Editing Geographic coordinate conversion (section)

== Datum transformations ==
{{further|Geodetic datum}}

[[File:Possible paths for datum transform.svg|400px|right|alt=coordinate transform paths|The different possible paths for transforming geographic coordinates from datum <math>A</math> to datum <math>B</math>]]
Transformations among datums can be accomplished in a number of ways. There are transformations that directly convert geodetic coordinates from one datum to another. There are more indirect transforms that convert from geodetic coordinates to ECEF coordinates, transform the ECEF coordinates from one datum to another, then transform ECEF coordinates of the new datum back to geodetic coordinates. There are also grid-based transformations that directly transform from one (datum, map projection) pair to another (datum, map projection) pair.

=== Helmert transformation ===

{{main|Helmert transformation}}

Use of the Helmert transform in the transformation from geodetic coordinates of datum <math>A</math> to geodetic coordinates of datum <math>B</math> occurs in the context of a three-step process:<ref name=HelmertNZ>{{cite web|title=Equations Used for Datum Transformations|url=http://www.linz.govt.nz/geodetic/conversion-coordinates/geodetic-datum-conversion/datum-transformation-equations/index.aspx|publisher=Land Information New Zealand (LINZ)|access-date=5 March 2014|archive-date=6 March 2014|archive-url=https://web.archive.org/web/20140306005832/http://www.linz.govt.nz/geodetic/conversion-coordinates/geodetic-datum-conversion/datum-transformation-equations/index.aspx|url-status=live}}</ref>

# Convert from geodetic coordinates to ECEF coordinates for datum <math>A</math>
# Apply the Helmert transform, with the appropriate <math>A\to B</math> transform parameters, to transform from datum <math>A</math> ECEF coordinates to datum <math>B</math> ECEF coordinates
# Convert from ECEF coordinates to geodetic coordinates for datum <math>B</math>

In terms of ECEF XYZ vectors, the Helmert transform has the form (position vector transformation convention and very small rotation angles simplification)<ref name=HelmertNZ/>

: <math>
  \begin{bmatrix} X_B \\ Y_B \\ Z_B \end{bmatrix} =
  \begin{bmatrix} c_x \\ c_y \\ c_z \end{bmatrix} + \left(1 + s \times 10^{-6}\right)
  \begin{bmatrix}
      1 & -r_z &  r_y \\
    r_z &    1 & -r_x \\
   -r_y &  r_x &    1
  \end{bmatrix} \begin{bmatrix} X_A \\ Y_A \\ Z_A \end{bmatrix}.
</math>

The Helmert transform is a seven-parameter transform with three translation (shift) parameters <math>c_x,\, c_y,\, c_z</math>, three rotation parameters <math>r_x,\, r_y,\, r_z</math> and one scaling (dilation) parameter <math>s</math>. The Helmert transform is an approximate method that is accurate when the transform parameters are small relative to the magnitudes of the ECEF vectors. Under these conditions, the transform is considered reversible.<ref name=OGP7_2>{{cite web|title=Geomatics Guidance Note Number 7, part 2 Coordinate Conversions and Transformations including Formulas|url=http://info.ogp.org.uk/geodesy/guides/docs/G7-2.pdf|publisher=International Association of Oil and Gas Producers (OGP)|access-date=5 March 2014|url-status=dead|archive-url=https://web.archive.org/web/20140306005736/http://info.ogp.org.uk/geodesy/guides/docs/G7-2.pdf|archive-date=6 March 2014}}</ref>

