPC to Approximate Ground Distances

One way to more or less remove scale distortion is to scale the XY data set by a particular scale factor, resulting in a close approximation of true ground distances. The scale factor that should be used is a by-product of the lat/long to XY conversion. For lack of a better term, this is referred to as “ground-scaled SPC.”

However, if such manipulation has occurred, the data set is no longer geo-referenced. Which means that if you attempt to merge geo-referenced data from another source into your ground-scaled SPC map, things will not line up correctly, probably by several hundred feet.

Also, if you attempt to convert a manipulated XY value to lat/long, the conversion will not be valid. The lat/long value may indeed look realistic, but it will not be correct unless the XY values used were pure (projected, non-scaled) values.

Minimizing Scale Distortion, Part 1:
Coordinate transformation to an arbitrary ground-measurable coordinate system

Many GPS practitioners use this approach. A pre-defined arbitrary local coordinate system is physically defined with points in the ground, based on ground measurements. A coordinate transformation is done using GPS to locate points in the local system; on-board software transforms the geo-referenced GPS coordinates to the local system.

This approach works well as long as (a) the user understands the software that reports the quality of the transformation; (b) the points located on the ground are the same points referenced on the survey; (c) the points remain undisturbed for future use; and (d) there is no need to merge this survey data set with other geo-referenced data.

Minimizing Scale Distortion, Part 2:
Designing a cartographic projection for a specific project area

The SPC system was designed decades ago to result in scale distortions not exceeding 1 part in 10,000, which was the typical accuracy range for surveys in the past. However, with the advent of GPS, computers, CAD software, and high-powered survey computational abilities in the field, that accuracy limit is easily surpassed, and may be totally unacceptable.

An alternate approach to dealing with projection scale error is to design a new projection to fit local project extents. The State of Wisconsin uses custom cartographic projections designed to fit specific counties. Williams & Works surveyors designed a test projection for Kent County resulting in minimal scale error of about 0.01 foot per half-mile, whereas SPC can result in scale distortion of about 0.3 foot per half-mile.

The end result of the custom projection approach is that both conventional terrestrial and GPS observations would be in agreement to well within equipment and measurement tolerance and the data is still geo-referenced. This means that a conversion to SPC (or other projection) would still be possible, simply by converting custom XY values back to lat/long, then applying another cartographic projection.

Which Way is North?

Essentially, there are three different north references. One is the arbitrary north shown on a previous survey or deed of record; another is a true or geodetic north; and the third, when dealing with a cartographic projection such as SPC, is grid north.

Grid north is the inherent orientation change generated by the cartographic projection process, as with the scale distortion mentioned above. This is unavoidable because of the convergence of the meridian (convergence at the poles). The north lines in an XY coordinate system are parallel with one another, whereas true north converges at the poles. The angular difference between grid north and true north is known as the convergence angle, or theta.

Artificial correction of this distortion is not as simple as dealing with scale error. Suffice it to say, if an angular correction is applied in some way, for instance, to match the bearing of a line on a survey of record, geo-referencing is lost, as with application of a scale factor.