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\baselineskip = 1 pc
\parskip = 1 pc
\parindent = 2 pc
\font\large = cmr10 scaled\magstep2
\setbox0 =\hbox{$''$\kern-.15cm{.}\kern.04cm}
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\write 0 {3.  CONVENTIONAL TERRESTRIAL REFERENCE SYSTEM
\noexpand\dotfill\the\pageno\noexpand\break}
\noindent{\large CHAPTER 3} {\bf CONVENTIONAL TERRESTRIAL REFERENCE
SYSTEM}
\bigskip
\write 0 {\indent Definition \noexpand\dotfill\the\pageno\noexpand\break}
\noindent{\bf Definition}
    
     The Terrestrial Reference System adopted for either the
analysis of individual data sets by techniques (VLBI, SLR, LLR,
GPS, DORIS, PRARE...) or the combination of individual solutions into a
unified set of data (station coordinates, Earth orientation parameters,
etc...) follows these criteria (Boucher, 1990):
\item{a)}  It is geocentric, the center of mass being defined for
     the whole Earth, including oceans and atmosphere.
\item{b)}  Its scale is that of a local Earth frame, in the meaning
     of a relativistic theory of gravitation.
\item{c)}  Its orientation was initially given by the BIH orientation at
     1984.0.
\item{d)}  Its time evolution in orientation will create no residual
     global rotation with regards to the crust.
\medskip
\write 0 {\indent Realization \noexpand\dotfill\the\pageno\noexpand\break}
\noindent{\bf Realization}

\nobreak

     A Conventional Terrestrial Reference System (CTRS) can be 
realized through a reference frame {\it i.e.} a set of coordinates for 
a network of stations. Such a realization will be specified by cartesian
equatorial coordinates $X$, $Y$, and $Z$, by preference. If geographical
coordinates are needed, the GRS80 ellipsoid is recommended 
(a=6378137.0 m, eccentricity$^2$ =0.00669438003).  The CTRS which is 
monitored by IERS is called the International Terrestrial Reference 
System (ITRS) and is specified by the IUGG Resolution no. 2 adopted 
at the 20th IUGG General Assembly of Vienna in 1991.

     Each analysis center should compare its reference frame to a
realization of the ITRS. Within IERS, each Terrestrial
Reference Frame (TRF) is either directly, or after transformation,
expressed as a realization of the ITRS.  The position of a point 
located on the surface of the solid Earth should be expressed by
$$\vec{X}(t)=\vec{X}_0+\vec{V}_0(t-t_0)+\sum_i\Delta\vec{X}_i(t),$$
where $\Delta\vec{X}_i$ are corrections due to various time changing
effects, and $\vec{X}_0$ and $\vec{V}_0$ are position and velocity at
the epoch $t_0$. The corrections to be considered are solid Earth tide
displacement (full correction including permanent effect, see Chapter 
7), ocean loading, post glacial rebound, and atmospheric loading.  
Further corrections could be added if they are at mm level or greater, 
and can be computed by a suitable model.

     Realizations of the ITRS are produced by IERS under the name 
International Terrestrial Reference Frames (ITRF), which consist of  
lists of coordinates (and velocities) for a selection of IERS sites 
(tracking stations or related ground markers).  Currently, ITRF-yy is 
published annually by the IERS in the {\it Technical Notes} ({\it cf.}
Boucher {\it et al.}, 1996).  The numbers (yy) following the 
designation ``ITRF" specify the last year whose data were used in the 
formation of the frame.  Hence ITRF94 designates the frame of 
coordinates and velocities constructed in 1995 using all of the IERS 
data available through 1994. More recently, since 1993, other special 
realizations have been produced, such as solutions for IGS core 
stations (ITRF-Py series) (IGS, 1995).  It is also anticipated that 
monthly series may be determined.

     Terrestrial reference frames may be defined within systems
which account for the effect of the permanent tide differently. 
In the terminology of Ekman (1989), Rapp (1991) and Poutanen {\it et
al.} (1996) the permanent or ``zero--frequency" tide is retained in
the ``zero--tide" system.  In this system the crust corresponds to
the realistic time average of the crust which varies because of
the action of the luni--solar tides.  In a different system
referred to as ``tide--free," all of the effects of the permanent
tide are removed.  This is not realistic since the crust in this
case cannot be ``observed," nor are the Love Numbers required to
describe the permanent tide known adequately.  No error is
associated with either convention as long as it is clear which
system is employed in the analysis of the data.  A third 
``mean--tide" system has been defined in which the crust is equivalent 
to that of the tide--free system and, in which, the effect of the
permanent tide is also removed from the geoid.

