From reading and speaking with surveyors: I understand I can use a similarity transformation using matrix algebra to arrive at a 2d scale factor from a set of control points, which I have done.?ÿ Or I can use the surveyors approach for small projects and use the grid-to-ground scale factor.?ÿ Both are representative of a single scale factor as an approximate average for the entire project.?ÿ These are two different methods to arrive at very nearly the same scale factor.?ÿ However, I also understand that "if SPCS reductions are done correctly (a different grid factor for each line based on height and latitude) can be obtained," the scale factor will be slightly different from the above average scale factor.?ÿ However, as a non-surveyor, I am trying to understand the language in the above underlined quoted text.?ÿ Can someone kindly explain what it means to have State Plane Coordinate System reductions and how elevation and latitude play into the process, a white paper is sufficient.
Latitude alone would only apply to a Lambert projection, I believe. For Transverse Mercator Latitude and Longitude would affect the scale factor.?ÿ
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Elevation would affect the elevation factor. Elevation here being the height of the point above the ellipsoid (not above seal level / geoid).?ÿ
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I would not recommend determining a scale factor by comparing observations because observations have errors. Today's observations (source) and Previous survey observations (destination) will each have errors. Those errors will affect the result of the determined scale factor. On top of survey measurement errors, it is also possible to have displacement, over time, of the marks being used. whether that be some gradual movement or sudden movement caused by local disturbance. The geodesy will be fairly consistent over time and the measurement errors that can affect the geodetic combined factor will have a much smaller affect, in my opinion.
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The magnitude of the Scale factor increases as the distance between the grid surface and the ellipsoid surface is greater. The magnitude of the Elevation factor increases as the height of the point of interest deviates from the ellipsoid surface. The product of the Scale factor and Elevation factor provides the combined factor. It is possible for the factors to cancel out (or nearly so) thus being so-called "low distortion".
Perhaps a numeric example will help. Consider the three NGS points AH3962, AH3964, and AH4018. These points are located in the North Carolina foothills in Surry County.
I used the simple average of pairs of combined factors. In the excellent article that @bill93 provided, a more complicated formula for the scale factor is given. In this example, the longest distance is less than 3 km, so little is lost by omitting the middle term in that formula.
Note that the difference in distances is millimeters. Over relatively small areas with little elevation change, this will likely be true.
In more cases where distances are longer and the more complicated average scale factor should be used, average scale factors and average elevation factors should be computed separately and then multiplied to get average combined factors. The scale factor is a defined mathematical continuous function while the elevation factor is not. The more complicated average formula comes from the calculus computation of the average value of a continuous function. Imposing that mathematics on values that are not from a continuous function leads to worse, not better, approximations.
For a theoretically sound computational methodology, consult this reference:
?ÿ https://www.ngs.noaa.gov/PUBS_LIB/ManualNOSNGS5.pdf
Be careful with this stuff; it??s not that easy to understand.
It's not that complicated. Inverse points around the project area, divide the ground by the grid for some lines and you will figure out a factor.?ÿ
Some programs require that you enter in the number as ground/grid, others require grid/ground.?ÿ
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