Tidal Deformation of the Solid Earth

F A C U L T Y O F S C I E N C E
U N I V E R S I T Y O F C O P E N H A G E N
Master Thesis in Geophysics, March 16, 2009
Tidal Deformation of the Solid Earth
- A Finite Difference Discretization
STINE KILDEGAARD POULSEN
Niels Bohr Institute, University of Copenhagen
SUPERVISORS:
Klaus Mosegaard
Carl Christian Tscherning

Abstract
Tidal forces are a source of noise in many geophysical field observations, but they are
useful in revealing information of the interior of the Earth. Tidal forces arise because
of the gravitational attraction from external bodies varying over the volume of a body.
The Earth’s response to the tide generating potential can be derived analytically by the
driving force, i.e. the knowledge of the orbital motion of the Earth and the Moon, the
Sun and other external objects, combined with the Love numbers, which are related to
the rheological properties of the Earth.
By using a two dimensional steady state version of Navier’s equation of motion,
the tidal deformation of the Earth is solved numerically. The Navier equation of motions
is discretized in polar coordinates using a finite difference approximation. The
linear system of equations proved to be ill-conditioned, and to subdue the obstacle a
Tikhonov regularization is applied.
Assuming the Moon being the only attracting body, and assuming a stationary
Earth-Moon relationship, two Earth models are developed. The first model, the homogeneous
Earth model, is assumed to be a pure Poisson solid, where the seismic
velocities and the elastic parameters are constant throughout the Earth. The second
model, the layered model, includes four layers representing an inner core, an outer
core, a mantle and a crust. The elastic properties are taken from the isotropic Preliminary
Reference Earth Model, [Dziewonski and Anderson, 1981], and averaged in each
layer. This gives a realistic Earth model with a fluid outer core.
The Earth models response to the tides are given by the radial and tangential displacement
fields. Though the compression of the radial displacement field not fully
fulfill the picture of the inner structures of the Earth, the results clearly reproduce the
physical picture of the solid Earth tides, by means of an expansive and compressive
Earth for the radial field and a tractive displacement for the tangential field. The radial
displacement field has a maximum expansion of 131 mm and 74 mm for the homogeneous
and layered Earth models, respectively, and the maximum compression are
-5 mm and -26 mm for each model. For the tangential displacement field the maximum
tractions are found to be 41 mm and 45 mm for the homogeneous model and the
layered Earth models, respectively. The layerd Earth model is similar to the analytical
derived, with a factor 1.4 to 4 too small, weathers the homogeneous Earth model shows
larger displacements, but as expected, it deviates more from the physical picture of the
analytic solution.

Dansk Resumé
Tidekræfter bliver ofte regnet som støj i forbindelse med geofysiske målinger og skal
udtrækkes fra data, men tidekræfterne kan afsløre vigtig information om Jordens indre.
Tidekræfter opstår som den varierende kraft på et legeme forårsaget af den gravitationelle
tiltrækning fra eksterne objekter. Jorden er ikke et stift legeme, men elastisk, og
kan derfor deformeres under påvirkning af eksterne kræfter. Tidejorden kan bestemmes
analytisk ved en kombination af bestemmelse af Jordens banebevægelse i forhold
til Månen, Solen og de andre eksterne objekter, og viden om Jordens elasticitet i form
af enhedsløse Love tal.
I dette studie bliver den to-dimensionalle Naviers ligning i ligevægts tilstand løst
numerisk ved at diskretisere Naviers ligning med endelige differencer i et polært koordinat
system. Lignings systemet viser sig at være dårligt stillet (ill-conditioned) og
derfor anvendes en Tikhonov regulering til at løse problemet.
Det antages at Månen er det eneste tiltrækkende legeme, og at Jord-Måne systemet
er stationært i forhold til hinanden. Der er udviklet to Jordmodeller, en homogen og
en lagdelt model. For den homogene model antages det, at Jorden er et fast Poisson
legeme, og de seismisk hastigheder og elastiske parametre antages at være konstante
gennem hele legemet. Den lagdelte model indeholder fire lag; en indre kerne, en ydre
flydende kerne, en kappe og en skorpe, hvilket gør den mere realistisk. Jordmodellen
bruger data fra den isotrope Preliminary Reference Earth Model, [Dziewonski and
Anderson, 1981], hvor parametrene er midlet over hvert lag.
Begge Jordmodellernes reaktion på tidekræfterne er givet i form af det radiale og
tangentielle forskydningsfelt. Resultatet er et klart fysiske billede af tidejorden, hvor
det radiale forskydningsfelt er ekspansiv med maksimum ved Ækvator og kompressiv
med maksimum ved Nordpolen, dog er kompressionen af de indre strukture af Jorden
ikke helt opfyldt. Den horisontale forskydning er rent trækkende med maksimum ved
45 ◦ og nul ved både den polære og den ækvatorielle akse.
Den homogene jordmodel resulterer i et radial forskydnings felt med maksimum
ved Ækvator med en 131 mm ekspansion og maksimum kompression ved Nordpolen
på -5 mm. Det tangentielle forskydnings felt for den homogene jordmodel har et maksimum
træk på 41 mm. Den lagdelte jordmodels radiale forskydnings felt er fundet
til ekspandere 74 mm ved Ækvator og blive presset sammen med en forskydning på
-26 mm ved Nordpolen. Det tangentielle forskydnings felt har et maksimalt træk på
45 mm. Den lagdelte jordmodel svarer overens med det analytisk beregnede resultat
med forskydninger på en faktor 1.4 til 4 for små. Den homogene jordmodel viser højere
forskydninger, men det fysiske billede afvigerer som ventet mere fra det analytiske
resultat.

Preface
This thesis in geophysics has been carried out at the Niels Bohr Institute, University
of Copenhagen and applies to a year’s work equivalent to 60 ECTS points. This work
will finish my master’s degree in geophysics.
The reader of this text is expected to have a basic geophysical knowledge with a
background in fundamental physics and mathematics.
Acknowledgments
First of all I would like to thank my supervisors Klaus Mosegaard and Carl Christian
Tcsherning for their guidance through the whole process and their patience with all my
questions. A special thanks goes to Klaus for your optimism in the frustrating process
of modeling and for all your constructive ideas. Thanks to Amir Khan for providing
me with the PREM data. I would also like to thank Sebastian Bjerregaard Simonsen
for proofreading, Thomas R. N. Jansson for proofreading and letting me login to your
home computer, when the server at NBI failed. Thanks to the office for the good
humor. Last, I would like to thank Andreas Rose for proofreding,your unconditional
support and for being my connection to the real world.
v
Stine Kildegaard Poulsen
Copenhagen, March 2009.

Contents
Contents vii
1 Introduction 1
1.1 Spherical Coordinate System . . . . . . . . . . . . . . . . . . . . . . 3
2 The Tides 5
2.1 Fundamentals of Gravity Field Theory . . . . . . . . . . . . . . . . . 5
2.1.1 Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Centrifugal Acceleration . . . . . . . . . . . . . . . . . . . . 7
2.1.3 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Tidal Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Tidal Acceleration and Potential . . . . . . . . . . . . . . . . 10
2.2.2 Observation of Tides . . . . . . . . . . . . . . . . . . . . . . 13
3 Theory of Elasticity 15
3.1 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Deformation and Strain . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Tidal Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4.1 Structural Effects on Earth Tides . . . . . . . . . . . . . . . . 22
4 Finite Difference Equations 25
4.1 The Centered Finite Difference Scheme . . . . . . . . . . . . . . . . 25
4.2 Error Estimates and Stability of FDE . . . . . . . . . . . . . . . . . . 28
4.3 The Finite Difference Formula . . . . . . . . . . . . . . . . . . . . . 30
4.3.1 Mesh Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4.1 At the Surface . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4.2 Center of Earth . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4.3 The Pole and Equator . . . . . . . . . . . . . . . . . . . . . . 37

viii Tidal Deformation of the Solid Earth
5 Regularization 39
5.1 Ill-conditioning of a System . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Tikhonov Regularization . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2.1 Singular Value Decomposition . . . . . . . . . . . . . . . . . 42
5.2.2 The Good Solution . . . . . . . . . . . . . . . . . . . . . . . 43
6 Model Description and Construction 45
6.1 Model Assumptions and Parameters . . . . . . . . . . . . . . . . . . 45
6.2 The Homogeneous Earth Model . . . . . . . . . . . . . . . . . . . . 46
6.3 The Layered Earth Model . . . . . . . . . . . . . . . . . . . . . . . . 47
6.3.1 Assumptions and Parameters . . . . . . . . . . . . . . . . . . 47
6.4 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.4.1 Grid Points and Meshes . . . . . . . . . . . . . . . . . . . . 48
6.4.2 The Coefficient Matrix . . . . . . . . . . . . . . . . . . . . . 49
7 Model Considerations and Tests 53
7.1 Model Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.2 Analyzing the System . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7.2.1 Singular Value Spectrum for the Homogeneous Model . . . . 54
7.2.2 Weighting of the Surface Boundary Conditions . . . . . . . . 55
7.2.3 Singular Value Spectrum for the Layered Model . . . . . . . 57
7.3 Choosing the Regularization Parameters . . . . . . . . . . . . . . . . 58
7.3.1 Tikhonov Filter Factors . . . . . . . . . . . . . . . . . . . . . 60
7.4 Testing the System . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.4.1 Test of Solving Methods . . . . . . . . . . . . . . . . . . . . 61
7.4.2 The General Solution . . . . . . . . . . . . . . . . . . . . . . 61
7.4.3 Testing the Weighting of Grid Numbers . . . . . . . . . . . . 62
7.4.4 Testing the Boundary Conditions . . . . . . . . . . . . . . . . 64
7.4.5 Testing the Tidal Acceleration . . . . . . . . . . . . . . . . . 65
8 The Tidal Deformation of the Earth 67
8.1 Deformation in the Homogeneous Earth Model . . . . . . . . . . . . 67
8.2 Deformation in the Layered Earth Model . . . . . . . . . . . . . . . . 70
8.3 Analytic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
8.3.1 Comparison Between Results . . . . . . . . . . . . . . . . . 73
9 Discussion 75
9.1 Results and Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
9.1.1 The Displacement Fields . . . . . . . . . . . . . . . . . . . . 75
9.1.2 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . 76
9.1.3 The Discretization . . . . . . . . . . . . . . . . . . . . . . . 78
9.1.4 Ill-conditioning . . . . . . . . . . . . . . . . . . . . . . . . . 78
9.1.5 Computer Errors . . . . . . . . . . . . . . . . . . . . . . . . 79
9.2 The Reliability of the Model . . . . . . . . . . . . . . . . . . . . . . 80
9.2.1 Elasticity Assumptions . . . . . . . . . . . . . . . . . . . . . 80
9.2.2 The Real Tides . . . . . . . . . . . . . . . . . . . . . . . . . 80

CONTENTS ix
9.3 Application of Tides . . . . . . . . . . . . . . . . . . . . . . . . . . 81
9.4 Model Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . 82
10 Conclusion 85
Bibliography 87
Appendices 94
A Hyperbolic Function 94
B Matlab Codes 95
B.1 FDE_premiso.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
B.2 Matlab Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
B.2.1 ftides_r.m . . . . . . . . . . . . . . . . . . . . . . . . . . 101
B.2.2 ftides_theta.m . . . . . . . . . . . . . . . . . . . . . . . 101
B.2.3 mean_alpha.m . . . . . . . . . . . . . . . . . . . . . . . . . 101
B.2.4 mean_beta.m . . . . . . . . . . . . . . . . . . . . . . . . . 102
C Averaging of PREM Layers 103
D Model Tests 104
D.1 Tikhonov Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
D.2 Regularization Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Chapter 1
Introduction
For most people tides are a synonym for the daily falling and rising of sea level. There
is a general understanding that it is due to the Moon’s attraction, but what about the
sea level rising, at the side turning away from the Moon and the influence of the Sun
and from other external objects?
Tidal forces are more complex than a one dimensional pull and causes many effects
in our solar system; from changing sea level to geological activity on distant moons,
galactic interactions and even the final faith of some moons. The gravitational force
that arise by the pull from external objects varies from one part of the effected object
to an other. These differential pulls produces, what are known as tidal forces.
Whenever we touch a body, internal stresses are set up in the material in contact.
Elastic materials can be deformed depending of the applied force and the properties of
the material. In short time scales the Earth can therefore be considered to be elastic
and can be deformed by an external force, the tidal force.
The tides have been of great discussion throughout history, and we know that it causes
many effects throughout the solar system. There has been a lot of misconception about
tides, and what causes them. In 1912 Alfred Wegener suggested that tidal forces broke
up the continents, [Wegener, 1966].
The discovery of tides dates back to the beginning of our calendar, when Pliny the
Elder in his Historia Naturalis states that near the temple of Hercules the ocean rises
and falls. The mathematical basis for the theory of tides was founded in 1687 by Isaac
Newton in Philosophiae Naturalis Principia Mathematica, (for a translated version
see [Newton, 1833]), when he developed the laws of motion and the universal law of
gravitation. This is shortly described in Section 2.1. In 1776, Pierre-Simon Laplace
formulated the first theoretical description of the ocean tides, by a system of partial
differential equation, known as the Laplace tidal equation (for a translated version see
[Laplace, 1832]). It was also this year the first experiment with a pendulum, to observe
the tides, was carried out. The Laplace tidal equation is still used today, and gives the
tidal acceleration on the surface in parts of long-, daily- and half daily periodic variations.
At this point in history, the Earth was assumed to be rigid, but about 1876,
Lord Kelvin stated that the Earth is not completely rigid, it can deform for example as

2 Tidal Deformation of the Solid Earth
a result of tidal forces. G. H. Darwin, assumed in 1883 that for a perfectly rigid Earth,
the observed amplitude of the ocean tide would equal the theoretical value. If the solid
part also was deformed, the measured amplitude would equal the difference between
oceanic and solid Earth tides. He applied it to observations of long periodic oceanic
tides, and found that the amplitude was only two-third of the theoretical tides, i.e. the
Earth is not completely rigid, but deformable. The Earth’s rigidity is often compared
with steel. Steel is not easy to bend, however under the right circumstances, it is possible.
Together with seismological data and polar motion, this new data led to the first
discoveries of the Earth’s elastic properties, [Melchior, 1978]. In 1909, E. H. Love
introduced the Love numbers, which are dimensionless constants of the Earth’s elastic
parameters, (see Love [1944]). Further definition of Love numbers will be given in
Section 3.4.
This thesis presents two simple models of the deformation of the Earth due to the
solid Earth tides, based on a finite difference discretization of the Navier equation of
motion. The two Earth models are a homogeneous Earth model and a layered Earth
model, where the later is an expansion of the first. The thesis opens with the basic
theory of tides in Chapter 2, followed by the theory of elasticity in Chapter 3. This
chapter also deals with the equation of motion and the tidal deformation of the Earth.
In Chapter 4, the finite difference equations are introduced and a discretization of
the equation of motion is developed in a polar coordinate system to match the tidal
acceleration. Solving the linear system was a cumbersome process and a lot of the
work in this master thesis has gone into regularizing the linear system of equations to
get the most suitable results. The regularization is described in Chapter 5.
Chapter 6 describes and sets up the assumption for the two Earth models and an
analysis of the system is performed. Chapter 7 dealing with the modeling of the system.
The construction of the model is described, the regularization parameters are
chosen and the system is tested in varies ways. In Chapter 8, the results from the modeling
are presented and compared to an analytic result. Chapter 9 discuss the given
results, where also suggestions for improvements are proposed, and Chapter 10 concludes
this master thesis.

Spherical Coordinate System 3
1.1 Spherical Coordinate System
Throughout the thesis several angles are defined and to keep a track of them from the
beginning, they are depicted in Figure 1.1.1. The figure shows the relation between the
Cartesian coordinates X, Y, Z placed in the Earth’s center of mass, where the XY-axes
spans the equatorial plane, and the spherical coordinates r, θ, λ. θ is the colatitude,
λ is the geocentric longitude. θ is the geocentric latitude, and ψ is simply the angle
between two points. The Tidal force is calculated using ψ and the polar coordinate
system (r, θ) is used in the Navier equation of motion.
X
0
Z
λ
θ
r
θ
Figure 1.1.1: Relation between coordinates. The Cartesian coordinates X, Y, Z is placed in
the Earth’s center of mass, where the XY-axes spans the equator plan. θ is the colatitude, θ
is the geocentric latitude, λ is the geocentric longitude and ψ is simply the angle between two
points. The Tidal force is calculated with ψ and the polar coordinate system used in the Navier
equation of motion, normally uses the colatitude θ.
ψ
r ′
P
P ′
Y

Chapter 2
The Tides
This chapter aims to describe the basic theory about tides. First, in Section 2.1, the
classic mechanics of gravitation and centrifugal force is shortly described. Secondly
followed a definition of tides in Section 2.2, where also the tidal acceleration in polar
coordinates is written out. The methods used to measure tides, rounds off this chapter.
2.1 Fundamentals of Gravity Field Theory
For a rotating body, the force of gravity (Section 2.1.3) is the combination of gravitational
attraction (Section 2.1.1) arising from the mass distributed in a planet for
example the Earth or other celestial bodies, as well as the centrifugal force (Section
2.1.2) arising from the body’s rotation about its own axis. The gravity field reveal
information of the Earth’s interior, (Section 2.1.1).
Gravity changes with time, appearing at recognizable frequencies, are called tides,
(Section 2.2). Gravity can also appear at time scales ranging from secular to abrupt.
Secular changes includes subduction of continental plates, deformation of the coremantle
boundary, post glacial rebound, melting of ice caps and glaciers. Abrupt
changes includes volcanic and earthquake activity and only gives local effects. Gravity
changes arise due to 1.) the time dependent gravitational constant (this is only a
theoretical consideration in cosmology, ant the time dependency is so fare not verified
for the Moon), 2.) variation and direction of the Earth’s rotation, which changes the
centrifugal force, 3.) variation of terrestrial mass displacements which includes vertical
crustal movements that changes the distance to the center of mass, and it includes
density variations in the crust and mantle and 4.) tidal accelerations, [Torge, 2001].
The first three point are out of scoop of this thesis and will not be discussed in further
details, however the fourth point, the tidal acceleration, is the emphasis of this thesis
and will be covered in the following, (Section 2.2).

6 Tidal Deformation of the Solid Earth
2.1.1 Gravitation
In an initial reference frame, two point masses m1 and m2, see Figure 2.1.1, with the
distance l between the masses, attract each other with the gravitational force given by
Newton’s law of gravitation
−11 m3
F = −G m1m2
l 2
l
l
, (2.1.1)
where G = 6.673 · 10
kgs2 is the universal constant of gravitation, l is the distance
between the masses, l = r − r ′ . The vectors F and l points in opposing directions, see
Figure 2.1.1.
Letting the mass, m1, at the attracting point P, go towards unity, the formula becomes
a = −G m2
l 2
The gravitation is now an acceleration, where a initiate at P, directed towards P ′ . The
gravitational field is invariant to rotation and can be represented by the gradient of the
potential V
where the scalar potential is given by
X
0
Z
l
l
.
a = ∇V , (2.1.2)
r ′
ψ
m2
Figure 2.1.1: Gravitation of two point masses m1 and m2 in a Cartesian coordinate system.
The points P and p ′ are separated by a distance l and angle ψ.
P ′
r
l
m1
P
Y

Fundamentals of Gravity Field Theory 7
V = GM
l
. (2.1.3)
At the surface of the Earth (l = r, where r is the radius of the Earth) the potential
for a point mass is 6.26 × 107 m2
s2 and the gravitation is on an average a = 9.82 m
s2 .
[Torge, 2001].
2.1.2 Centrifugal Acceleration
In an accelerated reference frame fictitious forces arise. The centrifugal force is one of
them, initiating from the rotation of the Earth, about its own axis. The centrifugal force
acts on every element of the Earth, in Figure 2.1.2 the centrifugal force is displaced for
an element of mass P near the Earth’s surface.
With a constant angular velocity, ω, the centrifugal force is given by
z = m(ω × r) × ω = m ω 2 ρ , (2.1.4)
where ρ = |ρ| = r cos θ, and r is the distance to the point in question at geocentric
latitude θ calculated positive from the Equator towards the Pole. See also Figure 1.1.1
in Section 1.1, where the different angles are depicted in a spherical coordinate system.
ω
0
Figure 2.1.2: Centrifugal force in a rotating reference frame. ω is the rotation vector, ρ is the
distance to the point P from the rotating axis. r is the distance to the point from the center of
mass and z is the centrifugal force, θ is the latitude.
ρ
θ
r
P
z

8 Tidal Deformation of the Solid Earth
The centrifugal potential written in the vector field yields
Z = 1
2 m ω2 ρ 2 . (2.1.5)
5 m2
At the Equator the centrifugal force has a value of 1.1 · 10
s2 and the centrifugal
acceleration is 0.03 m
s2 , which is 0.3% of the gravitation. The potential and the acceleration
are both zero at the Pole. The Earth’s rotation vector ω can change the centrifugal
acceleration because of its secular, periodic and irregular variations, [Knudsen
and Hjorth, 2000; Torge, 2001].
2.1.3 Gravity
The gravity g is the result of gravitation aG and centrifugal acceleration az
g = aG + az ,
and the same can be written for the potentials. For an ellipsoidal model, like the Earth,
gravitation decreases at equator and increases at the poles. This results in a varying
gravity between 9.78 m
s2 at the Equator to 9.83 m
s2 at the Poles, [Torge, 2001].
2.2 Tidal Forcing
As already mentioned the gravitational force on a body, arises by an external orbiting
object, working with different force throughout the body, see Figure 2.2.1(a). The
variation of the force over the volumes of material is defined as the differential force,
tide-generating force or simply the tidal force. Due to the acceleration of the external
object this is a non-inertial reference frame. In consequence of Newton’s 2. law,
fictitious forces have to be included, working against the gravitation, this is the centrifugal
force. These two forces cancels out in the center of mass of a homogeneous
sphere. The tidal acceleration is obtained by subtracting the centrifugal force from
the gravitation, see Figure 2.2.1(b). The total tidal force on a sphere is depicted on
Figure 2.2.1(c), and it shows how the force varies with power and direction all over
the volume. The coordinate system is placed at the center of mass in the body. The
tidal force pulls two bodies, with significant masses, from each other by the simple
approximation
Ft =
≈
G M m
R2 G M m
−
− r2 R2 (2.2.1)
2 G M m r
R3 . (2.2.2)
Where M and m are the mass of the body and the mass of the external object, respectively.
R is the distance between the two bodies and r is the radius of the body. R ≫ r,
[de Pater and Lissauer, 2001].
From Equation (2.2.1), the Moon is affected by 1.82 · 10 18 N from the Earth, and
the Earth is affected by 6.69·10 18 N from the Moon and 3.02·10 18 N from the Sun, i.e.