A fourteen-parameter Helmert transform, with linear time dependence for each parameter,{{r|OGP7_2|page1=131-133}} can be used to capture the time evolution of geographic coordinates dues to [[geomorphic]] processes, such as continental drift<ref name=Bolstad>{{cite book|last=Bolstad|first=Paul|title=GIS Fundamentals, 4th Edition|year=2012 |publisher=Atlas books|isbn=978-0-9717647-3-6|page=93|url=http://www.paulbolstad.net/4thedition/samplechaps/GISFundChap3.pdf|url-status=dead|archive-url=https://web.archive.org/web/20160202201558/http://www.paulbolstad.net/4thedition/samplechaps/GISFundChap3.pdf|archive-date=2016-02-02}}</ref> and earthquakes.<ref name=addend_8350_2>{{cite web|title=Addendum to NIMA TR 8350.2: Implementation of the World Geodetic System 1984 (WGS 84) Reference Frame G1150|url=http://gis-lab.info/docs/nima-tr8350.2-addendum.pdf|publisher=National Geospatial-Intelligence Agency|access-date=6 March 2014|archive-date=11 May 2012|archive-url=https://web.archive.org/web/20120511090551/http://gis-lab.info/docs/nima-tr8350.2-addendum.pdf|url-status=live}}</ref> This has been incorporated into software, such as the Horizontal Time Dependent Positioning (HTDP) tool from the U.S. NGS.<ref name=HTDP>{{cite web|title=HTDP - Horizontal Time-Dependent Positioning|url=https://www.ngs.noaa.gov/TOOLS/Htdp/Htdp.shtml|publisher=U.S. National Geodetic Survey (NGS)|access-date=5 March 2014|archive-date=25 November 2019|archive-url=https://web.archive.org/web/20191125025630/https://www.ngs.noaa.gov/TOOLS/Htdp/Htdp.shtml|url-status=live}}</ref>

=== Molodensky-Badekas transformation ===

To eliminate the coupling between the rotations and translations of the Helmert transform, three additional parameters can be introduced to give a new XYZ center of rotation closer to coordinates being transformed. This ten-parameter model is called the ''Molodensky-Badekas transformation'' and should not be confused with the more basic Molodensky transform.{{r|OGP7_2|page1=133-134}}

Like the Helmert transform, using the Molodensky-Badekas transform is a three-step process:
# Convert from geodetic coordinates to ECEF coordinates for datum <math>A</math>
# Apply the Molodensky-Badekas transform, with the appropriate <math>A\to B</math> transform parameters, to transform from datum <math>A</math> ECEF coordinates to datum <math>B</math> ECEF coordinates
# Convert from ECEF coordinates to geodetic coordinates for datum <math>B</math>

The transform has the form<ref name=MB_NGA>{{cite web|title=Molodensky-Badekas (7+3) Transformations|url=http://earth-info.nga.mil/GandG/coordsys/datums/molodensky.html|publisher=National Geospatial Intelligence Agency (NGA)|access-date=5 March 2014|archive-date=19 July 2013|archive-url=https://web.archive.org/web/20130719151529/http://earth-info.nga.mil/GandG/coordsys/datums/molodensky.html|url-status=live}}</ref>

: <math>
  \begin{bmatrix} X_B \\ Y_B \\ Z_B \end{bmatrix} =
  \begin{bmatrix} X_A \\ Y_A \\ Z_A \end{bmatrix} +
    \begin{bmatrix} \Delta X_A \\ \Delta Y_A \\ \Delta Z_A \end{bmatrix} +
    \begin{bmatrix}
         1 & -r_z &  r_y \\
       r_z &    1 & -r_x \\
      -r_y &  r_x &    1
    \end{bmatrix}
    \begin{bmatrix} X_A - X^0_A \\ Y_A - Y^0_A \\ Z_A - Z^0_A \end{bmatrix} +
    \Delta S \begin{bmatrix} X_A - X^0_A \\ Y_A - Y^0_A \\ Z_A - Z^0_A \end{bmatrix}.
</math>

where <math>\left(X^0_A,\, Y^0_A,\, Z^0_A\right)</math> is the origin for the rotation and scaling transforms and <math>\Delta S</math> is the scaling factor.

