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\write 0 {12. GENERAL RELATIVISTIC MODELS FOR PROPAGATION
\noexpand\dotfill\the\pageno\noexpand\break}
\noindent{\large CHAPTER 12}{\bf\ GENERAL RELATIVISTIC MODELS FOR
PROPAGATION}
\bigskip
\write 0 {\indent VLBI Time Delay \noexpand\dotfill\the\pageno\noexpand
\break}
\noindent{\bf VLBI Time Delay}
\nobreak
There have been many papers dealing with relativistic
effects which must be accounted for in VLBI processing; see
(Robertson, 1975), (Finkelstein {\it et al}., 1983), (Hellings, 1986),
(Pavlov, 1985), (Cannon {\it et al}., 1986), (Soffel {\it et al}.,
1986), (Zeller {\it et al}., 1986), (Sovers and Fanselow, 1987),
(Zhu and Groten, 1988), (Shahid-Saless {\it et al}., 1991), (Soffel
{\it et al}., 1991). As pointed out by Boucher (1986), the
relativistic correction models proposed in various articles are not
quite compatible. To resolve differences between the procedures
and to arrive at a standard model, a workshop was held at the U. S.
Naval Observatory on 12 October 1990. The proceedings of this
workshop have been published (Eubanks, 1991) and the model given
here is based on the consensus model resulting from that workshop.
Much of this chapter dealing with VLBI time delay is taken directly
from that work and the reader is urged to consult that publication
for further details. One change from that model has been made in
order to adopt the IAU/IUGG conventions for the scale of the
terrestrial reference system, in accord with 1991 resolutions
(see Appendix in McCarthy, 1992). Geodetic lengths should be
expressed in ``SI units," {\it i.e.}, be consistent with the
second as realized by a clock running at the TAI rate at sea
level. The only change needed to the 1992 formulation to satisfy
the IAU/IUGG Resolutions is to include the Earth's potential,
$U_\oplus$, in the total potential $U$ in the delay equation
(Equation 9). The only observable effect of this change will
be an increase of the VLBI terrestrial scale by 1.39385806 parts
in $10^9$.
As pointed out by Eubanks, the use of clocks running at the
geoid and delays calculated ``at the geocenter" ignoring the scale
change induced by the Earth's gravitational potential means that
terrestrial distances calculated from the consensus model will
not be the same as those calculated using the expressions in this
chapter which are equivalent to using meter sticks on the
surface of the Earth. The accuracy limit chosen for the consensus
VLBI relativistic delay model is $10^{-12}$ seconds (one picosecond)
of differential VLBI delay for baselines less than two Earth
radii in length. In the model all terms of order $10^{-13}$ seconds or
larger were included to ensure that the final result was accurate
at the picosecond level. Source coordinates derived from the
consensus model will be solar system barycentric and should have
no apparent motions due to solar system relativistic effects at
the picosecond level.
The consensus model was derived from a combination of five
different relativistic models for the geodetic delay. These are
the Masterfit/Modest model, due to Fanselow and Thomas (see
Treuhaft and Thomas, in (Eubanks, 1991), and (Sovers and
Fanselow, 1987)), the I. I. Shapiro model (see Ryan, in (Eubanks,
1991)), the Hellings-Shahid-Saless model (Shahid-Saless {\it et al}.,
1991) and in (Eubanks, 1991), the Soffel, Muller, Wu and Xu model
(Soffel {\it et al}., 1991) and in (Eubanks, 1991), and the Zhu-Groten
model (Zhu and Groten, 1988) and in (Eubanks, 1991). Baseline
results are expressed in ``local" or ``SI" coordinates appropriate
for clocks running at the TAI rate on the surface of the Earth,
to be consistent with the more general IERS conventions on the
terrestrial reference system scale. This means that the
gravitational potential of the Earth is now included in $U$; the
scale effects of the geocentric station velocities can still
be ignored as they will at most cause scale changes of order 1
part in $10^{12}$ (see Zhu and Groten, Soffel {\it et al.}, and
Fukushima, all in Eubanks (1991) and Shahid-Saless {\it et al.}
(1991) for further details on the implications of these choices).
