Geographic coordinate conversion#Datum transformations

Informally, specifying a geographic location usually means giving the location's latitude and longitude. The numerical values for latitude and longitude can occur in a number of different units or formats:{{cite web|title=Coordinate transformer|url=http://www.ordnancesurvey.co.uk/gps/transformation|publisher=Ordnance Survey Great Britain|access-date=4 March 2014|archive-date=12 August 2013|archive-url=https://web.archive.org/web/20130812112214/http://www.ordnancesurvey.co.uk/gps/transformation|url-status=live}}

• sexagesimal degree: degrees, minutes, and seconds : 40° 26′ 46″ N 79° 58′ 56″ W
• degrees and decimal minutes: 40° 26.767′ N 79° 58.933′ W
• decimal degrees: +40.446 -79.982

There are 60 minutes in a degree and 60 seconds in a minute. Therefore, to convert from a degrees minutes seconds format to a decimal degrees format, one may use the formula

: $\backslash rm\{decimal\backslash \; degrees\}\; =\; \backslash rm\{degrees\}\; +\; \backslash frac\{\backslash rm\{minutes\}\}\{60\}\; +\; \backslash frac\{\backslash rm\{seconds\}\}\{3600\}$.

To convert back from decimal degree format to degrees minutes seconds format,

: $\backslash begin\{align\}$

\rm{absDegrees} & = | \rm{decimal\ degrees} | \\

\rm{floorAbsDegrees} & = \lfloor \rm{absDegrees} \rfloor \\

\rm{degrees} & = \sgn ( \rm{decimal\ degrees} ) \times \rm{floorAbsDegrees} \\

\rm{minutes} & = \lfloor 60 \times (\rm{absDegrees} - \rm{floorAbsDegrees})\rfloor \\

\rm{seconds} & = 3600 \times (\rm{absDegrees} - \rm{floorAbsDegrees}) - 60 \times \rm{minutes} \\

\end{align}

where $\backslash rm\{absDegrees\}$ and $\backslash rm\{floorAbsDegrees\}$ are just temporary variables to handle both positive and negative values properly.

A coordinate system conversion is a conversion from one coordinate system to another, with both coordinate systems based on the same geodetic datum. Common conversion tasks include conversion between geodetic and earth-centered, earth-fixed (ECEF) coordinates and conversion from one type of map projection to another.

Image:Geodetic latitude and the length of Normal.svg

Geodetic coordinates (latitude $\backslash \; \backslash phi$, longitude $\backslash \; \backslash lambda$, height $h$) can be converted into ECEF coordinates using the following equation:{{cite book|title=GPS - theory and practice|author1=B. Hofmann-Wellenhof |author2=H. Lichtenegger |author3=J. Collins |isbn=3-211-82839-7|page=282|others=Section 10.2.1|year=1997 }}

: $\backslash begin\{align\}$

X & = \left( N(\phi) + h\right)\cos{\phi}\cos{\lambda} \\

Y & = \left( N(\phi) + h\right)\cos{\phi}\sin{\lambda} \\

Z & = \left( \frac{b^2}{a^2} N(\phi) + h\right)\sin{\phi} \\

& = \left( (1 - e^2) N(\phi) + h\right)\sin{\phi} \\

& = \left( (1 - f)^2 N(\phi) + h\right)\sin{\phi}

\end{align}

where

: $$

N(\phi) = \frac{a^2}{\sqrt{a^2 \cos^2 \phi + b^2 \sin^2 \phi }}

= \frac{a}{\sqrt{1 - e^2\sin^2\phi}} = \frac{a}{\sqrt{1 - \frac{e^2}{1 + \cot^2 \phi}}},

and $a$ and $b$ are the equatorial radius (semi-major axis) and the polar radius (semi-minor axis), respectively. $e^2\; =\; 1\; -\; \backslash frac\{b^2\}\{a^2\}$ is the square of the first numerical eccentricity of the ellipsoid. $f\; =\; 1\; -\; \backslash frac\{b\}\{a\}$ is the flattening of the ellipsoid. The prime vertical radius of curvature $\backslash ,\; N(\backslash phi)$ is the distance from the surface to the Z-axis along the ellipsoid normal.

