The geographic coordinate system is defined by the set of
three mutually orthogonal basis vectors (e
, e
, e
).
e
and e
are in the equatorial plane with e
intersecting
the Greenwich meridian, e
intersecting the 90 East meridian,
and e
intersecting the geographic north pole. Quantities
with subscripts 1, 2, and 3 are components along these axes. Earth
rotation variations are excited by the motion of air and water as
they exchange angular momentum with the solid Earth, while
conserving absolute angular momentum within the Earth system.
The linearized Liouville equations, expressing this conservation
of angular momentum are, following Gross (1992), and Barnes et
al, (1983)
with the usual convention that (m
, m
, 1 + m
)
is
the rotation vector of the Earth as reported by the International
Earth Rotation Service (IAU,1993), 
is the mean angular
velocity, and the quantities (m
, m
, m
) are all
small dimensionless numbers of the order of 
or so. In
simple terms, equation (1) describes the Chandler Wobble (free
Eulerian nutation) of the Earth when motion of matter causes the
greatest moment of inertia (principal) axis to be displaced from
the rotation axis. A similar wobble is readily seen in a poorly thrown
toy disk (Frisbee) when rotation and principal axes are
misaligned. Similarly, equation 2 describes changes in the Earth's spin
rate as axial angular momentum is exchanged between the solid Earth
and various constituents in the Earth system, such as the atmosphere
or oceans. The very simple form of equation (1) comes from the use
of complex notation to describe polar motion, in which the real axis
is identified with the Greenwich meridian and the imaginary axis with
90 degrees East longitude. In this notation, m is the quantity (m
i m
). Other terms in (1) and (2) are: relative angular momentum in
the Earth system due to winds and currents described by the vector
(h
, h
, h
); the complex quantity h = (h
+ ih
);
the polar moment of inertia of the Earth, C, and the equatorial moment
of inertia A, excluding the fluid core, which is assumed uncoupled from
the mantle; the complex quantity 
which describes fluctuations in products of
inertia associated with the (e
,e
) plane 
,
and the (e
,e
) plane, 
; 
,
which describes changes in the moment of inertia about e
; finally,
is 
, the complex Chandler
Wobble frequency with F near 0.843 cycles per year (cpy) and Q,
the dimensionless quality factor, near 175. Alternative expressions
for the left hand side can be given in terms of torques applied by air
and water to the Earth. There are a few interesting remarks to be
made concerning equations (1) and (2), pertaining to underlying theory,
and connections with other geodetic problems.
First, the left hand side of (1) has only recently been shown to be valid for polar motion observations reported in terms of the celestial ephemeris pole (Eubanks, 1993; Gross, 1992; Brzezinski, 1992; Brzezinski and Capitaine, 1993).
Second, (1) does not apply to PM near retrograde frequencies of 1 cycle per day (cpd), close to the free core nutation frequency. Such motion is best treated as nutation, (Herring and Dong, 1994; Watkins and Eanes, 1994; Sovers et al, 1993; Gross, 1993).
Third, quantities in brackets [] reflect the loading response of the Earth, with numerical values dependent upon an adopted physical model. These values may also be frequency dependent. In principle, they may be experimentally obtained, given accurate observations of quantities on both left and right hand sides, but this is difficult given available observations of the atmosphere and oceans (Chao, 1994).
Fourth, the complex Chandler frequency is also dependent upon
the physical properties of the Earth. Current estimates of F = 0.843 cpy
and Q = 175 were obtained assuming that the excitation process is
random, Gaussian, and stationary (Jeffreys,1940;Wilson and
Vicente,1990). Kuehne et al (1993) have shown that the excitation
is actually not stationary, showing strong seasonal variance
fluctuations. Thus, improved estimates of 
should be
possible, and continue to be of interest as a measure of global rheology
at a frequency well below the seismic band.
Fifth, the inertia terms on the left hand side of (1) and (2)
are proportional to changes in the spherical harmonic coefficients of
the global gravity field, commonly called the Stokes coefficients.
In particular, those in (1) are proportional to the degree 2-order
1 coefficients, and in (2) to the degree 2-order zero (zonal)
Stokes coefficient. Therefore, estimating inertia changes which cause
Earth rotation variations is a subset of a more general problem of
current interest, estimating time variations in the Earth's gravity
field and center of mass (Chao and Au, 1991; Mitrovica and Peltier,
1993; Peltier, 1994; Nerem et al, 1993; Trupin et al, 1990; Trupin,
1993; Chen et al, 1994; Dong et al, 1994; Vigue et al, 1992). One of
the principal data types used in the gravity field problem is
SLR observations of geodetic satellites (Gutierrez and Wilson, 1987).
This makes SLR important to Earth rotation studies in two separate ways,
by providing accurate observations of the right hand side of (1) and
(2), as is now routine, and by providing estimates of the inertia
tensor terms on the left hand side, which is a promise for the future.
In the case of LOD (equation 2), the prospects for success are
excellent, because changes in even zonal Stokes coefficients perturb
the precession rate of the satellite node (Lambeck, 1988, Chapter 6).
On the other hand, SLR data are unlikely to provide direct estimates
of 
at short periods because changes in the tesseral (non
zonal) spherical harmonics do not perturb satellite orbital elements in
a simple way. However, changes in 
may be inferred from PM
at long periods, because in this limit m(t) becomes directly proportional
to 
. This is a consequence of the rate term dm/dt
becoming small, and the likelihood that relative motion
contributions, h(t), also diminish at long periods.
Sixth, there is a developing synergy between geodetic technology and global numerical models of the atmosphere, oceans, and hydrologic cycle. The same air and water loads causing PM and LOD changes are now recognized as a major contributor to non-tidal, non-tectonic displacements at geodetic observatories (vanDam and Herring, 1994; Blewitt, 1994; vanDam et al, 1994). Another air-water connection to geodetic positioning is via GPS- and VLBI-determined delay corrections for tropospheric water vapor (MacMillan and Ma, 1994). With the impending proliferation of GPS receivers world-wide, these corrections should provide useful measures of atmospheric water vapor for assimilation into global hydrologic and atmospheric models. Thus, global summaries of air and water distribution, now used to explain PM and LOD changes, will eventually improve the space geodetic methods by which PM and LOD are observed, and, in turn, will benefit from new water vapor data provided by the geodetic stations.