As the time argument is now based on TAI, which is a quasi-local
time on the geoid, distance estimates from these conventions
will now be consistent in principle with ``physical" distances.
The model is designed for use in the reduction of VLBI
observations of extra-galactic objects acquired from the surface
of the Earth. The delay error caused by ignoring the annual
parallax is $>$ 1 psec for objects closer than several hundred
thousand light years, which includes all of the Milky Way galaxy.
The model is not intended for use with observations of sources in
the solar system, nor is it intended for use with observations
made from space-based VLBI, from either low or high Earth orbit,
or from the surface of the Moon (although it would be suitable
with obvious changes for observations made entirely from the
Moon).
It is assumed that the inertial reference frame is defined
kinematically and that very distant objects, showing no apparent
motion, are used to estimate precession and the nutation series.
This frame is not truly inertial in a dynamical sense, as included
in the precession constant and nutation series are the effects
of the geodesic precession ($\sim$ 19 milli arc seconds / year).
Soffel {\it et al}. (in Eubanks (1991)) and Shahid-Saless {\it et al}.
(1991) give details of a dynamically inertial VLBI delay equation. At
the picosecond level, there is no practical difference for VLBI
geodesy and astrometry except for the adjustment in the
precession constant.
Although the delay to be calculated is the time of arrival
at station 2 minus the time of arrival at station 1, it is the
time of arrival at station 1 that serves as the time reference
for the measurement. Unless explicitly stated otherwise, all
vector and scalar quantities are assumed to be calculated at $t_1$,
the time of arrival at station 1 including the effects of the
troposphere.
The notation follows that of Hellings (1986) and Hellings
and Shahid-Saless in Eubanks (1991) as closely as possible. It
is assumed that the standard IAU models for precession, nutation,
Earth rotation and polar motion have been followed and that all
geocentric vector quantities have thus been rotated into a nearly
non-rotating celestial frame. The errors in the standard IAU
models are negligible for the purposes of the relativistic
transformations. The notation itself is given in Table 12.1.
The consensus model separates the total delay into a classical
delay and a general relativistic delay, which are then modified
by relativistic transformations between geocentric and solar
system barycentric frames.
Table 12.1. Notation used in the model
\smallskip
\hrule
\item{$t_i$} the time of arrival of a radiointerferometric
signal at the $i^{th}$ VLBI receiver in terrestrial time (TAI)
\vskip-1pc
\item{$T_i$} the time of arrival of a radiointerferometric
signal at the $i^{th}$ VLBI receiver in barycentric time
(TCB or TDB)
\vskip-1pc
\item{$t_{g_i}$} the ``geometric" time of arrival of a
radiointerferometric signal at the $i^{th}$ VLBI receiver
including the gravitational ``bending" delay and the change in
the geometric delay caused by the existence of the atmospheric
propagation delay but neglecting the atmospheric propagation
delay itself
\vskip-1pc
\item{$t_{v_i}$} the ``vacuum" time of arrival of a
radiointerferometric signal at the $i^{th}$ VLBI receiver
including the gravitational delay
but neglecting the atmospheric propagation delay and the
change in the geometric delay caused by the existence of the
atmospheric propagation delay
\vskip-1pc
\item{$t_{i_J}$} the approximation to the time that the ray path
to station $i$ passed closest to gravitating body $J$
\vskip-1pc
\item{$\delta t_{atm_i}$} the atmospheric propagation delay
for the $i^{th}$ receiver $ = t_i - t_{g_i}$
\vskip-1pc
\item{$\Delta t_{grav}$} the differential gravitational time
delay, commonly known as the gravitational ``bending delay"
\vskip-1pc
\item{$\vec{x}_i(t_i)$} the geocentric radius vector
