Elevation Factor Calculation of a Control Station

You Are Mixing Apples And Oranges

Elevation Scale Factor has nothing to do calculating elevations. Elevation Scale Factor is a value derived from the elevation to determine a correction for horizontal location.

Elevation Factor is a correction determined by site specific observations. There is no formula that can adjust that factor for use at any other site.

Paul in PA

Hello,

For California I have seen accepted fixed radius values for each zone. See http://www.dot.ca.gov/hq/row/landsurveys/LSITWorkbook/11.pdf Appendix 3, the values are labelled r0, where 0 is subscript... I don't know if other states do this or not. I agree that the accepted radius of 6372km or the geometric mean are preferred since they are not hidden in some document.

Jacob

No - never said that

Paul, nowhere in my question did I say that I was using it to calc elevations ...

It will be used to scale total-station observations (ground) distances to grid at a site that uses State Plane Coordinates.

Howdy,

As you indicate in your message the impact of the small difference in elevation factor (e.g. 6 in the eighth place) on distances less than about 10 KM rounds to a millimeter. Not much to worry about.

The convention to use a single radius value is to simplify computations. To compute the value rigorously requires the computation of the radius at the point of interest. The 20906000 value (in feet) is a nominal value for North America and a legacy from NAD 27.

If you rearrange the equation EF = (R / (R+h)) to solve for R, you get R = (EF*h)/(1-EF). (If my algebra is right). This verifies that different sites have different radii.

Unfortunately, the NGS site file dsdata.txt that details the contents of the datasheet files does not detail how R is computed. Perhaps the State Plane Coordinate System of 1983 manual by Jim Stem has the answer. While available for free download at the NGS site, I am still recovering from Turkey Day and will pontificate only. My recollection is the R is the Gaussian radius at the point.

Enjoy,

DMM

Ground To Grid Requires Combined Scale Factor

Any SPC program should give you the Scale Factor and Elevation Scale Factor.

Like "Love and Marriage", "You can't have one without the other."

Paul in PA

Mike, I concur with the different radii proof.
Looked into the Stem reference, and from it, noticed that this might be enlightening:

NOAA Technical Memorandum NOA NGS-10, Use of Calibration Base Lines, Appendix I, The Geometrical Transformation of Electronically Measured Distances.

This is from 1977 and some of the equations are referenced from south, which to my recollection would relate to NAD27 calcs, not NAD83.

well using the same formula and knowing both their scale factor and the height, you can compute the radius they used.

There are several good sources for the computation of the Ellipsoid Reduction Factor.
USC&GS Special Publication 8, pg 19 "Formulae and Tables for the Computation of Geodetic Positions", USC&GS Special Publication 194, pg 7 "Manual of Traverse Computation on the Lambert Grid," and USC&GS Special Publication 195, pg 14. "Manual of Traverse Computation on the Transverse Mercator Grid.". The most likely reason for not matching exactly what NGS has on the datasheet is an improper computation of the radius of curvature perpendicular to the meridian and/or the radius of curvature in the prime meridian.

Good question and interesting discussion -

1. How "exact" do you need the answer? For example, an elevation factor to 8 decimal places will provide an answer within 0.01 feet for a 10,0000 foot distance. Nine decimal places will make it 0.001 feet.

2. Computation of elevation factor is sensitive to the elevation being used (is it elevation or ellipsoid height?) and less sensitive to the radius of the earth being used.

3. For a published discussion of same, see item #35 at my web site.

4. If you use the Global Spatial Data Model (GSDM) the issue of grid scale factor and elevation factor are both moot. The GSDM uses the distance you measure. It also provides the local tangent plane horizontal distance between points and the azimuth with respect to true north at the stand point with less work than using state plane coordinates. And, the GSDM is applicable all over the world.

While one can compute the R given and EF and h, the question was what type of radius is R?

Looking at base9geodesy's links, it appears that R = (M*N)^0.5 where M is the radius of curvature in the meridian and N is radius if curvature in the prime vertical. Values if M and N are dependent on the reference ellipsoid parameters and the site's geodetic latitude.

I have not run the numbers.

As Mr Burkholder states, these reductions are anachronistic in the sense that we mostly now use 3D measurements from 3D control. Plane coordinates are obviously 2D.

While alluded to in some of the comments, I would like to point out the factor is correctly computed for NAD83 using ellipsoid heights and not ortho heights (elevation), NOT all software does this correctly, I know for a fact that one of the major vendors used to do this wrong, hopefully not anymore. I use Leica LGO and that software is exact to the NGS data sheets at least out to 7-8 decimal places.

SHG

To duplicate a Data Sheet elevation factor, you have to use the mean radius of the ellipsoid at the geographic latitude of the point and the ellipsoidal height. Then the r/(r+h) works. A sample calculation for a point in the North Carolina mountains appears below.

In the formula for Mean Radius, a is the length of the major axis of the ellipsoid and e is the eccentricity. Both of these numbers are on page 13 of NGS Publication 5.