The following condition holds for the longitude in the same way as in the geocentric coordinates system:

:$\backslash frac\{X\}\{\backslash cos\backslash lambda\}\; -\; \backslash frac\{Y\}\{\backslash sin\backslash lambda\}\; =\; 0.$

And the following holds for the latitude:

:$\backslash frac\{p\}\{\backslash cos\backslash phi\}\; -\; \backslash frac\{Z\}\{\backslash sin\backslash phi\}\; -\; e^2\; N(\backslash phi)\; =\; 0,$

where $p\; =\; \backslash sqrt\{X^2\; +\; Y^2\}$, as the parameter $h$ is eliminated by subtracting

:$\backslash frac\{p\}\{\backslash cos\backslash phi\}\; =\; N\; +\; h$

and

:$\backslash frac\{Z\}\{\backslash sin\backslash phi\}\; =\; \backslash frac\{b^2\}\{a^2\}N\; +\; h.$

The following holds furthermore, derived from dividing above equations:

:$\backslash frac\{Z\}\{p\}\; \backslash cot\; \backslash phi\; =\; 1\; -\; \backslash frac\{e^2\; N\}\{N\; +\; h\}.$

The orthogonality of the coordinates is confirmed via differentiation:

:$\backslash begin\{align\}$

\begin{pmatrix} dX \\ dY \\ dZ \end{pmatrix} &=

\begin{pmatrix}

-\sin\lambda & -\sin\phi \cos\lambda & \cos\phi \cos\lambda \\

\cos\lambda & -\sin\phi \sin\lambda & \cos\phi \sin\lambda \\

0 & \cos\phi & \sin\phi \\

\end{pmatrix}

\begin{pmatrix} dE \\ dN \\ dU \end{pmatrix}, \\[3pt]

\begin{pmatrix} dE \\ dN \\ dU \end{pmatrix} &=

\begin{pmatrix}

\left(N(\phi) + h\right) \cos\phi & 0 & 0 \\

0 & M(\phi) + h & 0 \\

0 & 0 & 1 \\

\end{pmatrix}

\begin{pmatrix} d\lambda \\ d\phi \\ dh \end{pmatrix},

\end{align}

where

:$$

M(\phi) = \frac{a\left(1 - e^2\right)}{\left(1 - e^2 \sin^2 \phi\right)^\frac{3}{2}} = N(\phi) \frac{1 - e^2}{1 - e^2\sin^2\phi}

(see also "Meridian arc on the ellipsoid").

The conversion of ECEF coordinates to longitude is:

: $\backslash lambda\; =\; \backslash operatorname\{atan2\}(Y,X)$.

where atan2 is the quadrant-resolving arc-tangent function.

The geocentric longitude and geodetic longitude have the same value; this is true for Earth and other similar shaped planets because they have a large amount of rotational symmetry around their spin axis (see triaxial ellipsoidal longitude for a generalization).

The conversion for the latitude and height involves a circular relationship involving N, which is a function of latitude:

:$\backslash frac\{Z\}\{p\}\; \backslash cot\; \backslash phi\; =\; 1\; -\; \backslash frac\{e^2\; N\}\{N\; +\; h\}$,

:$h=\backslash frac\{p\}\{\backslash cos\backslash phi\}\; -\; N$.

It can be solved iteratively,A guide to coordinate systems in Great Britain. This is available as a pdf document at

{{cite web|url=http://www.ordnancesurvey.co.uk/oswebsite/gps/information/coordinatesystemsinfo/guidecontents |title=ordnancesurvey.co.uk |access-date=2012-01-11 |url-status=dead |archive-url=https://web.archive.org/web/20120211075826/http://www.ordnancesurvey.co.uk/oswebsite/gps/information/coordinatesystemsinfo/guidecontents/ |archive-date=2012-02-11 }} Appendices B1, B2Osborne, P (2008). [http://mercator.myzen.co.uk/mercator.pdf The Mercator Projections] {{webarchive|url=https://web.archive.org/web/20120118224152/http://mercator.myzen.co.uk/mercator.pdf |date=2012-01-18 }} Section 5.4 for example, starting with a first guess h≈0 then updating N.

More elaborate methods are shown below.

