The importance of the Earth's tides on Earth coordinates were examined for the following reasons: (1) the precision for obtaining the Earth's coordinates shows that the effects of Earth tides appear on the values obtained for the coordinates; (2) the possibility of determining the values of the Earth tides; and (3) the consideration of theoretical models that can compute the values of Earth tides. The astronomical and geodetic coordinates of a point at the Earth's surface are described

Vincente, R. O.

English

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The Influence of Earth Tides on Earth's Coordinates
R. O. Vicente
Department of Applied Mathematics, Faculty of Sciences
Lisbon, Portugal
The importance of Earth tides on Earth coordi-
nates has been considered only recently for several
reasons: 1) the precision that we are obtaining
nowadays for Earth's coordinates shows that the ef-
fects of Earth tides appear on the values obtained
for the coordinates; 2) the possibility of deter-
mining, by observations, the values of Earth tides;
3) the consideration of theoretical models that can
compute the values of Earth tides.
We are going to examine briefly some of these
reasons. First of all, it should be pointed out
that we have to be careful about the definitions of
what we mean by Earth coordinates.
When it was only possible to obtain a precision
of a few meters for the values of the Earth coordi-
nates, some of the systems of reference adopted in
the dynamics of the Earth rotation could not be dis-
tinguished within that precision. But when we are
aiming at precisions of a few centimeters, it is
very important to define carefully the systems of
reference employed for our coordinates on the Earth.
We can consider, for instance, astronomical and
geodetic coordinates of a point at the Earth's sur-
face.
The astronomical coordinates are referred to the
astronomical local vertical, and therefore, to the
instantaneous axis of rotation of the Earth, brief-
ly called the axis of rotation. The intersection of
this axis with the Earth's surface defines the geo-
graphic poles. The complement of the acute angle
between the astronomical vertical and the axis of
rotation of the Earth is called the astronomical
latitude. For the definition of astronomical longi-
tude we need a reference meridian passing by the
geographic poles [Woolard and Clemence, 1966].
Considering the case of the real Earth, there
are known irregularities in the direction of the
astronomical vertical from place to place over the
Earth, and, therefore, the astronomical meridians,
parallels of latitude and the equator are irregular
curves of double curvature, but they do not depart
very far from plane curves.
The instruments (visual zenith telescopes (VZT),
photographic zenith telescopes (PZT) and astro-
labes) , employed in the classical techniques of de-
termining polar motion, refer their observations to
the'astronomical vertical.
Depending on the adopted pole and reference mer-
idian, we can have different definitions of the co-
ordinates of a point on the Earth's surface. The
danger, nowadays, is that different people speak
about different poles and, therefore, the coordi-
nates they are speaking about are not the same.
The simplest Earth model considers the Earth as
a solid rigid body. Even this simple model shows
the very important distinction between free and
forced motions, originating periodic displacements
of the axes employed in the dynamics of the Earth
rotation. They have been called nutations because
Proc. of the 9th GEOP Conference, An International Symposium on the Applications of
Ceoilesy lo Giojymmiics, October 2-S, 1978. Dept. of Geodetic Science Rept. No. 280. The
Ohio State Univ., Columbus, Ohio 43210.
the displacements are periodic.
The equations of motion can be represented by
the vector equation
dt ~
showing that thg time derivative of the angular mo-
mentum vector H around the centre of mass is equal
to the vector moment G of the external forces,
mainly due to the Sun and Moon.
When there are no external forces acting on the
Earth G=0, therefore, !? is constant and fixed in
space. We have the so-called free nutation because
this type of motion exists even without external
forces. The important consequence is the fact that
any free nutation changes the Earth coordinates
showing, for instance, variations of latitude, but
it does not affect star places on the celestial
sphere. The name wobble has, unfortunately, been
recently employed to designate the free nutations.
When there are external forces Acting on the
Earth, G/0, and the position of H is not fixed
in space. We obtain the forced nutations that are
so important in any problem dealing with the trans-
formation of reference systems from one epoch to
another because they affect star places.
One aspect of the actions of the Sun and the
Moon on the Earth concerns the tidal attractions of
the luni-solar forces, which deform the Earth, giv-
ing rise to the tides of the solid Earth.
