Download body waves tomography from obs-recorded earthquakes

UNIVERSITÉ MOHAMMED V – AGDAL
FACULTÉ DES SCIENCES
Rabat
N° d’ordre : 2660
THESE DE DOCTORAT
Présentée par
ABDELJALIL LBADAOUI
Discipline : PHYSIQUE
Spécialité : SISMOLOGIE
BODY WAVES TOMOGRAPHY FROM OBS-RECORDED
EARTHQUAKES IN THE GULF OF CADIZ
Soutenue le 8 Juillet 2013
Devant le jury
Président :
Mohammed Ouadi BENSALAH
P.E.S
Faculté des Sciences, Rabat
P.E.S
P.H
P.E.S
P.E.S
Faculté des Sciences, Rabat
Faculté des Sciences, Rabat
CNRST Rabat
CNRST Rabat
Examinateurs :
Abdallah EL HAMMOUMI
Hamid BOUABID
Aomar IBEN BRAHIM
Azelarab EL MOURAOUAH
Faculté des Sciences, 4 Avenue Ibn Battouta B.P. 1014 RP, Rabat – Maroc
Tel +212 (0) 37 77 18 34/35/38, Fax : +212 (0) 37 77 42 61, http://www.fsr.ac.ma
ACKNOWLEDGEMENTS
This work was carried out in the Mechanics Laboratory under the direction of Professor
Abdallah EL HAMMOUMI of
the Faculty of Sciences of Rabat and in the National
Institute of Geophysics in the National Centre for Scientific and Technical Research of
Rabat, under supervision of Professor Aomar IBEN BRAHIM.
First and foremost, I want to thank my advisor professor Abdallah EL HAMMOUMI of the
faculty of sciences of Rabat, who guided me through all the difficulties in my research and
provided me huge supports and several ideas about how to thinks and to solve scientific
problems and how to be a real researcher.
My thank goes also to all members of the dissertation committee, professors Mohammed
Ouadi BENSALAH and Hamid BOUABID of the faculty of sciences of Rabat.
Special thank for Professors Aomar IBEN BRAHIM and Azelarab EL MOURAOUAH, for
their assistance and moral support, and encouragements for challenges. I learned from them
a great knowledge and patience. Their passion and persistence in science imprint in my mind
and will inspire me in the future.
2
TITLE
BODY WAVES TOMOGRAPHY FROM OBS-RECORDED EARTHQUAKES IN THE
GULF OF CADIZ
ABSTRACT
The gulf of Cadiz is a region considered as a complex seismic area, where strong
earthquakes occur and where the plate boundary between the African and Eurasian plates is
not exactly known. we use high resolution seismic data recorded by a network of ocean
bottom seismometers stations in the Gulf of Cadiz as well as eight Portugal land seismic
stations. The OBS network was deployed within an experiment of the NEAREST project.
Nearly 600 seismic events are extracted from the recorded data set and their analysis
revealed that most of them occur at 20 to 80 km depths, with clusters of seismicity that
occur mainly at the Gorringe Bank, within the SW segment of the Horseshoe fault and the
Marques de Pombal Plateau and the S. Vicente Fault. A new NW-SE trend of seismicity has
been revealed with depths that extend from 35 to 80 km. This seismicity trend is close and
nearly parallel to the SWIM faults lineament. We present the first regional-scale high
resolution P- and S-velocity distributions across the Gulf of Cadiz region. These velocity
models are obtained using 3D seismic tomography to invert the OBS data-set. The results
show that the patterns of anomalies in the Gulf of Cadiz are in general, oriented in NE-SW
and NW-SE directions. They also show the presence of a low velocity zone (LVZ) to the SE
of our study area. At shallow depth, this LVZ is interpreted as due to a large accumulation of
sediments within the accretionary wedge, while at a greater crustal depth, it may reflect a
continental crustal composition rather than an oceanic crust. Moreover, seismic velocity
profiles show that under this region of the Gulf of Cadiz, the Moho averages a 30-km depth.
The Gorringe Bank and the Marquise de Pombal plateau are found to be deeply rooted
features and represent expressions of mantle uplifting.
KEYWORDS
Seismic tomography, inversion, body wave, Gulf of Cadiz.
3
TITRE
TOMOGRAPHIE DES ONDES DE VOLUME DEPUIS DES TREMBLEMENTS DE
TERRE ENREGISTRES PAR DES OBS DANS LE GOLFE DE CADIX
RESUME
Le golfe de Cadix est une région considérée comme une zone sismique complexe. Dans ce
travail, nous utilisons les données sismiques enregistrées par un réseau de stations OBS
déployées dans cette région et par huit stations sismiques portugaises. 600 événements sont
détectés et analysés. La sismicité est principalement remarquée au Banc de Gorringe au sud
ouest de la faille Horseshoe, au niveau du plateau Marques de Pombal et de la faille Sao
Vicente. Une orientation nord-ouest/sud-est de la sismicité a été révélée avec des
profondeurs qui s'étendent de 35 à 80 km.
Nous avons présenté des modèles
tomographiques des ondes de volume. Ces modèles de vitesses sont obtenus en utilisant la
tomographie sismique 3D par inversion de l'ensemble des données enregistrées. Les résultats
montrent que les anomalies sont orientées dans des directions nord-ouest/sud-est et nordest/sud-ouest. Ils montrent également la présence d'une zone de faibles vitesses (LVZ) au
sud-est de la région étudiée. Les profils des vitesses obtenus montrent un soulèvement du
manteau au niveau de la faille Sâo Vicente coïncidant avec le plateau Marques de Pombal.
Ils montrent de plus que le Moho sous le golfe de Cadix est plutôt profond, avec une
profondeur moyenne d'environ 30 km en aval de la surface de l’océan. En outre, une zone
nord-ouest/sud-est de grandes vitesses est localisée au sud-ouest de notre zone d'étude.
MOTS CLEFS
Tomographie sismique, inversion, ondes de volume, Golfe de Cadix.
4
RESUME DETAILLE
Depuis une dizaine d'années, de nombreuses études ont été menées en tomographie sismique
pour établir une image de l'intérieur de la Terre et de comprendre la structure géologique de
certaines régions parmi les plus complexes du monde. Cependant, aucune étude d'imagerie
tomographique à base des ondes de volume avec des données OBS n’a été réalisée dans le
golfe de Cadix. Le principal objectif du travail présenté dans cette thèse est de mener une
étude détaillée permettant de déterminer un modèle de tomographie sismique locale dans le
golfe de Cadix. La zone étudiée est caractérisée par un aspect géologique très complexe,
résultant de la convergence des plaques africaine et eurasienne, où de violents séismes se
produisent et où la limite entre les plaques africaine et eurasienne n'est pas toujours
exactement connue. Ce travail a été réalisé en utilisant des données sismiques de très haute
résolution obtenues par un réseau sous-marin de stations OBS déployées dans le golfe de
Cadix ainsi que par huit stations sismiques terrestres situées au Portugal. Le réseau OBS a
été déployé dans le cadre du projet NEAREST.
Les images tomographiques des perturbations de vitesse en trois dimensions ont été réalisées
par inversion des temps d’arrivée des ondes sismiques de volume élaborées à partir de
sismogrammes, par détermination des différentes phases. Le système d'analyse sismologique
SEISAN a été utilisé pour localiser plus de 600 événements survenus pendant cette période.
Parmi lesquels le ML = 4.8 du 1er Janvier 2008 ,événement sismique qui présente la plus
grande magnitude enregistrée par le réseau OBS et les stations sismiques Portugaises
considérée durant cette période d'observation .Grace au réseau OBS, nous avons détecté un
nombre important d'événements locaux qui n'ont pas pu être identifiés précédemment avec
les réseaux sismiques Marocain, Espagnol et Portugais. Le grand nombre de tremblements
de terre enregistrés au cours de cette période d'une année montre que le golfe de Cadix est
une zone très active, en raison de la convergence entre les plaques et parce qu’il est réparti
sur une vaste zone de déformation.
Dans notre cas, la sismicité se situe
au nord de la zone des failles SWIM ; elle est
concentrée au niveau de trois principaux blocs : au sud du banc de Gorringe, la partie sud de
5
la plaine abyssale de Horseshoe, et au nord,nord-ouest de la faille de Horseshoe, le long du
plateau Marques de Pombal et la faille Sâo Vicente. Quelques tremblements de terre se sont
produits dans la zone du prisme d'accrétion, et une sismicité dispersée est observée au sud
du bassin d'Algarve et du banc de Portimâo. La distribution en profondeur des hypocentres
varie entre les catégories peu profonds et profondeur intermédiaire ; la majorité des
événements enregistrés ont entre 20 et 80 km de profondeur.
Nos résultats montrent une orientation NW-SE de la sismicité qui traverse la faille du
Horseshoe. Cette sismicité observée au sud de notre zone d'étude, parallèlement à la SFZ
peut donc être corrélée à l’ensemble des failles SWIM. La première section verticale montre
trois principaux blocs de tremblements de terre : le long du banc de Gorringe, la faille de
Sao Vicente et le sud du banc de Potimâo, tandis que le second profil montre que la
sismicité se concentre dans deux principaux domaines : au niveau banc de Gorringe et de la
faille du Horseshoe. La faille inverse du Horseshoe présente une pente d’environ 50 km de
longueur. L'inversion des temps d’arrivée est un processus itératif qui requiert un modèle de
référence. Dans ce travail, nous l’avons initialisé à l’aide du modèle proposé par l’équipe de
recherche du projet NEAREST , 2008. Ce modèle a été, par la suite, amélioré (Matias, et
al., 2009). Tous les événements enregistrés sont, ensuite, localisés par ce modèle de vitesse
initiale avec une amélioration en utilisant l’algorithme VELEST (Kissling, et al., 1994). Ce
dernier est constitué d'une croûte comportant trois couches avec une interface à environ 6
km de profondeur, et la discontinuité du Moho à environ 26 km de profondeur.
L’algorithme tomographique LOTOS-10 conçu pour l'inversion simultanée des ondes P et
S, a été utilisé dans ce travail. Nous avons défini les paramètres de l’amortissement et de
l’effet du bruit de fond en effectuant une série de tests synthétiques. De plus, un réglage
sélectif des données a été effectué, dans certaines régions. Ce test montre une excellente
zone de couverture entre 15 et 45 km de profondeur. Le paramétrage du domaine a été
réalisé avec la construction du maillage par le réglage de la distance entre les nœuds à 5 km.
Cet algorithme utilise la méthode dite de pseudo-bending pour le traçage des rais. Elle
permet l’estimation de la trajectoire la plus courte des rais : le rai initial est perturbé alors
que les points source et récepteur sont maintenus fixes (Nolet,. 2008). Le code LOTOS
6
utilise cette approche pour le traçage des rais avec une légère modification (Koulakov,
2009). Les rais construits de cette manière ont tendance à voyager à travers les anomalies de
grandes vitesses et éviter les modèles à faibles vitesses. Avant l’inversion, nous avons
effectué plusieurs séries de tests synthétiques afin d'obtenir un paramétrage optimal et
d'évaluer la fiabilité des modèles tomographiques. Le test synthétique est une technique qui
permet d'examiner la résolution des données utilisées, le test montre que les anomalies
périodiques sont reconstruites dans la plupart des régions de la zone d'étude, et la résolution
est beaucoup plus élevée dans la zone délimitée par le banc de Gorringe à l'ouest du Cap Sao
Vicente au nord, du banc de Portimao à l'Est et la ligne dorsale du Coral dans le sud. En
plus du test du damier, nous avons effectué le test odd/even pour donner plus de crédibilité
aux résultats tomographiques. Cet essai montre une bonne corrélation des résultats dans la
plupart des zones du golfe de Cadix. Les structures tridimensionnelles y révèlent une limite
marquante entre les différentes anomalies de haute et basse distribution de vitesse, soit dans
une direction NE-SO ou dans des directions NW-SE. Nous remarquons, en outre que les
tomogrammes des ondes P et S montrent une distribution de vitesses plus ou moins
semblables.
Délimitée en latitudes par 35,1 ° N et 37 ° N et en longitudes par 8 ° W et 10 ° W, une
grande anomalie négative dans la direction SE est mise en évidence. Elle peut être expliquée
par la présence d'une zone à forte concentration de sédiments. A de grandes profondeurs,
cette anomalie pourrait être interprétée comme une zone de forte réflexion. Cette partie du
golfe de Cadix est plutôt faite de la croûte continentale et donc, sa partie nord délimite plus
ou moins la limite entre les croûtes océanique et continentale. Nos résultats montrent une
ceinture NW-SE des anomalies de grande vitesse s'étendant à partir du banc de Gorringe
jusqu’au prisme d'accrétion traversant la plaine abyssale de Horseshoe. Cette ceinture suit
environ le SWIM (SFZ), présentant un ensemble de failles de décrochement comme il a été
prouvé précédemment. On constate en plus une zone de transition qui présente une
séparation claire de deux structures. Cela peut être interprété comme le résultat d'une
déformation de compression oblique de l’Ibérie par rapport à la plaque Nubie. Notre modèle
montre aussi une large anomalie positive, s'étendant sur 50 km à l'ouest de la profondeur du
Cap Sao Vicente. Cette anomalie est assez claire, la distribution des anomalies de vitesse des
7
ondes S y est presque similaire à celle des ondes P. Elle peut être associée à un soulèvement
du manteau profond. Des profils tomographiques verticaux ont été réalisés dans lesquels
nous présentons les vitesses réelles au lieu des leurs perturbations. Ces profils montrent un
soulèvement du manteau au niveau de la faille Sâo Vicente coïncidant avec le plateau
Marquis de Pombal. Ils montrent de plus que le Moho sous le golfe de Cadix est plutôt
profond, avec une profondeur moyenne d'environ 26 km en aval de la surface de l’océan.
Au NW de la zone d'étude, une anomalie distincte de grande vitesse s'étend à partir de 10
km à 35 km de profondeur et coïncide à peu près avec le soulèvement du banc de Gorringe.
En dessous de ces profondeurs, cette anomalie à grande vitesse continue à apparaitre mais
semble plutôt atténuée, ce qui confirme les résultats d’une récente étude (Jiménez, Munt et
al 2010), En outre, la coupe verticale de tomographie montre clairement que le banc de
Gorringe est une anomalie qui implique un soulèvement du manteau et que cette
caractéristique topographique émerge d’une profondeur considérable à partir du manteau.
L’orientation W-E du banc de Portimao où une anomalie de grande vitesse est observée, est
expliquée à la fois, par la convergence oblique des plaques et par l’existence d'une activité
sismique récente. La partie de la plaine abyssale Horseshoe que nous étudions est proche de
la frontière entre l’Eurasie et la plaque africaine. Cette région est fortement influencée par
un mouvement de compression. Les images tomographiques de la partie nord de la plaine
abyssale Horseshoe montrent une anomalie de faible vitesse séparant deux blocs d’anomalie
positive le long de l’alignement SWIM où l'activité sismique est faible. Cette anomalie peut,
sans aucun doute, être interprétée comme la zone de contact entre la faille Horseshoe et les
failles de
l’alignement SWIM. Certains auteurs soutiennent l'existence d'une zone de
subduction active sous le golfe de Cadix, les données GPS n'ont, par contre, montré aucun
mouvement différentiel à travers le détroit de Gibraltar (Stich et al, 2006; Serpelloni et al,
2007). En fait, pour des profondeurs inférieures 50 kilomètres, nos profils de vitesse ne
montrent aucune indication de la zone de subduction au SE de la zone d'étude.