The procedure is, however, sensitive to small accuracy due to $N$ and $h$ being maybe 10{{sup|6}} apart.[https://web.archive.org/web/20080920155754/http://www.ferris.edu/faculty/burtchr/papers/cartesian_to_geodetic.pdf R. Burtch, A Comparison of Methods Used in Rectangular to Geodetic Coordinate Transformations.]{{cite journal |last1=Featherstone |first1=W. E. |last2=Claessens |first2=S. J. |title=Closed-Form Transformation between Geodetic and Ellipsoidal Coordinates |journal=Stud. Geophys. Geod. |volume=52 |issue=1 |pages=1–18 |year=2008 |doi=10.1007/s11200-008-0002-6 |bibcode=2008StGG...52....1F |hdl=20.500.11937/11589 |s2cid=59401014 |hdl-access=free }}

The following Bowring's irrational geodetic-latitude equation,{{cite journal |last=Bowring |first=B. R. |title=Transformation from Spatial to Geographical Coordinates |journal=Surv. Rev. |volume=23 |issue=181 |pages=323–327 |year=1976 |doi=10.1179/003962676791280626 }} derived simply from the above properties, is efficient to be solved by Newton–Raphson iteration method:{{cite journal |last=Fukushima |first=T. |title=Fast Transform from Geocentric to Geodetic Coordinates |journal=J. Geod. |volume=73 |issue=11 |pages=603–610 |year=1999 |doi=10.1007/s001900050271 |bibcode=1999JGeod..73..603F |s2cid=121816294 }} (Appendix B){{cite book|first1=J. J. |title=Proceedings of the IEEE 1997 National Aerospace and Electronics Conference. NAECON 1997|volume=2|pages=646–650|last1=Sudano|doi=10.1109/NAECON.1997.622711|chapter=An exact conversion from an earth-centered coordinate system to latitude, longitude and altitude|year=1997|isbn=0-7803-3725-5|s2cid=111028929 }}

: $\backslash kappa\; -\; 1\; -\; \backslash frac\{e^2\; a\backslash kappa\}\{\backslash sqrt\{p^2\; +\; \backslash left(1\; -\; e^2\backslash right)\; Z^2\; \backslash kappa^2\}\}\; =\; 0,$

where $\backslash kappa\; =\; \backslash frac\{p\}\{Z\}\; \backslash tan\; \backslash phi$ and $p\; =\; \backslash sqrt\{X^2\; +\; Y^2\}$ as before. The height is calculated as:

: $\backslash begin\{align\}$

h &= e^{-2} \left(\kappa^{-1} - {\kappa_0}^{-1}\right) \sqrt{p^2 + Z^2 \kappa^2}, \\

\kappa_0 &\triangleq \left(1 - e^2\right)^{-1}.

\end{align}

The iteration can be transformed into the following calculation:

: $\backslash kappa\_\{i+1\}\; =\; \backslash frac\{c\_i\; +\; \backslash left(1\; -\; e^2\backslash right)\; Z^2\; \backslash kappa\_i^3\}\{c\_i\; -\; p^2\}\; =\; 1\; +\; \backslash frac\{p^2\; +\; \backslash left(1\; -\; e^2\backslash right)\; Z^2\; \backslash kappa\_i^3\}\{c\_i\; -\; p^2\},$

where $c\_i\; =\; \backslash frac\{\backslash left(p^2\; +\; \backslash left(1\; -\; e^2\backslash right)\; Z^2\; \backslash kappa\_i\; ^2\backslash right)^\backslash frac\{3\}\{2\}\}\{ae^2\}\; .$

The constant $\backslash ,\backslash kappa\_0$ is a good starter value for the iteration when $h\; \backslash approx\; 0$. Bowring showed that the single iteration produces a sufficiently accurate solution. He used extra trigonometric functions in his original formulation.

