# Journal of Geodetic Science article on a mathematical analysis of systematic errors in NAVD88 (Open Source)

(@geeoddmike)
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March 4, 2018 9:41 am

Li Xiapeng, an NGS contractor, is the author.   He states that using techniques like factor  analysis it will be possible to reduce the systematic errors in NAVD 88 to 3-7 cm precision. The new hybrid geoid model will benefit from his approach.

Lots of nice maths.  He provides good details on factor analysis and appendices describing additional approaches.

Enjoy,

DMM

“Religion is a culture of faith; science is a culture of doubt.”
― Richard Feynman

(@bill93)
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March 5, 2018 8:02 pm
Posted by: GeeOddMike

Lots of nice maths.

That's an understatement!  I can follow a lot of math but that gets far afield from anything I've waded before.

He makes a point that there is a correlation between NAVD88 error and mountainous terrain.  I can't work out the details, but it seems like the assumption made for density of rock between the geoid and the surface could be tweaked to account for that.

The higher you are above the geoid, the more likely you are to have densest materials in the path?

NGS uses Helmert Orthometric heights, which means you are estimating where the geoid is below the solid earth using a canned formula that assumes a density.  I'm looking at Hoffmann-Wellenhof & Moritz book Physical Geodesy2005.  I think it involves this book's eq 4-33 on p. 163.  The constant in that equation comes from eq 3-44 on p. 139, and seems to be derived there from an assumption of a density 2.67 grams per cubic cm.

That seems a little small for mountain bedrock, as I find a table with values like 2.6-2.7 for granite (pretty close), but 2.8-2.9 for dolomite, and 2.8-3.0 for basalt.  Limestone is only 2.3-2.7 but I would think the denser rock would predominate over large depths.

Can the geologists and geodesists comment on this idea?

(@geeoddmike)
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March 6, 2018 1:05 am

I recommend looking at articles resulting from a Google search using geoid  lateral density variations. When I did the search lots of good stuff showed up. Sjoberg and his former student Kiamehr have some worthwhile reads. Some of the older stuff by Candice, Zhang and Anderson are good as well. I did not notice anything by Featherstone who is always a good source. Check his page at Curtin University. Featherstone had an old paper on how geophysicists and geodesists differ in their approaches to these issues.

There was an interesting article on geoid modeling in the mountains of Taiwan in the Journal of Geodesy. It also references the non-NGS Mader who had a different approach to orthometric heights (from Helmert). As some old colleagues mentioned, the mountains in Taiwan are not the magnitude of the N.American Rockies.

Excuse the late night ramblings...

I should forgo the late night tacos.

DMM

“Religion is a culture of faith; science is a culture of doubt.”
― Richard Feynman

(@bill93)
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March 6, 2018 7:00 am

So is that last paper saying the problem isn't the value of a constant, but the assumption that it can be modeled as a constant?  Helmert Orthometric assumes the rate of change of gravity is constant versus height but the rate of change really varies with height.

I'm mostly getting an impression of what he is saying and not really able to do any calculations at my present state of ignorance.

(@yuriy-lutsyshyn)
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March 8, 2018 3:40 pm

"mountainous terrain"-I think it is nothing compared to the masses distribution "inside" the earth, look at the picture . none of the highest geoid variation takes place by the highest mount of Everest. Both the max and min are in the oceans.

(@geeoddmike)
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March 8, 2018 10:24 pm

My reply to Bill93’s 5 March posting probably confused more than elucidated the issues involved in improving NAVD88 via the modeling described by X.Li in the linked paper.

I seized upon Bill’s reference to the treatment of mass in generating orthometric heights to point out Hwang’s paper wherein he discussed approaches to orthometric heights like that of Mader (not the NGS one).

As we move to the combination of ellipsoid heights (h) and the models of ellipsoid-to-geoid separation (N) to yield physically meaningful heights (e.g. usable to determine water flow), we need to insure consistency between leveled heights and those derived via h - N = H.

Around the time of Hwang’s paper there was a project to validate the utility of GPS-derived orthometric heights and those determined by differential leveling at high elevations including mountain passes. Decimeter-level discrepancies were encountered.

I thought that the adequacy of the orthometric correction along the lines described in the paper should be explored. Unfortunately my maths skill and influence were once again proven inadequate.

Helmert orthometric heights are used by NGS and others to account for the non-parallelism of height surfaces. Given the advances in computing power, improved and much more data, improved approaches like those of Mader and Niethammer mentioned in the references should be considered by national agencies. As mentioned by X. Li there remain serious logistics and personnel deficiencies preventing the implementation.

The issue of improved orthometric corrections is not the same as the computation of a geoid model. See the ebook by Vermeer linked below.

For a good summary of various height systems, I recommend: http://geodesyattamucc.pbworks.com/f/HeightSystemsSneeuw.pdf from the a University of Calgary page written by Nico Sneeuw.

Also of potential interest to Bill93 is this free ebook:
https://users.aalto.fi/~mvermeer/mpk-en.pdf#page144 - Physical Geodesy digital text by Martin Vermeer of Aalto University in Finland 2016.

“Religion is a culture of faith; science is a culture of doubt.”
― Richard Feynman

(@geeoddmike)
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March 8, 2018 10:29 pm

On the matter of Helmert heights, see Sneeuw’s paper which discusses the Prey reduction.

“Religion is a culture of faith; science is a culture of doubt.”
― Richard Feynman

(@mathteacher)
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March 10, 2018 5:27 am

This is more my speed. I just finished "Gravity for Geodesy, part 1" and I think it was an hour well-spent.

As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality. Albert Einstein

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