Tidal Forcing 9
the Earth feels the tidal force 3.5 times stronger than the Moon feels from the Earth,
which is equivalent to the difference of the radius. This is due to the Earth’s greater
mass. The mass of the Sun, is much greater than the mass of the Moon, but it only
affect the Earth half of what the Moon does, which is the impact of the distance given
by the inverse cube. In fact the Earth tides are regulated by the orbits of the Moon and
the Earth-Moon system about the Sun and all other planetary components.
The tidal force affects the whole Earth system for example the atmosphere, the oceans
and the solid Earth. The atmospheric tide is a global periodic oscillation of the atmosphere
exited by insolation of the atmosphere, gravity field and non-linear interactions
between tides and planetary waves, [Torge, 2001]. Tides can also influence deep moonquakes,
which are triggered by the tides from the Earth [Lammlein, 1977], earthquakes
(see e.g. [Heaton, 1975]), the periodicity of volcanic activity (see e.g. [Sottili et al.,
2007]), and geyser activity (see e.g. [Rinehart, 1972]), ice quakes, where the seismicity
rate can be compared with the tidal forcing of an ice shelf, see [Anandakrishnan
and Alley, 1997], glaciers (see e.g. [Kulessa et al., 2003]), Climate, (see e.g. [Munk
and Dzieciuch, 2001]).
It is important to distinguish between ocean tides and body tides. Ocean tides are
variations of sea level strongly dependent of coast and bottom topography and ocean
currents. Body tides or solid earth tides is the deformation of ocean and land only
dependent of the gravitational forces of the Moon, Sun and other external objects, and
it works in all parts of the Earth and deforms the Earth to an ellipsoid aligned with the
Earth-Moon (Sun) axis, [Wang, 1997]. The tidal accelerations are within ±1µm/s 2 of
the Earth’s gravity, [Wenzel, 1997]. The ocean tides will not be covered in the thesis,
and from this point forward, the tides will only refer to the solid Earth tides, if nothing
else is mentioned.
B
(a)
A
B
A
(b) (c)
Figure 2.2.1: (a) The gravitational acceleration due to an external object is depicted for two
points A and B at the surface of a spherical body and in the center of the Earth. The external
object is placed to the right of the figure on the line from the bodies center of mass. (b) After
eliminating the fictive acceleration in the non-inertial reference frame, the tidal acceleration
arises in the points A and B. The forces try to deform the body along the body-external-objectline.
(c) the same as previous figure, but here the tidal acceleration is calculated for all point
in the Earth. It is seen that the tidal acceleration varies in power and direction over the sphere.
Figure (c) is adapted from [wikimedia.org, 2006].

10 Tidal Deformation of the Solid Earth
F2
Figure 2.2.2: The total torque of a planets north pole is shown in the figure. When a planet is
rotating about its own axis, rises an equatorial bulge around the Equator. The planet is also
affected by an external object or moon, which is orbiting the planet in the positive direction.
The planet experience an acceleration from the moon which creates an tidal bulge. The bulge
is pressed in front of the movement and it is lowering the rotation speed of the planet. F1 and
F2 are the gravitational force at the given points.
Deformable bodies rotating about their own axis, will experience a distortion of the
body. Looking at a spherical solid body entirely covered with water, rotating about its
own axis, it will experience an equatorial bulge of both solid and water. This is not
to be mistaken by the tidal bulge. The equatorial bulge is not created by the gravitational
force and has no periodic variation, but is only a consequence of rotation. The
equatorial bulge is the baseline, for which real tidal effects are measured. The tidal
bulge is on the other hand a result from gravitation alone. There arises two bulges due
to the conservation of angular momentum. The total torque of the sphere with a moon
moving prograde about the sphere is depicted in Figure 2.2.2. The sphere is shown
from the North Pole. The satellite’s orbiting period is longer than the sphere’s rotations
period, and due to the non perfect elasticity, the bulge would be pushed in front
of the movement. The gravitational force of the moon is greater on the near-side bulge
than the far-side bulge. This will lower the rotation speed of the sphere, [de Pater and
Lissauer, 2001; Tsantes, 1972].
The Earth moves prograde, as seen above the North Pole, and a little faster than the
Moon revolves around the Earth (also prograde). Therefore the Earth drags the high
tide bulge 3 ◦ ahead of its movement, [Tsantes, 1972].
2.2.1 Tidal Acceleration and Potential
Considering a revolving geocentric coordinate system without rotation, all points in
the same distance from the Earth’s center, experiences an equal orbital acceleration.
The gravitation of the celestial bodies have to cancel out in the center of the Earth.
The tidal acceleration is the difference between the gravitation a, depending on the
position of the point P, and the constant acceleration a0 at the Earth’s center
at = a − a0 = GMi
|r − ri| 2
|r − ri| GMi
−
|r − ri| r2 ri
ri
i
F1
, (2.2.3)
where Mi is the mass of the celestial body attracting the Earth. The calculations are
carried out for each two-body system with the Earth being one part and the Moon, the

Tidal Forcing 11
Sun or a planet are the second part. |r − ri| and ri are the distances to the celestial body
in question from the point P and the center of gravity, respectively. The total tidal
acceleration is the result of adding all the tidal accelerations together.
Using spherical harmonics the tidal potential is given by
Φn = GMi
∞ n r
Pn(cos ψ) , (2.2.4)
ri ri
n=2
where Pn(cos ψ) are the Legendre polynomials, n = 0 and n = 1 terms cancels out in
the center of the Earth. ψ is defined as the angle between the object and the point in
question. Looking only at the n = 2 term, which is the main contribution (≈ 98%), and
applying P2(cos ψ) = 3
2 cos2 ψ − 1
2 . The main tidal potential can be calculated from
Φ2 = GMi
2
r 2
r 3 i
3 cos 2 ψ − 1 , (2.2.5)
Equation (2.1.2) states that the tidal acceleration is the gradient of the tidal potential.
Assuming rotation symmetry and choosing the coordinate system with origin at the
center of the Earth and the radial axis pointing towards the Moon. The components of
the tidal acceleration, in vector coordinates atides = (ar(r, ψ, λ), aψ, (r, ψ, λ), aλ(r, ψ, λ)),
can then be written as
a tides
r (r, ψ, λ) = ∂Φ2
∂r
a tides
ψ (r, ψ, λ) = − ∂Φ2
r∂ψ
= GMi
r
r 3 i
= 3 GMi
2
3 cos 2 ψ − 1
r
r 3 i
(2.2.6)
sin 2ψ (2.2.7)
a tides
λ (r, ψ, λ) = − ∂Φ2
= 0 . (2.2.8)
r cos ψ∂λ
is the radial component and is both compressive and expansive in the Earth-Moon
is the latitudinal component parallel to
the tangent of the latitudinal direction towards south and atides λ is the longitudinal component
and is zero because of assumption of rotation symmetry. The two tangential
accelerations are tractive and dominates the large oceans, [Torge, 2001].
atides r
line and dominates the Solid Earth tides, atides ψ
The Periodic Tides
The position of the attacking body is dependent of time. The tides are periodic; diurnal
and semi-diurnal (12 hours) depending on the rotation of the Earth, fortnightly (14
days) and monthly due to the motion of the Moon, annual and semi-annual (6 months)
due to the orbit of the Earth. There is about a nineteen-yearly period called the Metonic
cycle, which is the time for recurrence of the full moon at the same date. Depending
on changes of the Moon, [Love, 1944].

12 Tidal Deformation of the Solid Earth
Following Torge [2001], the tidal potential in a geocentric system, also called
Laplace’s tidal equation, can be written as
Vt =
3
4 GMi
r 2
r 3 i
+ sin 2¯θ sin 2δi cos hi
+ cos 2 θ cos 2 δi cos 2hi
1
3 − sin2
¯θ (1 − 3 sin 2 δi)
, (2.2.9)
where θ is the geocentric latitude θ = 90 ◦ − θ, index i stands for the celestial body
generating the gravitational acceleration, ri is the distance from the attracting body to
the Earth’s center of gravity, δi is the declination and hi is the hour angle defined as.
hi = LAST − αi,
where LAST is the Local Apparent Sidereal Time referring to the observer’s local
meridian, and equals the hour angle of the true vernal equinox and αi is the right
ascension. The quantities ri, δi and hi in Equation (2.2.9) are all time dependent and
with different periods.
Laplace’s tidal equation is simply included because it explains the tidal periods.
The first term of Equation (2.2.9) is independent of the Earth’s rotation and show 14
days long-periodic variation for the moon and half a year for the sun. There is also
a non-periodic part
GMir2
, which only depends on latitude and which result in a
4r 3 i
permanent deformation of the level surface including the geoid. The second term is
due to the daily rotation of the Earth and oscillates with diurnal periods. The third term
represents semidiurnal periods where δ and α is controlling the long-periodic behaviors,
[Torge, 2001].
It is possible to derive the tidal deformation as spherical harmonics of second order,
this is especially fruitful when considering the disturbance of the two major external
bodies the Moon and the Sun. Each term in the tide-generating potential corresponds
to a deformation on a spherical harmonic function of second order and they can be
divided into three types, which is displayed in Figure 2.2.3. The three spheres are divided
into sectors, where the dark colour indicates a rising surface and white indicates
a lowering. The first sphere represents the zonal function and it is only dependent of
the latitude. It has a fortnightly period for the Moon and semi-annual for the Sun.
This function has a maximum at equator and gives a slight flattening of the Earth and
fluctuations in the rotation of the Earth. The second sphere in the figure represents the
tesseral function and the sectors are changing signs with the declination of the perturbing
body. The corresponding tides are diurnal and the amplitude reach maximum at
45 ◦ N and 45 ◦ S and is zero at the poles. This function also influence the inner core’s
rotation relative to the mantle. The last sphere in Figure 2.2.3 is the sectorial function.
The tides represented by this function is semi-diurnal and has a maximum at the
Equator when the declination of the perturbing body is zero, Melchior [1978].

Tidal Forcing 13
Figure 2.2.3: The tides on the Earth’s surface can be represented by three types of second
order spherical harmonics. The function are from left to right: zonal, tesseral and sectorial.
The spherical surfaces are divided in sectors, where white represents rising of the surface and
black lowering. The figure is adapted from Melchior [1978].
2.2.2 Observation of Tides
In many geophysical field observations the Earth tide is a source of noise, but it can also
be useful in revealing informations of the interior of the Earth. The advantage working
with Earth tide data is the exact knowledge of the source function, i.e. the gravitation,
thereby it is possible to detect small Earth tide signals by spectrum analysis, [Wang,
1997]. Studying the gravity field of the Earth is a widely used area to reveal the tidal
fields [Xu et al., 2004]. Earth tide observations compared with theoretical models may
give important constraints to the understanding of the Earth’s internal structures. The
theoretical tidal parameters (i.e. the dimensionless Love numbers which will be explained
in Section 3.4), differ only with a factor of ±10 −2 from observations. The tides
also effect the rotation and ellipticity of the Earth along with anelasticity and lateral
heterogeneities, [Wang, 1997]. The structural effects of tides will be elaborated in further
details in Section 3.4.1.
The solid earth tides are investigated by theoretical and numerical studies. These studies
have been confirmed by observable measurements which can be both ground-based
and space-based;
Ground-based The ground-based measurements includes gravimeters, tiltmeters and
strainmeters, and they can only observe small-scale terrestrial measurements.
Gravimeters are mainly elastic string gravimeters or superconducting gravimeters.
The common feature is to measure the gravity and its variation in space and
time. This is done by balancing the mean value of the gravity with a constant
force equal to the mean value, and then observe the small deviations. The elastic
gravimeters consist of a tensional string which stores the energy. A definite
force or torque is applied to support moving or vibrating masses or to indicate
and control a load or torque. Superconducting gravimeters have improved the
precision of tidal measurements, and is today the most used instrument, [Latycheva
et al., 2009]. The instrument consist of an aluminium sphere coated with
lead. The sphere is placed in a magnetic field of two coils in liquid helium temperature,
where the coils are carrying a current, and the force to keep the sphere
in place is measured, [Melchior, 1978]. Obviously the tiltmeter and strainmeter

14 Tidal Deformation of the Solid Earth
measures the tidal tilt and strain, which both are measurements of deformation,
that allow the flexure of the crust to be observed, [Agnew, 1986]. For more
information about modern instruments, see Boudin et al. [2008].
Space-based The space-based observations includes satellites and VLBI 1 . It is maybe
conceiving to call VLBI for space-based, because the telescope is based on the
ground, but it is anyways acting on space-based data. There exist three tidal effects
which have to be considered in precise satellite tracking. They are 1.) direct
perturbation of the satellite orbit, 2.) tidal deformation of oceans and solid Earth
and last 3.) gravitational effects of these deformations on the satellite orbit. 1.)
is the largest effect and can be model very accurate. The tidal deformation 2.) is
difficult due to the insufficient knowledge of the inner structures of the Earth. In
satellite geodesy is is a pure geometric problem, measured with GPS 2 satellites,
because of their precise positioning. They are very good at determine local displacements
of both the solid Earth tides and the ocean tides. The gravitational
effect 3.) can be calculated for near-Earth satellites. The degree of perturbation
by the solid Earth tides is much stronger for low orbiting satellites. The satellite
STARLETTE 3 is very suitable for studying the solid Earth tides, because of its
low orbit and small size compared to its mass, which makes it more sensitive
to gravitational attraction. GPS can also be used for this purpose, though because
of a larger orbit they are not perturbed as much as STARLETTE. In VLBI
data there are observed diurnal and semidiurnal variations due to the solid Earth
tides, [Seeber, 2003].
Analysis of the Earth tides are normally carried out to determine the tidal parameters
i.e. the frequency transfer function, see Section 3.4.1, between the Earth, station and
sensor. The frequency transfer function is usually found by a least squares adjustment
technique, which will not be discussed further in this thesis. Observations often
contain a drift, which is a long periodic signal caused by the instrument or by meteorological
effects, [Wenzel, 1997].
As it was elucidated in this chapter, the celestial bodies attractions or gravitation of
the Earth mainly from the Moon and the Sun changes with the inverse square of the
distance to the Moon or Sun. The lunar-solar attraction is responsible for the orbital
motion of the Earth as well as the almost periodic tidal motion. The tidal acceleration,
Equations (2.2.6) and (2.2.7), has been derived, and the assumption of rotation
symmetry was defined. The next chapter deals with the theory of elasticity and the
structural effects in the Earth due to the tidal force.
1 Very Long Baseline Interferometry
2 Global Positioning System
3 French passive satellite

Chapter 3
Theory of Elasticity
Elasticity theory is a mathematical framework to understand how materials deform
when they are squeezed, stretched or sheared by an external force. Materials can behave
elastic, plastic or viscose. Elastic materials deform when a force is applied and
return to original shape when the force is removed. A fluid’s resistance to deformation
by shear stress or normal stresses is called viscosity, and describes the fluid’s resistance
to flow. The study of flow and deformation is known as rheology. Viscoelasticity is
defined as the capacity of materials that both have viscous and elastic characteristics
when they deform. Viscoelastic materials behave as elastic solids on short times scale
and as viscous fluids on long time scales. Viscous materials resist shear and strain,
while elastic materials are affected by strain immediately when they are stretched, but
return to original state when the stress is removed. Viscoelastic materials have both
properties. The deformation of materials with plastic behavior have a permanent and
nonrecoverable deformation. Most materials are both elastic, plastic and viscose depending
on the time scales. All solids flow to a small extent by response of shear stress,
this suggest that solids are high viscose liquids, but even for small stresses solids are
elastic. On the other hand fluids are not elastic and they will always be characterized
as having low-stress behavior. In this work all materials in question is assumed to be
purely elastic, [Fowler, 1990].
There are two types of forces that applies to a solid body; body forces and surface
forces. The magnitude of the body force on an element is directly proportional to its
volume (for example the tidal force). In Earth, rocks become more dense with depth,
as the pressure increases, due to a pressure dependency. Surface forces acts across a
surface element. The magnitude is directly proportional to the area of the surface in
action. Surface forces are pressure (normal component) or stress forces (tangentially
component), [Turcotte and Schubert, 1982].
A body can undergo two types of motion. The first type is translation and rotation,
which changes the whole body. The second form is straining and internal deformation,
[Lay and Wallace, 1995]. It is only the second form that is of interest here, rotation
symmetry is assumed.

16 Tidal Deformation of the Solid Earth
3.1 Stress
The distribution of stress depends on the external force applied to the body as well as
the type of the material in the body. The internal elastic stresses on a solid body acts
both as a pressure-like force along the normal to a contact surface and as shear stress
tangential to the surface they act on. Together the forces keeps the material from being
pulled apart, [Lautrup, 2005].
Stress is defined as force per unit area. An arbitrary point in a material in space is
attracted by three planes making a total of nine component stress tensor. In matrix
notation
dFi =
3
j
τi jds j ,
where dF/ds is the force per unit area normal to the surface s, the first index denotes
the surface in which the stresses act, the second index indicates the positive direction in
which the stresses act. τi j is the stress field also called the stress tensor, it is symmetric
and defined as
⎛
⎞
τi j =
⎜⎝
τ11 τ12 τ13
τ21 τ22 τ23
τ31 τ32 τ33
⎟⎠
. (3.1.1)
The diagonal terms τ11, τ22 and τ33 are the principal or normal stresses. They are directed
outward, which mean that positive values are tensional and negative values are
τθr
τrλ
τθθ
τθλ
Figure 3.1.1: Geometry of the stress tensor components of a spherical element. τrr, τθθ, τλλ are
the normal stresses and are in the given figure compressive. τrθ = τθr, τrλ = τλr and τθλ = τθλ
are the shear stresses.
τrr
τrθ
τλr
τλθ
θ
τλλ
r
λ

Deformation and Strain 17
compressional. The off-diagonal terms are the shear stresses. The hydrostatic pressure
is given if the normal stresses equal each other, and then no shear stress will exist. In
spherical coordinates index 1 is the radius r, index 2 is the colatitude θ and index 3
is the longitude λ. In Figure 3.1.1 the geometry of the stress tensor components are
depicted for a spherical square element. The stresses are in this case compressional
and τrθ = τθr, τrλ = τλr and τθλ = τθλ, [Lay and Wallace, 1995; Lautrup, 2005].
3.2 Deformation and Strain
Deformation within a solid body involves variation of the displacement field u(x, t) =
ˆx ′ − ˆx, where x and t are the actual position and time, i.e. it is described with a
Lagrangian representation. Deformation is changes in both length and angular distortions,
this is expressed by spatial gradients of the displacement field, also called
strains.
Strains are as the stresses divided in normal strains and shear strains. Normal
strains are fractional changes in distance and measures how elongated the body has
become. Shear strains are rotations with respect to the surrounding material. The total
strain of a body must be described in three dimensions which is represented with a nine
component tensor. Just like the stress tensor. Three normal stresses in the diagonal and
six shear stresses in off-diagonal. In matrix notation, the strain tensor is given by
ei j = 1
2 (ui, j + u j,i) , (3.2.1)
where i and j goes from 1-3, ui, j = ∂ui
∂x j , and xi represent the coordinates, [Lay and
Wallace, 1995].
In spherical coordinates, see Figure 3.2.1, a deformation from point P to Q, can be
given as,
P(r, θ, λ) = Q(r + δr, θ + δθ, λ + δλ) .
The displacement tensor can be represented by three vectors u(r, θ, λ) = (u, v, w),
where u, v and w are the deformation in the r, θ, λ directions respectively. θ is the colatitude
calculated from the pole towards equator, and λ the geocentric longitude. Following
Hughes and Gaylord [1964] the spherical strain tensor components are given

18 Tidal Deformation of the Solid Earth
by
err = ∂u
∂r
(3.2.2)
eθθ = 1 ∂v u
+
r ∂θ r
(3.2.3)
eλλ
erλ
=
=
1 ∂w u cot θ
+ + v
r sin θ ∂λ r r
(3.2.4)
1
erθ =
1 ∂u w ∂v
− +
2 r sin θ ∂λ r ∂r
(3.2.5)
1
eλθ =
1 ∂u v ∂v
− +
2 r ∂θ r ∂r
(3.2.6)
1
1 ∂w w cot θ
− +
2 r ∂θ r
1
∂v
r sin θ ∂λ
.
(3.2.7)
Here err, eθθ, eλλ are the normal strains and adding them gives the dilatation, and
erλ, erθ, eλθ are the shear strains. The strain tensor in spherical coordinates can then
be derived to
⎡
e =
⎢⎣
∂u
∂r
1 ∂v/r 1 ∂u
2 r ∂r +
r ∂θ
1 1 ∂u ∂(w/r
2 r sin θ ∂λ + r ∂r
X
1 sin θ
2 r
1 ∂(v/r)
2 r ∂r
Z
+ 1
r
∂u
∂θ
1 ∂v u
r ∂θ + r
∂(w/ sin θ)
∂θ + 1
∂v
r sin θ ∂λ
λ
r sin θ
θ δθ
∂(w/r)
+ r
1 1 ∂u
2 r sin θ ∂λ ∂r
1 sin θ ∂(w/ sin θ)
2 r ∂θ + 1
∂v
r sin θ ∂λ
1 ∂w u cot θ
r sin θ ∂λ + r + r v
Figure 3.2.1: Deformation in spherical coordinates. The figure is adapted from Surfaces and
Contact Mechanics [2005], but slightly modified.
r
δλ
P
rδθ
Q
δr
Y
⎤
⎥⎦

Equation of Motion 19
3.3 Equation of Motion
The displacement for a continuous, homogeneous, isotropic, infinite, elastic medium
is given by the Navier equation of motion
(λ + µ)∇(∇ · u) + µ∇ 2 u + F = ρ ∂2 u
∂t 2
, (3.3.1)
where u is the displacement, λ and µ are the Lamé coefficients, F represent the forces
acting on a volume element also called the body forces, and ρ the density of the media
in question, [Lautrup, 2005]. For an isotropic media, a media with no directional
variation, the Lamé parameters are the only two independent elastic moduli, [Lay and
Wallace, 1995]. In terms of the P-wave velocity α2 λ + 2µ
= ρ which is a plane wave
of compression and shear and the S-wave velocity β2 = µ ρ which is pure shear with no
volume change, S-waves can not propagate in a fluid material. Equation (3.3.1) can be
written as
α 2 ∇(∇ · u) − β 2 ∇ × (∇ × u) + F
ρ = ∂2 u
∂t 2
. (3.3.2)
As described in Section 2, the attracting forces from the Moon and the Sun causes
the orbiting of the Earth and the tidal force as described in Section 2. The tidal time
variation is a very slow process compared to the deformation of the Earth and therefore
the time dependent term in Equation (3.3.2) disappears. The wave equation becomes
α 2 ∇(∇ · u) − β 2 ∇ × (∇ × u) = − F
ρ
. (3.3.3)
It is important to notice, that the right-hand side is a volume force over the density
which in fact is an acceleration, given in the units of m
s2 .
3.3.1 Boundary Conditions
The boundary conditions are describing the external influence on the Earth’s surface
and any boundaries inside the Earth. The boundary conditions are found by applying
continuity of stress and displacement on the internal boundaries and the free surface.
In this study, internal boundaries are disregarded. At the free surface (r = a), only the
tractions (τrr, τrθ, τrλ) are constrained, not the other components of stress, [Hurford
and Greenberg, 2002]. The boundary conditions are
τrr =
τrθ =
τrλ =
E
(1 + σ)(1 − 2 σ) [(1 − σ)err + σ(eθθ + eλλ)] = 0 (3.3.4)
E
1 + σ erθ = 2µerθ = 0 (3.3.5)
E
1 + σ erλ = 2µerλ = 0 , (3.3.6)
where e is the strain defined in Section 3.2, σ = λ
2(λ + µ)
is called the Poisson ratio
and is the ratio of the lateral contraction strain to the longitudinal extension strain.