The quartic equation of $\backslash kappa$, derived from the above, can be solved by Ferrari's solution{{cite journal|first1=H. |last1=Vermeille, H.|title=Direct Transformation from Geocentric to Geodetic Coordinates|journal= J. Geod.|volume=76|number=8|pages=451–454

|year= 2002|doi=10.1007/s00190-002-0273-6|s2cid=120075409 }}{{cite journal|first1=Laureano|last1=Gonzalez-Vega|first2=Irene|last2=PoloBlanco|title=A symbolic analysis of Vermeille and Borkowski polynomials for transforming 3D Cartesian to geodetic coordinates|journal=J. Geod.|volume=83|number=11|pages=1071–1081|doi=10.1007/s00190-009-0325-2|year=2009|bibcode=2009JGeod..83.1071G |s2cid=120864969 }} to yield:

: $$

\begin{align}

\zeta &= \left(1 - e^2\right)\frac{z^2}{a^2} ,\\[4pt]

\rho &= \frac{1}{6}\left(\frac{p^2}{a^2} + \zeta - e^4\right) ,\\[4pt]

s &= \frac{e^4 \zeta p^2}{4\rho^3 a^2} ,\\[4pt]

t &= \sqrt[3]{1 + s + \sqrt{s(s + 2)}} ,\\[4pt]

u &= \rho \left(t + 1 + \frac{1}{t}\right) ,\\[4pt]

v &= \sqrt{u^2 + e^4 \zeta} ,\\[4pt]

w &= e^2 \frac{u + v - \zeta}{2v} ,\\[4pt]

\kappa &= 1 + e^2 \frac{\sqrt{u + v + w^2} + w}{u + v}.

\end{align}

A number of techniques and algorithms are available but the most accurate, according to Zhu,{{cite journal|first1=J.|last1=Zhu|title=Conversion of Earth-centered Earth-fixed coordinates to geodetic coordinates|journal=IEEE Transactions on Aerospace and Electronic Systems|volume=30|issue=3|year=1994|pages=957–961|doi=10.1109/7.303772|bibcode=1994ITAES..30..957Z }} is the following procedure established by Heikkinen,{{cite journal|first1=M.|last1=Heikkinen|title=Geschlossene formeln zur berechnung räumlicher geodätischer koordinaten aus rechtwinkligen koordinaten.|journal=Z. Vermess.|volume=107|year=1982|pages=207–211|language=de}} as cited by Zhu. This overlaps with above. It is assumed that geodetic parameters $\backslash \{a,\backslash ,\; b,\backslash ,\; e\backslash \}$ are known

: $\backslash begin\{align\}$

a &= 6378137.0 \text{ m. Earth Equatorial Radius} \\[3pt]

b &= 6356752.3142 \text{ m. Earth Polar Radius} \\[3pt]

e^2 &= \frac{a^2-b^2}{a^2} \\[3pt]

e'^2 &= \frac{a^2 - b^2}{b^2} \\[3pt]

p &= \sqrt{X^2 + Y^2} \\[3pt]

F &= 54b^2 Z^2 \\[3pt]

G &= p^2 + \left(1 - e^2\right)Z^2 - e^2\left(a^2 - b^2\right) \\[3pt]

c &= \frac{e^4 Fp^2}{G^3} \\[3pt]

s &= \sqrt[3]{1 + c + \sqrt{c^2 + 2c}} \\[3pt]

k &= s + 1 + \frac{1}{s}\\[3pt]

P &= \frac{F}{3 k^2 G^2} \\[3pt]

Q &= \sqrt{1 + 2e^4 P} \\[3pt]

r_0 &= \frac{-Pe^2 p}{1 + Q} + \sqrt{\frac{1}{2} a^2\left(1 + \frac{1}{Q}\right) - \frac{P\left(1 - e^2\right)Z^2}{Q(1 + Q)} - \frac{1}{2}Pp^2} \\[3pt]

U &= \sqrt{\left(p - e^2 r_0\right)^2 + Z^2} \\[3pt]

V &= \sqrt{\left(p - e^2 r_0\right)^2 + \left(1 - e^2\right)Z^2} \\[3pt]

z_0 &= \frac{b^2 Z}{aV} \\[3pt]

h &= U\left(1 - \frac{b^2}{aV}\right) \\[3pt]

\phi &= \arctan\left[\frac{Z + e'^2 z_0}{p}\right] \\[3pt]

\lambda &= \operatorname{arctan2}[Y,\, X]

\end{align}

Note: arctan2[Y, X] is the four-quadrant inverse tangent function.