8
TABLE OF CONTENT
Acknowledgements ................................................................................................................................................. 3
abstract.................................................................................................................................................................... 4
Resumé .................................................................................................................................................................... 5
Résumé détaillé ....................................................................................................................................................... 6
Liste of figures ....................................................................................................................................................... 13
List of tables .......................................................................................................................................................... 18
General introduction ............................................................................................................................................. 19
PART 1
LITERATURE REVIEW
1. Introduction ....................................................................................................................................................... 22
2. THEORY OF ELASTICITY ...............................................................................................................................23
2.1 Displacement vector ........................................................................................................................................ 24
2.2 Strain tensor .................................................................................................................................................... 25
2.3 Stress tensor .................................................................................................................................................... 26
2.4 Equation of motion .......................................................................................................................................... 28
2.5 Stress strain relation ........................................................................................................................................ 29
2.5.1 Fluid material (zero viscosity) ....................................................................................................................... 29
2.5.2 Elastic material ............................................................................................................................................. 29
2.6 Wave equations ............................................................................................................................................... 30
2.7 Body waves ...................................................................................................................................................... 31
2.7.1 Longitudinal waves ....................................................................................................................................... 31
2.7.2 Transverse Waves ......................................................................................................................................... 32
2.8 Surface waves .................................................................................................................................................. 33
2.8.1 Rayleigh Waves ........................................................................................................................................... 35
2.8.2 Love waves ................................................................................................................................................... 41
2.8.3 Love waves in a layer on a half space ......................................................................................................... 41
2.7.4 Boundary conditions..................................................................................................................................... 43
3. EARTH STRUCTURE......................................................................................................................................45
3.1 The crust .......................................................................................................................................................... 47
3.2 The mantle....................................................................................................................................................... 48
3.3 The core ........................................................................................................................................................... 49
3.4 Tectonic plates................................................................................................................................................. 50
3.5 Convection and the earth’s mantle ................................................................................................................. 53
3.5.1 Where tectonic plates meet ......................................................................................................................... 54
4. EARTHQUAKES ............................................................................................................................................56
4.1 Where do earthquakes happen ? .................................................................................................................... 59
4.2 Seismographs .................................................................................................................................................. 61
4.3 Seismograms ................................................................................................................................................... 62
4.4 Phase nomenclature ........................................................................................................................................ 64
4.5 Teleseismic, Regional and Local earthquakes ................................................................................................. 66
9
4.5.1 Teleseismic earthquakes .............................................................................................................................. 66
4.5.2 Regional earthquakes ................................................................................................................................... 67
4.5.3 Local earthquakes ....................................................................................................................................... 68
5. WHAT IS SEISMIC TOMOGRAPHY? ..............................................................................................................69
5.1 The main steps to image earth interior ........................................................................................................... 72
5.2 Imaging the earth with seismic data ............................................................................................................... 74
5.2.1 Travel time tomography ............................................................................................................................... 74
5.2.2 Example of travel time tomography ............................................................................................................. 80
6. INVERSE PROBLEM ......................................................................................................................................82
6.1 Travel time inverse .......................................................................................................................................... 83
6.2 Why inverse problems are hard? .................................................................................................................... 87
6.3 Earthquakes location ....................................................................................................................................... 87
6.2.1 Example location of earthquake in homogenous medium .......................................................................... 92
7. CONCLUSION...............................................................................................................................................95
PART 2
BODY WAVES TOMOGRAPHY IN THE GULF OF CADIZ
1. INTRODUCTION ...........................................................................................................................................97
2. MOTIVATION AND RESEARCH OBJECTIVES..................................................................................................99
3. THE GULF OF CADIZ STUDY AREA ..............................................................................................................100
4. RAY TRACING AND TRAVEL-TIMES INVERSION ..........................................................................................105
4.1 The Eikonal equation ..................................................................................................................................... 106
4.1.1 Eikonal equation for fluid mediums ......................................................................................................... 106
4.1.2 Eikonal Equations in Isotropic Elastic Mediums ....................................................................................... 108
4.2 Ray geometry ............................................................................................................................................... 111
4.2.1 Ray solution in layered mediums ............................................................................................................... 112
4.2.2 Inversion of travel time .............................................................................................................................. 118
4.2.3 Shortest travel-time path ......................................................................................................................... 121
4.3 Bending method ............................................................................................................................................ 123
4.3.1 Pseudo bending method ............................................................................................................................ 125
5. SEISMICITY OF THE GULF OF CADIZ ................................................................................................................ 129
5.1 Nearest Project .............................................................................................................................................. 129
5.2 Data format ................................................................................................................................................... 134
5.3 Seisan analysis software ................................................................................................................................ 134
5.4 Seismicity ....................................................................................................................................................... 136
5.5 Inversion method and procedure .................................................................................................................. 140
6.5.1 Using the LOTOS code ................................................................................................................................ 145
5.5.2 One dimensional velocity optimization and preliminary source location algorithm ................................. 146
5.6 Starting velocity model .................................................................................................................................. 148
6. SYNTHETIC TESTS ............................................................................................................................................ 152
7. RESULTS AND DISCUSSION ............................................................................................................................. 158
7.1 Swim lineaments (SFZ) ................................................................................................................................. 165
7.2 Sâo Vicente Canyon ...................................................................................................................................... 167
7.3 Gorringe Bank ............................................................................................................................................... 172
10
7.4 Portimao-Bank............................................................................................................................................... 175
7.5 Horseshoe Abyssal Plain ............................................................................................................................... 176
8. conclusion ....................................................................................................................................................... 178
9. References ...................................................................................................................................................... 180
APPENDIX
1. Least squares method ...................................................................................................................................... 188
2. Snell’s law ....................................................................................................................................................... 192
3. Pane wave ........................................................................................................................................................ 194
4. Conversion of format seisan to lotos ............................................................................................................... 196
5. Surface waves tomography.............................................................................................................................. 200
11
LITERATURE REVIEW
LISTE OF FIGURES
Figure
Figure
Figure
Figure
Figure
1.1
1.2
1.3
1.4
1.5
Figure 1.6
Figure 1.7
Figure 1.8
Figure 1.9
Figure 1.10
Figure 1.11
Figure 1.12
Figure 1.13
Figure 1.14
Figure 1.15
Figure 1.16
Figure 1.17
Figure 1.18
Figure 1.19
Figure 1.20
Figure 1.21
Displacement of two neighboring point P and Q
Stress tensor
Displacement shape produced by compressional waves propagation.
Displacement shape produced by shear waves propagation.
Three component seismogram showing the surface waves phases, of
earthquake
, Rayleigh wave are observed in vertical and radial
component, whereas Love waves are shown in transverse component.
The particle motion for surface waves (Rayleigh waves), (P. Shearer, 2009).
The shape of displacement variation with depth shows that displacement is
exponentially decreasing with depth.
The particle motion for surface waves (Love waves), (P. Shearer, 2009).
Love waves in a half space.
A cross section through earth, showing the thicknesses of each layer,
dividing the earth in three main layers ( the crust, the mantle and the core).
A cross section through earth , showing detailed earth structure, (1)
Continental crust, (2) Oceanic Crust , (3) Subduction Zone, (4) Upper
Mantle, (5) Volcanic Eruption, (6) Lower Mantle ,(7) Panache Material
Warmer ,(8)Outer Core, (9) Inner Core, (10) Cells Mantle Convection, (11)
Lithosphere, (12) Asthenosphere, (13) Discontinuity Gutenberg, (14)
Discontinuity Mohorovicic.
Map show the crustal plates boundary (Stein and Wysession, 2003).
Picture shows the process of the Sea floor spreading.
Convection in the mantle drives plate tectonics .(www.geo .m tu.
edu/~hamorgan/ bigideas welc-o me.html).
Stress builds until it exceeds rock strength.
Agadir earthquake February 29, 1960, killed some 12,000 people and injured
12,000 others. Destruction of the old part of the city was complete, and some
70% of the new structures in the city were destroyed
(http://mimoun1.forumavie.com).
Comparison of frequency, magnitude, and energy release of earthquakes.
(Stein and Wysession., 2003).
30 years seismicity map of earthquakes magnitude
greater than four,
shows that most events occur along the boundaries between tectonic plates,
(Stein and Wysession, 2003).
First teleseismic record of earthquake of April 17 1889 in Japan , enregistred
by the Geodetic institute Potsdam (http://www.gfz-potsdam.de/portal/gfz).
Three component seismogram showing the body and the surface waves
phases of the earthquake occurred in Gulf of cadiz in August 1 , OBS 1.
Seismograms showing the differences in amplitudes and frequencies
between an earthquake occurred in India in April 4, 1995 of magnitude 4.8
(bleu signal) and an nuclear test occurred again in Indian in may 11, 1998
,magnitude 5.1 ( red signal), data are recorded at Nilore, Pakistan (Stein and
Wysession, 2003).
12
LITERATURE REVIEW
Figure 1.22 The 1994 Northridge earthquake recorded at station OBN in Russia. Some
of the visible phases are labeled (Shearer, 2010).
Figure 1.23 Examples of seismic rays and their nomenclature. The most commonly
identified phases used in earthquake location are the first arriving phases: P
and PKIKP (Stein and Wysession, 2003).
Figure 1.24 Teleseismic earthquake of may 2008 (China), M= 8.0, recorded by OBS 12
(all components). The seismic phases continued for more than 6000
Seconds. Long period surface waves (Rayleigh & Love) are also recorded.
Figure 1.25 Part of seismogram showing a regional earthquake of june 8/ 2008 (Greece)
recorded by OBS 12 (all components), M= 6.5.
Figure 1.26 Part of seismogram showing a local earthquake (all OBS’s vertical
components) of November 1st, 2008 (Greece), M=4.8 (SW Iberia).
Figure 1.27 Global Seismographic Network (IRIS).
Figure 1.28 Travel time picks for various body waves phases and travel time curves, the
data are 57655 travel times from 104 sources (earthquakes and explosions).
( Kennett and Engdahl, 1991).
Figure 1.29 An example ray path in a 3-D block velocity perturbation for Tomography
problems.
Figure 1.30 Travel time plot of the P seismic waves of the events occurred in morocco
between 1993 to 2003 recorded by Moroccan seismic station networks.
Figure 1.31 An example ray path and cell numbering scheme for a simple 2-D
tomography problem.
Figure 1.32 An example ray path and in 2-D dimension showing the blocs where the
basis function is none zero.
Figure 1.33 2-D block geometry velocity perturbation for an idealized tomography
problem, the model consists on identical blocks , traversed by 13 ray
paths.
Figure 1.34 Chart showing the differences between the inverse and forward problem.
Figure 1.35 Geometry for an earthquake location in earth with variance change in
velocity with depth.
Figure 2.1 Plate tectonic interactions between the southern Eurasia and the North
Africa plates with the main elements of plate boundaries superimposed:
AGL: Azores–Gibraltar Line; GC: Gulf of Cadiz; GF: Gloria Fault; MAR:
Mid-Atlantic Ridge; TR: Terceira Ridge. Solid yellow line: plate boundaries
(Zitellini et al 2009).
Figure 2.2 Gulf of Cadiz region offshore SW Iberia, showing the bathymetry map and
existing faults, SWIM is South West Iberian Margin faults lineament
(Duarte, et al 2009).
Figure 2.3 Geometry of the ray segment along a path from a surface source to a surface
receiver
Figure 2.4 Incidence angle of a ray
Figure 2.5 Incidence angle of a ray
Figure 2.6 Polar coordinate system for a ray in equatorial plan
Figure 2.7 Ray path in spherical earth model.
Figure 2.8 Ray and wavefront geometry.
13
LITERATURE REVIEW
Figure 2.9 An example of a shortest path followed by a seismic ray traveling from a
source S to receiver R.
Figure 2.10 Piece of the path shown in Figure (2.17).
Figure 2.11 Process of bending algorithm used to determinate the shortest path. Hatched
light grey patterns represent negative anomalies of -30%; dark grey patterns
are positive anomalies of +30%. (Koulakov, 2009).
Figure 2.12 Tangential, normal and anti-normal unit vectors along the ray path (Kazuki
et al,. 1997)
Figure 2.13 Three point perturbation scheme used in pseudo bending method, Um &
Thurber (1987).
Figure 2.14 Seismicity of the Gulf of Cadiz as recorded between august 2007 and July
2008 as shown by the red dots;, GB : Gorringe Bank , CP: Coral Pach, ,
SVC: Sâo vicente Canyon, RV : Rharb Valley , PB: Portimâo Bank, AB:
Algarve Bassin, AJB: Alentijo Bassin , Ocean Bottom Seismometers (OBS)
blue triangles and Portugal Land stations green triangles.
Figure 2.15 Ocean Bottom Seismometer’s on board, (Zitellini N., Carrara G. &
NEAREST Team. - ISMAR Bologna Technical Report, June 2009).
Figure 2.16 Location of the broad band stations used in this study, ocean Bottom
Seismometers (OBS) blue triangles and Portugal Land stations green
triangles.
Figure 2.17 Seismicity of the Gulf of Cadiz as recorded between august 2007 and July
2008 as shown by the black circles; AB: Algarve Bassin, AJB: Alentijo
Bassin the inclined blue line represents the SWIM faults zone (SFZ), and red
lines are the possible faults.CP: Coral Pach, GB : Gorringe Bank , GF :
Gorringe fault , HsF : horseshoe fault ,MPF : Marques de Pombal fault , PB:
Portimâo Bank, PSF: Pereira de Sousa fault, RV : Rharb Valley , SVC: Sâo
vicente Canyon, SVF: Sâo vicente fault,
Figure 2.18 Seismic profiles shown in the map Figure (2.9), showing events that
occurred within 40 km distance from the profile. This profile indicates a
more continuous pattern of seismicity. HsF: Horseshoe fault, GB: Gorring
Bank.
Figure 2.19 Seismic profiles shown in the map Figure (2.26), showing events that
occurred within 40 km distance from the profile, this shows three separate
clusters of seismicity. AW: Accretionary wedge, GB: Gorring Bank , SVF:
Sao Vicente fault.
Figure 2.20 Seismic profiles shown in the map Figure (2.26), showing events that
occurred within 40 km distance from the profile, this shows three separate
clusters of seismicity. AW: Accretionary wedge, GB: Gorring Bank , HsF:
horsechoe Fault.
Figure 2.21 Chart showing the General structure of the LOTOS code working process.
Figure 2.22 Ray paths in the map view at depth of 20, 40, 60 and 80 Km showing the
coverage paths, purple point are the stations.
Figure 2.23 Ray paths in the map view in a vertical cross section shown by grey dots.
Blue triangles are the stations.
Figure 2.24 Earthquakes location of more than 600 events recorded during NEAREST
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LITERATURE REVIEW
Figure 2.25.a
Figure 2.25.b
Figure 2.26
Figure 2.27
Figure 2.28
Figure 2.29
Figure 2.30
Figure 2.31
Figure 2.32
Figure 2.33
Figure 2.34
Figure 2.35
Figure 2.36
Figure 2.37
Figure 2.38
cruise survey.
Velocity model obtained by the VELEST algorithm , the black line plot
represent the model given by OBS location , the red line represent the model
given using both OBS’s and Portugal land station and the blue model is
given using only the Portugal land stations.
Different starting P-velocity models used for optimization of the initial
velocity model; model 1 is the velocity model proposed in the NEAREST2008 cruise report, model 3 is the model derived using VELEST, while
models 2,4,5,6 and 7 are initial-velocity models with slight modifications of
the previous ones.
Checkerboard test performed for P and S waves in horizontals sections.
Synthetic test performed for P and S waves in horizontals sections at 35 Km
depth.
Checkerboard test performed for P waves in vertical sections, AA’ and BB’
shown in Figure below.
Checkerboard test performed for S waves in vertical sections, AA’ and BB’
shown in Figure below.
Anomalies of P velocities distribution, test with inversion of two
independent data subsets (with odd/even numbers of events).
Anomalies of P velocities distribution, test with inversion of two
independent data subsets (with odd/even numbers of events).
P-velocity distribution at 10 km. Solid lines show the existing faults and
dotted lines show possible strike-slip faults, and black dots show the
epicenters location given by tomography program and inclined gray line
represent the SWIM fault zone (SFZ).
P-velocity distribution at 15 km. Solid lines show the existing faults and
dotted lines show possible strike-slip faults, and black dots show the
epicenters location given by tomography program and inclined gray line
represent the SWIM fault zone (SFZ).
P-velocity distribution at 25 km. Solid lines show the existing faults and
dotted lines show possible strike-slip faults, and black dots show the
epicenters location given by tomography program and inclined gray line
represent the SWIM fault zone (SFZ).
P-velocity distribution at 35 km. Solid lines show the existing faults and
dotted lines show possible strike-slip faults, and black dots show the
epicenters location given by tomography program and inclined gray line
represent the SWIM fault zone (SFZ).
P-velocity distribution at 50 km. Solid lines show the existing faults and
dotted lines show possible strike-slip faults, and black dots show the
epicenters location given by tomography program and inclined gray line
represent the SWIM fault zone (SFZ).
S-velocity distribution anomalies at 10 km. Solid lines show the existing
faults and dotted lines show possible strike-slip faults, and black dots show
the Epicenters location given by tomography program and inclined gray line
represent the SWIM fault zone (SFZ)
S-velocity distribution anomalies at 15 km. Solid lines show the existing
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LITERATURE REVIEW
Figure 2.39
Figure 2.40
Figure 2.41
Figure 2.42
Figure 2.43
Figure 2.44
faults and dotted lines show possible strike-slip faults, and black dots show
the Epicenters location given by tomography program and inclined gray
line represent the SWIM fault zone (SFZ).
S-velocity distribution anomalies at 25 km. Solid lines show the existing
faults and dotted lines show possible strike-slip faults, and black dots show
the Epicenters location given by tomography program and inclined gray
line represent the SWIM fault zone (SFZ)
S-velocity distribution anomalies at 35 km. Solid lines show the existing
faults and dotted lines show possible strike-slip faults, and black dots show
the Epicenters location given by tomography program and inclined gray line
represent the SWIM fault zone (SFZ)
S-velocity distribution anomalies at 50 km. Solid lines show the existing
faults and dotted lines show possible strike-slip faults, and black dots show
the Epicenters location given by tomography program and inclined gray line
represent the SWIM fault zone (SFZ)
Horizontal sections of P-velocity anomalies at 15,25,35,50 km depth.
(perspective view)
Horizontal sections of P-velocity anomalies at 15,25,35,50 km depth.
(perspective view)
NW-SE belt of high velocity anomaly (SWIM lineament)
Figure 2.45 Horizontal P-velocity distribution at 25, 35 and 50 km, showing high
velocity anomaly southwest Portuguese Margin.
Figure 2.46 Positions of the vertical cross-sections velocity profiles 1 and 2
Figure 2.47 P velocity distribution in vertical cross-sections 1, the position of vertical
section is shown in the Figure (2.39)., GB: Gorringe Bank. HsF: Horseshoe
Fault
Figure 2.48 P velocity distribution in vertical cross-sections 2, the position of vertical
section is shown in the Figure (2.39), SVF : Sao Vicente Fault, AW :
Accretionary wedge.
Figure 2.49 Horizontal P-velocity distribution at 25, 35 and 50 km, showing the
attenuation of anomaly A between the Gorringe bank and horsechoe abyssal
plain.
Figure 2.50 Density and temperature depth profiles at four positions identified in Fig
(2.26): Tagus Abyssal Plain (profile a), Gorringe Bank (profile b), and
Horseshoe Abyssal Plain (profiles c and d), (Jiménez, Munt et al 2010).
Figure 2.51 Figure (3.5) Anomalies of P velocities distribution, test with inversion of
two independent data subsets (with odd/even numbers of events).
Figure 2.52 Figure (3.4) Checkerboard test performed for S waves in vertical sections,
AA’ and BB’ shown in Figure below.
Figure 2.53 Model results showing the lithosphere structure. The map above show the
position of the profile. White dots are earthquake hypocenters, (Jiménez,
Munt et al 2010).
Figure 2.54 P- velocity distribution in vertical cross-sections 1, shows the layers going to
50 km depth , precising the direction in which the African oceanic
lithosphere thrust Eurasian plate.
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LITERATURE REVIEW
Figure 2.55 Horizontal P-velocity distribution at 25, 35 and 50 km, showing high
velocity anomaly in the Porimaô Bank.
Figure 2.56 Horizontal P-velocity distribution at 25, 35 and 50 km, showing low
velocity anomaly in the N-W of the horseshoe Abyssal Plain.
Figure 3.1 a projection in subspace, p is the projection in column space, d is the data
vector and e is the error vector. b) Illustrate the projection of the data
vector in terms of components, showing the error quantities and
is an
outlier.
Figure 3.2 A plane wave incident on a horizontal surface, is the incidence angle,
denote the length of the ray.
Figure 3.3 A plane wave crossing a horizontal interface between two homogeneous
half-spaces.
Figure 3.4 Propagation of plane wave.
Figure 3.5 Chart of processing method
Figure 3.6 Resolution Kernel (Badal et al 2003).
LISTE OF TABLE
Table 1.1
Earthquakes with 70 000 or more deaths (http://earthquake .usgs.gov).
Table 1.
