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\write 0 {6. GEOPOTENTIAL \noexpand\dotfill\the\pageno
\noexpand\break}
\noindent{\large CHAPTER 6} {\bf GEOPOTENTIAL}
\bigskip
The recommended geopotential field is the JGM-3 model (Tapley
{\it et al.}, 1995). The $GM_\oplus$ and a$_{\rm{e}}$ values reported
with JGM-3 (398600.4415 km$^3$/s$^2$ and 6378136.3 m) should be used as
scale parameters with the geopotential coefficients. The recommended
$GM_\oplus$ = 398600.4418 should be used with the two-body term when
working with SI units (398600.4415 or 398600.4356 should be used by
those still working with TDT or TDB units, respectively). Although
the JGM-3 model is given with terms through degree and order 70, only
terms through degree and order twenty are required for Lageos.
Values for the $C_{21}$ and $S_{21}$ coefficients are included
in the JGM-3 model. The $C_{21}$ and $S_{21}$
coefficients describe the position of the Earth's figure axis. When
averaged over many years, the figure axis should closely coincide
with the observed position of the rotation pole averaged over the
same time period. Any differences between the mean figure and mean
rotation pole averaged would be due to long-period fluid motions in
the atmosphere, oceans, or Earth's fluid core (Wahr, 1987, 1990).
At present, there is no independent evidence that such motions are
important. The JGM-3 values for
$C_{21}$ and $S_{21}$ give a mean figure axis that corresponds to
the mean pole position recommended in Chapter 3 Terrestrial Reference
Frame.
This choice for $C_{21}$ and $S_{21}$ is realized as follows. First,
to use the geopotential coefficients to solve for a satellite
orbit, it is necessary to rotate from the Earth-fixed frame,
where the coefficients are pertinent, to an inertial frame, where
the satellite motion is computed. This transformation between
frames should include polar motion. We assume the polar motion
parameters used are relative to the IERS Reference Pole. If $\bar{x}$
and $\bar{y}$ are the angular displacements of the Terrestrial
Reference Frame described in Chapter 3 relative to the IERS Reference
Pole, then the values
$\bar{C}_{21}=\sqrt{3}\bar{x}\bar{C}_{20}$,
$\bar{S}_{21}=-\sqrt{3}\bar{y}\bar{C}_{20}$,
\noindent where $\bar{x} = 0.223\times 10^{-6} $ radians
(equivalent to 0.046 arcsec) and $\bar{y} = 1.425\times 10^{-6} $
radians (equivalent to 0.294 arcsec) (Nerem {\it et al.}, 1994)
are those used in the geopotential model, so that the mean
figure axis coincides with the pole described in Chapter 3.
This gives normalized coefficients of
$\bar{C}_{21}({\rm IERS}) = -0.187\times 10^{-9}$,
$\bar{S}_{21}({\rm IERS}) = 1.195\times 10^{-9}$.
\noindent JGM-3 is available via ftp at ftp.csr.utexas.edu on the
directory pub/grav in file JGM3.GEO.Z. It can also be accessed by
World Wide Web at http://www.csr.utexas.edu by clicking the ``library
of data files" selection.
\write 0 {\indent Effect of Solid Earth Tides
\noexpand\dotfill\the\pageno\noexpand\break}
\noindent{\bf Effect of Solid Earth Tides}
\nobreak
The changes induced by the solid Earth tides in the free space
potential are most conveniently modeled as variations in the standard
geopotential coefficients $C_{nm}$ and $S_{nm}$ (Eanes {\it et al}.,
1983). The contributions $\Delta C_{nm}$ and $\Delta S_{nm}$ from
the tides are expressible in terms of the $k$ Love number. The effects
of ellipticity and rotation of the Earth on tidal deformations
necessitates the use, in general, of three $k$ parameters,
$k_{nm}^{(0)}$ and $k_{nm}^{(\pm)}$, to characterize the changes
produced in the free space potential by tides of spherical harmonic
degree and order $(nm)$ (Wahr, 1981). Within the diurnal tidal band,
for $(mn)=(21)$, these parameters have a strong frequency dependence
due to the Nearly Diurnal Free Wobble resonance. Anelasticity of
the mantle causes $k_{nm}^{(0)}$ and $k_{nm}^{(\pm)}$ to acquire
small imaginary parts (reflecting a phase lag in the deformational
response of the Earth to tidal forces), and also gives rise to a
further variation with frequency which is particularly pronounced
within the long period band. Though modeling of anelasticity at
the periods relevant to tidal phenomena (8 hours to 18.6 years) is
not yet definitive, it is clear that the magnitudes of the
contributions from anelasticity cannot be ignored (see below).
Consequently the anelastic Earth model is recommended for use
in precise data analysis.
The degree 2 tides produce time dependent changes in $C_{2m}$
and $S_{2m}$, through $k_{2m}^{(0)}$, which can exceed $10^{-8}$ in
magnitude. They also produce changes exceeding a cutoff of
$3\times 10^{-12}$ in $C_{4m}$ and $S_{4m}$ through $k_{2m}^{(+)}$.
(The direct contributions of the degree 4 tidal potential to these
coefficients are negligible.) The only other changes exceeding this
cutoff are in $C_{3m}$ and $S_{3m}$, produced by the degree 3 part
of the tide generating potential.
The computation of the tidal contributions to the geopotential
coefficients is most efficiently done by a two-step procedure. In
Step 1, the $(2m)$ part of the tidal potential is evaluated in the
time domain for each $m$ using lunar and solar ephemerides, and the
corresponding changes $\Delta C_{2m}$ and $\Delta S_{2m}$ are
computed using frequency independent nominal values $k_{2m}$ for the
respective $k_{2m}^{(0)}$. The contributions of the degree 3 tides
to $C_{3m}$ and $S_{3m}$ through $k_{3m}^{(0)}$ and also those of the
degree 2 tides to $C_{4m}$ and $S_{4m}$ through $k_{2m}^{(+)}$ may
be computed by a similar procedure; they are at the level of $10^{-11}$.
Step 2 corrects for the deviations of the $k_{21}^{(0)}$ of
several of the constituent tides of the diurnal band from the constant
nominal value $k_{21}$ assumed for this band in the first step. Similar
corrections need to be applied to a few of the constituents of the
other two bands also.
