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# Ellipsoid and Datum Equations

Posted by Martin F on December 14, 2013 at 1:18 amI just saw this question on GIS StackExchange

http://gis.stackexchange.com/questions/80456/what-do-different-realizations-of-the-same-datum-have-in-common*Firstly, am I correct in assuming that the process of determining the shape and the size of the spheroid is independent from the process of determining the origin and the orientation?**For example, for NAD27, it seems that the following equation was fitted to the surveyed points:**where the parameters to be determined were x_0, y_0, and z_0 instead of a general equation of an oblate spheroid:**Suppose we have the complete equation (with parameters already determined), and the origin and the orientation relative to some arbitrary standard coordinate system, of each different realization of all the datums. Further suppose we do not know the name of each. For example, NAD83(CORS94) may be called “1” and WGS84 (G730) perhaps “2”. So they are all jumbled up. Is it possible to sort them into different groups so that each group consists of the different realizations of the same datum?*It’s been too long since i studied geodesy and i cannot answer the Q confidently. Not sure how many geodesists are over there. Anyone care to go there and have a stab? You can answer there, or here if you prefer and i’d post it. If the question is not clear, we can ask for clarification.

base9geodesy replied 10 years, 5 months ago 6 Members · 14 Replies- 14 Replies

You are correct that the determination of the origin and orientation of a horizontal/geometric datum are independent of the reference ellipsoid. That being said I have never seen this solution used with respect to the determination of the ellipsoid. As per your example of NAD 27 it would be important to note that the U.S. Coast & Geodetic Survey had adopted the use of the CLARKE 1866 ellipsoid in all computations of horizontal positions in 1880, long before the development of NAD 27. In addition, the solution you show accounts for values in Earth-Center, Earth-Fixed (XYZ) coordinates which were seldom determined prior to the more contemporary capability of space-based positioning as the knowledge of true ellipsoid heights was extremely poor. Could you cite a reference source for your solution?

I am the one who posted the original question at stackexchange.

I am not familiar with the way you used the word “solution”. Is it equivalent to “equation”? The first equation in the question is the Clarke 1866 ellipsoid (reference 1).

References:

1. ARSITECH “Constants for Reference Ellipsoids used for Datum Transformations”

http://www.arsitech.com/mapping/geodetic_datum/Thanks, base9, that’s more or less what i suspected, but was not confident to state. I posted your answer, and am happy to continue to be the go-between, if necessary. :good:

I hope I have not misconstrued the post.

As originally stated “For example, for NAD27, it seems that the following equation — called the Clarke 1866 ellipsoid (reference 1) — was fitted to the surveyed points:” This is not the case. You can’t have the coordinates for the points until you’ve identified the reference ellipsoid to use (and of course the origin and orientation). I am a bit confused by the intention of the statement “For example, NAD83(CORS94) may be called “1” and WGS84 (G730) perhaps “2”. So they are all jumbled up. Is it possible to sort them into different groups so that each group consists of the different realizations of the same datum?”

The equations as stated will give the same results as long as the reference ellipsoid(s) are at least nearly identical such as the Geodetic Reference System 1980 (GRS80) and World Geodetic System 1984 (WGS 84) but it wouldn’t tell you anything about the relationship of either different realizations of the same datum or different datums that use the same or very close reference ellipsoids.

I may be mistaken about the way NAD27 was realized. Here is what I am imagining:

The first equation only tells me the size and the shape of the ellipsoid. I imagine it like a free floating ellipsoid. Then I am assuming each of the surveyed points had already been assigned coordinate values (x, y, z). (Did you mean in your last comment that, to do this, some pre-existing coordinate system had to be used?) Then the free floating ellipsoid was fixed by least square or some other method.I do not know if the earth-centered earth-fixed or a different coordinate system was used for the process above. I wanted to state the question free of a particular coordinate system but I wanted to give an example and I could not find the way without assuming a coordinate system. I think the main question is indifferent of a coordinate system.

I am also assuming that if we have the equation of an ellipsoid (e.g. Clarke 1866) in the ECEF coordinate system, going to the fixed (i.e. realized) ellipsoid can be done by translations (setting origin) and rotations (setting orientation).