of the $i^{th}$ receiver at the geocentric time $t_i$
\vskip-1pc
\item{$\vec{b}$} $\vec{x}_2(t_1) - \vec{x}_1(t_1)$ and is thus
the geocentric baseline vector at the time of arrival $t_1$
\vskip-1pc
\item{$\vec{b}_0$} the {\it a priori} geocentric baseline vector
at the time of arrival $t_1$
\vskip-1pc
\item{$\delta\vec{b}$} $\vec{b}(t_1) - \vec{b}_0(t_1)$
\vskip-1pc
\item{$\vec{w}_i$} the geocentric velocity of the
$i^{th}$ receiver
\vskip-1pc
\item{$\hat{K}$} the unit vector from the barycenter to the
source in the absence of gravitational or aberrational bending
\vskip-1pc
\item{$\hat{k}_i$} the unit vector from the $i^{th}$
station to the source after aberration
\vskip-1pc
\item{$\vec{X}_i$} the barycentric radius vector of the
$i^{th}$ receiver
\vskip-1pc
\item{$\vec{X}_\oplus$} the barycentric radius vector of the
geocenter
\vskip-1pc
\item{$\vec{X}_J$} the barycentric radius vector of the
$J^{th}$ gravitating body
\vskip-1pc
\item{$\vec{R}_{i_J}$} the vector from the $J^{th}$
gravitating body to the $i^{th}$ receiver
\vskip-1pc
\item{$\vec{R}_{\oplus_J}$} the vector from the
$J^{th}$ gravitating body to the geocenter
\vskip-1pc
\item{$\vec{R}_{\oplus_\odot}$} the vector from the
Sun to the geocenter
\vskip-1pc
\item{$\hat{N}_{i_J}$} the unit vector from the $J^{th}$
gravitating body to the $i^{th}$ receiver
\vskip-1pc
\item{$\vec{V}_\oplus$} the barycentric velocity of the geocenter
\vskip-1pc
\item{$U$} the gravitational potential at the geocenter plus the
terrestrial potential at the surface of the Earth. At the
picosecond level, only the solar and terrestrial potentials
need be included in $U$ so that $U = GM_\odot/|\vec{R}_
{\oplus_\odot}| c^2 + GM_\oplus/a_\oplus c^2$
\vskip-1pc
\item{$M_i$} the mass of the $i^{th}$ gravitating body
\vskip-1pc
\item{$M_\oplus$} the mass of the Earth
\vskip-1pc
\item{$\gamma$} a PPN Parameter = 1 in general relativity
\vskip-1pc
\item{$c$} the speed of light in meters / second
\vskip-1pc
\item{$G$} the Gravitational Constant in Newtons meters$^2$
kilograms$^{-2}$
\vskip-1pc
\item{$a_\oplus$} the equatorial radius of the Earth
\line{\dotfill}
Vector magnitudes are expressed by the absolute value sign $ [|x|
= (\Sigma x_i^2)^{1\over 2}]$. Vectors and scalars expressed
in geocentric coordinates are denoted by lower case ({\it e.g.}
$\vec{x}$ and $t$), while quantities in barycentric coordinates
are in upper case ({\it e.g.} $\vec{X}$ and $T$).
MKS units are used throughout. For quantities such as V$_\oplus$,
$\vec{w}_i$, and $U$ it is assumed that a table (or
numerical formula) is available as a function of TAI and that they
are evaluated at the atomic time of reception at station 1, $t_1$,
unless explicitly stated otherwise. A lower case subscript
({\it e.g.} $\vec{x}_i$) denotes a particular VLBI receiver,
while an upper case subscript ({\it e.g.} $\vec{x}_J$)
denotes a particular gravitating body.
\hrule
\bigskip
\write 0 {\indent \indent Gravitational Delay \noexpand\dotfill
\the\pageno\noexpand\break}
\noindent\underbar{Gravitational Delay}
\nobreak
The general relativistic delay, $\Delta t_{grav}$, is given
for the $J^{th}$ gravitating body by
$$\Delta t_{grav_J} = (1+ \gamma){{GM_J} \over {c^3}}
\ln{{|\vec{R}_{1_J}| + \vec{K}\cdot
\vec{R}_{1_J}}\over{|\vec{R}_{2_J}| +
\vec{K}\cdot\vec{R}_{2_J}}}.\eqno(1)$$
At the picosecond level it is possible to simplify the delay
due to the Earth, $\Delta t_{grav_\oplus}$ , which becomes
$$\Delta t_{grav_\oplus} = (1+ \gamma){GM_\oplus
\over {c^3}}\ln{{|\vec{x}_1| + \vec{K}\cdot
\vec{x}_1}\over{|\vec{x}_2| + \vec{K}\cdot\vec{x}_2}}.\eqno(2)$$
The Sun, the Earth and Jupiter must be
included, as well as the other planets in the solar system along
with the Earth's Moon, for which the maximum delay change is
several picoseconds. The major satellites of Jupiter, Saturn and
Neptune should also be included if the ray path passes close to
them. This is very unlikely in normal geodetic observing but may
occur during planetary occultations.