Table 1.1
2 P and S velocities in the reference 1-d model after optimization by the
Lotos software.
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LITERATURE REVIEW
GENERAL INTRODUCTION
Nowadays, the plate tectonics theory provides several models trying to explain the major
geological features of the present structure of our planet. This theory proposes that the
current deformation of the lithosphere is related to several internal processes inside the
earth. For few a decades, many research studies have been conducted in seismic
tomography to image the interior of the Earth and to understand the most complex
geological structure of the globe. Recently, the development and improvement of
equipments in several domains, as well as advances in computer studies (processing power,
memory capacity), in electronic (more sensitive sensors) have contributed significantly to
obtain more explicit models of the earth. However, it is known that the resolution of
tomographic models is limited by a number of factors, including the distribution of
seismogenic zones and seismological networks scattered around the world. The
seismotectonic context does not always allow the investigation of some complex areas by
non-destructive methods.
Before that, it was too difficult to investigate the oceans
underground, since the sensors are often located only on the continental zones; the nonuniform distribution location of these sensors makes the ray tracing to be a difficult task.
The direct result of these limitations is that the regions of low coverage have often lower
resolution models. Seismic tomography is a still widely used method of exploring the earth
interior, although the gravimetric measurements and morphological studies of bathymetry
(Zitellini et al, 2009), (Gutscher et al, 2002), (Gracia et al, 2010) begin to show their
importance, and the international community is naturally oriented towards tomographic
techniques for determining the basement velocity structure. These techniques have been
field proven in the continental, and marine seismic application did not require major
changes.
The main objective of the work presented in this thesis is to undertake a detailed study that
will help produce a local seismic tomography model of the south West region of the Gulf
of Cadiz, which is an area characterized by a complex geology, resulting from the
convergence of the African and Eurasian plates.
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LITERATURE REVIEW
The local seismic tomography studies of this offshore zone requires in general that the
stations must be in the same study area, thus, achieving a tomographic studies based on
the data collected by land stations can never reveal the detailed structure of the region. In
August 2007, a European research consortium team decided to make detailed studies of the
western zone of the Gulf of Cadiz, in the project nearest (integrated observations from near
shore sources of tsunamis: towards an early warning system) which is an eu-funded project
(goce, contract n. 037110) mainly addressed to the identification and characterisation of
large potential tsunami sources located near shore in the Gulf of Cadiz, and to realize a
quantitative understanding of lithosphere processes.
Since, it was necessary to carry out seismic measurements at sea; the broad band seismic
stations have been installed in the Atlantic Ocean, since, a glimmer of hope will begin to
emerge, then we were assured that we will use a worthwhile quality feedstock, the National
Center for Scientific and Technical Research (CNRST) in Rabat, participated to this
project by accomplishing a tomographic study of the region by exploiting the data
collected in the ocean. A tremendous amount of work was made, and incredible patience is
needed to extract all the seismic events that occurred in this period, thorough job was
carried out when analyzing earthquakes, knowing that good location leads to successful
tomographic models. Often seismic tomography need a large database to get an excellent
coverage, however, having a huge amount of data can be misleading sometimes about the
reality reality of the earth interior. We have exploited more than 600 events in this study,
which is an enormous database for an excellent coverage. After that, our database is
prepared for inversion process, containing all the epicenters and arrival times of all
recorded earthquakes; we examined several inversion methods, to choose the adapted
method for our data inversion and to avoid accumulation of errors by applying appropriate
algorithms, the most recent methods and systems have been used, for example we have
chosen ray tracing algorithms that use methods of integration instead of differentiation.
We showed results from 10 km depth, as we will see after, from this depth rays coverage
begin to be excellent, it means that there are still 5.2 km of the crust needs to be
investigated, because of deepest station is in about 4.8 km deep, we plan to study this layer
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LITERATURE REVIEW
applying a surface wave tomography rather than body waves, by determination of velocity
models for shallow structure, using the Rayleigh waveforms which depend strongly on the
shallow velocity structure of the medium, which is used to obtain the group-velocity
dispersion curve (Dziewonski et al., 1969), and applying the digital filtering with a
combination of Multiple Filter Technique (Dziewonski et al 1969), (Badal et al, 1992),
(Chourak et al, 2003).
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Part 1
Literature review
CHAPTER OUTLINE
1. Introduction
2. Theory of elasticity
3. Earth structure
4. Earthquakes
5. What is seismic tomography?
6. Inverse problem
7. Conclusion
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LITERATURE REVIEW
1. Introduction
Seismology is the multi-variant discipline that can never be understood without dealing with
the theory of elasticity. During the Master studies we have carried out several notions to
deeply understand this theory, which allows me to penetrate in theoretical concepts of
seismology. The theory of elasticity solved several seismological problems even if
approximations are sometime needed for simplifying. Accordingly, basic concepts and
principles, dealing with stresses and strains were used to establish relationships applicable to
different types of mediums; however, they do not allow the resolution of complex problems.
That is why scientists introduce other concepts to define cases of the ideal material, so even
if the continuum mechanics are not able to examine the true nature of matter, it’s still
establishing different laws of behavior of real materials. We know that the surface of the
earth, where we are living is constantly shaken, because of the movement of the plates,
sometimes is violently shaken during earthquakes, causes then a human disaster. We
describe in this thesis the main processes of the dynamics of the planet consists of various
internal layers that are in constant motion, and the causes of these disasters. Great progress
for the development of seismic instrumentation in recent years has greatly helped
geophysicists to perform great tomographic models. But it’s not always easy to estimate the
parameters of inverse problems, seismic tomography models are mainly resolved by
inverting the travel time of the different phases, and they need often a great effort in the
estimation of parameters, and powerful algorithms for matrix inversion, because often the
amount of data exceeds the number of model parameters, we are dealing here with an
inverse problem which consists of determining the internal state of unknown system, based
on a given quantities of observations, knowing the structure of the system. Inverse problems
are present in almost sciences and engineering, and they are applied when you search for
information on a system without being able to measure directly. In this part we will explain
how we can solve an inverse problem using a mathematical approach.
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LITERATURE REVIEW
2. Theory of elasticity
The theory of seismic waves is based on the theory of elasticity. This theory is closely
related to the development of the seismology. Regarded on Hook’s discovery, Navier was
the first how present the equations for equilibrium and motion of elastic solid, and
investigate the general equations of the theory of elasticity. Development of theory of
elasticity was largely due to the work of Cauchy and Poisson when they worked on the
propagation of light, in 1831 Poisson found that, at large distance from source of disturbance
(not the case at the present with sophisticated instruments) the motion transmitted by the fast
wave was longitudinal, followed by a transverse wave that is the slowest one. After that
Stokes demonstrate in 1845 that the displacement of matter was irrotational dilatation, and
the slower wave was a wave of equivoluminal distortion with elements rotations, he made
the observation that resistance to compression and resistance to shearing are the two
fundamental kinds of elastic resistance by introducing the modulus of compressibility and
rigidity.
Forty two years later, Lord Rayleigh discovered that a specific wave can be formed near the
free surface of homogeneous body propagating along the surface with a different velocity,
and decay with a depth, he found that this wave is elliptically polarized in the plane
determined by the normal to the surface and by the direction of propagation. Since, it is the
discovery of Rayleigh waves; 24 years later Love called them “Rayleigh waves” after he
discovered them theoretically. This entire exploit, has been discovered by several
mathematicians and scientists long before any seismic records were obtained, in this thesis
we will develop the theory for more understanding the process of wave propagation, and the
elastic constitutive equations and equations of motion in Cartesian coordinates. All these
results can be found in several textbooks, (Love 1944), (Fung 1965), (Takeuchi and Saito,
1972), (Aki and Richard, 1980), (Thorne lay terry c. Wallace, 1995), (Seth Stein and
Michael Wysession, 2003), (P. Shearer, 2009) and (Novotny, 1999).
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2.1 Displacement vector
Consider a particle at point P (original state) , which is moved to point P’ (deformed state,
the position of the vector y depends on the one of x, therefore
(1,1)
Under the condition of continuity, we notice the existence of the inverse function
According to Lagrange description of motion, the displacement is given by
–
In a neighborhood of the P point, let's consider another point Q, displaced to the point Q’ in
the deformed state Figure (1.1)
Figure 1.1 Displacement of two neighboring point P and
Q.
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LITERATURE REVIEW
Using the Taylor expansion we obtain
,
Where
(1.4)
and using the Einstein’s summation convention
So, we can get
2.2 Strain tensor
From the displacement field we can derive the strain field; we take the straightforward
approach, assuming that if only the small strains are considered (Ewing et al 1957). The
distance between P and Q is
and the square of this distance can be expressed as
From the quadrangle (PP’QQ’) and Equation (1.6)
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LITERATURE REVIEW
Can be expressed in term of components
Consequently
We consider:
with
and the quantity S
and
we introduce
, the strain tensor
can be expressed as
Strain is the formal description of the change in shape of a material, if we neglect the nonlinear terms we get the Cauchy’s infinitesimal strain tensor
2.3 Stress tensor
The stress vector is defined by
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LITERATURE REVIEW
Figure 1.2 Stress tensor.
Where
is the effect of all surface forces exerted across the element of surface
. Let
be the stress vector acting on this element, this vector can be
expressed on tensor notation as
where
,
are the components of
the stress tensor. Applying the condition of equilibrium yield
Using the Cauchy’s formula
Applying the Gauss theorem
And putting
, where A is a continuous vector with continuous derivatives and v is
the unit outward normal,
may be expressed as a volume integral
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LITERATURE REVIEW
If the integral assumed to be equal to zero, this yield
Eq.
is known as the equation of the equilibrium.
2.4 Equation of motion
The equation of motion can be obtained by adding the inertial force
, we define
is the density, v the velocity and t is the time, the total derivative with respect to time is
equal to the corresponding partial derivative
where u is the displacement. The equation of motion of a continuum mechanics can be
expressed in this form
Taking in account the approximation assuming that the products of the derivatives are small
we get
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LITERATURE REVIEW
2.5 Stress strain relation
2.5.1 Fluid material (zero viscosity)
The stress tensor is defined in a fluid material of zero viscosity as
2.5.2 Elastic material
The stress tensor is defined in elastic material of zero viscosity as
is the material constant.
In an isotropic Earth the elasticity tensor involves only two constants, the bulk modulus
and the shear modulus
where
where
(Nolet 2008).
so,
is the volume dilatation,
is the shear modulus and
has no immediate physical
interpretation. Such laws can be very complicated, but are greatly simplified when we ignore
the hysteresis caused by anelastic effects and when we confine ourselves to very small
displacements. In that case the medium deforms approximately linearly with the applied
stress. We replace (1.22) into the Equation of motion (1.18)
For a homogeneous isotropic medium, the elastic coefficients are constant, than (1.23) can
be written as
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LITERATURE REVIEW
It is known that
So (1.23) become
Representing the equation of motion in terms of components of displacement vector
Finally, if we replace
we deduce the equation of motion well-known in seismology,
from which we shall derive the wave equations
2.6 Wave equations
From the previous equation
, than we neglect the body forces F, this equation can take
the form
Applying the divergence operator yield
and
, equation (1.29), become
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LITERATURE REVIEW
Where
is the velocity of compressional waves, and if we applying the curl
operator to the equation (1.28), we get the vector wave equation
Where
is the velocity of shear waves.
The purpose of the following is to give a precise explanation of the theory of propagation of
elastic waves, it also defines essential mathematical formulas for understanding the physics
of waves propagations, and we will deal with the different types of waves that propagate in
different elastic media.
2.7 Body waves
It follows from the theory of elasticity that there are two principal types of elastic body
waves, Includes the P and S waves, and the wavefront makes a spherical shape when the
body waves propagate in elastic medium.
2.7.1 Longitudinal waves
Also called compressional or irrotational waves; in seismology we call them the primary
waves or just P waves, because they are the fastest waves, so they are the first appearing on
seismograms, longitudinal waves undergo an volume change, as the waves propagates, the
displacements in the direction of waves propagation cause material to be alternately
compressed and expanded. The irrotational waves are thus generated by a scalar potential.
The solutions for P and S waves like those given in equation
and
give the
locations of wave-fronts, when P-wave emerges from deep in the Earth to the surface, a
fraction of it is transmitted into atmosphere as sound waves. Such sounds, if frequency is
greater than 15 cycles per second, are audible to animals or human beings. These are known
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LITERATURE REVIEW
as earthquake sound (J.R. Kayal, 2008), for it cannot describe all wave phenomena. These
approximations are collectively known as geometric ray theory and are the standard basis for
seismic body-wave interpretation.
Figure 1.3 Displacement shape produced by compressional waves propagation.
2.7.2 Transverse Waves
also called shear, rotational or secondary waves (S waves), when these waves pass through
elastic medium they involve shearing and rotation of material without any volume changes,
the displacement associated with propagating shear wave is perpendicular to the direction of
wave propagation.
Figure 1.4 Displacement shape produced by shear waves propagation.
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LITERATURE REVIEW
The velocities of longitudinal waves
and the transverse waves
in homogeneous and
isotropic medium, satisfy the Hook’s law, are given by the formulas
,
where
is the shear modulus,
is the Lamé coefficient, and
relation assumes the fact where the coefficients are equals,
is the density. Poisson
this relation is often used
in seismology to describe many elastic materials, involve that
Both longitudinal and transverse waves can propagate in solid media, but only primary
waves can propagate in fluid medium when
. Elastic body waves are reflected and
transmitted at the discontinuities of elastic parameters; in fact we observe increases the
number of waves on seismograms.
2.8 Surface waves
We have seen that the body waves can propagate in a homogeneous, isotropic and
unbounded medium. If the medium is bounded, another kind of waves can be guided along
the surface of the medium. The Surface waves are the waves that propagate along a
boundary and whose amplitudes go to zero as the distance from the boundary goes to
infinity. There are two basic types of surface waves, Love and Rayleigh waves, named after
the scientists who studied them first. Love’s work was directed to the explanation of waves
observed in horizontal seismographs, while Rayleigh predicted the existence of the waves
with his name. The main difference between the two types of waves is that the motion is of
SH type for Love waves, and of P–SV types for Rayleigh waves. Love and Rayleigh waves
show a large surface wave train arriving on a seismometer’s Figure (1.5); transverse
component shows the arrival of Love waves, followed by Rayleigh waves on a vertical and
radial component, which are distinguished from each other by the types of particle motion in
their wave fronts. In the description of body waves, the motion of particles in the wavefront
was resolved into three orthogonal components – a longitudinal vibration parallel to the ray
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LITERATURE REVIEW
path (the P-wave motion), a transverse vibration in the vertical plane containing the ray path
(the vertical shear or SV-wave) and a horizontal transverse vibration (the horizontal shear or
SH-wave). These components of motion, restricted to surface layers, also determine the
particle motion Figure (1.6) and Figure (1.8) and character of these two types of surface
waves.
Figure 1.5 Three component seismogram showing the surface waves phases, of
earthquake
, Rayleigh wave are observed in vertical and radial component,
whereas Love waves are shown in transverse component.
After a large earthquake, contrary to the body waves whose energy spreads three
dimensionally, the surface waves can circle the globe many times, and their energy spreads
tow dimensionally and it’s concentrated near the earth surface.
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LITERATURE REVIEW
2.8.1 Rayleigh Waves
In 1885 Lord Rayleigh described the propagation of a surface wave along the free surface of
a semi-infinite elastic half-space. The particles in the wavefront of the Rayleigh wave are
polarized to vibrate in the vertical plane Figure (1.6). The resulting particle motion can be
regarded as a combination of the P- and SV-vibrations
Figure 1.6 The particle motion for surface waves (Rayleigh waves),
(Adapted from , Shearer, 2009).
It is well known that a general transient wave can be expressed as a superposition of
harmonic waves by means of the Fourier integral (Novotny, 1999). And by the superposition
of harmonic waves of different amplitude and frequencies, we can construct rather
complicated wave shapes as well as seismic surface waves. If we consider a wave field in
term of potential component, from the equation of motion for a homogeneous isotropic
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LITERATURE REVIEW
medium equation
, the displacement field can be decomposed into scalar potential
and vector potential .
And express components of displacement vector
in terms of potentials
In the case of the wave field independent of the y-coordinate
On the other hand we can obtain from 1
and 1 22 the components of the stresses acting
in the perpendicular plan of z axis are
And from the components of displacement vector
of propagating plan
contains
and
we see that the perpendicular plan
, whereas
is in
can be decomposed into two parts, and express the vector potential
36
thus, the wave field
as
LITERATURE REVIEW
where
represent the displacement in
represent the displacement in
plan and
plan.
Consider the potential for longitudinal waves,
, and the potential for transverse waves,
, in the form of plane harmonic waves, moving with constant velocity
and
without change of shape, propagating in the -direction. The displacement vector can then
be decomposed as
Where
and
and
, for
part of wave the components
equals zero, hence the displacement is
where
and
can be defined as
where
is a given angular frequency,
and
depth-dependent amplitudes, we now replace
are unknown functions, describing the
in the wave equations
and
we get
These two equations represent the equations of harmonic oscillator, which can be easily
resolved, the solutions take the form
,
where
,
,
So, equations
37
can be written as
LITERATURE REVIEW
Can be decomposed into,
Where
and
is the wave number.
Or in the this form
In witch
and
describes the decomposition of surface wave into two body waves,
(Novotny, 1999), ( C. A. Coulson, 1977), replacing
into
and
we can now express the stress components by inserting the displacement components into
stress formula
, the stress components can then be written as follow
We have calculated the stress component to use them, for those further boundary conditions,
the stress components equals zero in the free surface of
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The amplitudes diminish with the depth increase for
If the apparent velocity (see plane waves in the Appendix)
,
are real, thus
and
the
goes to infinity, which no agree
with the second boundary condition, we have to consider the case to avoid this problem and
take the imaginary part of
and .
The amplitudes now take the form
Or in this simple form
for that
and
increase to
infinity for infinite depth, this terms doesn’t represent any physical meaning, so we keep
The terms
for
putting
Become now
These equations show that the waves are exponentially decreasing with depth increase.
Figure (1.47) represents the vertical and horizontal displacements calculated for a given
value of poison coefficient, these displacements are normalized with respect to the vertical
displacement of the particle motion in the surface.
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LITERATURE REVIEW
Figure 1.7
The shape of displacement
variation with depth shows that displacement is
exponentially decreasing with depth.