As for the main question, let us imagine we plot all the realized ellipsoids of 2 datums (say, NAD27 and NAD83) in some coordinate system (say, ECEF). Let us say we use blue for realized ellipsoids of one of the datums and red for the other. Will it look like there are 2 groups of them, each group corresponding to either of the 2 datums? Or will they look mixed up?

*“The first equation only tells me the size and the shape of the ellipsoid. I imagine it like a free floating ellipsoid. Then I am assuming each of the surveyed points had already been assigned coordinate values (x, y, z)”*For what it’s worth,

As posted by base9geodesy, NAD 27 coordinates were not determined with respect to the geocenter. Prior to the modern era of surveying with the advent of space-based systems, national entities chose a reference ellipsoid which best fit their part of the world.

You might want to review the following document: http://earth-info.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf which details the relationship of WGS 84 to other national datums. Appendix 4.6 shows the United States.

As indicated in the posted wiki page (on MEADES RANCH), NAD 27 originated at a physical monument with certain assumptions about its deflections of the vertical, etc.

There is a lot of interesting history related to the determination of the earth’s dimensions. Weighing the Earth by Danson is one of my favorites.

As for geometric geodesy, I note that lecture notes for OSU’s Professor Rapp course is available for download. Professor Jekeli’s lecture notes on reference systems in geodesy at https://kb.osu.edu/dspace/bitstream/handle/1811/24301/Geom_Ref_Sys_Geodesy.pdf mentions how the traditional courses in geometric geodesy and geodetic astronomy are considered to be

~~obsolete~~“irrelevant, anachronistic and in need of revision.” and replaced by the reference systems course.Attempting to differentiate among different versions of NAD 83 by transformation parameters is not an approach I would recommend. As for differences between NAD 83 and ITRF implementations, they are embedded in the HTDP source code.

Hope this helps,

DMM

Jekeli’s phrasing corrected…

I could not find Appendix 4.6. The one with the closest content I found is 7.3 “Relationship of WGS 84 to the NAD 83”.

I still do not understand how NAD27 was determined after looking at the Wikipedia articles on Meades Ranch and NAD. They mention the assumptions about the location of Meades Ranch (Osbourne, KS) and the location of Waldo (also in Osbourne, KS) relative to Meades Ranch. It is not clear what other things were used.

I guess the example I included in the original question may be irrelevant, if not entirely wrong, since the question is about what the realized ellipsoids are and not how they came to be.

I did not mean to fixate on the idea of earth-centered earth-fixed. I was assuming a different coordinate system like (Lat, Long, Elev) can be converted to (x, y, z) in my previous posts.

I am not attempting to differentiate different versions of a datum. I was wondering if the different versions of a datum are “similar” to one another than to the different versions of a different datum. I am not sure what “similar” means exactly. I will think about how I can make the question clearer or if it even makes sense.

Sorry for the error. It is Appendix B.6.

At risk of misunderstanding your question, the way national agencies worked prior to the modern period was to create a regional datum consisting of observations (angles and distances) fitted to a chosen ellipsoid.

Some of the other posts have provided links to tables of ellipsoids. Geocentricity

**was**not a consideration. NAD 27 fitted to Clarke 1866 yields a difference at the origin of over 200 meters.As for the issue of transforming between XYZ and Lat, Lon and ellipsoid heights, the equations, of course, exist. However, remember that geodetic surveying prior to the modern era was generally 2D where work was reduced to the ellipsoid. Ellipsoid heights were not used in computations overall. The level network was a separate activity. If you examine NGS data sheets for classically-determined stations you will note that XYZ coordinates are not provided (unless subsequently positioned using GPS) and heights frequently based on scaling or vertical angles. Ellipsoid heights for these points were not obtained.

Further details can be found in the following lecture/presentation:

http://geodesyattamucc.pbworks.com/f/25Feb2010_NAD.pdf

The presentation draws extensively from NGS presentations. The blue bordered slides are copies. Presentation was not intended as a stand-alone but to support discussion. Feel free to ask for clarification (or corrections).

