About geographic transformations and how to choose the right one

By Aileen Buckley, Mapping Center Lead

The Earth as a sphere

You will often be prompted to select the geographic transformation when you are projecting data or setting the projection of a data frame in a map document. Here are some concepts that might help you understand what this is all about AND how to make the right selection.

First, “geographic coordinates” are expressed in terms of latitude and longitude. “Latitude” is the north-south angular measure from the equator to the point of interest. “Longitude” is the east-west angular measure along the equator from the prime meridian to the point of interest’s longitude. Assuming that the earth is a sphere, geographic coordinates are determined relative to the center of the sphere – these coordinates are called “geocentric latitude and longitude”. (See the figure at the right; all figures are from Map Use: Reading and Analysis, 6th edition, Esri Press.)

Second, we know the earth is not a perfect sphere – it is elongated along the equator due to centrifugal force – its “equatorial radius” is longer than its “polar radius”. (The figure below is drawn using the real difference in length between the two axes which is not noticable at this scale.)

Ellipsoid

Therefore, instead of assuming that the earth is a sphere, we can more accurately assume it is an “oblate ellipsoid of revolution” (a sphere that is slighted squashed in at the poles), which we usually just call an “ellipsoid”. Rather than use coordinates defined by the spheroidal approximation of the earth, we use geographic coordinates expressed as latitude and longitude that are referenced to a particular ellipsoid of a defined shape and size. These coordinates are called “geodetic latitude and longitude”, and they will be slightly different from the geocentric ones. (See the figures below; in the one at the right, the difference between the equatorial and polar axes is greatly exaggerated!)

Geocentric and Geodetic Latitude

Different ellipsoids are used to approximate the earth, based on slightly different definitions of polar and equatorial axes of the ellipsoid - that is, the size and shape of the ellipsoid.

Third, imagine that the ellipsoid that is used as the model of the earth is then “positioned” relative to the surface of the earth to fit a specific area or use. This “fitted ellipsoid” becomes the frame of reference for specifying point coordinate locations, and it is called a “datum”. You should now be able to see how coordinates are datum dependent!

We have said that the ellipsoid is positioned to fit a specific area or use.  If the ellipsoid is being used as the basis for a national or regional datum (i.e., frame of reference to define coordinates), it will be positioned to best fit the area of interest (like the North American Datum of 1927, or NAD27).  If it is used as the basis for a world datum, the ellipsoid will most likely be positioned relative to the center of the earth’s mass (like the World Geodetic System of 1984, or WGS84).

Fourth, the datum, along with the prime meridian and angular units comprise a “geographic coordinate system” (GCS) which is referenced to the 3D ellipsoidal approximation to the earth. style=”text-decoration: underline” This is used to specify locations defined by latitude and longitude and heights above or below the ellipsoid at any latitude, longitude location.

Fifth, a “projected coordinate system” (PCS) results when the GCS is flattened down onto a 2D surface (e.g., paper, computer screen, etc.)  A projected coordinate system is always defined relative to the GCS that it comes from, which in turn is based on a specified datum, which is defined in part by its ellipsoid.

SO – when you are asked to select the geographic transformation, you are being asked to choose which mathematical calculation will be used to convert coordinates referenced to one datum to coordinates referenced to another datum. This geographic or datum transformation is often embedded in the procedure to convert between coordinate systems, or in other words, the projection process. This process often involves more than one coordinate transformation. For example, let’s say you want to convert between two projected coordinate systems. This is what would happen in the projection process:

  1. Define the PCS that your data are currently in. The PCS includes the GCS.
  2. Unproject the data to geodetic latitude and longitude using the same GCS.
  3. Transform the data to geodetic latitude and longitude using the new GCS.
  4. Project the data to the new PCS using the new GCS.

You can see that when you select the PCS you want to use, you also need to select the geographic transformation (step 3 above) because there are multiple mathematical calculations that can be used to define how the coordinates will be converted to the new GCS. Note that even if you are converting from one GCS to another GCS, you still need to define the geographic transformation; however, the will be no need to convert from PCS to GCS and visa versa during this kind of projection process.

