What is a datum transformation?

A datum transformation is a set of math formulas that converts point coordinates from one datum to another.

Datum transformations take place in three-dimensional space, but you can get a sense of how they work from a two-dimensional example. Suppose you have two different maps of the same set of points. The differences between them can be expressed in terms of linear distance, spatial orientation, and scale. The challenge is to get the two sets of points into alignment—say, by aligning the points in Datum 2 with the points in Datum 1. The datum transformation is the math you use to do this. Two sets of points representing the same locations on earth.

The first step is to put the two sets of points into a common coordinate space. The systems are put into common coordinate space.

Next, you align the origins of the two systems by moving a certain number of units in the x-direction and a certain number of units in the y-direction. Then you rotate the axes into alignment. Finally, you apply a scale factor. Left: the origins are aligned, bringing the two sets of points into proximity. Center: the axes are rotated, and the points almost match up. Right: the axes are scaled and the transformation is complete.

The process is similar in three-dimensional space except that you are working with a z-axis as well as x- and y-axes. In a datum transformation, coordinates in both the "from" datum and the "to" datum are converted from latitude-longitude into three-dimensional cartesian coordinate space. Next, the math is done to align the coordinates. Finally, the coordinates are translated back into the latitude-longitude values of the "to" datum. In the x,y,z coordinate system, the positive x-axis lies in the plane of the equator and passes through 0° longitude; the positive y-axis lies in the plane of the equator and passes through 90°E longitude; the positive z-axis is parallel to the earth's rotation axis and passes through 90°N latitude.

Datum transformation methods

Depending on the "to" and "from" datums involved, different transformation methods are used. In both the three- and seven-parameter methods, the transformation aligns the x,y,z axes of the two datums in three-dimensional cartesian coordinate space.

Three-parameter methods
In a three-parameter transformation (also called a geocentric translation), the axes of the two datums are aligned using linear shifts of the x, y, and z axes of the datum being transformed. A three-parameter transformation is appropriate when the x, y, and z axes of the two datums are parallel and identically scaled.

Seven-parameter methods
A seven-parameter transformation is used when the axes of the two datums are not parallel and identically scaled. In addition to the three linear shift parameters, there are three rotation parameters (one for each axis) and a scale factor.

Another type of transformation, called a grid-based transformation, is available for converting between the NAD27 and NAD83 datums. This type of transformation is more accurate than the equation-based method described above. It relies on the fact that the exact latitude-longitude differences between the two datums have been calculated for thousands of control points.

Essentially what happens in this transformation is that point coordinates on the "from" datum (usually NAD27) are moved to the correct location on the "to" datum simply by looking up the latitude and longitude differences for that location in a table. The exact amount of shift hasn't been calculated in advance for every point, of course, but every point does fall within a grid cell which has bounding control points of known shift. The amount of shift for a point within that cell can then be estimated by interpolation.

Molodensky method
The Molodensky method transforms the latitude-longitude coordinates of the "from" datum directly to the latitude-longitude coordinates of the "to" datum, thus skipping the process of converting both datums into three-dimensional cartesian coordinates. It uses linear shifts of the x, y, and z axes of the "from" datum's spheroid, plus the differences in the semi-major axes and flattening ratios of the two spheroids.

Datum transformations in ArcGIS

When you work with spatial data in ArcMap, you add data sets as layers to a data frame. Every data set you add has a geographic coordinate system associated with it, describing the data's latitude-longitude coordinates. If the data set is already in a map projection, then it is also associated with a projected coordinate system, describing the projection and its parameters. The geographic coordinate system is still there, however—lurking below the surface, so to speak.

A data frame is not inherently associated with a coordinate system, but it adopts the coordinate system—the
GCS and the PCS, if any—of the first layer that is added to it. (You also have the option to set the data frame to the coordinate system of your choice at any time.) If the coordinate systems of subsequently-added layers do not match this system, then they must be changed to match it—this is how ArcMap ensures that all layers in a data frame are in correct spatial alignment.

If a layer has the same GCS as the data frame but a different PCS, ArcMap resolves the PCS mismatch through a process called on-the-fly projection. It simply "undoes" the layer's projection and uses the common GCS to reproject the layer to the specifications of the data frame's PCS.

If, however, the layer has a different GCS from the data frame, then a datum transformation is required. ArcMap™ will alert you to this, and it will even perform the transformation—but, except for one particular case, it doesn't do the transformation automatically. Instead, ArcMap presents you with a list of various possible transformations—sometimes just one, sometimes upwards of twenty—and it's up to you to pick one. You'll learn how to do that in the next example and in the final exercise of this module.

The process of matching the coordinate systems of layers to the coordinate system of the data frame has many permutations, summarized in the following table.

 Data frame's coordinate system Layer's coordinate system Operation needed GCS1 GCS1 None GCS1 GCS1 + PCS1 None; ArcMap "unprojects" the layer on the fly GCS1 GCS2 Datum transformation from GCS2 to GCS1 GCS1 GCS2 + PCS1 Datum transformation from GCS2 to GCS1 and on-the-fly unprojection of layer GCS1 + PCS1 GCS1 On-the-fly projection of layer to PCS1 GCS1 + PCS1 GCS1 + PCS1 None GCS1 + PCS1 GCS1 + PCS2 On-the-fly reprojection of layer to PCS1 GCS1 + PCS1 GCS2 Datum transformation from GCS2 to GCS1 and on-the-fly projection of layer to PCS1 GCS1 + PCS1 GCS2 + PCS1 Datum transformation from GCS2 to GCS1 GCS1 + PCS1 GCS2 + PCS2 Datum transformation from GCS2 to GCS1 and on-the-fly reprojection of layer to PCS1