With frequency-independent values $k_{nm}$ ({\bf Step 1}), changes
induced by the $(nm)$ part of the tide generating potential in the
normalized geopotential coefficients having the same $(nm)$
are given in the time domain by
$$\Delta\bar{C}_{nm}-i\Delta\bar{S}_{nm}= {k_{nm}\over {2n+1}}
\sum_{j=2}^3{GM_j\over GM_\oplus}\Bigl({R_e \over r_j}\Bigr)^{n+1}
\bar{P}_{nm} (\sin\Phi_j)e^{-im\lambda_j}\eqno(1)$$
\noindent (with $S_{n0}=0$), where
\item{$k_{nm}$} = nominal degree Love number for degree $n$ and order $m$,
\item{$R_e$} = equatorial radius of the Earth,
\item{$GM_\oplus$} = gravitational parameter for the Earth,
\item{$GM_j$} = gravitational parameter for the Moon ($j=2$) and
Sun ($j=3$),
\item{$r_j$} = distance from geocenter to Moon or Sun,
\item{$\Phi_j$} = body fixed geocentric latitude of Moon or Sun,
\item{$\lambda_j$} = body fixed east longitude (from Greenwich) of Moon or Sun,
\noindent and $\bar{P}_{nm}$ is the normalized associated Legendre function
related to the classical (unnormalized) one by
$$\bar{P}_{nm}= N_{nm} {P}_{nm},\eqno(2a)$$
\noindent where
$$N_{nm}=\sqrt{(n-m)!(2n+1)(2-\delta_{om})\over (n+m)!}.\eqno(2b)$$
\noindent Correspondingly, the normalized geopotential coefficients
($\bar{C}_{nm},\bar{S}_{nm}$) are related to the unnormalized
coefficients ($C_{nm}, S_{nm}$) by
$$C_{nm}=N_{nm}\bar{C}_{nm},\quad S_{nm}=N_{nm}\bar{S}_{nm}.\eqno(3)$$
Equation (1) yields $\Delta \bar{C}_{nm}$ and $\Delta \bar{S}_{nm}$
for both
$n=2$ and $n=3$ for all $m$, apart from the corrections for frequency
dependence to be evaluated in Step 2. (The particular case $(nm) =
(20)$ needs special consideration, however, because it includes a
time-independent part which will be discussed below in the section
on the permanent tide.)
One further computation to be done in Step 1 is that of the
changes in the degree 4 coefficients produced by the
degree 2 tides. They are given by
$$\Delta\bar{C}_{4m}-i\Delta\bar{S}_{4m}= {k_{2m}^{(+)}\over 5}
\sum_{j=2}^3{GM_j\over GM_\oplus}\Bigl({R_e \over r_j}\Bigr)^3
\bar{P}_{2m} (\sin\Phi_j)e^{-im\lambda_j},\quad (m=0,1,2),\eqno(4)$$
\noindent which has the same form as Equation (1) for $n=2$ except for
the replacement of $k_{2m}$ by $k_{2m}^{(+)}$.
The parameter values for the computations of Step 1 are given in Table
6.1. The choice of these nominal values
(which are complex for $m=1$ and $m=2$ in the anelastic case)
has been made so as to minimize the number of terms for which
corrections will have to be applied in Step 2.
The nominal value for $m=0$ has to be chosen real because there
is no closed expression for the contribution to $\bar{C}_{20}$
from the imaginary part of $k^{(0)}_{20}$. The frequency dependent
values for use in Step 2 are taken from the results of computations
by Mathews and Buffett (private communication) using the PREM
elastic Earth model with the ocean layer replaced by solid, and
for the evaluation of anelasticity effects, the Widmer {\it et al.}
(1991) model of mantle $Q$. As in Wahr and Bergen (1986), a power
law was assumed for the frequency dependence of $Q$ with 200 $s$
as the reference period; the value $\alpha=0.15$ was used for the
power law index. The anelasticity contribution (out of phase and
in phase) to the tidal changes in the geopotential coefficients is
at the level of one to two percent in-phase, and half to one percent
out-of-phase, {\it i.e.}, of the order of $10^{-10}$.
\vfil\eject
\noindent Table 6.1. Nominal values of solid Earth tide external
potential Love numbers.
\bigskip
\hrule
\medskip
\halign{\kern1pc#&\kern1pc#&\kern1pc#\hfil&\kern1pc#\hfil&\kern1pc#\hfil
&\kern1pc#\hfil&\kern1pc#\hfil\cr
\noalign{\medskip}
& & \hfil{Elastic Earth}\hfil\span\omit
& \hfil{Anelastic Earth}\hfil\span\omit\span\omit\cr
\noalign {\medskip}
$n$ & $m$ & \hfil$k_{nm}$&$\hfil k^+_{nm}$&{\rm Re\ }$k_{nm}$
&{\rm Im\ }$k_{nm}$&$k^+_{nm}$\cr
\noalign{\medskip}
2 & 0 & 0.29525 & $-$0.00087&0.30190&$-0.00000$&$-$0.00089\cr
2 & 1 & $0.29470$ & $-$0.00079&0.29830$$&$-0.00144$&$-$0.00080\cr
2 & 2 & 0.29801 & $-$0.00057&0.30102&$-$0.00130&$-0.00057$\cr
3 & 0 & 0.093&$\cdots$&&&\cr
3 & 1 & 0.093&$\cdots$&&&\cr
3 & 2 & 0.093&$\cdots$&&&\cr
3 & 3 & 0.094&$\cdots$&&&\cr}
\medskip
\hrule
The frequency dependence corrections to the $\Delta \bar{C}_{nm}$
and $\Delta \bar{S}_{nm}$ values obtained from Step 1 are computed
in {\bf Step 2} as the sum of contributions from a number of tidal
constituents belonging to the respective bands. The contribution
to $\Delta\bar{C}_{20}$ from the long period tidal constituents of
various frequencies $f$ is