20 Tidal Deformation of the Solid Earth
µ(3λ + 2µ)
E =
λ + µ
is Young’s modulus or the elastic modulus and is the stiffness of
an isotropic, elastic material and is defined as the ratio of extensional stress over the
extensional strain of for example a cylinder being pulled on both ends, [Hughes and
Gaylord, 1964; Shearer, 1999].
Because of the constrain of rotation symmetry, the displacement is confined to the
rθ-plane, i.e. w = 0 and therefore getting ∂i/∂λ = 0. τrr is the plane tangent to the
radius of the Earth and τrθ is the plane tangent to the latitude of the Earth. This can
also be expressed by the displacement coordinates and after some reduction, we get
τrr =
E
(1 − σ)
(1 + σ)(1 − 2 σ)
∂u
+ σ 2
∂r u θ
+ vcot
r r
+ 1
r
∂v
= 0
∂θ
(3.3.7)
1 ∂u v ∂v
τrθ = µ − + = 0 , (3.3.8)
r ∂θ r ∂r
where u = (u, v) is the displacement described in Section 3.2. u is the radial displacement
component and v is the tangential component of the displacement. Both
are working in the domain r = a, where a is the radius of the Earth and 0 ≤ θ ≤ 2π
is the colatitude. Notice that the angle, θ in Equations (3.3.7) and (3.3.8) is not the
same as the angle ψ in Equation (2.2.4), given in Section 2.2.1. θ is calculated positive
from the Pole axis towards the Equator, which is opposite the angle, ψ. ψ is the angle
between the tidal force’s attacking point and the line of the attracting body line. In the
Earth-Moon system, where the Moon is located in the equator line ψ = θ = 90 ◦ − θ.
All angles are depicted in Figure 1.1.1 in Section 1. There can be no normal stresses at
the surface, hence τrr = 0, and τrθ = 0 because no shear stress can exist along the free
surface of a distorted sphere, [Lay and Wallace, 1995]. Further note that the derivative
with respect to λ has vanished.
These equations can also be explained physically by looking at a simple example
where a traction is applied perpendicular on both ends of a rod. The rod will be
stretched out in the length and thinned in the width. The same is valid for a volume
element in the Earth, i.e. when the Earth is affected by a radial traction of the Moon, it
not only gives a radial displacement on the attracted volume element but also a lateral
displacement.

Tidal Deformation 21
3.4 Tidal Deformation
The solid Earth’s response to rotation and ellipticity was first described by Love [1911].
It was a simple model where the Earth was assume to be incompressible and homogeneous.
The tidal deformation in a homogeneous spherical symmetric Earth, can be
written as components of the displacement vector u = (u, v, w)
u(r, ψ, λ) =
v(r, ψ, λ) =
w(r, ψ, λ) =
1
g(r)
1
g(r)
1
g(r)
∞
Hn(r) Φn
n=2
∞
Ln(r)
n=2
∂Φn
∂ψ
∞ ∂Φn
Ln(r)
sin ψ ∂λ
n=2
(3.4.1)
(3.4.2)
. (3.4.3)
Φn is the tidal potential given in Equation (2.2.4), g(r) is the acceleration of gravity,
described in Section 2.1.3, and Hn and Ln are the amplitudes connecting the Love numbers
hn, kn, ln at the surface of the Earth, [Melchior, 1978]. The Love numbers h, k, l are
dimensionless numbers used as a representation of the deformation phenomena which
is produced by a potential perturbed by spherical harmonics. The Love numbers are
a measure of how solid the Earth is, and they change as the order of the potential
changes. h describes the elastic radial displacements of the crustal surface relative to
the ellipsoid, k describes the modification of the potential caused by the deformation
and l describes the elastic tangential displacement. Because of rotation and ellipticity
it is difficult to verify the latitudinal dependencies of the Love numbers. For a rigid
body h = k = l = 0, but the Earth is elastic and the Love numbers are in the range
0.304 ≤ k ≤ 0.312
0.616 ≤ h ≤ 0.624
0.088 ≤ l ≤ 0.084 .
This scale is of course depending on the Earth model. Normally the largest tidal contribution
n = 2 is on the surface averaged to h = 0.6, k = 0.6 and l = 0.08, [Varga, 1974].
Inserting the tidal potential and calculating the derivatives, the radial and tangential
displacement in polar coordinates, become
u(r, ψ) = h
g
GMr 2
2r 3 i
(3 cos 2 ψ − 1) (3.4.4)
v(r, ψ) = l 3GMr
g
2
2r3 sin 2ψ . (3.4.5)
i
The Earth’s radial displacement field is compressive and expansive. In a Earth-Moon
system, the maximum expansion happens at the Earth-Moon line, and for the Moon
aligned with the equatorial axis, the maximum expansion is at the Equator and maximum
compression at the Pole. The horizontal or tangential displacement field is tractive,
with maximum value at mid-latitude. By expanding the Legendre polynomials

22 Tidal Deformation of the Solid Earth
in the tidal potential, Equation (2.2.4), into spherical harmonics, the tides can be expressed
with the Love numbers. This is by far the most used form, when studying the
deformation of the Earth. The spherical harmonics can be computed to high degree.
This is among others described by Love [1944]; Takeuchi and Saito [1962]; Varga and
Grafarend [1996]; Wenzel [1997].
3.4.1 Structural Effects on Earth Tides
Longman [1962] and Farrell [1972] were some of the first to compute realistic Earth
models for the tidal deformation. The models applied to a non-rotating, spherically
layered, elastic and self-gravitating Earth. Self-gravitating states, that when a deformation
in the interior occurs, the mass is redistributed, and this causes a change in the
gravity field and influence the deformation again, [Wang, 1997]. Longman [1962] used
a Greens function, which is a method to solve inhomogeneous differential equations
with boundary problems, to model the surface deformation on Earth under a surface
mass load and the perturbations in the gravity field. The methods were made realistic
by applying the Love numbers. Farrell [1972] studied the elastic deformation in a
elastic stratified half-space.
The geophysical properties of the Earth’s interior are deduced mainly by seismology
and low frequency normal modes. Rheological properties are found by glacial
rebound. Rheology combine elements of elasticity, viscosity and plasticity, and it relate
the flow and deformation behavior of material and its internal structure. Earth
tides is the only tool to explore the frequencies in between.
In a model from the fifties it was shown by Jeffreys and Vicente [1957], that the
Earth’s fluid core responded with a resonance at diurnal frequencies. This is caused by
a normal mode of the rotating Earth, and can not be recovered from seismological interpretation.
This resonant behavior is called The nearly- diurnal free wobble-resonance,
[Zürn, 1997], because it is in frequency with the diurnal tides. The resonance arises
when the axis of the mantle and outer core is slightly separated. The reacting of the
realignment causes the wobbling effect. There do also exist an other wobbling effect
called the Chandler wobble named after the discoverer S. C Chandler (1891). It arises
because the rotational axes deviates by a small angle from the symmetry axis, the axis
of maximum moment of inertia. A prograde precession exert as the torque acting on
the angular momentum and causes the pole tides, the only tidal effect not arisen by
external bodies. The deformation arising from this effect, adjusts the equatorial bulge
towards symmetry, [Stacey and Davis, 2008].
The mantle is also influenced by the solid Earth tides, by impacting the heterogeneity
of the mantle’s convective flow. The viscoelasticity of the mantle causes a
frequency dependence of the tidal parameters. Lateral heterogeneities in the mantle
can be estimated by Love-Li or Molodensky-Wang’s perturbations, but that is beyond
the scoop of this thesis. A reliable mantle model by Wang [1997] illustrates that the
large scale mantle heterogeneity only gives small anomalies in the Earth tides, and
amplitude increase and phase delay of the Earth’s response functions.
Local influences of Earth tides are not fully understood. Strain and tilt measurements
are vigorously dominated by fault zones and cavities near the instruments. To
deduce the local inhomogeneities is a hard task because of the geology and compli-

Tidal Deformation 23
cated deformation.
This chapter rounds off the fundamental theory behind the thesis. The steady state
version of the Navier equation of motion, Equation (3.3.3) was introduced. Assumptions
about the elasticity of the Earth has been defined, and the analytic solution of
the tidal displacement at the surface layer, the Equations (3.4.4) and (3.4.5), has been
introduced.

Chapter 4
Finite Difference Equations
This Section describes the numerical methods of Finite Difference Equations (FDE)
which is used for solving differential equations. The basic theory is adapted from
Kreyszig [1999], but expanded from Cartesian to polar coordinates. The aim of this
section is to perturb the two-dimensional Navier equation of motion, Equation (3.3.3)
described in Section 3.3, into a FDE formulation.
4.1 The Centered Finite Difference Scheme
The Navier equation of motion is a hyperbolic partial differential equation, verified in
Appendix A, and applied on the spherical Earth, it leads to a boundary value problem.
With the vector identity ∇ × (∇ × u) = ∇(∇ · u) − ∇ 2 u, the Equation (3.3.3) can be
written as
− F
ρ = (α2 − β 2 )∇(∇·u) + β 2 ∇ 2 u . (4.1.1)
In this case the left-hand side equals the tidal acceleration. The two dimensional tidal
acceleration components are given by the Equations (2.2.6) and (2.2.7). When writing
out the coordinates (r, θ, λ), for the radial (u) and tangential (v) system, where λ = 0
and ∂λu = 0 (because of the rotation symmetry).

26 Tidal Deformation of the Solid Earth
− Fu
ρ
− Fv
ρ
= (α 2 − β 2
)
+β 2
− 1 1 ∂u
u +
r2 r ∂r + ∂2u 1
+
∂r2 r2 ∂u
∂θ
1 ∂u
r ∂r + ∂2u 1
+
∂r2 r2 ∂2u ∂θ2
= − α2 − β2 r2 u + α2
r ur + α 2 urr
+ α2 − β2 r2 uθ + β2
r2 uθθ + α2 − β2 uθr
r
= (α 2 − β 2
) − 1
r2 ∂v 1
+
∂θ r2 ∂2v 1
+
∂θ2 r
+β 2
1 ∂v
r ∂r + ∂2v 1
+
∂r2 r2 ∂2v ∂θ2
=
β 2
r vr + β 2 vrr − α2 − β 2
r 2
vθ + α2
vθθ
r2 ∂2v
∂r∂θ
+ 1
r
∂2u
∂θ∂r
(4.1.2)
(4.1.3)
(4.1.4)
+ α2 − β2 vrθ . (4.1.5)
r
The second equality sign is written with index notation, where subscripts r, rr denotes
the first and second derivatives of u and v with respect to r, subscripts θ, θθ denotes
the first and second derivatives of u and v with respect to θ and θr, rθ denotes the second
order mixed partial derivative of u and v with respect to r and θ. Fu
ρ = atides r and
Fv
ρ = atides θ .
The following will only be carried out for the radial system, u, because the deduction
of v would be trivial. Taylor expansion of u(r, θ) to the second order, gives for the
radial direction
u(r + hr, θ) = u(r, θ) + hr ur(r, θ) + 1
2 h2 r urr(r, θ) (4.1.6)
u(r − hr, θ) = u(r, θ) − hr ur(r, θ) + 1
2 h2 r urr(r, θ), (4.1.7)
where hr is an infinitesimal change in the radial direction. Subtracting Equation (4.1.6)
from Equation (4.1.7), and solving for the first derivative of r, ur gives
ur(r, θ) = u(r + hr, θ) − u(r − hr, θ)
2hr
. (4.1.8)
This is called a centered finite difference formula for the first order derivative of r. By
addition of Equation (4.1.6) and (4.1.7), the second order centered finite difference of

The Centered Finite Difference Scheme 27
r, resolves to
urr(r, θ) = u(r + hr, θ) + u(r − hr, θ) − 2 u(r, θ)
h 2 r
. (4.1.9)
The same procedure is carried out for the colatitudinal direction. This gives, the centered
finite differences for the first and second order derivatives of θ
uθ(r, θ) = u(r, θ + hθ) − u(r, θ − hθ)
2 hθ
uθθ(r, θ) = u(r, θ + hθ) + u(r, θ − hθ) − 2 u(r, θ)
h2 θ
(4.1.10)
. (4.1.11)
The mixed derivative urθ and uθr equals each other, [Spiegel and Liu, 1999], and this
continuity was assumed in Section 3.3. The mixed terms are derived by the Taylor
expansion at (r, θ) in the radial and the latitudinal direction. The second order Taylor
expansion for functions of two variables gives in total four equations
u(r + hr, θ + hθ) = u(r, θ) + hr ur(r, θ) + hθ uθ(r, θ)
+ 1
2 [h2 r urr(r, θ) + 2hrhθ urθ(r, θ)
+ h 2 θ uθθ(r, θ)] (4.1.12)
u(r + hr, θ − hθ) = u(r, θ) + hr ur(r, θ) − hθ uθ(r, θ)
+ 1
2 [h2 r urr(r, θ) + 2hrhθ urθ(r, θ)
+ h 2 θ uθθ(r, θ)] (4.1.13)
u(r − hr, θ + hθ) = u(r, θ) − hr ur(r, θ) + hθ uθ(r, θ)
+ 1
2 [h2 r urr(r, θ) + 2hrhθ urθ(r, θ)
+ h 2 θ uθθ(r, θ)] (4.1.14)
u(r − hr, θ − hθ) = u(r, θ) − hr ur(r, θ) − hθ uθ(r, θ)
+ 1
2 [h2 r urr(r, θ) + 2hrhθ urθ(r, θ)
+ h 2 θ uθθ(r, θ)] . (4.1.15)
Solving for the mixed derivative by adding Equations (4.1.12) and (4.1.15) and subtracting
Equations (4.1.13) and (4.1.14), gives
urθ = u(r + hr, θ + hθ) − u(r + hr, θ − hθ) − u(r − hr, θ + hθ) + u(r − hr, θ − hθ)
4 hr hθ
.
(4.1.16)

28 Tidal Deformation of the Solid Earth
This last centered finite difference scheme closes Section 4.1. Before Section 4.3,
where the finite difference equation formulas are discretized, Section 4.2 elucidate the
errors arising by the Taylor expansion.
4.2 Error Estimates and Stability of FDE
When writing up the central differences, an approximation was made to the actual
solution.
Truncation Error is defined as the difference between the governing differential
equation and the approximation to the governing equation. This is in contrast to the
solution error, which is the difference between the exact solution and the approximate
solution.
Following Holmes [2007], the local truncation error for the central finite difference
schemes for ur and uθ are
τ i r = − h2 r
6 urrr(ηi, θ j) (4.2.1)
τ j
θ = −h2 θ
6 uθθθ(ri, η j) . (4.2.2)
For urr and uθθ the truncation error are
τ i rr = − h2 r
12 urrrr( ¯ηi, θ j) (4.2.3)
τ j
θθ = − h2 θ
12 uθθθθ(ri, ¯η j) . (4.2.4)
For urθ and uθr, the truncation error is not known. It will in the following be called
i j
τ
rθ and τ
i j
θr
. (4.2.5)
The η’s are between ri−1 and ri+1 or between θ j−1 and θ j+1. The total truncation error
τu i of the inner system of u, is given by
τ u i
= −α2
r
− α2 − β 2
r 2
− α2 − β 2
h2 r
6 urrr(ηi, θ j) − α 2 h2r 12 urrrr( ¯ηi, θ j)
h 2 θ
6 uθθθ(ri, η j) − β2
r 2
i j
τrθ h 2 θ
12 uθθθθ(ri, ¯η j)
r
. (4.2.6)
The total truncation error τv i of the inner system of v, is given by
τ v i
= −β2
r
− α2 − β 2
r 2
− α2 − β 2
r
h2 r
6 vrrr(ηi, θ j) − β 2 h2r 12 vrrrr( ¯ηi, θ j)
h 2 θ
6 vθθθ(ri, η j) − α2
r 2
i j
τθr h 2 θ
12 uθθθθ(ri, ¯η j)
. (4.2.7)

Error Estimates and Stability of FDE 29
The truncation reduces when hr and hθ reduces. If the partial difference equation is
approximated consistently, it is not preposterous to think that the solution is stable,
and therefore converge to the exact solution in a certain time step. The finite difference
scheme is consistent if
lim τ(r, θ) = 0 , where l = max{hr, hθ} .
l→0
Therefore the discretization is consistent and second order accurate, but consistency is
not enough, the system also needs to be stable to get a reliable solution. The Equations
(4.3.5)-(4.3.6), can be written as the linear system
Ax = F.
A is the coefficient matrix, x is representing the displacements and the vector F contains
the body forces and the boundary constrains with the traction free surface and
the symmetry at the polar and equatorial axes. This equation will be discussed more
throughly in next chapter. If the solution is consistent and stable, the finite difference
equations is convergent, [Holmes, 2007].
It was easy to see that the solution was consistent, but the coefficient matrix A has
to be studied more thoroughly to determine the stability of the system, this is carried
out in Chapter 7. This does not describe the difference between the numerical and the
exact solution, which of course is the most direct measure, of how good the numerical
solution is. A complete numerical solution of this system is not derived, but the
computed results are compared with the surface displacements given by the Equations
(3.4.4) and (3.4.5) in Section 3.4.

30 Tidal Deformation of the Solid Earth
4.3 The Finite Difference Formula
In this section the newly achieved truncation error is dropped, and the system is
changing from being exact to being an approximation. Substituting Equations (4.1.8)-
(4.1.11) and (4.1.16) into the radial Equation (4.1.3), gives after some calculations
− Fu(r, θ)
ρ
=
1 α
r
2 (2r + hr)
2 h2
u(r + hr, θ)
r
1 α
+
r
2 (2r − hr)
2 h2
u(r − hr, θ)
r
⎡
+ ⎢⎣ 1
⎤
⎥⎦ u(r, θ + hθ)
+
r 2
⎡
⎢⎣ 1
2β 2 + (α 2 − β 2 ) hθ
2h 2 θ
r2 2β2 − (α2 − β2 ) hθ
2h2 θ
⎤
⎥⎦ u(r, θ − hθ)
+
1 α
r
2 − β2
[u(r + hr, θ + hθ) − v(r + hr, θ − hθ)
4 hr hθ
−
−
u(r − hr, θ + hθ) + u(r − hr, θ − hθ)]
⎡
⎢⎣ α2 − β2 r2 + 2 α2
⎤
⎥⎦ u(r, θ) . (4.3.1)
h 2 r
+ 2 β2 )
r 2 h 2 θ
The same is carried out for the tangential system, Equation (4.1.5)
− Fv(r, θ)
ρ
=
+
+
+
1 β
r
2 (2r + hr)
2 h2
v(r + hr, θ)
r
1 β
r
2 (2r − hr)
2 h2
v(r − hr, θ)
r
⎡
⎢⎣ 1
r2 2 α2 − (α2 − β2 )hθ
2h2 ⎤
⎥⎦ v(r, θ + hθ)
θ
⎡
⎢⎣ 1 2 α2 + (α2 − β2 ⎤
)hθ
⎥⎦ v(r, θ − hθ)
r 2
2h 2 θ
+
1 α
r
2 − β2
[v(r + hr, θ + hθ) − v(r + hr, θ − hθ)
4 hr hθ
−
−
v(r − hr, θ + hθ) + v(r − hr, θ − hθ)]
⎡
2 β2
⎢⎣ + 2 α2
⎤
⎥⎦ v(r, θ) . (4.3.2)
h 2 r
r 2 h 2 θ
These equations are differential equations corresponding to the Navier equation of
motion, Equation (4.1.1), where hr, and hθ are the mesh sizes in the radial and the
latitudinal planes, respectively. At the point (r, θ), u equals the mean values of
⎡
⎢⎣ α2 − β2 r2 + 2 α2
h2 +
r
2β2
⎤
⎥⎦
r 2 h 2 θ

The Finite Difference Formula 31
✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧
✡ ✡
✡
✡
✡
✡
✡
✡
✡ ✡
(i + 1, j + 1)
✈
✡
✈(i
+ 1, j)
(i, j + 1) ✈
(i j) ❢
(i − 1, j + 1) ✈
❨
✈ hθ
(i − 1, j)
✈ ❄ ✈
✈
(i − 1, j − 1)
✛
hr
(i, j − 1)
✲
(i + 1, j − 1)
Figure 4.3.1: The figure is showing a nine point stencil of the discretized Navier equation of
motion, Equation (4.1.1). The grid points are showed with the corresponding index notation.
The hollow dot indicates in which point the solution is calculated. This figure shows a central
finite difference scheme, where hr and hθ are the meshes for the radial and colatitudinal
direction.
and v equals
⎡
2 β2
⎢⎣
h2 +
r
2 α2
, at each of the eight neighboring points, the grid is pictured in Figure 4.3.1. This is
called a nine point stencil.
4.3.1 Mesh Points
In the last section the mesh points were introduced, now they are used to discrete
the equations. The mesh points and the corresponding values for the solution can be
written with the convenient notation
r 2 h 2 θ
Pi j = (ihr, jhθ), ui j = u(ihr, jhθ) .
By using a uniform grid and letting the radius cover the domain R0 ≤ r ≤ a, where r0
is the center of the sphere, a is at the surface of the sphere, and the colatitude is given
by θ0 ≤ θ ≤ 2π. With this new notation, the grid and meshes are
ri = R0 + i hr, hr =
a − R0
n
θ j = θ0 + jhθ, hθ = θn − θ0
m
⎤
⎥⎦
for i = 1, . . . , n
for j = 1, . . . , m .

32 Tidal Deformation of the Solid Earth
n and m are the grid points in the radial and the latitudinal direction, respectively. The
pole singularity (R0 = 0) is discussed later in Section 4.4.1. For any mesh point Pi j,
Equation (4.3.1) and Equation (4.3.2), can be discretized using the conservative form,
the so called central difference discretization, which can be visualized in Figure 4.3.1
again. The hollow dot indicates in which grid point, the solution is calculated. Note
that there are as many solutions as equations. With this notation, the Equations (4.3.1)-
(4.3.2), can be written for any mesh point Pi j in the form
and
− Fu(ri, θ j)
ρ
− Fv(ri, θ j)
ρ
=
=
1 α2 (2ri + hr)
ri
2 h 2 r
+ 1 α2 (2ri − hr)
ri
+ 1
r2 i
+ 1
r2 i
2 h 2 r
ui+1, j
ui−1, j
2β 2 + (α 2 − β 2 )hθ
2h 2 θ
2β 2 − (α 2 − β 2 )hθ
2h 2 θ
+ 1 α2 − β2 −
ri
4 hr hθ
α2 − β2 + 2 α2
r 2 i
ui, j+1
ui, j−1
(uui+1, j+1 − ui+1, j−1 − ui−1, j+1 + ui+1, j+1),
h 2 r
1 β2 (2ri + hr)
ri
2 h 2 r
+ 1 β2 (2ri − hr)
ri
+ 1
r2 i
+ 1
r2 i
2 h 2 r
+ 2β2
r 2 i h2 θ
vi+1, j
vi−1, j
2 α 2 − (α 2 − β 2 ) hθ
2h 2 θ
2 α 2 + (α 2 − β 2 ) hθ
2h 2 θ
+ 1 α
ri
2 − β2 ⎡
2 β2
− ⎢⎣ + 2 α2
ui, j , (4.3.3)
vi, j+1
vi, j−1
(vi+1, j+1 − vi+1, j−1 − vi−1, j+1 + vi−1, j−1)
4 hr hθ
⎤
h 2 r
r 2 i h2 θ
⎥⎦ vi, j , (4.3.4)
where i = 2, . . . , n − 1
j = 2, . . . , m − 1 .
The finite difference scheme at a grid point (ri, θ j) involves nine grid points north,

The Finite Difference Formula 33
northwest, west, southwest, south, southeast, east, northeast and the center, where
north and south are in the radial direction and east and the west are in the latitudinal
direction. The center (ui j and vi j) are called the master grid points and are the approximation
to the solution (u(ri, θ j) and v(ri, θ j)), respectively. The coordinates can been
seen in Figure 4.3.1, where north is the point in the direction of radius.
By defining
r i− 1 2
r i+ 1 2
= ri − hr
2
= ri + hr
2 ,
and separating the equations in radial, latitudinal and mixed terms, this more convenient
notation becomes
− Fu(ri, θ j)
ρ
− Fv(ri, θ j)
ρ
=
=
1
ri
α 2 (r i+ 1 2
− (α2 − β 2 ) ui j
r 2 i
+ 1
r 2 i
+ 1
ri
and
ui+1, j − (r 1 i+ 2 , j + ri− 1 ) ui j + r 1
2 i−
ui−1, j)
2
+ 1
r 2 i
(α 2 − β 2 ) (ui, j−1 − ui, j+1)
2 hθ
h2 r
β2 (ui, j+1 − ui j + ui, j−1)
(α 2 − β 2 )(ui+1, j+1 − ui+1, j−1 − ui−1, j+1 + ui−1, j−1)
4 hr hθ
1 β
ri
2 (r 1 i+
vi+1, j − (r 1
2
i+ 2
+ r 1 i− ) vi j + r 1
2 i−
vi−1, j)
2
h2 r
+ 1
r2 α
i
2 (vi, j+1 − vi j + vi, j−1)
h2 θ
+ 1
r 2 i
+ 1
ri
(α 2 − β 2 ) (vi, j−1 − vi, j+1)
2 hθ
(α 2 − β 2 ) (vi+1, j+1 − vi+1, j−1 − vi−1, j+1 + vi−1, j−1)
h 2 θ
4 hr hθ
,
.
(4.3.5)
(4.3.6)
These last two equations are the final FDE’s corresponding to the steady state version
of the Navier equation of motion, Equation (4.1.1), where ui j and vi j are the approximated
to the solution u(ri, θ j) and v(ri, θ j). In principal the constants should also have
been discretized, but is here neglected.

