Spheroid, Datum, Projections

Datum

Datums are made to correct human errors while positionning the spheroids on the earth. Basically, when you create a map of an island you don't care about where the island is, but you care where is the pub compared to your house. Well, that's the whole theory, you set a reference point somewhere, put a big piece of concrete, a nice copper plate, and ask the prime minister to put his name on it. From this reference point (primary trigonometric station), you call a team of surveyors, with their theodolytes and build a network of reference points all over the island. That's called a grid, or more precisely a map grid. For instance in Cook Island, Rarotonga, the reference point is next to the pub: Trader Jack's.

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And from there you build your grid:

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(It is just a sample, maybe not the reality)

Now you have to put this point on the earth. You look at the stars, you look at your reference point, you look at the stars again (with a sextant of course!), and set the point on the earth.

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But which representation of the earth have we used? How good was our work? If we have been using GPS we have set the point with 100m accuracy or cm accuracy if we had time to compute all the correction factors which disturb the GPS signal. In reality you will position the spheroid now using very stable and far reaching sources, such as quasar and other celestal objects. You will use precise radionometrics instruments, and find coordinates of points forming your primary grid (primary trigonometric stations).

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If you did it a few years ago, let say around 1924 using the International spheroid, you will position your points within about a few kms, while still using the same mathematical model of the earth. Whatever is your accuracy, at the time, you will decide of a set of coordinates for you primary trigonometric points on your grid.Don't forget that these points will move in time, 10cm a year sometimes, but you are fixing arbitrarily the coodinates for mapping purposes. From these coordinates all other points coordinates will be derivated. The survey department is fixing the grid in time and in location for all newly produced maps. They create a local datum. Now you redo the same exercise now using WGS84 spheroid, and you decide a new set of coordinates points in the WGS84 spheroid reference for eactly the same primary trigonometric stations. You create another local datum.

Station Name Lat(Fiji Geodetic Datum 1986) Lon(Fiji Geodetic Datum 1986) Lat(WGS84) Long(WGS84)




For instance the Fiji Geodetic Datum 1986 is based on a WGS72 spheroid but it is not the WGS72 Datum.

Outside the difference in shape of the 2 spheroid composing the datum (semi major axis a and eccentricity f), the centre of the datums and the axes of the datums are different by:

• Rotation Ox
• Rotation Oy
• Rotation Oz
• Translation dx
• Translation dy
• Translation dz
• Scale m
• Meridian of Origin

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These parameters are reffered as 7 parameters of the Molodensky transformation. The meridian of origin is not part of the parameters as it just defines which earth meridian as the 0 value. Because the British were the biggest navy in the world when accurate map were drawn, the Greenwich Meridian was adopted, but many were existing, such as the Paris Meridian, or the Spanish Meridian.

The transformation matrix from (x1,y1,z1) to (x2,y2,z2) is written as follow:

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All these parametres in MapInfo? must be expressed from the local datum to the WGS84 Datum. I will come back on why the WGS84 is a spheroid but also a datum. If you look at the MapInfo? list of datum, you will see a list of names, area of application, and the spheroid to which is based the datum. It looks like this:

Datum	Area	Spheroid
Viti Levu 1916	Viti Levu, Fiji	Clarke 1880
Wake-Eniwetok? 1960	Marshall Islands	Hough
Bellevue (IGN)	Efate and Erromango Islands	Australian National
WGS60	Worldwide	WGS60
WGS72	Worldwide	WGS72
WGS84	Worldwide	WGS84

You see in the case of the WGS (World Geodetic System) series, there were no mistake of location, therefore there is no need to correct it. The Datum/Spheroid is valid worldwide. Well, that's what the Americans think, but in reality the WGS spheorids are not perfectly positioned except for the WGS84 and need to be corrected if you want at least meter accuracy between your datum coordinates and GPS coordinates (WGS84).

If changing coordinates from one datum to another is not difficult using the 7 Molendensky parameters, finding these parameters is the art of the surveyor. I hope to present here, later on, how it is done. Stay tuned.

Datum	Ox	Oy	Oz
AustralianWGS72 to WGS84	-2.6858678E-06 rad	0	0
Australian WGS72 to AGD84	-1.0181087E-06 rad	1.8907734E-06 rad	1.1150715E-06 rad
Fiji Geodetic Datum(WGS72) to WGS84	-1.37101925914775"	0.841968640179647"	4.718005596536956"

above table Continued

dx(m)	dy(m)	dz(m)	m
0	0	4.5	1+0.2263E-06
116	50.47	-137.19	1+0.128E-06
35.17277931989845	-136.5712601696053	36.96429263506973	1.537002362494988ppm

For instance you will do the following operations to move from Datum1 to Datum2:

1. Convert Datum1 geodetic coordinates (Lat1, Lon1, H1) in Datum1 cartesian coordinates(x1,y1,z1).
2. Apply the transformation matrix from Datum1 cartesian cordinates to Datum2 cartesian coordinates
3. Convert Datum2 cartesian coordinates(x2,y2,z2) to Datum2 geodetic coordinates(Lat2,Lon2,H2).

Now let see how we represent, this datum on a piece of paper. We need to speak about projections.


Next

Franck Martin 1998
With thanks to Mike Poidevin for his inputs.