Mentor Software’s coordinate conversion products support nine different techniques for converting from one datum to another, often called a datum shift. Mentor Software did not invent any of these; they are just implementations of techniques commonly used in geodetic applications. Thus, you will see similar types of things in other software as well. In this article, we separate the various techniques into three categories:

1. Grid based interpolation;
2. Geocentric coordinates;
3. Analytical formulas.

Last month we briefly described the basic grid technique for computing datum shifts. This technique is based on a data file which contains the actual datum shift at each point in a grid network. For example, the CONUS.LAS data file which is a part of the NADCON system, contains the latitude shift of some 33,000 points represented by a 15 minute grid between 63° and 131° west longitude and 20° and 50° north latitude. Given a specific geographic coordinate, programs extract the shift value for the four points of the specific grid cell in which the point to be converted resides. An interpolation algorithm is then used to determine the shift at the specific point.

In the US, the "agency" is the National Geodetic Survey. This agency also provides a PC program (sorry, MS-DOS only) called NADCON which will actually do the grid file lookup and interpolation calculations. The entire package is in the public domain and available on the Internet at:

ftp.ngs.noaa.gov/pub/pcsoft/nadcon

The FORTRAN source code to the program is available in the SOURCE sub-directory.

Notes:

.LAS and .LOS files which have HPGN as the last four characters of their base file names have another purpose in life described below.

I do not know why the islands off the coast of Alaska have their own files. Unless you know, Mentor Software recommends that you ignore these files.

In the Canada, the "agency" is Geomatics Canada, and the conversion is referred to as the National Transformation. The original version, now called version one, was released about 1992. It was been replaced with version 2 in about 1996. The computer program is named INTGRID. Many software vendors, such as Mentor Software, have included the Geomatics algorithm into their products. Therefore, most Casual Cartographers do not have a need for the actual Geomatics program istself. However, all implementations rely on access to the grid data file. This file is NOT in the public domain, and its use must be licensed from Geomatics Canada. You can contact them at:

www.geod.nrcan.gc.ca or information@geod.nrcan.gc.ca  or (613)-995-4410

The Canadian grid data file structure is much more sophisticated than that of its US counterpart. Both latitude and longitude shifts (as well as accuracy values) are incorporated into a single data file. The file structure also supports the concept of multiple grids, and sub-grids within a major grid. The end result is that the grid density in the Canadian data file varies from a maximum of 5 minutes to as little as 30 seconds. Of course, this means that this file is pretty large; a skosh under 14 megabytes. Therefore, Geomatics Canada distributes the data file on CD-ROM, and includes a program for extracting, for example, a smaller sub-set which covers only the specific geography which is of interest to you.

Due to the orthogonal nature of the grids upon which the US and Canadian systems are based, there is substantial overlap between the two systems. In the overlapped areas, the two systems match perfectly, right? NOT!!! The two systems differ by several centimeters in the region on overlap. Since Mentor Software’s products automatically select the proper grid data file when converting between NAD27 and NAD83, this can, in rare cases, become an issue.

This is not a big issue for the following reasons. First, since the Canadian data file is not in the public domain and requires a non-trivial license fee, it is unlikely that non Canadian users will have the data file. Second, if the Canadian file is present, it is likely that the user is a Canadian and would prefer the Canadian results over the US results. Third, since the Canadian grid has a higher density, it is reasonable to assume that the Canadian results would be more accurate. Due to this reasoning, Mentor Software’s automatic selection feature will always prefer the Canadian data to the US data in the region of overlap if both data files are present. There is an obscure means for changing this. Making this technique a little less obscure is on our "to-do" list.

NAD83 was completed prior to the completion of the Global Positioning System (GPS). Since the completion of GPS in the early 1990’s, the NGS has been using GPS to refine the accuracy of NAD83. Originally, most folks referred to the refined result as NAD83/91. It later came to be called High Precision GPS Network (i.e. HPGN). Eventually, the term High Accuracy Reference Network (HARN) became the preferred name for these refinements. These refinements are rather small, the shift being the in 1 to 2 foot range.

In addressing the issue of a program which would calculate the shifts, the NGS choose to use the existing NADCON program, and related grid data file formats, which had worked so well for NAD83. To succeed in doing this, the convention was established to make the last four characters of the grid data file base name "HPGN"; and some minor user interface changes were made to the NADCON program. Thus, the official means of converting between NAD83 and HARN is virtually identical to that used for converting between NAD27 and NAD83.

