\magnification = 1095
\vsize = 9.0 true in
\hsize = 6.7 true in
\voffset = .5 true in
\baselineskip = 1 pc
\parskip = 1 pc
\parindent = 2 pc
\font\large = cmr10 scaled\magstep2
\setbox0 =\hbox{"\kern-.15cm{.}\kern.04cm}
\def\qd{\copy0\/}

\write 0 {7.   SITE DISPLACEMENT \noexpand\dotfill\the\pageno
\noexpand\break}
\noindent{\large CHAPTER 7}  {\bf \ SITE DISPLACEMENT}
\bigskip
\write 0 {\indent Ocean Loading \noexpand\dotfill\the\pageno\noexpand
\break}
\noindent {\bf Local Site Displacement due to Ocean Loading}

Local site displacement is understood as an effect of (visco-) elastic
deformation of the Earth in response to time-varying surface loads. 
The reference point of zero deformation is the joint mass center of the
solid Earth and the load, while the sites are attached to the solid 
Earth.  This convention implies that the rigid body translation 
of the solid Earth that counterbalances the motion of the load's mass 
center is not contained in the local displacement model.  This 
convention follows strictly Farrell (1972).

\write 0 {\indent Ocean Loading \noexpand\dotfill\the\pageno\noexpand
\break}
\noindent {\bf Ocean Loading}

\nobreak

Three dimensional site displacements due to ocean tide loading are 
computed using the following scheme. Let $\Delta c$ denote a 
displacement component (radial, west, south) at a particular site and 
time $t$. Let $W$ denote the tide generating potential ({\it e.g.} 
Tamura, 1987; Cartwright and Tayler, 1971; Cartwright and Edden, 1973),
$$W=g\sum_jK_jP_2^{m_j}(\cos\psi)\cos(\omega_jt+\chi_j+m_j\lambda),\eqno(1)$$
where only degree two harmonics are retained. The symbols designate 
colatitude $\psi$, longitude $\lambda$, tidal angular velocity 
$\omega_j$, amplitude $K_j$ and the astronomical argument $\chi_j$ at 
$t=0^h$. Spherical harmonic order $m_j$ distinguishes the fundamental 
bands, {\it i. e.} long-period ($m=0$), diurnal ($m=1$) and 
semi-diurnal ($m=2$). The parameters $K_j$ and $\omega_j$ are used to 
obtain the most completely interpolated form
$$\Delta c=\sum_ja_{cj}\cos(\omega_jt+\chi_j-\phi_{cj}), \eqno(2)$$
with
$$a_{cj}\cos\phi_{cj}=K_j\biggl[{A_{ck}\cos\Phi_{ck}\over \bar{K}_k}
(1-p)+ {A_{c,k+1}\cos\Phi_{c,k+1}\over \bar{K}_{k+1}}p\biggr],$$
$$a_{cj}\sin\phi_{cj}=K_j\biggl[{A_{ck}\sin\Phi_{ck}\over \bar{K}_k}
(1-p)+ {A_{c,k+1}\sin\Phi_{c,k+1}\over \bar{K}_{k+1}}p\biggr].$$

For each site, the amplitudes $A_{ck}$ and phases $\Phi_{ck}, 1\leq k
\leq 11,$ are taken from Table 7.1. For clarity symbols written with 
bars overhead designate tidal potential quantities associated with the 
small set of partial tides represented in the table. These are the 
semi-diurnal waves $M_2, S_2, N_2, K_2,$ the diurnal waves $K_1, O_1, 
P_1, Q_1,$ and the long-period waves $M_f, M_m,$ and $S_{sa}.$ 

Interpolation is possible only within a fundamental band, {\it i.e.} 
we demand
$$\bar{m}_k=m_j=\bar{m}_{k+1}.\eqno(3)$$ Then  
$$p={\omega_j-\bar{\omega}_k\over\bar{\omega}_{k+1}-\bar{\omega}_k},
\qquad \bar{\omega}_k\leq\omega_j \leq \bar{\omega}_{k+1}.$$
If no $\bar{\omega}_k$ or $\bar{\omega}_{k+1}$ can be found meeting 
(3), $p$ is set to zero or one, respectively. 

A shorter form of (2) is obtained if the summation considers only the 
tidal species of Table 7.1 and corrections for the modulating effect 
of the lunar node. Then,  
$$\Delta c=\sum_jf_jA_{cj}\cos(\omega_jt+\chi_j+u_j-\Phi_{cj}),\eqno(4)$$
where $f_j$ and $u_j$ depend on the longitude of the lunar node
according to Table 26 of Doodson (1928). The astronomical arguments 
needed in (4) can be computed with subroutine ARG below. 

