# Update on vanity station

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Attached is a picture of station Orville, my vanity survey mark, with forms still in place. I've used about 24 hours of averaged Garmin data sampling every 30 seconds to get a rough location, which will have to do for now until better affordable GPS technology becomes available. What I have is probably good enough to locate a point somewhere on the monument. Now on to creating a "datasheet", which is really the point of the whole exercise - a hand's on project to learn more about what all those numbers mean. I've started by finding the smallest triangle of genuine stations containing my mark inside of it, and computed the barycentric SPC coordinates of my approximate location. That should let me interpolate actual numbers from the station data sheets as a double check on the ones I compute.

A couple of questions if someone would be so kind as to give me a little guidance.

(1) I've been able to exactly reproduce the state plane coordinates and UTM coordinates of the nearby marks, but am having trouble figuring out the meaning and purpose of the elevation, scale, and combined factors. According to NYS law, my area uses Transverse Mercator with central meridian 76 degrees 35 minutes, origin at 40 degrees N, and false easting equal to 250000 meters. The law further stipulates that the "scale on the central meridian should be one part in 16000 too small", which I interpret to mean that the scale factor for the projection should be 15999/16000 = .9999375. Indeed, when I use that scale factor I get the exact X,Y values for the STCs of the marks around me. But this factor is different from anything listed on the data sheets, and when I use any of the scale factors listed there I get coordinates that are tens of meters off.

(2) I think I've finally figured out what the "Laplace correction" is, but the literature on this topic is incredibly confusing! The data sheets list a single number, which in my area is usually an angle of about 4 seconds give or take. This is described as being somehow "derived" from DEFLEC18, but DEFLEC18 gives you 3 numbers as output with no explanation of their meaning. The first two appear to be the two components (xi and eta) of the deflection of the gravitational vertical from the normal vector to the ellipsoid.  After some experimentation, I seem to get a number that agrees with the Laplace correction on the data sheets by computing -eta x tan(phi)  where phi is the geodetic latitude. This is the number you have to add to an (observed) astronomical azimuth to get a geodetic azimuth. Several docs on the web have you divide by tan(phi) but this doesn't make any sense to me unless phi is colatitude instead of latitude. (It would make the conversion singular at the equator rather than at the north pole.) So I guess my question is whether I have the definition of Laplace correction right.

Nice looking monument.

Sounds like you are making progress on figuring this stuff out. It appears you have found the NGS Toolkit, and may need to explore more of the tools. There is lots of tutorial info on the NGS site and elsewhere on line.

SPC and UTM are projections onto a plane slicing through the earth. As such the distortion changes as you move on the map away from center. That 1 part in 16000 is the chosen distortion near center, and it reduces toward zero distortion (1:1) as you move toward the line where the projection plane slices the ellipsoid model of the earth, and then increases with the opposite sign as you move farther out to the edge of the intended coverage area. 16000 balanced the worst positive and negative distortion, as measured by the scale factor.

So far we haven't considered elevation. As you move higher above the projection plane, the projected "image" of the coordinate grid stretches in proportion to the distance from the center of the earth. The Elevation Factor is the measure of this. A map distance corresponds to a different ground distance at different elevations.

The product of scale and elevation factors is the Combined Factor, also known as the grid to ground factor (or is it the inverse of it?).

Thanks, as usual, Bill! I think I understand all the numbers now! (One small nit: UTM and SPC are projections onto a cylinder slicing through the earth, not a plane. To get a plane you have to cut the cylinder along a generator and unroll it. Kind of like wrapping paper. This doesn't affect the reasoning about stretch with elevation change.)

I figured out that the elevation factor is given by a/(a+h) where a is the semi-major axis of the ellipsoid = equatorial radius, and h = orthometric height + geoid height = height above the ellipsoid. I had been really hung up by the "scale factors" since they differ from what you have enter as the scale factor in the apps (.9999375 for my STC and .9996 for UTM). I now realize these are the scale factors for the particular site, i.e., the local distortion in length, just as Converg = local distortion in angles. I think I'll just estimate both of these by interpolation.

Are you a surveyor by trade?

You're right about the unrolling.  Sorry 'bout that.

I'm not a surveyor.  I'm an electrical engineer, now retired, and thus comfortable with most of the math needed in surveying and geodesy as well as error statistics.  My interest in the intersection of surveying and history goes back to high school days.  For decades, I've kept a copy of one or more surveying textbooks around for curiosity about the application of math and the intricacies of the instruments.  If you're looking for a modern surveying textbook, I recommend a slightly out of date (cheaper) edition of Wolf & Ghilani's Elementary Surveying. If you are interested in the older methods, try to find a book from the 1960's or so.  The best of those I've seen is Davis, Foote, & Kelly.

When a friend told me about geocaching and then I discovered benchmarks on the  site I was hooked, and have learned much more about geodesy since. I joined the Surveyors Historical Society and have gone to a couple of their "Rendezvous" conferences with presentations on how surveying affected history.  I met benchmark hunter moser at the event in Philadelphia, but it's been a while since he has been a frequent poster here.

I follow www.surveyorconnect.com and seem to be accepted as a member of that community, where I try to limit my comments to what I really know (plus the members chit chat category). Most people don't realize that while geodesy is about measurement and modeling, land surveying is more about law and evidence than measurement, so there is a lot to learn on that site besides math. And there's a math teacher on that site who knows far more than I do about math and mapping.

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