$${\rm Re} \sum_{f(2,0)}(A_0\delta k_f H_f \,e^{i\theta_f})
= \sum_{f(2,0)}(A_0 H_f (\delta k^R_f\cos\theta_f -
\delta k^i_f\sin\theta_f), \eqno(5a)$$
while the contribution to $(\Delta\bar{C}_{21}-i\Delta\bar{S}_{21})$
from the diurnal tidal constituents and to
$\Delta\bar{C}_{22}-i\Delta\bar{S}_{22}$ from the semidiurnals are given by
$$\Delta\bar{C}_{2m}-i\Delta\bar{S}_{2m}= \eta_m \sum_{f(2,m)}
(A_m\delta k_f H_f) \,e^{i\theta_f}, \quad (m=1,2),\eqno(5b)$$
\noindent where
$$A_0 = {1\over R_e\sqrt{4\pi}} = 4.4228\times 10^{-8}\ {\rm m}^{-1},
\eqno(5c)$$
$$ A_m = {(-1)^m\over R_e\sqrt{8\pi}}= (-1)^m (3.1274 \times 10^{-8})
\ {\rm m}^{-1}, \qquad (m\neq 0), \eqno(5d)$$
$$ \eta_1 = -i, \eta_2 = 1, \eqno(5e)$$
\item{$\delta k_f$} = difference between $k_f\equiv k_{2m}^{(0)}$ at
frequency $f$ and the nominal value
$k_{2m}$, in the sense $k_f-k_{2m}$,
\item{$\delta k^R_f$} = real part, and $\delta k^I_f$ = imaginary part,
of $\delta k_f$,
\item{$H_f$} = amplitude (m) of the term at frequency $f$ from the
harmonic expansion of the tide generating potential, defined according
to the convention of Cartwright and Tayler (1971), and
\item{$\theta_f$} = $\bar{n}\cdot\bar{\beta}=\sum_{i=1}^6n_i\beta_i$,
\qquad or \qquad $\theta_f=m (\theta_g+\pi)-\bar{N}\cdot\bar{F}=m
(\theta_g+\pi)- \sum_{j=1}^5 N_jF_j $,
\noindent where
\item{$\bar{\beta}$} = six-vector of Doodson's fundamental arguments
$\beta_i$, ($\tau, s, h, p, N', p_s$),
\item{$\bar{n}$} = six-vector of multipliers $n_i$ (for the term
at frequency $f$) of the fundamental arguments,
\item{$\bar{F}$} = five-vector of fundamental arguments $F_j$
(the Delaunay variables $l,l',F,D,\Omega$) of nutation
theory,
\item{$\bar{N}$} = five-vector of multipliers $N_i$ of the Delaunay
variables for the nutation of frequency $-f+d\theta_g/dt$,
\noindent and $\theta_g$ is the Greenwich Mean Sidereal Time expressed
in angle units ({\it i.e.} $24^h = 360^\circ$; see Chapter 5).
\noindent ($\pi$ in $(\theta_g + \pi)$ is now to be replaced by 180.)
\noindent For the fundamental arguments ($l,l',F,D,\Omega$) of nutation
theory and the convention followed here in choosing their multipliers
$N_j$, see Chapter 5. For conversion of tidal amplitudes defined
according to different conventions to the amplitude $H_f$ corresponding
to the Cartwright-Tayler convention, use Table 6.4 given at the end of
this Chapter.
The correction due to the $K_1$ constituent, for example, is obtained
as follows, given that $A_m = A_1 = -3.1274\times 10^{-8}$,
$H_f=0.36871$, and $\theta_f = (\theta_g+\pi)$ for this tide. If
anelasticity is ignored, $(k_{21}^{(0)})_{K_1}=0.25377$, and the
nominal value chosen is 0.29470. Hence $\delta k_f$ is
$0.25377-0.29470 = -0.04093$, and $A_m (\delta k)_f H_f$
reduces to $472.0\times 10^{-12}$. The corrections to the (21)
coefficients then become
\smallskip
\halign{#\hfil&#\hfil\cr
${(\Delta\bar{C}_{21})}_{K_1}$&$= 472.0\times 10^{-12}\sin (\theta_g+\pi),$ \cr
${(\Delta\bar{S}_{21})}_{K_1}$&$= 472.0\times 10^{-12}\cos (\theta_g+\pi).$
\cr}
\noindent With anelasticity included, $(k_{21}^{(0)})_{K_1} =
0.25745-i\,0.00148 $, and on choosing the nominal value as
$(0.29830 - i\,0.00144)$ one obtains the corrections to the
coefficients by replacing $\delta k_f$ in the above calculation by
$(-0.04085-i\,0.00004)$.
In general, if $\delta k_f = \delta k_f^R+i\delta k_f^I$,
\smallskip
\halign{#\hfil&#\hfil\cr
${(\Delta\bar{C}_{2m})}_{K_1}$&$= A_m H_f \ (\delta k_f^R \sin \theta_f
+ \delta k_f^I \cos \theta_f),$ \cr
${(\Delta\bar{S}_{2m})}_{K_1}$&$= A_m H_f \ (\delta k_f^R \cos \theta_f
- \delta k_f^I \sin \theta_f).$ \cr}
Table 6.2a lists the results for all tidal terms which contribute
$10^{-13}$ or more, after round-off, to the $(nm)=(21)$ geopotential
coefficient. A cutoff at this level is used for the individual terms
in order that accuracy at the level of $3\times 10^{-12}$ be not
affected by the accumulated contributions from the numerous smaller
terms that are disregarded. The imaginary parts of the contributions
are below cutoff (except for $K_1$, as given above) and are not
listed. Results relating to the (20) and (22) coefficients are
presented in Tables (6.2b) and (6.2c), respectively.
\bigskip
\noindent Table 6.2a. Amplitudes ($A_1 \delta k_f H_f$) of the
corrections for frequency dependence of $k_{21}^{(0)}$, taking the
nominal value $k_{21}$ for the diurnal tides as 0.29470 for the
elastic case, and $(0.29830 -i\,0.00144)$ for the anelastic case.
Units: $10^{-12}$. Multipliers of the Doodson arguments identifying
the tidal terms are given, as also those of the Delaunay variables.
