The Geoid - An Equipotential Description with Gravity

The Geoid is a surface that is not often talked about on blogs or mentioned on the web - so this may be a first.

The Geoid surface is irregular, unlike reference ellipsoids (such as Clarke 1866, Bessel, Hayford, etc.) which have been used to approximate the shape of the physical Earth at a local point. The geoid is considerably smoother than Earth's physical surface.

In looking for a good description that makes sense to many people, I stumbled upon this one on Wikipedia:

"In geodetic surveying, the computation of the geodetic coordinates of points is commonly performed on a reference ellipsoid closely approximating the size and shape of the Earth in the area of the survey. The actual measurements made on the surface of the Earth with certain instruments are however referred to the geoid. The ellipsoid is a mathematically defined regular surface with specific dimensions. The geoid, on the other hand, coincides with that surface to which the oceans would conform over the entire Earth if free to adjust to the combined effect of the Earth's mass attraction (gravitation) and the centrifugal force of the Earth's rotation. As a result of the uneven distribution of the Earth's mass, the geoidal surface is irregular and, since the ellipsoid is a regular surface, the separations between the two, referred to as geoid undulations, geoid heights, or geoid separations, will be irregular as well."

and further it states:

"The geoid is a surface along which the gravity potential is everywhere equal and to which the direction of gravity is always perpendicular. The latter is particularly important because optical instruments containing levelling devices are commonly used to make geodetic measurements. When properly adjusted, the vertical axis of the instrument coincides with the direction of gravity and is, therefore, perpendicular to the geoid. The angle between the plumb line which is perpendicular to the geoid (sometimes called "the vertical") and the perpendicular to the ellipsoid (sometimes called "the ellipsoidal normal") is defined as the deflection of the vertical. It has two components: an east-west and a north-south component."

The reference surface for heights is traditionally taken as Mean Sea Level (MSL).

The geoid, as described above, is a surface of equal gravity potential which closely approximates mean sea level.

With GPS becoming more and more relevant in our daily lives, what is the height measurement we get?

The heights derived from GPS are relative to the GPS reference ellipsoid (WGS84). The separation between the geoid and an ellipsoid is known as the geoid-ellipsoid separation, or N value.

In a mathematical sense, we have the following then:

H = h - N

where H = Orthometric Height

h = Ellipsoidal Height (for example, the height above the ellipsoid WGS84)

N = Geoid-Ellipsoid Height (this is also called the Geoid Undulation)

Note that with N, that if the geoid is above the ellipsoid, N is positive. If the geoid is below the ellipsoid, N is negative.

How does mass effect the geoid and the ellipsoid?

Where a mass deficiency exists, the geoid will dip below the mean ellipsoid and where a mass surplus exists, the geoid will rise above the mean ellipsoid.

Where are the largest undulations?

Well, the largest undulations known, with the minimum in the Indian Ocean at a value of N = -100 metres and the maximum in the northern part of the Atlantic Ocean with N = +70 metres.

So how do we describe the shape and size of the Earth?

There are three surfaces to be considered:

• The topography - the physical surface of the earth.



• The Geoid - the level surface (also a physical reality).



• The Ellipsoid - the mathematical surface for computations.
  

Mean Sea Level (MSL) points, an approximation to the geoid, and can be used as reference surfaces for height measurements (i.e. orthometric heights).

Ellipsoidal heights (such as those derived by GPS) have to be adjusted before they can be compared to the orthometric heights given on topographic maps.

The deviation between the geoid and an reference ellipsoid is called Geoid undulation (N). Geoid undulations can be used to adjust the ellipsoidal heights (H = h +/- N).

This is an introduction to the Geoid and the science of Geodesy. It hopefully clears up some questions about this surface.

I'll explain in more detail some of the formulations of the Geoid and how geodesists over time have tried to model it and some of the efforts being conducted presently to come up with a global gravity model to aid in height determination at a later date.

There are some very interesting projects going on in Africa, South America, and Canada.

Google Maps: Is the Earth a Sphere or Ellipsoid?

Google Maps sees the Earth as a Spheroid, not an Ellipsoid. This came up through a discussion on the PROJ mailing list and I thought it was interesting to point out how Open Source can even handle projected lat/long systems (such as Google Maps) using a very familiar tool called cs2cs.

Christoper Schmidt wrote about it on his blog and also on his blog he points out the EPSG code to use. The magical number for the Google Mercator Projection (of a lat/long grid based on a sphere) is: 900913

Now onto the fun, showing how we can use Open Source to have our data show within a KML project and Google Maps. Quoting from Frank's FAQ, he provides an excellent example, we see the following the use of cs2cs:

"cs2cs +proj=latlong +datum=WGS84 +to +proj=merc +a=6378137 +b=6378137 +lat_ts=0.0 +lon_0=0.0 +x_0=0.0 +y_0=0 +k=1.0 +units=m +no_defs"

Notice the sphere is being used? a=b

Because we are dealing with a sphere, the Y values will be greatly different from those on an ellipsoid (30 to 100 metres or more).