Different components of the Earth tide produce
the nutations that are important in astronomy, and
the forced nutations correspond to diurnal tides,
so there is a close connection between nutations
and tides.
It is fortunate that, all the studies concerned
with the tides of the solid Earth employ the same
standard, that -is, the bodily .tide numbers h_, Ic
and £ are defined considering a statical theory
applied to an elastic solid Earth with spherical
symmetry, and the disturbing tidal potential is a
spherical harmonic of the second degree [Jeffreys,
1976].
The value of k is related to the period of the
free Eulerian nutation, and the value of A=l+k-£
is influenced by disturbances that affect the
position of the astronomical vertical of a place on
the Earth, that is, there are variations in the
geographic coordinates of a place with periods de-
pending on the periods of the tidal forces.
The actions of the Sun and Moon deflect the
position of the astronomical vertical, and the
maximum deviation is of the order of O'.'OS; they are
the most regular of the variations in direction of
the astronomical vertical. There are remaining
variations due to geophysical causes, not yet very
well known, and the tidal variations are very much
modified by local geophysical factors.
In practice, the determination of the values of
the bodily tide numbers is more difficult because
there are local corrections.
337
https://ntrs.nasa.gov/search.jsp?R=19790013339 2020-03-21T23:58:39+00:00Z
TABLE 1. Diurnal Waves
Jeffreys-Vicente
1957
Central Particle
1.183
1.209
1.221
0.714
0.676
0.658
Roche
1.185
1.172
1.211
0.693
0.696
0.658
Molodensky
1961
2
6=l+h - -| k
1.151
1.161
1.166
Y=l+k-h
0.721
0.695
0.686
Shen-Mans inha
1976
6
1.1380
1.1539
1.1596
0.7340
0.7019
0.6895
Observations
1.1406 A.M.
1.1507 W.M.
1.1699 A.M.
1.1664 W.M.
1.1522 A.M.
1.1676 W.M.
0.7422 A.M.
0.7501 W.M.
0.7068 A.M.
0.7167 W.M.
. 0.6785 A.M.
0.6752 W.M.
Kl
Pl
°1
1.120
1.149
1.160
1.162
1.180
1.183
1.1689
1.2033
1.2055 . ;
A.M. Arithmetic Mean
W.M. Weighted Mean
It is convenient to have the possibility of de-
termining the values of ^  and _fc by processes that do
not depend on any'geophysical assumptions. This
fact shows the importance of having good determi-
nations of the period of the free Eulerian nutation,
called the Chandler period, that permit the deter-
mination of the value of Ic, and the advantage of
careful analysis of the observations of variation of
latitude that give the possibility of computing
(l+k-2.). Some of the modern techniques that can ob-
serve polar motion have an important role to play in
these studies.
The observations of Earth tides are interpreted
in amplitude ratio form (observed to theoretical am-
plitude), and we can consider, for instance, the
following linear combinations for the bodily tide
numbers:
for the horizontal component =l-h+k -
for the vertical component =l+h - -~k
The computed values of Earth tides show that the
maximum effect will correspond to a radial defor-
mation of about 30 cm. We can see the importance
of Earth tides for the present day goal of achiev-
ing precisions of a few centimeters on the Earth's
coordinates.
It has been demonstrated that the application of
a statical theory for the semi-diurnal, fortnightly
and semi-annual tides does not introduce appreci-
able errors in theoretical models [Jeffreys, 1949].
But for diurnal tides we have to consider a dynami-
cal theory and the importance of the liquid core of
the Earth has been proved. The main waves are:
Doodson's code
number
165.555
163.555
145.555
Tidal
component
Nutation
component
Precession
Semiannual
Fortnightly
The observations of Earth tides show a very ir-
regular distribution over the Earth. The majority
of stations are concentrated in Europe and North
America. Local effects have been detected, for in-
stance, due to the proximity of oceans and big
rivers.
It. is fortunate that .there is an "International
Centre for Earth Tides".where all the observations
are collected and the computations performed, there-
fore, we have a consistent treatment of the data
which is very important.
Some results of theoretical and observed values
for S, y and A are indicated in Table 1.
The observations correspond to 21 stations with
very irregular distribution on the Earth [Melchior,
1971]. We can see that the consideration of arith-
metic mean or weighted mean even alters some values.
This is a good example about the difficulties encoun-
tered in the combination of observations which become
important for the precision we are trying to achieve.
The consideration of the fourth decimal place has
probably no physical meaning at the present time.
The theoretical models indicated correspond to
different approximations that were made. The values
computed by Jeffreys-Vicente [1957] and Molodensky
[1961] were based on models of the internal struc-
338
ture of the Earth that appeared about 25 years ago,
taking account of the liquid core.
Jeffreys-Vicente employed two models (central
