DATUM TRANSFORMATIONS

Historically, a goal of geodesy has been to obtain one common datum for coordinates. However realistically, each country or region has often developed their own datum (reference frame) independently. Today we can determine the transformation between reference frames by simply observing satellite coordinates on points of known position in a particular datum and performing a transformation to a geocentric coordinate system. Thus we would determine the geocentric coordinates of a point using both the satellite coordinate system and the datum. The difference in this pair of geocentric coordinates would represent a shift between the satellite reference system, and the regional reference system. Knowing these shifts, other points in the regional reference system can be similarly transferred. This process is depicted in the figure to the right.

If we assume that our reference coordinate systems have different centers, then the transformation process would be to simply (1) transfer the geocentric coordinates to geocentric coordinates using equations developed in the Properties of Biaxial Ellipsoids lesson, (2) add the determined differences between the coordinates in the satellite (S) and the regional (R) reference coordinate systems, and then (3) transfer the geocentric coordinates back into their geodetic coordinates, or proceed into some other known reference datum. 

Step (2) in the above procedure can be mathematically represented as

X_S = X_R + TX
Y_S = Y_R + TY               (1)
Z_S = Z_R + TZ

Given a sufficient number of stations were the coordinates are known in both reference systems, the datum shifts of TX, TY, and TZ can be determined. 

The above datum conversion model assumes that the axes of the two systems are parallel, the systems have the same scale, and the geodetic network has been consistently computed. Reality is that none of these assumptions occurs, and thus TX, TY, and TZ can vary from point to point. A more general transformation involves seven parameters: a change in scale factors (DS) between the two systems, the rotation of the axes between the two systems (R_X, R_Y, R_Z), and the three translation factors (TX, TY, TZ). The following set of equations known as the Helmert transformation utilize seven parameters and can be written as

                                     (2)

The equations, transformation parameters, and software available for transforming coordinates between the NAD 83 and ITRF 96 reference frames are discussed in a 1999 article by Richard A. Snay entitled "Using the HTPD Software to Transform Spatial Coordinates Across Time and Between Reference Frames." which appeared in Surveying and Land Information Systems 59 (No.1): 15-25. The NGS CORS site contains links to both this paper and the HTPD software at http://www.ngs.noaa.gov/CORS/utilities3/.

The parameters are given as

• TX(t) = 0.9910 m, TY(t) = -1.9072 m, TZ(t) = -0.5129 m,
• R_X(t) = [125033 + 258(t - 1997.0)] (10^-12) radians
• R_Y(t) = [46785 - 3599(t - 1997.0)] (10^-12) radians
• R_Z(t) = [56529 - 153(t - 1997.0)] (10^-12) radians
• DS(t) = 0.0

The IERS site at http://lareg.ensg.ign.fr/ITRF/index.html gives both equations and transformation parameters to transforms between the various ITRF reference frames and WGS 84.

where T1, T2, and T3 are TX, TY, and TZ, respectively, and R1, R2, and R3 are R_X, R_Y, R_Z, respectively. The transformation parameters for each ITRF XX system to ITRF 2000 are

TRANSFORMATION PARAMETERS AND THEIR RATES FROM ITRF2000 TO PREVIOUS FRAMES
(See Note Below)

-------------------------------------------------------------------------------------
SOLUTION T1 	T2 	T3 	D 	R1 	R2 	R3 	EPOCH 	Ref. 
UNITS--->cm 	cm 	cm 	ppb 	.001" 	.001" .	001" 	IERS 	Tech.
. . . . . . . Note #
RATES 	T1 	T2 	T3 	D 	R1 	R2 	R3
UNITS> 	cm/y 	cm/y 	cm/y 	ppb/y 	.001"/y .001"/y .001"/y
-------------------------------------------------------------------------------------
ITRF97 	0.67 	0.61 	-1.85 	1.55 	0.00 	0.00 	0.00 	1997.0 	27
rates 	0.00 	-0.06 	-0.14 	0.01 	0.00 	0.00 	0.02 
ITRF96 	0.67 	0.61 	-1.85 	1.55 	0.00 	0.00 	0.00 	1997.0 	24
rates 	0.00 	-0.06 	-0.14 	0.01 	0.00 	0.00 	0.02 
ITRF94 	0.67 	0.61 	-1.85 	1.55 	0.00 	0.00 	0.00 	1997.0 	20
rates 	0.00 	-0.06 	-0.14 	0.01 	0.00 	0.00 	0.02 
ITRF93 	1.27 	0.65 	-2.09 	1.95 	-0.39 	0.80 	-1.14 	1988.0 	18
rates 	-0.29 	-0.02 	-0.06 	0.01 	-0.11 	-0.19 	0.07 
ITRF92 	1.47 	1.35 	-1.39 	0.75 	0.00 	0.00 	-0.18 	1988.0 	15
rates 	0.00 	-0.06 	-0.14 	0.01 	0.00 	0.00 	0.02 
ITRF91 	2.67 	2.75 	-1.99 	2.15 	0.00 	0.00 	-0.18 	1988.0 	12 
rates 	0.00 	-0.06 	-0.14 	0.01 	0.00 	0.00 	0.02 
ITRF90 	2.47 	2.35 	-3.59 	2.45 	0.00 	0.00 	-0.18 	1988.0 	9
rates 	0.00 	-0.06 	-0.14 	0.01 	0.00 	0.00 	0.02 
ITRF89 	2.97 	4.75 	-7.39 	5.85 	0.00 	0.00 	-0.18 	1988.0 	6
rates 	0.00 	-0.06 	-0.14 	0.01 	0.00 	0.00 	0.02 
ITRF88 	2.47 	1.15 	-9.79 	8.95 	0.10 	0.00 	-0.18 	1988.0 	IERS An. Rep.
rates 	0.00 	-0.06 	-0.14 	0.01 	0.00 	0.00 	0.02 		for 1988

The necessary parameters to transform between reference frames can be found on the NIMA web site at http://164.214.2.59/GandG/wgs84dt/. The values for the three parameters solution are shown below for your convenience. 

Molodensky Transformation Formulas

This transformation can also be done in geodetic, curvilinear coordinates using the Molodensky Transformation equations where

f_WGS84 = f_Local + Df
l_WGS84 = l_Local + Dl
h_WGS84 = h_Local + Dh

where Df, Dl, Dh are provided by the standard Molodensky transformation formulas of

• Df" = {-DX sinf cosl - DY sinf sinl + DZ cosf + Da(R_N e² sinf cosf)/a + Df [R_M(a/b) + R_N(b/a))] 
              sinf cosf} [(R_M+h)sin 1"]

• Dl" = [-DX sinl + DY cosl] × [(R_N+h)cosf sin 1"]^-1

• Dh  = DX cosf cosl + DY cosf sinl + DZ sinf - Da(a/R_N) + Df(b/a)

where f, l, h are geodetic coordinates of the point in the local datum.

There are other datum transformation procedures, but these are the big three. Note that when you find a site with the required transformation parameters, the required equations are generally listed also.

... P.S. and that's the rest of the story!!! Good Day!

REFERENCE ELLIPSOIDS

Last Updated: August 18, 2006
Created by Dr. Charles Ghilani
Copyright © 2000, All rights reserved