The effect on the bending delay of the motion of the
gravitating body during the time of propagation along the ray path is
small for the Sun but can be several hundred picoseconds for
Jupiter (see Sovers and Fanselow (1987) page 9). Since this
simple correction, suggested by Sovers and Fanselow (1987) and
Hellings (1986) among others, is sufficient at the picosecond
level, it was adapted for the consensus model. It is also
necessary to account for the motion of station 2 during the
propagation time between station 1 and station 2. In this model
$\vec{R}_{i_J}$, the vector from the $J^{th}$
gravitating body to the $i^{th}$ receiver, is iterated once, giving
$${t}_{1_J} = \min[t_1,t_1-\hat{K}\cdot
(\vec{X}_J(t_1)-\vec{X}_1(t_1))],\eqno(3)$$
so that
$$\vec{R}_{1_J}(t_1) = \vec{X}_1(t_1)-\vec{X}_J(t_{1_J}),\eqno(4)$$
and
$$\vec{R}_{2_J} = \vec{X}_2(t_1) - {{\vec{V}_\oplus}\over{c}}
(\hat{K}\cdot\vec{b}_0) - \vec{X}_J(t_{1_J}).\eqno(5)$$
Only this one iteration is needed to obtain picosecond level
accuracy for solar system objects. If more accuracy is required,
it is probably better to use the rigorous approach of Shahid-Saless
{\it et al}. (1991). $\vec{X}_1(t_1)$ is not tabulated,
but can be inferred from $\vec{X}_\oplus(t_1)$ using
$$\vec{X}_i(t_1) = \vec{X}_\oplus(t_1) +\vec{x}_i(t_1),\eqno(6)$$
which is of sufficient accuracy for use in equations 3, 4,
and 5, when substituted into equation 1 but not for use in computing
the geometric delay. The total gravitational delay is the sum over
all gravitating bodies including the Earth,
$$\Delta t_{grav} = \sum_J \Delta t_{grav_J}.\eqno(7)$$
\write 0 {\indent \indent Geometric Delay \noexpand\dotfill\the
\pageno\noexpand\break}
\noindent\underbar{Geometric Delay}
\nobreak
In the barycentric frame the vacuum delay equation is, to a
sufficient level of approximation:
$$T_2-T_1 = -{1\over c} \hat{K}\cdot(\vec{X}_2(T_2)-\vec{X}_1(T_1))+
\Delta t_{grav}.\eqno(8)$$
This equation is converted into a geocentric delay equation using
known quantities by performing the relativistic transformations
relating the barycentric vectors $\vec{X}_i$ to the corresponding
geocentric vectors $\vec{x}_i$, thus converting equation 8 into an
equation in terms of $\vec{x}_i$. The related transformation between
barycentric and geocentric time can be used to derive another
equation relating $T_2-T_1$ and $t_2- t_1$, and these two equations
can then be solved for the geocentric delay in terms of the
geocentric baseline vector $\vec{b}$.