34 Tidal Deformation of the Solid Earth
4.4 Boundary Conditions
To apply the FDE equations on areal Earth, there have to be applied seven boundary
conditions two at the free surface, depending of the surface tractions, one at the center
of Earth, two at the polar axis and last two at the equatorial axis.
4.4.1 At the Surface
To apply the traction free conditions at the free surface, i = a, where a is the surface
of Earth, a first order backward difference scheme is used for for the r-derivative, to
avoid dealing with fictitious points. A centered second order scheme is used for the
θ-derivatives. This is an implicit method. See the stencil for this problem in Figure
4.4.1, the figure is true for both both displacements u and v. The boundary conditions
at the free surface was described in Section 3.3.1 with the Equations (3.3.7)-(3.3.8).
After discretization, the equations becomes
0 = τrr
E
=
(1 − σ)
(1 + σ)(1 − 2 σ)
ua j − ua−1, j
hr
+ σ
2 ui j +
a
va,
j+1 −
va, j−1
+ cot θ j va j
2 hθ
(4.4.1)
0 = τrθ
1 ua, j+1 − ua, j−1
= µ
+
a 2 hθ
va j − va−1, j
−
hr
1
a va
j . (4.4.2)
The constants are not canceled out because of the computer precision. When values are
close to zero the constants have an important contribution to the solution. The equations
(4.4.1) and (4.4.2) are both dependent of the radial and tangential displacement,
therefore the two FDEs, Equations (4.3.5)-(4.3.6), have to be combined and solved as
one system.
The local truncation errors for the backward scheme for ur and uθ are
τ i r = − hr
2 urr( ˜ηi, θ j) (4.4.3)
τ j
θ
= −hθ
2 uθθ(ri, ˜η j) . (4.4.4)
These truncation errors add to the total truncation error at the boundaries.

Boundary Conditions 35
i
i − 1
i − 2
j − 1
❣
✲ ❣ ✛
✇ ✇
✒
✻
j j + 1
❣
✇
✇ ✇ ✇
✛ i = a, Free surface
Figure 4.4.1: The diagram is showing the free surface at i = a. This is an implicit method. In
the radial direction a backward finite difference scheme is used, and in the latitudinal direction
a central finite difference scheme is used.
4.4.2 Center of Earth
The case of the boundary condition R0 = 0 needs special attention due to the pole singularity.
By using a staggered grid the coefficient matrix gets a satisfactory structure.
ri =
i − 1
hr, where hr =
2
a
n − 1
2
for i = 1, 2, . . . , n
[Li, 2004]. Using the non-conservative form of the discretization at i = 1
− Fu(r1, θ j)
ρ
=
1 α2 (u0, j − 2 u1, j + u2, j)
r1
h 2 r
+ 1 α
r1
2 (u2, j − u0, j)
2hr
+ (α2 − β2 ) u1, j
r2 i
+ 1
r2 β
1
2 (u1, j+1 − 2 u1 j + u1, j−1)
h2 θ
+ 1
r2 (α
1
2 − β2 ) (u1, j−1 − u1, j+1)
2 hθ
(α2 − β2 ) (u2, j+1 − u2, j−1 − u0, j+1 + u0, j+1)
+ 1
r1
4 hr hθ
(4.4.5)
, (4.4.6)

36 Tidal Deformation of the Solid Earth
and
− Fv(r1, θ j)
ρ
=
1 β2 (v0, j − 2v1, j + v2, j)
r1
h 2 r
+ 1 β
r1
2 (v2, j − v0, j)
2hr
+ 1
r2 1 h2 θ
α 2 (v1, j+1 − 2 v1 j + v1, j−1)
+ 1
r2 (α
1
2 − β2 ) (v1, j−1 − v1, j+1)
2 hθ
(α2 − β2 ) (v2, j+1 − v2, j−1 − v0, j+1 + v0, j+1)
+ 1
r1
4 hr hθ
(4.4.7)
By definition of i it is determined that U0 j = 0. The finite difference equation for i = 1
is simplified to
The radial field
− Fu(r1, θ j)
ρ
and for the tangential field
− Fv(r1, θ j)
ρ
=
=
1 α2 (u2, j − 2 u1, j)
r1
h 2 r
+ 1 α
r1
2 u2, j
2hr
+ (α2 − β2 ) u1, j
r2 i
+ 1
r2 β
1
2 (u1, j+1 − 2 u1 j + u1, j−1)
h2 θ
+ 1
r2 (α
1
2 − β2 ) (u1, j−1 − u1, j+1)
2 hθ
(α2 − β2 ) (u2, j+1 − v2, j−1)
+ 1
r1
4 hr hθ
1 β
r1
2 (v2, j − 2 v1, j)
h2 +
r
1 β
r1
2 v2, j
2hr
+ 1
r2 α
1
2 (v1, j+1 − 2 v1 j + v1, j−1)
h2 θ
+ 1
r2 (α
1
2 − β2 ) (v1, j−1 − v1, j+1)
2 hθ
(α2 − β2 ) (v2, j+1 − v2, j−1)
+ 1
r1
4 hr hθ
(4.4.8)
, (4.4.9)
(4.4.10)

Boundary Conditions 37
4.4.3 The Pole and Equator
The displacement governed from the tidal acceleration applied on a sphere is equally
sized in the four quadrants, i.e. the boundary problem in the latitudinal direction is
periodic in the domain [0, 2π], and because of axes symmetry, we are only looking at
one quadrant, the interval [0, π
2 ] i.e. from the polar axis to the equatorial axis.
For the radial displacement the most simple choice, to apply boundary conditions
on the free Pole and Equator axes, is to exploit the symmetry of the displacements on
both sides of the axes, the displacement vector is horizontal at the Pole and vertical at
the Equator. In practice this is carried by smoothing the partial derivatives
ui,2 − ui,1 = 0 (4.4.11)
ui,m−1 − ui,m = 0 , (4.4.12)
where i = 1, 2, . . . , n. Equation (4.4.11) is the boundary condition for the polar axis and
Equation (4.4.12) is for the equatorial axis. The latitudinal displacement is symmetric
about the center of mass and therefore zero at the polar and equatorial axes. This yields
vi,1 = 0 (4.4.13)
vi,m = 0 . (4.4.14)

Chapter 5
Regularization
This chapter deals with the problem of solving the two dimensional Navier equation
of motion and how to treat an ill-conditioned system. The chapter is not in chronological
order with the rest of the thesis, but the theory of regularization was important to
outline, as the linear system of equations proved to be ill-conditioned.
Let A ∈ R n×m be a rectangular matrix, F a vector with n data points and x a vector
with m model parameters. We are aiming to solve the linear system
Ax = F. (5.0.1)
This equation corresponds to the FDE, Equations (4.3.5)-(4.3.6), derived in Section
4.3. A is the coefficient matrix with the coefficient of the differentials, F is the lefthand
side of the Equations (4.3.5) and (4.3.6). This is the tidal acceleration given in the
Equations (2.2.6)-(2.2.7) in Section 2.2.1 together with the boundary constrains given
in Section 4.4. x corresponds to the displacements u and v from Chapter 4, which are
going to be modeled.
If A is nonsingular, Equation (5.0.1) has a unique solution for every F. On the other
hand if A is singular the equation has zero or infinite solutions, [Aster et al., 2005].
This is not a regular inverse problem, because the data vector F is exactly known from
theory. The connection and stability between data and model spaces and the solution
uniqueness and ability to fit data is examined in the next section.
5.1 Ill-conditioning of a System
An ill-conditioned system is one without a solution, i.e. a system that does not have
a unique solution or where the solution is discontinuously dependent of the data. A
common characteristic of the problem is a large condition number. A condition number
states a measure of instability of the solution, [Aster et al., 2005]. When there is no
errors in the data vector, the condition number can more accurately be defined as the
maximum ratio of the relative error in the true solution x, that is x − ˆx/x, divided
by the relative error in A, and that is A − Â/A. ˆx is the actual and  is the actual

40 Tidal Deformation of the Solid Earth
data. The relative error bound of the model can be computed to
x − ˆx
x ≤ AA−1 A − Â
A
The constant is the condition number of A
.
cond(A) = AA −1 . (5.1.1)
[Parker, 1994]. A condition number of A near 1 indicates a well-conditioned matrix,
and a matrix with high condition number is said to be ill-conditioned. The rank of A
is the dimension of the column space or range of A. The column space is the set of all
vectors F that can be written as a linear combination of the columns in A. A has full
rank if k = rank(A) = min(m, n). If the rank is smaller than min(m, n) the matrix is
rank-deficient, [Aster et al., 2005]. When measurement errors or approximation errors
are present or discretization is applied on the system, there do not, in a mathematical
frame work, exist a rank of matrix A. If there exist some columns of A (with some
error level), which are practically linearly independent, there exist a pseudo rank. This
very loose formulation is the definition of a numerical rank, [Hansen, 1996].
According to Hansen [1996], ill-conditioned systems can be divided into two classes
of problems based on the properties of the coefficient matrix.
Rank deficient problems The coefficient matrix has the problems
• Cluster of small singular values.
• Large well-defined gap between large and small singular values.
This implies that there exist one or more linear combinations in the coefficient
matrix. There usually exist a reformulation that eliminate the ill-conditioning by
extracting the linearly independent information. There exist a numerical rank,
and the problem involves matrices which are exactly or almost rank deficient.
This definition do not contemplate how little a small singular value is.
Discrete ill-posed problems Come from the discretization of continuous ill-posed
problems, and the properties of their coefficient matrices are:
• Large cluster of small singular values which increases with the dimension
of the problem.
• The singular values decay sequentially to zero with no gap, and the Picard
condition is satisfied, see Section 5.2.2.
Usually there is no reformulation of the problem, which can eliminate the illconditioning.
Because of the lack of gap in the singular values, there do not
exist a notion for the numerical rank. The goal of the solution is to find a good
balance of the residual norm and the solution norm.

Tikhonov Regularization 41
If an ill-conditioned matrix does not apply to either of the classes, then the regularization
is not possible and the system has to be solved as good as possible without regularization
e.g. by iterative clarification. A well-conditioned problem is one where the
components of the solution’s SVD, increases on an average and the condition number
can be a measure of the solution’s sensitivity towards the data space, [Hansen, 1996].
Rank deficient problems arises in applications where noise or other unwanted disturbance
are suppressed for example in signal processing. Discrete ill-posed problems
normally arises doing the numerical treatment of inverse problems. Oil exploration is a
typical field, where ill-posed problem arises when trying to determine the composition
of the subsurface with seismic inversion techniques, other fields are image restoration
and tomography.
5.2 Tikhonov Regularization
In order to solve and stabilize the ill-conditioned problem, a Tikhonov regularization
program is used. The Matlab code is to be downloaded from the Matlab Central at
Hansen [2008a], and the code is a part of a larger regularization package, regutools,
documented in Hansen [2008b].
Tikhonov regularization is one of the oldest and probably most used regularization
methods for solving inverse problems. The idea is to find the best regularized solution
xε that minimizes the damped least square problem
xε = min{Ax − F 2 2 + ε2Lx 2 2 } , (5.2.1)
where ε is the damping or regularization parameter, which controls the regularization,
a large ε yields a large damping on the system. See more about choosing the good
solution in Section 5.2.2. L is in the standard form the identity matrix or in the generalized
form L Im where L is a p × m matrix with full rank. · 2 denotes the second
norm, l2. Tikhonov regularization is a direct method, because the solution xε given by
Equation (5.2.1) is the solution to
min A
ε I
F
x −
0
2
, (5.2.2)
where L = I. When using Equation (5.2.1) it is expected that the errors in the righthand
side is unregularized and that their covariance matrix is identical to the identity
matrix, [Hansen, 2008b].

42 Tidal Deformation of the Solid Earth
5.2.1 Singular Value Decomposition
In Hansen [2008b]’s Tikhonov regularization a singular value decomposition (SVD)
is used to analysis the ill-posed problem.
Letting A ∈ R n×m be a rectangular matrix with n ≥ m. The singular value decomposition
of A can be written as
A = UΣV T =
k
i=1
ui σi v T . (5.2.3)
U = (u1, . . . , un) and V = (v1, . . . , vm) are orthonormal matrices i.e. the transposed
matrix multiplied with the non-transposed matrix gives the identity matrix. p is the
number of nonzero singular values, the rank of A. U spans the data space R n and V
spans the data space R m . The vectors ui and vi are called left and right singular values
of A, respectively. Σ = diag(σ1, . . . , σk) is a diagonal matrix with non-negative
entries. σ1 ≥ σ2 ≥ . . . ≥ σk ≥ 0 and the numbers σi is called the singular value of A.
k = min(m, n). If p = k, A has full rank.
In l2 the condition number of A is given by the ratio
cond(A) = σ1
σk
. (5.2.4)
This is exactly true for A, if it has full rank and all the singular values are used in the
pseudoinverse solution (p = k). The SVD is unique for a given matrix A, aside from
singular vectors connected to multiple singular values, [Aster et al., 2005; Hansen,
2008b]. This can be stated because SVD is coupled to the eigenvalue decomposition
by the relation
and
A T A = VΣ T U T UΣV T = VΣ 2 V T
(5.2.5)
AA T = UΣV T VΣ T U T = UΣ 2 U T . (5.2.6)
The right-hand side describe the eigenvalue decomposition of the left-hand side, [Trefethen
and Bau, 1997].
SVD is effective in the Tikhonov regularization because it smoothens the solution by
less weighting of small singular values, as mentioned small singular values implies an
almost rank deficient matrix, [Hansen, 2008b]. With SVD it can be shown
p uT i F
vi , (5.2.7)
xLSQ =
i=1
where p is the rank of the matrix. From this equation it is very clear how very small
singular values in the dominator can give large values in the model space, which will
dominate the solution and it can become very unstable. By dropping the small singular
values the system is regularized and the solution is less sensitive to noise from the data
space. The cost of the regularization, is the fact that the solution is no longer unbiased
and the resolution is reduced, [Aster et al., 2005] .
σi

Tikhonov Regularization 43
5.2.2 The Good Solution
The essence about choosing the good solution is a question of balance. Because the
system is ill-conditioned, it has to be smoothed to give a suitable solution, but it is not
wanted that to smoothen to solution too much that you lose important information.
Filter Factor and Discrete Picard Condition
As mentioned, regularization is a processes of damping or filtering out small singular
values. The Tikhonov regularized solution in terms of SVD, Equation (5.2.7) gets the
form
xreg =
k
i=1
u
fi
T i F
σi
vi , (5.2.8)
where fi is the filter factors. In the used Tikhonov regularization the filter factor for
(L = I) is
fi =
σ 2 i
σ 2 i
+ ε2 . (5.2.9)
The filter factor is especially effective when σi > ε. This also shows if ε > σ1, then
the problem is unregularized. The filter factor is usually close to one for large singular
values. For values in between, the filter factors decreases monotonic when the singular
values decreases.
Equation (5.2.8) shows that the decay of |uT i F| compared to the decay of σi plays
an important role in regularization. In real world the right-hand side F is always con-
taminated with an error. The discrete Picard condition states that if |u T i
F| in a discrete
ill-posed problem, on an average, decay faster to zero than the singular values, then
the unperturbed right-hand side satisfy the Picard condition. If this is not true, the
Tikhonov regularization cannot give a useful solution, [Hansen, 1996].
Regularization Parameter
ε controls the sensitivity of the regularized solution to perturbations in A and F. The
threshold of the perturbation is proportional to ε −1 . There exist several methods to
estimate the best regularization parameter. There are the L-curve criterion, generalized
cross-validation (GCV), discrepancy principle or the Quasi-optimality criterion. In
most cases, the L-curve gives the best estimate of the regularization parameter, [Rojas,
1996; Aster et al., 2005; Hansen, 2008b].
The L-curve criterion plots the solution norm ||Lxreg|| versus the misfit norm ||Axreg−
F|| on a double logarithmic scale. It is called the L-curve because it almost always has
a characteristic L shaped curve, with a distinct corner, see Figure 5.2.1. The L-curve
is piecewise linear with break points at
(xi, yi) = (log ||Axi − F||, log ||Lxi||),
where i = 1, . . . , p. The optimal value ɛ, the regularization parameter, is considered to
be at the corner. The corner of the L-curve marks this transition, since it represents a

44 Tidal Deformation of the Solid Earth
compromise between the minimization of the norm of the residual and the semi-norm
of the solution. The horizontal branch of the L-curve is dominated by the regularization
error and the vertical branch shows the sharp increase of the semi-norm caused by
propagation errors. The ill-conditioning of A causes a strong growth in the weighted
semi-norm when ɛ exceeds a certain threshold, which is in fact the numerical rank of
A for a well-chosen ɛ.
In the Regularization package, [Hansen, 2008a], the Matlab codes lcurve.m and
lcorner.m plots respectively the L-curve and calculates the L-corner in relation to
the Tikhonov regularization where the filter factor is incorporated. For Tikhonov regularization
the L-curve is a smooth function, and this method selects the value, ɛ which
maximizes the curvature of the L-curve, [Hansen, 2008b]. If the discrete Picard condition
do not hold, the L-curve method may fail to give a good estimate of the regularization
parameter, [Vogel, 1996].
Figure 5.2.1: Example of the L-curve. Adapted from Hansen [2008b]

Chapter 6
Model Description and Construction
The following two chapters are concerning the development of two Earth models. The
two models in question are a completely homogeneous Earth and a layered Earth. They
are designed to determine the displacement field of the Earth due to the solid Earth
tides. This is done by solving the steady state Navier equation of motion, Equation
(3.3.3) described in Section 3.3. The problem is to discretize the Navier equation by
the numerical method of finite difference outlined in Chapter 4 with the main Equations
(4.3.5) - (4.3.6) and the seven boundary conditions given in Section 4.4. Finally the
displacement is untangled by the Tikhonov regularization, see Section 5.2. In next
chapter the models are analyzed and tested, but first this chapter starts by describing the
common model assumptions, then homogeneous Earth model is introduced followed
by the layered Earth model and in the end of this chapter the structure and development
of the models are outlined
6.1 Model Assumptions and Parameters
The models are defined to be continuous, homogeneous, isotropic, infinite and an elastic
medium, and from this the Navier equation of motion, Equation (3.3.3) is valid.
The right side of the equation are the tidal accelerations given by the Equations (2.2.6)-
(2.2.7) in Section 2.2.1. Further the Moon is assumed to be the only attracting body,
thereby neglecting the perturbations from the Sun and other planetary objects. The
Earth is assumed to be spherical symmetric about the center of the Earth and rotation
symmetry is applied, hence the Earth-Moon system is stationary in relation to each
other, and the Earth-Moon line is chosen going through the Equator. In Table 6.1.1,
the universal constants used in the model is given.

46 Tidal Deformation of the Solid Earth
Name Symbol Size Unit
Mean radius of the Earth r⊕ 6.3710 · 10 6 m
Orbital distance of the Moon R 3.8440 · 10 8 m
Mass of the Earth M⊕ 5.9742 · 10 24 kg
Mass of the Moon M 7.3483 · 10 22 kg
Constant of gravitation G 6.6726 · 10 −11 m 3 kg −1 s −2
Table 6.1.1: Universal constants with five significant numbers given by Woan [2003].
6.2 The Homogeneous Earth Model
The homogeneous Earth model is assumed to be a Poisson solid. For a Poisson solid
the Lamé parameters µ and λ equals each other, which is close to be true for rocks, [Lay
and Wallace, 1995]. The Lamé parameters are determined from the seismic velocities
given in Section 3.3, and can be derived as
µ = ρ β 2
and λ = ρ α 2 − 2 ρ β 2 . (6.2.1)
Poisson’s ratio σ and Young’s modulus are given in Section 3.3.1. The model is assuming
a P-wave velocity, α, of 8000 ms −1 , which corresponds to the top of the mantle
or at the core-mantle boundary, [Dziewonski and Anderson, 1981]. The ratio of the
seismic velocities in a Poisson solid is α/β = √ 3. The S-wave velocity, β, is calculated
from this ratio.
Since the Earth is homogeneous, the density of the Earth model is given by the
simple expression ρ = M⊕/(4πr 3 ⊕ ). M⊕ and r⊕ are the mass and the mean radius of
the Earth, respectively. The seismic velocities, density and the elastic parameters are
given in Table 6.2.1. Young’s modulus, the bulk modulus and the Lamé parameters for
rocks are in the magnitude of 20 − 120 GPa, [Fowler, 1990], which correspond to the
table values.
Name Symbol Size Unit
P-velocity α 8.0000 · 10 4 ms −1
S-velocity β 4.6188 · 10 4 ms −1
Density ρ 5.5153 · 10 3 kg m −3
Shear modulus µ 1.1766 · 10 11 Pa
Lamé Parameter no. 2 λ 1.1766 · 10 11 Pa
Poisson ratio σ 0.2500 Pa
Young’s modulus E 2.9419 · 10 11 Pa
Table 6.2.1: Seismic velocities, density and elastic parameters of the homogeneous Earth
model, given with five significant digits. The P-wave velocity is an assumption and the bottom
part of the table is calculated from this value and from the universal constants given by the
previous table, Table 6.1.1.

The Layered Earth Model 47
6.3 The Layered Earth Model
In this section, the homogeneous model in previous Section 6.2, is expanded to apply
for a layered Earth, this results in some new assumptions and new parameters have to
be introduced.
6.3.1 Assumptions and Parameters
The layered model uses the same universal constants as given in Table 6.1.1. The
difference between the two models are that the layered Earth model, as the name suggest,
is composed of four layers; inner core, outer core, mantle and crust. There are
not implemented any other boundaries within the layers such as the transition zone,
Moho or core-mantle boundary. The seismic velocities and the density in each layer
are assumed to be constant through out the layer.
The model uses data from the isotropic Preliminary Reference Earth Model (PREM)
by Dziewonski and Anderson [1981]. The geophysical properties from PREM are averaged
over each layer. In Figure 6.3.1, the seismic body wave velocities and the
densities through the Earth are depicted as functions of the radius. The different layers
in the Earth are characterized by abrupt changes in these values. The inner core is defined
in the range of 0-1221 km. The outer core is given from 1221-3480 km, and the
materials within the outer core are fluids. This is apparent from the zero shear velocity,
remember from Section 3.3 shear velocity can not exist in fluids, also the Poison ratio
of 0.5 states a fluid material. The mantle is ranging from 3480-5701 km and the outer
layer or the crust is defined from 5701-6371 km.
Figure 6.3.1: Plot of the seismic body wave velocities and the density through the Earth, as a
function of radius (m). Data is adapted from the isotropic PREM by Dziewonski and Anderson
[1981]. The blue curve shows the P-wave velocity and green is the S-wave velocity both in m s .
The density is represented by the red curve, given in kg
3 . The different layers in the Earth are
m
characterized by abrupt changes in these values.

