Generalization of Farrell's loading theory

1
2005 AGU fall meeting, 5-9 December 2005, San
Francisco, USA
Generalization of Farrell's loading theory for
applications to mass flux measurement using
geodetic techniques
J. Y. Guo(1,2), C.K. Shum(1)
(1) Laboratory for Space Geodesy and Remote
Sensing, School of Earth Sciences, The Ohio State
University, Columbus, Ohio, USA (2) The Key
Laboratory of Geospace Environment and Geodesy,
Ministry of Education, School of Geodesy and
Geomatics, Wuhan University, China.
ABSTRACT  Surface mass loading deforms the Earth
by the action of gravitation and pressure. In the
classical ocean tide loading theory Longman,
1962, 1963 Farrell, 1972, several assumptions
have been made (1) The Earth is assumed to be
spherically symmetrical, non-rotating, elastic
and isotropic (SNREI) (2) The mass of load is
approximately considered to be confined in a thin
shell of negligible thickness located at the
surface of the spherical Earth (3) The pressure
at the Earths surface and the gravitational
force that the Earth exerts on the mass of load
are in balance. In this work we generalize the
loading theory in two aspects (1) In the case of
atmospheric loading, we take into account of the
atmospheric thickness (2) In the case of tsunami
loading, we take into account the fact that the
pressure at the Earths surface and the
gravitational force that the Earth exerts on the
mass of load are not in balance. For both these
cases, two sets of load Love numbers need to be
defined, since the effects of gravitation and
pressure can no longer be treated together like
in the classical loading.
Introduction Modern geodetic techniques,
including the Global Positioning System (GPS),
superconducting gravimeter (SG), absolute
gravimeter (AG) and the Gravity Recovery and
Climate Experiment (GRACE) satellite mission, are
providing accurate data that requires knowledge
of various loading effects. In some
applications, the loading effects need to be
removed from the data, for example, when using
GPS data to monitor tectonic movement and
postglacial rebound, when using AG data to detect
gravity variations caused by vertical crustal
movement, and when using SG data to detect
gravity variations caused by oscillatory flow in
the core. In some other applications the data are
inverted to recover mass transfer at the Earths
surface, of which the underlying principle is
also the loading theory. Surface mass loading
deforms the Earth by the action of gravitation
and pressure. In classical ocean tide loading
theory for spherically symmetrical, non-rotating,
elastic and isotropic (SNREI) Earth model
Longman, 1962, 1963 Farrell, 1972, 2
assumptions are made on the mass of load. The
first assumption is that the mass of load is
considered to be confined in a thin shell of
negligible thickness located at the surface of
the spherical Earth. The second one is that the
pressure at the Earths surface balances the
gravitational force exerted on the mass of load.
Let us denote the ocean surface height relative
to the average position by , the density of
ocean water by , and the gravity at the
Earths surface by . The Earth is then
deformed by the gravitation of a thin layer of
mass with a surface density , and a
pressure at the surface of the
spherical Earth. In this work we generalize the
loading theory for two problems. In the case of
the atmospheric loading, we recognize the fact
that the atmospheric pressure is exerted at the
Earths surface, but the atmospheric mass is
distributed over the elevation from the Earths
surface to the top of the atmosphere. In the case
of tsunami loading, we take into account the fact
that pressure at the Earths surface and
gravitational force exerted on the mass of load
are not in balance. In both cases, two sets of
load Love numbers need to be defined, since the
effects of gravitation and pressure can no longer
be treated together as in classical loading
theory. The characteristic of the atmospheric
and tsunami loading in comparison to the
classical loading theory is shown in Fig. 1.
The possibly observable quantities of loading
effects include the vertical and horizontal
displacements, gravitational potential (geoid)
and gravity variations, tilt and strain.
Gravitational potential (geoid), gravity and tilt
can be divided into two parts the first is the
direct effect that is related to the
gravitational attraction of the load itself and
the second involves the indirect effect related
to the resulting deformation of the Earth. The
other quantities are only related to the
deformation of the Earth caused by the load and
thus are all indirect effects. In this work we
focus on the indirect effect. Formulation As the
classical loading theory of Longman 1962, 1963
and Farrell 1972 is well known, we write out
the generalized theories in comparison to the
classical theory. In the classical theory, only
one set of load Love numbers are defined that
include the effects of both pressure and
gravitation. For the two generalized cases we are
studying, the effects of pressure and gravitation
should be considered separately, thus two sets of
load Love numbers should be defined. All the
three sets of load Love numbers should be
computed by solving a set of six ordinary
differential equations with appropriate boundary
conditions. The differential equations to be
solved are the same
, The notations are quite standard
in the literature, and is not explained here. The
boundary conditions and the definition of the
various load Love numbers are listed in Table 1.
Atmospheric
Tsunami
Classical
Volume mass density
Surface mass density
Surface mass density
Pressure
Pressure
Pressure
Fig. 1 Comparison of three types of loading
Table 1 Boundary condition and definition of the
various load Love numbers
Boundary conditions
Classical
Pressure
Gravitation
Load Love numbers
Classical
Pressure
Gravitation
model Merriam, 1992 that the consideration of
the atmospheric thickness practically leads to
the same result as the classical loading theory
Guo et al., 2004. Conclusions Increasingly
accurate geodetic data require finer theoretical
loading model for improved geophysical
interpretation.  This work is an attempt to
categorize different kinds of load to interpret
modern geodetic data. Certainly, the next
potential major advance in loading theory might
be the consideration of lateral heterogeneity in
the Earth structure, which may be particularly
important to interpreting GPS displacement
measurements in the form of mass transfer at the
Earth's surface, since displacement is purely an
indirect effect related to the deformation of the
Earth.
Table 2 Greens Function
Classical
Pressure
Gravitation
Only is given for illustrative purpose.
The various load Love numbers are then used to
compute the relevant Greens functions. For
illustrative purpose, we have listed in Table 2
the expression of the Greens function of .
The loading effect is computed by convolving
the Greens function with the respective source.
In the case of classical loading theory, the
Greens function should be convolved with the
surface density of mass at the Earths surface.
In the case of atmospheric loading, the Greens
function of the effect of pressure should be
convolved with the atmospheric pressure at the
Earths surface, and the Greens function of the
effect of gravitation should be convolved with
the volume mass density of the atmosphere in the
space that the atmosphere occupy. The total
indirect effect is their sum. The computation of
tsunami loading is different from the atmospheric
loading only in one aspect the effect of
gravitation should be computed by convolving the
Greens function with the surface mass
density. For atmospheric loading, if the period
concerned is long enough, the atmosphere may be
approximately considered to be at equilibrium
state, and the pressure-density
relation holds. In this case, we have shown
using a simple atmospheric

• References
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