So how do you choose the geographic transformation that should be used? Here are two Esri Knowledge Base articles that can help you:

HowTo: Select the correct geographic (datum) transformation when projecting between datums. This article contains links to downloadable zip files (for different versions of the software) that contain a list of all available datum transformations and their appropriate geographic areas of use.

HowTo: Determine which NAD_1983_To_WGS_1984 transformation to use.  The gist of this article is summarized below:

  1. NAD_1983_To_WGS_1984_1 – for the entire North American continent.
  2. NAD_1983_To_WGS_1984_2 – for the Aleutian islands.
  3. NAD_1983_To_WGS_1984_3 – for Hawai’i.
  4. NAD_1983_To_WGS_1984_4 – superseded by _5; this transformation method should no longer be used!
  5. NAD_1983_To_WGS_1984_5 – for the 48 contiguous United States.
  6. NAD_1983_To_WGS_1984_6 – for the Canadian province of Quebec.
  7. NAD_1983_To_WGS_1984_7 – for the Canadian province of Saskatchewan.
  8. NAD_1983_To_WGS_1984_8 – for the Canadian province of Alberta.

Note that geographic transformations work in either direction. For example, the transformation listed as NAD_1983_To_WGS_1984_5 transforms from NAD 1983 to WGS 1984, as well as from WGS 1984 to NAD 1983. When using the Project Tool, the geographic transformation is recorded in the metadata.

To learn more about geographic transformation methods, check out:

Thanks to Melita Kennedy, David Burrows, and Rob Juergens of the ArcGIS Projection Engine Team for helping me to make sure I said things correctly!

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18 Comments

  1. mattfindley says:

    Thanks. The graphics distinguishing geocentric and geodetic concepts was helpful. Any ideas on how to choose between lists of other transformations like NAD83 WGS84 that have several varieties? For example, the differences between the following transformation functions aren’t real clear (at least between 2 and 3):

    NAD_1983_HARN_To_WGS_1984
    NAD_1983_HARN_To_WGS_1984_2
    NAD_1983_HARN_To_WGS_1984_3

    Almost scared to ask, but what’s the significance of all the dx, dy, dz parameters that appear at the bottom of the dialog box when loading a transformation function?

  2. maphew says:

    this article needs just a few more pictures, these ones here for example: http://yukongis.ca/bin/view/Main/WebForum#nabble-p12172118 Since they came from ESRI literature to begin with, it is only natural they should be here too!

    Since posting the above thread, I’ve also been asked several times for similar images for transformations for other parts of the globe.

  3. Helenak says:

    Dear Dr. Buckley,

    Your explanation is much better than my GIS teacher.
    Thank you very much.

    Recently, I have a problem transferring GCS 1984 (unit: degree) to Behrmann (unit: km).

    The projection was all right, but the unit was not.
    Do you have any advice ?

    Thank you in advance.

    Elaine

  4. abuckley says:

    Regarding the comment about the graphics that mattfindley wrote in about, you might be happy to find that the graphics for this article came from the Map Use book published last year, and that you can order the book with a fuller explanation and more graphics through ESRI Press. You might be even happier to find that you can download right off the web the set of PowerPoint presentations that contain EVERY graphic in the book on a separate slide! This is a fantastic resource for teachers, students and others — kudos to ESRI Press for sharing this fabulous resource with all of us!

  5. abuckley says:

    I’m copying this response from Margaret to Elaine’s question:

    Projection files for projected coordinate systems that are installed with ArcGIS Desktop use units of foot_us, foot (the international foot), or meters. Using units of kilometers in a projection file is not advised, because the unit of measure is so large you can lose accuracy in your data. You can customize a projection file to use that unit, following the instructions in Knowlege Base articles 30583 or 32939, but we don’t recommend it. Instead, use the standard units of meters. You can then easily calculate length in the attribute table of the data with units of kilometers using the Calculate Geometry tool. You can also add a new field to the attribute table named LENGTH_KM for example, as a DOUBLE field, then divide the existing length in meters by 1000 to give lengths in kilometers.