Quoting Frank again:

"In this case, and many other cases using spherical projections, the desired approach is to actually treat the lat/long locations on the sphere as if they were on WGS84 without any adjustments when using them for converting to other coordinate systems. The solution is to "trick" PROJ.4 into applying no change to the lat/long values when going to (and through) WGS84. This can be accomplished by asking PROJ to use a null grid shift file for switching from your spherical lat/long coordinates to WGS84.

cs2cs +proj=latlong +datum=WGS84 +to +proj=merc +a=6378137 +b=6378137 +lat_ts=0.0 +lon_0=0.0 +x_0=0.0 +y_0=0 +k=1.0 +units=m +nadgrids=@null +no_defs

Note the strategic addition of +nadgrids=@null to the spherical projection definition"

As you can see the value of Open Source and the mailing list and Open Source software is that people are sharing knowledge - whether it be via blogs, lists, or some other means of communication. There is a community out there that supports each other. These are actual users facing everyday problems and looking for solutions. The answers do exist, just the question has to be asked, and the community comes together to help.

Is WGS84 really WGS84? Is it correctly defined?

While doing some research on things that geodesists like - questioning MSL and the geoid, questioning how exact our definitions of ellipsoids and the earth are, I stumbled upon a very interesting article, written by Muneendra Kumar, PhD (Retired, National GeoSpatial Intelligence Agency) and James P Reilly, PhD (New Mexico State University) and they were asking:

Is definition of WGS 84 correct?

The following is written by these two distinguished gentlemen and I hope it proves to be as interesting to everyone as it was to me.

In the end, though, for most mapping purposes, WGS84 is still WGS84 and your latitude and longitude, or your Northing and Easting will remain the same.

As a side note, the North Pole is moving south and Sea Level is not level!

More on that little bit of geodesy in a later blog.

Enjoy and feel free to write me if you have any questions or comments.

Is Definition of WGS84 Correct?

by

MUNEENDRA
KUMAR, PHD AND JAMES P REILLY, PHD

A Geodetic Analysis

The first and original version of the "WGS 84", defined by a special committee of the Defense Mapping Agency (DMA), was released in September 1987.

As this task of updating the WGS 72 was concurrent with development of the North
American Datum (NAD) 1983, the committee members always had many in-depth
discussions with the members of the special committee of the National Geodetic
Survey (NGS). This approach ensured the; correct geodetic definition both for
WGS 84 and NAD 83. Around 1992, it was decided by DMA that, in future update(s)
of the "WGS 84" for accuracy enhancement, the academia and other satellite
geodesy experts would be associated. However, that scientific participation was
not followed and three subsequent updates were carried out without in-depth
discussions of satellite geodetic theory and/or correct statistical evaluation.
The non-scientific procedure(s) allowed definition deficiencies to creep in.
This paper outlines the geodetic details of the three updated versions of 1994,
1996, and 2001 and brings out in "open" the definition deficiencies in the
current version WGS 84 (G1150), which otherwise will remain hidden within the
National Geospatial- Intelligence Agency (NGA).The correctly defined "WGS 84",
the coordinate system used in GPS, is a critical requirement for the geodetic
integrity and accurate GPS positioning.

1984 "Original" Definition

The WGS 84 was originally defined with BIH Conventional Terrestrial System (CTS) for Reference Epoch "RE (84.0)". The main satellite data sets used were from the
Navy Navigation satellite System (NNSS). At the time of release in 1987, the
accuracy achieved was in the order of ± 1 - 2 meter and as such the tidal
effects, as specified in the International Association of Geodesy (IAG)
Resolution 16 of 1983 were not considered.

The "Three" WGS 84 Updates

1994 "WGS 84 (G730)"

This version was updated with the International Earth Rotation Service(IERS) realized International Terrestrial Reference Frame (ITRF) 19921, RE (88.0). During this update, NGA moved the RE (88.0) of the defining
ITRF to RE (94.0), which is incorrect. For this "change", DMA geodesists did not
have the capability and expertise. And, they did not have the authority to
override IERS. Note: With a new origin and orientation of its three axes, WGS 84
(G730) is geodetically a different coordinate system than the original WGS 84.
For mapping, the two could be considered the same. 1 First six ITRF solutions,
viz., ITRF 1988, ITRF 1989, ITRF 1990, ITRF 1991, ITRF 1992, and ITRF 1993, were realized for the RE (88.0). As the ITRF 1993 was based on all the data sets
available up to the end of year 1993 and thus realized in 1994, it would not
have been possible for DMA to define the WGS 84 (G730), which was realized using
the GPS data for the week starting 2 January 1994.

1996 "WGS 84 (G873)"

At the time of this update, the ITRF 1994, RE (93.0) was used (Note: ITRF96 (93.0) was not available). But, DMA geodesists again incorrectly moved the epoch of the
defining "RF" to RE (97.0). And, for geodetic application, they created the
third WGS 84. In addition, ignoring IAG Resolution No. 16 of 1983 and bypassing
IERS Conventions (IERS, 96), which recommend the "Zero-tide" model, National
Imagery and Mapping Agency (NIMA) geodesists adopted an "arbitrary" practice to
use "Tide-free" model. Note: According to IERS, the positions in the "Tide-free"
environ are non-realistic and not observable.