\bigskip
\hrule
\bigskip
{\tt
\settabs\+C\ \ &99&.&\     \  &99&.&\cr
\+C&    SUBROUTINE ARG(IYEAR,DAY,ANGLE)\cr
\+C\cr
\+C&  COMPUTES THE ANGULAR ARGUMENT WHICH DEPENDS ON TIME FOR 11\cr
\+C&  TIDAL ARGUMENT CALCULATIONS\cr
\+C\cr
\+C\cr
\+C&    ORDER OF THE 11 ANGULAR QUANTITIES IN VECTOR ANGLE\cr
\+C\cr
\+C&  01-M2 &&&&  02-S2\cr
\+C&  03-N2 &&&&  04-K2\cr
\+C&  05-K1 &&&&  06-O1\cr
\+C&  07-P1 &&&&  08-Q1\cr
\+C&  09-Mf &&&&  10-Mm\cr
\+C&  11-Ssa\cr
\+C\cr
\+C&  TAKEN FROM 'TABLE 1 CONSTANTS OF MAJOR TIDAL MODES'\cr
\+C&       WHICH DR. SCHWIDERSKI SENDS ALONG WITH HIS TAPE OF TIDAL\cr
\+C&       AMPLITUDES AND PHASES\cr
\+C\cr
\+C\cr
\+C&  INPUT--\cr
\+C\cr
\+C&     IYEAR - EX. 79 FOR 1979\cr
\+C&     DAY - DAY OF YEAR GREENWICH TIME\cr
\+C&                      EXAMPLE 32.5 FOR FEB 1   12 NOON\cr
\+C&                              1.25 FOR JAN 1    6 AM\cr
\+C\cr
\+C&  OUTPUT--\cr
\+C\cr
\+C&     ANGLE - ANGULAR ARGUMENT FOR SCHWIDERSKI COMPUTATION\cr
\+C\cr
\+C***********************************************************\cr
\+C\cr
\+C\cr
\+C&              C A U T I O N\cr
\+C\cr
\+C&  OCEAN LOADING PHASES COMPUTED FROM SCHWIDERSKI'S MODELS\cr   
\+C&  REFER TO THE PHASE OF THE ASSOCIATED SOLID EARTH TIDE\cr
\+C&  GENERATING POTENTIAL AT THE ZERO MERIDIAN ACCORDING TO\cr  
\+C\cr
\+C&      OL\_DR = OL\_AMP X COS (SE\_PHASE" - OL\_PHASE)\cr
\+C\cr
\+C&  WHERE OL = OCEAN LOADING TIDE,\cr
\+C&        SE = SOLID EARTH TIDE GENERATING POTENTIAL.\cr
\+C\cr
\+C&  IF THE HARMONIC TIDE DEVELOPMENT OF CARTWRIGHT, ET AL.\cr 
\+C&  ( = CTE) (1971, 1973) IS USED, MAKE SURE THAT SE\_PHASE"\cr 
\+C&  TAKES INTO ACCOUNT\cr
\+C\cr
\+C&  (1) THE SIGN OF SE\_AMP IN THE TABLES OF CARTWRIGHT ET AL.\cr
\+C\cr
\+C&  (2) THAT CTE'S SE\_PHASE REFERS TO A SINE RATHER THAN A\cr
\+C&      COSINE FUNCTION IF (N+M) = (DEGREE + ORDER) OF THE\cr
\+C&      TIDE SPHERICAL HARMONIC IS ODD.\cr
\+C\cr
\+C&  I.E. SE\_PHASE" = TAU(T)   N1 + S(T)   N2 + H(T)   N3\cr
\+C&                 + P(T)   N4 + N'(T)   N5 + PS(T)   N6\cr
\+C&                 + PI   IF CTE'S AMPLITUDE COEFFICIENT < 0\cr
\+C&                 + PI/2 IF (DEGREE + N1) IS ODD\cr
\+C\cr
\+C&  WHERE TAU ... PS = ASTRONOMICAL ARGUMENTS,\cr
\+C&         N1 ... N6 = CTE'S ARGUMENT NUMBERS.\cr
\+C\cr
\+C&  MOST TIDE GENERATING SOFTWARE COMPUTE SE\_PHASE" (FOR\cr
\+C&  USE WITH COSINES).\cr
\+C\cr
\+C&  THIS SUBROUTINE IS VALID ONLY AFTER 1973.\cr
\+C\cr
\+C******************************************************************\cr
\+&&&      SUBROUTINE ARG(IYEAR,DAY,ANGLE)\cr
\+&&&      IMPLICIT DOUBLE PRECISION (A-H,O-Z)\cr
\+&&&      REAL ANGFAC(4,11)\cr
\+&&&      DIMENSION ANGLE(11),SPEED(11)\cr
\+C\cr
\+C&  SPEED OF ALL TERMS IN RADIANS PER SEC\cr
\+C\cr
\+&&&  EQUIVALENCE (SPEED(1),SIGM2),(SPEED(2),SIGS2),(SPEED(3),SIGN2)\cr
\+&&&  EQUIVALENCE (SPEED(4),SIGK2),(SPEED(5),SIGK1),(SPEED(6),SIGO1)\cr
\+&&&  EQUIVALENCE (SPEED(7),SIGP1),(SPEED(8),SIGQ1),(SPEED(9),SIGMF)\cr
\+&&&      EQUIVALENCE (SPEED(10),SIGMM),(SPEED(11),SIGSSA)\cr
\+&&&      DATA SIGM2/1.40519D-4/\cr
\+&&&      DATA SIGS2/1.45444D-4/\cr
\+&&&      DATA SIGN2/1.37880D-4/\cr
\+&&&      DATA SIGK2/1.45842D-4/\cr
\+&&&      DATA SIGK1/.72921D-4/\cr
\+&&&      DATA SIGO1/.67598D-4/\cr
\+&&&      DATA SIGP1/.72523D-4/\cr
\+&&&      DATA SIGQ1/.64959D-4/\cr
\+&&&      DATA SIGMF/.053234D-4/\cr
\+&&&      DATA SIGMM/.026392D-4/\cr
\+&&&      DATA SIGSSA/.003982D-4/\cr
\+&&&      DATA ANGFAC/2.E0,-2.E0,0.E0,0.E0,4*0.E0,\cr
\+&&     .            2.E0,-3.E0,1.E0,0.E0,2.E0,3*0.E0,\cr
\+&&     .            1.E0,2*0.E0,.25E0,1.E0,-2.E0,0.E0,-.25E0,\cr
\+&&     .            -1.E0,2*0.E0,-.25E0,1.E0,-3.E0,1.E0,-.25E0,\cr
\+&&     .            0.E0,2.E0,2*0.E0,0.E0,1.E0,-1.E0,0.E0,\cr
\+&&     .            2.E0,3*0.E0/\cr
\+&&&      DATA TWOPI/6.28318530718D0/\cr
\+&&&      DATA DTR/.174532925199D-1/\cr
\+C\cr
\+C&  DAY OF YEAR\cr
\+C\cr
\+&&&      ID=DAY\cr
\+C\cr
\+C&  FRACTIONAL PART OF DAY IN SECONDS\cr
\+C\cr
\+&&&      FDAY=(DAY-ID)*86400.D0\cr
\+&&&      ICAPD=ID+365*(IYEAR-75)+((IYEAR-73)/4)\cr
\+&&&      CAPT=(27392.500528D0+1.000000035D0*ICAPD)/36525.D0\cr
\+C\cr
\+C& MEAN LONGITUDE OF SUN AT BEGINNING OF DAY\cr
\+C\cr
\+&&&      H0=(279.69668D0+(36000.768930485D0+3.03D-4*CAPT)*CAPT)*DTR\cr
\+C\cr
\+C& MEAN LONGITUDE OF MOON AT BEGINNING OF DAY\cr
\+C\cr
\+&&&      S0=(((1.9D-6*CAPT-.001133D0)*CAPT+481267.88314137D0)*CAPT\cr
\+&&     .  +270.434358D0)*DTR\cr
\+C\cr
\+C& MEAN LONGITUDE OF LUNAR PERIGEE AT BEGINNING OF DAY\cr
\+C\cr
\+&&&      P0=(((-1.2D-5*CAPT-.010325D0)*CAPT+4069.0340329577D0)*CAPT\cr
\+&&     .  +334.329653D0)*DTR\cr
\+&&&      DO 500 K=1,11\cr
\+&&&      ANGLE(K)=SPEED(K)*FDAY+ANGFAC(1,K)*H0+ANGFAC(2,K)*S0\cr
\+&&     .  +ANGFAC(3,K)*P0+ANGFAC(4,K)*TWOPI\cr
\+&&&      ANGLE(K)=DMOD(ANGLE(K),TWOPI)\cr
\+&&&      IF(ANGLE(K).LT.0.D0)ANGLE(K)=ANGLE(K)+TWOPI\cr
\+\ \ 500 CONTINUE\cr
\+&&&      RETURN\cr
\+&&&      END\cr}
\smallskip
\hrule
\bigskip
Table 7.1 is available electronically by anonymous ftp to 
{\tt maia.usno.navy.mil} or \break
{\tt ftp://gere.oso.chalmers.se/}$\sim${\tt pub/hgs/oload/README}.
For sites not contained in the list the following is recommended: 
If the distance to the nearest site contained in the table is less 
than ten km, its data can be substituted. In other cases, coefficients 
and/or software can be requested from {\tt hgs@oso.chalmers.se 
(Hans-Georg Scherneck)}. The Tamura tide potential is available from the
International Centre for Earth Tides, Observatoire Royal de Belgique, 
Bruxelles.

The coefficients of Table 7.1 have been computed according to Scherneck 
(1983, 1991).  Tangential displacements are to be taken positive in 
west and south directions.  The ocean tide maps adopted are due to 
LeProvost {\it et al.} (1994), Schwiderski (1983), Schwiderski and 
Szeto (1981), and Flather (1981).  Refined coastlines have been 
derived from the topographic data sets ETOPO5 and Terrain Base (Row 
{\it et al.}, 1995) of the National Geophysical Data Center, Boulder, 
CO.  Ocean tide mass budgets have been constrained using a uniform 
co-oscillating oceanic layer.  Load convolution employed a 
disk-integrating Green's function method (Farrell, 1972; Zschau, 
1983; Scherneck, 1990).  An assessment of the accuracy of the loading 
model is given in Scherneck (1993).  Adoption of more recent ocean 
tide results drawing from TOPEX/POSEIDON results is currently under 
consideration.

\noindent Table 7.1  Sample of ocean loading table file. Each 
site record shows a header with the site name, the CDP monument
number, geographic coordinates and comments.  First three rows
of numbers designate amplitudes (meter), radial, west, south,
followed by three lines with the corresponding phase values (degrees). 
\hrule
Columns designate partial tides
$M_2, S_2, N_2, K_2, K_1, O_1, P_1, Q_1, M_f, M_m,$ and $S_{sa}$. 

{\parindent = 0 pc
 \parskip = 0 pc
 \baselineskip = 1 pc
 \obeylines
 \tt
\$\$
\quad  ONSALA60  7213
\$\$
\$\$ Computed by H.G. Scherneck, Uppsala University, 1989
\$\$ ONSALA  7213  \quad             lon/lat:   11.9263   57.3947
$\vbox{\settabs 11 \columns
\+ \cr
\+  .00384 &.00091 &.00084 &.00019 &.00224 &.00120 &.00071 &.00003 
    &.00084 &.00063 &.00057\cr
\+  .00124 &.00034 &.00031 &.00009 &.00042 &.00041 &.00015 &.00006 
    &.00018 &.00010 &.00010\cr
\+  .00058 &.00027 &.00021 &.00008 &.00032 &.00017 &.00009 &.00004 
    &.00007 &.00001 &.00020\cr
\+\hfill   -56.0 &\hfill -46.1 &\hfill -90.7 &\hfill -34.4 
  &\hfill -44.5 &\hfill -123.2 
   &\hfill -49.6 &\hfill 178.4 &\hfill  14.9  &\hfill 37.3   &\hfill 24.6 &\cr
\+\hfill    75.4 &\hfill  97.6 &\hfill  40.8 &\hfill 94.8  
  &\hfill 119.0 &\hfill  25.4  
   &\hfill 98.7  &\hfill -14.1 &\hfill -177.0 &\hfill -126.7 &\hfill -175.8 &\cr
\+\hfill    84.2 &\hfill 131.3 &\hfill  77.7 &\hfill 103.9 
  &\hfill  17.2 &\hfill -55.0  
   &\hfill  25.2 &\hfill-165.0 &\hfill 173.3 &\hfill 121.8 &\hfill  91.3 &\cr
{\bigskip}   
}$
}
\hrule
\bigskip
\write 0 {\indent Effects of the Solid Earth Tides \noexpand\dotfill
\the\pageno \noexpand\break}
\noindent{\bf Effects of the Solid Earth Tides}

\nobreak

\medskip 

Site displacements caused by tides of spherical harmonic degree and 
order $(nm)$ are characterized by the Love number $h_{nm}$ and the 
Shida number $l_{nm}$. The effective values of these numbers depend 
on station latitude and tidal frequency (Wahr, 1981). This dependence 
is a consequence of the ellipticity and  rotation of the Earth, and 
includes a strong frequency dependence within the diurnal band due to 
the Nearly Diurnal Free Wobble resonance. A further frequency 
dependence, which is most pronounced in the long period tidal band, 
arises from mantle anelasticity which leads to corrections to the 
elastic Earth Love numbers; these corrections have a small imaginary 
part and cause the tidal displacements to lag slightly behind the 
tide generating potential. All these effects need to be taken into 
account when an accuracy of 1 $mm$ is desired in determining station 
positions. 

