Graduado

A note on frame transformations with applications to geodetic datums
Tomás Soler & John Marshall
National Geodetic Survey, NOS, NOAA
N/NGS22, #8825
1315 East-West Highway
Silver Spring, MD 20910-3282, USA
Tom.Soler@noaa.gov
Tel: +1-301-7133205 ext. 157
Fax: +1-301-7134324
Abstract. Rigorous equations in compact symbolic matrix notation are introduced to
transform coordinates andvelocities between ITRF frames and modern GPS-based
geocentric geodetic datums. The theory is general but after neglecting higher than
second-order terms it is shown that the equations revert to the formulation currently
applied in most major continental datums. We discuss several examples: the North
American Datum of 1983 (NAD 83), European Terrestrial Reference System of 1989
(ETRS89), GeodeticDatum of Australia of 1994 (GDA94), and the South American
Geocentric Reference System (SIRGAS).
Introduction
Modern-day frame transformations have become increasingly complex to better
accommodate time-dependent processes such as plate tectonics and other geophysical
phenomena. In fact, many modern frame transformations extend the classical 7-parameter
Helmert transformation to complex14-parameter formulations, which augment the
original 7 parameters with their time derivatives. In practice, this augmented formulation
is used to transform GPS densification results from one epoch to another epoch. Although
the treatment of 14-parameter transformations between geocentric terrestrial reference
frames has been published in geodetic literature with various degrees of rigor (e.g. Soler1998; Sillard et al. 1998; Boucher et al. 1999; Altamimi et al. 2002; Soler and Marshall
2002), very little attention has been devoted to the extension of this formulation, in a
straight didactical manner, to the geodetic datum problem. This article complements the
theory previously given in Soler and Marshall (2002) and provides a direct practical
solution to the transformation betweengeocentric frames and geodetic datums; we focus
on four major continental datums currently in use. The discussion presented here is
intended to clear up the prevalent confusion about rotation of vectors (rigid body
rotation) used in plate kinematics vs. rotation of coordinate frames when both operations
are applied in the same transformation.
Theoretical concepts
In Soler and Marshall (2002) thegeneral formulation to transform coordinates between
two arbitrary frames was given. Nevertheless, higher than second order terms were
neglected. Although the contribution of many of these terms can be ignored, some
readers are interested in the most rigorous approach possible and a complete set of

1

equations is presented here. The mapping of our transformation is denoted
ITRF 00(t D) → ITRFyy (t D ) where t D denotes an epoch associated with a datum and
ITRF is an abbreviation for International Terrestrial Reference Frame. In this particular
case, the equation to transform coordinates given in the ITRF00 frame to the ITRFyy
frame under the condition that frame ITRF00 is not changing with time, and that the
coordinates of the stations are fixed in space (no velocities areinvolved) may be written
in compact matrix form as the well-known classical Helmert (similarity) transformation:
{x(t D )}ITRFyy = {Tx } + (1 + s )[δℜ]{x(t D )}ITRF 00

(1)

where the differential rotation matrix denoted by [δℜ] in the above equation is given
explicitly by:
0

[δℜ] = [ I ] + [ε ] = [ I ] +  −ε z
 εy

t

εz
0
−ε x

−ε y 

εx .
0


(2)

Thesuperscript t stands for transpose, [I] denotes the 3 × 3 unit matrix, and [ε ] is a skewsymmetric (anti-symmetric) matrix containing the rotation parameters. To complete the
description of Eq. (1), it should be mentioned that all 3 × 3 matrices are represented
between brackets, 3 × 1 column vectors between braces, and scalars between parentheses.
Equation (1) is consistent with counter-clockwise...