The Molodensky-Badekas transform is used to transform local geodetic datums to a global geodetic datum, such as WGS 84. Unlike the Helmert transform, the Molodensky-Badekas transform is not reversible due to the rotational origin being associated with the original datum.{{r|OGP7_2|page1=134}}

=== Molodensky transformation ===

The Molodensky transformation converts directly between geodetic coordinate systems of different datums without the intermediate step of converting to geocentric coordinates (ECEF).<ref name=esri_eq_based>{{cite web|title=ArcGIS Help 10.1: Equation-based methods|url=http://resources.arcgis.com/en/help/main/10.1/index.html#//003r00000012000000|publisher=ESRI|access-date=5 March 2014|archive-date=4 December 2019|archive-url=https://web.archive.org/web/20191204151744/http://resources.arcgis.com/en/help/main/10.1/index.html#//003r00000012000000|url-status=live}}</ref>  It requires the three shifts between the datum centers and the differences between the reference ellipsoid semi-major axes and flattening parameters.

The Molodensky transform is used by the [[National Geospatial-Intelligence Agency]] (NGA) in their standard TR8350.2 and the NGA supported GEOTRANS program.<ref name=NGA_Datum>{{cite web|title=Datum Transformations|url=http://earth-info.nga.mil/GandG/coordsys/datums/index.html|publisher=National Geospatial-Intelligence Agency|access-date=5 March 2014|archive-date=9 October 2014|archive-url=https://web.archive.org/web/20141009125117/http://earth-info.nga.mil/GandG/coordsys/datums/index.html|url-status=live}}</ref> The Molodensky method was popular before the advent of modern computers and the method is part of many geodetic programs.

=== Grid-based method ===

[[File:Datum Shift Between NAD27 and NAD83.png|thumb|Magnitude of shift in position between NAD27 and NAD83 datum as a function of location.]]

Grid-based transformations directly convert map coordinates from one (map-projection, geodetic datum) pair to map coordinates of another (map-projection, geodetic datum) pair. An example is the NADCON method for transforming from the North American Datum (NAD) 1927 to the NAD 1983 datum.<ref name=ESRI_grid>{{cite web|title=ArcGIS Help 10.1: Grid-based methods|url=http://resources.arcgis.com/en/help/main/10.1/index.html#//003r00000013000000|publisher=ESRI|access-date=5 March 2014|archive-date=4 December 2019|archive-url=https://web.archive.org/web/20191204151744/http://resources.arcgis.com/en/help/main/10.1/index.html#//003r00000013000000|url-status=live}}</ref> The High Accuracy Reference Network (HARN), a high accuracy version of the NADCON transforms, have an accuracy of approximately 5 centimeters. The National Transformation version 2 ([[NTv2]]) is a Canadian version of NADCON for transforming between NAD 1927 and NAD 1983. HARNs are also known as NAD 83/91 and High Precision Grid Networks (HPGN).<ref name=nadcon_harn>{{cite web|title=NADCON/HARN Datum ShiftMethod|url=http://www.bluemarblegeo.com/knowledgebase/geocalc/classdef/datumshift/datumshifts/nadcon.html|publisher=bluemarblegeo.com|access-date=5 March 2014|archive-date=6 March 2014|archive-url=https://web.archive.org/web/20140306000427/http://www.bluemarblegeo.com/knowledgebase/geocalc/classdef/datumshift/datumshifts/nadcon.html|url-status=live}}</ref> Subsequently, Australia and New Zealand adopted the NTv2 format to create grid-based methods for transforming among their own local datums.