\bigskip
\hrule
\bigskip
\halign{\hfil#\hfil&\hfil#\hfil&\hfil#\hfil\kern.7pc&
\hfil#\kern.2pc&\hfil#\kern.2pc&\hfil#\kern.2pc&\hfil#\kern.2pc&
\hfil#\kern.2pc&\hfil#\kern.7pc&\hfil#\kern.2pc&\hfil#\kern.2pc&
\hfil#\kern.2pc&\hfil#\kern.2pc&\hfil#\kern.5pc&\hfil#\kern.5pc&
\hfil#\kern.5pc&\hfil#\kern.5pc&\hfil#\cr
Name & deg/hr & Doodson & $\tau$& $s$& $h$& $p$& $N'$& $p_s$&
$\ell $& $\ell'$&$F$& $D$& $\Omega$& $\delta k_f^{el}$\ \ & Amp.&
$\delta k_f^{anel}$& Amp.\cr
& & No. & & & & & & & & & & & & & elas.& & anel.\cr
\noalign{\medskip}
& 13.39645 & 135,645 & 1&-2& 0& 1&-1& 0& 1& 0& 2& 0& 1& -0.00044& -0.1& -0.00045& -0.1\cr
Q$_1$ & 13.39866 & 135,655 & 1&-2& 0& 1& 0& 0& 1& 0& 2& 0& 2& -0.00044& -0.7& -0.00046& -0.7\cr
$\rho_1$ & 13.47151 & 137,455 & 1&-2& 2&-1& 0& 0&-1& 0& 2& 2& 2& -0.00047& -0.1& -0.00049& -0.1\cr
& 13.94083 & 145,545 & 1&-1& 0& 0&-1& 0& 0& 0& 2& 0& 1& -0.00081& -1.2& -0.00082& -1.3\cr
O$_1$ & 13.94303 & 145,555 & 1&-1& 0& 0& 0& 0& 0& 0& 2& 0& 2& -0.00081& -6.6& -0.00082& -6.7\cr
N$\tau_1$ & 14.41456 & 153,655 & 1& 0&-2& 1& 0& 0& 1& 0& 2&-2& 2& -0.00167& 0.1& -0.00168& 0.1\cr
LK$_1$ & 14.48741 & 155,455 & 1& 0& 0&-1& 0& 0&-1& 0& 2& 0& 2& -0.00193& 0.4& -0.00193& 0.4\cr
NO$_1$ & 14.49669 & 155,655 & 1& 0& 0& 1& 0& 0& 1& 0& 0& 0& 0& -0.00196& 1.3& -0.00197& 1.3\cr
& 14.49890 & 155,665 & 1& 0& 0& 1& 1& 0& 1& 0& 0& 0& 1& -0.00197& 0.2& -0.00198& 0.3\cr
$\chi_1$ & 14.56955 & 157,455 & 1& 0& 2&-1& 0& 0&-1& 0& 0& 2& 0& -0.00231& 0.3& -0.00231& 0.3\cr
$\pi_1$ & 14.91787 & 162,556 & 1& 1&-3& 0& 0& 1& 0& 1& 2&-2& 2& -0.00834& -1.9& -0.00832& -1.9\cr
& 14.95673 & 163,545 & 1& 1&-2& 0&-1& 0& 0& 0& 2&-2& 1& -0.01114& 0.5& -0.01111& 0.5\cr
P$_1$ & 14.95893 & 163,555 & 1& 1&-2& 0& 0& 0& 0& 0& 2&-2& 2& -0.01135& -43.3& -0.01132& -43.2\cr
S$_1$ & 15.00000 & 164,556 & 1& 1&-1& 0& 0& 1& 0& 1& 0& 0& 0& -0.01650& 2.0& -0.01642& 2.0\cr
& 15.03886 & 165,545 & 1& 1& 0& 0&-1& 0& 0& 0& 0& 0&-1& -0.03854& -8.8& -0.03846& -8.8\cr
K$_1$ & 15.04107 & 165,555 & 1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& -0.04093& 472.0& -0.04085& 471.0\cr
& 15.04328 & 165,565 & 1& 1& 0& 0& 1& 0& 0& 0& 0& 0& 1& -0.04365& 68.3& -0.04357& 68.2\cr
& 15.04548 & 165,575 & 1& 1& 0& 0& 2& 0& 0& 0& 0& 0& 2& -0.04678& -1.6& -0.04670& -1.6\cr
$\psi_1$ & 15.08214 & 166,554 & 1& 1& 1& 0& 0&-1& 0&-1& 0& 0& 0& 0.23083& -20.8& 0.22609& -20.4\cr
$\phi_1$ & 15.12321 & 167,555 & 1& 1& 2& 0& 0& 0& 0& 0&-2& 2&-2& 0.03051& -5.0& 0.03027& -5.0\cr
$\theta_1$& 15.51259 & 173,655 & 1& 2&-2& 1& 0& 0& 1& 0& 0&-2& 0& 0.00374& -0.5& 0.00371& -0.5\cr
J$_1$ & 15.58545 & 175,455 & 1& 2& 0&-1& 0& 0&-1& 0& 0& 0& 0& 0.00329& -2.1& 0.00325& -2.1\cr
& 15.58765 & 175,465 & 1& 2& 0&-1& 1& 0&-1& 0& 0& 0& 1& 0.00327& -0.4& 0.00324& -0.4\cr
SO$_1$ & 16.05697 & 183,555 & 1& 3&-2& 0& 0& 0& 0& 0& 0&-2& 0& 0.00198& -0.2& 0.00195& -0.2\cr
OO$_1$ & 16.13911 & 185,555 & 1& 3& 0& 0& 0& 0& 0& 0&-2& 0&-2& 0.00187& -0.7& 0.00184& -0.6\cr
& 16.14131 & 185,565 & 1& 3& 0& 0& 1& 0& 0& 0&-2& 0&-1& 0.00187& -0.4& 0.00184& -0.4\cr}
\bigskip
\hrule
\bigskip
\noindent Table 6.2b.
Corrections for frequency dependence of $k_{20}^{(0)}$ of the zonal
tides due to anelasticity. Units: $10^{-12}$. The nominal value $k_{20}$
for the zonal tides is taken as $0.30190$. The real and imaginary
parts $\delta k_f^{R}$ and $\delta k_f^{I}$ of $\delta k_f$ are listed,
along with the corresponding in phase ($ip$) amplitude $(A_0 H_f \delta
k_f^R)$ and out of phase ($op$) amplitude $(-A_0 H_f \delta k_f^I)$ to
be used in equation (5a). In the elastic case, $k_{20}^{(0)}= 0.29525$
for all the zonal tides, and no second step corrections are needed.
