“Modified to Ground” Mapping Projections
We know that published mapping projections contain computable distortion, which is compensated by the varying scale factors throughout the zone. These mapping projections can be modified (scaled) to “ground” to obtain parity between positions determined by GPS and those distances measured with an EDM (electronic distance measuring) instrument. Ground projections, however, are based on a single scale factor and are limited in geographical size. Determining the extent to which a modified to ground projection can be “pushed” is not a difficult task, being entirely dependent on the amount of error (variance from EDM distance) one is willing to accept for a predetermined baseline length. When this error becomes unacceptable and one still wishes to survey or map on ground values, it necessitates the creation of sequential ground projections. Let’s perform a brief review of the Transverse Mercator Mapping Projection and the steps required for computation of a combined scale factor. Then I’ll explain GPS ground projections and sequential ground projections.
The Zone Scale FactorA mapping projection is simply the mathematical process of converting geographical positions (points) from a horizontal ellipsoid datum to a plane. Otherwise stated, it is the transformation of a three-dimensional (3-D) surface to a two-dimensional (2-D) flat plane. The vertical components of field derived points are reduced to a common elevation to accommodate the conversion. In the past this common plane was mean sea level; today it is more frequently reduced to an ellipsoid. All mapping projections contain distortion, but it is computable throughout the zone. Depending the type of mapping projection, this distortion occurs in distance, shape, direction or area.
Geodetic coordinates are reduced to a 2-D Cartesian Coordinate System (a plane). The system consists of an x-(east) and a y-(north) axis. Any point position in the zone is then determined by an ordered x,y coordinate pair. To insure all coordinates have a positive algebraic sign, the zone always lies north and east of the x,y axis intersection, where both x and y have values of zero. The values of x and y increase when traveling east and north respectively from the x,y axis intersection. The central meridian of the zone is assigned a large x constant (i.e., x = 500,000 units), also known as a false easting. The southerly limit of the zone is coincident with the x axis, insuring positive y coordinate values. Thus, y = 0, or a false northing of 0 .
Reduction to Ellipsoid (Elevation factor)An elevation factor is determined by using a constant value for the mean radius of the earth. A good approximation of the earth’s radius throughout the latitudes of the continental United States is 6,372,000 meters, which is sufficiently accurate for most computations.
Figure 4 represents the equation R/(R+H+N) for ellipsoid reduction to NAD 83 Datum; where R = mean radius of the earth, N = geoid height or separation, H = height above sea level, and h = ellipsoid height. It is appropriate to note that the NAD83/GRS80 ellipsoid in the continental United States is always above the geoid. This separation (represented as a negative value) ranges from -8 to -53 meters, with the resultant formula more clearly shown as R/(R+(-N)+H).
Combined FactorThe product of the zone scale factor times the elevation factor is known as the combination factor, and is used to convert ground distances to grid values.
scale factor x elevation factor = combined factor
The reciprocal value of the combined factor is then used to convert from grid to ground values.
__________1_____________ scale factor x elevation factor
Ground ProjectionsSurveyors and engineers most often elect to work with “ground coordinates.” In order to easily accomplish this with GPS, a relationship between the ellipsoid and average project elevation must be established. As reviewed above, “grid coordinates” are reduced to a common ellipsoid or sea level surface. A ground projection is merely the rescaling of a published “grid” mapping projection, up to average project elevation, to produce “near ground values.”
The procedures are as follows:
- The latitude and longitude of a single point near the center of a given “project” is determined with the average project elevation. The reciprocal of the combined scale factor of this “centroid point” is computed and held as the new scale factor for the ground projection.
- The Central Meridian (y-axis) and Latitude Origin (x-axis) of the published zone are not affected by the rescaling and are held.
- The false northings and false eastings of the ground projection might and should be modified so they are significantly discernable from grid values. Common practice is to add 100,000 units of measurement to both the northings and the eastings. The false easting, however, requires an additional calculation, since its origin value is not zero and it is, in fact, affected by the new scale factor. The original false easting is multiplied by the reciprocal of the combined factor prior to adding the 100,000 units.
- The resulting coordinates (created by the zone’s defining parameters) will be identical to the manual exercise of individually multiplying the grid values of any coordinate pair within the zone by the new scale factor. This practice facilitates a very simple and accurate check of the new ground coordinates.
- The bearing basis is grid azimuth projected to the ground projection. Bearings established with by GPS are extremely consistent and only a single radial check on another known control point is necessary.
- No calibration is necessary! No wasted time!