48 Tidal Deformation of the Solid Earth
Parameters Inner core Outer core Mantle Crust
Radius range (km) r 0-1221 1221-3480 3480-5701 5701-6371
P-velocity (km s −1 ) α 11.1835 9.4203 12.4495 4.6833
S-velocity (km s −1 ) β 3.6128 0 6.7792 3.6508
Density (10 3 kg m −3 ) ρ 12.9792 11.2955 4.9835 3.2160
Shear modulus (10 11 Pa) µ 1.6941 0 1.6644 0.4284
Lamé no. 2 (10 11 Pa) λ 16.2329 9.9706 7.7230 0.7053
Young’s modulus (10 11 Pa) E 4.9222 0 4.6982 1.1234
Poisson ratio σ 0.4528 0.5000 0.4114 0.31102
Table 6.3.1: Physical properties of the different layers used in the analysis of the layered
Earth. The seismic velocities, the density and the elastic parameters are averaged over each
layer, with calculated from PREM.
The blue curve in Figure 6.3.1 is the P-wave velocity given in m s , green is the Swave
velocity also in m kg
s , and the red curve is the density given in
m3 . The radius is
measured from the center of the Earth to the surface in m. The values are written in
Table 6.3.1, and plotted in Appendix C.
6.4 Model Development
The two model are constructed in a similar fashion only the constants differ from each
other. The idea with the modeling is to make it as simple as possible. The equations
are complex enough, without using any smart solving method. The complexity arises
due to the chosen approach with the polar coordinate system, and the cross derivative
terms urθ and vrθ. When solving cross derivatives in finite difference equations, it is
not possible to use the simple method of dimension by dimension, which is a simple
and widely used method to solve FDE’s. This is because the partial difference equation
have no diagonal dominance as systems without cross derivative terms. In general
these terms make the modeling of the problem more difficult, [Li, 2004; Holmes,
2007], thereby the model uses a simple iterative scheme. The model is developed in
Matlab and the program code for the layered Earth model can be found in Appendix
B. This section is not meant to go through the whole modeling process but just to point
out and explain selected parts.
6.4.1 Grid Points and Meshes
The radial (u) and tangential (v) inner equations, from the Equations (4.3.5)-(4.3.6)
in Section 4.3, are discretized for each direction separately. Each FDE is made with
n = 66 grid points in the radial direction and m = 56 grid points in the co-latitudinal
direction, this were the limit allowed by Matlab. The mesh spans a part of a circle
sector, as described in Section 4.3, and a simple 4 × 4 grid is depicted in Figure 6.4.1,
where the starts with (1,1) in the center of the Earth.
The mesh sizes are respectively hr = 96.530 km and hθ = 0.028000 radians or
1.6043 ◦ , this gives arcs of 178.34 km at the surface, 120.98 km at a radius of 4413 km

Model Development 49
n
(4, 1)
(3, 1)
(2, 1)
(1, 1)
(2, 2)
(2, 3)
(2, 4)
(4, 2)
(3, 2)
n
(3, 3)
(3, 4)
m
(4, 3)
(n, m) = (4, 4)
Figure 6.4.1: The figure is displaying a 4 × 4 grid, where (1,1) starts at the center of the Earth,
and the numbering is working out from the center counting from the Pole axis towards the
Equator axis, where the Equator is the grid point (n, m). n corresponds to the radius and m
corresponds to the colatitude.
(mantle), 64.88 km at a radius of 2313 km (outer core) and 10.00 km at a radius of 392
km (inner core). Because of the uniform grid in the polar coordinate system, the grid
points are allocated much closer in the center of the Earth than at the surface. In stead
of using the staggered grid to deal with the pole singularity, as described in Section
4.4.2, a simple constrain of zero displacement is put on the center boundary condition.
Due to the coupled surface boundary conditions, as described in Section 4.4.1,
the radial and tangential linear system of equations have to be solved in one system.
Figure 6.4.2 shows a sketch of the linear system of equations Ax = F. As it is seen
both the radial and the tangential systems are evaluated together. u and v are the radial
and tangential displacements, respectively. Au and Av are the coefficients of the radial
and tangential problem, given in Equation (4.3.5)-(4.3.6). Ac represent the boundary
condition in the center of the Earth, Aue and Aup are the radial boundary conditions for
the equatorial and polar axes. Ave and Avp represent the boundary conditions for the
tangential problem. Last Ars and Ats are the radial and tangential boundary conditions
at the surface. The right-hand side represents the tidal acceleration and the boundary
constrains, where Fr = atides r and Fθ = atides θ , from the Equations (2.2.6)-(2.2.7). The
boundary constrains are the constrain of no shear at the surface and the symmetry
constrains of the polar and equatorial axes. The diagonal dotted lines in Au, Av, Ars
and Ats shows the tridiagonal form of the matrices.
6.4.2 The Coefficient Matrix
The tricky part is to construct the total coefficient matrix for both the radial and tangential
system. It is crucial to know the structure of the coefficient matrix, to keep track
of the entries. The two systems are connected through a row ordering matrix, with the

50 Tidal Deformation of the Solid Earth
Au
0-matrix
Ac
Aue
Aup
Ave
Avp
Ars
Ats
0-matrix
Av
u
v
=
Fr
Fθ
0 − vector
Figure 6.4.2: Sketch of the linear system of equations, Ax = F. u and v are the radial and tangential
displacement, respectively. Au and Av are the coefficients of the radial and tangential
problem, given in the Equations (4.3.5) and (4.3.6). The coefficient matrices are dominated by
tridiagonal lines, where the rest of the entries are zero. Ac represent the boundary condition
in the center of the Earth, Aue and Aup are the radial boundary conditions for the equatorial
and the polar axes. Ave and Avp represents the boundary conditions for the tangential problem.
Last Ars and Ats are the radial and tangential boundary conditions at the surface, constrained
of both v and u. The right-hand side represents the tidal acceleration, where Fr = atides r and
, and containing the boundary constrains.
Fθ = a tides
θ
following relation
k(i, j) = j + n(i − 1), for i = 1, 2, . . . , n, j = 1, 2, . . . , m.
This is called natural row ordering. k is a one dimensional array counting from 1 in the
top right corner to m on the top left corner. The last number is given in the right bottom
corner and is n · m. k(i, j) is a two dimensional array counting from (1, 1) to (n, m), and
it is very useful in determine the location of a given value. The whole model is build
on the row ordering matrix k(i, j). i and j are the same index as given in Section 4.3.1,
where i represent the radial direction from the center of the Earth to the surface, and j
represent the co-latitudinal direction from the polar axis to the equatorial axis.
The coupled coefficient matrix A is a sparse 7824 × 7392 matrix i.e. any nonzero
element is squeezed out and the system is discretized with 57,833,333 grid points,
where 54,641,660 are inner points and 3,193,344 grid points at the boundary. The data
vector (or the tidal acceleration vector) has the dimension 7824 × 1, where Fr and Fθ
both have the dimension 3696×1. The numerical rank of the full matrix A is calculated
to 7338. A table of the system values are given in Table 6.4.1.
A sketch of the coefficient matrix applied on the quarter Earth, is depicted in Figure
6.4.3. It shows the Earth in polar coordinates (r, θ) from the center to the surface of
the Earth and from the North Pole to the Equator. The different parts of the coefficient
matrix are applied on the sketch to visualize and explain the boundary conditions. To
evaluate the final results, the resulting vector is divided in the radial and tangential

Model Development 51
Model Parameters Symbol Size
Radial mesh size (m) hr 9.6530 · 10 4
Latitudinal mesh size (rad) hθ 0.0280
Number of radial grid points n 66
Number of latitudinal grid points m 56
Number of grid points 3696
Size of u n × m 66 × 56
Size of v n × m 66 × 56
Size of sparse coefficient matrix N × M 7824 × 7392
Total number of grid points 57,835,008
Number of inner grid points 54,641,660
Number of boundary grid points 3,193,344
Rank (numerical) rank(A) 7338
Table 6.4.1: Summarizing the model parameters. The first part of the table contains the finite
difference discretization parameters for the radial and the latitudinal problem, where the
number of grid points referred to the radial or the latitudinal problem. The second part are
the parameters for the coupled system. The bottom part gives the matrix properties of the
coefficient matrix.
The North Pole
Pole axis
Aup/Avp
Ac
Center of Earth
Au/Av
The free surface
Ars/Ats
r
Aue/Ave
Equatorial axis
θ
The Equator
Figure 6.4.3: Sketch of the coefficient matrix, applied on one quadrant. Ac represent the
boundary condition in the center of Earth, Aue and Aup are the radial boundary conditions
for the equatorial and polar axes. Ave and Avp represent the boundary conditions for the
tangential problem. Last Ars and Ats are the radial and tangential boundary condition at the
surface, constrained of both v and u.
components and the row ordering array is transfered back to only containing i’s and
j’s. Finally the dimension of u and v becomes 66 × 56.

Chapter 7
Model Considerations and Tests
In last chapter the models were constructed and in this chapter the problems which
have occurred in the modeling are described. This chapter stats out by analyzing the
linear system of equations, followed by a determination of the regularization parameter
and description of the different test which have been made on the system.
7.1 Model Considerations
As revealed in the end of Chapter 4, it was not possible to get a reliable result by
solving the linear system of equations with matrix inversion or even a damped least
square solution.
By using picard.m from Hansen [2008a], the singular values, σi, and the Fourier
coefficients from Equation (5.2.7) are plotted. From Figure 7.1.1, it is seen, that the
ill-conditioned problem does not satisfy the discrete Picard condition. Following Section
5.2.2, this states that the Tikhonov regularization can not give a reliable result for
a discrete ill-posed problem. This could indicate, that the system is not a discrete illposed
problem.
As mentioned in Section 6.4.2, the numerical rank of the coefficient matrix A is 7338.
The coefficient matrix has more rows (7824) than columns (7392), which makes it a
thin matrix and an overdetermined system. There are more equations than unknowns,
this means, the system is ill-conditioned and the coefficient matrix is both over- and
underdetermined, [Aster et al., 2005].
Regularization of the system is necessary, and as described in Chapter 5, choosing
the type of regularization method is a cumbersome process and needs analysis of the
system in relation to errors and physical interpretation. The regularization choice fell
on the Tikhonov regularization based on trial and error, this is described in further
details in Section 7.4.1.

54 Tidal Deformation of the Solid Earth
10 15
10 10
10 5
10 0
10 −5
10 −10
10 −15
10 −20
σ
i
T
|u F|
i
T
|u F|/σi
i
10
0 1000 2000 3000 4000 5000 6000 7000 8000
−25
i
Figure 7.1.1: The Picard plot showing the singular values, σi, (blue line), |uT i | (green x) and
their ratio (red o) , i are the entries.
7.2 Analyzing the System
To go into a deeper analysis of the systems ill-conditioning, the singular values of the
coefficient matrix A are analyzed. This is done separately for the homogeneous and
the layered Earth models.
7.2.1 Singular Value Spectrum for the Homogeneous Model
The condition number of the homogeneous Earth system is calculated by Equation
(5.2.4) to be 2.5130 · 10 23 . The large condition number of matrix A, states a very illconditioned
system, that means it is very sensitive to perturbations in the data space,
vector F.
In Figure 7.2.1, the singular value spectra for the homogeneous model is depicted.
Figure 7.2.1(a) shows the whole singular value spectrum. From this plot it is reasonable
to think that the system is rank deficient, because of the large gap between
the large and small singular values. Notice the large value of the singular values on
5.69 · 10 6 . The other three figures, Figure 7.2.1(b)-(d) are zoom levels of the singular
value spectrum. Figure 7.2.1(b) is from the beginning of the spectrum, and shows the
first cluster of large singular values, It is most of all shown because, this part looks
much like a discrete ill-posed problem with the sequently decay of singular values
without any significant gap. Figure 7.2.1(c) is a plot of the small singular values. The
lower values start at i = 113 with a singular value of 33.85. This i states the singular
value entries and is not to be compared with the FDE index representing the radius.
The last plot, Figure 7.2.1(d) shows the end of the spectrum, where the singular values
goes to zero, or very close to (10· −16 ) , which is most likely due to round-off errors,
this happens at i = 7339, this also agree with the rank of the system of 7338.

Analyzing the System 55
Singular values
Singular values
x 106
6
5
4
3
2
1
0
0 1000 2000 3000 4000 5000 6000 7000 8000
i
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
(a)
1000 2000 3000 4000 5000 6000 7000
i
(c)
Singular values
Singular values
5.4
5.35
5.3
5.25
x 10 6
5.2
15 20 25 30 35
i
40 45 50
x 10
5
−4
4
3
2
1
0
−1
(b)
7290 7300 7310 7320 7330 7340 7350
i
Figure 7.2.1: Singular spectra of the homogeneous model. (a) shows the whole singular
spectrum. Notice the large gap between the high and small singular values. (b) shows the
spectrum from i = 14 to i = 52. This is the second cluster of high singular values. (c) shows
a section of the low singular values, they clearly goes towards zero. (d) here is an other break
point, where the singular values goes from values of 10 −4 to values of 10 −16 .
7.2.2 Weighting of the Surface Boundary Conditions
The inner points are valued much more in the Tikhonov regularization, due to the larger
amount of points. Remember the 54, 641, 660 inner points against the 3, 193, 344
boundary points, where 827, 904 of the points are connected to the boundary. Multiplying
a factor, f fBC, on the surface boundary conditions, assign a weight on the
surface boundary conditions. This factor is not to be confused with the filter factor
applied by the Tikhonov regularization to the whole system. The boundary factor is
chosen to f fBC = 10 −5 , on the basis of the error plot in Figure 7.2.2 and physical interpretation,
surprisingly the best value was less than 1, i.e. the factor puts less weight
on the surface boundary condition. The figure show the residual error against different
factor values in a double logarithm representation.
The result of this weighting is very clear when plotting the new singular spectrum, see
the plot in Figure 7.2.3. Figure 7.2.3(a) shows the whole singular spectrum with the
applied boundary factor. Notice how the large singular values have gone from a factor
of 10 6 to 56.94. Looking at the spectrum from i = 1050 to i = 1850 given in Figure
7.2.3(c), it is not clear, that the spectrum is rank deficient, but zooming at i = 300 to
(d)

56 Tidal Deformation of the Solid Earth
|F−A*X|
10 −4
10 −5
10 −6
10 −7
10 −8
10 −9
10 −15
10 −10
Filter factor on Surface BC
10 −10
10 −5
10 0
Surfece factor, ffBC
Figure 7.2.2: Logarithmic error plot of different surface boundary factors. f fBC = 10 −5 is
chosen after physical interpretation of the results and the errors of the system.
Singular values
Singular values
60
50
40
30
20
10
0
0 1000 2000 3000 4000 5000 6000 7000 8000
i
0.25
0.2
0.15
0.1
0.05
0
(a)
1100 1200 1300 1400 1500 1600 1700 1800
i
(c)
Singular values
Singular values
1.5
1.4
1.3
1.2
1.1
1
x 10
14
−4
12
10
8
6
4
2
0
10 5
10 10
350 400 450 500
i
(b)
7150 7200 7250 7300 7350
i
Figure 7.2.3: Singular spectra for the homogeneous Earth model with surface boundary factor
of f fBC = 10 −5 . (a) shows the whole spectrum for the new singular values. (b) is the spectrum
from 300 to 550, the characteristic gap for a rank deficient problem is very clear. (c) shows the
spectrum from 1050 to 1850. (d) shows the end part of the spectrum, where the cluster of very
small singular values begin. This spectrum goes from 7100 to 7392.
(d)

Analyzing the System 57
i = 550, 7.2.3(b) we see the characteristic gaps. The last subfigure, Figure 7.2.3(d)
shows the spectrum going to zero at i = 7339 or very close to (2.6125 · 10 −16 ).
As a consequence of lowering the large singular values, the condition number has
decreased. The new condition number of the system is 1.2848 · 10 18 , which is a clear
improvement of the earlier result of 2.5130·10 23 . The condition number is still to large
to rely on the solution i.e. regularization is necessary!
7.2.3 Singular Value Spectrum for the Layered Model
The layered Earth model is analyzed like the homogeneous model in previous section,
and the same factor f fBC = 10 −5 on the surface boundary conditions are applied. It has
been chosen not to show the singular spectrum for the unfiltered boundary condition
because it resembles the homogeneous plot, Figure 7.2.1 in Section 7.2.1.
The singular spectrum for the layered model with the boundary factor is plotted in
Figure 7.2.4. Figure 7.2.4(a) shows the whole spectrum from i = 1 to i = 77329. At
first sight the large gap between the large and small singular values are there no more,
this will be discussed in more details in Section 9.1. The largest singular value is 66.14.
In Figure 7.2.4(b) the spectrum ranges from i = 20 to i = 44. The singular values
decay almost gradually. Figure 7.2.4(c) shows the singular spectrum from n = 88 to
n = 148, and it is a sequence with a clear gap in the values. The last figure, Figure
7.2.4(d), shows how the singular values go towards zero at i = 7339, just like in the
homogeneous model. The condition number has changed from 1.6340·10 23 to 1.7363·
10 18 with applying the boundary condition factor, which is clear an improvement of
the system stability, but still a too large value to relay on the system. This means that
regularization is still needed to solve the system.

58 Tidal Deformation of the Solid Earth
Singular values
Singular values
70
60
50
40
30
20
10
0
0 1000 2000 3000 4000 5000 6000 7000 8000
i
22
20
18
16
14
12
(a)
10
90 100 110 120
i
130 140
(c)
Singular values
Singular values
51
50.5
50
49.5
49
48.5
8
7
6
5
4
3
2
1
0
x 10 −5
22 24 26 28 30 32 34 36 38 40 42
i
(b)
7320 7325 7330 7335 7340 7345 7350
i
Figure 7.2.4: Singular value spectra for the layered Earth model with a filter factor on the
surface boundary conditions. (a) shows the whole spectrum, the first and largest value is
66.14. (b) shows the singular values from i = 20 − 44, the values decay almost sequently. (c)
the spectrum from i = 88 − 147, it shows large gaps between the values. (d) is a zoom of the
spectrum going toward zero at i = 7339.
7.3 Choosing the Regularization Parameters
In this section the regularization parameters for both the homogeneous and layered
Earth model are found, and the Tikhonov regularization parameters are described. The
regularization parameter is found with the L-curve method described in Section 5.2.2,
where the residual norm is plotted against the solution norm. Choosing various values
of ε ranging from 1 to 10 −20 the L-curve is generated with l_curve.m from Hansen
[2008a].
The Homogeneous Earth
The program calculated the best regularization parameter to be ε = 4.4802 · 10 −9 , the
plot can be seen in Figure 7.3.1. I should be mentioned, that the numbers in the figure
are generated by the l_curve.m and do not necessarily show the regulation parameter,
which is the case in this figure. The parameter is found by the program l_corner.m.
For testing the regularization effects, regularization parameters deflecting from the
L-curve generated value, have been tested the model. The main characteristics were a
(d)

Choosing the Regularization Parameters 59
solution norm || x || 2
10 5
10 0
10 −5
10 −10
10 −8
2.5994e−12
9.1764e−14
4.4802e−09
1.6738e−06
10 −7
10 −6
10 −5
residual norm || A x − b ||
2
4.7414e−05
0.0013431
0.038046
Figure 7.3.1: The L-curve for the homogeneous model. The regularization parameter is ε =
4.4802 · 10 −9 . The curve is generated from l_curve.m from Hansen [2008a], and shows the
residual norm against the solution norm.
larger displacement when more regularization was applied, for ε = 10 −6 the displacement
of the Earth was 130.7 mm and for ε = 10 −12 the maximum radial displacement
was 143.1 mm. For 10 −13 < ε < 10 −5 the physical interpretation does not appear
satisfied any more. In Appendix D.1 some early test can be found.
The Layered Earth
The regularization parameter is determine like the homogeneous model, Section 7.3,
by the L-curve method, described in Section 5.2.2. See the L-curve plot in Figure
7.3.2, the regularization parameter is calculated to ɛ = 1.0157 · 10 −8 .
solution norm || x || 2
10 5
10 0
10 −5
10 −10
10 −10
10 −15
2.3496e−13
10 −9
10 −8
6.6557e−12
1.0157e−8
10 −7
residual norm || A x − b || 2
10 −6
4.2857e−06
10 −5
1.0778
2.7595
10 −4
0.0001214
Figure 7.3.2: The L-curve for the layered model, with corner at 1.0157 · 10 −8 , which is the
chosen regularization parameter.
0.003439
0.097416
10 −4

60 Tidal Deformation of the Solid Earth
7.3.1 Tikhonov Filter Factors
The filter factors generated by the Tikhonov regularization, Equation 5.2.9 adds a filter
factor corresponding to all the singular values, σi, and the regularization parameter
ε. The filter factors can be generated by fil_fac.m from Hansen [2008a]. In Figure
7.3.3, the filter factors are plotted for the homogeneous and layered models.
For the homogeneous Earth model, the first 1402 entries are one, hereafter the filter
factors begins decaying, i.e. the regularization is applied. Until the 7338th value the
entries are very close to zero, i.e. these values are only smoothen very little. At a value
of i = 7338 a filter factor of 10 −8 is applied, it decays towards 9.91 · 10 −17 to the last
singular value, i = 7392. Notice that the regularization begins at i = 7338 the same
value as the rank of the system, and where the singular values goes towards zero, i.e.
the filter factor, filter out the small singular values.
The filter factor applied by the Tikhonov regularization to the layered model starts
filtering at i = 910, and at i = 7338 the filter factor decreases rapidly towards 1.44 ·
10 −17 , i.e. the values after i = 7338 are more smoothed.
Filter Factor
10 0
10 −5
10 −10
10 −15
homo
layered
−20
10
0 1000 2000 3000 4000 5000 6000 7000 8000
Index, n
(a)
Filter Factor
10 −14
10 −15
−16
10
Filter Factor
10 0
10
−1e−06
homo
layered
7340 7350 7360 7370 7380 7390
Index, n
(c)
7310 7315 7320 7325 7330 7335 7340
Index, n
homo
layered
Figure 7.3.3: Semilogarithmic plot of the filter factors for the homogeneous and layered models.
(b)

Testing the System 61
7.4 Testing the System
This section aims to show some of the test, which have been applied on the system,
besides the test of the regularization parameters, which have already been covered. The
testing includes, boundary test of the resulting solution method, regularization steps,
other solving methods and last the applied tidal acceleration is tested. In the following
a physical meaningful result, should be understood as a result that fulfill the theoretical
knowledge of the Earth’s tidal displacement field, described in Section 3.4. The tests
are carried out for both models, but the results are trivial, therefore only the results for
the layered Earth are shown.
7.4.1 Test of Solving Methods
When it was clear that the solution could not be solved with a simple inversion, X =
A −1 F, because A was not invertible and badly scaled, the first test applied of the model
was a regular Tikhonov solution , or damped least square solution, Equation 5.2.1.
For approximately 10 −6 > ε > 10 −8 , there were no physical meaningful solution,
for values in between, the results were physical acceptable. The maximum radial
displacements yield 69.0 ± 0.5 mm, the minimum displacement yield −18.5 ± 0.5 mm,
and the maximum tangential displacement was 45.1 ± 0.5 mm. This method were
less computational intensive and could therefore handle larger arrays, but the damped
least square solution was rejected because of an infinitely large condition number. The
largest achievable coefficient matrix was 10579 × 10082, with hr = 89.732 km and
hθ = 1.2676 ◦ . For the best result, with ε = 10 −5 , the residual norm was F − AX =
7.575 · 10 −6 .
Though the Truncated Singular Value Decomposition (TSVD) is the most advisable
approach of rank deficient problems, [Hansen, 1996; Aster et al., 2005], the
Tikhonov regularization gave better results for this study. The TSVD method is advised
for rank-deficient problems, because it handles the numerical rank of the coefficient
matrix. The singular values are cut of at the chosen truncation factor, normally
this factor is the rank. The filter factors in TSVD decay slower than the Tikhonov filter
factors, [Hansen, 1996] . The TSVD have been carried out for different truncations, but
no satisfying results have been found. Also QR regularization with Householder and
a damped singular value decomposition (DSVD) have been tested, and also without
the fruitful results. In short, QR decomposition factorize the coefficient matrix into an
orthogonal and a upper-triangular matrix, where householder is a computational tool
to make a matrix transformation. The DSVD method is quite similar to TSVD, but the
DSVD does not cut away any singular values, [Aster et al., 2005]. Examples of the
tests can be found in Appendix D.2.
7.4.2 The General Solution
The general damped least square method has been tested, and been carried out by
changing the roughening matrix L in Equation (5.2.1). L is changed from the identity
matrix by implementing a finite difference approximation to the Laplacian i.e. second
order derivatives in both directions. When solving the general Tikhonov, Matlab has