Now shall we determine the quantities
the equations
and, from the displacement expression given by
, displacement vector can take the form
And the stress component become
Inserting expressions of
and
into
, when the boundary conditions are satisfied
for stresses , we finally arrive at system of linear equations
where
and
40
with unknowns
and ,
LITERATURE REVIEW
The solution of this system of linear equation yield the Rayleigh velocity
For
typical values of Poisson solid, for more details see (Novotny, 1999) and (Stein and
Wysession, 2003).
2.8.2 Love waves
The boundary conditions which govern the components of stress at the free surface of a
semi-infinite elastic half space prohibit the propagation of SH-waves along the surface.
However, A.E.H. Love showed in 1911 that if a horizontal layer lies between the free
surface and the semi-infinite half-space.
Figure 1.8 The particle motion for surface waves (
Love waves), (P. Shearer, 2009).
2.8.3 Love waves in a layer on a half space
Love wave are a surface waves result from interaction of SH waves, this type of waves
require an increasing velocity with depth, if not the case Love waves cannot exist.
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LITERATURE REVIEW
Figure 1.9 Love waves in a half space.
The displacement of the particle motion associated with propagation of Love waves can be
expressed as
where
42
LITERATURE REVIEW
Where
,
and
is the wave number.
2.7.4 Boundary conditions
We consider the following boundary conditions, the stress components equals zero in the
free surface of
in
when
, all stresses and displacements are continuous;
the displacement go to zero
is real, thus
, if the apparent velocity
go to infinity, which no agree with the boundary condition,
we have to consider the case to avoid this problem and take the imaginary part and
where
43
.
LITERATURE REVIEW
The first boundary condition is satisfied when
So,
The second and third boundary conditions yield again a system of linear equations
with unknowns
and .
where
and
44
LITERATURE REVIEW
3. Earth structure
The earth is composed of several layers. The inner core in the extreme center is a solid iron
and nickel ball located about 6370 km from the surface of the earth. And outer core in form
of the ring, formed of a fluid mixture of molten iron and nickel. The core is surrounded by
the asthenospher composed of a portion of the mantle and the crust (oceanic or continental)
Figure (1.10). It is well known that the Earth has a molten core, what is now general
knowledge was slow to develop. In order to explain the existence of volcanoes, some
nineteenth century scientists postulated that the Earth must consist of a rigid outer crust
around a molten interior. It was also known in the last century that the mean density of the
Earth is about 5.5 times that of water (Lowrie, 2007). From this it was inferred that density
increased towards the Earth’s center under the effect of gravitational pressure.
Figure 1.10 A cross section through earth, showing the thicknesses of each layer, dividing the
earth in three main layers ( the crust , the mantle and the core )
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LITERATURE REVIEW
The key to modern understanding of the interior of the Earth is density, pressure and
elasticity was provided by the invention and improvements of sensitive seismographs. The
progressive refinement of this instrument and its systematic employment world-wide led to
the rapid development of the modern science of seismology. Important results were obtained
early in the twentieth century. The Earth’s fluid core was first detected seismologically in
1906 by R. D. Oldham. Resent geophysical studies have revealed that the Earth has several
distinct layers. Each of these layers has its own properties Figure (1.11).
Figure 1.11 A cross section through earth , showing detailed earth structure, (1)
Continental crust, (2) Oceanic Crust , (3) Subduction Zone, ,(4) Upper Mantle, (5)
Volcanic Eruption. , (6) Lower Mantle ,(7) Panache Material Warmer ,(8) Outer Core,
(9) Inner Core, (10) Cells Mantle Convection, (11) Lithosphere , (12) Asthenosphere,
(13) Discontinuity Gutenberg , (14) Discontinuity Mohorovicic. (Wikipedia).
46
LITERATURE REVIEW
3.1 The crust
Earth outermost layer, the crust preserves a memory of the Earth’s evolution that extends
back more than 3.4 Gy, while today it is well established that silicic material extracted from
the mantle forms an outer crust and it’s well documented now a compromise aspects of the
physical state of the planet and its evolution and the Earth's crust when many seismological
studies performed early from the beginning of the last century brought important elements of
the overall appearance and global view of the structure of the crust. Recently, using high
frequency resolutions of deep seismic reflection profiling (Meissner,1973. Oliver et al,
1976, Clemperer 1989) and wide-angle reflection profiling investigations (Healy et al.,
1982, W.D. Mooney, 1985) understanding the crustal structure began to change radically
when all studies showed that the crust is highly heterogeneous in composition and physical
properties, furthermore, the large number of collected data and the increasing improvements
of seismic station network in the last 40 years played a key role in determining the first
model of the crust, an important works mentioning soil crusts have made, and regional
crustal thickness models were created by several scientists (Mohorovicic, 1910), (Conrad
1925), (Byerly 1926), (Byerly and Dyk, 1932), (Gutenberg 1932), (DeGolyer, 1935),
(Heiland, 1935), (Press, 1964), (Crampin, 1964), (Kovach and Anderson, 1964),
(Christensen, 1965), (Dix, 1965), (Audry and Rossetti, 1962), (Aubert and Maignien, 1948),
(Leprun, 1978), (Pindell 1985)… Seismology began to have several successful discovery,
for example when has succeed to discern the petrological boundary between ultramafic and
silicic rocks, where the wave velocity increases by about 6.5 km/s (in the lower crust) to
more than 7.7 km/s (upper mantle), (James and Steinhart, 1966). The crust varies in
thickness. It is thin (about 7 km) under oceans, thicker (about 40 km) under continents, and
thickest (as much as 70 km) under high mountains. The continental crust is made up mostly
of low-density granitic rocks and that no granite exists on the floor of the deep ocean, the
crust there consists entirely of basalt and gabbro overlay by sediments (Grotzinger et al.,
2007). As a result of plate tectonics Figure (1.12), the oceanic crust and continental crust
differ systematically in their main physical properties, including density, thickness, age and
composition. Continental crust has an average thickness of
, and an average age of
, the density of
, whereas the oceanic crust has an average
47
LITERATURE REVIEW
thickness of about 6 km, the density of
. Is everywhere younger than
,
oceanic crust composed mainly of basalt theoleiitic, which has a dark and fin grain texture as
consequence of rapid cooling magma, by contrast, the continental crust has a felsic
composition than oceanic crust (Dziewonski and Romanowicz,. 2007).
3.2 The mantle
The mantle is the layer between the crust and the core, is the widest section of the Earth. The
most volume of the earth is taken by the mantle, and it has a thickness of approximately
2,900 km. The mantle is made up of semi-molten rock called magma. In the upper parts of
the mantle the rock is hard, but lower down the rock is soft and beginning to melt, made of
magma (melted rocks) and is around 3000 °C. The mantle is fundamentally different than
the crust based on its composition that it’s made up , the mantle itself can be subdivided into
tow layers, upper mantle and lower mantle, Upper part of the mantle is called lithosphere , is
the highest part of the earth, and is cooling enough to be solid ( it’s the solid portion of the
mantle). From 40 to 400 km depth there consists of the upper mantle, containing the olivine
and pyroxene (iron, magnesium and silicate) as well as calcium and aluminum. The upper
mantle is divided into three parts: the hydrosphere, is all the compartments containing water
on Earth (lakes, rivers, seas, oceans), the lithosphere is the group consisting of the crust and
the upper part of the mantle, is between 0 and 100 to 200 km, and Asthenosphere located
between 200 and 400 km depth is in the viscous state, with an average temperature would be
of the order of 1500 °C; the Transition Area ranges from about 400 to 670 km, basaltic
magma source; From 670 to 2960 km the lower mantle there is probably consisting of
silicon, magnesium and oxygen with iron, calcium and aluminum. The crust is divided into
several plates which float on the liquid upper mantle. They are called tectonic plates. They
reflect the current liquid mantle convection: from the deepest parts of the mantle magma
rising currents pushing his way to the surface. These currents have broken the solid crust of
the earth into several large plates that move slowly separate from each other transported by
movements of the mantle. This is what is known as continental drift. It is characterized by
two processes: dislocation and wrinkling. Dislocation occurs when two plates move away
from each other, allowing the new crust from forming through magma mantle. The folding
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LITERATURE REVIEW
occurs when two plates move towards the edge and that one of the plates immersed below
the other features to merge into the mantle. There are also transverse movements between
two plates like San Andreas Fault in California.
3.3 The core
The deep interior of the earth is currently inaccessible and perhaps would be forever.
However, through a large quantity of data collected at the surface, and contains information
of inside the earth transported to the surface, by several way, such as seismic waves and free
oscillations, topography and of gravity, magnetic and electric fields, heat fluxes and
hydrothermal circulation and chemical and isotopic variations in volcanic rocks,
geophysicists interpret as they can what earth interior looks like. But until now, they know
quite a bit about the composition of the core. Seismology tells us that the core below the
mantle is a fluid, but it is much fluid in the center of the earth when the pressure are
immense, over 4 million times the atmospheric pressure (Grotzinger et al., 2007), and
temperature; The Models produced up to date are consistent and provides a probable picture
of the core and lower mantle, although a number of questions remain unanswered, the
thermal history calculations for the earth were based upon the assumption that heat is
transported radially outward by conduction (Lubimova, 1969). However, with the guidance
of (Tozer 1965), opinion shifted to the view that thermal convection is the dominant
mechanism of heat transport in the interior of the earth, since that time, the model of thermal
convection has been adopted for a long time, recently, it has become clear that the best
model for the energy source is gravitational energy released by the growth of the solid inner
core, (Braginsky, 1963), (Gubbins, 1976), (Loper, 1978), ( Gubbins et al, 1979), (Gubbins
and Masters, 1979), (Loper and Roberts, 1983), the central feature of the gravitationally
powered dynamo is that the inner core is growing by freezing of outer-core material as the
earth gradually cools (Barry Saltzman, 1984), both seismic and theoretical studies indicate a
complicated structure for the inner core. First, the inner core is not completely solid, but is
an intimate mixture of solid and liquid, with the mass fraction of solid increasing
monotonically with depth. It was also demonstrated, that by the effect of the decreasing of
the radioactivity (released from the inside) with time, the earth cannot remain at a constant
temperature, and so that the earth is cooling at a given rate, and it can be related to the
49
LITERATURE REVIEW
growth of the inner core (Gubbins et al. 1979), and the growth of the solid inner core is
caused by the cooling of the core from above, with the core coolest at the top, freezing
occurs first at the bottom provided the liquidus gradient with pressure, finally some
discovers said that the earth's core is composed principally of iron, with a small but
significant percentage of some light constituent. The nature and amount of this constituent
are uncertain, (Brett, 1976), (Ringwood, 1977), (Stevenson, 1981), (Brown and McQueen,
1982).
3.4 Tectonic plates
The mantle can be divided into three subsections; the Mesosphere, Asthenosphere and
Lithosphere. The Mesosphere or lower mantle is a thick liquid, it’s the closest one to the
core, the Asthenosphere is a plastic like consistency, it’s a kind of rock not quite solid, but
it’s not quite liquid to, this is the cooling part of the mantle, and it’s basically the substances
that move very slowly. And finally, the Lithosphere is the upper most part of the mantle, and
it’s very rigid and divided into pieces of plates.
Figure 1.12
Map shows the crustal plates boundary (Adapted from Michael Allaby, (2009)).
50
LITERATURE REVIEW
In 1910, Alfred Wegener published his works where he proposed some of the core ideas of
the modern plate tectonic concept, Wegener proposed that the continents were drifting about
on the surface of the planet, and that they once fit together to form one great supercontinent,
known as Pangaea. After that a revolution shook up the Earth sciences that result in the
acceptance of the plate tectonic model which states that the Earth’s outer shell, or
lithosphere, is broken into several rigid pieces, called tectonic plates Figure (1.12). The
lithosphere is divided then into tectonic plates, the lithosphere is the part of the earth that is
basically all cracked and divided into deferent sections. Figure shows the pieces of the
lithosphere broken down into pieces, and they float on top of the Asthenosphere, so, they
move. But! How they all this giant pieces of floating rocks move?
The tectonic plates move because the Sea- floor spreading, see below the Figure (1.13)
where two plates separated by ocean, the red dark color indicate a blend of molten and semimolten rock called a magma, and what happen exactly ? the magma pushes up creating
under water a mountain ranges, called ridges, when the magma come up in top reaching the
bottom of the oceans, its cooling carrying out a new rocks forming, and going to push the
two older rocks, there is three ways allowing tectonic plates moving
A. Ridge push
The ridges formed during sea-floor spreading are higher than where it sinks into the mantle,
These are the areas from which the earth breathes, they Spanning about 70,000 km, the
oceanic ridges are the largest geological structure of the globe. These are huge relief
elongated submarine can reach a height of 3000 m and 2500 m wide.
B. Convection
Inside the earth, radioactive elements provides some of the thermal energy that causes
convection, the convection currents form in the mantle when thermal energy transfers from
the core to the mantle; Hotter magma closer to the core rises because it is less dense as the
51
LITERATURE REVIEW
denser cooler magma closer to the crust, this creates a current., the same thing can happen in
the ocean, the warmer water go to the top of the ocean towards the surface and a cold water
on the bottom and push the plates.
C. Slab pull
Oceanic crust denser than the continental crust sinks and pulls the rest of the plate with it.
The slab pull is a direct phenomenon result of the subduction process driving a tectonic plate
under another plate of lower density, usually an oceanic plate under a continental plate or in
a more recent oceanic plate. In the oceanic divergence oceanic lithosphere, its when two
plates collide, the most common to this detachment is when oceanic plate sink into
continental plates (driven by its own weight), then the convergence begins to characterized
by appearance of slabs.
Figure 1.13 Picture shows the process of the Sea floor spreading spreading
involving the upwelling and melting of the underlying mantle.
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LITERATURE REVIEW
3.5 Convection and the earth’s mantle
The main heat transfer mechanism in the Earth’s mantle is convection. It is a thermally
driven process in which heating at depth causes material to expand and become less dense,
causing it to rise while being replaced by complementary cool material that sinks. This
moves heat from depth to the surface in a very efficient cycle since the material that rises
gives off heat as it rises and cools, and the material that sinks gets heated only to eventually
rise again. Convection is the most important mechanism by which the Earth is losing heat,
with other mechanisms including conduction, radiation, and advection. However, many of
these mechanisms work together in the plate tectonic cycle. Mantle convection brings heat
from deep in the mantle to the surface where the heat release forms magmas that generate
the oceanic crust. The midocean ridge axis is the site of active hydrothermal circulation and
heat loss, forming black smoker chimneys and other vents. As the crust and lithosphere
move away from the midocean ridges, it cools by conduction, gradually subsiding
(according to the square root of its age) from about 1.5–2.5 miles (2.5–4.0 km) below sea
level. Heat loss by mantle convection is therefore the main driving mechanism of plate
tectonics, and the moving plates can be thought of as the conductively cooling boundary
layer for large-scale mantle convection systems.
Very early in the history of the planet at least part of the mantle was molten, and the Earth
has been cooling by convection ever since. It is difficult to estimate how much the mantle
has cooled with time, but reasonable estimates suggest that the mantle may have been up to
a couple of hundred degrees hotter in the earliest Archean. The rate of mantle convection is
dependent on the ability of the material to flow. The resistance to flow is a quantity
measured as viscosity, defined as the ratio of shear stress to strain rate. Fluids with high
viscosity are more resistant to flow than materials with low viscosity. The present viscosity
of the mantle is estimated to be 1020–1021 Pascal seconds (Pa/s) in the upper mantle, and
1021–1023 Pa/s in the lower mantle, which are sufficient to allow the mantle to convect and
complete an overturn cycle once every 100 million years. The viscosity of the mantle is
temperature dependent, so it is possible that in early Earth history the mantle may have been
able to flow and convectively overturn much more quickly, making convection an even
more efficient process and speeding the rate of plate tectonic processes (T. Kusky, 2005).
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Figure 1.14 Convection in the mantle drives plate tectonics. (www.geo.mtu.edu/~hamorgan/
bigideas welc-o me.html)
3.5.1 Where tectonic plates meet
When they are moving, the plates are always bumping into each other, pulling away from
each other, or past each other. The plates usually move at about the same speed that your
fingernails grow. They run into each other or sliding past each other, at present, we know
how this plates are actually moving and how fast (GPS data measurements), we have three
main types of plate’s boundary.
A. Divergent boundaries
It’s where we have the plates moving away from each other, they correspond to the axis of
mid-ocean ridges, African plates and South American plate are example of this boundaries.
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LITERATURE REVIEW
B. Convergent boundaries
It’s where we have the plates converge upon each other, they can correspond to the
subduction zones characterized by the presence of slabs, when oceanic – continental crusts
convergence, the Himalaya’s mountain is an example of convergence between two plates in
continental domain.
C. Transform boundaries (horizontal sliding)
They are characterized by transform faults which are breaks in the plates and which allow
sliding between two plate portions, San Andreas Fault in California is a result of horizontal
sliding of two boundaries.
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LITERATURE REVIEW
4. Earthquakes
Although seismology is a science with a long history, earthquakes were not understood in
their original. The Chinese were the first to ask questions related to the origin of
earthquakes, Chang Heng invent in 132 AD, the first functional Seismoscope, it was an
primitive device, of not characterized by the delicacy and sensitivity of instruments of our
day, but a beautiful design able to detected the arrival of seismic waves, when the origins of
earthquakes were not at all understood. In 1892, seismologists can finally make a
quantitative description of earthquakes at great distances due to a revolutionary invention of
a sensitive seismograph by John Milne.
Did you know that thousands of earthquakes occur every day? Fortunately, most are very
small and cannot be felt. Earthquakes occur when tectonic plates push, pull or slip past each
other, But this is not a fast process, this can take a million of years for plates to move
centimeters even millimeters of the movement every years, but as they are moving, a lot of
stress built up, it’s the stress increases near the plate’s edges, resulting an elastic deformation
of the rocks, when the stress exceeds the strength of the rocks along the fault, the fault slips,
releasing the stress suddenly and causing an earthquakes, Figure (1.15), big earthquakes are
not nearly so common, but when one strikes, it can bring real disaster. Earthquakes can
originate from sudden motion along a fault, from a volcanic eruption, or from bomb blasts.
(T. Kusky, 2005).