For small {{math|e²}} the power series

:$\backslash kappa\; =\; \backslash sum\_\{i\backslash ge\; 0\}\; \backslash alpha\_i\; e^\{2i\}$

starts with

:$\backslash begin\{align\}$

\alpha_0 &= 1; \\

\alpha_1 &= \frac{a}{\sqrt{Z^2 + p^2}}; \\

\alpha_2 &= \frac{aZ^2\sqrt{Z^2 + p^2} + 2a^2 p^2}{2\left(Z^2 + p^2\right)^2}.

\end{align}

To convert from geodetic coordinates to local tangent plane (ENU) coordinates is a two-stage process:

1. Convert geodetic coordinates to ECEF coordinates
2. Convert ECEF coordinates to local ENU coordinates

To transform from ECEF coordinates to the local coordinates we need a local reference point. Typically, this might be the location of a radar. If a radar is located at $\backslash left\backslash \{X\_r,\backslash ,\; Y\_r,\backslash ,\; Z\_r\backslash right\backslash \}$ and an aircraft at $\backslash left\backslash \{X\_p,\backslash ,\; Y\_p,\backslash ,\; Z\_p\backslash right\backslash \}$, then the vector pointing from the radar to the aircraft in the ENU frame is

: $$

\begin{bmatrix}x \\ y \\ z\end{bmatrix} =

\begin{bmatrix}

-\sin\lambda_r & \cos\lambda_r & 0 \\

-\sin\phi_r\cos\lambda_r & -\sin\phi_r\sin\lambda_r & \cos\phi_r \\

\cos\phi_r\cos\lambda_r & \cos\phi_r\sin\lambda_r & \sin\phi_r

\end{bmatrix}

\begin{bmatrix}

X_p - X_r \\

Y_p - Y_r \\

Z_p - Z_r

\end{bmatrix}

Note: $\backslash \; \backslash phi$ is the geodetic latitude; the geocentric latitude is inappropriate for representing vertical direction for the local tangent plane and must be converted if necessary.

This is just the inversion of the ECEF to ENU transformation so

: $$

\begin{bmatrix}X_p \\ Y_p \\ Z_p\end{bmatrix} =

\begin{bmatrix}

-\sin\lambda_r & -\sin\phi_r\cos\lambda_r & \cos\phi_r\cos\lambda_r \\

\cos\lambda_r & -\sin\phi_r\sin\lambda_r & \cos\phi_r\sin\lambda_r \\

0 & \cos\phi_r & \sin\phi_r

\end{bmatrix}

\begin{bmatrix}x \\ y \\ z\end{bmatrix} +

\begin{bmatrix}X_r \\ Y_r \\ Z_r\end{bmatrix}

Conversion of coordinates and map positions among different map projections reference to the same datum may be accomplished either through direct translation formulas from one projection to another, or by first converting from a projection $A$ to an intermediate coordinate system, such as ECEF, then converting from ECEF to projection $B$. The formulas involved can be complex and in some cases, such as in the ECEF to geodetic conversion above, the conversion has no closed-form solution and approximate methods must be used. References such as the DMA Technical Manual 8358.1{{cite web|title=TM8358.2: The Universal Grids: Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS)|url=http://earth-info.nga.mil/GandG/publications/tm8358.2/TM8358_2.pdf|publisher=National Geospatial-Intelligence Agency|access-date=4 March 2014|archive-date=3 March 2020|archive-url=https://web.archive.org/web/20200303184711/https://earth-info.nga.mil/GandG/publications/tm8358.2/TM8358_2.pdf|url-status=live}} and the USGS paper Map Projections: A Working Manual{{cite book|last=Snyder|first=John P.|title=Map Projections: A Working Manual|year=1987|publisher=USGS Professional Paper: 1395|url=https://pubs.er.usgs.gov/publication/pp1395|access-date=2017-08-28|archive-date=2011-05-17|archive-url=https://web.archive.org/web/20110517082057/http://pubs.er.usgs.gov/publication/pp1395|url-status=live}} contain formulas for conversion of map projections. It is common to use computer programs to perform coordinate conversion tasks, such as with the DoD and NGA supported GEOTRANS program.{{cite web|title=MSP GEOTRANS 3.3 (Geographic Translator)|url=http://earth-info.nga.mil/GandG/geotrans/|publisher=NGA: Coordinate Systems Analysis Branch|access-date=4 March 2014|archive-date=15 March 2014|archive-url=https://web.archive.org/web/20140315075748/http://earth-info.nga.mil/GandG/geotrans/|url-status=live}}

{{further|Geodetic datum}}

File:Possible paths for datum transform.svg

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.