62 Tidal Deformation of the Solid Earth
to make a general singular value decomposition, gsvd. This function needs to handle
an even larger data set than the svd function, because it is storing two orthonormal
matrices, a non singular matrix and two diagonal matrices, where the svd function
only stores three matrices, described in Section 5.2.1.
The maximum grid numbers achieved with this method were (40 × 40) corresponding
to mesh sizes of hr = 155.39 km and hθ = 2.1951 ◦ . The results were
physical meaningful, with a best fit of ε = 1.7453 · 10 −7 . The maximum radial displacements
was 76.4365 mm and the minimum was −23.7960 mm, and the maximum
tangential displacement was 47.5471 mm. These results gave the residual norm
F − AX = 7.1012 · 10 −6 . It could have been possible to use the general form, but
it has been chosen not use this method, because of the grid number limitations, and
because the general solution did not elucidate better results.
7.4.3 Testing the Weighting of Grid Numbers
The layered Earth model has been tested for different grid numbers. In Figure 7.4.1,
there are shown four selected tests for both the radial and tangential displacement
fields. In Figure 7.4.1(a)-(b) the grid numbers are (6 × 6) with mesh sizes of hr =
1061 km and hθ = 15.0 ◦ , Figure 7.4.1(c)-(d) are a (16 × 16) grid with mesh sizes of
hr = 398 km and hθ = 5.6 ◦ , Figure 7.4.1(e)-(f) displays a (36 × 36) grid , with the
mesh sizes hr = 177 km and hθ = 2.5 ◦ and last in Figure 7.4.1(g)-(g) a (66 × 56) grid
is shown, with mesh sizes of hr = 97 km and hθ = 1.6 ◦ . As it is shown from the figure,
the model fails to give a physical meaningful result with only (6 × 6) grid numbers.
The radial displacement field, Figure 7.4.1(a), becomes a purely expansive field, and
the tangential field has developed a negative traction at the center of the Earth. Figure
7.4.1(e)-(e) gives a nicer picture, the radial displacement field has a compression at the
North Pole, though it only extend to about θ = 15 ◦ . The result from Figure 7.4.1(g)-(g)
will be described in more details in Section 8.2.
Comparing the three tests, displayed in Table 7.4.1, it is clear how the magnitude of
the displacements are dependent of the grid numbers. The magnitude increases as the
number of grid points increases, the growths goes very fast until about a grid number
of 30 × 30, where the growth increases to a smaller degree, it could appear as the
growth of the displacement stabilizes. In Figure 7.4.2 a plot of the radial displacement
growth is displayed. Notice also how the displacement for the grid 50 × 50 actually is
larger than the 66 × 56 grid.

Testing the System 63
R
R
R
R
R
R
R
R
(a)
(c)
(e)
(g)
θ
θ
θ
θ
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
−0.005
−0.01
−0.015
0.06
0.05
0.04
0.03
0.02
0.01
0
−0.01
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
−0.01
Radial displacement, u(r, θ) in m .
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
−0.01
−0.02
Radial displacement, u(r, θ ) in m .
Radial displacement, u(r, θ ) in m .
Radial displacement, u(r, θ ) in m .
Figure 7.4.1: Test of four different grid sizes given by n × m where n is the grid in the radial
direction and m is the grid in the co-latitudinal direction. (a)-(b) are a 6 × 6 grid (c)-(d) are a
16 × 16 grid, (e)-(f) are a 36 × 36 grid and (g)-(h) are a 66 × 56 grid.
R
R
R
R
R
R
R
R
(b)
(d)
(f)
(h)
θ
θ
θ
θ
x 10 −3
0
16
14
12
10
8
6
4
2
0
−2
0.05
0.04
0.03
0.02
0.01
−0.01
0.05
0.04
0.03
0.02
0.01
0
−0.01
0.05
0.04
0.03
0.02
0.01
0
Tangential displacement, v(r, θ ), in m .
−0.01
Tangential displacement, v(r, θ ), in m .
Tangential displacement, v(r, θ ), in m .
Tangential displacement, v(r, θ), in m .

64 Tidal Deformation of the Solid Earth
Grid size
(n, m)
u(n, m)
in mm
u(n, 1)
in mm
v(n, m/2)
in mm
Cond(A)
·10
ε
F − AX
F
18 ·10−8 %
6 x 6 33.9553 2.3392 · 10−4 7.4574 0.1723 180.0938 81.1490
16 x 16 64.1823 -12.7884 38.7450 5.5237 22.0989 53.3956
36 x 36 73.3261 -17.8062 44.9401 1.9434 5.6186 36.4931
66 x 65 74.3154 -26.2925 45.3977 1.6218 2.7670 28.1446
Table 7.4.1: Table of the parameters in the grid size test. The columns are: grid size, where
n is the radial direction and m is the co-latitudinal direction, u is the radial displacement
given at the maximum magnitude (n, m), v is the maximum tangential displacement, ε is the
F − AX
regularization parameter and is the error given by the residual norm over the data
F
norm and Cond(A) is the condition number of the system before regularization.
Radial displacement displacement in mm
80
75
70
65
60
55
50
45
40
35
30
6x6
16X16
16X16
21X21
26x26
31x31
Number of grid points
Figure 7.4.2: plot of the growth of the radial displacement due to increase of grid numbers.
7.4.4 Testing the Boundary Conditions
The boundary conditions are tested by applying five point sources of −10 −3 m s 2 in the
radial direction instead of the tidal acceleration, and there is no tangential acceleration.
The negative sign makes it attracting point sources, as the tidal acceleration. The five
point sources are placed at the grid points (3, 3), (n/2, m/2), (n − 1, 3), (n − 1, m/2) and
(n − 1, m − 3), where n and m are the number of grid points in the radial and latitudinal
directions, counting from the center of Earth and from the pole respectively. n/2 and
m/2 corresponds respectively to half of the radius and quarter of the circle, review the
grid in Figure 6.4.1.
This results in the displacement field depicted in Figure 7.4.3. The plot shows that
when a point source is applied, no points in the center of the homogeneous Earth are
displaced. There is a little deflection in the central point, where the point source attack
on the half radius and quarter circle. The displacement is in this point 170 mm. At the
surface of the homogeneous model the displacement is the greatest, with a maximum
36X36
51X51
66X56

Testing the System 65
R
R
Figure 7.4.3: Testing the surface conditions with point sources applied at the grid points (3, 3),
(n/2, m/2), (n − 1, 3), (n − 1, m/2) and (n − 1, m − 3), where n and m are the number of grid
−3 points. The point sources has a values of −10 m
2 . The figure show the resultant displacement
s
field after the point sources are applied.
displacement of 634 mm. The behavior of the point source reacts just like if somebody
pulled at a single point. The displacement are largest at the attacking point and become
smaller, but still positive, and only in a local area away from the pulling direction.
It is remarkable to notice how the displacement is largest at the attacking point,
and how it become smaller moving farther away, to finally not being influenced by the
point source anymore.
7.4.5 Testing the Tidal Acceleration
The radial and tangential components of the tidal acceleration are plotted in Figure
7.4.4(a)-(b), as functions of radius and colatitude on a quarter Earth. Note the different
color scales. The radial component of the tidal acceleration is positive from the North
Pole to θ = 36◦ and negative from this point on to the equator axis. The acceleration
applied on the surface at Equator is Fr = −1.0999m s2 . The tangential component of
the tidal acceleration has the largest negative acceleration at θ = 45 ◦ with a value of
Fθ = −0.8224m s2 . These tidal component values are consistent with Torge [2001].
Figure 7.4.4(c) shows the radial displacement field in m of the quarter Earth, where
only a radial dependent tidal acceleration is applied. As excepted the Earth expands
outwards over the whole volume.
θ
0. 6
0. 5
0.4
0.3
0.2
0.1
0
−0.1
−0.2
Radial displacement in m, u(r, θ ) , in m.

66 Tidal Deformation of the Solid Earth
R
R
(a)
R
θ
x 10
5
−7
0
−5
−10
Radial comp. of tidal acc. in m/s 2 .
R
(c)
Figure 7.4.4: (a) The radial component and (b) tangential component of the tidal accelerations
applied on a quarter Earth. (c) The radial displacement field of the Earth due to a radial
dependent tidal acceleration alone. Note the different color scales.
R
θ
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
−0.005
−0.01
θ), in m .
R
(b)
Radial displacement, u(r,
θ
x 10
0
−7
−1
−2
−3
−4
−5
−6
−7
−8
Tangential comp. of tidal acc. in m/s 2
.

Chapter 8
The Tidal Deformation of the Earth
This chapter aims to describe the resulting deformation of the homogeneous and layered
Earth Models due to the tidal acceleration. Also a analytic solution of the displacement
at the surface is elaborated, and compared to the numerical results.
8.1 Deformation in the Homogeneous Earth Model
In this section, the final results from the modeling is given. The results are given as the
displacement of the homogeneous Earth due to the solid tides. There are no particular
difference in the results for the homogeneous Earth models with and without boundary
factor. The boundary factor f fBC only changed the displacement by 10 −4 mm, it is
chosen not to show the results for the model without the boundary factor, because of
triviality.
The results for the homogeneous Earth model are plotted in Figure 8.1.1, and a representation
of the displacements are summarized in Table 8.1.1. The figure shows the
displacements of the Earth in polar coordinates, displayed on a quarter sphere, with the
center of the Earth at the bottom left corner, the North Pole at the top left corner and
the Equator at the bottom right corner. The radius are ranging from the center to the
surface of the Earth. . The colatitude from the Pole to the Equator. The displacement
is visualized by a color scale ranging from -0.05 m to about 0.13 m. The colours in
the colour bar are not exactly alike in all figures, but there is emphasis on that green
symbolizing zero displacement, blue negative displacement and the yellow-reddish
colour symbolizing positive displacement. Notice also the contour lines showing the
displacement values.
The table columns are given as follows: 1.) grid points (i, j) where i and j represents
the radius and colatitude, respectively, 2.) the radius coordinate, r, in kilometer
and 3.) the colatitudinal coordinate, θ in degrees corresponding to the grid. 4) is the
radial displacement, v, 5.) the tangential displacement u, 6.) the total displacement, u,
all the displacements are given in millimeters.
The three figures, displayed in Figure 8.1.1, are the radial displacement field, Figure

68 Tidal Deformation of the Solid Earth
8.1.1(a), the tangential displacement field, Figure 8.1.1(b), and the total displacement,
Figure 8.1.1(c). The total displacement field is the squared sum of the radial and tangential
displacement fields.
The displacement is negative at the North Pole with a value of u = −5 mm, i.e.
the point is compressed towards the center of the Earth. Going along the surface from
the Pole to approximately θ = 15 ◦ , the displacement becomes positive, and continuing
towards the Equator the radial displacement gets larger until reaching its maximum
value at the Equator, here the displacement is u = 131 mm expanding towards the
Moon. In the direction from the surface of the Earth to the center of the Earth the
displacement looses strength until it becomes zero at the center of the Earth. At the
equatorial axis the displacement is u = 125 mm in a radius of 4313 km, u = 86 mm at
2353 km and u = 21 mm at 392 km. Notice the contour lines being perpendicular to
the polar and the equatorial axes, but not at the surface boundary.
The tangential displacement field is characterized by being zero at the polar and
the equatorial axes with maximum at the surface at θ = 45 ◦ . The model parameters are
not completely zero, but this is probably a numerical error. The maximum tangential
displacement is v = 41 mm, which is a tractive movement in the tangential plane. At
θ = 45 ◦ and a radius of 4313 km the tangential displacement is v = 39 mm, at 2353
km a point is stretched v = 11 mm and at 393 km from the center the displacement
is v = 0.075 mm. The contour lines are not completely tangent to the surface. The
total displacement field has a maximum displacement at the Equator, θ = 90 ◦ of u =
131 mm.

Deformation in the Homogeneous Earth Model 69
(a) Radial displacement
(b) Tangential displacement
(c) Total displacement
Figure 8.1.1: The displacement (in meter) for the homogeneous Earth model with surface
boundary factor and Tikhonov regularization with ɛ = 4.4802 · 10 −9 . (a) shows the radial displacement
field, (b) shows the tangential displacement field and (c) shows the total displacement
field. The displacements are given for a quarter sphere, where the bottom left corner is
the center of the Earth, top left corner is the North Pole, and the bottom right corner is the
Equator.

70 Tidal Deformation of the Solid Earth
Grid points r (km) θ ( ◦ ) u (mm) v (mm) u (mm)
(1,1) 0 0 −6.6723 · 10 −15 1.2442 · 10 −13 1.2460 · 10 −13
(5,1) 392 0 19.2574 2.4379 · 10 −14 19.2574
(5,28) 392 45 20.5049 0.075 20.5051
(5,55) 392 90 20.7787 3.9377 · 10 −14 20.7787
(25,1) 2353 0 36.562 −5.9404 · 10 −14 36.562
(25,10) 2353 15 43.319 6.4133 43.7912
(25,28) 2353 45 71.0693 10.86 71.8942
(25,45) 2353 72 84.794 5.2619 84.9571
(25,56) 2353 90 86.1656 4.9518 · 10 −14 86.1656
(45,1) 4313 0 -0.4317 −2.7243 · 10 −12 14.4317
(45,10) 4313 15 25.0779 18.6913 31.2773
(45,25) 4313 39 61.6813 38.0125 81.1367
(45,28) 4313 45 81.5458 38.7285 90.2752
(45,45) 4313 72 119.0393 23.4878 121.3344
(45,56) 4313 90 125.4658 1.6143 · 10 −13 125.4658
(66,1) 6371 0 -4.5777 −6.0341 · 10 −9 4.5777
(66,10) 6371 15 -1.5318 13.5711 14.3078
(66,20) 6371 31 35.4273 31.3569 47.3112
(66,28) 6371 45 67.2545 41.2706 78.9078
(66,40) 6371 72 121.2756 30.4703 125.0449
(66,50) 6371 80 128.4876 18.3878 129.7967
(66,56) 6371 90 131.2487 5.5113 · 10 −9 131.2487
Table 8.1.1: Displacement table for the homogeneous Earth model. u, v, u are the radial,
lateral and total displacement respectively. The displacements are given in millimeters for
different locations within the Earth, displaced as grid points (i, j) and the corresponding radius
in meter and colatitude in degrees.
8.2 Deformation in the Layered Earth Model
This section describes the resulting tidal deformation of the layered Earth model. In
Figure 8.2.1, the radial, tangential and total displacements are displayed for the layered
model with boundary factor. The blue arcs represent the boundaries between the four
layers; inner core, outer core, mantle and crust. In Table 8.2.1 a representation of the
model results are shown. The overall picture of the displacements are the same as
described in Section 8.1 for the homogeneous model, but the size of the displacements
are quite different.
For the layered Earth model, the radial displacement, Figure 8.2.1(a), has a maximum
expansion at the Equator with a displacement of u = 74 mm, and a maximum
compressional displacement of u = −26 mm at the North Pole. The displacements are
changing signs, or going from a compression to an expansion, around a colatitude of
θ = 28 ◦ . Moving in a line from the Equator towards the center of the Earth, the displacement
decays, and at a radius of 4313 km, which is in the mantle, the displacement
become u = 68 mm, when moving further towards the center of the Earth, to a radius
of 2353 km (the other core), the displacement is u = 44 mm, for finally having a radial

Deformation in the Layered Earth Model 71
displacement value of u = 9 mm, at 392 km (the inner core) from the center.
The tangential displacement field, Figure 8.2.1(b), has a maximum traction at θ =
45 ◦ with a displacement of about v = 45 mm. In the mantle, at a radius of 4313, km
the tangential displacement is v = 33 mm, in the outer core (2353 km) the value is
v = 2 mm and in the inner core ( 393 km) the displacement is v = 0.0011 mm, all
these values are for θ = 45 ◦ . The total displacement field, 8.2.1(c) has a maximum at
θ = 74 ◦ , with a value of u = 77 mm.
Grid points u (km) θ ( ◦ ) u (mm) v (mm) u (mm)
(1,1) 0 0 -3.1101 · 10 −14 −3.02321 · 10 −14 4.3373 · 10 −14
(5,1) 39 0 8.3851 −3.6523 · 10 −15 8.3851
(5,28) 393 45 8.4905 0.0113 8.4905
(5,55) 393 90 8.5009 −6.3158 · 10 −14 8.5009
(25,1) 2353 0 17.5184 −7.1280 · 10 −12 17.5184
(25,10) 2353 15 27.2964 0.6479 27.3041
(25,28) 2353 45 41.3682 1.6261 41.4002
(25,45) 2353 72 43.5761 2.4750 43.6463
(25,56) 2353 90 43.7002 8.1170 · 10 −12 43.7002
(45,1) 4313 0 -10.0585 −3.4946 · 10 −11 10.0585
(45,10) 4313 15 3.8140 17.4486 17.8606
(45,25) 4313 39 42.3912 32.5096 53.4218
(45,28) 4313 45 48.9089 31.7561 58.3141
(45,45) 4313 72 66.8801 10.053 67.6315
(45,56) 4313 90 67.8094 2.6334 · 10 −14 67.8094
(66,1) 6371 0 -26.2925 −3.5447 · 10 −9 26.2925
(66,10) 6371 15 -21.1048 13.0612 24.8195
(66,20) 6371 31 3.4461 33.7783 33.9537
(66,28) 6371 45 28.2988 45.3977 53.4955
(66,40) 6371 72 68.8496 33.9149 76.7495
(66,50) 6371 80 73.2741 20.1883 76.0044
(66,56) 6371 90 74.3154 −7.2879 · 10 −11 74.3154
Table 8.2.1: Displacement table for the layered Earth model. u, v, u are the radial, lateral
and total displacement respectively. The displacements are given in millimeters for different
locations within the Earth, displaced as grid points (i, j) and the corresponding radius in meter
and colatitude in degrees.

72 Tidal Deformation of the Solid Earth
(a) Radial displacement
(b) Tangential displacement
(c) Total displacement
Figure 8.2.1: The displacement fields (in meter) for the layered Earth model with a surface
boundary factor of f fBC = 10 −5 and Tikhonov regularization with ɛ = 2.8137 · 10 −8 . (a)
shows the radial displacement, (b) shows the tangential displacement and (c) shows the total
displacement. The displacements are given for a quarter sphere, where the bottom left corner
is the center of the Earth, top left corner is the North Pole, and the bottom right corner is the
Equator.

Analytic Solution 73
8.3 Analytic Solution
In Section 3.4 the displacements were given by a function of the tidal potential and the
Love numbers. This analytic solution for the surface displacement was given by the
Equations (3.4.1) and (3.4.2), and the results of the calculated values are displayed in
Table 8.3.1.
The analytic solution resembles the general picture of the displacement fields from
the following sections. The radial displacement field is expansive and compressive,
and the tangential field is tractive. The maximum radial expansion is about a factor
of 2.5 larger than the maximum tractive tangential displacement. The magnitude of
the displacement components on the other hand differentiate from the analytic and numerical
models. The maximum radial displacement is calculated to u = 214 mm and
the maximum compression is u = −107 mm. The change from compressive to expansive
displacement happens at θ = 35 ◦ . The tangential displacement has a maximum
traction of v = 86 mm at θ = 45 ◦ .
Grid points r(km) θ( ◦ ) u (mm) v (mm)
(66,1) 6371 0 -107.0426 0.0000
(66,10) 6371 15 -86.2891 42.1091
(66,20) 6371 31 -21.4086 75.7376
(66,28) 6371 45 48.9362 85.5599
(66,40) 6371 72 183.32025 50.3345
(66,50) 6371 80 204.7475 28.8777
(66,56) 6371 90 214.0852 0.0000
Table 8.3.1: Values of the analytic solution given by the Equations (3.4.1) and (3.4.2).
8.3.1 Comparison Between Results
For the homogeneous Earth model the radial displacement field is about a factor 1.6
times less compressive than the analytic model, and about 23 times less expansive than
the analytic model. The tangential displacement is between 1.6 to to 3.1 times smaller
than the analytic results, with a smaller difference at the Equator.
The displacement field for layered Earth model, does also show a smaller deformation
compared to the analytic results. For the radial displacement field it is about
2.6 to 4.0 less expansive than the analytic solution, and for the tangential displacement
field it is about a factor 1.4-3.2 too small.
The surface displacements for the three results are depicted in Figure 8.3.1, where
Figure 8.3.1(a) are showing the radial displacement fields and 8.3.1(b) are showing the
tangential displacement fields. The displacements are given as a function of colatitude,
where θ = 0 ◦ is at the Pole and θ = 90 ◦ is at the Equator. The figure clearly shows the
large difference in the results, especially in the radial displacement fails.

74 Tidal Deformation of the Solid Earth
(a)
(b)
Figure 8.3.1: Comparing the radial and tangential displacements at the surface as a function
of the colatitude in degrees.

Chapter 9
Discussion
In this chapter the finite difference discretization, the developed models and the final
results are discussed. The discussion is divided into four parts: First the results
and errors are discussed, second the reliability of the model, third some applications
of tidal deformation are outlined and fourth suggestions to future developments and
improvements of the numerical modeling are discussed.
9.1 Results and Errors
In the Chapters 4, 6 and 7 an Earth model is developed with the numerical methods of
finite difference equations and treated with the Tikhonov regularization to stabilize the
solution. In the Chapter 8 the final results are given.
9.1.1 The Displacement Fields
Looking at the overall picture of the radial and tangential deformation fields, the numerical
results of the displacement models are at the surface consistent with the normal
physical interpretation. The radial displacement field expands with a maximum at the
Equator and contracts with a maximum at the Pole. The tangential displacement field
shows only tractive displacements, with a maximum at θ = 45 ◦ , and there is no displacement
at the polar and the equatorial axes.
Compared to the analytic results, the numerical models did not accomplish to generate
sufficiently large displacements. The analytic result was dependent of the averaged
Love numbers, but the magnitude of the analytic result is verified by, among
others Melchior [1978] and Xing et al. [2007]. Xing et al. [2007] have made a 3dimensional
finite element analysis and found a total displacement of 260 mm at the
side closest to the Moon. The Earth model was build much like the layered model in
this thesis, with four homogeneous layers, but the outer core was not assumed to be
completely fluid and the outer layer included both the crust and the transition zone.
The most crucial result is the lack of compression in the homogeneous Earth
model, it was 23 times too small! In general the compressions are fare too small in the
numerical results compared to the analytic result, and extending to an insufficient area.

