Not much difference!

I think the proposition of doing a "datum shift" between NAD83 and WGS84 is a waste of time. The difference between the two is the philosophy of how an inertial datum should be defined, and is ignored by all but a handful of geodesists that are working at extraordinary levels of accuracy.

The ellipsoid of the World Geodetic System of 1984 is defined by four constants, one being the "bar C sub 20" Normalized Second Degree Zonal Harmonic Coefficient of the Gravitational Potential - the "J2" term is computed. The ellipsoid of the Geodetic Reference System of 1980 is defined by four constants, one being the "J2" Dynamical Form Factor - the "bar C sub 20" term is computed.

Geoid models (EGM96 versus GEOID96) will yield meaningful differences in ellipsoid/geoid separations (vertical) for North America, horizontal positions will not reflect anything beyond micro-fractions of a millimeter at best.

The differences between the semi-minor axes are computational (secondary DEPENDENT) variables based on different definitions of the gravity field. They are called "Derived Geometric Constants." The semi-major axis of the WGS84 ellipsoid is known to an accuracy of plus or minus two meters.

Transformations based on a derived difference of a tenth of a millimeter with an constant's accuracy of two meters is, IMHO, silly.

"Therefore, as long as the preceding is recognized, it can be stated that WGS84 and NAD83 are based on the same ellipsoid."

by: Clifford J. Mugnier cjmce@uno.edu 2/7/98

If you examine DMA Technical Report 8350.2, DOD WGS 84: Its Definition and Relationships with Local Geodetic Systems, dated 30 Sep 1987 you will note that the transformation parameters between NAD 83 and WGS 84 (deltaX, deltaY and deltaZ) are all zero (0). There is a difference in the flattening of -0.00000016. (See page 7-24).

The classified portions of the document are blank in the version available from either the National Geodetic Survey (NGS) Info Center at 301.713.3242 or from the commercial firm NAVTECH at 1.800.NAV.0885 (and elsewhere)

Complicating the statement that more most users NAD 83 and WGS 84 are equivalent is the decision by NIMA (formerly DMA) to redefine WGS84 to make it and keep it in line with the ITRF. The information below was extracted from an NGS summary of precise satellite ephemeris reference frames.

(Codes are used in NGS software.)

The Naval Surface Warfare Center in Dahlgren, Va had a technical report detailing the most recent WGS84 realignment on their web site. I don't have the web site address handy. If you are unable to find it yourself, please advise and I'll forward the link.

by: Donald M. Mulcare email: dmulcar@ibm.net 3/12/98

Much of the information below was based on a draft version of a paper relating NAD 83 to the ITRF written by Stephen Frakes of the National Geodetic Survey.

NAD 83 was implemented in 1986 with an adjustment combining terrestrial data with space-based data such as 3-D positions from satellite Doppler and 3D vectors from VLBI. The global parameters used in NAD83, X,Y,Z shifts and X, Y rotations plus scale were fixed to a priori values based on Doppler. The Z rotation was defined by astronomical azimuths. VLBI rotations and scale were solved as a means to compare systematic differences between the VLBI and NAD 83 coordinate systems. The NAD 83 adjustment did not include GPS. Positions determined in this adjustment are labeled NAD 83 (1986). The adoption of GPS revealed inadequacies in the system which prompted upgrade surveys (described below).

If your question is how the US National Geodetic Survey Continuously Operating Reference Stations (CORS) station positions indicated as "NAD 83 Precise" were derived, they were determined by a seven-parameter transformation which relates the NAD 83 system to the ITRF.

There are however, another version of NAD 83 coordinates which are part of an upgrade program. In 1988 the NGS began to upgrade the NAD 83 system by performing high accuracy GPS surveys. In order to establish control stations for these upgrade surveys the following steps were performed:

The NGS datasheets show the station position with its datum. States with a high accuracy reference network (HARN) will show the datum as NAD 83 (####) the #### indicating the date of the adjustment. A NAD 83 (1986) position is also provided on the data sheet. NAD 83 (1986) refers to the original NAD adjustment.

Nope. You have to first go to Geodetic Cooordinates, perform a Datum Shift in Geocentric Coordinates with either a 3-parameter or a 7-parameter transformation, then transform back to Geodetic Coordinates, then to whatever Grid system you want on the new Datum.