     Coordinates of the ITRF are given in a conventional frame
where the effects of all tides are removed as recommended in IERS
Technical Note 13 (McCarthy, 1992).  There, it is recommended
that equation 6 on p. 57 be used to account for the permanent
tide.  Note that this is in contradiction to Resolution 16 of the
General Assembly of the IAG in 1983.  The corresponding equation
in this publication is equation 8 of Chapter 7.  To place the
coordinates in a zero-tide system it is necessary to apply
equation 8 of IERS Technical Note 13 (McCarthy, 1992) or the
corresponding version, equation 17 of Chapter 7 in this
publication.

     In data analysis, $\vec{X}_0$ and $\vec{V}_0$ should be considered
as solve-for parameters.  In particular, if a non-linear change occurs
(earthquake, volcanic event ...), a new $\vec{X}_0$ should be
adopted.  When adjusting parameters, particularly velocities, the
IERS orientation should be kept at all epochs, ensuring the
alignment at a reference epoch and the time evolution through a
no net rotation condition.  The way followed by various analysis
centers depends on their own view of modelling, and on the
techniques themselves.  For the origin, only data which can be
modelled by dynamical techniques (currently SLR, LLR, GPS or DORIS for
IERS) can determine the center of mass.  The VLBI system can be
referred to a geocentric system by adopting for a station its
geocentric position at a reference epoch as provided from
external information.

     The scale is obtained by appropriate relativistic modelling. 
Specifically, according to IAU and IUGG resolutions, the scale is
consistent with the TCG time coordinate for a geocentric local 
frame.  A detailed treatment can be found in Chapter 11.  The 
orientation is defined by adopting IERS (or BIH) Earth orientation 
parameters at a reference epoch. In the case of dynamical 
techniques, an additional constraint in longitude is necessary to 
remove ill-conditioning.

     The unit of length is the meter (SI).  The IERS Reference
Pole (IRP) and Reference Meridian (IRM) are consistent with the
corresponding directions in the BIH Terrestrial System (BTS)
within $\pm 0\qd005$.  The BIH reference pole was adjusted to the
Conventional International Origin (CIO) in 1967; it was then kept
stable independently until 1987.  The uncertainty of the tie of
the IRP with the CIO is $\pm 0\qd03$.  The time evolution of the
orientation will be ensured by using a no-net-rotation condition
with regards to horizontal tectonic motions over the whole Earth.

\write 0 {\indent Transformation Parameters of World Coordinate
Systems and Datums \noexpand\dotfill\the\pageno\noexpand\break}
\noindent{\bf Transformation Parameters to World Coordinate Systems 
and Various Datums}

\nobreak

     The seven-parameter similarity transformation between any
two Cartesian systems, {\it e.g.}, from $(x,y,z)$ to $(xs,ys,zs)$
can be written as
$$\pmatrix{xs\cr ys\cr zs\cr}=\pmatrix{x\cr y\cr z\cr} + 
\pmatrix{T1\cr T2\cr T3\cr} + 
\pmatrix{D&-R3&R2\cr R3&D&-R1\cr-R2&R1&D\cr}\pmatrix{x\cr y\cr
z\cr},\eqno(1)$$
where $T1$, $T2$, $T3$ = coordinates of the origin of the frame 
$(xs,ys,zs)$ in the frame $(x,y,z)$; $R1, R2, R3$= differential 
rotations (expressed in radians) respectively, around the axes 
$(xs,ys,zs)$ to establish parallelism with the $(x,y,z)$ frame; and 
$D$ = differential scale change.  

     The use of the previous algorithm to transform coordinates
from one datum to another must be done with care.  At some level of 
accuracy, each reference system has multiple realizations.  In many 
cases, two such realizations of the same system have nonzero 
transformation parameters.  The usual reason is that such parameters 
are currently adjusted between two sets of coordinates.  In this case, 
any global systematic error which can be mapped into a scale, shift 
or rotation will go into these seven parameters.