In order to account for the latitude dependence of the effective 
Love and Shida numbers, the representation in terms of multiple $h$ 
and $l$ parameters employed by Mathews {\it et al.} (1995) is used.  
In this representation, parameters $h^{(0)}$ and $l^{(0)}$ play the 
roles of $h_{2m}$ and $l_{2m}$, while the latitude dependence is 
expressed in terms of additional parameters $h^{(2)},h'$ and 
$l^{(1)},l^{(2)},l'$. These parameters are defined through their 
contributions to the site displacement as given by equations (5) 
below.   Their numerical values as listed in Mathews {\it et al.} 
(1995) have since been revised, and the new values, presented in 
Table 7.2 are used here.  These values pertain to the elastic Earth 
and anelasticity models referred to in Chapter 6. 


 The vector displacement due to a tidal term of frequency $f$ is 
given in terms of the several parameters by the following expressions 
that result from evaluation of the defining equation (6) of Mathews 
{\it et al.} (1995): 

\smallskip
\noindent For a long-period tide of frequency $f$:

$$\eqalign{
\Delta \vec{r}_f = \sqrt {{5\over 4\pi}} H_f
&\Biggl
\{
\left [
h(\phi) \left(
{3\over 2} \sin^2\phi-{1\over 2}
\right) 
+ \sqrt { 4\pi\over 5} h'
\right] 
 \cos\theta_f \,\hat {r}
  + 3 l(\phi)\sin\phi \cos\phi \cos \theta_f\, \hat {n}\cr 
 &+ \cos\phi\, \left [
 3 l^{(1)} \sin^2\phi -\sqrt { 4\pi\over 5} l'
 \right ] 
 \sin \theta_f \,\hat {e} 
\Biggr 
 \}.
\cr
 }\eqno(5a)$$

\noindent For a diurnal tide of frequency $f$:

$$\eqalign {
\Delta \vec{r}_f = -\sqrt {{5\over 24\pi}}H_f 
& \Biggl\{
h(\phi) 3 \sin\phi \cos\phi \sin(\theta_f +\lambda) \,\hat {r} 
\cr&
+  \left[
3 l(\phi) \cos 2\phi  - 3 l^{(1)}  \sin^2 \phi 
+ \sqrt {{24\pi\over 5}}l'
\right]
\sin (\theta_f +\lambda) \,  \hat {n} 
\cr &  + \left[
  \left(
  3 l(\phi) - \sqrt {{ 24\pi\over 5}}l'
  \right)
   \sin \phi - 3 l^{(1)} \sin \phi \cos 2\phi
   \right] 
  \cos (\theta_f +\lambda)\, \hat {e}
  \Biggr \}.
  \cr
  }
  \eqno(5b)$$

\noindent For a semidiurnal tide of frequency $f$:

$$\eqalign{
 \Delta \vec{r}_f = \sqrt {{5\over 96\pi}} H_f \{ &[h(\phi) 3\cos^2\phi  
\cos(\theta_f + 2\lambda) \,\hat {r}- 6 \sin \phi\cos\phi [l(\phi) + l^{(1)}]
\cos(\theta_f + 2\lambda) \, \hat {n} \cr
 &-6 \cos\phi [l(\phi)  + l^{(1)}\sin^2 \phi] \sin (\theta_f + 2\lambda)
 \,\hat {e} \}.\cr
 }\eqno(5c)$$
 
\noindent In the above expressions,

$$h(\phi) = h^{(0)} + h^{(2)} [(3/2) \sin^2\phi -1/2], \qquad 
l(\phi) = l^{(0)} + l^{(2)} [(3/2) \sin^2\phi -1/2],\eqno(6)$$

\item{$H_{f}$} = amplitude (m) of the tidal term of frequency $f$, 

\item{$\phi$} = geocentric latitude of station,

\item{$\lambda$} = east longitude of station,

\item{$\theta_{f}$} = tide argument for tidal constituent with 
frequency $f$,

\item{$\hat{e}$} = unit vector in the east direction,

\item{$\hat{n}$} = unit vector at right angles to $\hat{r}$ in the 
northward direction.

The convention used in defining the tidal amplitude $H_f$ is as in 
Cartwright and Tayler (1971). To convert amplitudes defined according 
to other conventions that have been employed in recent more accurate 
tables, use the conversion factors given in Chapter 6, Table 6.4.

Equations (5) assume that the Love and Shida number parameters are all 
real. Generalization to the case of complex parameters (anelastic 
earth) is done simply by making the following replacements for the 
combinations $L \cos(\theta_f+m\lambda)$ and  $L \sin (\theta_f+m
\lambda)$, wherever they occur in those equations:
 
$$L \cos(\theta_f+m\lambda)\to L^R \cos(\theta_f+m\lambda) - 
L^I \sin (\theta_f+m\lambda), \eqno(7a)$$

$$L \sin (\theta_f+m\lambda)\to  L^R \sin (\theta_f+m\lambda) 
+ L^I \cos(\theta_f+m\lambda), \eqno(7b)$$

\noindent where $L$ is a generic symbol for $h(\phi), h', l(\phi), 
l^{(1)},\ {\rm and\ }
l'$, and $L^R$ and $L^I$ stand for their respective real and imaginary 
parts. Table 7.2 lists the  values of the Love and Shida number 
parameters.  The Earth model on which they are based is the 1 sec 
PREM, modified by replacement of the ocean layer by solid (Dehant, 
1987, Wang, 1994) and by adjustment of the fluid core ellipticity to 
make the FCN period = 430 sidereal days as inferred from the resonance 
in nutations.  The tidal frequencies shown in the table are in cycles 
per sidereal day ($cpsd$). Periods, in solar days, of the nutations 
associated with the diurnal tides are also shown. 

\bigskip

 \noindent Table 7.2. Displacement Love number parameters for degree 2
 tides. Quantities with subscripts $elas$ ($anelas$) are computed ignoring
 (including) anelasticity effects. Superscripts $R$ and $I$ identify the real
 and imaginary parts, respectively.
\bigskip
 \hrule
\bigskip            
\halign{\hfil#\kern.5pc&\hfil#\kern.5pc&\hfil#\kern.5pc&\hfil#\kern.5pc&
\hfil#\kern.5pc&\hfil#\kern.9pc&\hfil#\kern.9pc&\hfil#\cr
Name & Period & Frequency & $h^{(0)}_{elas}$ & $h^{(0)R}_{anelas}$ & $h^{(0)I}_{anelas}$ & 
$h^{(2)}$ & $h'$\cr
  \cr 
  Semidiurnal & & -2 $cpsd$     & .6026& .6078& -.0022 & -.0006\cr
  \cr
  Diurnal  \cr  
Q$_1$   &    9.13 & -0.89080  & .5971& .6033& -.0025 & -.0006\cr
145,545 &   13.63 & -0.92685  & .5964& .6026& -.0025 & -.0006\cr
O$_1$   &   13.66 & -0.92700  & .5964& .6026& -.0025 & -.0006\cr
NO$_1$  &   27.55 & -0.96381  & .5941& .6003& -.0025 & -.0006\cr
$\pi_1$ &  121.75 & -0.99181  & .5813& .5876& -.0025 & -.0007\cr
P$_1$   &  182.62 & -0.99454  & .5753& .5816& -.0025 & -.0007\cr
165,545 & 6798.38 & -0.99985  & .5214& .5280& -.0027 & -.0007\cr
K$_1$   & infinity& -1.00000  & .5166& .5232& -.0027 & -.0008\cr
165,565 &-6798.38 & -1.00015  & .5112& .5178& -.0027 & -.0008\cr
$\psi_1$& -365.26 & -1.00273  &1.0582&1.0534&  .0020 & -.0001\cr
$\phi_1$& -182.62 & -1.00546  & .6589& .6644& -.0022 & -.0006\cr
\cr \cr

Long period \cr
55,565  & 6798.38 &   .000015 & .5998& .6344& -.0093 & -.0006& .0001\cr
$S_{sa}$&  182.62 &   .000546 & .5998& .6182& -.0054 & -.0006& .0001\cr
$M_m$   &   27.55 &   .003619 & .5998& .6126& -.0041 & -.0006& .0001\cr
$M_f$   &   13.66 &   .007300 & .5998& .6109& -.0037 & -.0006& .0001\cr 
75,565  &   13.63 &   .007315 & .5998& .6109& -.0037 & -.0006& .0001\cr
      \cr \cr}
\halign{\hfil#\kern.2pc&\hfil#\kern.2pc&\hfil#\kern.5pc&\hfil#\kern.5pc&
\hfil#\kern.2pc&\hfil#\kern.9pc&\hfil#\kern.9pc&\hfil#\kern.9pc&
\hfil#\cr
Name & Period & Frequency & $l^{(0)}_{elas}$ & $l^{(0)R}_{anelas}$ & $l^{(0)I}_{anelas}$ & 
$l^{(1)}$ & $l^{(2)}$ & $l'$ \cr
  \cr 
  Semidiurnal &    & -2 $cpsd$   & .0831& .0847& -.0007 & .0024& .0002\cr
  \cr
  Diurnal  \cr 
Q$_1$   &    9.13 & -0.89080  & .0829& .0848& -.0007 & .0012& .0002& -.0002\cr
145,545 &   13.63 & -0.92685  & .0829& .0848& -.0007 & .0012& .0002& -.0002\cr
O$_1$   &   13.66 & -0.92700  & .0829& .0848& -.0007 & .0012& .0002& -.0002\cr
M$_1$   &   27.55 & -0.96381  & .0830& .0849& -.0007 & .0012& .0002& -.0002\cr
$\pi_1$ &  121.75 & -0.99181  & .0834& .0853& -.0007 & .0012& .0002& -.0002\cr
P$_1$   &  182.62 & -0.99454  & .0836& .0855& -.0007 & .0012& .0002& -.0002\cr
165,545 & 6798.38 & -0.99985  & .0853& .0871& -.0007 & .0011& .0002& -.0003\cr
K$_1$   & infinity& -1.00000  & .0854& .0872& -.0007 & .0011& .0002& -.0003\cr
165,565 &-6798.38 & -1.00015  & .0856& .0874& -.0007 & .0011& .0002& -.0003\cr
$\psi_1$& -365.26 & -1.00273  & .0684& .0710& -.0010 & .0019& .0002&  .0001\cr
$\phi_1$& -182.62 & -1.00546  & .0810& .0829& -.0008 & .0013& .0002& -.0002\cr
  \cr 
Long period \cr
55,565  & 6798.38 &  .000015   & .0831& .0936& -.0028 & .0000& .0002\cr
$S_{sa}$&  182.62 &  .000546   & .0831& .0886& -.0016 & .0000& .0002\cr
$M_m$   &   27.55 &  .003619   & .0831& .0870& -.0012 & .0000& .0002\cr
$M_f$   &   13.66 &  .007300   & .0831& .0864& -.0011 & .0000& .0002\cr
75,565  &   13.63 &  .007315   & .0831& .0864& -.0011 & .0000& .0002\cr
\cr}
\medskip
\hrule
\bigskip

