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
MolodenskyBadekas methods.
Other methods, such as NADCON and NTv2 use a grid of differences and convert the
longitude–latitude values directly.
Equationbased methods
Equationbased 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.
Threeparameter methods
The simplest datum transformation method is a geocentric, or threeparameter,
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 sevenparameter 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.
Gridbased methods
Gridbased transformation methods include the following:
NADCON and HARN methods
The United States uses a gridbased method to convert between geographic
coordinate systems. Gridbased 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.
NADCON EXTRACTED REGION NADGRD
37 21 1 122.00000 .25 32.00000 .25 .00000
.007383 .004806 .002222 .000347 .002868
.005296 .007570 .009609 .011305 .012517
.013093 .012901 .011867 .009986 .007359
.004301 .001389 .001164 .003282 .004814
.005503 .005361 .004420 .002580 .000053
.002869 .006091 .009842 .014240 .019217
.025104 .035027 .050254 .072636 .087238
.099279 .110968 