76 Tidal Deformation of the Solid Earth
The compression of the radial displacement would have been expected to expand from
the North Pole down to the center of the Earth and extending to θ = 38 ◦ as the analytic
result showed. The tangential field should, at the maximum displacement, have been
about two and a half times the strength of the maximum radial field. The numerical
result give a three times smaller tangential displacement field for the homogeneous
Earth model and one and a half smaller in the layered Earth model. Again the layered
model show better results though the homogeneous model has a larger displacement
It is not surprising that the homogeneous Earth model is more deformed at the crustal
layer than the layered Earth model, because of the smaller shear modulus. Neither
is it surprising that the layered model shows better results compared to the analytic
solution, it is in fact more a reassemble to the real Earth.
The total displacement field fails to give a conceiving picture of the reel Earth
picture, due to the small compressions in the radial displacement fields. The total
displacement should have had a maximum displacement at the Equator.
From these results with less fluctuations in the values from the Pole to the Equator
and the smaller magnitude of the displacements it could seem to reflect a too large
damping of the system due to the regularization, but it was showed in Section 7.3 that
the regularization parameter did not have that great of a weight on the system to be the
reason for the bad scaled results.
When regarding the inner structures of the Earth, it is uncertain how large the real
displacement fields are, except that they becomes smaller moving down the Earth until
reaching the center of the Earth, where there is no displacement at all. This is in fact
due to the symmetry of the spherical Earth. This is indeed accomplished in the two
models.
The surface boundaries were tested (Figure 7.4.4) by applying five attacking point
sources, and they seemed to be working. The model behaved as if a head of a pin was
affixed on a piece of leader and somebody pulled the pin from the opposite side. The
leader would be displaced in the direction of the pull, depending on the force of the
pulling. The surrounding areas would also be displaced, but not in the same degree.
The radial contour lines are perpendicular to the polar and the equatorial axes, due
to the constrains of symmetry. The contours are not perpendicular to the surface of the
Earth, this is due to the large contribution of the partial derivatives with respect to θ,
from Equations (3.3.7) and (3.3.8). This can be shown by the simple calculation of the
grid point (n, 40), with the discretization given by the Equations (4.4.1) and (4.4.2),
where n is the last grid point in the radial direction, i.e. the surface n = a, we get
1
a
∂u
∂r = ua,40 − ua−1,40
hr
∂u
∂θ
9.1.2 Regularization
1 ua,39 − ua,41
=
a 2hθ
= −7.50 · 10 −09
= −3.06 · 10 −09 .
It is essential to investigate how good the regularized solution xreg is compared to the
result of the true solution. The bias put on the system by the Tikhonov regularization,
means that the solution would never be the true solution for a non zero regularization
parameter, ε, even with noise free data, [Aster et al., 2005]. It is not possible to

Results and Errors 77
compare with the true solution of the Earth’s displacement field, because such solution
does not exist. The knowledge of the inner structures of the Earth are only based on
theoretical assumptions, though it is possible with seismology and airborne satellites
to give a pretty good picture of of the displacement field within the Earth.
It is possible to examine the stability and errors of the models. In Table 9.1.1, the
condition numbers and regularization parameters are summarized with and without the
boundary factor.
When the boundary condition factor is applied, it approves the solution, especially
the stability of the system i.e. the condition number, has been enhanced by this step.
The stability has been approved, but it would be false to call the system stable, the
condition number is still much lager than one. The difference on ε shows that the
layered Earth model is less regularized compared to the homogeneous model, but the
regularization parameter are relative high, i.e. a large damping is embedded on the
system.
The boundary factor did not effect the results to give an observable change in the
displacement fields, but changed the condition number remarkable. Because of the
smaller number of grid points at the surface compared to inner points, a strengthening,
i.e. a greater weight of the surface boundary conditions were expected, but this was
not the case.
Cond(A) ε Cond(A) f fBC ε f fBC
Homogeneous 2.1635 · 10 23 4.4802 · 10 −9 2.5534 · 10 18 2.2531 · 10 −9
Layered 1.7301 · 10 23 2.8137 · 10 −8 1.7363 · 10 18 1.10157 · 10 −8
Table 9.1.1: Comparison of the condition number and the regularization parameters for the
homogeneous Earth model and the layered Earth model. The values are given both with and
without the surface boundary factor applied.
The error of the system is calculated with the residual norm divided by the data norm
F − AX
F − AX
and the solution norm . The results are shown in Table 9.1.2,
F
X
For the residual norm it does only shows a difference in the fifth decimal. In general
the layered model is more accurate than the homogeneous model. The residual norm
and data norm ratio, show a 10% error for the layered model and a 28% error for the
homogeneous model. These values are still too large for being acceptable.
Homogeneous
F − AX
8.6236 · 10
F − AX
X
F − AX
F
−6 1.90152 · 10−6 28.1462%
Layered 8.6234 · 10−6 5.4212 · 10−6 10.2352%
Table 9.1.2: Errors of the homogeneous and layered Earth models given by the residual norm,
the residual norm over the solution norm and the residual norm over the data norm.

78 Tidal Deformation of the Solid Earth
9.1.3 The Discretization
The Earth was discretized into 66 × 56 grid points for both the radial and latitudinal
systems, which gave mesh sizes of 96.53 km × 1.6043 ◦ , spanning a part of a circle
sector. Even after the important values were packed in a temporary directory, and the
unimportant values were deleted on the go, this were the finest discretization, possible
by Matlab due to limited memory. Closer grid would have been preferable, as the
regularization parameter, ε, gets smaller as the number of grid points increases, as
described in Section 7.4.3, Table 7.4.1. The computational limit was due to the use
of the Matlab function svd, the singular value decomposition. The function stores
the unitary matrices Up and Vp, containing the left and right singular value vectors,
and Matlab needs a lot of memory for this step. The server used did not have enough
memory to allow an increase of the grid numbers.
From Figure 7.4.1 it can be seen how the compression of the radial displacement
field is dependent of the grid sizes. For very small grid the displacement field fails
to give any compression at all. It is also clear from the table, Table 7.4.1 and Figure
7.4.2 that the magnitude of the displacement grow as the grid points increases. It is
hard to say what would have happened with a finer grid, and if the magnitude of the
displacements have reached a stable level, as described in Section 4.2. Though it is
hard to believe the displacements would have grown sufficient to match the analytic
results. The displacement field with 66×56 grid points is smaller than the displacement
field with 51 × 51 grid points, in Aster et al. [2005] it is briefly mentioned, that square
systems could give a better stability. Even though this is true, it would not have given
the extra needed displacement.
The body force applied by finite difference equations, only attacks at a single point
not a volume element, as in reality. It is possible that a normalization of the tidal
acceleration by adding a volume element should have been applied. This enhances the
importance of the mesh sizes, and it could be causing the errors, but as it is seen, the
mesh sizes do not change the displacement to a large degree, so this is probable not
the case. Durran [1999] states, that stability of pole singularity problems requires very
small step sizes.
9.1.4 Ill-conditioning
Determination of the systems ill-conditioning by the singular value spectra is a cumbersome
process, and the literature is very sparse. Nevertheless has there in this study
been a great focus on the process. It is the belief, that a great deal of information can
be extracted from these singular value spectra.
Ill-conditioning of this system is not due to measurement errors, but is caused
because the system is underdetermined, i.e. there is too many solutions. The question
whether the singular value spectrum for the layered Earth model shows a rank-deficient
problem, can be discussed. The definition of rank-deficient problems and discrete illposed
problems are very weak. Compared to Hansen [1996], also stated in Section 5.1,
there has to be a well-defined gap between the large and small values, what precisely
a well defined means, is not clear, neither is it stated how little a small singular values
is. Rojas [1996] makes a slightly different definition, that is, rank-deficient problems

Results and Errors 79
(a) Discrete ill-posed (b) Rank-deficient
Figure 9.1.1: Example of discrete ill-posed and rank-deficient problem by Rojas [1996]. The
singular values are given for (a) a discrete ill-posed problem (b) a rank deficient problem.
have a clear gap between large and small singular values and there is a (usually) small
cluster of small singular value. Compared to a plot from Rojas [1996] of a discrete
ill-posed problem and rank-deficient problem, see Figure 9.1.1, it still looks more like
a rank-deficient problem than a discrete ill-posed problem. The definition of small is
still uncertain, but it is believed to be values close to zero.
Actually in this case it does not make the greatest difference what type of illconditioning
the system had. Different solving methods have been tested, and the
choice of regularization method fell on Tikhonov based on trial an error. As mentioned
in Section 7.4.1, it is usual to solve rank deficient problems by the normal or
truncated singular value decompositions, because they are handling the rank of the
coefficient matrix. But even though the Tikhonov regularization do not involve the
numerical rank, it can be used with great success. There is of course the third option
that the system is neither rank deficient nor discrete ill-posed. Hansen [1996] states
such solutions have to be solved as accurate as possible, without regularization, and he
suggest iterative refinement techniques.
Comparing the filter factor plot in Figure 7.3.3, and the singular value spectra,
Figure 7.2.3 and 7.2.4 shows how the singular values much larger than zero have filter
factors close to or exactly zero. From the figure it can also be seen how the singular
values much smaller then ε, are very close to zero. Though Tikhonov regularization do
not involve numerical rank directly, it clearly puts a larger filter on the singular values
larger than the rank, (Rank(A) = 7338).
9.1.5 Computer Errors
When dealing with errors in numerical methods, one can not ignore the round-off
errors in numerical differentiation, and of course the machine precision. The errors
in numerical differentiation have already been dealt with in Section 4.2, where the
truncation errors was introduced. Because the computer is only able to use a finite
number of digits in its calculations, the computational errors actually gets worse, when

80 Tidal Deformation of the Solid Earth
the mesh sizes, hr and hθ decrease and it could be a fatal error with values very close
to zero, [Holmes, 2007]. The large mesh sizes in this study, allows us to disregard this
problem. The machines precision varies depending on the computer, system and the
program used, it is a full-study alone.
9.2 The Reliability of the Model
Throughout the thesis, there have been given a lot of physical assumptions for the
purpose of creating a simple physical system in which mathematics could be applied.
Some assumptions do not play a major role in the physical expression of the Earth,
when others changes the Earth remarkable.
9.2.1 Elasticity Assumptions
In order to apply the Navier equation of motion, in Section 3.3, the Earth was assumed
to be continues, homogeneous, isotropic, infinite and elastic. The assumptions of continuity
and infinity of the medium greatly simplify the mathematic framework, and are
valid in the large scale we are looking at. Inhomogeneity is a local change in physical
parameters within a larger area, [Fowler, 1990]. The homogeneity assumption is not
true for the Earth, but when looking at the rough global picture the assumption is valid
to some degree, and will not cause any trouble. Studies concerning tides or elasticity
theory in an inhomogeneous Earth can be found in Chakravorty [1972] and Fu and
Sun [2007]. The assumption of an isotropic media, i.e. the stress-strain relationship
is independent of the orientation of the material, is true for most Earth material, but
the upper mantle have layers which are anisotropic and also the inner core indicates
anisotropy, [Lay and Wallace, 1995].
The Earth is not completely elastic, as already mentioned in Chapter 3, in fact
the mantle of the Earth can be seen as viscoelastic. Though considering the tides, the
mantle could be regarded as a solid body, but may react as a fluid depending on the
timescale applied, [Wang, 1999]. Because the outer core is fluid it is deformed more
than the mantle by the external gravitational potential, [Hinderer et al., 1987].
9.2.2 The Real Tides
In Chapter 2, the Earth tides was described by a completely solid Earth. The real Earth
has a solid crust with thin layers of ocean bounded by continents. The ocean tides
plays a significant role in the crustal deformation. The oceans have a large effect and
can be calculated by ocean models, see for example Farrell [1972], which describes
the ocean loading by the elastic Greens function, or Khan and C.C.Tscherning [2001],
which studied the semi-diurnal ocean tide loading using GPS data. In reality the solid
Earth tides are dominated by the compressive and expansive radial component of the
tidal force. The large oceans are dominated by the tangential components of the tidal
forces. The tidal bulges are small compared with the radius of Earth, but it still raises
a huge amount of water. When the continents are added, the ocean reflect on the
shorelines setting up currents and standing waves. Also the coastal topography plays

Application of Tides 81
an significant role in intensifying water level. This is only due to the tidal bulge,
[Jentzsch, 1997].
The Earth is not spherical symmetric and it rotes, this means in reality the tidal
force is more complicated than described. The tidal displacement is actually larger at
the side closest to the Moon, than on the side turning away, [de Pater and Lissauer,
2001]. The tidal forces causes a precession of the Earth’s rotation axis by applying a
torque attempting to pull the equatorial bulge into the plane of the ecliptic. The Earth’s
nutation is a result of the diurnal frequency band. The precession and nutation are true
even for a symmetric sphere, [Broche and Schuh, 1998]. Also the tidal deformation
is influenced by the rotation of the Earth. On short times scales less than a month,
the core is decoupled from the mantle rotation, and this causes the love number k
to decrease by approximately 8%, [Wang, 1997]. Atmospheric tides, due to diurnal
heating of the atmosphere induces surface loading, also has to be considered, [Petrov
and Boy, 2003].
9.3 Application of Tides
The solid Earth tides influence the entire universe and the tides can tell us about Earth’s
interior, earthquake occurrence and estimation of rock parameters in bore holes of connected
aquifer, [Kümpel, 1997]. The global tidal response is not possible to recover
from tilt and strain tide observations, they can only determine local effects. Seismological
investigation is by far the best method to extract information about the Earth’s
interior, but also studies of the Earth’s magnetic field and gravity anomalies can reveal
information of the interior. The nearly-diurnal free wobble-resonance, caused by a
resonance between the mantle and the outer core, short described in Section 3.4.1, is
unrecoverable from seismic interpretation but can be recovered from VLBI data and
gravity models from for example superconducting gravimeters, [Zürn, 1997].
The Earth’s response to the tides have been used to study other stress induced
phenomenas in the Earth such as earthquake triggering and volcanic eruptions. The
Earth’s response to tectonic stress are much lager than the tidal induced stress level,
but the stress rates, i.e. stress in a given period, are comparable, [Heaton, 1982; Steacy
et al., 2005]. Earthquake triggering due to the solid Earth tides have among others
been studied by Schuster [1897]; Heaton [1982]; Emter [1997]; Tananka et al. [2002,
2004]; Beeler and Lockner [2003]; Cochran et al. [2004]; Métivier et al. [2009].
Another approach to the deformation of the Earth is by looking at the elastic energy.
Getino and Ferrandez [1991]; Barkin and Ferrandiz [2002]; Barkin et al. [2006]
have studied the tension of the Earth’s elastic mantle as a perturbation of a combined
rotational and tidal forces from the Moon and the Sun. The strain components are the
partial divided of the displacement.
Even though tidal deformation have been studied in many years, no comprehensive
studies of the global displacement field have been deduced. There have been few
investigations of tidal deformation throughout the whole spherical Earth. Tidal deformations
is normally carried out for the surface of the Earth, and if the mantle or core
are investigated it is with the emphasis to determine the displacement at the surface,
see for example Métivier et al. [2007]. This is because the known theory about the de-

82 Tidal Deformation of the Solid Earth
formation of the deeper structures may only be deviating little from the true Earth, due
to the small displacements in depths, also the crust is most relevant to our lives. Xing
et al. [2007] have studied the global deformation, but they are only concerned about
the crustal layer, not the inner Earth. There have been several local investigations of
the mantle and crust and a lot of those are concerning the studies of tidal triggering
of earthquakes. Varga and Grafarend [1996] have made a study of the stress tensor
components in the mantle due to perturbations from the Moon and the Sun, the emphasis
were to determine the tidal influences of mantle viscoelasticity and the lateral
heterogeneities for a rotation Earth. They find, that the mantle viscoelasticity causes
a amplitude decrease and phase delay of the Earth’s response function. Takeuchi and
Saito [1962] gives the lateral and radial displacements in the Mantle and a homogeneous
core for various rigidity values of the core by spherical harmonics. Tidal bulges
influences GPS and is important in the satellite calibration and VLBI measurements,
[Torge, 2001].
Tides influence nutation and angular rotation and therefore knowledge of tides
is important in astronomy to make precise angular measurements, [Melchior, 1978].
The particle accelerator in Cern also take the tides into account, for making precise
operations [Takao and Shimada, 2000].
9.4 Model Improvements
In the previous discussion, it has been described how studies of the tidal deformation
of the Earth, can be carried out in much more details, than have been done in this
thesis. It have never been the intention, with this one year study, to develop a model
that could match those who have been in development for years. In future perspectives
a lot of improvements can be carried out, but keeping it in a realistic perspective and
without loosing the fundamental approaches, it is chosen only to suggest the most apparent
improvements.
If the work could have been done all over again, it would have been preferable to
use a programming language such as Fortran instead of Matlab. A lot of the trouble
which has been encountered doing the study, would probable have been avoided by
taken the trouble of writing the code in Fortran, because Fortran would speed up the
calculations and most important the memory problem would not have bene as big an
issue. The model is programmed with for loops which are not very efficient within
Matlab, instead the arrays should have been written with inner and outer matrices,
[Acklam, 2003].
One of the most obvious improvements is the finite difference discretization. First
of all a higher order of accuracy in the finite difference formula could have been applied.
In this study a second order centered finite difference formula is used. For an
example, the third order accurate for ur can easily be written as
ur(r, θ) = 2u(r + hr, θ) + 3u(r, θ) − 6u(r − hr, θ) + u(r − 2hr, θ)
6hr
.

Model Improvements 83
This led to an other FDE improvement, instead of using the centered finite difference
a bunch of other methods could be used to discretized the Navier equation. Different
approximations have different truncation errors, Li [2004] suggest these simple
methods to be favorable for handling hyperbolic problems: Crank-Nicholson scheme,
Lax-Wendroff method, Leap-frog method, Up-wind scheme, Lax-Freidrichs method
and the Beam-Warming method.
The general limitation with finite FDE concerns the use of grid points, and by this
approach the model do not care about what lying in between the grid points. Finite
element methods on the other hand is concerned about the defined elements instead of
the grid points.
Other improvements to the FDE concerning the boundary conditions. To avoid the
pole singularity a staggered grid can be used for r = 0. This was described in Section
4.4.2 and solves the singularity problem by avoiding grid points directly at the pole.
Also the clustering of grid points at the center can be avoided, [Mosheni and Colonius,
2000; Vasilyev et al., 2004]. At the free surface, the traction free conditions could have
been handled with the ghost point method, where fictive grid points are added to keep
the second order accuracy at the boundaries, [Pérez-Ruiz et al., 2007]. Finally the grid
sizes have to be mentioned in this context, but it have been discussed earlier.
Next step in improving the model would be to incorporate some assumptions handling
the fluid core of the Earth and going from two dimensional problem to a three
dimensional problem with the Earth rotating about its own axis.

Chapter 10
Conclusion
The study of tides is an important field in geophysics, there are many factors to deal
with and many approaches to the problem. The behavior of the solid Earth tides is well
understood, because it is determined by the known orbital motions of the Earth and the
Moon, and also by the Sun and other external objects. Combining this knowledge
and with the elasticity of Earth, given by the Love numbers, the deformation due to the
solid Earth tides can be found. Most studies are calculating an analytic solution, where
the leading approach is to expand the tidal potential into spherical harmonics and using
Love numbers to determine the rigidity of the inner structures. The Earth’s responds to
the tidal force, in a particular point, can be deduced by Green’s function. The tides are
often studied in terms of the different tidal periods, and the Earth’s deformation due to
the tidal force is often limited to the surface or the crust of the Earth. Also a coupling
of advanced Earth, ocean and atmospheric models are widely used. The Earth is a
complex body, and a good knowledge of the driving force is not enough to observe the
Earth’s deformation, because of the many other factors which also deform the Earth.
The main purpose of this thesis, has been to determine the tidal deformation of the
solid Earth by using a steady state version of the Navier equation of motion, verified
by the slow deformation processes. The approach was meant to be as simple as possible
and the equations were discretized with the numerical method of finite difference
equations.
There has been developed two Earth models. The first model, called the homogeneous
Earth model, was assumed to be a pure Poisson solid, where the seismic
velocities and the elastic parameters were constant throughout the Earth. In the second
model, called the layered model, the homogeneous model was expanded from being
a complete Poisson solid to including four layers representing an inner core, an outer
core,a mantle and a crust. The properties of each layer were taken from the isotropic
Preliminary Reference Model and averaged over each layer. This resulted in a more
realistic Earth model with a fluid outer core.
A dead end was reached, in the numerical process of solving the linear system of
equations, when the system, to a first approach, proved to be unsolvable. The system
was rank deficient, and both over- and underdetermined. Especially because of too

86 Tidal Deformation of the Solid Earth
many best fitted solutions (underdetermined) the best physical result was hard to find.
To overcome the obstacle a Tikhonov regularization was applied.
The displacement field induced by the solid Earth tides, were presented for the
radial, the tangential and the total displacement in a two dimensional quarter Earth.
In the view of physical interpretations the displacement fields have gone from not
showing anything meaningful at all, to (after regularization) give a solution which in
broad terms provided a physical meaningful result.
The magnitude of the displacement fields were too small. For the homogeneous
model, the maximum expansive displacement was calculated to 131 mm, the maximum
compressive displacement was −5 mm. For the layered model these displacements
were 74 mm and −26 mm. These results compared to the analytic result were
2.6 to 4 times too small for the layered model and for the homogeneous model the
expansive radial displacement field was 1.6 times too small, and the compressive part
of the radial displacement field was 23 times too small. The displacement became
smaller moving further towards the center of the Earth, which of course is due to the
tidal force losing strength until it cancels out in the center of the Earth. The tangential
displacement fields were also determined. For the layered Earth model the maximum
traction was 45 mm, and the results for the homogeneous model gave a maximum traction
of 41 mm. For the layered Earth model this deviated with a traction with a factor
of 1.4 to 3.2 times too small. For the homogeneous model the tractions where a factor
1.6 to 3.1 times too small compared to the analytic results.
Except for the radial displacement field that did not provide enough compression,
probably due to the rough grid, the modeled displacement fields gave a physical interpretable
pictures of the tidal deformation of the solid Earth.

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Appendices

Appendix A
Hyperbolic Function
Considering a general partial difference equation in the domain Ω of the form
a(r, θ)urr + 2b(r, θ)urθ + c(r, θ)uθθ + d(r, θ)ur + e(r, θ)uθ + g(r, θ)u(r, θ) (A.0.1)
To verify that the Navier equation of motion, Equation (3.3.3), is a hyperbolic function
has to be valid.
b 2 − ac > 0 for all (r, θ) (A.0.2)
In the figure below the determinant, b 2 − ac, has been plotted, and as the figure shows,
the determinant is at all points larger than zero. This prove, that the partial difference
equation is a hyperbolic function.