Computation of the variations of station coordinates due to solid Earth 
tides, like that of geopotential variations, is done most efficiently 
by the use of a two-step procedure. The evaluations in the first step  
use the expression in the time domain for the full degree 2 tidal 
potential or for the parts that pertain to particular bands  
$(m = 0,1,\ {\rm or}\ 2)$. Nominal values used for the Love and Shida 
numbers $h_{2m}$ and $l_{2m}$ are common to all the tidal constituents 
involved in the potential and to all stations.  They are chosen with 
reference to the values in Table 7.2 so as to minimize the computational 
effort needed in Step 2.  Along with expressions for the dominant 
contributions from $h^{(0)}$ and $l^{(0)}$ to the tidal displacements,
relatively small contributions from some of the other parameters are
included in Step 1 for reasons of computational efficiency. The 
displacements caused by the degree 3 tides are also computed in the 
first step,  using constant values for $h_3$ and $l_3$. 

 Corrections to the results of the first step are needed to take 
account of the frequency dependent deviations of the Love and Shida 
numbers from their respective nominal values, 
and also to compute the out of phase contributions from the zonal tides. 
The scheme of computation is outlined in the chart below.   

\bigskip
\noindent \centerline {CORRECTIONS FOR THE STATION TIDAL DISPLACEMENTS}
\bigskip
\hrule
\medskip
\noindent {\bf Step 1} : Corrections to be computed in the time domain
\halign{&#\hfil&#\hfil&#\hfil \cr
&& \cr
 in phase & for degree 2 and 3 &  Nominal values \cr
          &   . for degree 2 $\to$ eq (8) & $h_2$ $\to$ $h(\phi) = h^{(0)} + h^{(2)} [(3 \sin^2\phi -1)/2]$ \cr 
          &                            & $l_2$ $\to$ $l(\phi) = l^{(0)} + l^{(2)} [(3 \sin^2\phi -1)/2]$ \cr
          &   $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$  elastic     & $h^{(0)}=0.6026$, $h^{(2)}=-0.0006$; $l^{(0)}=0.0831$, $l^{(2)}=0.0002$ \cr
          &   $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$  anelastic   & $h^{(0)}=0.6078$, $h^{(2)}=-0.0006$; $l^{(0)}=0.0847$, $l^{(2)}=0.0002$ \cr
          &   . for degree 3 $\to$ eq (9) & $h_3=0.292$ and $l_3=0.015$\cr
&& \cr
 out-of-phase & for degree 2 only & Nominal values  \cr
              &   . diurnal tides      $\to$ eq (13) $\;$ & $h^I=-0.0025$ and $l^I=-0.0007$  \cr
              &   . semi-diurnal tides $\to$ eq (14) $\;$ & $h^I=-0.0022$ and $l^I=-0.0007$ \cr
&& \cr
 contribution & from latitude dependence & Nominal values  \cr
              & . diurnal tides $\to$ eq (11)  $\;$     & $l^{(1)}=0.0012$ \cr
              & . semi-diurnal tides $\to$ eq (12) $\;$ & $l^{(1)}=0.0024$ \cr
&& \cr}
\noindent {\bf Step 2} : Corrections to be computed in the frequency domain and 
to be added to results of Step 1
\halign{&#\hfil&#\hfil&#\hfil \cr
&& \cr
 in phase & for degree 2 \cr
          &      . diurnal tides      $\to$ eqs (15)& $\;\rightarrow$ Sum over all the components of Table 7.3a  \cr  
          &      . semi-diurnal tides             & negligible \cr
 in phase & and out of phase for degree 2  \cr
          &      . long-period tides  $\to$ eqs (16)& $\;\rightarrow$ Sum over all the components of Table 7.3b  \cr
&& \cr}
\medskip
\hrule
\bigskip

\write 0{\indent \indent Displacement due to degree 2 tides \noexpand 
\dotfill\the\pageno\noexpand\break}
\noindent Displacement due to degree 2 tides, with nominal values for 
$h_{2m}^{(0)}$ and $l_{2m}^{(0)}$
\vskip-1.7pc
\noindent\vrule width26.6pc height0pt depth0.4pt

The first stage of the  {\bf Step 1} calculations employs real nominal 
values $h_2$ and $l_2$ common to all the degree 2 tides for the Love 
and Shida numbers. It is found to be computationally most economical 
to choose these to be the values for the semidiurnal tides (which have 
very little intraband variation). For the same reason, the nominal 
values used when anelasticity is included are different from the 
elastic case. (Anelasticity contributions are at the one percent 
level, which is about 4 $mm$ in the radial displacement due to the full 
degree 2 tide.) The out of phase contributions due to anelasticity are 
dealt with separately below. 
 
On using the nominal values, the vector displacement of the station 
due to the degree 2 tides is given by

$$\Delta\vec{r}=\sum_{j=2}^3{GM_jR_e^4\over GM_\oplus R_j^3}
\biggl\{h_2\,\hat{r}\biggl({3\over 2}(\hat{R}_j\cdot\hat{r})^2-{1\over 2} 
\biggr)+3 l_2(\hat{R}_j\cdot\hat{r})[\hat{R}_j-(\hat{R}_j\cdot\hat{r})
\,\hat{r}]\biggr\}, \eqno(8)$$

\noindent where $h_{22}^{(0)}$ and $l_{22}^{(0)}$ of the semidiurnal tides are 
chosen as the nominal values $h_2$ and $l_2$.  These values depend on 
whether anelasticity is included in the computation or not, but no 
imaginary parts are included at this stage. In equation (8), 

\item{$GM_j$} =  gravitational parameter for the Moon ($j=2$) or the 
             Sun ($j=3$),

\item{$GM_\oplus$} =  gravitational parameter for the Earth,

\item{$\hat{R}_j,R_j$} =  unit vector from the geocenter to Moon or
Sun and the magnitude of that vector,

\item{$R_e$} = Earth's equatorial radius,

\item{$\hat{r},r$} =  unit vector from the geocenter to the station and 
             the magnitude of that vector,

\item{$h_2$} =  nominal  degree 2 Love number,

\item{$l_2$} =  nominal degree 2 Shida number.

\noindent 
Note that the part proportional to $h_2$ gives the radial (not 
vertical) component of the tide-induced station displacement, and the 
terms in $l_2$ represent the vector displacement transverse to the 
radial direction (and not in the horizontal plane). 

The computation just described may be generalized to include the 
latitude dependence arising through  $h^{(2)}$ by simply adding 
$h^{(2)}[(3/2) \sin^2 \phi -(1/2)]$ to the constant nominal value 
given above, with $h^{(2)} = -0.0006$. The addition of a similar term 
(with $l^{(2)}= 0.0002$) to the nominal value of $l_2$ takes care of 
the corresponding contribution to the transverse displacement.  The 
resulting incremental displacements are small, not exceeding 0.4 $mm$ 
radially and 0.2 $mm$ in the transverse direction. 


\write 0{\indent \indent Displacement due to degree 3 tides \noexpand 
\dotfill\the\pageno\noexpand\break}
\noindent\underbar{Displacements due to degree 3 tides}

The Love numbers of the degree 3 tides may be taken as real and 
constant in computations to the degree of accuracy aimed at here.  
The vector displacement due to these tides is then given by 

$$\Delta\vec{r}=\sum_{j=2}^3{GM_jR_e^5\over GM_\oplus R_j^4}
\biggl\{h_3 \hat{r}\biggl({5\over 2}(\hat{R}_j\cdot\hat{r})^3-
{3\over 2} (\hat{R}_j\cdot\hat{r})\biggr)+l_3 \biggl({15\over 2}
(\hat{R}_j\cdot\hat{r})^2-{3\over 2}\biggr) [\hat{R}_j-(\hat{R}_j
\cdot\hat{r})\hat{r}]\biggr\}.  \eqno(9)$$

Only the Moon's contribution ($j=2$) need be computed, the term due to 
the Sun being quite ignorable.  The transverse part of the displacement
(9) does not exceed 0.2 $mm$, but the radial displacement can reach 1.7
$mm$. 