There are a number of "free" packages that can do this for you without the need to bother with the details if your accuracy requirements will tolerate a black-box transform. Common packages are NADCON from the National Geodetic Survey and CorpsCon from the Topographic Engineering Center IF YOU ARE WORKING IN THE UNITED STATES. Foreign datum shift stuff is available from the NIMA site under the general topic of "MUSE," the specific package is MADTRAN. Note that the NIMA package is for modest accuracy requirements, it generally is not suitable for scales larger than 1:50,000. The NADCON/CorpsCon packages are suitable for larger scale applications but still fall short for geodetic survey accuracy. You, the user, need to access the accuracy requirements for your application. If you do not know, then you need to get help before you proceed further.

This stuff gets deep quick. I would recommend a bit of research before "jumping into" Datum Shifts. I publish a (nearly) monthly column in "Photogrammetric Engineering and Remote Sensing" on "Grids and Datums." Each installment is a different continent and country.

by: Clifford J. Mugnier cjmce@uno.edu 2/25/98

You'r going the right way! ECEF is essential. Convert the lat/long to ECEF (using the correct parameters for the spheroid they are on), then apply the dX,Y,Z, scale factor (and rotations if you have them). Getting the signs right here is vital! In some conversions, reversing the signs can put you +1 km from the correct position. Use the resulting ECEF value to compute lat & long on the new spheroid, once again using the correct parameters.

To give you an example, let us assume we have a point at 25d 00m 00s South Latitude, 141d 00m 00s East Longitude with a spheroidal height of 65.459m. This is on the Australian Geodetic Datum 1984 (AGD84) (6378160, 1/298.25) We want to convert to WGS84 (6378137.00, 1/298.257223563) The datum shift FROM ADG84 TO WGS84 is dX -116.00 dY -50.47 dZ141.690. Scale & rotations have been ignored.

For anyone trying to repeat this, please don't email me if its slightly out!!! This is simply a fast and dirty approximation done with software designed for a 7 parameter transform. ;-)

For anyone who wants to look seriously at transforms, I can recommend Ausligs calc page (URL below)

http://www.auslig.gov.au/geodesy/calcs.htm

There are a number of Excel spreadsheets to perform the above calcs etc.

by: Doug Hall dwh@fl.net.au 3/8/98

Maps using the NAD 27 datum also generally use the National Geodetic Vertical Datum of 1929. Elevations are measured against Mean Sea Level (MSL). The two datums (horizontal and vertical) are, essentially independent of one another.

Maps using the more recent NAD 83 datum (very similar to WGS 84.. within a meter or two) will still have essentially the same elevations above MSL.

It is interesting how the GPS (based on WGS 84) works. Four or more satellites can be used to determine three position dimensions and time. Position dimensions are computed by the receiver in Earth-Centered, Earth-Fixed X, Y, Z (ECEF XYZ) coordinates. When more than "the best four" satellites are used, this is called an over-determined solution.

Most receivers compute (and store) the geodetic (in WGS 84) coordinates of latitude, longitude and height above the ellipsoid (HAE). WGS 84 coordinates are transformed and displayed in whatever datum you choose. Always choose the datum of the map you are working with.

The height displayed on most consumer handheld GPS receivers is orthometric height, the height above mean sea level (MSL). It is straight forward to approximate MSL world wide by interpolation of the GEOID model (table) and making the simple calculation:

Orthometric Height (MSL) = HAE - Geoid Undulation

The relative offsets in terms of a 3-Parameter Datum Shift is because the single board computer (sbc) in the old Magnavox 1502 Transit Satellite Doppler Receivers could only accomodate that simple model in 1972 when they were put into production. When the WGS72 Datum was declassified in 1976, the 1502's were declassified and sold to the public. THAT's when people started to learn what Datums were all about. Considering that before then, the fanciest (absolute) positioning hardware was a First-Order Astronomical Theodolite with a radio chronometer and impersonal micrometer that after two evenings of observations got you a position to +/- 100 meters.

It is enough for small regions and countries if you're delivering 155 mm Howizer Cannon fire or sending Cruise missiles with High Explosive Warheads. It is not generally suitable for geodetic applications and many GIS applications if you need an accuracy better than +/- 25 meters.

