# High-Accuracy Datum Transformations

The standard Molodensky transformation is one of the most widely used methods for transforming geodetic coordinates from one datum to another. It is also one of the least accurate, due in large part to the fact that it does not account for rotation or scaling between datums. 1-sigma errors on the order of meters or tens of meters are the norm with Molodensky.

This document does not represent an exhaustive list; it includes only those transformations of which I'm aware that have generally better accuracy than standard Molodensky.

# Multiple Regression Equations

NIMA TM 8350.2, Appendix D, lists multiple regression equations for transforming coordinates from several local datums to WGS84, with accuracies of generally 2 meters. Reverse transformations are not supplied.

• Australian Geodetic Datum 1966

• Australian Geodetic Datum 1984

• Campo Inchauspe (Argentina)

• Corrego Algere (Brazil)

• European Datum 1950 (Western Europe)

• North American 1927 (Conterminous U.S. and mainland Canada)

• South American 1969 (South American mainland)

See the bibliography for information on TM 8350.2.

# NATO Helmert Transformation Parameters

NATO favors the 7-parameter Helmert transformation in STANAG 2211. The Helmert transformation differs from the Molodensky transformation in that it has rotation and scale factors, and it performs the transformation on Cartesian coordinates rather than shifting geodetic coordinates. Transformations between local datums and WGS84 for which NATO has all seven parameters are generally estimated to be accurate to within a few meters in all directions. See http://earth-info.nga.mil/GandG/coordsys/datums/.

If your browser can present MathML, you can see the 7-parameter Helmert transformation here:

${\left[\begin{array}{c}X\\ Y\\ Z\end{array}\right]}_{\mathrm{WGS84}}=\left[\begin{array}{c}\mathrm{\Delta X}\\ \mathrm{\Delta Y}\\ \mathrm{\Delta Z}\end{array}\right]+\left(1+\mathrm{\Delta S}\right)\left[\begin{array}{ccc}1& {\mathrm{-R}}_{Z}& {R}_{Y}\\ {R}_{Z}& 1& {\mathrm{-R}}_{X}\\ {\mathrm{-R}}_{Y}& {R}_{X}& 1\end{array}\right]{\left[\begin{array}{c}X\\ Y\\ Z\end{array}\right]}_{\mathrm{local}}$

(Browsers that can't handle MathML will usually present an approximate representation, but you're better off with a browser or plugin that can support it properly.)

Note that in the 7-parameter transformation, the rotation parameters can be (and are) interpreted in two different ways. The Helmert transformation supported by NATO is also referred to as the similarity transformation in other places. The rotation parameters in this case describe the rotation of the coordinate frame: the description on the NGA Web site states that "two three-dimensional coordinate systems in space are related to each other by [the transformation formula]," clearly indicating the rotation convention observed by STANAG 2211.

The seven-parameter Bursa-Wolf transformation, which is widely applied in Europe, uses the same formula, but the meaning of the rotation parameters is different. It is a position vector transformation, meaning the rotation parameters describe the rotation of the point position with respect to a fixed coordinate frame. This is essentially the opposite of the coordinate frame rotation.

Another way of contrasting these two transformations, from the perspective of transforming a point, is this: in the Helmert or similarity transformation, the rotation of a point is counterclockwise around the axes. In the Bursa-Wolf transformation, the rotation of a point is clockwise.

To use parameters defined for one rotation convention in the other, changing the signs of the rotation parameters is sufficient. You must take care, however, to know for certain the convention for which a given set of parameters is defined before using them in critical work.

# WGS72

NIMA TM 8350.2, Appendix E, describes a transformation from WGS72 to WGS84, with lots of caveats.

# Transformations By Country Or Region

## Australia

• Australian Geodetic Datum of 1966 to Geocentric Datum of Australia 1994: a grid-based solution is available at Australia's ICSM Web site. The technical manual located there explains the 7-parameter and 3-parameter transformations as well.

• Australian Geodetic Datum of 1984 to Geocentric Datum of Australia 1994: GDAit

## Great Britain

• Ordnance Survey Great Britain 1936 to European Terrestrial Reference System 1989: GridInQuest (free but commercial product; MS Windows only; no reverse transformation; see end of document at this location for a link to the software installer).

## New Zealand

• New Zealand Geodetic Datum 1949 to New Zealand Geodetic Datum 2000: both seven-parameter and grid-based solutions are available at the Land Information New Zealand Web site.

## North America

• North American 1927 (conterminous United States and Alaska), Old Hawaiian, and Old Puerto Rican datums to North American 1983: NADCON or CORPSCON. CORPSCON extends NADCON by including vertical datum transformations.

• North American 1927 (Canada) to North American 1983: NTv2.

• WGS84 or ITRF to North American 1983: HTDP.

A recent article about this tool, "Horizontal Time-Dependent Positioning" by R. A. Snay, appeared in Professional Surveyor, 23(11) (November 2003), and is available in PDF via this link.