Grid to Ground / Combined Scale Factor

You're going to the the full gambit of instruction on this. I'll yield to whatever Kent, Loyal or Mighty Moe say for exact though.

However, my little diagram I used to keep till I memorized it said this.

CSF = GF * EF

Grid factor = where you are on the grid, no "Z" involved.

EF = mean radius of the earth/(mean radius of the earth + your orthometric height)

The hiccup is that NAD 27 uses 20,906,000 and NAD 83 uses something similar, but not exact (but I've NEVER seen enough to change me from using the NAD 27 version)

So, assuming for your project (this is where it's really going to get different) you have a grid factor of 0.999981 (Dave Doyle said no more than 6 digits was necessary) and you have an average elevation of 500', then your combined scale factor is
0.999981 * (20,906,000/[20,906,000+500]) = 0.999957

At least that's how I do it.

Scale Factor and Convergence

Three optional output fields for Interactive and Point Database Conversions are Scale Factor, Convergence, and Orthometric Height Scale. These fields only apply when a projected coordinate system is chosen as the output system. Orthometric height scale will only display when a Vertical Reference is selected. Points in a Geodetic systems do not have a scale or convergence.

Grid Scale Factor

Grid Scale Factor, often simply called "Scale Factor" is a measure of distortion at a given point on a projected map. The scale factor is not cartographic scale, but a factor used to calculate actual ellipsoidal distances rather than distances on the projected surface.

Convergence

Convergence is the angle of difference in direction from Grid North to True North. This will vary across a projected coordinate system and can be used as a measure of accuracy of angular measurements at a given point on the map.
Note: In Transverse Mercator projected coordinate systems, convention is to specify the convergence angle from True North to Grid North.

Orthometric Height Scale

Also known as Elevation factor, Orthometric Height scale represents a factor of elevation that can be used to calculate geodetic distances above or below the ellipsoid (also known as reducing to the ellipsoid). This scale is determined using a constant radius for the earth in the area of the calculation, but this is typically considered accurate enough for most applications.

Combined Factor

Combined Factor is simply Grid scale multiplied by the Orthometric Height Scale. This factor is used to calculate ellipsoid distance from a grid distance above or below the ellipsoid.

Example: The Grid Scale for two points is 0.999689, and the Orthometric Height scale for the points is 0.999999123, the combined factor is: 0.999689 x 0.999999123= 0.999688123272747
If the grid surface distance between the two points is 1000 meters, to calculate ellipsoid distance between the two: 0.999688123272747 x 1000meters = 999.69 meters

Point of order

When playing in NAD83, you should be using the “average ELLIPSOID Height” NOT the average orthometric height to do these calculations. Even at [only] 6 decimal places, this is a significant (non-trivial) difference in many parts of CONUS.

Kinda busy right now, but I'll get back to this a little later this morning.

Loyal

Loyal

I knew you'd kick in. I think you like geodesy more than beer and women sometimes. 🙂

Kris

Welllll...the geodesy PAYS for the beer and women!

🙂
Loyal

Grid to Ground / Combined Scale Factor vs Light Squared

Don't forget that robots & other total stations think the world is still flat.

Once again, life will be good.... 😛

Here's a link to the PowerPoint presentation I created when I presented this topic at Autodesk University:

http://www.ejsurveying.com/downloads/CustomCoordinateSystems.pptx

And here's a link to the paper I wrote a few years back on the subject:

http://www.ejsurveying.com/Articles/Working_with_Grid_Coordinates.pdf

If you don't like multiplying those six- or eight- or ten-place decimals just add them together and discard the 1 to the left of the decimal place.

Richard - Love your paper, it is so nice of you to share with the surveying community, a great service.

Thanks !

T.W.

Thank you for the links to the Power point and Paper. i have had a copy of the paper for a few years, and the power point is a huge help. I don't get much of a chance to work with State Plane Coords with my projects.

There is no such thing as a combined scale factor.

If you don't like to inverse the combined factor once you are done with the above addition, just subtract the 0.99998765 from 2 and use 1.00001234 which is so close to 1.00001235 no one will ever know the difference.

Paul in PA

Excellent Powerpoint presentation Richard.

I'm glad people find it useful!

Unfortunately, the AU people failed to record my presentation. I have some improvements I want to make to those Powerpoint slides, then I'm planning on recording the whole presentation in a series of Camtasia videos, which I'll get posted on either the Edward-James or the Quux website. I've got some other things I'm working on first, so it won't happen for a while yet, but I'll post a notice about it when the videos and revised Powerpoint slides are ready.

I too love your paper! Thank you very much for posting it! Just found it. You are an excellent writer!

> There is no such thing as a combined scale factor.

lol.....I hear you. I can tell you're "old school". Teachers and more "purists" always taught to keep the grid scale factor and the elevation scale factors separate. I have always kept that in mind. The term "Combined scale factor" slowly became a "legitimate" factor which is taught in the NGS courses and everywhere else. It is now published in people's "metadata" or whatever you want to call it as well.

That is the problem also with "not needing" to go beyond six places. That is true if you are using the scale facors correctly. The theory is that you use the scale factor to scale up or down a distance from or to the grid not a whole coordinate table. It is the problem of people scaling up the sp coordinates with an 'average' combined factor that has introduced the need to be precise on how many decimal places you use (Or you'll get different coordinate values than someone who used only six places). When you start using numbers that start in the millions, you get a noticably different coordinate value depending on the factor you use. If you are scaling up a state plane coordinate project with an average combined factor, you should use as many digits (or one more) than is in the value you are multiplying. 3,100,123.12 would require a multiplier with at least 9 digits.

Anyway, these two items....combining two scale factors into one, and scaling a whole coordinate project by multiplying the coordinates by one combined factor has been a lot like the (formerly) improper words ain't and irregardles becoming legitimate words in the dictionaries. It all appears to be acceptable methods throughout the country now, except for by people like Loyal or Dave Doyle, who actually understand what they are doing.

Tom

Kris

In that order, or reverse? 🙂

> That is the problem also with "not needing" to go beyond six places.

I would say that if you're using a "combined scale factor", you really DON'T need to worry about going beyond six decimal places. The whole key in such a system is that everyone is using the same scale factor. But that's also true if you're defining an LDP... The most important thing, with either system, is that there's published metadata that everyone uses.

For the record, though, I tend to agree that anything that uses a CSF is undesirable... But that's more because I have enough experience in the industry to discover how many problems can arise from this approach... It's not that a CSF "doesn't exist", it's just that there's so many variables involved that any single miscommunication on any one project can result in a nightmare.

Grid to Ground / Combined Scale Factor vs Light Squared

>> Don't forget that robots & other total stations think the world is still flat.

Not exactly...

You should be able to configure your data collector so that it is working in a grid projection such as State Plane or an LDP. When you do so, it should scale the ground distances measured by your robot/ts to grid in the appropriate fashion, based on your instrument location (as defined by your grid coordinates and elevation). Over the usable range of a robot/ts, this works fine, and accounts for earth curvature in quite an adequate fashion for most surveying, assuming you have good grid coordinates on all your control points. So I suppose the robot doesn't know what's going on, but robot+dc does.