    This was also answered on this page: http://blogs.esri.com/Support/blogs/mappingcenter/archive/2009/06/21/Angular-and-linear-units-in-the-projection-file.aspx#comments

  6. xienchen says:

    Regarding the question from mattfindley about differences between the transformation methods, the transformations with parameter values and areas of use are in the geographic_transformations.pdf file in ArcGIS installation > Documentation folder. You can also look in the “Geographic Transformations” section in the desktop help, online here: http://webhelp.esri.com/arcgisdesktop/9.3/index.cfm?TopicName=Geographic_transformation_methods

    The section on Equation-based methods might also be useful.

  7. abuckley says:

    Regarding the question from mattfindley about differences between the transformation methods, the transformations with parameter values and areas of use are in the geographic_transformations.pdf file in ArcGIS installation > Documentation folder. You can also look in the “Geographic Transformations” section in the desktop help, online here: http://webhelp.esri.com/arcgisdesktop/9.3/index.cfm?TopicName=Geographic_transformation_methods

    The section on Equation-based methods might also be useful.

  8. heavymetal says:

    Regarding the question from mattfidley about “what’s the significance of all the dx, dy, dz parameters”, the related concepts and illustrations are in the online help at this URL:

    http://webhelp.esri.com/arcgisdesktop/9.3/index.cfm?TopicName=Equation-based_methods

  9. abuckley says:

    Regarding the question from mattfidley about “what’s the significance of all the dx, dy, dz parameters”, the related concepts and illustrations are in the online help at this URL:

    http://webhelp.esri.com/arcgisdesktop/9.3/index.cfm?TopicName=Equation-based_methods

  10. nisha says:

    Hello, Dr. Buckley

    I found your article to be very helpful. Thank you.

    I live and work in Minnesota.

    I now know which method to use for NAD_1983_To_WGS_1984, but I am still unsure of which method to use for NAD_1983_HARN_To_WGS_1984. There are three options for my state.

    I have a similar issue with NAD_1927_To_WGS_1984. For this method, I have two different options for my state.

    Would you say, generally speaker, we should use the one with the biggest last digit number for the applicable area?

    For instance, for NAD_1927_To_WGS_1984, I use NAD_1927_To_WGS_1984_5 instead of NAD_1927_To_WGS_1984_4?

    Please advise. Thank you again.

  11. cmcglone75 says:

    How can I determine which method of transformation and parameters are used in the geographic transformation files?
    Where are they stored?
    Wouldn’t it be a lot more useful to have this information available when actually doing the reprojection, ie, within the project dialog box?

  12. abuckley says:

    Nisha: Wondering if you can post your specific question on our Ask a Cartographer page — this is a great place to send us questions exactly like this. Here’s the link:
    http://mappingcenter.esri.com/index.cfm?fa=ask.gateway

  13. dbmcalpin says:

    Some of the information in this blog is outdated. Be sure to check the hyperlinks text where available. The list of NAD83 to WGS84 conversions, for instance, is very different from a 1 June 2011 update:
    http://support.esri.com/en/knowledgebase/techarticles/detail/24159.

  14. yvonne1986 says:

    HELOOO WHICH IS THE BEST GEOGRAPHIC TRANSFORMATION METHOD NAD27 TO NAD83 OR I WANT TO TRANSFORM well locations from NAD27 TO NAD83

    • abuckley says:

      There are a number of NAD27 to NAD83 transformations. You want to pick one that has been created specifically for the part of the world that your data are location in. On Mapping Center, click the ArcGIS Resources tab and on the More tab (http://mappingcenter.esri.com/index.cfm?fa=arcgisResources.more), download the Geographis Transformations zip file. Unzip it and you will find a Word document taht lists all the geographic transformation. Use the Find tool to search for NAD_1927. You will see all the NAD27 to NAD83 transformations and a short description of the area of the world that each was created for. From this list, you will know which geographic transformation to specify.