\bigskip
\hrule
\bigskip
\halign{\hfil#\hfil&\hfil#\hfil\kern.7pc&\hfil#\hfil\kern.7pc&
\hfil#\kern.2pc&\hfil#\kern.2pc&\hfil#\kern.2pc&\hfil#\kern.2pc&
\hfil#\kern.2pc&\hfil#\kern.7pc&
\hfil#\kern.2pc&\hfil#\kern.2pc&\hfil#\kern.2pc&\hfil#\kern.2pc&
\hfil#\kern.5pc&
\hfil#\kern.5pc&\hfil#\kern.5pc&\hfil#\kern.5pc&\hfil#\cr
Name & Doodson & deg/hr & $\tau$& $s$& $h$& $p$& $N'$& $p_s$&
$\ell $& $\ell'$&$F$& $D$& $\Omega$&$\delta k_f^{R}$\ \ \ & Amp.&
$\delta k_f^{I}$\ \ \ & Amp.\cr
& No. & & & & & & & & & & & & & & ($ip$)& & ($op$)\cr
\noalign{\medskip}
& 55,565 & 0.00221 & 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 1& 0.01347 & 16.6 & -0.00541 & -6.7\cr
& 55,575 & 0.00441 & 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 2& 0.01124 & -0.1 & -0.00488 & 0.1\cr
$S_a$ & 56,554 & 0.04107 & 0& 0& 1& 0& 0&-1& 0&-1& 0& 0& 0& 0.00547 & -1.2 & -0.00349 & 0.8\cr
$S_{sa}$ & 57,555 & 0.08214 & 0& 0& 2& 0& 0& 0& 0& 0&-2& 2&-2& 0.00403 & -5.5 & -0.00315 & 4.3\cr
& 57,565 & 0.08434 & 0& 0& 2& 0& 1& 0& 0& 0&-2& 2&-1& 0.00398 & 0.1 & -0.00313 & -0.1\cr
& 58,554 & 0.12320 & 0& 0& 3& 0& 0&-1& 0&-1&-2& 2&-2& 0.00326 & -0.3 & -0.00296 & 0.2\cr
$M_{sm}$ & 63,655 & 0.47152 & 0& 1&-2& 1& 0& 0& 1& 0& 0&-2& 0& 0.00101 & -0.3 & -0.00242 & 0.7\cr
& 65,445 & 0.54217 & 0& 1& 0&-1&-1& 0&-1& 0& 0& 0&-1& 0.00080 & 0.1 & -0.00237 & -0.2\cr
$M_m$ & 65,455 & 0.54438 & 0& 1& 0&-1& 0& 0&-1& 0& 0& 0& 0& 0.00080 & -1.2 & -0.00237 & 3.7\cr
& 65,465 & 0.54658 & 0& 1& 0&-1& 1& 0&-1& 0& 0& 0& 1& 0.00079 & 0.1 & -0.00237 & -0.2\cr
& 65,655 & 0.55366 & 0& 1& 0& 1& 0& 0& 1& 0&-2& 0&-2& 0.00077 & 0.1 & -0.00236 & -0.2\cr
$M_{sf}$ & 73,555 & 1.01590 & 0& 2&-2& 0& 0& 0& 0& 0& 0&-2& 0& -0.00009 & 0.0 & -0.00216 & 0.6\cr
& 75,355 & 1.08875 & 0& 2& 0&-2& 0& 0&-2& 0& 0& 0& 0& -0.00018 & 0.0 & -0.00213 & 0.3\cr
$M_f$ & 75,555 & 1.09804 & 0& 2& 0& 0& 0& 0& 0& 0&-2& 0&-2& -0.00019 & 0.6 & -0.00213 & 6.3\cr
& 75,565 & 1.10024 & 0& 2& 0& 0& 1& 0& 0& 0&-2& 0&-1& -0.00019 & 0.2 & -0.00213 & 2.6\cr
& 75,575 & 1.10245 & 0& 2& 0& 0& 2& 0& 0& 0&-2& 0& 0& -0.00019 & 0.0 & -0.00213 & 0.2\cr
$M_{stm}$& 83,655 & 1.56956 & 0& 3&-2& 1& 0& 0& 1& 0&-2&-2&-2& -0.00065 & 0.1 & -0.00202 & 0.2\cr
$M_{tm}$ & 85,455 & 1.64241 & 0& 3& 0&-1& 0& 0&-1& 0&-2& 0&-2& -0.00071 & 0.4 & -0.00201 & 1.1\cr
& 85,465 & 1.64462 & 0& 3& 0&-1& 1& 0&-1& 0&-2& 0&-1& -0.00071 & 0.2 & -0.00201 & 0.5\cr
$M_{sqm}$& 93,555 & 2.11394 & 0& 4&-2& 0& 0& 0& 0& 0&-2&-2&-2& -0.00102 & 0.1 & -0.00193 & 0.2\cr
$M_{qm}$ & 95,355 & 2.18679 & 0& 4& 0&-2& 0& 0&-2& 0&-2& 0&-2& -0.00106 & 0.1 & -0.00192 & 0.1\cr}
\bigskip
\hrule
\bigskip
\noindent Table 6.2c. Amplitudes ($A_2\delta k_f H_f$) of the
corrections for frequency dependence of $k_{22}^{(0)}$, taking the
nominal value $k_{22}$ for the sectorial tides as 0.29801 for the
elastic case, and $(0.30102 -i\,0.00130)$ for the anelastic case.
Units: $10^{-12}$. The corrections are only to the real part, and are
the same in both the elastic and the anelastic cases.
\bigskip
\hrule
\bigskip
\halign{\hfil#\hfil&\hfil#\hfil\kern.7pc&\hfil#\hfil\kern.7pc&
\hfil#\kern.2pc&\hfil#\kern.2pc&\hfil#\kern.2pc&\hfil#\kern.2pc&
\hfil#\kern.2pc&\hfil#\kern.7pc&
\hfil#\kern.2pc&\hfil#\kern.2pc&\hfil#\kern.2pc&\hfil#\kern.2pc&
\hfil#\kern.5pc&
\hfil#\kern.5pc&\hfil#\cr
Name & Doodson No.& deg/hr & $\tau$& $s$& $h$& $p$& $N'$& $p_s$&
$\ell $& $\ell'$&$F$& $D$& $\Omega$&
$\delta k_f^{R}$\ \ \ & Amp.\cr
\noalign{\medskip}
$N_2$ & 245,655 & 28.43973 & 2&-1& 0& 1& 0& 0& 1& 0& 2& 0& 2& 0.00006 & -0.3\cr
$M_2$ & 255,555 & 28.98410 & 2& 0& 0& 0& 0& 0& 0& 0& 2& 0& 2& 0.00004 & -1.2\cr}
\bigskip
\hrule
\bigskip
The total variation in geopotential coefficient $\bar C_{20}$ is
obtained by adding to the result of Step 1 the sum of the contributions
from the tidal constituents listed in Table 6.2b computed using
equation (5a). The tidal variations in $\bar{C}_{2m}$ and
$\bar{S}_{2m}$ for the other $m$ are computed similarly, except that
equation (5b) is to be used together with Table 6.2a for $m=1$ and
Table 6.2c for $m=2$.