Appendix B
Matlab Codes
In this appendix a selection of the Matlab codes developed doing this thesis are displayed.
B.1 FDE_premiso.m
FDE_premiso.m is the main code for the layered Earth model. It is chosen not to show
the code for the homogeneous Earth model, because of triviality.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% This program s o l v e s r a d i a l u and t a n g e n t i a l v d i s p l a c e m e n t from
% t h e N a v i e r e q u a t i o n of Motion .
% In t h e domain : r = [ R_0 , a ] , t h e t a = [ t h e t a _ 0 , theta_m ] .
% R_0=0 , a =3671000 , t h e t a _ 0 =0 , theta_m=p i / 2 ,
% with seven boundary c o n d i t i o n s r=R_0 , two a t r=a and
% two a t t h e t a _ 0 =0 and two a t theta_m=p i / 2
%
% Master t h e s i s by S t i n e k i l d e g a a r d P o u l s e n .
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
t i c
c l e a r a l l , c l o s e a l l
opengl neverselect
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% CONSTANTS ( a l l i n SI − u n i t s )
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
g l o b a l rho G Mm rm
%F _ t i d e s
G = 6.6725985 e −11; % G r a v i t a t i o n a l c o n s t . m^ 3 / ( kg * s ^ 2)
Mm = 7.3483 e22 ; % Mass of t h e Moon
Me = 5.9742 e24 ;
rm = 3.84400 e8 ; % O r b i t a l d i s t a n c e of t h e Moon
a = 6 . 3 7 1 e6 ;
% E l a s t i c p a r a m e t e r s d e r i v e d from t h e i s o t r o p i c PREM:
l o a d . . / premiso / meanvalues_alpha . dat
l o a d . . / premiso / meanvalues_beta . dat

96 Tidal Deformation of the Solid Earth
l o a d . . / premiso / meanvalues_rho . dat
lam = meanvalues_alpha ( 4 ) ^2 * meanvalues_rho ( 4 ) −2 * meanvalues_beta ( 4 ) ^ 2 ;
mu = meanvalues_rho ( 4 ) * meanvalues_beta ( 4 ) ^ 2 ;
sigma=lam / ( 2 * ( lam+mu ) ) ; % u n i t l e s s
Emodulus = mu * (3 * lam+2 * mu ) / ( lam+mu ) ; % kg / (m * s ^ −2)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% S e t t i n g up t h e m a t r i c e s
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% −−−−−−−−−−−−−R a d i a l d i r e c t i o n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−
% R a d i a l C o n s t a n t s
r0 = 0 ; % Radius of t h e e a r t h
dr=a / 6 5 ; % loop s t e p
istart = 1 ; % loop s t a r t
radius = ( r0 : dr : a ) ' ; % G e n e r a t i n g t h e f i n a l r v e c t o r − s i z e : hr x 1
iend = l e n g t h ( radius ) ; % # g r i d p o i n t s i n r and end of l oop
hr = ( a−r0 ) / iend ; % Mesh s i z e i n r − d i r e c t i o n
% −−−−−−−−−−−−The l a t i t u d i n a l d i r e c t i o n :−−−−−−−−−−−−−−−−−−−−−−−−−−−−
% L a t i t u d i n a l C o n s t a n t s
theta0 = 0 ; thetaend = p i / 2 ; % l a t i t u d i n a l d i r e c t i o n
dtheta =( thetaend−theta0 ) / 5 5 ; % loop s t e p s i z e
jstart =1; % s t a r t of loop
theta = ( theta0 : dtheta : thetaend ) ' ; % G e n e r a t i n g l a t . S i z e : ( h t h e t a +1) x 1
jend= l e n g t h ( theta ) ; % # g r i d p o i n t i n t h e t a and end of loop
htheta =( thetaend−theta0 ) / jend ; % Mesh s i z e i r a d i a n s
%−−−−−−−−−−−Making t h e row o r d e r i n g matrix −−−−−−−−−−−−−−−−−−−−
k= z e r o s ( iend , jend ) ;
f o r i=istart : iend
f o r j=jstart : jend
k ( i , j ) = j + jend * ( i −1) ;
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The C o e f f i c i e n t m a t r i x A and t h e d a t a k e r n e l F ( t i d a l acc . )
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
M = iend * jend ; % Dimension
%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
% BOUNDARY CONDITIONS
%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
%−−−−− BC a t c e n t e r of t h e E a r t h ( Ac and f c )−−−−−−−−−−−−−−−−
Ac= s p a r s e ( jend , 2 * M ) ;
fc = s p a r s e ( jend , 1 ) ' ;
i =1;
f o r j=jstart : jend
Ac ( j , k ( i , j ) ) = 1 ;
fc ( j ) = 0 ;
end
%−−−BC a t e q u a t o r f o r t h e r a d i a l component ( Aue and f u e )−−−−
Aue = s p a r s e ( iend , 2 * M ) ;
fue = s p a r s e ( iend , 1 ) ' ;
f o r i=istart : iend
Aue ( i , k ( i , jend −1) ) = 1 ;
Aue ( i , k ( i , jend ) ) = −1;
fve ( i ) = 0 ;
end

FDE_premiso.m 97
%−−−BC a t t h e p o l e f o r t h e r a d i a l komponent ( Aup and fup )−−−−
Aup = s p a r s e ( iend , 2 * M ) ;
fup = s p a r s e ( iend , 1 ) ' ;
j=jend ;
f o r i=istart : iend
Aup ( i , k ( i , 1 ) ) = −1;
Aup ( i , k ( i , 2 ) ) = 1 ;
fup ( i ) = 0 ;
end
%−−BC a t E q u a t o r and t h e P ole f o r l a t . component ( Aue , Aup and fuep )−−
Aue = s p a r s e ( iend , 2 * M ) ;
Aup = s p a r s e ( iend , 2 * M ) ;
fuep = s p a r s e ( iend , 1 ) ' ;
f o r i=istart : iend
Aue ( i , k ( i , jend )+M ) = 1 ;
Aup ( i , k ( i , 1 ) +M ) = 1 ;
fuep ( i ) = 0 ;
end
%−−BC a t t h e s u r f a c e of t h e E a r t h − r a d i a l d i r e c t i o n ( Ars , f r s )−−−−−
taukonst=Emodulus / ( ( 1 + sigma ) * (1 −2 * sigma ) ) ;
Ars = s p a r s e ( jend , 2 * M ) ;
frs = s p a r s e ( jend , 1 ) ' ;
% With no g h o s t p o i n t
i=iend ;
f o r j =2: jend
Ars ( j , k ( i −1 ,j ) ) = −taukonst * (1 − sigma ) / hr ; % Au_ ( a −1 , j )
Ars ( j , k ( i , j ) ) =taukonst * (1 − sigma ) / hr . . .
+ 2 * taukonst * sigma / a ; % Au_aj
Ars ( j , k ( i , j −1)+M ) = −taukonst * sigma / ( htheta * a ) ; % Av_ ( a , j −1)
Ars ( j , k ( i , j )+M ) = taukonst * sigma / ( htheta * a ) . . .
+ taukonst * sigma * c o t ( theta ( j ) ) / a ; % Av_ ( a , j )
frs ( j ) = 0 ; % F ( a )
end
Ars ( 1 , k ( iend , 1 ) )=Ars ( 2 , k ( iend , 2 ) ) ;
Ars ( 1 , k ( iend , 1 ) +M )=Ars ( 2 , k ( iend , 1 ) +M ) ;
Ars ( 1 , k ( iend , 2 ) +M )=Ars ( 2 , k ( iend , 2 ) +M ) ;
Ars ( 1 , k ( iend −1 ,1) )=Ars ( 2 , k ( iend −1 ,2) ) ;
frs ( 1 ) = 0 ;
%−−BC a t t h e s u r f a c e of t h e E a r t h − ( r , t h e t a ) d i r e c t i o n ( Ats , f t s )−−−−−
Ats = s p a r s e ( jend , 2 * M ) ;
fts = s p a r s e ( jend , 1 ) ' ;
i=iend ;
f o r j =2: jend
Ats ( j , k ( i , j ) ) = mu / ( a * htheta ) ; % Au_ ( a , j )
Ats ( j , k ( i , j −1) ) = −mu / ( a * htheta ) ; % Au_ ( a , j −1)
Ats ( j , k ( i , j )+M ) = mu / hr . . .
− mu / a ; % Av_ ( a , j )
Ats ( j , k ( i −1 ,j )+M ) = −mu / hr ; % Av_ ( a −1 , j )
fts ( j ) = 0 ; % F ( a )
end
Ats ( 1 , k ( iend , 1 ) )=−Ats ( 2 , k ( iend , 2 ) ) ;
Ats ( 1 , k ( iend , 2 ) )=Ats ( 2 , k ( iend , 2 ) ) ;
Ats ( 1 , k ( iend , 1 ) +M )=Ats ( 2 , k ( iend , 2 ) +M ) ;
Ats ( 1 , k ( iend −1 ,1)+M )=−Ats ( 2 , k ( iend , 2 ) +M ) ;
fts ( 1 ) = 0 ;
%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
% The main Coef . m a t r i x Au and fu ( r a d i a l d i r e c t i o n )

98 Tidal Deformation of the Solid Earth
%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
fu = z e r o s ( M , 1 ) ' ; % Making t h e s i z e of t h e d a t a k e r n e l
%−−−−−−−−−−−−−−−− The t i d a l a c c e l e r a t i o n ( r a d i a l ) −−−−−−−−−−−−−−
f o r i =1: iend
f o r j =1: jend
fu ( k ( i , j ) ) = ftides_r ( radius ( i ) , theta ( j ) ) ; % c a l l i n g −Fu ( r , t h e t a )
end
end
% P r e p a r i n g t h e Av m a t r i x
Au = s p a r s e ( M , M ) ; % Making t h e m a t r i x
f o r i =2: iend−1
f o r j =2: jend−1
% −−−−−−−−−−−−−−− The main FDE e q u a t i o n s :−−−−−−−−−−−−−−−−−−−−−−−−
Au ( k ( i , j ) , k ( i , j ) ) = − 2 * mean_beta ( i ) ^2 . / ( radius ( i ) . ^ 2 * . . .
htheta ^2)− 2 * mean_alpha ( i ) ^2 . / ( hr ^ 2) − ( mean_alpha ( i ) ^2 − . . .
mean_beta ( i ) ^ 2) . / ( radius ( i ) . ^ 2 ) ; % Au_ij
%−−−−−−−−−−−−−−−− The r a d i a l d i r e c t i o n : −−−−−−−−−−−−−−−−−−−−−−−−−
Au ( k ( i , j ) , k ( i −1 ,j ) ) = mean_alpha ( i ) . ^ 2 * (2 * radius ( i ) + hr ) . . .
. / (2 * radius ( i ) * hr . ^ 2 ) ; % Au_ ( i −1 , j )
Au ( k ( i , j ) , k ( i+1 ,j ) ) = mean_alpha ( i ) . ^ 2 * (2 * radius ( i ) − hr ) . . .
. / (2 * radius ( i ) * hr . ^ 2 ) ; % Au_ ( i +1 , j )
%−−−−−−−−−−−−−−−− The l a t i t u d i n a l d i r e c t i o n :−−−−−−−−−−−−−−−−−−−−−
Au ( k ( i , j ) , k ( i , j −1) ) = (− ( mean_alpha ( i ) . ^ 2 − mean_beta ( i ) . ^ 2 ) . . .
* htheta + 2 * mean_beta ( i ) ^ 2) . . .
. / (2 * radius ( i ) . ^ 2 * htheta ^ 2) ; % Au_ ( i , j −1)
Av ( k ( i , j ) , k ( i , j+1) ) = ( ( mean_alpha ( i ) . ^ 2 − mean_beta ( i ) . ^ 2 ) . . .
* htheta + 2 * mean_beta ( i ) ^ 2) . . .
. / (2 * radius ( i ) . ^ 2 * htheta ^ 2) ; % Au_ ( i , j +1)
% −−−−−−−−−−The mixed t e r m s:−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Au ( k ( i , j ) , k ( i+1 ,j+1) ) = ( mean_alpha ( i ) .^2 − mean_beta ( i ) . ^ 2 ) . / . . .
(4 * radius ( i ) * hr * htheta ) ; %Au_ ( i +1 , j −1)
Au ( k ( i , j ) , k ( i+1 ,j −1) ) = −( mean_alpha ( i ) .^2 − mean_beta ( i ) . ^ 2 ) . / . . .
(4 * radius ( i ) * hr * htheta ) ;%Au_ ( i +1 , j −1)
Au ( k ( i , j ) , k ( i −1 ,j+1) ) = −( mean_alpha ( i ) .^2 − mean_beta ( i ) . ^ 2 ) . / . . .
(4 * radius ( i ) * hr * htheta ) ;%Au_ ( i −1 , j +1)
Au ( k ( i , j ) , k ( i −1 ,j −1) ) = ( mean_alpha ( i ) .^2 − mean_beta ( i ) . ^ 2 ) . / . . .
(4 * radius ( i ) * hr * htheta ) ; %Au_ ( i −1 , j −1)
end
end
%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
% The main Coef . m a t r i x Av and fv ( t a n g e n t i a l d i r e c t i o n )
%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
% P r e p a r i n g t h e Av m a t r i x
Av = s p a r s e ( M , M ) ; % Making t h e m a t r i x
fv = s p a r s e ( M , 1 ) ' ; % Making t h e s i z e of t h e d a t a k e r n e l
%−−−−−−−−−−−−−−−− The t i d a l a c c e l e r a t i o n ( a n g u l a r ) −−−−−−−−−−−−−−
f o r i =1: iend
f o r j =1: jend
fv ( k ( i , j ) ) = ftides_theta ( radius ( i ) , theta ( j ) ) ; % t i d a l acc . −Fv
end
end
f o r i =2: iend−1
f o r j =2: jend−1

FDE_premiso.m 99
end
end
% −−−−−−−−−−−−The main FDE e q u a t i o n s −−−−−−−−−−−−−−−−−−−−−−−−−−−−
Av ( k ( i , j ) , k ( i , j ) ) = − 2 * mean_beta ( i ) . ^ 2 / ( hr ^ 2) . . .
− 2 * mean_alpha ( i ) . ^ 2 . / ( radius ( i ) . ^ 2 * htheta ^ 2) ;% v _ i j
%−−−−−−−−−−−−The r a d i a l d i r e c t i o n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Av ( k ( i , j ) , k ( i −1 ,j ) ) = mean_beta ( i ) . ^ 2 * (2 * radius ( i ) − hr ) . . .
. / (2 * radius ( i ) * hr ^2) ; % v_ ( i −1 , j )
Av ( k ( i , j ) , k ( i+1 ,j ) ) = mean_beta ( i ) . ^ 2 * (2 * radius ( i ) + hr ) . . .
. / (2 * radius ( i ) * hr ^2) ; % v_ ( i +1 , j )
%−−−−−−−−−−−−−The l a t i t u d i n a l d i r e c t i o n :−−−−−−−−−−−−−−−−−−−−−−−
Av ( k ( i , j ) , k ( i , j −1) ) = (2 * mean_alpha ( i ) . ^ 2 + ( mean_alpha ( i ) . ^ 2 . . .
− mean_beta ( i ) . ^ 2 ) * htheta ) . . .
. / (2 * radius ( i ) . ^ 2 * htheta ^2) ;
Av ( k ( i , j ) , k ( i , j+1) ) = (2 * mean_alpha ( i ) . ^ 2 − ( mean_alpha ( i ) ^2 . . .
− mean_beta ( i ) . ^ 2 ) * htheta ) . . .
. / (2 * radius ( i ) . ^ 2 * htheta ^ 2) ;
% −−−−−−−−−−The mixed t e r m s:−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Av ( k ( i , j ) , k ( i+1 ,j+1) ) = ( mean_alpha ( i ) . ^ 2 − mean_beta ( i ) . ^ 2 ) . / . . .
(4 * radius ( i ) * hr * htheta ) ;
Av ( k ( i , j ) , k ( i+1 ,j −1) ) = −( mean_alpha ( i ) . ^ 2 − mean_beta ( i ) . ^ 2 ) . / . . .
(4 * radius ( i ) * hr * htheta ) ;
Av ( k ( i , j ) , k ( i −1 ,j+1) ) = − ( mean_alpha ( i ) . ^ 2 − mean_beta ( i ) . ^ 2 ) . / . . .
(4 * radius ( i ) * hr * htheta ) ;
Av ( k ( i , j ) , k ( i −1 ,j −1) ) = ( mean_alpha ( i ) . ^ 2 − mean_beta ( i ) . ^ 2 ) . / . . .
(4 * radius ( i ) * hr * htheta ) ;
%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
% Assembling t h e c o e f f i c i e n t m a t r i x and f o r c e v e c t o r
%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
zero= z e r o s ( s i z e ( Au ) ) ;
A1 = [ Au zero ] ;
A2 = [ zero Av ] ;
ff=1e −5; %BC f a c t o r
% F i n a l Coef . M a t r i x
A = [ A1 ; A2 ; Ac ; Aue ; Aup ; Ave ; Avp ; ff * Ars ; ff * Ats ] ;
% F i n a l f o r c e v e c t o r
F = [ fu , fv , fc , fue , fup , fvep , fvep , ff * frs , ff * fts ] ' ;
A= s p a r s e ( A ) ;
F= s p a r s e ( F ) ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SOLVING THE SYSTEM OF EQUATIONS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Tikhonov r e g u l a r i z a t i o n f u n c t i o n from P . Hansen
[ reg_corner , rho , eta , reg_param ]= l_curve ( U , s , F , ' Tikh ' ) ;
epsilon=reg_corner ;
[ U , s , V ] = csvd ( A ) ; % Compact SVD
[ X , rho , eta ] = tikhonov ( U , s , V , F , lambda ) ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Transform back t o ( i , j ) form t o p l o t t h e s o l u t i o n
% Making t h e f i n a l c o e f i c i e n t m a t r i x
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ur=X ( 1 : M , 1 ) ;
Utheta=X ( M +1:2 * M , 1 ) ;
ur= s p a r s e ( iend , jend ) ; % R a d i a l d i s p l a c e m e n t
utheta= s p a r s e ( iend , jend ) ; % Lat . d i s p l a c e m e n t

100 Tidal Deformation of the Solid Earth
fr= s p a r s e ( iend , jend ) ; % R a d i a l t i d a l f o r c e
ftheta= s p a r s e ( iend , jend ) ; % Lat . t i d a l f o r c e
f o r l =1: M
i = f l o o r ( ( l −1) / jend ) + 1 ;
j = mod ( l −1 , jend ) +1;
ur ( i , j ) = Ur ( l ) ;
utheta ( i , j )=Utheta ( l ) ;
fr ( i , j )= fu ( l ) ;
ftheta ( i , j )=fv ( l ) ;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Analyze and V i s u a l i z e t h e r e s u l t .
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
NORMcalc=norm ( F−A * X ) / norm ( F ) ;
R = ( 0 : dr : a ) ' ;
T= p i / 2 −theta ' ;
XX=R * cos ( T ) ;
YY=R * s i n ( T ) ;
% R a d i a l d i s p l a c e m e n t
f i g u r e ( 1 )
a x i s equal tight
p c o l o r ( XX , YY , ur ) , s h a d i n g flat
hcb = c o l o r b a r ( ' WestOutside ' ) ;
s e t ( g e t ( hcb , ' Y l a b e l ' ) , ' S t r i n g ' , ' R a d i a l d i s p l a c e m e n t i n m. ' )
hold on
ContourHandle= c o n t o u r ( XX , YY , ur , 2 0 , ' k−− ' ) ;
c l a b e l ( ContourHandle , ' F o n t S i z e ' , 1 2 , ' Color ' , ' r ' , ' R o t a t i o n ' , 0 )
hold on
a x i s off
%[ px , py ] = g r a d i e n t ( ur , 0 . 2 , 0 . 2 ) ;
%q u i v e r (XX,YY, px , py )
% T a n g e n t i a l d i s p l a c e m e n t
f i g u r e ( 2 )
a x i s equal tight
p c o l o r ( XX , YY , utheta ) , s h a d i n g flat
x l a b e l ( 'R (m) ' , ' F o n t S i z e ' , 1 2 )
y l a b e l ( 'R (m) ' , ' F o n t S i z e ' , 1 2 )
%t i t l e ( ' u_ \ t h e t a , A l l c o n s t a n t s \ lambda =0.5 e − 5 ' )
hold on
ContourHandle= c o n t o u r ( XX , YY , utheta , 1 0 , ' k−− ' ) ;
c l a b e l ( ContourHandle , ' F o n t S i z e ' , 1 2 , ' Color ' , ' r ' , ' R o t a t i o n ' , 0 )
hcb = c o l o r b a r ( ' WestOutside ' ) ;
s e t ( g e t ( hcb , ' Y l a b e l ' ) , ' S t r i n g ' , ' T a n g e n t i a l d i s p l a c e m e n t i n m. ' )
hold on
a x i s off
%hold on
%[ qx , qy ] = g r a d i e n t ( u t h e t a , 0 . 2 , 0 . 2 ) ;
%q u i v e r (XX,YY, qx , qy )
t o c

Matlab Functions 101
B.2 Matlab Functions
This section contains Matlab functions called from the main program FDE_premiso.m.
B.2.1 ftides_r.m
The radial tidal acceleration.
f u n c t i o n f = ftides_r ( radius , theta )
g l o b a l G Mm rm
f = − G * Mm * radius . / ( rm ^ 3) * (3 * ( cos ( p i /2 − theta ) ' ) . ^ 2 − 1) ;
B.2.2 ftides_theta.m
The horizontal tidal acceleration.
f u n c t i o n f = ftides_theta ( radius , theta )
g l o b a l G Mm rm
f = − 3 / 2 * G * Mm * radius . / ( rm ^ 3) * s i n ( 2 * ( p i /2 − theta ) ' ) ;
B.2.3 mean_alpha.m
The seismic P-wave velocity for the four layers.
f u n c t i o n f = mean_alpha ( radius )
l o a d premiso / meanvalues_alpha . dat
i f ( radius >= 0) && ( radius < 1221491)
f = meanvalues_alpha ( 1 ) ; % i n n e r c o r e
e l s e i f ( radius >= 1221491) && ( radius < 3479958)
f = meanvalues_alpha ( 2 ) ; % o u t e r c o r e
e l s e i f ( radius >= 3479958) && ( radius < 5701000)
f = meanvalues_alpha ( 3 ) ; % m a n t l e
e l s e
f = meanvalues_alpha ( 4 ) ; % Outer l a y e r
end

102 Tidal Deformation of the Solid Earth
B.2.4 mean_beta.m
The seismic S-wave velocity for the four layers.
f u n c t i o n f = mean_beta ( radius )
l o a d premiso / meanvalues_beta . dat
i f ( radius >= 0) && ( radius < 1221491)
f = meanvalues_beta ( 1 ) ;
e l s e i f ( radius >= 1221491) && ( radius < 3479958)
f = meanvalues_beta ( 2 ) ;
e l s e i f ( radius >= 3479958) && ( radius < 5701000)
f = meanvalues_beta ( 3 ) ;
e l s e
f = meanvalues_beta ( 4 ) ;
end

Appendix C
Averaging of PREM Layers
This appendix shows how the P- and S-wave velocities and the density is averaged in
each layer of the layered Earth model. The data is adapted from PREM, [Dziewonski
and Anderson, 1981]. Blue is the P-wave velocity given in m s , green is the S-wave
velocity also in m kg
s , and the red curve is the density given in
m3 . The radius is counting
from the center of Earth to the surface.
(a) Inner core (b) Outer core
(c) Mantle (d) Outer layer

Appendix D
Model Tests
In this chapter some of the results from the testing of the model is shown.
D.1 Tikhonov Tests
Test of the Tikhonov regularization with λ = 10 −1 − λ = 10 −5 , where lambda is the
regularization parameter. A point source is applied at (25,25) in the radial direction.
The grid numbers are 50 × 50.

Tikhonov Tests 105
(e) λ = 10 −1 (f) λ = 10 −2
(g) λ = 10 −3 (h) λ = 10 −4
(i) λ = 10 −5

106 Tidal Deformation of the Solid Earth
D.2 Regularization Tests
Surface plots of different regularization methods, only the best are showed. The grid
size is 50 × 50. A point source is applied at (25,25) in the radial direction. Here λ is
the regularization parameter.
50
45
40
35
30
25
20
15
10
5
QR m. householder
10 20 30 40 50
(j) QR med householder,
10−14 |F−A·X|
|F|
(l) TSVD, k = 3000, |F−A·X|
|F|
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
= 5.407 ·
(n) TSVD, k = rank(A) = 4853, |F−A·X|
|F|
1.94
(k) DSVD λ = 10 −2 , |F−A·X|
|F|
= 0.6115 (m) Tikhonov, lambda = 1e − 4,
0.0341
=
= 0.409
|F−A·X|
|F|
(o) TSVD, k = rank(A) = 5000, |F−A·X|
|F|
2.29 · 10−6 =
=