     Therefore we give here (Table 3.1) only parameters from ITRF94
to previous ITRF series.  These numbers are the responsibility of the 
IERS.  For other transformations, either to global systems (such as 
WGS84) or regional (such as NAD83 or ETRS89), or even local systems, 
users should contact relevant agencies.  For WGS84, the main result 
is that the most recent realizations of WGS84 based on GPS are 
consistent with the ITRF realization at the 0.1 meter level (Malys 
and Slater, 1994).  Otherwise, one can refer to IAG bodies, such as 
Commission X on Global and Regional Geodetic Networks (Geodesist's 
Handbook, 1996).

\noindent Table 3.1. Transformation parameters from ITRF94 to past
ITRFs.  ``ppb" refers to parts per billion ($10^9$).  Rates must be
applied for ITRF93.  The units for rate are understood to be ``per
year."
\hrule
\smallskip
\halign{#\hfil&\kern1pc\hfil#&\kern1pc\hfil#&\kern1pc\hfil#&
\kern1pc\hfil#&\kern1pc\hfil#&\kern1pc\hfil#&\kern1pc\hfil#&\kern1pc\hfil#\cr
Coordinate\cr
  System  &  T1  &   T2 &   T3 &    D &  R1   &   R2  &  R3   \cr
 (datum)  & (cm) &  (cm)&  (cm)& (ppb)& (mas)& (mas)& (mas)& Epoch\cr
 \hrulefill&\kern-1pc\hrulefill&\kern-1pc\hrulefill&\kern-1pc
 \hrulefill&\kern-1pc\hrulefill&\kern-1pc\hrulefill&\kern-1pc
 \hrulefill&\kern-1pc\hrulefill&\kern-1pc\hrulefill\cr
   ITRF88 & 1.8 &  0.0 & -9.2 &  7.4 & 0.1 & 0.0 & 0.0 & 1988.0\cr
   ITRF89 & 2.3 &  3.6 & -6.8 &  4.3 & 0.0 & 0.0 & 0.0 & 1988.0\cr
   ITRF90 & 1.8 &  1.2 & -3.0 &  0.9 & 0.0 & 0.0 & 0.0 & 1988.0\cr
   ITRF91 & 2.0 &  1.6 & -1.4 &  0.6 & 0.0 & 0.0 & 0.0 & 1988.0\cr
   ITRF92 & 0.8 &  0.2 & -0.8 & -0.8 & 0.0 & 0.0 & 0.0 & 1988.0\cr
   ITRF93 & 0.6 & -0.5 & -1.5 &  0.4 &-0.39& 0.80&-0.96& 1988.0\cr
\kern1pc rates  &-0.29&  0.04&  0.08&  0.00&-0.11&-0.19& 0.05&\cr}
\vskip .07pc
\hrule
\bigskip
Once the Cartesian coordinates $(x,y,z)$ are known, they can be
transformed to ``datum" or curvilinear geodetic coordinates $(\lambda,
\phi,h)$ referred to an ellipsoid of semi-major axis $a$ and flattening
$f$, using the following code (Borkowski, 1989). First compute 
$\lambda=\tan^{-1}({y\over x})$ properly determining the quadrant 
from x and y ($0 \le \lambda < 2\pi$) and $r=\sqrt{x^2 + y^2}$ .
\bigskip
{\tt
\settabs\+C\ \ &99&\&&\cr
\+&&&      subroutine GEOD(r,z,fi,h)\cr
\+c& Program to transform Cartesian to geodetic coordinates\cr
\+c&     based on the exact solution (Borkowski,1989)\cr
\+c& Input :  r, z = equatorial [m] and polar [m] components\cr
\+c& Output: fi, h = geodetic coord's (latitude [rad], height [m])\cr
\+&&&      implicit real*8(a-h,o-z)\cr
\+c& GRS80 ellipsoid: semimajor axis (a) and inverse flattening (fr)\cr
\+&&&      data a,fr /6378137.d0,298.257222101d0/\cr
\+&&&      b = dsign(a - a/fr,z)\cr
\+&&&      E = ((z + b)*b/a - a)/r\cr
\+&&&      F = ((z - b)*b/a + a)/r\cr
\+c& Find solution to: t**4 + 2*E*t**3 + 2*F*t - 1 = 0\cr
\+&&&      P = (E*F + 1.)*4.d0/3.d0\cr
\+&&&      Q = (E*E - F*F)*2.d0\cr
\+&&&      D = P*P*P + Q*Q\cr
\+&&& \kern2pc   if(D.ge.0d0) then\cr
\+&&&       s = dsqrt(D) + Q\cr
\+&&&       s = dsign(dexp(dlog(dabs(s))/3.d0),s)\cr
\+&&&       v = P/s - s\cr
\+c& Improve the accuracy of numeric values of v\cr
\+&&&       v = -(Q + Q + v*v*v)/(3.d0*P)\cr
\+&&& \kern2pc   else\cr
\+&&&       v = 2.d0*dsqrt(-P)*dcos(dacos(Q/P/dsqrt(-P))/3.d0)\cr
\+&&& \kern2pc   endif\cr
\+&&&      G = .5d0*(E + dsqrt(E*E + v))\cr
\+&&&      t = dsqrt(G*G + (F - v*G)/(G + G - E)) - G\cr
\+&&&      fi = datan((1.d0 - t*t)*a/(2.d0*b*t))\cr
\+&&&      h = (r - a*t)*dcos(fi) + (z - b)*dsin(fi)\cr
\+&&&      return\cr
\+&&&      end\cr}