Refining NAD83 using GPS is being accomplished on a state by state, or region, basis. Thus, instead of having a single grid data file for the 48 conterminous states, there are several grid data files. (Keep in mind that since the same basic algorithm is in use, the grid data for HARN still comes in .LAS and .LOS file pairs.) File names often use the two character postal code for the state which is covered by the file. For example, COHPGN.LAS and COHPGN.LOS are the file names used for the state of Colorado. However, this convention is not rigorously followed. For example, the files named MDHPGN.L?S cover both Maryland and Delaware. The files named WOHPGN.L?S cover both Washington and Oregon. Originally there was a CAHPGN.L?S file pair which covered northern California, but now there are CNHPGN.L?S and CSHPGN.L?S for northern and southern California respectively. Figuring all of this out lead to the creation of last month’s freebie which displays the extents of the coverage of any such file.

As of this writing, HARN is still a work in progress. Every so often, the NGS releases new .L?S file pairs for new regions. You may wish to periodically check the NADCON Internet site given above for new data files. Mentor Software’s products are programmed to automatically pick up on the presence of new grid data files; thus the new data files simply need to be copied into the appropriate directory on your system. Other vendors may have different procedures.

Techniques in this category are based on the concept of the geocentric coordinate system. Such a system is a 3 dimensional cartesian coordinate system with its origin at the center of mass of the earth. That is, a coordinate system with three orthogonal axes X, Y, and Z where each axis is perpendicular to the plane formed by the other two. The positive Z axis starts at the center of the earth and protrudes through the north pole; implying that the negative Z axis intersects with the south pole. The X axis intersects the earth at the point where the Prime Meridian (i.e. Greenwich) intersects with the equator, and the Y axis completes a right handed coordinate system. Right handed? In this case, that means the Y axis comes out somewhere in the Indian Ocean, about halfway between Sri Lanka and Indonesia.

This is important as there are analytical formulas that, given an ellipsoid definition, will convert from latitude and longitude to X, Y, and Z geocentric coordinates; and vice versa. This is the manner in which geographic coordinates (i.e. latitudes and longitudes) can be converted from one ellipsoid to another. That is, the original ellipsoid is used to convert the geographic coordinates to X, Y, and Z geocentric coordinates, and the new ellipsoid is used when converting back to the new geographic coordinates. In and of itself, this calculation causes a small change in the latitude.

Presumably named after the individual who developed it, the Molodensky transformation is an efficient means of performing the above described procedure of converting geographic coordinates from one ellipsoid to another. However, the Molodensky adds one more feature. Effectively, while in geocentric coordinate form, the Molodensky transformation enables a translation of the X, Y, and Z coordinates by a certain prescribed amount. In practise, the ability to perform this shift enables to Molodensky to become a reasonably good general datum shift algorithm.

This technique, therefore, requires that the mathematics performing the datum shift know the original ellipsoid, the target ellipsoid, and exactly how much the intermediate X, Y, and Z geocentric coordinates are to be shifted. (Note, the Molodensky Transformation does not actually develop the geocentric coordinates. We present the transformation in these terms to provide a means to visualize what the mathematics is accomplishing.) Thus, Mentor Software products and the products of other vendors usually require you to provide a delta X, a delta Y, and a delta Z.

Molodensky only supports the translation of geocentric coordinates. Could not better results be achieved if we also could do some scaling? How about some rotation? Essentially, the Seven Parameter Transformation provides this capability. As a result, a general scale factor and three rotation angles are added to the delta X, delta Y, and delta Z translation values to produce the seven required parameters. The three rotation angles represent rotation about each of the three axes.

Rigorous implementation of the Seven Parameter Transformation requires some pretty heavy duty calculations. Since these calculations are basically an approximation, it has been common practise to simply approximate the rigorous seven parameter transformation with an approximation; an approximation which produced very good results since the rotation angles are usually very small. Mentor Software has chosen, perhaps erroneously, to use the term Bursa/Wolf Transformation for this approximation and distinguish it from the more rigorous Seven Parameter Transformation.

Analytical Formulas

Given a reasonable number of points at which both the source datum and the target datum coordinates are known, mathematical techniques can be used to develop analytical formulas which can convert between the two datums. Perhaps the term Least Squares is familiar to you; this is just one of the several techniques which can be used. A general term for such techniques is regression analysis, or when applied to more than one dimension, multiple regression analysis.

This method is supported by Mentor Software products as the United States Defense Mapping Agency (DMA) has developed and published many such formulas derived using this technique. While some criticisms have been directed at the usefulness of these formulas, Mentor Software has chosen to provide access to these and let users make their own choice.