Modified to Ground - Mapping Projections

Tying projects to known datums with GPS provides better accuracy
By Jack Gnipp, P.L.S.


We know that published mapping projections contain computable distortion, which is compen-sated by the varying scale factors throughout the zone. These mapping projections can be modi-fied (scaled) to "ground" to obtain parity between positions determined by GPS and those dis-tances 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 Factor

A 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 trans-formation 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 dis-tortion, but it is computable throughout the zone. Depending on the type of mapping projection, this distortion occurs in distance, shape, direction or area.

For simplicity, we will address only the Transverse Merca-tor Mapping Projection (see Figure 1). The basis of this map-ping projection is a central meridian, of which there is a scale factor with a value of less than one. The scale factor is consistent along the entire length of the meridian. Scale fac-tors throughout the zone vary as a function of longitude, or vary proportionally to the distance traveled east or west of the zones Central Meridian. At the zones "lines of intersec-tions" with the controlling ellipsoid the scale factor is "exact" or equal to one. Extending further east or west beyond these "lines of intersection," the scale factor becomes greater than one (see Figure 2).

State plane coordinate mapping projections are restricted to a total width of 158 miles to limit the error ratio between grid and ground distances to less than 1/10,000 (see Figure 3). With the advent of modern survey equipment and tech-niques, this tolerance is weak by today standards.

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 c o o r-dinate values. Thus, y=0, or a false northing of 0.

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