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# GRS 80 not merely geometric

GeeOddMike is correct that GRS80 is not merely an ellipsoid

but is a rotating, equipotential surface that is an ellipsoid.

As such, it generates not only an ellipsoidal figure, but

also a normal (that is "generic") gravity field. GRS80 can be

used to compute gravity anomalies from gravity measurements

(by subtraction of the normal field).

There are 4 defining parameters:

a =6378137.q0 semimajor axis, meters

gm =3986005.q+8 G*mass Earth, m**3/s**2

dj2 = 108263.q-8 j2 -- tide free

omega=7292115.q-11 rotation rate of Earth, rad/sec

the computation of other parameters, both geometric and

geophysical is rather exacting, and benefits from use

of quadruple precision (that is 128 bit floating point).

The iteration loop for first eccentricity squared, e2, is slowly

convergent -- and yields:

e2 = 6.694380011821693349263024628781072E-0003

e2 = 6.694380022928091730238403447816178E-0003

e2 = 6.694380022903360802879867466743127E-0003

e2 = 6.694380022903415871926288600370417E-0003

e2 = 6.694380022903415749302505441666407E-0003

e2 = 6.694380022903415749575555243028413E-0003

e2 = 6.694380022903415749574947235607143E-0003

e2 = 6.694380022903415749574948589061952E-0003

e2 = 6.694380022903415749574948586516432E-0003

e2 = 6.694380022903415749574948586143422E-0003

e2 = 6.694380022903415749574948586144959E-0003

e2 = 6.694380022903415749574948586290942E-0003

e2 = 6.694380022903415749574948586548475E-0003

e2 = 6.694380022903415749574948586457761E-0003 <<<-- answer

With e2 in our pocket, the remaining values are readily computed:

f = 3.352810681183637418165046184849002E-0003

finv = 298.257222100882711243162836600094

e2 = 6.694380022903415749574948586457761E-0003

b = 6356752.31414034743838861704682171

E = 521854.009700354411772065745653499

c = 6399593.62586403164801394233560317

m = 3.449786003077674246389384933960420E-0003

ep2 = 6.739496775481621906223307129611649E-0003

ep = 8.209443815193342259764018499876038E-0002

q0 = 7.334625841081868876618940446801025E-0005

q0p = 2.688041313666664981318347214790298E-0003

ge = 9.78032677153489285793472943427557

gp = 9.83218636851957475228545044898622

cay = 1.931851353260676360748688677805263E-0003

u0 = 62636860.8500461186518037764982839

Note: all of these are *derived* quantities. They are

consistent among themselves, and with the defining quantities.

It is certainly true that if you truncate one or more of

these quantities, then it matters where you truncate to

keep the same truncated accuracy level. These quantities

are interrelated, and have quite different units, so it is

not safe to truncate at the same number of digits. The values

above should be accurate for all purposes.

The reference equations for this can be found in a reprint

from the "Geodesist's Handbook" :

By the way, GRS80 is not a Reference Frame, whereas any of the

ITRF's or any of the NAD83's are. Hence, there will be transformations

between Reference Frames. There are expressions for transformations

between different ellipsoid references, but these are seldom used

in practice. One just goes from lat/long to XYZ, then back to lat/long

with the target ellipsoid.

Best wishes....

I post this “oldie but goodie” to supplement discussion in the WGS - ITRF thread.

The particular issue was the computation of the first eccentricity squared. Here Dr Milbert shows the result of his iterative results using “... quadruple precision (that is 128-bit floating point).

The link to the Geodesist’s Handbook in Dr Milbert’s near six-year old post does not work. My review of recent copy’s of the Geodesist’s Handbook do not show the equations. Those here look good:

Note that the author of the linked document (R.E. Dealing) acknowledges our own Mr. Nolton’s correction to his determination of 1/f. I confess that I was unaware of the program Maxima. I don’t know if I’m up to explore another calculation tool.