The "free" stuff from NIMA is intended for destroying large geographic areas. Oil companies use it for rough reconnaissance positioning. Ignorant technicians misuse it for precise mapping.

What's enough? I like the 7-parameter Bursa-Wolfe Datum Shift and I like the 7-parameter Molodensky Datum Shift. Those accommodate rotations in all three geocentric axes as well as an overall scale factor. That is usually enough for most small countries.

EGM96 is the latest geoid for the world. GEOID96 is the latest geoid specifically for the United States of America. It is a smidgen different, but worth it if you're diddling with vertical stuff from GPS observations.

Those geoids are NOT the same as NAVD88, certainly NOT NGVD29.

The latest thing is using GPS geodetic-quality receivers and GEOID96 to obtain reliable vertical in the U.S. However, it does require a number of very specific procedures to obtain reliable results within a few centimeters.

I am refering a copy of this to Mr. Dave Doyle of the National Geodetic Survey in Silver Spring, Maryland. Mr. Doyle can help you better than I can, and he can also refer you to specialists within NGS regarding specifications of hardware, software options, publications, etc.

If the distance is less than about 20 km and the locations of the 2 points in Cartesian coordinates are X1,Y1 and X2,Y2 then the Pythagorean Theorem

d = sqrt((X2-X1)^2 + (Y2-Y1)^2)

will result in an error of: less than 30 meters for latitudes less than 70 degrees; less than 20 meters for latitudes less than 50 degrees; less than 9 meters for latitudes less than 30 degrees.

Otherwise, presuming a spherical Earth with radius R (see below), and the locations of the 2 points in spherical coordinates (longitude and latitude) are lon1,lat1 and lon2,lat2 then Sinnott's Formula (from R.W. Sinnott, "Virtues of the Haversine", Sky and Telescope, vol. 68, no. 2, 1984, p. 159):

dlon = lon2 - lon1

dlat = lat2 - lat1

a = sin^2(dlat/2) + cos(lat1) cos(lat2) sin^2(dlon/2)

c = 2 arcsin(sqrt(a))

d = R c

will give mathematically and computationally exact results. The intermediate result c is the great circle distance in radians.

The Pythagorean flat-Earth approximation assumes not only that great circles are negligibly different from straight lines, but that the parallels of latitude are negligibly different from great circles. Close to the poles, the parallels of latitude are not only shorter than great circles, but indispensably curved, as well. Taking this into account leads to the use of polar coordinates and the planar law of cosines for computing short distances near the poles: The Polar Coordinate Flat-Earth Formula

a = pi/2 - lat1

b = pi/2 - lat2

d = R sqrt(a^2 + b^2 - 2 a b cos(lat2 - lat1)

will give smaller errors than the Pythagorean Theorem for high latitudes, but even at 88 degrees the error can be as large as 20 meters when the distance between the points is 20 km.

An UNRELIABLE way to calculate distance on a spherical Earth is the Law of Cosines for Spherical Trigonometry

** NOT RECOMMENDED **

a = sin(lat1) sin(lat2) + cos(lat1) cos(lat2) cos(lon2 - lon1)

c = arccos(a)

d = R c

although this formula is mathematically exact, it is unreliable for small distances because the inverse cosine is ill-conditioned. Sinnott (in the article cited above) offers the following table to illustrate the point:

A computer carrying seven significant figures cannot distinguish the cosines of any distances smaller than about one minute of arc.

Q5.1a: What value should I use for the radius of the Earth, R?

The shape of the Earth is well approximated by an oblate spheroid with a polar radius of 6357 km and an equatorial radius of 6378 km. Any value in that range will do, such as

R = 6378 - 21 sin(lat) See the WARNING below!

where lat is the latitude in which the bulk of your calculations occur, PROVIDED a spherical approximation is satisfactory.

WARNING: The above formula for R gives but a rough approximation to the radius of curvature. According to Snyder ("Map Projections - A Working Manual", by John P. Snyder, U.S. Geological Survey Professional Paper 1395, United States Government Printing Office, Washington DC, 1987, p24), the radius of curvature varies with direction and latitude; in the plane of the meridian it is given by

R' = a (1 - e^2) / (1 - e^2 sin^2(lat))^(3/2)

where a is the equatorial radius; b is the polar radius; and e is the eccentricity of the ellipsoid = (1 - b^2/a^2)^(1/2).

