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.

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:

[ X Y Z ] WGS84 = [ ΔX ΔY ΔZ ] + ( 1 + ΔS ) [ 1 -RZ RY RZ 1 -RX -RY RX 1 ] [ X Y Z ] 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.


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

Transformations By Country Or Region


Great Britain

New Zealand

North America

Other Information Sources

POSC also has a discussion on transformation methods and theory on its Web site.

Chuck Taylor -- (Copyright) -- (Contact)