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 Earth’s 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
Earth’s 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 Earth’s surface, of which the
underlying principle is also the loading theory.
Atmospheric
Classical
Surface mass density:
Volume mass density:


Surface mass density:
p  g R
Pressure:
Tsunami
Pressure:
H

p  (or )  dh  g R
 0

Pressure:

p  g R
Fig. 1: Comparison of three types of loading
Table 1: Boundary condition and definition of the various load Love numbers
Boundary conditions
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 Earth’s surface balances the gravitational force
exerted on the mass of load. Let us denote the ocean surface
h of
height relative to the average position by , the density
 w gravity at the Earth’s surface
ocean water by
, and the
g R Earth is then deformed by the gravitation of a
by
. The
   w h, and a
thin layer of mass with a surface density
pressure p  g R 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 Earth’s surface, but the atmospheric mass is distributed
over the elevation from the Earth’s surface to the top of the
atmosphere. In the case of tsunami loading, we take into
account the fact that pressure at the Earth’s 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.
Classical:
Pressure:
Gravitation:
Load Love numbers
Classical:
Pressure:
Gravitation:
Table 2: Green’s Function
Classical:
Conclusions
Pressure:
The characteristic of the atmospheric and tsunami loading in
comparison to the classical loading theory is shown in Fig. 1.
Gravitation:
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:
6
dyi
  Aij (r ) y j ,
i  1,  ,6
dr
Only
is given for illustrative purpose.
The various load Love numbers are then used to compute the
relevant Green’s functions. For illustrative purpose, we have
listed in Table 2 the expression of the Green’s function of u r .
The loading effect is computed by convolving the Green’s
function with the respective source. In the case of classical
loading theory, the Green’s function should be convolved with
the surface density of mass at the Earth’s surface. In the case
of atmospheric loading, the Green’s function of the effect of
pressure should be convolved with the atmospheric pressure at
the Earth’s surface, and the Green’s 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 Green’s 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
j 1
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.
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].
H

p    dh  g R
 0

holds. In this case, we have shown using a simple atmospheric
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.
References
• Farrell, W.E., 1972. Deformation of the earth by
surface loads, Rev. Geophys. Space Phys., 10, 761797.
• Guo, J. Y., Li, Y. B., Huang, Y., Deng, H. T., Xu, S.
Q. & Ning, J. S., 2004. Green's function of the
deformation of the Earth as a result of atmospheric
loading. Geophys. J. Int. 159 (1), 53-68. doi:
10.1111/ j.1365-246X.2004.02410.x.
• Guo, J.Y., Shum, C.K., 2005. On the Green's
function of tsunami loading, Geophys. J. Int., In
review.
• Longman, I.M., 1962. A Green’s function for
determining the deformation of the Earth under
surface mass loads, 1. Theory, J. Geophys. Res., 67,
845-850.
• Longman, I.M., 1963. A Green’s function for
determining the deformation of the Earth under
surface mass loads, 2. Computation and numerical
results, J. geophys. Res., 68, 485-496.
• Merriam, J.B., 1992. Atmospheric pressure and
gravity, Geophys. J. Int., 109, 488-500.