\vfil\eject
\write 0{\indent \indent Contributions to the transverse \noexpand 
\dotfill \the\pageno\noexpand\break}
\noindent\underbar{Contributions to the transverse displacement due to 
the $l^{(1)}$ term}

The imaginary part of $l^{(1)}$ (anelastic case) is completely 
ignorable, and so is the difference between the real part and the 
elastic Earth value, as well as the intra-band frequency dependence; and 
$l^{(1)}$ is effectively zero in the zonal band. 

In the expressions given below, and elsewhere in this Chapter,


\item{$\Phi_j$} = body fixed geocentric latitude of Moon or Sun, and 
\item{$\lambda_j$} = body fixed east longitude (from Greenwich) of Moon or Sun.

The following formulae may be employed when the use of Cartesian 
coordinates $X_j, Y_j, Z_j$ of the body relative to the terrestrial 
reference frame is preferred:

$$P_2^0(\sin\Phi_j) = {1\over R_j^2} \left ({3\over 2} Z_j^2 -{1\over 2}
R_j^2 \right ),\eqno(10a)$$
$$P_2^1(\sin\Phi_j) \cos \lambda_j = {3X_j Z_j\over R_j^2}, \qquad 
P_2^1(\sin\Phi_j) \sin \lambda_j = {3Y_j Z_j\over R_j^2},\eqno(10b)$$
$$P_2^2(\sin\Phi_j) \cos 2\lambda_j = {3\over R_j^2} (X_j^2-Y_j^2) , 
\qquad P_2^2(\sin\Phi_j) \sin 2\lambda_j = {6\over R_j^2} X_j Y_j.
\eqno(10c)$$


\noindent Contribution from the diurnal band (with $l^{(1)}= 0.0012$): 

$$\delta \vec {t}  = -l^{(1)} \sin\phi 
\sum_{j=2}^3 {GM_jR_e^4\over GM_\oplus 
R_j^3}  P_2^1(\sin \Phi_j) [\sin \phi \cos (\lambda -\lambda_j) 
\,\hat n -\cos 2\phi \sin (\lambda - \lambda_j) \,\hat e]. \eqno(11)$$

\noindent Contribution from the semidiurnal band (with $l^{(1)}= 0.0024$):

$$\delta \vec {t}  = -{1\over 2} l^{(1)}\sin \phi \cos \phi  
 \sum_{j=2}^3 {GM_jR_e^4\over GM_\oplus R_j^3}  P_2^2(\sin \Phi_j) 
[\cos 2(\lambda - \lambda_j) \,\hat n 
+ \sin\phi \sin 2(\lambda - \lambda_j) \,\hat e].
\eqno(12)$$

The contributions of the $l^{(1)}$ term to the transverse displacements
caused by the diurnal and semidiurnal tides could be up to 0.8 mm and 
1.0 $mm$ respectively. 

\write 0{\indent \indent Out of phase contributions  \noexpand \dotfill 
\the\pageno\noexpand\break}
\noindent Out of phase contributions from the imaginary parts
of $h_{2m}^{(0)}$ and $l_{2m}^{(0)}$
\vskip-1.7pc
\noindent\vrule width25pc height0pt depth0.4pt

 In the following, $h^I$ and $l^I$ stand for the imaginary parts of   
$h_{2m}^{(0)}$ and $l_{2m}^{(0)}$, which do not exist in the elastic 
case. 

Contributions $\delta r$ to radial and $\delta \vec{t}$ to transverse 
displacements from diurnal tides (with $h^I = -0.0025,\ l^I = -0.0007$):

$$\delta r  = -{3\over 4} h^I  \sum_{j=2}^3 {GM_jR_e^4\over 
GM_\oplus R_j^3}\sin 2\Phi_j \sin 2\phi \sin (\lambda-\lambda_j),\eqno(13a)$$

$$\delta \vec{t} =  -{3\over 2} l^I  \sum_{j=2}^3 {GM_jR_e^4\over GM_\oplus R_j^3}
\sin 2\Phi_j [\cos 2\phi \sin (\lambda-\lambda_j) \hat n + \sin \phi  
\cos (\lambda-\lambda_j) \hat e].\eqno(13b)$$

Contributions from semidiurnal tides (with $h^I = -0.0022$, $l^I = -0.0007$):

$$\delta r = -{3\over 4} h^I  \sum_{j=2}^3 {GM_jR_e^4\over 
GM_\oplus R_j^3}\cos^2\Phi_j  \cos^2\phi\sin 2(\lambda-\lambda_j),\eqno(14a) $$

$$\delta \vec{t} =  {3\over 4} l^I  \sum_{j=2}^3 {GM_jR_e^4\over 
GM_\oplus R_j^3}\cos^2\Phi_j
 [\sin 2\phi \sin 2(\lambda-\lambda_j) \hat n - 2\cos \phi  
\cos 2(\lambda-\lambda_j) \hat e].\eqno(14b)$$
The out of phase contributions from the zonal tides has no closed expression 
in the time domain. 

Computations of {\bf Step 2} are to take account of the intraband 
variation of $h_{2m}^{(0)}$ and $l_{2m}^{(0)}$. Variations of the 
imaginary parts (anelastic case) are ignorable except as stated below. 
For the zonal tides, however, the contributions from the imaginary part have
to be computed in Step 2. 

A FORTRAN program for computing the various corrections is available at 
ftpserver.oma.be, subdirectory /pub/astro/dehant/IERS (anonymous-ftp system). 

\write 0{\indent \indent Correction for frequency \noexpand \dotfill 
\the\pageno\noexpand\break}
\noindent\underbar{Correction for frequency dependence of the Love and 
Shida numbers}
\nobreak


\noindent (a) Contributions from the diurnal band
 
Corrections to the radial and transverse station displacements 
$\delta r$ and $\delta \vec t $ due to a diurnal tidal term of 
frequency $f$ are obtainable from equation (5b): 

$$\delta r =\delta R_f\sin 2\phi \sin(\theta_{f}+\lambda),\eqno(15a)$$
$$\delta \vec t = \delta T_f [\sin\phi \cos(\theta_{f}+\lambda)
\,\hat{e} + \cos 2\phi \sin(\theta_f+\lambda)\,\hat{n}],\eqno(15b)$$
 where
$$\delta R_f = -{3\over 2}\sqrt{5\over 24\pi}\delta h_{f}H_{f}
\qquad {\rm and} \qquad 
\delta T_f = -3\sqrt{5\over 24\pi} \delta l_{f}H_{f}, \eqno(15c)$$
and 
\item{$\delta h_{f}$} = difference of $h^{(0)}$ at frequency $f$ from 
the nominal value of $h_2$,

\item{$\delta l_{f}$} = difference of $l^{(0)}$ at frequency $f$ from 
the nominal value of $l_2$.


Values of $\Delta R_{f}$ and $\Delta T_{f}$ listed in Tables 7.3a and 
7.3b  are for the constituents that must be taken into account to 
ensure an accuracy of 1 $mm$. $\Delta R_{f}^{el}$ and $\Delta T_{f}^{el}$
are for the elastic case, and $\Delta R_{f}^{anel}$ and $\Delta T_{f}^
{anel}$ are for use when anelasticity effects are included. It should 
be noted that different nominal values are used for the two cases in 
order to minimize the number of terms for which corrections are needed. 

Corrections to the out-of-phase parts, arising from variation of the 
imaginary parts of $h_{21}^{(0)}$ and $l_{21}^{(0)}$, are very small. 
The only one with an amplitude exceeding the cutoff of 0.05 $mm$ 
used for the table is 0.06 $mm$ in the vertical component due to the 
K$_1$ tide. Its contribution to the displacement is $0.06 \sin 2\phi 
\cos(\theta_g+\pi+\lambda)$ $mm$.
 