HTH,

DMM

I see you give a link to a paper by Prof. Dr. R.E. Deakin on **ECCENTRICITY OF THE NORMAL ELLIPSOID.**

I don't think you have read this paper. Please read it and look at the Notes on page 3 ;

(**Notes on minor corrections to original version)**

**Then on page 4 you will find the code that both Dr. Deakin and I used to calculate some of the parameters.**

**I did my calculations in 1995.**

**Please look at his values on page 4 (near the bottom of the page) e^2,f and 1/f; THEY ARE CORRECT**

**JOHN NOLTON**

While I do not usually participate in pointless disputations I feel the need to address your post. I use the term “pointless” as the level of difference between the values of the first eccentricity squared (e2) provided by Milbert and Deakin is 1.7 x 10^(-31). In the Moritz paper he shows the derived value to fifteen decimal places.

I did read the Deakin paper as well as two previously linked.

As computations of the flattening (f) and its inverse (1/f) depend on e2 (and the formulas are the same in both Moritz and Deakin), I only concerned myself with the calculation of e2. Deakin usefully provides a good starting point for the iterations. As Milbert states the formula is slowing converging.

As the link provided in the post by Milbert is no longer valid, I did a search and found a similar paper Deakin. It has useful detail about the iterations needed to derive e2. A comparison of the formulas in Moritz and Deakin show them to be the same.

As you have stated that you used the code in Deakin for your calculations did you also use Maxima?

I note that the program is described on page 3 of Deakin thusly:

“Maxima is a computer algebra system that yields high precision numerical results by using exact fractions, arbitrary precision integers, and variable precision floating point numbers. http://maxima.sourceforge.net/ “.

While Milbert did not detail his computation process, I assume he used the formulas in Moritz (and in Physical Geodesy) as well a using quadruple precision in his work. I cannot explain the differences but wouldn’t quadruple precision trump “variable precision floating point numbers”? Remember I am unfamiliar with Maxima and its syntax.

As for the differences in the two results (Milbert and Deakin), they amount to 1.7 x 10^(-31). Not many tools work at that level of precision. Like the calculation of ever more precise values of pi, at what point is it useful in applied sciences like Geodesy?

In closing, I do not know what explanatory point is served by telling me to look at his calculation results. Assertions that “THEY ARE CORRECT” shows confidence in your work (and Deakin) not much else. Making your statements in bold doesn’t make them any more convincing.

Tools available to me will not allow me to work at the precision of your results.

You use the word **"pointless" (see your post above); well it not pointless. **If you print a number then it should be correct to the number of digits printed. Would you want a logarithmic table (in the old days) to say its is 7 places but only correct to 5?

I would hope not.

In 1995 (20 Jan.) I used MACSYMA to calculate geodetic guantitys to many digits (90). I would **NOT** have put in print all 90 digits because of computer roundoff etc. FYI the Newton iteration package in MACSYMA was not working (had a flaw and I reported it to the company). I did use T. Vincenty's value for 1/f 298.25722210088 to start with.

Another question from you; "wouldn't quadruple precision trump "variable precision floating point numbers"

answer: **NO**

Final statement: my calculations in 1995, Prof. Deakin in 2019 and Dr. C.F.F. Karney circa 2011 all agree and are correct, **period.**

**JOHN NOLTON**

Not wishing to extend this thread further, I nonetheless close my participation with the following comments:

1. Having spent some time trying to install Maxima on my MacBook (unsuccessfully) and then starting to develop a script using the equations in Mortitz’s GRS 80 paper in Matlab, renewed my appreciation for those able to wean high precision results in their computations of the derived GRS80 parameters like you and Deakin.

2. I too harshly characterized work to obtain the highest levels of precision possible as “pointless.” Perhaps I should have used the formulation in Deakin’s paper on transforming Cartesian to Geographic coordinates that appeared in the Australian Surveyor (1999) from which this excerpt is taken:

In another post on this thread I provided a link to a listing of Prof. Deakin’s papers. I enjoyed those that I read.

I consider this thread closed,

DMM

Missed this spell check error while trying to make a larger point.

I thought it might be something like that. Sometimes I try to type too fast and then hit the wrong key and then I have a bad habit of not checking what I typed. Plus just being a bad speller leads to a bad post on my part.

Prof. Deakin is a friend and he is **Retired (last year).** I hope people interested in geodesy on this board will look at his stuff.

Hope you are feeling better

JOHN NOLTON

For those interested in Prof Deakin’s papers, many are linked here:

https://scholar.google.com/citations?user=M6keBAkAAAAJ&hl=en

thank you

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