\bigskip
\write 0 {\indent Plate Motion Model \noexpand\dotfill\the\pageno
\noexpand\break}
\noindent{\bf Plate Motion Model}

\nobreak

     One of the factors which affect Earth rotation results is
the motion of the tectonic plates which make up the Earth's
surface.  As the plates move, fixed coordinates for the observing
stations become inconsistent with each other.  The rates of
relative motions for some observing sites are 5 cm per year or
larger.  The observations of plate motions by modern methods
appear to be roughly consistent with the average rates over the
last few million years derived from the geological record and
other geophysical information.  Thus, the NNR-NUVEL1A model for 
plate motions given by DeMets {\it et al.} (1994) is recommended.

     If a velocity has not yet been determined in the ITRF for a
particular station, the velocity $\vec{V}_0$ should be expressed as
$$\vec{V}_0 = \vec{V}_{plate} + \vec{V}_r$$
\noindent where $\vec{V}_{plate}$ is the horizontal velocity computed 
from the NNR-NUVEL1A model (DeMets {\it et al.}, 1994) and $\vec{V}_r$
a residual velocity.

     The Cartesian rotation vector for each of the major plates is
given in Table 3.2.  A subroutine called ABSMO\_NUVEL is also included
below.  It computes the new site position from the old site position
using the recommended plate motion model.  Fig. 3.1 shows the plates
on a map of the world.

\vfil\eject
\special {psfile="map2.ps" voffset = -425 hscale = 69 vscale = 69}
\noindent Figure 3.1.  Map of the tectonic plates.

\vskip 4in
\noindent Table 3.2.  Cartesian rotation vector for each plate using the
NNR-NUVEL1A kinematic plate model (no net rotation).
\hrule
\smallskip
\halign{#\hfil\kern2pc&\hfil#\kern2pc&\hfil#\kern2pc&\hfil#\cr
	  &$\Omega_x$\kern1pc&$\Omega_y$\kern1pc&$\Omega_z$\kern1pc\cr
Plate Name     &           rad/My.&         rad/My.&       rad/My.\cr
\hrulefill&\kern-2pc\hrulefill&\kern-2pc\hrulefill&
\kern-2pc\hrulefill\cr
Pacific        &           -0.001510&       0.004840&     -0.009970\cr
Cocos          &           -0.010425&      -0.021605&      0.010925\cr
Nazca          &           -0.001532&      -0.008577&      0.009609\cr
Caribbean      &           -0.000178&      -0.003385&      0.001581\cr
South America  &           -0.001038&      -0.001515&     -0.000870\cr
Antarctica     &           -0.000821&      -0.001701&      0.003706\cr
India          &            0.006670&       0.000040&      0.006790\cr
Australia      &            0.007839&       0.005124&      0.006282\cr
Africa         &            0.000891&      -0.003099&      0.003922\cr
Arabia         &            0.006685&      -0.000521&      0.006760\cr
Eurasia        &           -0.000981&      -0.002395&      0.003153\cr
North America  &            0.000258&      -0.003599&     -0.000153\cr
Juan de Fuca   &            0.005200&       0.008610&     -0.005820\cr
Philippine     &            0.010090&      -0.007160&     -0.009670\cr
Rivera         &           -0.009390&      -0.030960&      0.012050\cr
Scotia         &           -0.000410&      -0.002660&     -0.001270\cr}
\hrule
\bigskip