The papers by Soffel {\it et al}. in Eubanks (1991), Hellings and
Shahid-Saless in Eubanks (1991), Zhu and Groten (1988) and
Shahid-Saless {\it et al}. (1991) give details of the derivation of the
vacuum delay equation. To conserve accuracy and simplify the
equations the delay was expressed as much as is possible in terms
of a rational polynomial. In the rational polynomial form the
total geocentric vacuum delay is given by
$$t_{v_2}-t_{v_1}={{\Delta t_{grav}-{{\hat{K}\cdot\vec{b}_0}\over
{c}}[1-(1+\gamma)U-{{|\vec{V}_\oplus|^2}\over{2c^2}}-{{\vec{V}_\oplus
\cdot\vec{w}_2}\over{c^2}}]-{{\vec{V}_\oplus\cdot\vec{b}_0}\over
{c^2}}(1+\hat{K}\cdot\vec{V}_\oplus/2c)}\over{1+{{\hat{K}\cdot
(\vec{V}_\oplus+\vec{w}_2)}\over{c}}}}.\eqno(9)$$
Given this expression for the vacuum delay, the total
delay is found to be
$$t_2-t_1=t_{v_2}-t_{v_1}+(\delta t_{atm_2}-\delta t_{atm_1})+
\delta t_{atm_1}{{\hat{K}\cdot(\vec{w}_2-\vec{w}_1)}\over c}.\eqno(10)$$
For convenience the total delay can be divided into separate
geometric and propagation delays. The geometric delay is given by
$$t_{g_2}-t_{g_1}=t_{v_2}-t_{v_1}+
\delta t_{atm_1}{{\hat{K}\cdot(\vec{w}_2-\vec{w}_1)}\over c},\eqno(11)$$
and the total delay can be found at some later time
by adding the propagation delay:
$$t_2-t_1=t_{g_2}-t_{g_1}+(\delta t_{atm_2}-\delta t_{atm_1}).
\eqno(12)$$
The tropospheric propagation delay in equations 11 and 12
need not be from the same model. The estimate in equation 12
should be as accurate as possible, while the $\delta t_{atm}$ model in
equation 11 need only be accurate to about an air mass
($\sim$ 10 nanoseconds). If equation 10 is used instead, the model
should be as accurate as is possible.
If the difference, $\delta\vec{b}$, between the {\it a priori}
baseline vector $\vec{b_0}$ used in equation 9 and the true baseline
vector is less than roughly three meters, then it suffices to add
$-(\hat{K}\cdot\delta\vec{b})/c$ to $t_2-t_1$.
If this is not the case, however, the delay must be modified by
adding
$$\Delta(t_{g_2}-t_{g_1})=-{{{\hat{K}\cdot\delta\vec{b}_0}\over c}\over
{1+{{\hat{K}\cdot(\vec{V}_\oplus+\vec{w}_2)}\over c }}}-{{\vec{V}
_\oplus\cdot\delta\vec{B}}\over{c^2}}\eqno(13)$$
to the total time delay $t_2-t_1$ from equation 10 or 12.
\write 0 {\indent \indent Observations Close to the Sun \noexpand
\dotfill\the\pageno\noexpand\break}
\noindent\underbar{Observations Close to the Sun}
\nobreak
For observations made very close to the Sun, higher order
relativistic time delay effects become increasingly important. The
largest correction is due to the change in delay caused by the
bending of the ray path by the gravitating body described in
Richter and Matzner (1983) and Hellings (1986). The change to
$\Delta t_{grav}$ is
$$\delta t_{grav_i}={{(1+\gamma)^2G^2M_i^2}\over{c^5}}{{\vec{b}
\cdot(\hat{N}_{1_i}+\hat{K})}\over{(|\vec{R}|_{1_i}+\vec{R}_{1_i}
\cdot\hat{K})^2}},\eqno(14)$$
which should be added to the $\Delta t_{grav}$ in equation 1.
\write 0 {\indent \indent Summary \noexpand\dotfill\the\pageno
\noexpand\break}
\noindent\underbar{Summary}
\nobreak
Assuming that time $t_1$ is the Atomic (TAI) time of reception
of the VLBI signal at receiver 1, the following steps are
recommended to correct the VLBI time delay for relativistic effects.
\item{1.} Use equation 6 to estimate the barycentric station vector
for receiver 1.
\item{2.} Use equations 3, 4, and 5 to estimate the vectors from the
Sun, the Moon, and each planet except the Earth to receiver
1.
\item{3.} Use equation 1 to estimate the differential gravitational
delay for each of those bodies.