Figure 1.15 Stress builds until it exceeds rock strength
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LITERATURE REVIEW
Earthquakes cause many problems; ground shaking can knock buildings down and destroy
roads and bridges, vibrating earth can cause entire mountainsides to give way, creating
landslides that destroy everything in their paths, Figure (1.16). Undersea earthquakes
sometimes bring about monster waves called tsunamis, Tsunamis can travel over the ocean
for thousands of miles causing death and destruction far from the earthquake, therefore
seismological studies based on the observation of significant effects, concentrated in areas in
a big frequency occurrence, where these effects are most intense.
Figure 1.16 Agadir earthquake February 29, 1960, killed some 12,000 people and injured
12,000 others. Destruction of the old part of the city was complete, and some 70% of the new
structures in the city were destroyed. (http://mimoun1.forumavie.com)
When an earthquake occurs releasing enormous energy (the same as millions of explosives
being set off at the same time) causes the ground to shake and vibrate, associated with
passage of waves of energy released at its source. Earthquakes can be extremely devastating
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LITERATURE REVIEW
and costly events, sometimes killing tens or even hundreds of thousands of people and
leveling entire cities in a matter of a few seconds or minutes. Recent earthquakes have been
covered in detail by the news media and the destruction and trauma of those affected is
immediately apparent. A single earthquake may release the energy equivalent to hundreds or
thousands of nuclear blasts Figure (1.17), and may cost billions of dollars in damage.
Figure 1.17 Comparison of frequency, magnitude, and energy release of earthquakes. (Stein
and Wysession., 2003)
Earth’s surface is always on the move. Earth’s crust is broken into tectonic plates, these are
large pieces of crust that shift and move. They are always joining together and pulling apart.
When tectonic plates move, the edges grate or scrape against each other. This movement
causes Earth to tremble and the place where tectonic plates meet is called a fault line.
Scientists begin with an understanding of how the planet Earth formed and how its internal
heat engine drives tectonic plates to move around on the surface. Geologists and natural
philosophers have speculated on the origin of continents, oceans, mountain ranges, and
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LITERATURE REVIEW
earthquakes for hundreds of years. Early geologists recognized and classified many of the
major surface and tectonic features of the continents and oceans (T. Kusky,. 2008).
4.1 Where do earthquakes happen ?
Most earthquakes occur along the edge of the oceanic and continental plates. The seismicity
of the world Figure (1.18) tracks the boundary of the plates Figure (1.12), we see that most
earthquakes occur along the plate’s boundary, and the largest earthquakes occur along the
Pacific Ocean with 80% of the annual recent seismic activity, each year, one or two
earthquakes of
magnitude greater than 8 take place somewhere in the world. We note for
example that some regions were shaken several times during the last century; this is the case
for example in Chile, Mexico, and Japan, it's important to know that more than 90% of
earthquakes in the world occur at shallow depth less than 60 km, when the plates move
against each other three types of boundaries that are the seat of an intense seismic world
activity.
Figure 1.18 30 years seismicity map of earthquakes magnitude
greater than four, shows that
most events occur along the boundaries between tectonic plates, (Stein and Wysession, 2003).
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LITERATURE REVIEW
We then notice that epicenters are distributed along closed curves which divide the world
into large oceanic and continental regions, more or less devoid of seismic activity, or
"plates", thus satisfying the theory of plate tectonics. There is a diffuse seismicity outside
these limits (inter plate seismicity), but as much volcanic activity, almost all earthquakes
occur at the boundaries of these plates, and his is called tectonic earthquakes. Because
catastrophic earthquakes occur rarely in any particular region, humanity often forgets how
devastating these events can be. However, history should remind us of their power to
suddenly kill tens of thousands of people Tab (1.1).
Table ( 1.1) Earthquakes with 70 000 or more deaths (http://earthquake.usgs.gov)
YEAR
LOCATION
MAG
DEATHS
893
Ardabil, Iran
---------
150 000
1138
Aleppo, Syria
---------
230 000
1290
Chihli, China
---------
100 000
1667
Shemakha,Caucasia
---------
80 000
1755
Lisbon, Portugal
8.7
70 000
1920
Gansu,China
7.8
200 000
1923
Kanto, Japan
7.9
143 000
1927
Tsinghai, China
7.9
200 000
1932
Gansu,China
7.6
70 000
1948
Ashgabat,Turkmenistan
7.3
110 000
1976
Tangshan, China
7.5
255 000
2004
Sumatra
9.1
283 106
2005
Pakistan
7.6
86 000
2008
Eastern Sichuan, China
7.9
87 652
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4.2 Seismographs
The first seismological instrument was constructed in second century AD by a Chinese
philosopher Chian-hen, but the really big development increase in seismology happened
from around 1900 and onwards and was mainly due to improvement in making more
sensitive seismographs, to detect even small earthquakes magnitude. Later, the
measurement of the ground motion allows studying seismic wave attenuation. in 1750
scientist began to use pendulums to make more sensitive instruments, however, in 1889 in
Potsdam (Germany), Ernst von Rebeur-Paschwitz invented the first instrument able to
record signal from a teleseismic event from an earthquake in Japan Figure (1.19), in 1915
Shaw and Milan produced Milan-Shaw long period seismograph, it was characterized by his
low weight, after that a significant progress in seismological instrument was made and
several seismographs were created in Japan, Russia, and Italy. In 1922 Wood Anderson
created an instrument records on photographic paper, after that, several electromagnetic
seismographs were developed, and since 1970, a new generation of seismographs appeared
with electronic amplification of the electric signal.
Figure 1.19 first teleseismic record of earthquake
of April 17 1889 in Japan , recorded by the
Geodetic institute Potsdam
(http://www.gfz-potsdam.de/portal/gfz).
Nowadays, sophisticated seismological instrument created in particular for the purpose of
detecting ground motion and nuclear explosions, able to record the very small movements of
the soil. "Networks survey". The development of networks of seismographs around the
world quickly showed that the seismicity was not randomly distributed, but it was structured
along major seismic continuous lines on the surface of the globe; installation of broadband
world network stations, and Ocean Bottom Seismometer, have strongly promote recovery
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LITERATURE REVIEW
and collection of a huge quantities of seismic data in both regional and global scale, we can
now use a hard disks with a larger memory storage can be used for recording, the recorder
memory is not a problem today and recording is easy and a easy to share, the control of
different acquisition parameters become easy and it’s mostly done with development of
computational techniques , this instrumental progress helped many geophysicists to study
several area that were mysterious. Since, the seismic instrumentation has contributed
significantly to saving lives around the world, and helped to investigate the earth interior.
4.3 Seismograms
When an earthquake happened, its recorded by a different sensors, and graphical recording
is called seismogram, thus, the record signal can be analogical on smoked photographic, or
plain paper, in as digital data stored in hard discs. The velocities of seismic waves is big,
several kilometers per second, sensors coupled with very accurate clocks are used and
perfectly synchronized to the position of the ground return in time with a precision of a few
milliseconds, The sensors must record the signal, and the time of the signal, and time, it was
difficult to get a good external time reference, availability of GPS (Global Positioning
System) signals make the end of all timing problems, the movements of the Earth are
continuously recorded. Thus, seismograph records background noise generated by
atmospheric changes, marine activity, humane activities, and microseismic, as P-waves are
faster than S-waves on the seismogram they appear at first, followed by the S wave and
surface waves at the end Figure (1.20).
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Figure 1.20 Three component seismogram showing the body and the surface waves phases of
the earthquake occurred in Gulf of cadiz in August 1 , OBS 1.
Figure 1.21 Seismograms showing the differences
in
amplitudes
and
frequencies
between
an
earthquake occurred in India in April 4, 1995 of
magnitude 4.8 ( bleu signal) and an nuclear test
occurred again in Indian in may 11, 1998 ,
magnitude 5.1 ( red signal), data are recorded at
Nilore, Pakistan (Stein and Wysession, 2003).
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Seismograms exhibit a number of wave trains some of which travel into the globe
completely before arriving at a station, examining seismograms is a very important step in
tomographic studies of any region in the globe. Excellent analysis of seismograms often
leads to a good resolution of a tomographic problem often solved by inverse methods;
indeed, data comes from seismograms and data are treasure of seismologists, data are the
key contribute answers to fundamental questions in geophysics. Permanently seismograms
are recordings ground motion, recordings made by seismometers, and it is easy to identify
from signal the differences between earthquakes and nuclear test Figure (1.21).
4.4 Phase nomenclature
The different layers in the Earth (e.g., crust, mantle, outer core, and inner core), combined
with the two different body-wave types (P, S), result in a large number of possible ray
geometries, termed seismic phases. The following naming scheme has achieved general
acceptance in seismology. Earthquakes causes propagation of seismic body and surface
waves, these waves travel with different velocities, the time taken by each wave to reach the
station is different from other waves, on seismograms, we see different phases, each distinct
phase can be associated with a certain travel path “Travel of certain seismic phases” Each
distinct phase has a certain travel time, Figure (1.22).
Figure 1.22 The 1994 Northridge earthquake recorded at
station OBN in Russia. Some of the visible phases are
labeled (Shearer, 2010).
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LITERATURE REVIEW
Seismic-wave energy can travel multiple paths from a source to a receiver at a given
distance Figure (1.23), to help sort out the various phases, seismologists have developed a
nomenclature to describe each phase in terms of its general ray path. The various portions of
the path a ray takes, for example, between the source and the free surface, are known as legs.
Each leg of a ray is designated with a letter indicating the mode of propagation as a P or S
wave, and the phase is designated by stringing together the names of legs. Thus, there are
four possible depth phases that have a single leg from the surface reflection point to the
receiver: pP, sS, pS, and sP. The main layers constricting earth are the mantle, the fluid outer
core, and the solid inner core.
P- and S-wave legs in the mantle and core are labeled as follows
Some examples of these ray paths and their names are shown in Figure (1.23). Notice that
surface multiple phases are denoted by PP, PPP, SS, SP, and so on. For deep focus
earthquakes, the up going branch in surface reflections is denoted by a lowercase p or s; this
defines pP, sS, sP , etc. (see Figure. (3.8)). These are termed depth phases, and the time
separation between a direct arrival and a depth phase is one of the best ways to constrain the
depth of distant earthquakes. P-to-S conversions can also occur at the CMB; this provides
for phases such as PcS and SKS. Ray paths for the core phase PKP are complicated by the
Earth's spherical geometry, leading to several triplications in the travel time curve for this
phase. Often the inner-core P phase PKIKP is labeled as the df branch of PKP. Because of
the sharp drop in P velocity at the CMB, PKP does not turn in the outer third of the outer
core. However, S-to-P converted phases, such as SKS and SKKS, can be used to sample this
region, (Shearer, 2010).
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Figure 1.23 Examples of seismic rays and their nomenclature. The
most commonly identified phases used in earthquake location are the
first arriving phases: P and PKIKP. (Stein and Wysession, 2003).
4.5 Teleseismic, Regional and Local earthquakes
Earthquakes may be further classified into three types depending on the distance from
source to the seismograph station. Since the character of seismograms depends on epicentral
distance, the nomenclature for seismic phases is also distant dependent.
4.5.1 Teleseismic earthquakes
The earthquakes, which are recorded by a seismograph station at a greater distance, are
called teleseismic earthquakes. These are very often called teleseisms. By international
convention the epicentral distance is required to be more than 1000 km for a teleseism. Lay
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and Wallace (1995), however, define teleseismic distance as
. An example of a tele-
seismogram is shown in Figure (1.24). Depending on the magnitude of the earthquake,
teleseismic amplitudes can range from barely perceptible to those that saturate the
instrument. These earthquakes provide very useful seismic phases for the crust as well as for
the interior of the Earth. The direct P and S-wave arrivals recorded at teleseismic distances
between 30° and 95° are relatively simple.
Figure 1.24 Teleseismic earthquake of may 2008 (china), M= 8.0, recorded by
OBS 12 (all components). The seismic phases continued for more than 6000
Seconds. Long period surface waves (Rayleigh & Love) are also recorded.
4.5.2 Regional earthquakes
The earthquakes, which occur beyond say 500 km but within 1000 km of a seismograph
station, are called regional earthquakes. An example of a regional earthquake, recorded by a
micro earthquake recorder at a distance of about 700 km, is shown in Figure (1.25). Like
teleseismic events, amplitudes of regional earthquakes can range from barely perceptible to
large, but their periods are less than those of the teleseismic events. These earthquakes also
provide seismic wave data for the Earth’s crust and mantle.
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Figure 1.25 Part of seismogram showing a regional earthquake of
june 8/ 2008 (Greece), recorded by OBS 12 (all components), M=
6.5.
4.5.3 Local earthquakes
Earthquakes occurring within a distance of few hundred km, say 500 km, from a seismic
station are called local earthquakes. Local earthquakes are often characterized by impulsive
onsets and high frequencies. A seismogram showing local earthquakes recorded by a microearthquake station at a distance ranging from 50 to 250 km is illustrated in Figure (1.26).
The local earthquake signal has typically an exponential decreasing tail. The seismic-wave
data are very useful to study the local geological structure/velocity structure of the Earth’s
crust and upper mantle.
Figure 1.26 Part of seismogram showing a local earthquake (all
OBS’s vertical components) of November 1st, 2008 (Greece),
M=4.8 (SW Iberia).
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5. What is seismic tomography?
Tomography is a name that comes from the Greek word TOMOS meaning slice or section
and, graph meaning drawing or imaging. Tomography now is used in different areas, in
geophysics to investigate the earth’s interior in many different directions and construct a
tree-dimensional image of what’s inside, in medicine to sweep human body in many
directions. Nowadays we know that the principles of tomography are applicable to
seismology, as well. For example, earthquake seismologists applied tomographic methods to
produce a velocity model for the earth’s mantle. But how is this done with earthquakes? , in
this case we call it seismic tomography and it represent a data inference technique that
exploits information contained in seismic records to constrain 2D or 3D models of earth
interior (Rawlinson et al, 2010). Seismic tomography can be designed as a powerful process
of estimating the properties of a medium from measurements of sound energy propagating
inside the earth. The Progress in seismic tomography has been growing up rapidly during the
last 25 years, and substantial numbers of review papers have been published, nevertheless,
Imaging of the Earth’s interior based on seismological observation goes back to the
beginning of the 20th century. Recently, geophysicists have successfully used seismic
tomography to image velocity variations of the earth, the Moho discontinuity (Mohorovicic,
1909) and the existence of the inner core (Lehmann, 1936) were rapidly identified, the body
wave travel-times were tabulated by Jeffreys and Bullen (1940). In the following years,
Keiiti Aki published a seminal paper in 1976 on 3D velocity determination beneath
California from local earthquakes (Aki and Lee, 1976). (Aki et al., 1977; Dziewonski et al.,
1977) have solved for 3-dimentional velocity structure under the NORSAR array,
(Dziewonski et al. 1977) derive a low resolution model of 3-D velocity perturbations, when
he succeed to show a significant correlation between velocity anomalies in the lowermost
mantle. Ellsworth and Koyagani imaged the structure beneath the Kilauea volcano on
Hawaii in 1977, in this same year, the first attempts to retrieve the lateral variations in the
Earth’s structure were done again by Aki et al. (1977), four years later, Dziewonski &
Anderson (1981) constructed an average model, known as preliminary Reference Earth
Model. In 1984, (Thurber et al,. 1984) imaged velocities at Kilauea that were low enough to
be interpreted as the underlying magma complex. Recently, seismic tomography has
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revealed features in the mantle clearly associated with mantle convection. Researchers
succeed to construct a tomographic model of S-wave velocity variation in the mantle that
shows clearly the structure of the plate tectonics.
Before less than forty years, observing the internal structure of the Earth was impossible,
now seismology is a scientific branch which is only at the beginning stage of progress,
seismologists know well that they can never determine the structure of the earth with
absolute precision, even if, at present they are applying the most powerful principles of
seismic tomography to construct the underground structure. But, the development of
powerful computer technology and sensitive sophisticate instruments helped a lot in
improving earthquake location and in more precise determination of travel-times of seismic
body waves, giving more credibility to tomographic models recently determined.
Figure 1.27 Global Seismographic Network (IRIS).
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The biggest seismology feat is to produce the first maps of the interior of the Earth through
the establishment of global seismological networks Figure (1.27) , to develop images of
individual slices through the deep Earth, understand the composition of Earth’s interior, and
explain geologic mysteries along the world, it has the major advantage of not needing to
destroy the earth, this technique need only Sources (data) and receivers (seismometers
placed on the Earth’s surface), these sources are either explosions (active) or earthquakes
(passive).
The technique is to visualize the changes in seismic velocity (expressed in percentage)
called seismic velocity anomalies. These abnormalities are related to speed of temperature
variations rocks. If a seismic wave travel through a medium whose physical properties
(density, elastic modulus) are different from those of the average model, those waves arrive
late or ahead of the predictions of this model, we then deduce a map variations of different
kinds of seismic wave from the model, the establishment of these maps requires the data of a
large number of travel time, we retrieve the records provided by each station, seismic station
recorded the arrival of different seismic waves over time as a time series called seismogram,
after that seismologist, can operate using a several methods to analyze the data. All require
some knowledge of the physics of waves and their propagation. These data can be processed
by an optimization process also called inverse problem (see the next section) to recover the
properties of the basement. Inverse modeling or inversion is a method of estimating
propagation or interval velocities from seismic data. It uses ray tracing over finite depth
intervals and to simulate the actual travel path of waves by updating a readably starting
velocity earth models, seismic data acquisition parameters, and theoretical models of
physical processes to generate synthetic data that match actual seismic data. Inversion is a
process used to predict observed data. By contrast, the direct problem (modeling) associated
with this inverse problem is when calculate the travel time of seismic waves in a priori
known model. These travel times are the solution of the eikonal equation (expressing the
equality of the squared modulus of the gradient of the travel time and the square of the
slowness). It is the use of these two problems which will optimize the model established by
the least squares method, in the most case the method of tomographic inversion of travel
time is used to obtain three-dimensional velocity perturbations in the Earth's, travel time of
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seismic waves can be determined from seismograms, is a graph of the time that it takes for
seismic waves to travel from the epicenter of an earthquake seismograph stations varying
distances away.