\write 0 {\indent Solid Earth Pole Tide \noexpand\dotfill\the\pageno
\noexpand\break}
\noindent {\bf Solid Earth Pole Tide}
The pole tide is generated by the centrifugal effect of polar motion,
characterized by the potential
$$\Delta V = -(\Omega^2 R_e^2/2) \sin 2 \theta (x_p \cos\lambda-y_p \sin
\lambda).$$
(See the section on Deformation due to Polar Motion in Chapter 7 for
further details). The deformation which constitutes this tide
produces a perturbation $k_2\Delta V$ in the external potential which
is equivalent to changes in the geopotential coefficients $C_{21}$ and
$S_{21}$. Using for $k_2$ the elastic Earth value 0.2977
appropriate to the polar tide yields
$$\Delta\bar{C}_{21} = -1.290 \times 10^{-9}(x_p),$$
$$\Delta\bar{S}_{21} = 1.290 \times 10^{-9}(y_p),$$
\noindent where $x_p$ and $y_p$ are in seconds of arc as defined in
Chapter 7. For the anelastic Earth, $k_2$ has real and imaginary parts
$k_2^R = 0.3111$ and $k_2^I = -0.0035$, leading to
$$\Delta\bar{C}_{21} = -1.348 \times 10^{-9}(x_p+0.0112 y_p),$$
$$\Delta\bar{S}_{21} = 1.348 \times 10^{-9}(y_p-0.0112 x_p).$$
\bigskip
\write 0 {\indent Treatment of the Permanent Tide \noexpand\dotfill
\the\pageno\noexpand\break}
\noindent{\bf Treatment of the Permanent Tide}
\nobreak
The degree 2 zonal tide generating potential has a mean (time average)
value which is nonzero. This permanent (time independent) potential
produces a permanent deformation which is reflected in the static
figure of the Earth, and a corresponding time independent contribution
to the geopotential which can be considered as part of the adopted
value of $\bar{C}_{20}$, as in the JGM-3 model. Therefore, for
$(nm)=(20)$, the zero frequency part should be excluded from the
expression (1). Hereafter the symbol $\Delta \bar{C}_{20}$ is reserved
for the temporally varying part of the tidal contribution to
$\bar{C}_{20}$; the expression (1) for $(mn)=(20)$ will be redesignated
as $\bar{C}^*_{20}$.
$$\Delta\bar{C}^*_{20}={k_{20}\over {5}}
\sum_{j=2}^3{GM_j\over GM_\oplus}\left({R_e\over r_j}\right)^3
\bar{P}_{20}(\sin\Phi_j).$$
\noindent Its zero frequency part is
$$ \langle\Delta\bar{C}_{20}\rangle=
A_0 H_0 k_{20} = (4.4228\times 10^{-8})(-0.31460) k_{20}. \eqno(6)$$
\noindent To represent the tide induced changes in the (20)
geopotential, one should then use only the time variable part
$$\Delta\bar{C}_{20}= \Delta\bar{C}^*_{20}- \langle\Delta\bar{C}_{20}
\rangle. \eqno(7)$$
In evaluating it, the same value should be used for $k_{20}$ in both
$\Delta\bar{C}^*_{20}$ and $\langle\Delta\bar{C}_{20}\rangle$. If the
elastic Earth value $k_{20} = 0.29525$ is used, $ \langle\Delta\bar{C}_
{20} \rangle= -4.108\times 10^{-9}$, while with the value $k_{20} =
0.30190$ of the anelastic case, $ \langle\Delta\bar{C}_{20} \rangle=
-4.201\times 10^{-9}$.
The restitution of the indirect effect of the permanent tide is done
to be consistent with the XVIII IAG General Assembly Resolution 16;
but to obtain the effect of the permanent tide on the geopotential,
one can use the same formula as equation (6) using the fluid limit
Love number which is $k=0.94$.
\write 0 {\indent Effect of the Ocean Tides \noexpand\dotfill
\the\pageno\noexpand\break}
\noindent{\bf Effect of the Ocean Tides}
\nobreak
The dynamical effects of ocean tides are most easily incorporated
by periodic variations in the normalized Stokes' coefficients. These
variations can be written as
$$\Delta\bar{C}_{nm}-i\Delta\bar{S}_{nm}=F_{nm}\sum_{s(n,m)}\sum_+^-
(C_{snm}^\pm\mp iS_{snm}^\pm)e^{\pm i\theta_f},\eqno(8)$$
where
$$F_{nm}={4\pi G\rho_w\over g}\sqrt{(n+m)!\over(n-m)!(2n+1)
(2-\delta_{om})}\Bigg({1+k'_n\over2n+1}\Bigg),$$
\noindent $g$ and $G$ are given in Chapter 4,
{$\rho_{w}$} = density of seawater = 1025 kg m$^{-3}$,
{$k'_n$} = load deformation coefficients ($k'_2= -0.3075,
k'_3= -0.195, k'_4= -0.132, k'_5= -0.1032, \
\break \hphantom {-0.08}
k'_6= -0.0892)$,
{$C_{snm}^\pm, S_{snm}^\pm$} = ocean tide coefficients (m) for
the tide constituent $s$
{$\theta_s$} = argument of the tide constituent $s$ as defined
in the solid tide model (Chapter 7).
The summation over + and - denotes the respective addition of
the retrograde waves using the top sign and the prograde waves using
the bottom sign. The $C_{snm}^\pm$ and $S_{snm}^\pm$ are the
coefficients of a spherical harmonic decomposition of the ocean
tide height for the ocean tide due to the constituent $s$ of the
tide generating potential.
For each constituent $s$ in the diurnal and semi-diurnal tidal bands,
these coefficients were obtained from the CSR 3.0 ocean tide height
model (Eanes {\it et al.}, 1996), which was estimated from the TOPEX/
Poseidon satellite altimeter data. For each constituent $s$ in the
long period band, the self-consistent equilibrium tide model
of Ray and Cartwright (1994) was used. The list of constituents
for which the coefficients were determined was obtained from
the Cartwright and Tayler (1971) expansion of the tide raising
potential.
These ocean tide height harmonics are related to the Schwiderski
convention (Schwiderski, 1983) according to
$$C_{snm}^\pm-iS_{snm}^\pm= -i\hat{C}_{snm}^\pm e^{i(\epsilon_{snm}^
\pm+\chi_s)},\eqno(9)$$
where
\item{$\hat{C}_{snm}^\pm$} = ocean tide amplitude for constituent $s$
using the Schwiderski notation,
\item{$\epsilon_{snm}^\pm$} = ocean tide phase for constituent $s$,
\noindent and $\chi_s$ is obtained from Table 6.3, with $H_s$ being
the Cartwright and Tayler (1971) amplitude at frequency $s$.