Hope this helps,

DMM

DMM’s description is spot on. In the specific case of NAD 27 the coordinates for the origin station MEADES RANCH were actually derived by U.S. Coast & Geodetic Survey in 1901 by computing the values through the network of triangulation from the station PRINCIPO in Maryland which had been used as the origin point for the New England Datum. MEADES RANCH was selected as a datum origin because it was close to the geographic center of the country (lower 48). The datum at that time was called the U.S. Standard Datum. It was changed in name only to the North American Datum in 1913 and readjusted as NAD 27 in 1927. The coordinates for MEADES RANCH and the corresponding azimuth to station WALDO remained unchanged for all three datums. As DMM indicated the CLARKE 1866 ellipsoid was forced fit to MEADES RANCH with a resulting ellipsoid height and deflection of the vertical components of 0. The coordinates for the rest of the NAD 27 stations were computed by network least squares adjustments of the distance and direction observations. This was a standard practice for all horizontal datums around the world regardless of which ellipsoid they used.

CLARKE 1866 had been chosen as the most appropriate ellipsoid for the country by USC&GS in 1880 replacing the BESSEL 1841 which more closely approximated the east coast but did not fit so well in other parts of the country.Please check:

The coordinates of all the other control stations were calculated from the knowledge of their positions (directly or indirectly) relative to MEADES RANCH using “network least squares adjustments”.

For NAD27, Clarke 1866 ellipsoid was fit so that:

(a) MEADES RANCH is exactly on the ellipsoid;

(b) the vertical deflection at MEADES RANCH is 0 though it did not come out exactly 0 according to the lecture slides;

(c) some measure of error between the control stations (other than MEADES RANCH) and the ellipsoid surface was minimized; and

(d) ellipsoid minor axis and earth’s rotation axis are parallel.Questions:

1. Does condition (a) imply condition (b) or vice versa? Or are they separate?

2. Is condition (d) required only for regional datums and not the global ones?

3. Is a point that is to be exactly on the ellipsoid called a “datum point”?

4. Were the coordinates of MEADES RANCH determined “through the network of triangulation from the station PRINCIPO in Maryland which had been used as the origin point for the New England Datum” or as “its astronomic latitude and longitude”, or are they the same thing?

5. Was WALDO only used for determining the positions of the control stations (other than MEADES RANCH)? It was not used to fit the ellipsoid?

6. “Choice of MEADES RANCH based on an

analysis of the minimization of sum of squares

of differences between astronomic and

geodetic azimuths.”Does this mean different DATUM POINTS were tried and it was found that MEADES RANCH minimize “the minimization of sum of squares of differences between astronomic and geodetic azimuths”?

7. “Adustment was NOT simultaneous least squares.”

Was it an iterative sort of least squares such as this? Was the fitting method not simultaneous for NAD83 as well?

8. Does adjustment of a datum mean shifting from the existing realization rather than fitting anew to all the control stations? If so, what happens if the coordinates of a control station are changed?

9. “NAD 27 fitted to Clarke 1866 yields a difference at the origin of over 200 meters.”

You mean the distance between the center of the earth and the center of the realized ellipsoid is over 200 meters?

10. I wonder how they knew Clarke 1866 approximated the whole country better than BESSEL 1841 and the opposite for approximating the east coast.

11. “geodetic surveying prior to the modern era was generally 2D where work was reduced to the ellipsoid. Ellipsoid heights were not used in computations overall. The level network was a separate activity. If you examine NGS data sheets for classically-determined stations you will note that XYZ coordinates are not provided (unless subsequently positioned using GPS) and heights frequently based on scaling or vertical angles. Ellipsoid heights for these points were not obtained.”

I am not sure what the paragraph means. Does it mean the surveyed points are in space (dimension = 3) but they were mapped onto an ellipsoid which is a surface (dimension = 2)? I guess it does not mean the surveyed points were recorded in (some sort of) 2-dimensional coordinate system but they were recorded in (some sort of) 3 dimensional coordinate system and then subsequently mapped onto an ellipsoid, a 2-dimensional surface.