     The NNR-NUVEL1A model should be used as a default, for stations
which appear to follow reasonably its values.  For some stations,
particularly in the vicinity of plate boundaries, users may benefit
by estimating velocities or using specific values not derived from
NNR-NUVEL1A.  This is also a way to take into account now some non-
negligible vertical motions.  Published station coordinates should
include the epoch associated with the coordinates.

The original subroutine was a coding of the AM0-2 model from J. B.
Minster.  Changes have been made to represent NNR-NUVEL1A model 
(DeMets {\it et al.}, 1994).
\bigskip
{\tt
\settabs\+C\ \ &99&\&&\cr
\+&&& SUBROUTINE ABSMO\_NUVEL(PSIT,T0,X0,Y0,Z0,T,X,Y,Z)\cr
\+C\cr
\+C&   ABSMO\_NUVEL takes a site specified by its initial coordinates\cr
\+C&   X0,Y0,Z0 at time T0, and computes its updated positions X,Y,Z\cr
\+C&   at time T, based on the geological "absolute". \cr
\+C&   \cr
\+C&   \cr
\+C\cr
\+C&   Original author: J.B. Minster, Science Horizons.\cr
\+C&   DFA: Revised by Don Argus, Northwestern University\cr
\+C&   DFA: uses absolute model NNR-NUVEL1\cr
\+C\cr
\+C&   Transcribed from USNO Circular 167 "Project Merit Standards"\cr
\+C&   by Tony Mallama with slight modification to the documentation\cr
\+C&   and code.\cr
\+C\cr
\+C&   Times are given in years, e.g. 1988.0 for Jan 1, 1988.\cr
\+C\cr
\+C&   PSIT is the four character abbreviation for the plate name,\cr
\+C&   if PSIT is not recognized then the new positions are returned\cr
\+C&   as zero.\cr
\+C\cr
\+&&&     IMPLICIT NONE\cr
\+&&&     CHARACTER*4        PSIT,PNM(16)\cr
\+&&&     REAL*8             OMX(16),OMY(16),OMZ(16)\cr
\+&&&     REAL*8             X0,Y0,Z0\cr
\+&&&     REAL*8             X,Y,Z,T,T0\cr
\+&&&     INTEGER*2          IPSIT,I\cr
\+C\cr
\+C&   DFA: NNR-NUVEL1A\cr
\+C\cr
\+&&&      DATA    (PNM(I),     OMX(I),     OMY(I),     OMZ(I),\cr
\+&&    \&         I =  1,16)\cr
\+&&    \&        /'PCFC',    -0.001510,   0.004840,  -0.009970,\cr
\+&&    \&         'AFRC',     0.000891,  -0.003099,   0.003922,\cr
\+&&    \&         'ANTA',    -0.000821,  -0.001701,   0.003706,\cr
\+&&    \&         'ARAB',     0.006685,  -0.000521,   0.006760,\cr
\+&&    \&         'AUST',     0.007839,   0.005124,   0.006282,\cr
\+&&    \&         'CARB',    -0.000178,  -0.003385,   0.001581,\cr
\+&&    \&         'COCO',    -0.010425,  -0.021605,   0.010925,\cr
\+&&    \&         'EURA',    -0.000981,  -0.002395,   0.003153,\cr
\+&&    \&         'INDI',     0.006670,   0.000040,   0.006790,\cr
\+&&    \&         'NAZC',    -0.001532,  -0.008577,   0.009609,\cr
\+&&    \&         'NOAM',     0.000258,  -0.003599,  -0.000153,\cr
\+&&    \&         'SOAM',    -0.001038,  -0.001515,  -0.000870,\cr
\+&&    \&         'JUFU',     0.005200,   0.008610,  -0.005820,\cr
\+&&    \&         'PHIL',     0.010090,  -0.007160,  -0.009670,\cr
\+&&    \&         'RIVR',    -0.009390,  -0.030960,  -0.012050,\cr
\+&&    \&         'SCOT',    -0.000410,  -0.002660,  -0.001270/\cr
\+C\cr
\+C\cr
\+C&   Initialize things properly\cr
\+C\cr
\+&&&      IPSIT = -1\cr
\+&&&      X = 0.0D0\cr
\+&&&      Y = 0.0D0\cr
\+&&&      Z = 0.0D0\cr
\+C\cr
\+C&   Look up the plate in the list.\cr
\+C\cr
\+&&&      DO 20 I = 1,16\cr
\+&   20&& IF (PSIT .EQ. PNM(I)) IPSIT = I\cr
\+C\cr
\+C&   If plate name is not recognized return the new plate position\cr
\+C&   as zero.\cr
\+C\cr
\+&&&      IF (IPSIT .EQ. -1) RETURN\cr
\+C\cr
\+C&   Compute the new coordinates\cr
\+C\cr
\+&&&      X = X0 + (OMY(IPSIT)*Z0 - OMZ(IPSIT)*Y0) * (T-T0)/1.0D+6\cr
\+&&&      Y = Y0 + (OMZ(IPSIT)*X0 - OMX(IPSIT)*Z0) * (T-T0)/1.0D+6\cr
\+&&&      Z = Z0 + (OMX(IPSIT)*Y0 - OMY(IPSIT)*X0) * (T-T0)/1.0D+6\cr
\+C\cr
\+C&   Finish up\cr
\+C\cr
\+&&&      RETURN\cr
\+&&&      END\cr}