particle and Roche) which were considered as repre-
senting extreme cases of the possible behaviour of
the core. This way of looking at the problem cor-
responds to setting up an upper and a lower bound
to the behaviour of the core, considering that the
knowledge about the structure of the core was not
very detailed at the time. More recent models,
based on a better knowledge of the structure of the
core, have confirmed the general trends found by the
central particle and Roche models.
Molodensky model 2 considers an inner core. It
should be pointed out that the existence of the in-
ner core introduces difficulties, and it has been
shown that the partial differential equations to be
satisfied are hyperbolic and the boundary conditions
have to be considered carefully [Jeffreys and
Vicente, 1966, p. 24].
Shen-Mansinha [1976] employed a model 6 cor-
responding to an elliptical rotating Earth with a
liquid outer core. The consideration of the other
models, designated by 8=-0.2 and 8=+0.2, does not
alter the conclusions.
The theoretical values obtained for the vertical
component 6, written on Table 1, show that model
6 presents slightly smaller values for all waves
in comparison with the other models.
Considering the wave 0., we see that the verti-
cal component 6 is more affected by different
types of models than the horizontal component Y-
For the horizontal component y> the model B
shows slightly greater values than Molodensky model
2, being more in agreement with the observations.
A few theoretical values of A are computed and
there is no comparison with the observations because
it is very difficult, at the present time, to obtain
reliable observational values for these waves.
The models show general agreement with the ob-
servations because we cannot rely too much on the
geophysical meaning of the third and fourth decimal
places. The differences among the theoretical
models are within five per cent.
Another important aspect is the frequency de-
pendence of the bodily tide numbers which was re-
vealed by the earlier investigations [Jeffreys and
Vicente, 1957]. This result means that the behav-
iour of the Earth is very much conditioned, for the
diurnal tides, by the period of the waves consider-
ed, as it is shown by the values written on Table 1.
The existence of damping and some recent deter-
minations of Q values, depending on the period of
the motion considered, show some of the difficul-
ties encountered in the researches about Earth
tides.
The global values of the bodily tide numbers have
been employed to allow for the influence of Earth
tides on the coordinates of a point on the Earth,
but the improvement in the precision of the deter-
mination of coordinates would benefit from local
observations of Earth tides by the appropriate
techniques.
The waves with greater amplitudes appear in the
following sequence: K.. (due to the Moon and Sun) ,
0, (due to the Moon) and P.. (due to the Sun and
Moon); and the frequencies are well defined by the
theory of the diurnal tides. The great advantage of
the bodily tides is the fact of their well-known
periodicities.
The analyses of the observations, made by some of
the modern techniques, should reveal these periodi-
cities if they correspond to an adequate interval of
time. It is nevertheless necessary that the obser-
vational programme is well planned and does not suf-
fer from discontinuities. The modern techniques,
employing artificial satellites, the Moon and radio
interferometry, offer the advantage of providing
global values.
We are attempting to obtain precisions of the
order of a few centimeters and that raises the ques-
tion if we are nowadays approaching the noise level
of the system. In this case, we mean by noise level
of the system, the possible range of values for the
coordinates of a point on the Earth which are af-
fected by numerous phenomena, some of them not very
well known, and corresponding to the more general
case of an Earth model considered as a collection of
particles subject to so many forces.
References
Jeffreys, H., Dynamic effects of a liquid core,
Monthly Not. R. Astr. Soc. , 109, 670-687, 1949.
Jeffreys, H., The Earth, Camb. Univ. Press, 1976.
Jeffreys, H. and R. 0. Vicente, The theory of nu-
tation and the variation of latitude, Monthly
Not. R. Astr. Soc., 117, 142-173, 1957.
Jeffreys, H. and R. 0. Vicente, Comparison of forms
of the elastic equations for the earth, Mem. Cl.
Sci. Acad. Roy. Belgique, 2e. ser., 37, fasc. 3,
1966.
Melchior, P., Precession-nutations and tidal po-
tential, Celestial Mech. , 4_, 190-212, 1971.
Molodensky, M. S., Communications Obs. Roy.
Belgique. No. 188, 1961.
Shen, P. Y. and L. Mansinha, Oscillation, nutation
and wobble of an elliptical rotating earth with
liquid outer core, Geophys. J. R. Astr. Soc., 46,
467-496, 1976.
Woolard, E. W. and G. M. Clemence, Spherical Astro-
nomy, Academic Press, 1966.
339


The influence of earth tides on earth's coordinates

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