5.1 The main steps to image earth interior
The first main step is to acquire the observations, although either their nature, they may
come from natural origin (faulting, volcanic eruption, landslide) as they can also be of
artificial origin (mine collapse, burst, excavation, shooting exploration or nuclear test),
differences in their origin can be identified from signal records Figure (1.21). In which there
are differences in the frequencies and amplitudes. Numerous seismologists have compiled
large arrival-time data sets like that shown in Figure (1.28). Average fits to the various
families of arrivals are known as travel-time curves or charts, these represented painstaking
data-collection efforts over the first four decades of the century, curves that seismologists
are trying to inverse to get the velocities variances in deep structure, Tomography of first
arrival seeks to estimate velocity model wave propagation from seismic first arrival times
picked on seismograms. The velocity model obtained can then allow a structural
interpretation of the environment or serve as a model for other initial treatment imaging
seismic. The areas of application of this method lie in different scales of the geotechnical
seismology through petroleum geophysics.
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Figure 1.28 Travel time picks for various body waves phases and travel time curves, the data
are 57655 travel times from 104 sources (earthquakes and explosions), ( Kennett and Engdahl,
1991).
The theory of ray tracing (see chapter 2) is the theoretical concept used to infer the seismic
waves travel time, is one of the powerful mathematics tool needed to get help calculating
travel time curves, expressions for ray tracing are determined from a known velocity
structure in a one-dimensional (1-D) velocity model in which velocity varies only with
depth. For three-dimensional (3-D) structures the ray theory is more complex even if follows
similar principles. Studding structure of the earth began with examining seismograms and
calculating travel time curves, the case where we are given travel times obtained from
observations and wish to invert for a velocity structure that can explain the data. In general,
the inversion is much more complicate than the forward problem. The inversion for travel
time is often an iterative process where a reference model is needed to start inversion, in this
stage, travel time residuals are computed for each datum by subtracting the predicted time
from the observed time, then we change the model to minimize the difference between
observed and computed travel times. After minimizing all the differences error between
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predicted and observed travel time, a 3-D model is obtained by inverting the travel time
residuals for velocity perturbations relative to the reference model; this is the main basis for
tomographic inversion techniques, we can also look at the surface waves tomography using
harmonic modes of Rayleigh waves and love waves, this type of imaging of earth interior at
shallow depth level, however, the surface wave tomography is part of another theory
process. It is not easy to combine with that of the body waves, the assumptions are different
in each case, and they are generally treated separately.
5.2 Imaging the earth with seismic data
5.2.1 Travel time tomography
Travel time Tomography seeks to estimate velocity propagation of seismic waves from the
time of first arrivals, generally, data rarely fit a straight line exactly. Usually, we must be
satisfied with rough predictions. Typically, we have a set of data whose scatter plot appears
to "fit" a straight line; this is called a line of best fit. It is important to fit the data to have an
initial velocity model of wave propagation, which can then allow preliminary structural
interpretation of the environment and serve as initial velocity model to use for inversion.
When the 1-D model is available, the next step is to parameterize the 3-D velocity
perturbations into uniform blocks velocity perturbations, we illustrate the block
parameterization considering a three dimension geometry with the model divided into
similar blocks Figure (1.29), the model contain in general all paths connecting the sources
and receiver, The next challenge is to find the exact ray path, and it is not always
straightforward to do so, there is several methods for solving the tow point ray tracing, the
most known one is shooting method, in which slightly the source are sampled in order to
converge on the correct receiver location and
bending method, where the source and
receiver are kept fixed and try to bend the ray, or finite difference or graph theory techniques
that require a grid of points.
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Figure 1.29 An example ray path in a 3-D block velocity
perturbation for Tomography problems
From seismograms we obtain the observations Figure (1.30) by picking different arriving
waves, at least P and S phases, at this stage we have to generate linear regression to fit the
observations data and use an inversion technique to derive an initial velocity model.
Obviously the observed data can not exactly looks like the predicted one, due to the errors in
picking phases, effect of instruments measurements and site, we define the travel time
residual as the sum of three terms
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Temps de parcours (secondes )
120
100
80
60
40
20
0
0
100
200
300
400
500
600
700
800
900
1000
Distance (kilomètres )
Figure 1.30 Travel time plot of the P seismic waves of the events occurred in morocco between
1993 to 2003 recorded by Moroccan seismic station networks (Lbadaoui Master thesis, 2008)
Almost all seismic tomography methods involve subdividing the medium into blocks or
other spatial functions, such as spherical harmonic expansions, and solving for slowness
perturbations that cause predicted times to match observed times better than an initial
(usually homogeneous or one-dimensional) model. The idea is that the path integral through
the medium perturbations should equal the observed travel-time residual. The objective is to
seek for a model that minimize the travel time residual
, before we have to compute the
observed travel time for each ray path. The tomographic process in a nonlinear inverse
problem, for which the travel time for the
where
represent the observations,
ray can be expressed by the vector equation
is the model parameter and
is the frechet
derivatives. The travel time tomography is an iterative process, so, an initial model
parameter
must be used, and is can be created from the best fit of the travel time curve.
we make than a little change in the initial model parameter the get a new model, expected to
looks like our observations when we will proceed to inversion.
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where
represent the perturbation in model parameter and
is the new model, then
we compute the Taylor expansion of
Where
are the Freshet derivatives, ( represent the number of the rays and
represent the number of cells), the little change in the model (slowness), we’ll end up with a
small change in travel time, according to the Fermat’s principal we ignore the change in ray
path, thus the observations can be expressed as
and
where
represent the length of the
perturbation. Equation
ray path in the
block and
is the slowness
is in the form of a linear system of equation which can be
solved by the matrix inversion methods, in this case, the path length of each ray in
corresponding block,
is the partial derivatives
of the travel time with respect to the
slowness of that block, in general we have more raypaths than model parameters, yielding an
over determined system of equation, so, the generalized inverse solution is
where
,
,
, (see the inverse theory in the next chapter). the
Figure below chow a seismic ray path passing through a medium divided one similar blocs,
with small red segment we define the length cover by the path in each bloc, obviously the
majority of the segment are not cover by this path, set to zero elements in operator matrix
(see the following example), and the geometry of the ray is determined using ray tracing
methods, in this work , we have used the bending method (see ray tracing in Part 2).
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Figure 1.31 An example ray path and cell
numbering scheme for a simple 2-D tomography
problem.
To solve the tomographic problem we have to liberalize about the reference model, the
relationship between the observation and model parameter is given by
where
is the sensitivity kernel and
where
is the model perturbation;
is defined by
denote the basis function and
Figure (1.32).
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is the weight
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Figure 1.32 An example ray path and in 2-D
dimension showing the blocs where the basis
function is none zero.
by replacing
Where
in the equation
we get
is the sensitivity matrix kernel, projected on the basis functions and the value of
the weight is
length of the
. The element of the basis function represent the
ray in the
block, with these element we construct the matrix
. In
general, the number of observation is deferent than the number of blocs, so, the operator
matrix
is not square, in this case we get the solution by applying the generalized least
square (see Appendix).
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5.2.2 Example of travel time tomography
To illustrate this idea, we consider the ideal case model consisting on a number of blocks of
unit length
in tow dimension case Figure (1.33), in which 13 ray paths
traverse the
model in different directions.
Figure 1.33 2-D block geometry velocity perturbation
for an idealized tomography problem, the model consists
on identical blocks , traversed by 13 ray paths.
The travel time perturbation along the ray path is then given by the sum of the product of
each block travel time with the fractional velocity perturbation and can be expressed as
All the blocks not encountered by the ray are set to zero in the operator matrix, so, for the
ray path the travel time residual can be expressed as
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and
The reference slowness model is assumed to be appropriate outside the blocks, so the entire
travel time residual for each path is attributed to slowness perturbations in the blocks. Thus
the problem looks like
Where
denotes the total number of blocks in the given model. To reach the receiver, it is
known that the rays pass through only some blocks of the setting model, so most of the
elements of the operator matrix are set to zero. For
measurements it can be written in
matrix notation as
where
is a vector of observations or differences between observations and
predictions,
is a matrix of partial derivatives
In our case we have
determined problem and
, and
observations and
has rank
, finding
generalized inverse solution in the next section).
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model
is the vector of model parameters.
model unknowns, is an over
is dealing with an inverse problem (see the
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6. Inverse problem
In seismology a known incident wave is given and it’s perturbed by a variation of the
nominal features of the medium, if these variations are known, the calculation of the
perturbation undergone by the incident wave (forward problem) represent a classic methods.
Reciprocally, the measure of the perturbation of signal (seismograms) provides the
supplementary data permitting reconstruction of an unknown variation of the nominal
features (inverse problem). In general inverse problems are hard to solve, we need to be
concerned with far more than simply finding mathematically acceptable answers to
parameter estimation, One reason is that there may be many models that adequately fit the
data, and the process of computing an inverse solution can be, and often is, extremely
unstable in that a small change in measurement then can lead to an enormous change in the
estimated model. From the mathematical point of view we can represent many physical
experiments with a model space consisting of model parameters
a data space with data
and
Both the model parameters and the data are
physical magnitudes whose functional relationship are given by a set of computational rules
(the function f). If we assume that the fundamental physics are adequately understood, so a
function may be specified relating
and
as
Inverse problem cannot solve the system exactly, but have to minimize the misfit between
data
and model parameter
. Generally formulated, the data
are a
series of observations in the form of measured values, whereas the model parameters
stand for the physical properties of the research object. The latter are
not necessarily directly measurable. The function
stands for a method (usually a
mathematical representation of a physical theory), which relate the model parameters
data
to
.
If the model parameters are given and the accompanying data calculated, then we speak of a
forward problem (unique solution), on the other hand if we want to determine unknown
model parameters based on the data (for example, physical measurements, then we speak of
the inverse problem (not unique solution).
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Figure 1.34 Chart showing the differences between the
inverse and forward problem.
In geophysics the term inversion usually refers to making inferences from observations
(mainly on the Earth's surface) about the material parameters (density, wave velocity, etc.)
and geometric parameters (depth, earthquake parameters, etc.) in the subsurface.
6.1 Travel time inverse
The travel time for a given ray can be expressed as
The travel time changes with a small perturbation in the path
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Discrete form of the travel time perturbation
where
is the distance the
ray and
is the slowness in
block or node.
Observations of properties integrated along the number of paths through the medium are
used to infer the distribution of the physical property within a medium.
Is the expression for estimating slowness perturbation from observed travel time
perturbation and
denote the partial derivative of the rays travel time with respect to the slowness in the
block, it is an operator relating the data and model vectors, the dimension of the model
vectors
is the same as the number of blocks or nodes elements, and the data vector has
the dimension of the number of the ray paths. The operator
the data function
with respect to the model parameters
is also the partial derivative of
,
can be expressed as
In general, the system of equation is over-determined because in the most cases we have
more equation (ray paths, or observations) than the number of the blocks (model parameter),
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the travel time thus, is predicted using an appropriate starting model, and travel time
residuals are computed for each ray using the error vector
The error vector illustrates the quality of initial model, where
is the observed data and
denote the predicted, in most cases, the error vector magnitude is defined by the
normalized term called Euclidian length
The inverse problem for estimating slowness perturbation from observed travel time
perturbation can be expressed in the form
is in general not square matrix, so, we form
Where
is
transpose, to get the generalized inverse solution, we invert
operator,
and the analysis of the inverse problem use the decomposition in singular values of one
matrix. See (Marc Bonnet, 2008) for more detail.
Where
, used in specific case when we need not to apply Lanczos
decomposition. The model parameter
because,
, so the
inversion correctly resolve the inversion, likewise when one has the same quantity of the
data as for blocks (vector parameter), and
linearly independent. In general case when
can be solved easily if the rows of
is not square matrix,
then will be square
symmetric Matrix that can be decomposed using its eigenvectors and eigenvalues.
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are
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where
comprise the eigenvectors of
eigenvalues of the matrix
matrix, and
diagonal matrix contains all the
.
The eigenvalues are distinct, so the eigenvectors are orthogonal
Similarly
and
and it’s possible to decompose
can be decomposed as
only eigenvectors with non zero-eigenvalues, where
eigenvectors associated with non-zero eigenvalues, and
using
is the matrix
is the matrix
with
with
eigenvectors associated with zero eigenvalues. So
known as Lanczos decomposition, (Stein and Wysession,.2003), generalized inverse is
rather than
who can then be written as
So the solution to the inverse problem is
Finally we find that,
and
, where
is the true model for which we are seeking
is called the resolution matrix. The columns of the resolution matrix indicate
how much the true model is smeared into the various parameters of the inversion model.
Ideally, one would obtain a diagonal resolution matrix, recovering the full model.
Calculation of the resolution matrix is essential for assessing an inversion result.
We define also the information density matrix given by
And the covariance matrix is given by
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These equations give us physical insight into the inversion solution, for a perfect model
matrix
is equal to identity matrix, diagonal elements that are close to one
correspond to parameters for which we can claim good resolution, it happen when all the
model parameters are associated with nonzero singular values. Conversely, if any of the
diagonal elements are small, then the corresponding model parameters will be poorly
resolved. The resolution and stability can be controlled by using the catoff on the ratio of a
given eigenvalue to the largest eigenvalue, the cutoff condition number to determine the
number of singular values to retain is defined by :
See the example of locating an earthquake by using the generalized inversion.
6.2 Why inverse problems are hard?
The difficulty of the inverse problem is a combination of several factors; In general the Goal
function is not convex, so this leads to the existence of local minima, and process of the
convergence does not stop at the right target. The inverse problem can be ill-conditioned
problem, because of a big quantity of data, which makes the existence of multiple solutions,
ie several parameters producing the same observations, leads to a non uniqueness of the
solution. Discontinuity producing instability in inversion process, Even if you can (in
theory) solve the problem for accurate observations.
6.3 Earthquakes location
One of the most important tasks in observational seismology is locating seismic sources.
The location of an earthquake is among the one of the classic inverse problems in
seismology, involving determination of both the hypocentral coordinates and the source
origin time. In addition, improving the accuracy of localization is one of the major concerns
of seismologists, were believes that clearly identified structure can be derived from well
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localized earthquakes, this requires identification of seismic phases and measuring their
arrival times. Currently sensitive seismographs help scientists to determine the arrival times
of the different seismic phases, nearly 3000 seismic stations distributed worldwide have
been systematically reporting major seismic phase arrival times to the International
Seismological Centre (ISC) since 1964 (Lay and Wallace,1995). Thus the travel time of the
wave is evaluated using these arrival times at several stations.
An earthquake happened generally at unknown time caused by a fault rupture, the waves
propagate in all directions and travel long enough distances before they are detected by
seismometers scattered across the globe. The hypocenter, Figure (1.35), or focus, is the
breaking point or point source, this is the point where the seismic waves begins to spread,
the projection of the point source on the surface of the earth when
is called epicenter.
Figure 1.35 Geometry for an earthquake location in earth with variance change in
velocity with depth.
earthquakes are usually detected by relatively dozens or hundreds of stations of coordinates
, each station specifies the arrival time of the earthquake, hence there is
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record for
stations, so
arrival times
for an earthquake occurs at time
Arrival time
which depends only on the travel time
in a position x.
can be expressed according to travel time and origin time as
Knowledge of the structure, allows knowledge of both, information source coordinates
and origin time , this can plays a crucial role in determining data, this is the forward
problem, however seismologists are much more interested in inverse problems since the
main goal is estimation of model parameter
and not the data.
or
The inverse problem is then to find a model that "looks like" real model, ie find the
particular model for which the predicted and observed data were in best agreement. The
perfect model which reflects the reality of the structure can never be determined because the
inverse problem cannot led to a unique solution, in fact the inverse problem can be seen as a
process of elimination of any information that cannot deal with corresponding observed
data. After collection of a large number of data, the first step that must begin with is to
estimate an initial model
(see chapter). Note that even the perfect model can not reflect
the data observed in seismograms, because there are always errors in reading the various
phases. Estimated data
often do not reflect the observed data, an explanation of
the observed data then require an initial model preferably close to real model to satisfy the
linear approximation (Stein and Wysession, 2003), it is then a temptation to predict what we
have observed, it can lead by effecting a little change of
in the initial model
The vector data do not depend linearly on the model, we have to linearize
of
, the model can be written as:
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.
in the vicinity
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The difference between the predicted and observed data is given by
which can also be written as a system of linear equations:
where
,
is the operator matrix that contains all the partial derivatives of the
vector data times model vector
, an application of the operator
after perturbation
of
model parameter leads to a change in the vector data, the simplest case to solve is when we
have
when
operator is a square matrix. But in general this is not the case, because
we often have a large amount of data, thus characterizing an overdetermined problem, in this
case the operator is not square matrix, hence it require the generalized inverse solution.
In practice, the arrival time of a wave still contains errors, uncertainties in instrumental
measurements or errors reading phases, choosing a simple problem with only four
observations cannot reach as well the reality of the structure, it is the fact that the model
itself contains errors and is laterally discontinuous, thereby seismologists solve the problem
by taking into account a plenty of data, they are carried to solve over-determined and
inconsistent problems, and try to minimize the quantity
, One of the most common
ways to do this is to write an equation for the squared error
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and force
to be a minimum by taking the derivative of
parameters
where
quantity
with respect to the model
and setting it equal to zero
is the standard deviations which describe how widely the data is spread. The
represents the prediction errors and the determination of the partial derivatives
of the equation
with respect to a small perturbation lead to minimize the misfit.
where
where
Or collecting terms
In the case of equal variance for all data (Stein and Wysession, 2003),
in matrix notation as
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can be written
LITERATURE REVIEW
This is very useful form called the normal equations.
take the form of a square matrix,
thus can easily be inversed (as long as it is not singular!). Furthermore,
is symmetric
operator, which means that its eigenvalues are all real and nonnegative values. Therefore, by
inverting
we can write a system of linear equation of the form
Where
is the generalized inverse of, the
will only serve as a
correction to a starting model, and we must repeat the inverse as an iterative process, with
updated for the new model.
6.2.1 Example location of earthquake in homogenous medium
Locating earthquake with consideration of an homogenous media is straightforward, because
the velocity is constant, thus the ray paths connecting the source and receiver are just a
straight lines, when an event is occurred at time at location
stations of coordinates
the arrival time at the
, Let and
, it’s recorded by a
be the origin time of the earthquake and
station, respectively. Then
Where
denote the travel time and
the velocity of wave propagation in the medium.
For the land stations we set
Ocean Bottom Seismometers
, and the equation become (not that in the case of the
is always a negative number, because the elevation is taken
from the sea level).
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LITERATURE REVIEW
If we clearly see this expression, we infer that
related with model
represent an element of the data vector
by the equation
Where x, y, z, and are the elements of the model vector m that we wish to determine. Now
we have to form the matrix
respect to the model parameter
, and calculate all of its elements by referencing
.