\item{4.} Use equation 2 to find the differential gravitational delay
due to the Earth.
\item{5.} Sum to find the total differential gravitational delay.
\item{6.} Add $\Delta t_{grav}$ to the rest of the {\it a priori} vacuum
delay from equation 9.
\item{7.} Calculate the aberrated source vector for use in the
calculation of the tropospheric propagation delay:
$$\vec{k}_i=\hat{K}+{{\vec{V}_\oplus+\vec{w}_i}\over c}-
\hat{K}{{\hat{K}\cdot(\vec{V}_\oplus+\vec{w}_i)}\over c}.\eqno(15)$$
\item{8.} Add the geometric part of the tropospheric propagation delay
to the vacuum delay, equation 11.
\item{9.} The total delay can be found by adding the best estimate of
the tropospheric propagation delay
$$t_2-t_1=t_{g_2}-t_{g_1}+[\delta t_{atm_2}(t_1-{\hat{K}\cdot
\vec{b}_0\over c},\vec{k}_2)-\delta t_{atm_1}(\vec{k}_1)].\eqno(16)$$
\item{10.} If necessary, apply equation 13 to correct for ``post-model"
changes in the baseline by adding equation 13 to the total
time delay from equation step 9.
\write 0 {\indent Propagation Correction for Laser Ranging \noexpand
\dotfill\the\pageno\noexpand\break}
\noindent{\bf Propagation Correction for Laser Ranging}
The space-time curvature near a massive body requires a
correction to the Euclidean computation of range, $\rho$. This
correction in seconds, $\Delta t$, is given by (Holdridge, 1967)
$$\Delta t={(1+\gamma)GM\over c^3}
\ln\biggl({R_1+R_2+\rho\over R_1+R_2-\rho}\biggr),\eqno(17)$$
\noindent where
\item{$c$} = speed of light,
\vskip-1pc
\item{$\gamma$} = PPN parameter equal to 1 in General Relativity,
\vskip-1pc
\item{$R_1$} = distance from the body's center to the beginning of the
light path,
\vskip-1pc
\item{$R_2$} = distance from the body's center to the end of the light
path,
\vskip-1pc
\item{$GM$} = gravitational parameter of the deflecting body.
\noindent For near-Earth satellites, working in the geocentric frame of
reference, the only body to be considered is the Earth (Ries
{\it et al.}, 1989). For lunar laser ranging, which is
formulated in the solar system barycentric reference frame, the
Sun and the Earth must be considered (Newhall {\it et al.}, 1987).
In the computation of the instantaneous space-fixed positions
of a station and a lunar reflector in the analysis of LLR
data, the body-centered coordinates of the two sites are affected
by a scale reduction and a Lorentz contraction effect (Martin
{\it et al.}, 1985). The scale effect is about 15 cm in
the height of a tracking station, while the maximum value of the
Lorentz effect is about 3 cm. The equation for the transformation
of $\vec{r}$, the geocentric position vector of a station expressed
in the geocentric frame, is
$$\vec{r}_b=\vec{r}\biggl(1-{\gamma\Phi\over c^2}\biggr)+{1\over 2}
\biggl({\vec{V}\cdot\vec{r}\over c^2}\biggr)\vec{V},\eqno(18)$$
\noindent where
\item{$\vec{r}_b$} = station position expressed in the barycentric
frame,
\vskip-1pc
\item{$\Phi$} = gravitational potential at the geocenter (excluding the
Earth's mass),
\vskip-1pc
\item{$\vec{V}$} = barycentric velocity of the Earth,
\noindent A similar equation applies to the selenocentric reflector
coordinates; the maximum value of the Lorentz effect is about 1 cm
(Newhall {\it et al.}, 1987).
\noindent{\bf References}
\nobreak
\item{}\kern-2pc
Boucher, C., 1986, ``Relativistic effects in Geodynamics" in
{\it Relativity in Celestial Mechanics and Astrometry}, J.
Kovalevsky and V. A. Brumberg (eds), pp. 241--253.
\item{}\kern-2pc
Cannon, W. H., Lisewski, D., Finkelstein, A. M., Kareinovich, and V.