\medskip
\noindent{\bf References}

\nobreak

\item{}\kern-2pc
Borkowski, K. M., 1989, ``Accurate Algorithms to Transform Geocentric
     to Geodetic Coordinates," {\it Bull. Geod.}, {\bf 63}, pp. 50--56.

\item{}\kern-2pc
Boucher, C., 1990, ``Definition and Realization of Terrestrial
     Reference Systems for Monitoring Earth Rotation,"  {\it Variations
     in Earth Rotation}, D. D. McCarthy and W. E. Carter (eds), pp.
     197--201.

\item{}\kern-2pc
Boucher, C., Altamimi, Z., Feissel, M., and Sillard, P., 1996, {\it 
     Results and Analysis of the ITRF94}, IERS Technical Note 20, 
     Observatoire de Paris, Paris.

\item{}\kern-2pc
DeMets, C., Gordon, R. G., Argus, D. F., and Stein, S., 1994, ``Effect
     of recent revisions to the geomagnetic reversal time scale on 
     estimates of current plate motions," {\it Geophys. Res. Let.}, 
     {\bf 21}, pp. 2191--2194.

\item{}\kern-2pc
Ekman, M., 1989, ``Impacts of Geodetic Phenomena on Systems for
     Height and Gravity," {\it Bull. Geod.}, {\bf 63}, pp. 281--296.

\item{}\kern-2pc
{\it Geodesist's Handbook}, 1996, International Association of
     Geodesy, in press.

\item{}\kern-2pc
IGS, 1995, International GPS Service for Geodynamics 1994 Annual
     Report.  IGS Central Bureau, JPL, Pasadena.

\item{}\kern-2pc
Malys, S. and Slater, J. A., 1994,``Maintenance and enhancement of
     the World Geodetic System 1984," ION GPS 94, Salt Lake City,
     Utah.

\item{}\kern-2pc
Poutanen, M., Vermeer, M., and Mekinen, J., 1996, ``The Permanent
     Tide in GPS Positioning," {\it J. of Geod.}, {\bf 70}, pp.
     499--504.

\item{}\kern-2pc
Rapp, R. H., 1989, ``The treatment of permanent tidal effects in the
     analysis of satellite altimeter data for sea surface topography,"
     {\it Man. Geod.}, {\bf 14}, pp. 368--372.

\item{}\kern-2pc
Soler, T. and Hothem, L. D., 1989, ``Important Parameters Used in
     Geodetic Transformations," {\it J. Surv. Engrg.}, {\bf 115},
     pp. 414--417.

\vfil\eject