And for the fourth unknown t we have
For example, the first element of the operator
is
Represent the first observation given by the first station. And,
Represent the second observation given by the first station.
93
With
LITERATURE REVIEW
Not that, determination of
operator
and
which is the diagonal matrix with the eigenvalues of the
the operator containing the eigenvectors (see inversion theory section
in chapter I) can provide precious information about stability of our inversion process, thus,
inversion estimates for the change in
are most stable for the large values of
Wallace, 1995), this is best illustrated by an example, If we consider example of
(Lay and
operator
with or nonzero singular values.
For this example the inversion process estimates for the change in elevation associated with
the
parameter are the most stable, and estimate of changes to the travel time are the least
stable part of the inversion process, so we have tactfully determining true travel time given
the starting model we have chosen. To start the inversion process, we "guess" a solution
from which we can calculate an estimated or predicted data
vector. If the initial guess is not very good, which quickly resulted from the travel time
residuals a priori computed for each observation using the error vector
The process is repeated making a little change in the initial model, and so on until
is
minimized. The process of the inversion for location earthquakes for more complex
geometries is obtained from the follow of proceeding along similar method, with the only
real differences being in the calculation of the partial derivatives needed to constrain the
different operators required for inversion.
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LITERATURE REVIEW
7. Conclusion
In this chapter we have given an overview of the structure of the earth and the mathematical
background necessary to solve inverse problems. In Seismology the problems are often are
characterized by their large size due to huge quantity of data. , we often need to use
powerful algorithms and simple examples for a perfect homogeneous structure are
introduced.
We are dealing with a passive seismic tomography, which is based on the joint inversion of
the earthquake location and difference waves arrival times of the seismic body waves. An
effort is made to use the waveform and in particular, the seismic rays tracing, to build a
model of the corresponding interfaces with adequate methods. In Chapter II we explained
the choice of optimization algorithms and how to use it, we will give the general formulation
of the ray theory that characterizes the path followed by the seismic waves from the source
to reach a receiver, These waves undergo multiple reflections or refractions by discontinuity
surfaces, the determination of the parameters of our model were determined rigorously, for
inversion of the data collected by the stations. We have also well seen the literature and the
research work that have been madepreviously for a good choice of the initial model used in
the conversion process.
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LITERATURE REVIEW
Part 2
CHAPTER OUTLINE
Body waves tomography
of the Gulf of Cadiz
1. Introduction
2. Motivation and research
objectives
3. The Gulf of Cadiz study area
4. Ray tracing and travel-times
inversion
5. Seismicity of the Gulf of Cadiz
6. Synthetic tests
7. Results and discussion
8. Conclusion
9. References
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
1. Introduction
The gulf of Cadiz region connects the Betic-Rif orogenic arc to the oceanic plate boundary
between the Africa and Eurasia plates and the Gloria Fault (Zitellini et al , 200). The nature
and exact location of this plate boundary are still uncertain. It is a large area, with complex
deformations and diffuse seismicity that extends over 200 km from north to south (JimenezMunt et al, 2004), (Sartori et al, 1994), ( Tortella et al , 1997). The 1755 Lisbon earthquake
is still considered the most destructive event in the western Mediterranean. This event was
followed by a large tsunami that caused several thousand deaths and many localities were
largely destroyed. In particular, all localities on the Atlantic coast of Morocco from Tangier
to Agadir, were severely affected by the combined effects of the earthquake and the tsunami
that followed (Kaabouben et al, 2009). The tsunami flooded about 2 km inland along the
coast of El Jadida, Safi and Essaouira (El Hammoumi et al, 2009), It is worth noting that
this event occurred in the Atlantic Ocean, but its precise location is still a matter of debate
(El Mrabet et al, 2005),(Baptista et al, 1998), (Zitellini et al , 2000).
Some authors attempted to explain the tectonics and the formation of this zone by
suggesting a delamination of the continental lithosphere (Buforn et al, 1998), (Platt et al,
1989), while others suggested an oceanic subduction (Calvert et al, 2000), (Duggen et al,
2004), (Gutscher et al , 2002) that led to the formation of an accretionary wedge to the west
of the Gibraltar arc. Some of these studies used data recorded by seismic reflection or
refraction surveys. However, such studies usually have a limited depth penetration, which
limits their tectonic interpretations, especially that most of the seismic activity in the Gulf of
Cadiz occurs at more than 20 km depth as will be shown in this work. Other studies used
data recorded by land stations in Morocco and Iberia to invert for the three-dimensional
velocities underneath the Gulf of Cadiz. However, since the used stations are far away from
the Gulf of Cadiz, the results are not quite reliable and remain of limited resolution.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
In this work, we show the results of a body-waves seismic tomography study based on local
earthquakes recorded by a network of 24 broadband ocean bottom seismometers (OBS)
deployed directly on top of the sea floor of the Gulf of Cadiz. The data used were collected
within the framework of the European project NEAREST (Integrated observation from
NEAR shore sourcES of Tsunamis: towards an early warning system), in which the 24
OBSs were deployed by the German DEPAS instrument pool coordinated by the Alfred
Wegener
Institute
for
Polar
and
Marine
Research,
Bremerhaven,
and
the
GeoForschungsZentrum, Potsdam, Germany. The data were collected during one year of
recording, beginning August 2007, and augmented by seismic data recorded by eight land
seismic stations that belong to the Portugal permanent seismic network. The objective of this
work is thus, to use these high quality data to help constrain a 3D velocity model of the earth
interior beneath the Gulf of Cadiz and to draw implications for the tectonics of the region.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
2. Motivation and research objectives
Seismology is at an extreme of the whole spectrum of earth sciences, first it is concerned only
with mechanical properties and dynamics of the earth, and second, it offers a means by which
investigation of the earth’s interior can be carried out to the greatest depths, with resolution
and accuracy higher than are attainable in any other branch of geophysics.
Using the Nearest Network, it was possible to detect a large number of local earthquakes with
small magnitudes that we could not otherwise be detected using the existing land stations
network. These data will certainly help to improve both the geological and the geophysical
knowledge of the tectonic architecture of the area which is presumed to be the source of the
great 1755 Lisbon Earthquake which caused several thousand of deaths. A large effort has
been made in the last decade to determine the shallow and deep crustal structure of the
Atlantic margins of SW Iberia and NW Africa (e.g., Maldonado et al., 1999; Gràcia et al.,
2003; Medialdea et al., 2004;, zitellini et al 2009, Gutsher et al 2004, Stich et al, 2006,
Terrinha et al, 2009 ).
The research subjects in seismic tomography have been strongly associated with the
emergence and progress of deep seismic imaging since the first applications of this approach
in three dimensional scales. Today, it become a part of the "toolbox" of petroleum geophysics
Offshore, and is used almost routine for exploring parts of the world. In this thesis the work is
focused on two main steps:
 Study of the seismicity of the region of the Gulf of Cadiz throughout analyzing the seismic
NEAREST
network records (all local earthquakes, even small ones) that will allow us to
better assess the seismic activity in this region, as well as the maximum possible
magnitude, recurrence of earthquakes, and tsunami risk, etc..
 Second step is a construction of three dimensional models which describes the velocities of
the waves in the subsurface using the arrival times of body seismic waves, and after
achieving this work, we will focusing on construction of model velocity using surface
waves to better investigate the earth interior at shallow depth.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
3. The Gulf of Cadiz study area
The Gulf of Cadiz is located in the eastern part of the Atlantic Ocean between Africa and
Europe. It extends from the Strait of Gibraltar (Spain) to Sâo Vicente Cap (Portugal) with an
orientation NW / SE controlled by recently active tectonic pattern (Maldonado et al., 1999).
This region undergoes complex deformations due to plate tectonic interactions between the
southern Eurasia and the North Africa plates; Figure (2.1) shows the displacement these
different plates, due to its complex geology it is characterized by irregular bathymetry and
concave morphology towards the SW.
The plate boundary is still poorly defined until recent day, and the convergence between the
African and Eurasian plates is accommodated by a zone of active extensive deformation
(Sartori et al., 1994). The Gulf of Cadiz had a complex geological history in response to
changes in the location and movement of the plate boundary (Srivastava et al., 1990).
Figure 2.1 plate tectonic interactions between the southern Eurasia and the North Africa plates with the main
elements of plate boundaries superimposed: AGL: Azores–Gibraltar Line; GC: Gulf of Cadiz; GF: Gloria
Fault; MAR: Mid-Atlantic Ridge; , SVC: Sao Vicente Cap. TR: Terceira Ridge. Solid yellow line: plate
boundaries (Zitellini et al 2009).
In the last decade, several geophysical studies have been undertaken in the Gulf of Cadiz,
since this region is a source of many earthquakes and the historical devastating tsunami that
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
followed the great Lisbon earthquake of 1st November 1755th with an estimated M=8.5 to 8.7
magnitude (Solares, et al ,2004). The evolution this area is associated with the collision
between the Iberian Peninsula and northwest Africa (Nubia) tectonic plates. This collision is
the main cause of the formation of the Betic Rif orogenic arc in the Miocene epoch and the
formation of submarine accretionary wedge in the Gulf of Cadiz. The Iberian margin area is
more complex than most of the North Atlantic margins. It is characterized by several phases
of rifting, convergence and strike-slip motions (Srivastava et al, 1990., 1997; Maldonado et
al.1997, Duarte et al 2010). Iberia has moved independently from the Eurasian plate and the
African plate. this displacement allowed a tectonic sticking of Iberia and the African plate.
Therefore the plate boundary between the Iberian Peninsula and Africa has become active
again.
Recently, scientific research provides several morpho-tectonic information on the study area,
for instance, seismic reflection profiles helped to understand and locate the plate boundary
Africa and Eurasia. Furthermore, bathymetric data reveal the existence of a series of subparallel WNW-ESE lineaments that extend from the Hirondelle Seamount to the Moroccan
continental shelf, across the Horseshoe abyssal Plain and the Gulf of Cadiz accretionary
wedge (Zitellini et al,. 2010). The present tectonics of the region is dominated by the
reactivation of these faults lineaments as thrust or strike slip faults, which trend from the
horseshoe Abyssal Plain to the Rharb Valey and are in places associated with fault scars and
important mass wasting deposits (Gracia et al. 2003), (Zitellini et al. 2004). Gutscher et al
(2008) show from bathymetric data and seismic reflection profiles, the location of the
different active processes shaping the accretionary wedge. The boundary between the Iberian
Peninsula, Eurasia and Africa is conditioned by the relative movement of these plates.
Recetelly; data indicate the existence of an intense mechanical coupling between Iberia and
Africa during the Tertiary, In the western part, initial displacement between these plates was
divergent and progressively changed to strike-slip (Vicente et al,. 2008).
In this region, scientists have successfully identified some traces of the lithospheric
converging plates. In addition, bathymetric and tomography studies suggested the existence of
a subduction zone west of the Gibraltar arc (Srivastava, et al 1990) and seem to indicate that
the tectonic processes leading to the formation of an accretionary prism and the morphology
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
of the southern Gulf of Cadiz are influenced by processes associated with tectomorphic
deformation of the accretionary prism of the Gulf of Cadiz (Maldonado, et al 1999) as well as
a gravitational process (Gutscher, et al 2009). Based on an analysis of seismic data, it was
suggested that the base of the accretionary wedge is dipping eastward and has a primarily
tectonic origin (Gutscher, et al 2002). The majority of the accretionary wedge was
constructed by offscraping of deep sea sediments during Miocene due to westward motion of
the Gibraltar arc (Thiebot, et al 2006). Other geodynamic models suggest however, a
delamination of continental lithosphere process (Platt,et al, 1989),(Calvert, et al, 2000).
Recent detailed bathymetric mapping in the Gulf of Cadiz shows the existence of major
inverse faults that trend NE-SW, Figure (2.2) mainly perpendicular to the principal stress
direction, as well as sub-parallel WNW-ESE strike-slip trending lineaments (Stich, et al
2005),(Zitellini, et al., 2009); the SWIM Faults, that extend from the Hirondelle seamount to
the Moroccan continental shelf. Based on this, it was suggested that the SWIM Faults zone
(SFZ) constitute the plate boundary between Africa and Eurasia along a 600 km span between
the Gloria Fault and the Rif-Tell plate boundary (Zitellini, et al., 2009).
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.2 Gulf of Cadiz region offshore SW Iberia, showing the bathymetry map and
existing faults, SWIM is South West Iberian Margin faults lineament (Duarte, et al 2009).
Zitellini, et al (2009) further suggest the existence of active thrust structures in the Gulf of
Cadiz to explain the main clusters of seismicity. While some authors support that subduction
is still active and poses significant seismic hazard to the region, others based on the low level
of seismicity in the accretionary wedge and the undeformed package of deposited sediments,
find that the subduction zone is not active at present time or is rather dying out and has largely
ceased (Zitellini et al., 2009). Grevemeyer et al (2008) confirm the existence of an oblique
collision between the Nubia and the Iberia plates. Some studied velocity model indicates that
the Betic crustal root might be underlined by a low velocity anomaly in southeastern Iberian
Peninsula, a variation in the uppermost mantle velocities that coincides with the structural
complexity of the Eurasia and Nubian plate boundary in the Gulf of Cadiz. When a clear lowvelocity anomaly offshore from Cape San Vicente is shown, and high velocities distribution
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
are shown along the coast in the Gulf of Cadiz (Serrano et al 2005). Neves et al (2009) show
that stresses are focused in the ocean-continent transition, between the base of the continental
slope and the Horseshoe Abyssal Plain.
Recently, based on focal mechanisms of seismic events occurred in the Gulf of Cadiz, it has
been shown a compressional acting in the northern part of the Horseshoe fault (Geissler et al.,
2010), While a strike-slip regime acts in its southern part. Trending parallel to the plate
boundary, the seismic data reveal that the compressional deformation trend over a circle shape
of about 100 km wide, delimited by the Pereira de Susa fault and the Tagus Abyssal Plainthrust, and 150 km length region between the Gorringe Bank and the Setúbal Peninsula
(Cunha et al 2010). And a small relative motion between Iberia and the Maghrebian region is
shown, proposing a model explain that Iberia is moving together with the Nubian plate
(Cunha et al 2012).
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
4. Ray tracing and travel-times inversion
Ray theory is an integral part of many seismological techniques, including body wave
tomography, reflection data, and earthquake relocation, and has played a prominent role in the
history of natural science, as a practical theory to explain the propagation of light. For both
the rectilinear propagation of light and the law of reflection, throughout the development of
geometric ray theory, it was understood that the theory was limited and that it could not
explain a large number of phenomena observed in experiments with light (Kraaijpoel, 2003).
However, finding correct geometry of a ray requires to trace rays following Snell’s law. This
is comparatively easy in the case of layered or spherically symmetric media. On the other
hand, if the seismic velocity is also a function of one or two horizontal coordinates, it may be
very difficult. Fortunately, Fermat’s Principle allows us often to use background models with
lateral homogeneity. Many approximations are used to predict wave propagation in smoothly
varying media, for which length scale variations of Lamé coefficients are much larger than the
seismic wavelength (the high frequency assumption).
The application of the high frequency assumption to the wave equation immediately yields the
eikonal equation as the basic equation to calculate travel-time and rays. In order to find the
correct ray geometry between a given source and receiver location, we not only need an
accurate solver for the differential equations, but also a way to determine which initial
condition (ray orientation at the source) satisfies the end condition (ray arriving in the
receiver), (Nolet,2008). In this chapter, we show how we can derive the eikonal equations for
body waves propagation, defining rays as characteristics of this equation, called the system of
ray equations. The eikonal equation is assessed for two waves which may propagate
separately: P and S waves. The relevant travel times and rays of these waves are, however,
again controlled by a similar eikonal equation with a different velocity (Serveny, 2000). More
after much recent work has been done on computing travel times by solving the Eikonal
equation.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
4.1 The Eikonal equation
4.1.1 Eikonal equation for fluid mediums
In fluid areas in the earth, the stress tensor, in the case of zero viscosity fluid, is given by
where
If
denotes the Kronecker symbol, and P is the hydrostatic pressure.
is the bulk modulus and
is the density of the medium, we have, according to the
Hook’s law
Inserting
By dividing by
From
in
we get
and differentiating according to
, and
We assume a harmonic solution of
Where
, we obtain
, then
becomes
in the form
is function of angular frequency
106
and position vector .
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Replacing
in
Where the spectrum of the source
everything that follows we set
, is modeled as the divergence of a stress tensor. For
outside of the source.
Let’s now consider
Where
is the amplitude which varies with location and the signal (Fourier transform of
(2.5)) denoted by,
is a constant and takes the shape of a delta function, with a delay
, (Nolet,.2008).
The amplitude
only depends on the position so,
It is known that
As a consequence
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
We substitute
in
and keeping only terms up to order
Ray theory is strictly valid for media with high frequency body waves assumption. At low
frequencies, diffraction and scattering can be significant, and ray theory is not generally valid.
The terms at zero order
in
yield the eikonal equation for the wavefront
is known as the Eikonal equation. It describes the kinematic propagation (with velocity
) of high frequency waves in a zero viscosity fluid. It describes a nonlinear partial
differential equation which solution requires numerical methods, basically, the Eikonal
equation expresses how the local solution wave number is related to the local material
properties.
4.1.2 Eikonal Equations in Isotropic Elastic Mediums
In an isotropic elastic earth, the elasticity tensor takes into account only the Lamé coefficients
and
. For such a medium the stress tensor is given by
with
Referring to
and inserting the expression of
108
, we get
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
In the frequency domain, we have
Then
This equation can also be expressed in term of vector equation
For a constant and , when individual waves propagate independently in a smoothly varying
structure
Analogously to
the displacement, solution of
Let us now insert expression
will take the form
in the differential equation
, as we are looking for
high-frequency solutions, we can only consider the terms with the highest powers of
obtain
109
. We
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Or in vector notation form
This simple approach is used to derive the most important equations and related wave
phenomena from the seismological point of view. It is shown how the elastic wave field is
approximately separated into individual elementary waves (P,S), equation
is true
either by
1. Choosing
Where
to be parallel to
give
defines the velocity of compressional waves. The P-wave is linearly polarized, the
particle motion has the same direction as
, and the slowness vector can also be expressed as
Where n is a unite vector parallel to the ray and perpendicular to the wavefront.