When is it NOT okay to assume the Earth is a sphere?

A quick test is: Compute the values of R produced by the above equation when you use the highest and lowest latitudes that occur in your analysis. Use these two values in your analysis. If the different results are different enough to cause you to change your action (or your recommendation, or your interpretation of the implication of the results, etc.), then assuming the Earth is spherical is NOT okay.

For most purposes, it is quite satisfactory to treat the Earth as a sphere. If necessary, an oblate spheroid can provide a better approximation.

The shape the Earth would assume if it were all measured at mean sea level is called the geoid. The geoid varies no more than about a hundred meters above or below a well-fitting ellipsoid, a variation far less than the ellipsoid varies from the sphere. Terrain relief is reported relative to the geoid. (Paraphrased from p. 11 of the book by Snyder cited above.) Distances on the surface of the geoid are not particularly meaningful.

Distances on the surface of the terrain, whether geodesic, on roads, cross-country, or straight-line, depend on relief and on the status of engineering projects. Hence, computation is idiosyncratic and not well suited to simple approximations.

You will find distance formula at this place: THE MATH FORUM

Subject: Distance Between Two Points on the Earth

Mathematically, the Lambert Conical Orthomorphic projection is synonomous with the Lambert Conformal Conic. There is NO DIFFERENCE. However, the term, "orthomorphic," is classical British and implies a considerably different problem to the unfortunate user of the aforementioned software package.

For the ellipsoidal case,

The "American Definition" defines a secant zone by two standard parallels, and that is the way the General Cartographic Transformation Program (GCT P) is programmed and is used by many near-illiterate (geodetically unwashed) software houses world wide, particularly the large dunder-headed GIS houses in North America. Works great in the U.S. for the State Plane Coordinate Systems on NAD 1927 or NAD 1988 !!! Does not do diddly elsewhere ...

The "British Definition" defines a secant zone by a Latitude of Origin and a Scale Factor at Origin. This is a very different way of doing things and applies, for the most part, to every Lambert Zone in the entire world outside of the United States and its possessions. If the brilliant software gurus used GCTP as the transformation engine for your application package, too bad for you. Get in line and maybe someone will be willing to develop a "work around" for your project because you pay enough to warrant VIP treatment. (Their "work-arounds" are the joke of the industry!)

Both definitions yield EXACTLY the same projection and Grid System if you know what you're doing.

Now, for the Island of Borneo, there are two British Lamberts. One is defined for Papua, New Guinea on the AS120 Tokang, Balik Papan Datum. It is a true Lambert Conical Orthomorphic. (The Latitude of Origin is below the equator, and probably is listed in the marginal information on your paper map.) The other is defined for Indonesia's portion of the Island and uses Paga Hill 1939 Datum. For that particular Lambert Conical Orthomorphic the Latitude of Origin is defined at the equator (zero degrees) - at which point a Lambert instantaneously converts into a Normal Mercator! Therefore, if you're diddling with Indonesian Borneo, GCTP will accomodate a Mercator. If, on the other hand you're diddling with Papua, New Guinea Borneo; you're probably in deep doo-doo.

Gauss Krueger Coordinate System

There is a site in Geneva for the European Petroleum Studies Group (EPSG) that gives free data out on many Grid Systems in Europe. Note that Switzerland does not use a Transverse Mercator; it uses a Rosenmund Oblique Cyllindrical that just by chance is already programmed in PROJ4. European use of the Transverse Mercator has been 100% Gauss-Kruger (u with an umlaut on Kruger) since the Second World War. Prior to WWII, there's some pretty weird stuff (in Europe) that is currently found only in the various colonies settled by the European Nations.

