Scale, Elevation, Grid, and Combined Factors Used in Instrumentation

Surveyors deal with small but important adjustments that their modern instrumentation can automatically do for them, but are not always sure exactly how to deal with this. Some of these adjustments are for variations in the speed of light or radio waves through the atmosphere due to temperature, barometric pressure and relative humidity, prism constants, antenna constants, horizontal and vertical collimation errors, height of standards (also called trunnion axis) error, and errors in horizontal and vertical angles due to the instrument not being perfectly level.

One adjustment that is often ignored, but shouldn't be, is what is often referred to as "scale error." As it stands "scale error" may mean different things in different situations, geographic locations, and even instrumentation. Not only must the theory of how scale error is computed and applied be understood, but that understanding must then be applied to all the work the surveyor does in the field and in the office, particularly when trying to match field results to inversed valued computed from grid coordinates.

Part of the reason for needing to be concerned about knowing how scale factors are developed and applied comes from using State Plane Coordinates. Whether one uses a total station or GPS or both, scale factor cannot be ignored. In addition, we have the added complication that GPS positioning (at its core) is computed on an ellipsoid, which means that some understanding of geodetic surveying is required by practitioners who may have only practiced plane surveying, and who may not have not been exposed to geodetic concepts and ways to implement them in field procedures. Many people think that "geodetic surveying" refers to surveying that is extremely accurate; that may be so, but it doesn't have to be, so long as it takes the curvature of the earth's surface and the ellipsoid into account.

The challenge for local surveyors is to use GPS without misunderstanding what the scale factors are, so that errors aren't compounded. This only gets more complicated when combining an instrument that produces "plane" measurements at ground level if no corrections are applied with an instrument that generates ellipsoidal measurements. This is the situation with the combination of total station and GPS.

In Figure 1 you see one step of the scale factor process, assuming that one is converting a ground distance to a grid distance. Using the mean radius of the earth and the elevation of the end points of the ground distance, the sea-level or geodetic distance may be computed. This factor can be called the elevation factor, although other names may be used.

In Figure 2, this sea-level distance is then projected onto a developable surface; that is, a geometric shape that can be made into a plane. A sphere cannot be made into a plane. An ellipsoid cannot be made into a plane, but a cylinder or cone, if you imagine it to be made of paper, can be cut and laid out flat.

Regardless of whether it is State Plane or Universal Transverse Mercator (UTM) coordinates, this conversion of individual distances to grid distances must be done. To get from sea-level distance to grid distance, the constants applicable to the particular State Plane Coordinate or Universal Transverse Mercator System zone must be used to determine what is often called the grid factor, though other names may be used.

Then, using conventional coordinate geometry, the grid coordinates of the end points of the lines can be determined. The two-step conversion may be combined into one by multiplying the elevation and grid factors together to produce what is often called a "combined" factor. This number must be used guardedly as it only applies for similar elevations and X and Y values in the grid system. Of course a tie to points whose grid coordinates are known must be done. Even if the coordinate system used is an arbitrary one, these reductions must be done to properly combine total station and GPS measurements.

It gets more problematic when doing stake-out, or checking the positions of existing monuments whose grid coordinates are known. If a measurement is made, for example, with an EDM, the EDM distance must either be adjusted to grid and then compared with the inversed grid coordinates. Alternatively, the inversed distance can be converted to ground by dividing by the combined factor and compared with the ground distance.

It is particularly important that in the process of doing these calculations that surveyors not multiply coordinates by these scale factors; ONLY distances must be multiplied by these factors.

Geomatics Industry Association of America

(GIA) is an organization of manufacturers, suppliers, and distribution partners, en-compassing the present and emerging technologies which address customer needs in surveying, GPS, engineering, construction, GIS/LIS, and related fields by providing leadership in training and education to enhance the efficiency and effectiveness of their related business.

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