Ya, 1986, ``Relativistic Effects in Earth Based and Cosmic
Long Baseline Interferometry," {\it ibid}., pp. 255--268.
\item{}\kern-2pc
Eubanks, T. M., ed, 1991, {\it Proceedings of the U. S. Naval
Observatory Workshop on Relativistic Models for Use in Space
Geodesy}, U. S. Naval Observatory, Washington, D. C.
\item{}\kern-2pc
Finkelstein, A. M., Kreinovitch, V. J., and Pandey, S. N., 1983,
``Relativistic Reductions for Radiointerferometric Observables,"
{\it Astrophys. Space Sci}., {\bf 94}, pp. 233--247.
\item{}\kern-2pc
Hellings, R. W., 1986, ``Relativistic effects in Astronomical
Timing Measurements," {\it Astron. J.}, {\bf 91}, pp. 650--659.
Erratum, {\it ibid}., p. 1446.
\item{}\kern-2pc
Holdridge, D. B., 1967, {\it An Alternate Expression for Light Time
Using General Relativity}, JPL Space Program Summary 37-48,
III, pp. 2--4.
\item{}\kern-2pc
Martin, C. F., Torrence, M. H., and Misner, C. W., 1985, ``Relativistic
Effects on an Earth Orbiting Satellite in the Barycentric
Coordinate System," {\it J. Geophys. Res.}, {\bf 90}, p. 9403.
\item{}\kern-2pc
McCarthy, D. D., 1992, IERS Standards, IERS Technical Note 13,
Observatoire de Paris, Paris.
\item{}\kern-2pc
Newhall, X X, Williams, J. G., and Dickey, J. O., 1987, ``Relativity
Modelling in Lunar Laser Ranging Data Analysis," in
{\it Proceedings of the International Association of Geodesy
(IAG) Symposia}, Vancouver, pp. 78--82.
\item{}\kern-2pc
Pavlov, B. N., 1985, ``On the Relativistic Theory of Astrometric
Observations. III. Radio Interferometry of Remote Sources,"
{\it Soviet Astronomy}, {\bf 29}, pp. 98--102.
\item{}\kern-2pc
Richter, G. W. and Matzner, R. A., 1983, ``Second-order Contributions
to Relativistic Time Delay in the Parameterized Post-Newtonian
Formalism," {\it Phys. Rev. D}, {\bf 28}, pp. 3007--3012.
\item{}\kern-2pc
Ries, J. C., Huang, C., and Watkins, M. M., 1988, ``The Effect of
General Relativity on Near-Earth Satellites in the Solar
System Barycentric and Geocentric Reference Frames," {\it Phys.
Rev. Let.}, {\bf 61}, pp. 903--906.
\item{}\kern-2pc
Robertson, D. S., 1975, ``Geodetic and Astrometric Measurements
with Very-Long-Baseline Interferometry," Ph. D. Thesis,
Massachusetts Institute of Technology.
\item{}\kern-2pc
Shahid-Saless, B., Hellings, R. W., and Ashby, N., 1991, ``A Picosecond
Accuracy Relativistic VLBI Model via Fermi Normal Coordinates,"
{\it Geophys. Res. Let.}, {\bf 18}, pp. 1139--1142.
\item{}\kern-2pc
Soffel, M., Ruder H., Schneider, M., Campbell, J., and Schuh, H.,
1986, ``Relativistic Effects in Geodetic VLBI Measurements,"
in {\it Relativity in Celestial Mechanics and Astrometry}, J.
Kovalevsky and V. A. Brumberg (eds), pp. 277--282.
\item{}\kern-2pc
Soffel M. H., Muller, J., Wu, X., and Xu, C., 1991, ``Consistent
Relativistic VLBI Theory with Picosecond Accuracy," {\it Astron.
J.}, {\bf 101}, pp. 2306--2310.
\item{}\kern-2pc
Sovers, O. J. and Fanselow, J. L., 1987, {\it Observation Model and
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\item{}\kern-2pc
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\item{}\kern-2pc
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\vfil\eject