The various formulae that comprise the family of Transverse Mercator truncations are rarely found in English. Most of the stuff is German and Eastern European languages; occasionally French, Spanish, Italian and Portuguese. A little bit is in Mandarin.

by: Clifford J. Mugnier cjmce@uno.edu 2/25/98

I don't know are there available simple spreadsheet for converting ECEF X,Y,Z to UTM X,Y over WGS84 ellipsoid but using 2 programs it possible:

a) ECEF X,Y,Z to Lat, Long is available to download as MS Excel spreadsheet at: http://www.auslig.gov.au/geodesy/calcs.htm

The Datum and Grid on Curacao was established in 1951 and uses a local Transverse Mercator. The origin of the datum is "DP 8" ("Dienst Punt 8" or Survey Point 8). The stuff I have there for WGS84 datum shifts looks really, really good. (sub-centimeter accuracy)

The projection and datum of St.Lucia, West Indies: The current classical geodetic datum for your country is "DOS 3 Lighthouse 1955" established by the Directorate of Overseas Surveys. (They are now part of the Ordnance Survey of Great Britain.) I do not have the coordinates of the origin or the azimuth of orientation, but the ellipsoid is the "Clarke 1880, (Modified)" where a = 6,378,249.145 meters, 1/f = 293.465. The associated Grid is the British West Indies "BWI Grid" that is based on the Gauss-Kruger Transverse Mercator projection with no special truncations. The Central Meridian is 62W, False Easting is 400 Km, scale factor at origin is reduced to 1/2000. Eastings for your country are always greater than 500 Km., the Northings are always greater than 1,500 Km. (That's a lot of digits for your island!!)

The "Vieux Fort" is an old Astro Datum known to exist, I think it's now obsolete. It's probably "T 1 1949", which is a Tidal bench Mark in that town.

The 1:50,000 mapping of your country is on the North American Datum of 1927 and the Grid is UTM. (That's for NATO use and is ignored by the civilians.) Isn't it comforting to know that they're prepared to "call in" accurate artillery fire? Modern civilization ...

There are four collocated "A" Order GPS points on St. Lucia, two are in your city of Castries. One is known by Mr. Paul Boland (LS 61 G), the other is on Mons Fortune (DCS 33). The orientation to WGS84 is particularly good, but no current mapping or local surveys are known to use it. The Ordnance Survey is rumored to be contemplating changing the West Indies to WGS84, but that will probably be implemented over a very long period of time. When your current 1:2,500 mapping becomes hopelessly obsolete, then maybe.

In regard to sunny Yemen, there are a number of datums found there. The Aden Datum of 1925 uses the Aden Lambert Zone (Clarke 1880). The Sanaa Datum uses the Yemen Azimuthal Equidistant Grid System (International), and datums with unknown grids exist as:

Kamaran Datum of 1926-27, Ras Karma Datum, Socotra, Socotra Datum of 1957, and Socotra Datum of 1964-65. These weird local datums are all on islands, and are probably Astro stations based on the Clarke 1880 ellipsoid or the International ellipsoid - particularly for the more recent ones.

UTM for the area is based on the European Datum of 1950 (International Ellipsoid). Some UTM may be on Clarke 1880, but if so, it's probably stuff that is over 30 years old. Rumors exist that some stuff was produced on WGS72, but if true, was probably only extremely large scale seismic stuff for geophysical exploration. That stuff would probably only show shot points ...

For 1:100,000 mapping, the British have produced several series of maps through several different agencies. The Russians have also, using the Krassovsky ellipsoid which presumably is referenced to the Pulkovo System of 1942 Datum. The Russian stuff, then, is probably based on the standard Russia Belts which are identical to UTM except that the scale factor at origin is NOT 0.9996 but rather 1.0

The Puerto Rico Datum of 1901 is based on its origin at Cardona Island Lighthouse with a defining azimuth to Ponce Southwest Base. The Datum is referenced to the same ellipsoid as NAD 1927 which is the Clarke 1866. The island is now on the NAD83, and the Grid parameters have changed slightly.

The Palestine Datum of 1928 is referenced to the Clarke 1880 ellipsoid (modified). Three grids are found on this datum in Israel, one is the Palestine 1923 Civil Grid (Cassini-Soldner projection) another EXTREMELY SIMILAR is the Palestine 1942 Belt (Gauss-Kruger Transverse Mercator) - the northings are identical, BUT the eastings change with distance from the Central Meridian. The Levant Lambert Zone (1920) uses the French Army Truncated Cubic, also on the Palestine 1928 Datum. There exists some more modern military mapping that is UTM, European Datum 1950 referenced to the International ellipsoid, but I was under the impression that such 1:50,000 scale stuff was quite military-sensitive.