Setting up a ground projection as described above can be expedited by copying and editing a published grid coordinate system from the geodetic library found within many GPS software packages. It is imperative, though, to understand exactly what parameters are being applied in the coordinate transformation. Some other small actions or tricks may be necessary. For example, in TG Office Software (Trimble Navigation Ltd., Sunnyvale, Calif.), it is necessary to multiply the ground zones centroid scale factor (reciprocal of the combined factor) by the grid zones central meridian scale factor. The product of this formula is the actual value entered as the ground projections defining scale factor. Caution must be exercised here because this is not an intuitive act.
Ground Projection LimitsIn determining the extents to which a ground projection can be “pushed,” the amount of error, (variance from an EDM distance) for a predetermined baseline length must be ascertained.
The combined factors of a pair of points on opposite sides of a survey projects must be analyzed. For a modified Transverse Mercator Projection the two selected points should lie opposite one another in nearly an east-west direction. For a modified Lambert Conformal Conical Projection the points should be nearly inWhen these combined factors vary by one digit in the fifth significant decimal place, a baseline between them of 20,000 meters (12.4 miles) will appear to contain a 10 PPM (parts per million) error ratio. This ratio represents 0.2 meters (0.66 feet) variance from an EDM measured distance corrected for curvature and refraction.
The procedure, when reducing long EDM measured ground distances to “grid lengths” is to multiply the mean of the combined scale factors, calculated at the ends of the measured line, by the ground distance. This same logic is applicable to modified ground projections.
The apparent error is actually reduced by one-half, to 0.1 meters (0.33 feet), when comparing either one of the point’s combined factors to the already predetermined zone “centroid” combined factor. The “centroid” combined factor value should be nearly equal to the mean combined factor of the two opposing points.
Analyzing the 0.1 meter error statistic for 10,000 meters, we can now easily calculate that the inherent error in one mile is only 0.016 meters (0.05 feet)—an acceptable tolerance for most surveying applications.
Sequential Ground ProjectionsCorridor surveys, large-scale mapping projects or large control networks will regularly exceed the limits to which a ground projection can be “pushed.” This situation is generally remedied by working on grid coordinates, reducing all values to the common ellipsoid plane, i.e., State Plane Coordinate Zones. Many clients, however, are often not comfortable with grid coordinates and would like to have their project data remain on ground values, necessitating the creation of sequential ground projections.
It must first be understood that the true essence of a GPS position is its latitude, longitude and height, the first two of which are of immediate concern here. All other attached records (i.e., coordinates) are merely attributes. As discussed earlier, published mapping projections contain false northing and false easting parameters, and the values of these false coordinates are randomly selected.
It is not uncommon to reduce coordinates from a known zone to latitude and longitude and then recalculate new coordinates for the same geographic position in another adjacent zone. The coordinate values will change because of the adjacent zones defining parameters, but the geodetic position on the same ellipsoid will definitely not.
Ground projection coordinates react in an identical manner to grid coordinates when shifted from zone to zone. The coordinate values will change when shifted, but the geodetic position remains the same. Two sets of coordinate values will then be associated with control points and defining alignment geometry points near the limits of each zone. An analogy to this is the National Geodetic Survey (NGS) common practice of publication of two sets of coordinate pairs for adjacent State Plane Coordinate Zones. Furthermore, the need for grid to ground distance equations along geometric alignments is eliminated.
Currently, in some states and in the near future in others, surveys large and small will be required to be conducted on known datums, not only by controlling government agencies but by clients and other professionals as well. It is not insightful in today’s world for any professional to think that the need for one project tying to another does not exist. Any point or monument located from known datums using GPS will be perpetuated as long as the U.S. Department of Defense (DoD) satellites are healthy. It is my contention that an accurate GPS position on a monument is a more enduring record than the monument itself, as long as the datum’s adjustment date is noted.
There are cynics who argue that continental drift (approximately 2 cm per year) will degrade the quality of record positions. This is a concern only in areas on active seismic faults, but commonly all monuments within the limits of a network will maintain the same velocity (drift) and direction. Remember, the surveying community achieves centimeter level accuracies by differential GPS (DGPS) survey techniques, holding fixed other known relatively close published monument positions. Also, at a velocity of two centimeters per year, it would take 500 years to introduce a 1 PPM error in baseline solutions from an incorrect geodetic base point position.
My advice is to base GPS equipment purchasing decisions on more than a vendor’s sales pitch, which flaunts the ease of a local coordinate system calibration. True, there are some advantages to calibrating, but in the long term, local coordinate system surveys will leave you with nothing but disjointed projects based on various azimuths with no relationship to one another. Tying projects to known datums with GPS requires a lot less effort than one may think, needing only a basic understanding of the relationship between grid and ground mapping projections. The survey community’s level of professionalism would be significantly enhanced and spatial data more readily shared if we all found ourselves executing GPS surveys on the same datums.