{{main|Helmert transformation}}

Use of the Helmert transform in the transformation from geodetic coordinates of datum $A$ to geodetic coordinates of datum $B$ occurs in the context of a three-step process:{{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}}

1. Convert from geodetic coordinates to ECEF coordinates for datum $A$
2. Apply the Helmert transform, with the appropriate $A\backslash to\; B$ transform parameters, to transform from datum $A$ ECEF coordinates to datum $B$ ECEF coordinates
3. Convert from ECEF coordinates to geodetic coordinates for datum $B$

In terms of ECEF XYZ vectors, the Helmert transform has the form (position vector transformation convention and very small rotation angles simplification)

: $$

\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}.

The Helmert transform is a seven-parameter transform with three translation (shift) parameters $c\_x,\backslash ,\; c\_y,\backslash ,\; c\_z$, three rotation parameters $r\_x,\backslash ,\; r\_y,\backslash ,\; r\_z$ and one scaling (dilation) parameter $s$. 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.{{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}}

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{{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}} and earthquakes.{{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}} This has been incorporated into software, such as the Horizontal Time Dependent Positioning (HTDP) tool from the U.S. NGS.{{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}}

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:

1. Convert from geodetic coordinates to ECEF coordinates for datum $A$
2. Apply the Molodensky-Badekas transform, with the appropriate $A\backslash to\; B$ transform parameters, to transform from datum $A$ ECEF coordinates to datum $B$ ECEF coordinates
3. Convert from ECEF coordinates to geodetic coordinates for datum $B$

The transform has the form{{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}}

: $$

\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}.

where $\backslash left(X^0\_A,\backslash ,\; Y^0\_A,\backslash ,\; Z^0\_A\backslash right)$ is the origin for the rotation and scaling transforms and $\backslash Delta\; S$ 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}}

The Molodensky transformation converts directly between geodetic coordinate systems of different datums without the intermediate step of converting to geocentric coordinates (ECEF).{{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}} 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.{{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}} The Molodensky method was popular before the advent of modern computers and the method is part of many geodetic programs.

File:Datum Shift Between NAD27 and NAD83.png

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.{{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}} 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).{{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}} 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.{{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}}{{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 }}

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.{{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 }} The standard NIMA TM 8350.2, Appendix D,{{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}} lists MRE transforms from several local datums to WGS 84, with accuracies of about 2 meters.{{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}}

The MREs are a direct transformation of geodetic coordinates with no intermediate ECEF step. Geodetic coordinates $\backslash phi\_B,\backslash ,\; \backslash lambda\_B,\backslash ,\; h\_B$ in the new datum $B$ are modeled as polynomials of up to the ninth degree in the geodetic coordinates $\backslash phi\_A,\backslash ,\; \backslash lambda\_A,\backslash ,\; h\_A$ of the original datum $A$. For instance, the change in $\backslash phi\_B$ could be parameterized as (with only up to quadratic terms shown){{r|IHO|page1=9}}

:$\backslash Delta\; \backslash phi\; =\; a\_0\; +\; a\_1\; U\; +\; a\_2\; V\; +\; a\_3\; U^2\; +\; a\_4\; UV\; +\; a\_5\; V^2\; +\; \backslash cdots$

where

: $a\_i,$ parameters fitted by multiple regression

: $\backslash begin\{align\}$

U &= K(\phi_A - \phi_m) \\

V &= K(\lambda_A - \lambda_m) \\

\end{align}

: $K,$ scale factor

: $\backslash phi\_m,\backslash ,\; \backslash lambda\_m,$ origin of the datum, $A.$

with similar equations for $\backslash Delta\backslash lambda$ and $\backslash Delta\; h$. Given a sufficient number of $(A,\backslash ,\; B)$ 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.

• Gauss–Krüger coordinate system
• List of map projections
• Spatial reference system
• Topocentric coordinate system
• Universal polar stereographic coordinate system
• Universal Transverse Mercator coordinate system
• Geographical distance

{{reflist|30em}}

*Conversion

Category:Geodesy