\bigskip

\noindent Table 7.3a.  Corrections due to frequency variation of Love 
and Shida numbers for diurnal  tides.  Units: $mm$. All terms with radial 
correction $\ge$ 0.05 $mm$ are shown. Nominal values are $h=0.6026$ and 
$l=0.0831$ for the elastic case, and $h^R=0.6078$ and $l^R=0.0847$ for the real 
parts in the anelastic case. Frequencies shown are in degrees per hour.
\bigskip
\hrule
\hrule
\medskip
\halign{#\hfil & #\hfil & \hfil# & 
\hfil# & \hfil# & \hfil# & \hfil# & \hfil# & \hfil# & 
\hfil# & \hfil# & \hfil# & \hfil# & \hfil# & 
\hfil# & \hfil# & \hfil# & \hfil# \cr 
Name & Frequency & Doodson & $\;\tau$& $\;s$& $\;h$& $\;p$& $N'$& $p_s$&
$\;\;\;\; \;\ell $& $\ell'$& $F$& $D$& $\Omega$& $\Delta R_{f}^{el}$ & 
$\Delta T_{f}^{el}$ 
& $\Delta R_{f}^{anel}$ & $\Delta T_{f}^{anel}$\cr
\noalign{\bigskip}
Q$_1$     & 13.39866 & 135,655 & 1&-2& 0& 1& 0& 0& 1& 0& 2& 0& 2& -0.11& -0.01& -0.09&  0.00\cr
          & 13.94083 & 145,545 & 1&-1& 0& 0&-1& 0& 0& 0& 2& 0& 1& -0.12& -0.01& -0.10&  0.00\cr
O$_1$     & 13.94303 & 145,555 & 1&-1& 0& 0& 0& 0& 0& 0& 2& 0& 2& -0.63& -0.04& -0.53&  0.02\cr
NO$_1$    & 14.49669 & 155,655 & 1& 0& 0& 1& 0& 0& 1& 0& 0& 0& 0&  0.07&  0.00&  0.06& -0.00\cr
$\pi_1$   & 14.91787 & 162,556 & 1& 1&-3& 0& 0& 1& 0& 1& 2&-2& 2& -0.06&  0.00& -0.05&  0.00\cr
P$_1$     & 14.95893 & 163,555 & 1& 1&-2& 0& 0& 0& 0& 0& 2&-2& 2& -1.29&  0.05& -1.23&  0.07\cr
          & 15.03886 & 165,545 & 1& 1& 0& 0&-1& 0& 0& 0& 0& 0&-1& -0.23&  0.01& -0.22&  0.01\cr
K$_1$     & 15.04107 & 165,555 & 1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 12.25& -0.65& 12.04& -0.72\cr
          & 15.04328 & 165,565 & 1& 1& 0& 0& 1& 0& 0& 0& 0& 0& 1&  1.77& -0.09&  1.74& -0.10\cr
$\psi_1$  & 15.08214 & 166,554 & 1& 1& 1& 0& 0&-1& 0&-1& 0& 0& 0& -0.51&  0.03& -0.50&  0.03\cr
$\phi_1$  & 15.12321 & 167,555 & 1& 1& 2& 0& 0& 0& 0& 0&-2& 2&-2& -0.11&  0.01& -0.11&  0.01\cr
& & & & & & & & & & & & & & & & & \cr 
\noalign{\hrule}
\noalign{\bigskip}}

\noindent (b) Contributions from the long period band 


Corrections $\delta r$ and $\delta \vec t$ due to a zonal tidal term 
of frequency $f$ include both in phase ($ip$) and out of phase ($op$) 
parts.  From equations (5a) and (7) one finds that 

$$\delta r = \left ( {3\over 2} \sin^2 \phi - {1\over 2} \right ) \,
(\delta R^{(ip)}_f\cos\theta_{f} + \delta R^{(op)}_f\sin\theta_{f}),\eqno(16a)$$

\noindent and 
$$\delta \vec t = \sin 2\phi\, (\delta T^{(ip)}_f\cos\theta_{f} + \delta 
T^{(op)}_f\sin\theta_{f})\,\hat{n}, \eqno(16b)$$
\noindent where
$$\delta R{(ip)}_f = \sqrt{5\over 4\pi}\delta h^R_{f}H_{f}\qquad {\rm and} 
\qquad \delta R{(op)}_f = -\sqrt{5\over 4\pi}\delta h^I_{f}H_{f},
\eqno(16c)$$
$$\delta T{(ip)}_f = {3\over 2} \sqrt{5\over 4\pi} \delta l^R_{f}H_{f} 
\qquad {\rm and} \qquad \delta T{(op)}_f = -{3\over 2} \sqrt{5\over 4\pi} 
\delta l^I_{f}H_{f} . \eqno(16d)$$
 
\vfil\eject
\noindent Table 7.3b.  Corrections due to frequency variation of Love 
and Shida numbers for zonal tides.  Units: $mm$. All terms with radial correction 
$\ge$ 0.05 $mm$ are shown. Nominal values are $h=0.6026$ and $l=0.0831$
for the elastic case, and $h^R=0.6078$ and $l^R=0.0847$ for the real 
parts in the anelastic case. For each frequency, the in phase amplitudes 
$\Delta R_f^{(ip)}$ and $\Delta T_f^{(ip)}$ are shown on the first line, and 
the out of phase amplitudes  $\Delta R_f^{(op)}$ and $\Delta T_f^{(op)}$ on 
the second line. Frequencies shown are in degrees per hour.
 
\bigskip
\hrule
\hrule
\medskip
\halign{#\hfil & \hfil# & \hfil#\hfil & 
\hfil# & \hfil# & \hfil# & \hfil# & \hfil# & \hfil# & 
\hfil# & \hfil# & \hfil# & \hfil# & \hfil# & 
\hfil# & \hfil# & \hfil# & \hfil# \cr 
Name & Frequency & Doodson & $\;\tau$& $\;s$& $\;h$& $\;p$& $N'$& $p_s$&
$\;\;\;\;\ell $& $\ell'$& $F$& $D$& $\Omega$& $\Delta R_{f}^{elas}$ & $\Delta T_{f}^{elas}$ 
& $\Delta R_{f}^{anel}$ & $\Delta T_{f}^{anel}$\cr
        & 0.00221 & 55,565 &0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 1& -0.05&  0.00&  0.47&  0.23\cr
&&&&&&&&&&&&&&&&0.16&0.07\cr
$S_{sa}$& 0.08214 & 57,555 &0& 0& 2& 0& 0& 0& 0& 0&-2& 2&-2&  0.05&  0.00& -0.20& -0.12\cr
&&&&&&&&&&&&&&&&-0.11&-0.05\cr
$M_m$   & 0.54438 & 65,455 &0& 1& 0&-1& 0& 0&-1& 0& 0& 0& 0&  0.06&  0.00& -0.11& -0.08\cr
&&&&&&&&&&&&&&&&-0.09&-0.04\cr
$M_f$   & 1.09804 & 75,555 &0& 2& 0& 0& 0& 0& 0& 0&-2& 0&-2&  0.12&  0.00& -0.13& -0.11\cr
&&&&&&&&&&&&&&&&-0.15&-0.07\cr
        & 1.10024 & 75,565 &0& 2& 0& 0& 1& 0& 0& 0&-2& 0&-1&  0.05&  0.00& -0.05& -0.05\cr
&&&&&&&&&&&&&&&&-0.06&-0.03\cr
& & & & & & & & & & & & & & & & & \cr 
\noalign{\hrule}
\noalign{\bigskip}}


\write 0{\indent \indent Permanent deformation \noexpand \dotfill 
\the\pageno\noexpand\break}
\noindent\underbar{Permanent deformation}
\nobreak


The zonal part of the degree 2 potential contains a time-independent 
constituent of amplitude ($-0.31460$) $m$. A permanent deformation due 
to this constituent forms a part of the site displacement computed 
from Equation (8) using the nominal values $h_2 = 0.6026$ and $l_2 = 
0.0831$ when ignoring anelasticity. The permanent part of radial 
displacement thus computed is 
$$%\delta r =
  \sqrt{5\over 4\pi}(0.6026)(-0.31460)
  \biggl({3\over 2}\sin^2\phi-{1\over 2}\biggr)=-0.1196
\biggl({3\over 2}\sin^2\phi-{1\over 2}\biggr)\ {\rm meters},\eqno(17a)$$
and the transverse component, which is in the northward direction, is
$$%\delta\vec{t} = 
\sqrt{5\over 4\pi}(0.0831)(-0.31460) 3\cos\phi\sin\phi 
=-0.0247 \sin 2\phi\  {\rm meters}.\eqno(17b)$$

This permanent deformation  must be removed from the displacement 
vector computed by the procedure described above in order to obtain 
the temporally varying part of the tide-induced site displacement.  If 
the nominal values for the anelastic case ($h_2=0.6078, l_2= 0.0847$)
were used in  the first step computation, one should have $-0.1206$ 
and $-0.0252$ respectively in the expressions for the radial and 
northward components (instead of $-0.1196$ and $-0.0247$).  

If the $h^{(2)}$-latitude dependence of the Love numbers were 
accounted for in the first step, {\it i.e.} if the change $h_2$ $\to$ 
$h(\phi) = h^{(0)} + h^{(2)} [(3 \sin^2\phi -1)/2]$ and $l_2$ $\to$ 
$l(\phi) = l^{(0)} + l^{(2)} [(3 \sin^2\phi -1)/2]$ were made,
the values will change accordingly.

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 get the permanent tide at the station, one should use the 
same formula (equations (17a) and (17b)) replacing the Love numbers by 
the fluid limit Love numbers which are $l=0$ and $h=1+k$ with $k=0.94$.

\write 0 {\indent Rotational Deformation Due to Polar Motion \noexpand
\dotfill\the\pageno\noexpand\break}
\noindent{\bf Rotational Deformation Due to Polar Motion}

\nobreak

     The variation of station coordinates caused by the pole
tide is recommended to be taken into account.  Let us choose $\hat{x},
\hat{y}$ and $\hat{z}$ as a terrestrial system of reference. The 
$\hat{z}$ axis is oriented along the Earth's mean rotation axis, the 
$\hat{x}$ axis is in the direction of the adopted origin of longitude
and the $\hat{y}$ axis is orthogonal to the $\hat{x}$ and $\hat{z}$ 
axes and in the plane of the $90^\circ$ E meridian.

     The centrifugal potential caused by the Earth's rotation is
$$V={1\over 2}[r^2|\vec{\Omega}|^2-(\vec{r}\cdot\vec{\Omega})^2],
\eqno(18)$$
where $\vec{\Omega}=\Omega(m_1\hat{x}+m_2\hat{y}+(1+m_3)\hat{z})$. 
$\Omega$ is the mean angular velocity of rotation of the Earth, $m_i$
are small dimensionless parameters, $m_1$, $m_2$ describing polar
motion and $m_3$ describing variation in the rotation rate, $r$ is the
radial distance to the station.