\bigskip
\noindent Table 6.3. Values of $\chi_s$ for long-period, diurnal and
semidiurnal tides.
\bigskip
\hrule
\bigskip
\halign{#\hfil\kern1pc&\hfil#\hfil\kern1pc&\hfil#\hfil\cr
$\underline{\rm Tidal\ Band}$&$\underline{H_s>0}$&$\underline{H_s<0}$\cr
Long Period & $\pi$ & $0$\cr
Diurnal & $\pi\over 2$ & $-{\pi\over 2}$\cr
Semidiurnal & $0$ & $\pi$\cr}
\bigskip
\hrule
\bigskip
For clarity, the terms in equation 1 are repeated in both conventions:
$$\Delta\bar{C}_{nm}=F_{nm}\sum_{s(n,m)}[(C_{snm}^++C_{snm}^-)
\cos\theta_s+(S_{snm}^++S_{snm}^-)\sin\theta_s]\eqno(10a)$$
or
$$\Delta\bar{C}_{nm}=F_{nm}\sum_{s(n,m)}[\hat{C}_{snm}^+\sin(\theta_s+
\epsilon_{snm}^++\chi_s)+\hat{C}_{snm}^-\sin(\theta_s+\epsilon_{snm}^-
+\chi_s)],\eqno(10b)$$
$$\Delta\bar{S}_{nm}=F_{nm}\sum_{s(n,m)}[(S_{snm}^+-S_{snm}^-)
\cos\theta_s-(C_{snm}^+-C_{snm}^-)\sin\theta_s]\eqno(10c)$$
or
$$\Delta\bar{S}_{nm}=F_{nm}\sum_{s(n,m)}[\hat{C}_{snm}^+\cos(\theta_s+
\epsilon_{snm}^++\chi_s)-\hat{C}_{snm}^-\cos(\theta_s+\epsilon_{snm}^-
+\chi_s)].\eqno(10d)$$
The orbit element perturbations due to ocean tides can be loosely
grouped into two classes. The resonant perturbations arise from
coefficients for which the order ($m$) is equal to the first Doodson's
argument multiplier $n_1$ of the tidal constituent $s$ (See Note), and
have periodicities from a few days to a few years. The non-resonant
perturbations arise when the order $m$ is not equal to index $n_1$. The
most important of these are due to ocean tide coefficients for which
$m=n_1+1$ and have periods of about 1 day.
Certain selected constituents ({\it e.g.} S$_{\rm a}$ and S$_2$) are
strongly affected by atmospheric mass distribution (Chapman and
Lindzen, 1970). The resonant harmonics (for $m=n_1$) for some of
these constituents were determined by their combined effects on the
orbits of several satellites. These multi-satellite values then
replaced the corresponding values from the CSR3.0 altimetric ocean
tide height model.
Based on the predictions of the linear perturbation theory outlined in
Casotto (1989), the relevant tidal constituents and spherical harmonics
were selected for several geodetic and altimetric satellites. For
geodetic satellites, both resonant and non-resonant perturbations were
analyzed,whereas for altimetric satellites, only the non-resonant
perturbations were analyzed. For the latter, the adjustment of
empirical parameters during orbit determination removes the errors in
modeling resonant accelerations. The resulting selection of ocean tidal
harmonics was then merged into a single recommended ocean tide force
model. With this selection the error of omission on TOPEX is
approximately 5 $mm$ along-track, and for Lageos it is 2 $mm$
along-track. The recommended ocean tide harmonic selection is
available via anonymous ftp from ftp.csr.utexas.edu.
For high altitude geodetic satellites like Lageos, in order to reduce
the required computing time, it is recommended that out of the complete
selection, only the constituents whose Cartwright and Tayler amplitudes
$H_s$ is greater than 0.5 $mm$ be used, with their spherical harmonic
expansion terminated at maximum degree and order 8. The omission errors
from this reduced selection on Lageos is estimated at approximately 1 cm
in the transverse direction for short arcs.
\noindent NOTE:
The Doodson variable multipliers ($\bar{n}$) are coded
into the argument number (A) after Doodson (1921) as:
$$A = n_1(n_2+5)(n_3+5) . (n_4+5)(n_5+5)(n_6+5).$$
\noindent {\bf Conversion of tidal amplitudes defined according to
different conventions}
The definition used for the amplitudes of tidal terms in the recent
high-accuracy tables differ from each other and from Cartwright and
Tayler (1971). Hartmann and Wenzel (1995) tabulate amplitudes in
units of the potential (m$^2 $s$^{-2}$), while the amplitudes of
Roosbeek (1996), which follow the Doodson (1921) convention, are
dimensionless. To convert them to the equivalent tide heights $H_f$
of the Cartwright-Tayler convention, multiply by the appropriate
factors from Table 6.4. The following values are used for the
constants appearing in the conversion factors: Doodson constant $D_1$
= 2.63358352855 m$^2$\ s$^{-2};\ $$g_e\equiv g$ at the equatorial
radius = 9.79828685 (from $GM=3.986004415 \times 10^{14}$ m$^3
$\ s$^{-2}$, $ R_e=6378136.55$ m).
