Editing Geographic coordinate conversion (section)

=== 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.