     Neglecting the variations in $m_3$ which induce displacements
that are below the $mm$ level, the $m_1$ and $m_2$ terms give a first
order perturbation in the potential $V$ (Wahr, 1985)
$$\Delta V(r,\theta,\lambda)=-{\Omega^2r^2\over 2}\sin2\theta\,
(m_1\cos\lambda+m_2\sin\lambda).\eqno(19)$$

     The radial displacement $S_r$ and the horizontal displacements 
$S_\theta$ and $S_\lambda$ (positive upwards, south and east 
respectively in a horizon system at the station) due to $\Delta V$ are 
obtained using the formulation of tidal Love numbers (Munk and 
MacDonald, 1960):
$$S_r=h_{2}{\Delta V\over g},$$
$$S_\theta={\ell_{2}\over g}\partial_\theta\Delta V,\eqno(20)$$
$$S_\lambda={\ell_{2}\over g}{1\over\sin\theta}\partial_\lambda
\Delta V.$$

     In general, these computed displacements have a non-zero
average over any given time span because $m_1$ and $m_2$, used to find
$\Delta V$, have a non-zero average.  Consequently, the use of these
results will lead to a change in the estimated mean station
coordinates.  When mean coordinates produced by different users
are compared at the centimeter level, it is important to ensure
that this effect has been handled consistently.  It is recommended
that $m_1$ and $m_2$ used in Equation 19 be replaced by parameters 
defined to be zero for the Terrestrial Reference Frame discussed in 
Chapter 3.

     Thus, define
$$x_p=m_1-\bar{m_1},\eqno(21)$$
$$y_p=-(m_2-\bar{m_2}),$$
where $\bar{m_1}$ and $\bar{m_2}$ are the values of $m_1$ and $m_2$ for
the Terrestrial Reference Frame.  Then, using Love number values 
appropriate to the pole tide ($h = 0.6027$, $l= 0.0836$) and $r = a = 
6.378 \times 10^6$m, one finds 
$$S_r=-32\sin2\theta(x_p\cos\lambda-y_p\sin\lambda)\ {\rm mm},$$
$$S_\theta=-9\cos2\theta(x_p\cos\lambda-y_p\sin\lambda)\ {\rm mm},
\eqno(22)$$
$$S_\lambda=9\cos\theta(x_p\sin\lambda+y_p\cos\lambda)\ {\rm mm}.$$
for $x_p$ and $y_p$ in seconds of arc.

     Taking into account that $x_p$ and $y_p$ vary, at most, 0.8 arcsec,
the maximum radial displacement is approximately 25 $mm$, and
the maximum horizontal displacement is about 7 $mm$.

     If $X$, $Y$, and $Z$ are Cartesian coordinates of a station in a
right-handed equatorial coordinate system, we have the displacements
of coordinates
$$[dX, dY,dZ]^T=R^T[S_\theta,S_\lambda,S_r]^T,\eqno(23)$$
where
$$R=\pmatrix{\cos\theta\cos\lambda&\cos\theta\sin\lambda&-\sin\theta\cr
-\sin\lambda&\cos\lambda&0\cr
\sin\theta\cos\lambda&\sin\theta\sin\lambda&\cos\theta\cr}.$$

     The deformation caused by the pole tide also leads to time 
dependent perturbations in the $C_{21}$ and $S_{21}$ geopotential 
coefficients (see Chapter 6).  

\write 0 {\indent Antenna Deformation \noexpand\dotfill\the\pageno
\noexpand\break}
\noindent{\bf Antenna Deformation}
 
\nobreak

Changes in antenna height and axis offset due to temperature
changes can be modeled simply.

Let $V$ be the antenna height and $A$ the axis offset. changes
due to temperature, $T$, are then given by
$$dV = k V(T-T_0),\ {\rm and}$$
$$dA = k'A(T-T_0),$$
\noindent where $k$ and $k'$ are estimated constants and $T_0$
is the reference temperature. Typically $k$ and $k'$ are of the 
order of $10^{-5}$.

\write 0 {\indent Atmospheric Loading \noexpand\dotfill\the\pageno
\noexpand\break}
\noindent{\bf Atmospheric Loading}
 
\nobreak

   Temporal variations in the geographic distribution of atmospheric 
mass load the Earth and deform its surface. Displacement variations are
dominated by effects of synoptic pressure systems; length scales of
1000-2000 km and periods of two weeks.  Pressure loading effects
are larger at high latitudes due to the larger storms found there.
Effects are smaller at low latitude sites (35S to 35N) and at sites
within 300 km of the sea or ocean.  Theoretical studies by Rabbel
and Zschau (1985), Rabbel and Schuh (1986), vanDam and Wahr (1987),
and Manabe {\it et al.} (1991) demonstrate that vertical displacements 
of up to 25 $mm$ are possible with horizontal effects of one-third this
amount.
 
   All pressure loading analyses make the assumption that 
the response of the ocean to changes in air pressure is inverse
barometric.  It is likely that the ocean responds to pressure as an
inverted barometer at periods of a few days to a few years. (See Chelton
and Enfield (1986) or Ponte {\it et al.} (1991) for a summary of 
observational evidence for the inverted barometer response.)  A local 
inverted barometer response is probably appropriate for periods as 
short as 3 or 4 days.  On the other hand, the decidedly nonequilibrium 
diurnal ocean tides imply that the global response is certainly not 
an inverted barometer at periods close to a day.

   There are many methods for computing atmospheric loading 
corrections.  In contrast to ocean tidal effects, analysis of the 
situation in the atmospheric case does not benefit from the presence 
of a well-understood periodic driving force.  Otherwise, estimation
of atmospheric loading via Green's function techniques is
analogous to methods used to calculate ocean loading effects. 
Rabbel and Schuh (1986) recommend a simplified form of the dependence
of the vertical crustal displacement on pressure distribution. 
It involves only the instantaneous pressure at the site in
question, and an average pressure over a circular region C
with a 2000 km radius surrounding the site.  The expression
for the vertical displacement ($mm$) is
$$\Delta r=-0.35p-0.55\bar{p}\eqno(15)$$  
where $p$ is the local pressure anomaly with respect to the
standard pressure of 101.3 kPA (equivalent to 1013 mbar), and $\bar{p}$
the pressure anomaly within the 2000 km circular region mentioned
above.  Both quantities are in $10^{-1}$ kPA (equivalent to
mbar).  Note that the reference point for 
this displacement is the site location at standard pressure. 
Equation 15 permits one to estimate the seasonal displacement
due to the large-scale atmospheric loading with an error less
than $\pm$1 $mm$ (Rabbel and Schuh, 1986).
  
   An additional mechanism for characterizing $\bar{p}$ may be applied.
The two-dimensional surface pressure distribution surrounding a site is
described by
$$p(x,y) = A_0 + A_1x + A_2y + A_3x^2 + A_4xy + A_5y^2,$$
where $x$ and $y$ are the local East and North distances of the
point in question from the VLBI site.  The pressure anomaly $\bar{p}$
may be evaluated by the simple integration
$$\bar{p}={\int\int_C dx\ dy\ p(x,y)\over\int\int_C dx\ dy}$$
giving
$$\bar{p} = A_0 + (A_3 + A_5)R^2/4,$$
where $R^2=(x^2 + y^2)$.

It remains the task of the data analyst to perform a quadratic
fit to the available weather data to determine  the coefficients
$A_{0-5}$.  

   vanDam and Wahr (1987) computed the displacements due to
atmospheric loading by performing a convolution sum between barometric 
pressure data and the mass loading Green's Function.  They found that 
the corrections based on Equation 15 are inadequate for stations 
close to the coast.  For these coastal stations, Equation 15 can be 
improved by extending the regression equation.

   A few investigators (Manabe {\it et al.}, 1991; MacMillan and 
Gipson, 1994; vanDam and Herring, 1994; vanDam {\it et al.}, 1994) 
have attempted to calculate site dependent pressure responses by 
regressing local pressure changes with theoretical, VLBI or GPS height 
variations.  For example vanDam and Herring (1994) find significant 
site displacements in some VLBI stations which can be modeled by a 
simple linear regression.  Table 7.4 shows the relationship between 
variations in local atmospheric pressure and theoretical radial surface 
displacements for these sites as found in their analysis.  This 
rate is available by anonymous ftp to {\tt gracie.grdl.noaa.gov}.

\vfil\eject
\noindent Table 7.4.  Regression Coefficients
\hrule
Station  \kern 1pc Slope (mm/mbar)
\halign{\kern 2pc#\hfil&\kern 2pc#\hfil\cr
GILC  &  -0.45 $\pm$ .001\cr
ONSA  &  -0.32 $\pm$ .001\cr
WETT  &  -0.46 $\pm$ .002\cr
WEST  &  -0.43 $\pm$ .002\cr
HATC  &  -0.40 $\pm$ .002\cr
NRAO  &  -0.41 $\pm$ .002\cr
KASH  &  -0.04 $\pm$ .020\cr
MOJA  &  -0.42 $\pm$ .004\cr
GRAS  &  -0.55 $\pm$ .002\cr
RICH  &  -0.35 $\pm$ .002\cr
KAUA  &  -0.26 $\pm$ .005\cr}
\hrule
\medskip
\write 0 {\indent Postglacial Rebound \noexpand\dotfill\the\pageno
\noexpand\break}
\noindent{\bf Postglacial Rebound}

\nobreak

     The current state-of-the-art model of the global process
of postglacial rebound is that described in Peltier (1994).
The model consists of a history of the variations in ice thickness
from the time of the last glacial maximum (LGM) until the present
coupled with a radial profile of viscosity in the planetary mantle.
The ice model, called ICE-4G, is a significant improvement over the
previous model of Tushingham and Peltier (1991) called ICE-3G in
that it has been constrained to fit the detailed history of relative
sea level rise at Barbados that is derivative of dated coral
sequences that extend from the LGM to the present.