2001 Current "WGS 84 (G1150)"

During the updating of this version, the ITRF00, RE (97.0) was used.
But, like the 1994 and 1996 versions, NIMA geodesists incorrectly moved the RE
of the defining RF from (97.0) to (01.0), They also kept the 1996 practice for
"Tide-free" model, even after being alerted that the world's eminent geodesists
support the "Zerotide" of the IAG' standing Resolution No. 16 of 1983.
Furthermore, during the adjustment of the GPS tracking stations network, about
65% stations were held fixed (Note: An objection by the first author was not
even discussed). This over constrained adjustment is statistically incorrect and
not acceptable. Note: For geodetic positioning, this is the fourth version of
WGS 84.

The "Version" Identifiers

The "G730", "G873", and "G1150" indicate the GPS-week, of which the data sets were used to realize the three updates. As these "identifiers" do NOT specifically identify any definite time epoch, they do NOT have any geodetic significance.

Important "Contrast" To Note In SIRGAS 2000, the "RE" of the defining ITRF has NOT been "moved".

Analytical Conclusion

The current "WGS 84 (G1150)" is incorrectly defined, does not
comply with IAG Resolution No. 16 of 1983, and its time epoch is not definitive.
Furthermore, the adjustment of the GPS tracking station network is statistically
incorrect.

Reference

IERS, 96 IERS Conventions, Tech Note 21, July 1996.

cs2cs & BeTA2007 - Open Source, Germany & NTv2

Recently in the PROJ.4 Mailing List there was a discussion about working with the DHDN / Gauss Krueger to WGS84 conversion.

One of the posters listed a site and document that covered BeTA2007 (which means Bundeseinheitliche Transformation fur ATKIS) and allowed me to brush up on my German.

In reviewing the document, we find out that the method of Operation for the datum conversion is based on NTv2 and the coordinate reference system is ETRS89/UTM.

Within the document listed above, they even demonstrate how to do a coordinate coversion using cs2cs and how to use the +nadgrids parameter.

An example from the document shows that the +nadgrids does not have to be specific to North America, only that the data file containing the grid has the same format. This format has been specified by the Canadian Government and has been adopted by many countries (such as South Africa and Australia, to name a few).

This is the published example, using the BETA2007.gsb file and a data file containing these two points:

(1) 2490000.00 5652000.00
(2) 2504000.00 5628000.00


>> cs2cs \
+proj=tmerc +lat_0=0 +lon_0=6 +k=1.0000000 \
+x_0=2500000 +y_0=0 +ellps=bessel +units=m \
+nadgrids=./BETA2007.gsb
+to \
+proj=utm +ellps=GRS80 +zone=32

Resulting in:

2490000.00 5652000.00 <-- Input (1)
279488.01 5654871.71 0.00 <-- UTM 32/GRS80 Answer for (1)

2504000.00 5628000.00 <-- Input (2)
292503.36 5630318.18 0.00 <-- UTM 32/GRS80 Answer for (1)

So cs2cs works with other grid systems as long as they are defined the same as the NTv2 standard.

So all the above makes more sense, let me define, from Wikipedia, what ETRF is: "The European Terrestrial Reference System 1989, usually referred to as ETRS89, is a three-dimensional geodetic frame of reference - a mapping coordinate system used as the standard high accuracy system for GPS in Europe. It coincided with the World Geodetic System 1984 in 1989, hence the name, and is based on the same GRS80 ellipsoid. Unlike WGS84 or ITRS it is centred on Europe and diverges from them with the movements of the tectonic plates associated with this landmass."

Please see: http://en.wikipedia.org/wiki/ETRS89 for ETRS89 and http://en.wikipedia.org/wiki/World_Geodetic_System_1984 for WGS84

The parameters for a datum shift (dx, dy, dz, rx, ry, rz, etc.) between ETRS89 and WGS84 are not constant, due to the movement of the Eurasian geophysical plate with respect to WGS84. The differences between both datums can grow by several centimetres a year. Currently they are a couple of decimetres in difference. For many applications, these differences are not relevant. Coordinates or positions in WGS84 have usually been obtained by GPS and this results in an accuracy at the level of several metres. However, satellite positioning techniques continuously improve in accuracy, also without using differential stations. So the differences between the datum's will grow. Note that many nations are using WGS84 to define boundaries nowadays - so WGS84 is relevant.

One poster from France, to the mailing list, replied that they solved the problem of moving plates by adding a date to the location, and they stated their method as follows:

"We have solved this problem with moving plates by adding a simple date with the coordinates of WGS84. That way you can always go back to the original position of that particular date in WGS84 (or any). The WGS84 datum (with date) will stay accurate for ever, since it is always possible to trace back where the plates were at that specific date.

latitude,
longitude,
(WGS84),
date"

There are always solutions and answers to questions out there. Just feel free to ask, whether it be to a mailing list or your local expert, people are always willing to help and pass on knowledge.