Moving your data between coordinate systems sometimes includes transforming between the geographic coordinate systems.

Because the geographic coordinate systems contain datums that are based on spheroids, a geographic transformation also changes the underlying spheroid. There are several methods, which have different levels of accuracy and ranges, for transforming between datums. The accuracy of a particular transformation can range from centimeters to meters depending on the method and the quality and number of control points available to define the transformation parameters.

A geographic transformation is always defined in a particular direction. When working with geographic transformations, if no mention is made of the direction, the command will handle the directionality automatically. For example, if converting data from WGS 1984 to NAD 1927, you can pick a transformation called NAD_1927_to_WGS_1984_3 and the software will apply it correctly.

A geographic transformation always converts geographic (longitude-latitude) coordinates. Some methods convert the geographic coordinates to geocentric (X,Y,Z) coordinates, transform the X,Y,Z coordinates, and convert the new values back to geographic coordinates.

These include the Geocentric Translation, Molodensky, Coordinate Frame, and Molodensky-Badekas methods.

Other methods, such as NADCON and NTv2 use a grid of differences and convert the longitude-latitude values directly.

Equation-based methods

Equation-based transformation methods can be classified into the following four method types. Usually the transformation parameters are defined as going from a local datum to WGS 1984 or another geocentric datum.

Three-parameter methods

The simplest datum transformation method is a geocentric, or three-parameter, transformation. The geocentric transformation models the differences between two datums in the X,Y,Z coordinate system. One datum is defined with its center at 0,0,0. The center of the other datum is defined at some distance (DX,DY,DZ) in meters away.

The three parameters are linear shifts and are always in meters.

A more complex and accurate datum transformation is possible by adding four more parameters to a geocentric transformation. The seven parameters are three linear shifts (DX,DY,DZ), three angular rotations around each axis (rx,ry,rz), and a scale factor.

The rotation values are given in decimal seconds, while the scale factor is in parts per million (ppm). The rotation values are defined in two different ways. It's possible to define the rotation angles as positive either clockwise or counterclockwise as you look toward the origin of the X,Y,Z systems.

The United States, Australia, New Zealand, and a few other countries define the equations such that the rotation values are positive counterclockwise. This method is called the Coordinate Frame Rotation transformation. Europe uses a different convention called the Position Vector transformation. Both methods are sometimes referred to as the Bursa-Wolf method. In the Projection Engine, the Coordinate Frame and Bursa-Wolf methods are the same. Both Coordinate Frame and Position Vector methods are supported, and it is easy to convert transformation values from one method to the other simply by changing the signs of the three rotation values. For example, the parameters to convert from the WGS 1972 datum to the WGS 1984 datum with the Coordinate Frame method are (in the order DX,DY,DZ,rx,ry,rz,s):

(0.0, 0.0, 4.5, 0.0, 0.0, -0.554, 0.227)

To use the same parameters with the Position Vector method, change the sign of the rotation so the new parameters are:

(0.0, 0.0, 4.5, 0.0, 0.0, +0.554, 0.227)

It's impossible to tell from the parameters alone which convention is being used. If you use the wrong method, your results can return inaccurate coordinates. The only way to determine how the parameters are defined is by checking a control point whose coordinates are known in the two systems.

The Molodensky-Badekas method is a variation of the seven-parameter methods. It has an additional three parameters that define the XYZ origin of rotation. Sometimes this point is known as the origin of the datum, or geographic coordinate system. Given the XYZ origin of rotation point, it is possible to calculate an equivalent Coordinate Frame transformation. The DX, DY, and DZ values will change but the rotation and scale values will remain the same.

The Molodensky method converts directly between two geographic coordinate systems without actually converting to an X,Y,Z system. The Molodensky method requires three shifts (DX,DY,DZ) and the differences between the semimajor axes (Da) and the flattenings (Df) of the two spheroids. The Projection Engine automatically calculates the spheroid differences according to the datums involved.

Abridged Molodensky method

The Abridged Molodensky method is a simplified version of the Molodensky method.

Grid-based methods

Grid-based transformation methods include the following:

NADCON and HARN methods

The United States uses a grid-based method to convert between geographic coordinate systems. Grid-based methods allow you to model the differences between the systems and are potentially the most accurate method. The area of interest is divided into cells. The National Geodetic Survey (NGS) publishes grids to convert between NAD 1927 and other older geographic coordinate systems and NAD 1983. These transformations are grouped into the NADCON method. The main NADCON grid, CONUS, converts the contiguous 48 states. The other NADCON grids convert older geographic coordinate systems to NAD 1983 for:

• Alaska
• Hawaiian islands
• Puerto Rico and Virgin Islands
• St. George, St. Lawrence, and St. Paul Islands in Alaska

The accuracy is approximately 0.15 meters for the contiguous states, 0.50 for Alaska and its islands, 0.20 for Hawaii, and 0.05 for Puerto Rico and the Virgin Islands. Accuracies can vary depending on how good the geodetic data in the area was when the grids were computed (NADCON, 1999).

The Hawaiian islands were never on NAD 1927. They were mapped using several datums that are collectively known as the Old Hawaiian datums.

New surveying and satellite measuring techniques have allowed NGS and the states to update the geodetic control point networks. As each state is finished, the NGS publishes a grid that converts between NAD 1983 and the more accurate control point coordinates. Originally, this effort was called the High Precision Geodetic Network (HPGN). It is now called the High Accuracy Reference Network (HARN). Four territories and 46 states have published HARN grids as of January 2004. HARN transformations have an accuracy approximately 0.05 meters (NADCON, 2000).

The difference values in decimal seconds are stored in two files: one for longitude and the other for latitude. A bilinear interpolation is used to calculate the exact difference between the two geographic coordinate systems at a point. The grids are binary files, but a program, NADGRD, from the NGS, allows you to convert the grids to American Standard Code for Information Interchange (ASCII) format. Shown at the bottom of the page is the header and first row of the CSHPGN.LOA file. This is the longitude grid for Southern California. The format of the first row of numbers is, in order, the number of columns, number of rows, number of z-values (always one), minimum longitude, cell size, minimum latitude, cell size, and not used.

The next 37 values in this case are the longitude shifts from -122° to -113° at 32° N in 0.25°, or 15 minute, intervals in longitude.