Multi-Parameter Transformation

There are two different transformations that are classified together as Multi-Parameter due to the similar expression of the transformation terms.

A 3D Similarity transformation is usually performed on geocentric Cartesian XYZ coordinates, hence the original coordinates (either geographic or grid, and ellipsoidal heights) are converted first to XYZ values, transformed, and then converted back to the geographic or grid coordinates as required in the target system. A 3D Similarity Transformation can also be applied to simple Cartesian X, Y and Height coordinates in a job whose Coordinate System is defined as Plane.

The Molodensky transformation is performed on geographic coordinates, requiring only a grid-geographic conversion when the points are defined in terms of grid projection coordinates. Molodensky is the lesser of the two methods as there is no provision to account for scale differences and rotations between the source and target datums.

IMPORTANT - ALWAYS verify the accuracy and reliability of a transformation before applying to critical data - always test first with points whose coordinates are known in both systems.

General Transformation Principles

General transformation principles will be found under the main topic for this section, Data Transformations - General.


Multi-Parameter transformations are defined in the Transformations option under the Multi-parameter tab; the sub-type is set to either 3D-Similarity or Molodensky. For specific details of 3D-Similarity or Molodensky transformations for particular regions, refer to the appropriate government agency or controlling body.

Transformation direction

The direction of the transformation is as defined by the Definitions library (in Transformations) but may be applied in the reverse direction if required - the signs of the transformation parameters are automatically reversed. Usually, the direction will be determined according to the selected transformation and the datum defined for the current job.

Point heights

Points must have valid heights for the 3D Similarity or Molodensky transformations - points without valid heights will be omitted. If valid heights are not available, the usual practise is to set the heights to zero (0.000) so that a transformation can be achieved for the 2D coordinates, in which case the transformed heights would be ignored - the effect on the horizontal coordinates may be minimal, even for large height errors (depending on the accuracy of the transformation). A more reliable approach to minimise errors in the horizontal coordinates, particularly in elevated areas well above the height of the spheroid, would be to assign approximate heights according to the terrain, but for preference, accurate heights (and geoid-spheroid separations if the vertical datum is geoidal) should be used.

The Multi-parameter transformations require spheroidal heights - if the vertical datum is geoidal in the old system, the heights are first converted to spheroidal heights (on the old spheroid, using each point's existing N-value) prior to conversion of the point coordinates to XYZ values. The only vertical datums listed for the target system will be spheroidal datums - if the new vertical datum was set to a geoidal datum, the specified default N-value (Default Geoid-Spheroid sep'n field) could be applied to each point's new spheroidal height to derive its geoidal height in the target system but this is not recommended; rather, the target system's vertical datum should be set to a spheroidal datum (i.e., matching the new horizontal datum), in which case no application of a default N-value is necessary, and if required, new precise N-values may be interpolated later.

3D Similarity

This transformation is also known as a 7-Parameter or Helmert transformation. Simplification for small rotation angles between datums is often used in algorithms for this method but in Geoida, the rigorous formulae are used. There are also two different rotation conventions in use and one or the other may be defined accordingly:

The only difference between the two variations of the method is the sign applied to the rotation parameters. If there is doubt about which method should be used with a given set of parameters, apply the transformation to a known set of points. If the solution is incorrect, redefine the rotation convention to the other option and recompute the transformation on the original points - the correct solution should result.

For projects of regional scale, the 3D-Similarity transformation may be considered to be of medium accuracy but may be of better accuracy in smaller areas where the parameters have been well established relative to surrounding known points.

The Localisation function within the Least Squares Network Adjustment option (Coord. Observations tab) derives the parameters for a 3D-Similarity transformation. These parameters may be used immediately or recalled within Least Squares Network Adjustment for the localisation of observed points and (optionally) GNSS Vector observations, and may also be optionally saved to Transformations in the Definitions menu for later use in the Points menu Import, Export and Transform Job options. Refer to Localisation for more detail.


The Molodensky transformation, similar to a 3D Similarity/Helmert transformation with rotations of zero and unit scale, is used to convert between geodetic coordinates on different ellipsoids. The XYZ Cartesian axes of the two ellipsoids are assumed to be parallel, and the differences between the two ellipsoids are expressed in terms of the difference in semimajor axis (Da) and flattening (Df).

The Molodensky transformation in Geoida applies 'Abridged' formulae - these formulae are not quite as accurate as the complete algorithms. For projects of regional scale, the Molodensky transformation is usually considered to be of low accuracy but may be of acceptable accuracy in localised areas where the parameters have been well established relative to surrounding known points. It is now primarily of historical use only.

In selecting a suitable transformation, the source and target datums must match the datums for which the transformation is defined.

Further Details

Further details of the transformation options will be found in the Transform Job topic.

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