Like the multiple regression equation transform, grid-based methods use a low-order interpolation method for converting map coordinates, but in two dimensions instead of three. The [[NOAA]] provides a software tool (as part of the NGS Geodetic Toolkit) for performing NADCON transformations.<ref name=NOAA_NADCON>{{cite web|title=NADCON - Version 4.2|url=http://www.ngs.noaa.gov/PC_PROD/NADCON/|publisher=NOAA|access-date=5 March 2014|archive-date=6 May 2021|archive-url=https://web.archive.org/web/20210506162736/https://www.ngs.noaa.gov/PC_PROD/NADCON/|url-status=live}}</ref><ref name=Mulcare>{{cite web|last=Mulcare |first=Donald M. |title=NGS Toolkit, Part 8: The National Geodetic Survey NADCON Tool |url=http://www.profsurv.com/magazine/article.aspx?i=1193 |publisher=Professional Surveyor Magazine |access-date=5 March 2014 |url-status=dead |archive-url=https://web.archive.org/web/20140306001134/http://www.profsurv.com/magazine/article.aspx?i=1193 |archive-date=6 March 2014 }}</ref>

=== Multiple regression equations ===

Datum transformations through the use of empirical [[multiple regression]] methods were created to achieve higher accuracy results over small geographic regions than the standard Molodensky transformations. MRE transforms are used to transform local datums over continent-sized or smaller regions to global datums, such as WGS 84.<ref name=IHO>{{cite report |title=User's Handbook on Datum Transformations Involving WGS 84 |date=August 2008 |edition=3rd |series=Special Publication No. 60 |publisher=International Hydrographic Bureau |location=Monaco |url=https://www.iho.int/iho_pubs/standard/S60_Ed3Eng.pdf |access-date=2017-01-10 |archive-date=2016-04-12 |archive-url=https://web.archive.org/web/20160412230130/http://www.iho.int/iho_pubs/standard/S60_Ed3Eng.pdf |url-status=live }}</ref> The standard NIMA TM 8350.2, Appendix D,<ref name=tr8350_2>{{cite web|title=DEPARTMENT OF DEFENSE WORLD GEODETIC SYSTEM 1984 Its Definition and Relationships with Local Geodetic Systems|url=http://earth-info.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf|publisher=National Imagery and Mapping Agency (NIMA)|access-date=5 March 2014|archive-date=11 April 2014|archive-url=https://web.archive.org/web/20140411101805/http://earth-info.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf|url-status=live}}</ref> lists MRE transforms from several local datums to WGS 84, with accuracies of about 2 meters.<ref name=taylor_high>{{cite web|last=Taylor|first=Chuck|title=High-Accuracy Datum Transformations|url=http://home.hiwaay.net/~taylorc/bookshelf/math-science/geodesy/datum/transform/high-accuracy/|access-date=5 March 2014|archive-date=4 January 2013|archive-url=https://web.archive.org/web/20130104235158/http://home.hiwaay.net/~taylorc/bookshelf/math-science/geodesy/datum/transform/high-accuracy/|url-status=live}}</ref>

The MREs are a direct transformation of geodetic coordinates with no intermediate ECEF step. Geodetic coordinates <math>\phi_B,\, \lambda_B,\, h_B</math> in the new datum <math>B</math> are modeled as [[polynomial]]s of up to the ninth degree in the geodetic coordinates <math>\phi_A,\, \lambda_A,\, h_A</math> of the original datum <math>A</math>. For instance, the change in <math>\phi_B</math> could be parameterized as (with only up to quadratic terms shown){{r|IHO|page1=9}}

:<math>\Delta \phi = a_0 + a_1 U + a_2 V + a_3 U^2 + a_4 UV + a_5 V^2 + \cdots</math>

where
: <math>a_i,</math> parameters fitted by multiple regression
: <math>\begin{align}
    U &= K(\phi_A - \phi_m) \\
    V &= K(\lambda_A - \lambda_m) \\
  \end{align}</math>
: <math>K,</math> scale factor
: <math>\phi_m,\, \lambda_m,</math> origin of the datum, <math>A.</math>

with similar equations for <math> \Delta\lambda</math> and <math>\Delta h</math>. Given a sufficient number of <math>(A,\, B)</math> coordinate pairs for landmarks in both datums for good statistics, multiple regression methods are used to fit the parameters of these polynomials. The polynomials, along with the fitted coefficients, form the multiple regression equations.