This post is getting more and more confusing as I follow along. I’m not sure what you’re trying to get at Black Cat. Based on your post, I think you’re not an American Surveyor. But regardless of that.

Back in the day before there were Satellites most regions simply chose an Ellipsoid which worked and best fit it to their part of the World.

NAD 27 Best fit the North American Continent, it used Clarke 1866 (as stated previously)ED50 (European Datum 1950) Best fit the European Continent, it used “Hayford-Ellipsoid” of 1909.

Here’s a picture, note where the 2 ellipsoids are with respect to the Earth’s center of Mass and how they lay within the Continents they were intended to serve. (Courtesy of Jan Van Sickles’ Book)

I believe I got it right 🙂

Excuse the formatting problems. I timed out and copied this from the site.

>

> The coordinates of all the other control stations were calculated from the knowledge of their positions (directly or indirectly) relative to MEADES RANCH using “network least squares adjustments”.You infer that the control station positions were used in the network adjustment, this is not so. The observations (angles and distances) were used. The MEADES RANCH (hereafter MR) monument was chosen as the starting point from which positions to other points were determined. In addition to the observations an orientation must be provided. Scale is derived from precise baselines.

Network least squares is “merely” the process of combining the observations to account for geometry, redundancy and varying quality of the data.

The reason the deflection of the vertical is important is the basic fact that terrestrial measurements are made with respect to the gravity field. We want these observations to agree with geodetic observations made in the local ellipsoid. In addition to MR additional Laplace stations where the deflection of the vertical was determined were included in the 1927 adjustment. The knowledge of these deflections allows determination of the divergence between the geoid and ellipsoid. Closer agreement between the surface measurements and the reference ellipsoid makes computations more accurate. See also http://www.cage.curtin.edu.au/~will/SCONG99b.pdf

BTW, Laplace corrections were based on astronomical observations. It may be interesting to note that astronomical observations still play a role in geoid modeling as can be seen in the NGS Geoid Slope Validation Survey of 2011 (in Texas). In addition to the materials in the NGS web site, the current issue of the Journal of Geodesy includes a nice detailed description of the camera-based DOV system used for the project.

Without attempts to make the orientation of the datum coincident with the orientation if the Earth even more error would result.

Global datums are greatly different from regional datums.

The terminology used by national agencies often vary. I would not limit the use of the term datum point to one located on the surface of the ellipsoid.

The astronomical latitude and longitude at a points differ from its geodetic values by the Xi and Eta terms see the NGS web page for the Deflection of the Vertical tool. The knowledge of these differences is what Laplace stations provide.

WALDO was used for orientation.

As base9geodesy stated, MR was chosen for its central location. I will let him address the points he made. But remember we are talking about NAD 1927 when a computer was actually a human being working with books of logarithms and working through these problems by hand.

I encourage you to read the NOAA Professional Paper on NAD 1983 to appreciate the significant new methods and thinking it represented.

As can be seen by reviewing documentation for the NAD 83 (NSRS 2007) and subsequent NAD 83 2011 adjustments, they were computed using measurements (GPS only) and using the 3D coordinates of CORS sites as fixed control. New positions were determined for all points for which connections were available in terms of quality GPS vectors. Likewise, there are good technical papers on the NGS site about these adjustments.

The magnitude of the difference between the geocenter used for NAD 83 (1986) and the computed value for NAD 27 is over 200 meters. As indicated previously geocentricity was NOT a goal in NAD 27.

As stated above, geodesists knew the differences between measurements in the local gravity field and the values at the surface of the mathematical surface.

Observations made using triangulation were two-dimensional in the sense that the horizontal angle between two points even at greatly different heights does not account for height differences. Observations in NAD 27 were reduced to the surface of the ellipse. As you know, the geodetic latitude and longitude can be computed without height information.My reference to the NGS data sheets illustrates this fact. Classically determined (e.g. Triangulation) points do not have heights associated with them. Pre-NAD 83 we had horizontal points and vertical heights and NOT 3-D coordinates like the XYZ Cartesian coordinates common today.

Perhaps my terminology is confusing. When I mention a 2-D system I mean one where only latitudes and longitudes are determined and not heights with respect to the reference ellipsoid. Of course a latitude/longitude system is not a planar surface but determined with respect to a reference ellipsoid.

