Scale Factor
I think tha tin your example, you compared a ground distance to a grid distance. If I take your state plane distance of 3711.92 a multiply it by your DAF of 1.000308, I get 3713.06.
That's the same answer that you got with your DAF adjusted coordinates.
If the two things weren't the smae thing expressed differently, we'd be in a world of hurt.
No in theory it's not the same number, the point of the publication I posted was that the DAF will simulate CSF within 20ppm. That's the 1/50,000.
So near the edges of the site the most you would see in 10,000' between a measured ground distance and a surface grid distance would be 0.20'.
Usually where I work the difference between ground and state coordinates inversed distances at 10,000' would range between 2.5 and 4'. 10 to 20 times less accurate than the worst the DAF produces.
The problem is that a site 10x4 miles will generate an infinite number of CSF. It's unwieldy to figure them all out so a Site or Project or Datum factor is picked.
It's applied to the coordinates and only one number is used.
That's why I call them surface coordinates. Any 3d point will have a unique CSF. But each point along that Lat, Long pair will have an infinite number of CSF as the elevation changes.
The distance between two 3d points will have a unique CSF different than either point connected to the end of the line. So instead of the massive amount of calculations necessary to "get on" ground perfectly at any location the DAF is used one time.
You could move to western Kansas or Eastern Colorado and use the same number on everything.
I have driven across Kansas. Not a lot of vertical relief for sure. Flat as a pancake.
See, here's where we're missing each other. Each point has a combined factor and each point has a DAF as well. This DAF should be the reciprocal of the point's combined factor. That has to be true or else the DAF is something that doesn't follow the mathematical development of map from ellipsoid.
You use some sort of a single DAF for the entire project. Some sort of a single combined factor can also be calculated for the entire project.
If the assumptions are the same for calculating both single factors, then one will be the reciprocal of the other and they will produce identical results.
But you're correct in that inversing between two points using their individual combined factors will produce a different answer than inversing with either a site-wide DAF or a site-wide combined factor. If that difference is too big, then there is a defect in either the single factor or its application.
An apples to apples comparison is site-wide DAF to a site-wide combined factor. If there's a significant difference, then something is wrong.
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That's pretty much it.
To figure out a DAF I never even look at scale factors. My only interest is to match Ground and Grid distances as closely as possible.
With modern software all that's needed is to inverse and compare the two since the software will display them. If you want to find a DAF for a site then figure out the limits of the project, plot it on a quad sheet, a county GIS, Google, any data source that will match lat, long with elevation.
Then put some random points that cover the site into the software and inverse, you will soon find out the DAF. It's a simple ground/grid calculation and the mean or close to the mean normally jumps out at 1ppm.
That's the DAF.
Enter it into the software and it will do the rest.
Print out a data sheet, put it into the file and your metadata is covered, it gets scanned into the job when it's finished.
Every time you survey, you're creating a surface. When a TS is set over a point and is used to locate points the surface is the Total Station height. As you move to different points, a new surface is created. This doesn't crop up as a calculation concern unless you're doing huge traverses. But it's always there. To close out a subdivision survey with a dozen set-ups you ignore the dozen surfaces and calculate essentially with a DAF of 1.
No one thinks about that unless you're an old surveyor that did big jobs with conventional traversing, then you needed to consider it.
You're describing process, not defining DAF. That's like saying that to create carbon dioxide, mix vinegar and baking soda. The process works but why it works is not mentioned.
However, you did define DAF when you said that ir's a grid/ground conversion factor. That makes it a form of a combined factor.
The reason that your process works is that state plane coordinates are determined by the rigorous mathematical relationship between an ellipsoid and a mapping plane and our ability to measure elevations. Those elements give us a combined factor for each point.
Those combined factors can be averaged to produce a site-wide combined factor.
Your DAF is the reciprocal of that site-wide combined factor. My guess is that the reciprocal is used because multiplication was much easier to perform in pre electronic computer days.
Suppose that you have state plane coordinates for two points. Suppose that you also have the ground distance between those points.
If you divide the grid distance between them by the ground distance, you get the average combined factor for the line that joins them. If you divide the ground distance by the grid distance, you get the average DAF.
But, you say, your process for DAF never uses distances. No matter. Just average the state plane combined factors and, if your ground measurement is as accurate as the state plane coordinates, you'll get the same answer as before. The methods are mathematical equivalents. Take the reciprocal of that number and you'll get the same DAF as before.
But, you say, you use more than two points. Fine. Just average as many combined factors as you want. It's the underlying math that controls, not the process nor the arithmetic.
The DAF is not mysticism applied to surveying. Instead, it's a number that converts grid to ground by multiplication rather than division. And it relies on the mathematics of Gauss, Lambert, Mercator and others, along with the insights of Oscar Adams in order to work.
All true, but the real world smacks you in the face, surveying in the Madison isn't useful when you're building or defining legal areas in the real world. Efficiently making the world work needs to be done.
Which is why this process is used.
I couldn't agree more, but there are at least a dozen equivalent processes that work, too. And the thing that makes them work, and also makes them equivalent, is that they all build on the mathematical principles that define the combined factor.
Knowing why a process works and what it is equivalent to is invaluable when things don't go according to plan. Otherwise, all we have is the excuse that we've always done it that way.
We have always done it that way is a saying that is very much heard in the surveying community. If I had a nickel for every time I heard someone say that I would be wealthier than Jeff Bezos or Bill Gates. lol. You are correct though having understanding is invaluable for sure.