     The planet's response to the model deglaciation event is computed
by first solving an integral equation to determine the site dependent
variation of ocean bathymetry since the LGM in order to ensure that
the ocean surface remains a gravitational equipotential.  Time
dependent, three dimensional displacement fields are then determined by
spectral convolution of the visco-elastic impulse response Green
function for the planet with the complete history of surface mass 
loading (with its distinct ice and ocean components).  A succinct 
review of the complete theory for the three dimensional displacement 
calculation and a recent application to the computation of VLBI 
baseline variations will be found in Peltier (1995).

     A file containing a listing of the radial and horizontal 
displacements for a number of sites is available by anonymous ftp.  ftp 
to maia.usno.navy.mil.  Change directories to conventions (cd 
conventions).  The file is called pgr.model.  Any comments or 
corrections to this file should be directed to Prof. Richard 
Peltier (peltier@rainbow.physics.utoronto.ca).
  
\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
Cartwright, D. E. and Edden, A. C., 1973, ``Corrected Tables of
       Tidal Harmonics," {\it Geophys. J. Roy. Astron. Soc.}, {\bf 33},
       pp. 253--264.

\item{}\kern-2pc
Chelton, D. B. and Enfield, D. B., 1986, ``Ocean Signals in Tide 
       Gauge Records," {\it J. Geophys.}, {\bf 91}, pp. 9081--9098.

\item{}\kern-2pc
Dehant, V., 1987, ``Tidal parameters for an inelastic Earth," {\it Phys.
       Earth Plan. Int.}, {\bf 49}, pp. 97--116.

\item{}\kern-2pc
Doodson, A. T., 1928, ``The Analysis of Tidal Observations,"
       {\it Phil. Trans. Roy. Soc. Lond.}, {\bf 227}, pp. 223--279.

\item{}\kern-2pc
Farrell, W. E., 1972, ``Deformation of the Earth by Surface
       Loads," {\it Rev. Geophys. Space Phys.}, {\bf 10}, pp. 761--797.

\item{}\kern-2pc
Flather, R. A., 1981, {\it Proc. Norwegian Coastal Current Symp.
       Geilo}, 1980, Saetre and Mork (eds), pp. 427--457.

\item{}\kern-2pc
Le Provost, C., Genco, M. L., Lyard, F., Incent, P., and Canceil, P.,
       1994, ``Spectroscopy of the world ocean tides from a finite
       element hydrological model," {\it J. Geophys. Res.}, {\bf 99},
       pp. 24777--24798.

\item{}\kern-2pc
MacMillan, D.S. and J.M. Gipson, 1994, ``Atmospheric Pressure Loading 
       Parameters from Very Long Baseline Interferometry Observations,"
       {\it J. Geophys. Res.}, {\bf 99}, pp. 18081--18087.

\item{}\kern-2pc
Manabe, S.T., T. Sato, S. Sakai, and K. Yokoyama, 1991, ``Atmospheric  
       Loading Effects on VLBI Observations," in Proceedings of the AGU
       Chapman Conference on Geodetic VLBI:  Monitoring Global Change, 
       {\it NOAA Tech. Rep. NOS 137 NGS 49}, pp. 111--122.

\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
Munk, W. H. and MacDonald, G. J. F., 1960, {\it The Rotation of the
     Earth}, Cambridge Univ. Press, New York, pp. 24--25.

\item{}\kern-2pc
Peltier, W. R, 1994, ``Ice Age Paleotopography," {\it Science}, 
     {\bf 265}, pp. 195--201.

\item{}\kern-2pc
Peltier, W.R., 1995, ``VLBI baseline variations from the ICE-4G model
     of postglacial rebound," {\it Geophys. Res. Lett.}, {\bf 22},
     pp. 465--468.

\item{}\kern-2pc
Ponte, R. M., Salstein, D. A., and Rosen, R. D., 1991, ``Sea Level 
       Response to Pressure Forcing in a Barotropic Numerical Model," 
       {\it J. Phys. Oceanogr.}, {\bf 21}, pp. 1043--1057.

\item{}\kern-2pc
Rabbel, W. and Schuh, H., 1986, ``The Influence of Atmospheric
       Loading on VLBI Experiments," {\it J. Geophys.}, {\bf 59},
       pp. 164--170.

\item{}\kern-2pc
Rabbel, W. and Zschau, J., 1985, ``Static Deformations and Gravity
       Changes at the Earth's Surface due to Atmospheric Loading,
       " {\it J. Geophys.}, {\bf 56}, pp. 81--99.

\item{}\kern-2pc
Row, L. W., Hastings, D. A., and Dunbar, P. K., 1995, ``TerrainBase
       Worldwide Digital Terrain Data," NOAA, National Geophysical
       Data Center, Boulder CO.

\item{}\kern-2pc
Scherneck, H. G., 1983, {\it Crustal Loading Affecting VLBI Sites},
       University of Uppsala, Institute of Geophysics, Dept. of
       Geodesy, Report No. 20, Uppsala, Sweden.

\vfil\eject
\item{}\kern-2pc
Scherneck, H. G., 1990, ``Loading Green's functions for a continental
       shield with a Q-structure for the mantle and density constraints
       from the geoid," {\it Bull. d'Inform. Mar\'ees Terr.}, {\bf 108},
       pp. 7757--7792.

\item{}\kern-2pc
Scherneck, H. G., 1991, ``A Parameterized Solid Earth Tide
       Model and Ocean Tide Loading Effects for Global Geodetic
       Baseline Measurements," {\it Geophys. J. Int.}, {\bf 106},
       pp. 677--694.

\item{}\kern-2pc
Scherneck, H. G., 1993, ``Ocean Tide Loading:  Propagation of
       Errors from the Ocean Tide into Loading Coefficients,"
       {\it Man. Geod.}, {\bf 18}, pp. 59--71.

\item{}\kern-2pc
Schwiderski, E. W., 1983, ``Atlas of Ocean Tidal Charts and
       Maps, Part I: The Semidiurnal Principal Lunar Tide M$_2$,"
       {\it Marine Geodesy}, {\bf 6}, pp. 219--256.

\item{}\kern-2pc
Schwiderski, E. W. and Szeto, L. T., 1981, NSWC-TR 81-254,
       Naval Surface Weapons Center, Dahlgren Va., 19 pp.

\item{}\kern-2pc
Tamura, Y., 1987, ``A harmonic development of the tide-generating
       potential," {\it Bull. d'Inform. Mar\'ees Terr.}, {\bf 99},
       pp. 6813--6855.

\item{}\kern-2pc
Tushingham, A. M. and Peltier, W. R., 1991, ``Ice-3G:  A New Global
       Model of Late Pleistocene Deglaciation Based Upon Geophysical
       Predictions of Post-Glacial Relative Sea Level Change," 
       {\it J. Geophys. Res.}, {\bf 96}, pp. 4497--4523.

\item{}\kern-2pc
Tushingham, A. M. and Peltier, W. R., 1991, ``Validation of the ICE-3G
       Model of Wurm-Wisconsin Deglaciation Using a Global Data Base
       of Relative Sea Level Histories," {\it J. Geophys. Res.}, 
       {\bf 97}, pp. 3285--3304.

\item{}\kern-2pc
vanDam, T. M. and Wahr, J. M., 1987, ``Displacements of the
       Earth's Surface due to Atmospheric Loading:  Effects on
       Gravity and Baseline Measurements," {\it J. Geophys. Res.},
       {\bf 92}, pp. 1281--1286.

\item{}\kern-2pc
vanDam, T. M. and Herring, T. A., 1994, ``Detection of Atmospheric
       Pressure Loading Using Very Long Baseline Interferometry
       Measurements," {\it J. Geophys. Res.}, {\bf 99}, pp. 4505--4517.

\item{}\kern-2pc
vanDam, T. M., Blewitt, G., and Heflin, M. B., 1994, ``Atmospheric 
       Pressure Loading Effects on Global Positioning System Coordinate 
       Determinations," {\it J. Geophys. Res.}, {\bf 99}, pp. 
       23,939--23,950.

\item{}\kern-2pc
Verheijen and Schrama, 1995, personal communication.

\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. M., 1985, ``Deformation Induced by Polar Motion," {\it J.
     Geophys. Res.}, {\bf 90}, pp. 9363--9368.

\item{}\kern-2pc
Wang, R., 1994, ``Effect of rotation and ellipticity on earth tides,"
      {\it Geophys. J. Int.}, {\bf 117}, pp. 562--565.

\item{}\kern-2pc
Zschau, J., 1983, ``Rheology of the Earth's mantle at tidal and Chandler
       Wobble periods," {\it Proc. Ninth Int. Symp. Earth Tides}, New
       York, 1981, J. T. Kuo (ed), Schweitzerbart'sche
       Verlagsbuchhandlung, Stuttgart, pp. 605--630.

\vfil\eject
