Understanding Spheroids and Datums

Spheroids

A spheroid is an oblate ellipsoid of revolution (basically an ellipsoid) that is used to model the surface of the earth for making maps. A spheroid is defined by an earth radius, which is the major axis of the ellipsoid and the flattening (f), which can also be expressed as the eccentricity (e) or ellipticity (l).

l = 1/f

e = sqrt((2/l) - 1/(l*l))

e = sqrt(2f - f*f )

In the past, as cartographers created maps of different parts of the earth, spheroids were chosen to best approximate the shape of the earth in the region of the map. This led to a number of different spheroids or ellipsoids (such as Clarke 1866, Hayford 1910, etc.) that are in common use for different parts of the world.

In some cases, maps of a particular region of the earth have been created with different spheroids, and so it is necessary to convert coordinates from one spheroid to another. This only works if both spheroids share the same earth centre, which they normally will for older maps. This is because map survey work has normally used gravity as the reference for the earth centre, and maps of the same area will clearly have used the same gravity field.

Datums

The problem with using spheroids alone to define an earth model becomes apparent when working with satellite locations from GPS receivers. GPS systems commonly use base locations on a spheroid known as World Geodetic System 1984 (WGS 1984), which naturally places the centre of the earth at the true centre of gravity. Unfortunately this differs from the earth centre that has been used for most local maps of the world because these maps use an assumed earth centre based on the local gravity field, which is perpendicular to the geoid at that location.

To account for this we need a datum. A datum includes a spheroid and an earth centre offset from WGS 1984. Some datums may also include a rotation of the minor axis of the spheroid relative to WGS 1984. Datums for various countries and regions of the world have been compiled and defined in the mapproj.csv file (located in the user-specific folder %USERPROFILE%\Documents\Geosoft\Desktop Applications\csv).

What does this mean?

The longitude/latitude in one datum is not necessarily the same as in another datum (there may be an offset and rotation). Before the earth centre was introduced, all we needed was a spheroid (i.e. Hayford 1910). This worked fine for projections within the same country or region of the earth because the earth centre was the same. Note that Hayford 1910 is not a datum, it is a spheroid. If the spheroid alone is specified, an earth offset (X,Y,Z) of (0,0,0) with no rotation (see mapproj.csv) is required.

Working with Spheroids and Datums

A "datum" defines the earth model used to represent a "geoid", which is effectively a surface that would be defined by the sea level throughout an area being mapped. The "geoid" undulates according to the gravitational field; hence it cannot be defined perfectly by a simple mathematical expression. The business of cartography is to map features that exist on a local geoid to a flat piece of paper.

Arriving at a Map from a Geoid

1. The first step is the "datum". The datum consists of an ellipsoid, a prime meridian, and a specific part of the earth for which the datum applies. The datum ellipsoid is a perfect mathematical surface that best approximates the shape of the earth over the area of a datum. Latitude, longitude locations on a datum are the closest match of the geoid location to the datum ellipsoid surface. For example, latitude, longitude coordinates on the NAD27 datum use the Clarke 1866 ellipsoid with Greenwich as the prime meridian. One should note that the mapping of the geoid shape to the ellipsoid is not perfect because of the imperfections in the geoid.

2. At this point it is worth pointing out that many datums share the same ellipsoid. The difference between them is that a specific datum only applies to a specific area of the earth, and the geoid of that area is implied by the datum name. For example, "Luzon 1911", "Mound Dillon" and "NAD27" are all datums based on the "Clarke 1866" ellipsoid, but they are used to map the geoid from the Philippines, the island of Tobago, and North (and central) America respectively. Once we have a longitude, latitude on a specific datum, it is the job of a map projection to convert the latitude, longitude to a Cartesian X,Y coordinate for a flat piece of paper.

3. The next piece of the puzzle is to understand why we need a refinement of the datum into what the Geosoft applications call "local datums". The advent of GPS and satellite mapping required the definition of a single datum that best approximates the entire earth. After a bit of evolution, we have settled on WGS 84, which is truly a "perfect" datum that represents an exact ellipsoid, and the centre of that ellipsoid is at the gravitational centre of the earth. A fundamental problem in modern cartography is how to convert a latitude, longitude on the WGS 84 datum to NAD27 for example, so that a location can be used on NAD27 maps, or vice versa.

Because datums like NAD27 represent a geoid, and WGS represents a perfect ellipsoid, we need a way to convert the imperfect geoid shape of NAD27 to the perfect WGS27. The best way to do this is to measure the difference between known latitude, longitude locations on the NAD27 datum and the WGS84 location that one receives from a GPS measurement. This has been done throughout Canada to produce the NTv2 model of NAD27, and throughout the United States to produce the NADCON model. This process is also being carried out at continental scale in other parts of the world. Both NTv2 and NADCON are implemented as gridded corrections models, that given a location, one can look up the correction (both are supported in Geosoft applications).

An Alternative Approach

A second way to make the correction is to force the surface of the WGS 84 ellipsoid to lie as closely as possible to the surface of the datum ellipsoid (which in turn approximates the geoid), such that a simple mathematical conversion is within acceptable accuracy. This can be done by adjusting the location of the centre of the earth of the datum ellipsoid relative to WGS84, which is the basis of the Molodenski and Bursa Wolf corrections (parameters are in datumtrf.csv). However, such simple conversions are only accurate over a relatively small part of the datum, depending, of course, on the complexity of the geoid of that datum. For datums that cover a large region, such as NAD27, one needs many different adjustments depending on which part of the NAD27 datum you are on. For example, the very large area approximations such as "MEAN Canada" are much less accurate in the Yukon than "Canada (Yukon)". Both are less accurate than the NTv2 correction lookup.

Scenario

When I buy a topographic map of Canada (of say B.C.) it says it uses the NAD27 ellipsoid. It doesn't say what local transformation it uses?

First, NAD27 is a datum, not an ellipsoid, and the NAD27 datum uses the Clarke 1866 ellipsoid. You must choose which local datum transform is most appropriate for your needs. I would recommend always selecting the local transform that is most specific to your area, in this case "[NAD27] Canada Alberta; British Columbia". You could also choose "[NAD27] Canada NTv2 (20 min)" if the highest accuracy is required, but this is slower and more demanding of system resources.

In our "Datum" list we also include the names of all the common earth ellipsoids (with a "*" prefix). This is because practitioners have commonly confused ellipsoids with datums, and one often only knows the ellipsoid. By including the ellipsoid names in the list we make life a bit easier for you when you receive a map and the information that it is "Clarke 1850, UTM 42S". However, if you need to do a local datum transform, you must determine (or guess) the real datum name, which is also why we list the local transforms by area of use.

A common mistake that is made when using projections in Oasis montaj is to mix projections based on a spheroid with projections that use a datum.

For example, you may be in South America and you have data that is defined as using the Hayford 1910 spheroid, and you wish to convert this to a local map datum, say the Brazil Corrego Alegre datum, which is based on the International 1924 spheroid with an earth centre offset of (206,-172,6). In this case, your input projection datum should be HAYF1910 (Hayford 1910), and your output projection datum should be INT1924 (International 1924), NOT 55INT924 (Brazil Corrego Allegre). This is because you only know the spheroid of the input, not the full datum, so you must only use the spheroid of the output system and assume that both coordinates use the same earth centre, which is usually the case.

The only exception is when dealing with GPS locations based on WGS 1984 as the input coordinates. Here, in fact you know the full datum of the input because WGS 1984 has a (0,0,0) offset. In this case, specify the input projection as WGS1984 and the output projection as 55INT924.