In a recent thread that descending into a train wreck, [USER=677]@GeeOddMike[/USER] referenced this quote:
When we talk of geodetic leveling I always recollect the statement in "Physical Geodesy" by Heiskanen and Mortiz who state that: "Leveling without gravity measurements, although applied in practice, is meaningless from a rigorous point of view, for the use of leveled heights as such leads to contradictions..."
This has been on my mind for a couple of years now. I've been thinking of it more in terms of reducing terrestrial slope distance to horizontal than applying it to elevations. If I understand the concept, this is a contrast between dynamic heights and orthometric height differences. Dynamic heights are the difference in height between two points measured normal to gravity*. Orthometric height is the height of a point above the geoid (equipotential surface of the Earth), and the orthometric height difference would be the difference between the orthometric heights of two points. At first glance it would be tempting to say that the dynamic height and the orthometric height difference between two points would be the same, and I'm sure in many cases the difference between the two is trivial. However when we consider a leveling run to determine the elevation difference between two points, we must recognize that "level" is determined by gravity. We understand that normal (perpendicular) to gravity does not point to the center of a spherical Earth, but that variations in the mass of the Earth changes the direction of "level". This variation in mass occurs everywhere (with varying significance). Because of this, each setup of the level on a level run will have a unique plane for what is "level" or normal to gravity.
Imagine two benchmarks on a plain. Between them is a mountain. A level run begins at one benchmark, extends across the plain, over the mountain, down the mountain and across the plain to the other benchmark. A second level run begins at one benchmark, extends across the plain, around the base of the mountain to the other benchmark. Assuming no error in the level run, the two runs would not give the same elevation at the end because the impact of the mass of the mountain affected the two runs differently. Each time the level was setup the plane defining "level" was different. These are two distinct dynamic heights.
The orthometric height difference of the two points may be different from either of the two dynamic heights from the level runs. The orthometric height difference is not affected by the variation in gravity between the two points. We have an ellipsoid height, minus a geoid height, at the first benchmark and an ellipsoid height, minus a geoid height, at the second benchmark and a difference between them.
Which is right? All of them, depending on the intended use. The dynamic height makes sense when one is concerned with how water will behave along the route of the level loop. However, the dynamic height will not be perfectly consistent with other routes between the two points (which I believe is what the quote from GeeOddMike addresses).
If the vertical is affected by these differences, I must also believe that reducing horizontal differences would also be affected by dynamic heights. Imagine a precise traverse between two points. Just like with the level run, the horizontal distance between the points will vary depending on the route taken. Of course, the effect to a horizontal distance will be much, much less, but it would still have an effect.
I believe there is quite a bit of information on the NGS website regarding this, but I thought Mike's passing comment was mentioning.
I look forward to fully digesting your post when I have more time.
A key concept is that dynamic height differences do not track with orthometric height differences. Dynamic heights are a fiction that translates gravity potential into more familiar units of feet or meters. Ortho is what you measure on the level rod. A dynamic height follows the surface of constant gravity. Dynamic heights are compressed together or further apart in a physical sense depending on changes in gravity as you move around or even as you change elevation over one spot.
Shawn Billings said: " I've been thinking of it more in terms of reducing terrestrial slope distance to horizontal"
You might want to look at Survey Review, XXVII, 175, Jan. 1975,
"A note on the reduction of measured distances to the Ellipsoid" by T. Vincenty
JOHN NOLTON
The best compilation of information about different height systems is this: http://geodesyattamucc.pbworks.com/f/HeightSystemsSneeuw.pdf
The quote was intended as a reminder that all geometric heighting is non-unique meaning that it is route dependent. I will determine a different height for a point if I choose a different path as gravity will differ along the path. Orthometric heights published by the US NGS are Helmert orthometric heights and are computed using geopotential numbers.
Read the linked document.
BTW, the issue of dynamic v orthometric heights has been an object of discussion for years. Of course, being retired, I have not been part of them. Note that dynamic heights use the gravity at 45 degrees N Latitude one reason for their use in the International Great Lakes Datum.
Hope this helps,
DMM
I am traveling and view postings intermittently. I have chosen to ignore the Vertical Datum thread given its rancor.
These three papers will make everything clear. Print them, put them in a binder, and read them with highlighter in hand.
http://digitalcommons.uconn.edu/thmeyer_articles/2/
http://digitalcommons.uconn.edu/thmeyer_articles/3/
http://digitalcommons.uconn.edu/cgi/viewcontent.cgi?article=1001&context=nrme_articles
Thank you for sharing, Teach.
gschrock, post: 420895, member: 556 wrote: Sort of hijack. [daydream on]
I've always kind of dreamed about a gravity meter in the pole that would take measurements and relate them back to a real-time modeling system. Impractical for a typical spring type meter, and aspirations for particle fountain style meters (have been looking into that arena for fun) would be too sensitive (at this time) to move around... but despite those challenges such a thing might not be too far fetched in a few decades... [daydream off]
You better Patent that idea before Javad is trying to sell it to you...
GeeOddMike, post: 420833, member: 677 wrote: The quote was intended as a reminder that all geometric heighting is non-unique meaning that it is route dependent. I will determine a different height for a point if I choose a different path as gravity will differ along the path.
Not only route dependent, but time dependent as well, it would seem.
These articles are blowing my mind. Excellent stuff on a rainy day like today.
MathTeacher, post: 420884, member: 7674 wrote: These three papers will make everything clear. Print them, put them in a binder, and read them with highlighter in hand.
http://digitalcommons.uconn.edu/thmeyer_articles/2/
http://digitalcommons.uconn.edu/thmeyer_articles/3/
http://digitalcommons.uconn.edu/cgi/viewcontent.cgi?article=1001&context=nrme_articles
MathTeacher, where is Part IV?
Here's a link to a monograph (pdf) with the same title and authors: http://www.ceri.memphis.edu/people/smalley/ESCI7355/what%20height%20really%20mean.pdf
Part IV is here.
mkennedy, post: 420918, member: 7183 wrote: Here's a link to a monograph (pdf) with the same title and authors: http://www.ceri.memphis.edu/people/smalley/ESCI7355/what height really mean.pdf
Part IV is here.
Thank you!
I think the effects of gravity variations are usually negligible in terms of reducing slope measurements to gravitational horizontal versus going parallel to the ellipsoid, because you are usually working with a difference of a few arc seconds. The sine of a zenith angle of 89* 59' 30 (for instance) is awfully close to 1.
Bill93, post: 420928, member: 87 wrote: I think the effects of gravity variations are usually negligible in terms of reducing slope measurements to gravitational horizontal versus going parallel to the ellipsoid, because you are usually working with a difference of a few arc seconds over some miles. The sine of a zenith angle of 89* 59' 30 (for instance) is awfully close to 1.
I think you are right, Bill. That part was just a musing as I was thinking about what "horizontal" distance really is. It snakes along parallel to the undulating geoid, with inflections wherever the instrument happened to be setup along the traverse. The practical difference, as you say, is negligible.
[USER=249]@Jim in AZ[/USER] Sorry about that. The broccoli soup needed stirring. Here's Part IV:
http://digitalcommons.uconn.edu/cgi/viewcontent.cgi?article=1004&context=nrme_articles
Bill93 (by the way is that your age? , not being funny just wondering) you are talking about short distances and flat land surveying. If you and
Shawn Billings will read Vincenty paper above that I gave reference to, Shawn won't be musing and you will know.
JOHN NOLTON