\bigskip
\noindent Table 6.4 Factors for conversion to Cartwright-Tayler
amplitudes from those defined according to Doodson's and Hartmann and
Wenzel's conventions
\bigskip
\hrule
\bigskip
\halign{#\hfil\kern1pc&#\hfil\kern1pc\cr
From Doodson &From Hartmann \& Wenzel \cr
\noalign {\medskip}
$ f_{20}=- {{\sqrt{4 \pi}} \over{\sqrt{5}}} {{D_1} \over{g_e}} = -0.426105 $ &
$ f'_{20}= {{2 \sqrt{\pi}} \over{g_e}} = 0.361788 $ \cr
$ f_{21}=- {{2 \sqrt{24 \pi}} \over{3 \sqrt{5}}} {{D_1} \over{g_e}} = -0.695827 $ &
$ f'_{21}=- {{\sqrt{8 \pi}} \over{g_e}} = -0.511646 $ \cr
$ f_{22}= {{\sqrt{96 \pi}} \over{3 \sqrt{5}}} {{D_1} \over{g_e}} = 0.695827 $ &
$ f'_{22}= {{\sqrt{8 \pi}} \over{g_e}} = 0.511646 $ \cr
$ f_{30}=- {{\sqrt{20 \pi}} \over{\sqrt{7}} } {{D_1} \over{g_e}} = -0.805263 $ &
$ f'_{30}= {{2 \sqrt{\pi}} \over{g_e}} = 0.361788 $ \cr
$ f_{31}=- {{\sqrt{720 \pi}} \over{8 \sqrt{7}}} {{D_1} \over{g_e}} = -0.603947 $ &
$ f'_{31}=- {{\sqrt{8 \pi}} \over{g_e}} = -0.511646 $ \cr
$ f_{32}= {{\sqrt{1440 \pi}} \over{10 \sqrt{7}}} {{D_1} \over{g_e}} = 0.683288 $ &
$ f'_{32}= {{\sqrt{8 \pi}} \over{g_e}} = 0.511646 $ \cr
$ f_{33}=- {{\sqrt{2880 \pi}} \over{15 \sqrt{7}}} {{D_1} \over{g_e}} = -0.644210 $ &
$ f'_{33}=- {{\sqrt{8 \pi}} \over{g_e}} = -0.511646 $ \cr}
\noindent{\bf References}
\nobreak
\item{}\kern-2pc
Cartwright, D. E. and Tayler, R. J., 1971, ``New Computations of
the Tide-Generating Potential," {\it Geophys. J. Roy. Astron.
Soc.}, {\bf 23}, pp. 45--74.
\item{}\kern-2pc
Casotto, S., 1989, ``Ocean Tide Models for TOPEX Precise Orbit
Determination," Ph.D. Dissertation, The Univ. of Texas at Austin.
\item{}\kern-2pc
Chapman, S. and Lindzen, R., 1970, {\it Atmospheric Tides}, D. Reidel,
Dordrecht.
\item{}\kern-2pc
Doodson, A. T., 1921, ``The Harmonic Development of the Tide-Generating
Potential," {\it Proc. R. Soc. A.}, {\bf 100}, pp. 305--329.
\item{}\kern-2pc
Eanes, R. J., Schutz, B., and Tapley, B., 1983, ``Earth and Ocean
Tide Effects on Lageos and Starlette," in {\it Proceedings of the
Ninth International Symposium on Earth Tides}, J. T. Kuo (ed),
E. Sckweizerbart'sche Verlagabuchhandlung, Stuttgart.
\item{}\kern-2pc
Eanes, R. J. and Bettadpur, S. V., 1996, ``The CSR 3.0 global ocean
tide model," in preparation.
\item{}\kern-2pc
Hartmann, T. and Wenzel, H.-G., 1995, ``The HW95 Tidal Potential
Catalogue," {\it Geophys. Res. Lett.}, {\bf 22}, pp. 3553--3556.
\item{}\kern-2pc
Mathews, P. M., Buffett, B. A., and Shapiro,. I. I., 1995, ``Love
numbers for a rotating spheroidal Earth: New definitions and
numerical values," {\it Geophys. Res. Lett.}, {\bf 22}, pp.
579--582.
\item{}\kern-2pc
Nerem, R. S., Lerch, F. J., Marshall, J. A., Pavlis, E. C., Putney,
B. H., Tapley, B. D., Eanes, R. J., Ries, J. C., Schutz, B. E.,
Shum, C. K., Watkins, M. M., Klosko, S. M., Chan, J. C., Luthcke,
S. B., Patel, G. B., Pavlis, N. K., Williamson, R. G., Rapp, R. H.,
Biancale, R., Nouel, F., 1994, ``Gravity Model Development for
TOPEX/POSEIDON: Joint Gravity Models 1 and 2," {\it J. Geophys.
Res.}, {\bf 99}, pp. 24421--24447.
\item{}\kern-2pc
Ray, R. D. and Cartwright, D. E., 1994, ``Satellite altimeter
observations of the $M_f$ and $M_m$ ocean tides, with simultaneous
orbit corrections," {\it Gravimetry and Space Techniques
Applied to Geodynamics and Ocean Dynamics}, Geophysical
Monograph 82, IUGG Volume 17, pp. 69--78.
\item{}\kern-2pc
Roosbeek, F., 1996, ``RATGP95: a harmonic development of the
tide-generating potential using an analytical method,"
{\it Geophys. J. Int.}, {\bf 126}, pp. 197--204.
\item{}\kern-2pc
Schwiderski, E., 1983, ``Atlas of Ocean Tidal Charts and Maps,
Part I: The Semidiurnal Principal Lunar Tide M2," {\it Marine
Geodesy}, {\bf 6}, pp. 219--256.
\item{}\kern-2pc
Tapley, B. D., M. M. Watkins, J. C. Ries, G. W. Davis, R. J. Eanes,
S. R. Poole, H. J. Rim, B. E. Schutz, C. K. Shum, R. S. Nerem,
F. J. Lerch, J. A. Marshall, S. M. Klosko, N. K. Pavlis,
and R. G. Williamson, 1996, ``The Joint Gravity Model 3,"
{\it J. Geophys. Res}, {\bf 101}, pp. 28029--28049.
\item{}\kern-2pc
Wahr, J. M., 1981, ``The Forced Nutations of an Elliptical, Rotating,
Elastic, and Oceanless Earth," {\it Geophys. J. Roy. Astron.
Soc.}, {\bf 64}, pp. 705--727.
\item{}\kern-2pc
Wahr, J., 1987, ``The Earth's C$_{21}$ and S$_{21}$ gravity
coefficients and the rotation of the core," {\it Geophys. J.
Roy. Astr. Soc.}, {\bf 88}, pp. 265--276.
\item{}\kern-2pc
Wahr, J., 1990, ``Corrections and Update to `The Earth's C$_{21}$ and
S$_{21}$ gravity coefficients and the rotation of the core',"
{\it Geophys. J. Int.}, {\bf 101}, pp. 709--711.
\item{}\kern-2pc
Wahr, J. and Z. Bergen, 1986, ``The effects of mantle elasticity on
nutations, Earth tides, and tidal variations in the rotation rate"
{\it Geophys. J. R. Astr. Soc.}, 633--668.
\item{}\kern-2pc
Widmer, R., G. Masters, and F. Gilbert, 1991, ``Spherically symmetric
attenuation within the Earth from normal mode data'', {\it Geophys.
J. Int.}, {\bf 104}, pp. 541--553.
\vfil\eject