Hope this is responsive.

DMM

By the way this is the last time I’ll try a lengthy response in this damned iPAD.

>

> For NAD27, Clarke 1866 ellipsoid was fit so that:

> (a) MEADES RANCH is exactly on the ellipsoid;

> (b) the vertical deflection at MEADES RANCH is 0 though it did not come out exactly 0 according to the lecture slides;

> (c) some measure of error between the control stations (other than MEADES RANCH) and the ellipsoid surface was minimized; and

> (d) ellipsoid minor axis and earth’s rotation axis are parallel.

>

>

> Questions:

>

> 1. Does condition (a) imply condition (b) or vice versa? Or are they separate?

>

> 2. Is condition (d) required only for regional datums and not the global ones?

>

> 3. Is a point that is to be exactly on the ellipsoid called a “datum point”?

>

> 4. Were the coordinates of MEADES RANCH determined “through the network of triangulation from the station PRINCIPO in Maryland which had been used as the origin point for the New England Datum” or as “its astronomic latitude and longitude”, or are they the same thing?

>

> 5. Was WALDO only used for determining the positions of the control stations (other than MEADES RANCH)? It was not used to fit the ellipsoid?

>

> 6. “Choice of MEADES RANCH based on an

> analysis of the minimization of sum of squares

> of differences between astronomic and

> geodetic azimuths.”

>

> Does this mean different DATUM POINTS were tried and it was found that MEADES RANCH minimize “the minimization of sum of squares of differences between astronomic and geodetic azimuths”?

>

> 7. “Adustment was NOT simultaneous least squares.”

>

> Was it an iterative sort of least squares such as this? Was the fitting method not simultaneous for NAD83 as well?

>

> 8. Does adjustment of a datum mean shifting from the existing realization rather than fitting anew to all the control stations? If so, what happens if the coordinates of a control station are changed?

>

> 9. “NAD 27 fitted to Clarke 1866 yields a difference at the origin of over 200 meters.”

>

> You mean the distance between the center of the earth and the center of the realized ellipsoid is over 200 meters?

>

> 10. I wonder how they knew Clarke 1866 approximated the whole country better than BESSEL 1841 and the opposite for approximating the east coast.

>

> 11. “geodetic surveying prior to the modern era was generally 2D where work was reduced to the ellipsoid. Ellipsoid heights were not used in computations overall. The level network was a separate activity. If you examine NGS data sheets for classically-determined stations you will note that XYZ coordinates are not provided (unless subsequently positioned using GPS) and heights frequently based on scaling or vertical angles. Ellipsoid heights for these points were not obtained.”

>

> I am not sure what the paragraph means. Does it mean the surveyed points are in space (dimension = 3) but they were mapped onto an ellipsoid which is a surface (dimension = 2)? I guess it does not mean the surveyed points were recorded in (some sort of) 2-dimensional coordinate system but they were recorded in (some sort of) 3 dimensional coordinate system and then subsequently mapped onto an ellipsoid, a 2-dimensional surface.

}*3. Is a point that is to be exactly on the ellipsoid called a “datum point”?*The latitude and longitude of all points in a horizontal/geometric datum are exactly on the ellipsoid. Only the ellipsoid height component is provided above or below the ellipsoid and as previously noted NAD 27 and all other classically determined datums did not have a good knowledge of the ellipsoid heights.

*4. Were the coordinates of MEADES RANCH determined “through the network of triangulation from the station PRINCIPO in Maryland which had been used as the origin point for the New England Datum” or as “its astronomic latitude and longitude”, or are they the same thing?*While station PRINCIPO can be referred to as an astronomic station because USC&GS did perform astronomic latitude and azimuth observations there in 1866, those data were used as part of the effort to compute the deflection of the vertical which GeeOddMike explained so well.

*7. “Adustment was NOT simultaneous least squares.”*NAD 27 and all other horizontal and vertical datums of that time were computed by hand. As long as the proper procedures are followed the results of an adjustment should not be significantly different if computed by hand or “simultaneously” through a computer program.

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