2. Or choosing
Where
perpendicular to
which implies
defines the velocity of shear waves. And the slowness vector of the S wave is
given by the relation
Likewise n is a unite vector parallel to the ray and perpendicular to wavefront.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
4.2 Ray geometry
For waves propagating with velocity c, the Eikonal equation is given by
where
is called the slowness and the travel time function,
, is called the phase
factor.
Shows that, the phase factor
slowness. The function
perpendicular to
gradient of
Equation
has a gradient whose amplitude is equal to the local
, defines surfaces called wavefronts. Lines
or parallel to
are termed rays. The ray direction is defined by the
, (Shearer, 2009).
allow us to define the unit normal vector
Now let’s consider a ray, and
is the tangent along this ray with length
straightforward to see that we can define the slowness
And it’s obvious that
and
So,
Then
111
, thus
it is
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
4.2.1 Ray solution in layered mediums
Flat Earth
Considering a wave which propagates in the x-z plan of flat earth, in which the velocity
varies with depth z.
Figure 2.3 Geometry of the ray segment along a path from a surface
source to a surface receiver.
The velocity of the medium varies only along the z direction, so there will be symmetry of
down-going and up-going legs of the ray path. In this case
, from
and according to
we find
Where is the angle of incidence and it gives the inclination of a ray measured from the
vertical (z direction) at any given depth Figure (2.3), The constant
(not that, if lateral
velocity gradients are present, then p will change along the ray path) is called the ray
parameter, or horizontal slowness,
represents the apparent slowness of the wavefront in a
horizontal direction and varies from 0 (vertical travel path) to 1/c (horizontal travel path),
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
equation
is also known as Snell's law, which can also be derived from Fermat's
principle. Fermat's principle states that a ray- path is a path of stationary time. Thus travel
time along a ray path is a minimum time, and is the familiar expression from optics called
Snell's law after Willebrod Snell (1591-1626), the generalization of Snell's law is
is also called the seismic parameter,
is constant for the entire travel path of a ray. The
consequence of a ray traversing material of changing velocity, c, is a change in inclination
angle with respect to a reference plane. As a ray enters material of increasing velocity, the
ray is deflected toward the horizontal. Conversely, as a ray enters material of decreasing
velocity, it is deflected toward the vertical. If the ray is traveling vertically, then
, and
the ray will experience no deflection as velocity changes,), for the deepest point the ray
parameter is given by
at the maximum depth reached by the ray Figure
(2.3), (Lay and Wallace 1995).
Figure 2.4 Incidence angle of a ray
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
From the Figure (2.5), the ray parameter P can be determined directly from the travel time
T(x), we see that
, corresponding to the relation
, and we have
Using Figure (2.5), and integrating over the depth range, the distance
travelled by the
ray is given by
Figure 2.5 Incidence angle of a ray
When we specifies the integration over z varying between
, Z is the maximum depth of
penetration. The factor of 2 arises from the symmetry of the down-going and up-going
portions of the ray path (see Figure (2.1)), and equation
114
is now given by
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
This is the where of ray equations; given the angle at which a ray leaves the source, we can
calculate where it will arrive. The time it takes for the ray to arrive is obtained similarly
Where
is the travel time along the ray path to the distance
.we note also that change
in travel time with distance is equal to the ray parameter and is given by
.
Spherical Earth
Let’s consider, in the equatorial plan a ray path in spherical earth Figure (2.6). If we denote by
and
the polar coordinates of a point on the ray and by
coordinates of its vicinity on the ray.
Figure 2.6 Polar coordinate system for a
ray in equatorial plan
115
and
the polar
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
If
is the length between those two points, measured along the ray, we may write
Taking in account spherically symmetric case, the velocity is radial dependence only
, then
If
is parallel to the unit vector position and
is the epicentral distance and T is the correspondent travel time, we have
The formulas for travel time
and epicentral distance
note that an element of ray length is related to a change in radius by
the incidence angle.
By using
we obtain
116
are easily obtained if we
where is
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
And
In the case when the source and receiver are located on the surface see Figure (2.5), we give
the expression of the travel time
radius and
and epicentral distance
, where
is the smallest distance from the ray to the center of the earth.
And
Figure 2.7 Ray path in spherical earth
model.
117
is the earth
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
4.2.2 Inversion of travel time
Flat Earth
When the travel time data are interpreted in terms of ray theory, velocities inside the earth can
be inferred (Aki and Richards, 2002).
Consider the equation
And
for a given ray
By integrating over
At the surface, the ray leaves the source with a velocity
point from the source, the integration range between
, and
and
, at the deepest
. Equation
is written
in the standart form of Abel’s integral equation (Gorenflo and Vessella,.1991). The solution
of inverse problem can be written as (Aki and Richards,. 2002)
Where
Integrating again over and using
, we obtain:
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
By integrating
and using the fact that,
and
, we
finally get the expression of depth- velocity dependence:
Spherical Earth
Inversion of travel-time in the case of spherical Earth is obtained from
:
This expression is known as the Herglotz-Wiechert formulas, its gives the value of
given value of
which lead to the determination of one point velocity profile
for a
.
The Herglotz-Wiechert formula is used for inverting a travel-time curve to find velocity as a
function of depth. It was used extensively in the development of the earliest P-wave and Swave velocity models for the deep Earth structure. The procedure is stable as long as
continuous, with
is
decreasing with . If a low-velocity zone is present at depth, the
formula cannot be used directly, although it is possible to "strip off' layers above the lowvelocity zone and then use the contracted travel- time curve to construct smoothly increasing
velocities at greater depth. Thus, one could build an Earth model for the mantle, and then strip
this off before determining the velocity structure of the low-velocity core (Lay terry.
Wallace,1995). Among the most common problems in seismology is the prediction of the path
taken by the seismic rays from the source to reach the receiver. The presence of lateral
velocity variations of the seismic waves makes it even more complicated task. Difficulties
related to the precision of a path between two points resulting from non-linear relationship
between velocity and path geometry. In a homogeneous medium, the path would be simple
and not undergoing any deviation. Unfortunately it is not what we have in reality.
The Methods that have been adopted are known as ray tracing methods (Julian and Gubbins,
1977; Cervený 1987; Virieux and Farra, 1991; Cervený, 2001), in which the trajectory of the
path corresponding to the wavefront is calculated between two points. This approach is often
very precise and efficient, and lends itself naturally to the prediction of various properties of
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
seismic waves. However, it may not always converge especially in highly heterogeneous
medium. In this thesis, we describe two main techniques for ray tracing to solve the boundary
value problem of locating rays between the source and receiver. The first method computes
the shortest path and is known as a shooting method, the second method is a bending tracing
method. The shortest path method is relatively simple, considered as an initial value problem
which allows a ray path to be traced given an initial trajectory of the path (with appropriate
application of Snell's law), taking into account its initial trajectory. The challenge is to find
the initial direction of the ray source, the bending method is a technique in which we apply an
iterative process in order to adjust the geometry of the path that connects the initial arbitrary
source and receiver until it becomes a true ray path. The LOTOS-10 code (Koulakov,.2009) is
a ray tracing algorithm, similar to the Um and Thurber (1987) algorithm, in which they
attempt to minimize directly the travel-time as a function of the ray curve. For this purpose,
they represent the ray by a number of points, connected by straight line segments Figure (2.9),
based on the Fermat’s principle of travel time minimization. Because rays are the normals to
the wavefront Figure (2.8), the rays will also change with time. Fermat's principle governs the
geometry of ray paths. This usually means that the ray will follow a minimum-time path, the
technique guarantees for the path to be the shortest one in term of timing, a very efficient
algorithm to determine the shortest path in networks was proposed by (Dijkstra., 1959), used
after to compute a tomographic model by (Nakanishi and Yamaguchi 1986), broadly modified
in network ray tracing (Moser, 1991) trying to “bend” the ray , in order to obtain a more
precise estimate of travel time (Moser et al, 1992), called bending ray tracing or just bending
method. It’s used to solve boundary-value ray tracing problems. In our case, the bending
method may be applied as a postprocessor, correcting the preliminary trajectories. This
correction procedure is also known as bending the rays (Cerveny, 2000).
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.8 Ray and wavefront geometry.
4.2.3 Shortest travel-time path
To compute the shortest travel-time path, Figure (2.9) we consider the rays to be composed on
finite segments with a set of discrete angles, to get shortest travel time in general involves
changing the segments coordinates
of each node, in order to enhance the accuracy
of travel time determined, this is in the method of shooting, rays are computed that leave point
S in different directions by solving the ray tracing equations until one ray happens to arrive in
R.
Figure 2.9 An example of a shortest path followed by a seismic ray
traveling from a source S to receiver R.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
We discuss here the calculation of one selected ray, corresponding to specified positions of
the source S and receiver R, we consider the ray traveling from S to R, and we are searching
for the shortest path followed by the ray to reach the receiver R coming from a source S,
Figure (2.9), we note by
the length of segments or roads between the neighboring nodes,
it’s obvious the travel time performed between S and R is a function of velocities fixed
beforehand as initial velocity model ( note that each node can only be visited one time).
Let’s now consider
locations
the average velocity between two nodes; the nodes
satisfy a set of linear equations for a minimum travel time. The total travel
time thus can be expressed as
where N is the total number of segments of length
, and for vertex
and
According to Fermat’s principle, say that the travel time of a ray between two given points in
space must be stationary for small perturbations in the path followed by the ray, thus
for
varies between 2 and
.
where
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
and
,
Figure 2.10 Piece of the path shown
in Figure (2.17).
Finally this gives a system of equations of the form
4.3 Bending method
Great advances were made in seismic tomography when bending ray tracing has been
introduced for the first time by (Wesson,. 1971), there are two types of the bending method,
the first one applied by (Julian and Gubbins, 1977), in which the ray tracing equations are
approximated by discritezing the ray path into a number of points along the path, and the
second one introduced by (Um and Thurber., 1987), when they attempts to minimize directly
the travel time as a functional of the ray curve. One of the drawbacks of the Julian’s approach
is that applying the algorithm makes lost precision by using the finite-difference approach; by
contrast the second variant of the bending method makes the precision of the integration
increasing with the number of segments. In the bending method, some initial guess of the ray
is perturbed while source and receiver points are kept fixed until it satisfies the ray equations
or minimizes the travel time (Nolet ,. 2008), so that, velocity discontinuities are easy to handle
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
with bending method algorithm, with a considerable precision, which can be considered
necessary for many applications in seismic tomography, furthermore the method of ray
bending is preferred over shooting method because it’s yields the travel time of the diffracted
ray, even if the receiver is in the shadow zone where no "physical" rays arrive from the source
(Moser et al., 1992).
LOTOS code use a similar approach of bending tracing method with a slight modification,
(Koulakov, 2009), the ray constructed in this way tends to travel through high-velocity
anomalies and avoids low velocity patterns Figure (2.11), it can use any parameterization of
the velocity distribution and can define uniquely one positive velocity values at any point of
the study area.
Figure 2.11 Process of bending algorithm used to determinate the shortest path.
Hatched light grey patterns represent negative anomalies of -30%; dark grey
patterns are positive anomalies of +30%. (Koulakov, 2009).
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
4.3.1 Pseudo bending method
If
represents the position vector of a point on a wavefront, and
the path length of the curve
traced out by this point as the wavefront evolves, then
The travel time can be expressed as
Where
represents the path, the rate of change of travel time along the path is simply the
slowness, so
and by taking the gradient of both sides , we can derive the following expression
From differential geometry, the tangential and normal unit vectors
position Q indicated by the vector q Fig (2.20) are given as
The normal vector to the ray is given by
125
and
of a ray at the
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.12 Tangential, normal and anti-normal unit
vectors along the ray path (Kazuki et al,. 1997)
Knowing and
If we now let
, we derive the relation that express
in terms of
and , so
represent the anti-normal unit vector to the ray path at point , and substitute
velocity for slowness , then
Vector
defines the direction of path curvature. To update points along a path, we need three
points along this path Fig (2.20),
,
,
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.13 Three point perturbation scheme used in pseudo
bending method, Um & Thurber (1987).
The aim is to replace the initial guess point
with an improved estimate
improved estimate is obtained by considering a perturbation to the point
the midpoint between
and
. The
, which lies at
. The vector can then be simply approximated by
and the anti-normal unit vector , which specifies the bending direction, is computed from
, The next step is to find the distance
in the direction
which results in an improved
estimate of the path. An approximate analytic expression for the travel time
and
between
can be obtained using the trapezoidal rule (Rawlinson et al,. 2008)
The appropriate value for R can be obtained by appealing directly to Fermat’s principle of
stationary time, which in this case equates to setting
127
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Where
128
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
5. Seismicity of the Gulf of Cadiz
The Gulf of Cadiz, located at the southwestern Iberian margin and NW of the Moroccan
margin Atlantic, is characterized by widespread seismicity, compressional and strike-slip fault
plane solutions (Gràcia et al 2003),(Geissler et al 2010), Instrumental seismicity in the Gulf
of Cadiz, can be considered of moderate magnitudes and shallow to intermediate depth,
mostly above 60 Km. Most events occurring between
W and
W are located at shallow
depths between 5 and 25 Km, and can sometimes reach 90 Km in the oceanic domain
(Grandin., et al,. 2007), Since the 1969 earthquake, no other event has exceeded magnitude 6
in the region. Seismic catalogues report seismicity down to 100 km depth. According to
structural models for the region, this includes earthquakes that occur in the uppermost
continental mantle (Stich, et al 2005). Seismicity in the west is located around the Horseshoe
Abyssal Plain in offshore lithosphere; it’s restricted to a relative narrow zone of 100 Km and
the Moho depth of about 15 km (Tortella et al., 1997; Rovere et al., 2004). The Seismicity is
concentrated along the Azores–Gibraltar Fracture Zone, in the Azores and Gloria Fault, it
becomes disperse in the Gulf of Cadiz and Gibraltar Arc, indicating the more diffuse nature of
the contact in this area (Maldonado et al ,.1997) and a complex behavior of seismicity was
shown in the Gulf of Cadiz, where shallow events (h < 30 km) occur (Buforn et al ,. 2004).
5.1 Nearest (integrated observations from near shore sources of tsunamis: towards an
early warning system)
Nearest (integrated observations from near shore sources of tsunamis: towards an early
warning system) is a scientific project that was sponsored by the european conncil (EC) and
which aimed to study the Gulf of Cadiz region, which very complex area in the Atlantic
Ocean in this project which 24 ocean-bottom seismometer’s (OBS) were deployed, for a
period of one year in 2007-2008, the NEAREST consortiums consisted of the partnership of
several institutions:
1) Centre National pour la Recherche Scientifique et Technique, Morocco.
2) Consiglio Nazionale delle Ricerche, Istituto Scienze Marine, Dipartimento di Bologna,
Italy.
3) Fundação da Faculdade de Ciências da Universidade de Lisboa - Centro de Geofísica da
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Universidade de Lisboa, Portugal.
4) Consejo Superior de Investigaciones Cientificas – Unitat de Tecnologia Marina - Centre
Mediterrani d’Investigacions Marines i Ambientals, Barcelona, Spain.
5) Alfred-Wegener-Institute fur Polar-und Meeresforschung, Geophysics section, Germany.
6) Université de Bretagne Occidentale, UMR 6358 Domaines Océaniques, France.
7) Istituto Nazionale Geofisica e Vulcanologia, Roma 2 section, Roma, Italy.
8) Technische Fachhochschule Berlin - FB VIII - Maschinenbau, Verfahrens- und
Umwelttechnik AG Tiefseesysteme, Germany.
9) Instituto Andaluz de Geofísica - Universidad De Granada, Spain.
10) Instituto de Meteorologia, Divisão de Sismologia, Lisbon, Portugal.
11) XISTOS Développement S.A, Paris, France.
The scientific OBS deployment whiting the nearest project was performed between 16th
August and 4th September 2007 offshore Cap Sao Vicente and in the Gulf of Cadiz, in the
Portuguese and international waters. The main goals of the cruise were the deployment of a
multiparameter seafloor observatory (GEOSTAR), its communication buoy and an array of
the bottom seismeters. In addition, subbottom profiles, multibeam data and seafloor sampling
were collected. These data will improve both the geological and the geophysical knowledge
of the tectonic architecture of this area, presumed to be the source of the 1755 Lisbon
Earthquake.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.14 Seismicity of the Gulf of Cadiz as recorded between august 2007 and July 2008 as shown by
the red dots;, GB : Gorringe Bank , CP: Coral Pach, , SVC: Sâo vicente Canyon, RV : Rharb Valley , PB:
Portimâo Bank, AB: Algarve Bassin, AJB: Alentijo Bassin , Ocean Bottom Seismometers (OBS) blue
triangles and Portugal Land stations green triangles.
The cruise was split in two legs cause of the large volume of instruments to be deployed: the
first leg was from 16th of August until the 27th of August and the second one from 28th of
August until 4th of September 2007.
Tectonic structures in the transition from the Azores fracture zone to the postulated
subduction zone in the area of the Strait of Gibraltar will be localized and characterized. The
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
structure have the potential to cause Tsunamis. For this purposes, we deployed 22 broadband
ocean bottom seismometers (OBS) Figure (2.15) from the German DEPAS instrument pool
coordinated by the Alfred Wegener Institute for Polar and Marine Research, Bremerhaven and
the GeoForschungsZentrum, Potsdam. Seismicity studies and passive seismic imaging
techniques will be performed after 12 months recording, when the OBS have been recovered.
During the transfer among the OBSs Chirp and Multibeam data were collected. Despite the
project to build an array of 24 OBSs, cause technical problems during the second leg, it was
possible to deploy only 22 instead of 25 planned initially.
Figure 2.15 Ocean Bottom Seismometer’s on board, (Zitellini N., Carrara G. & NEAREST
Team. - ISMAR Bologna Technical Report, June 2009).
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Thus, within the NEAREST project, 24 Ocean Bottom Seismometers (OBS) were deployed in
the Gulf of Cadiz to collect seismic data during this project experiment for a period of one
year from August 2007 to July 2008. Figure (2.16) shows the location of the seismometers.
They were equipped each with a Güralp CMG-40T broadband seismometer incorporated in
titanium pressure housing, a hydrophone, and a GEOLON MCS (Marine Compact
Seismocorder). The electric power supply for the recorder and the seismometer is driven by
132 lithium power cells. Each sensor channel is sampled with 100 Hz, preamplifier gain of the
hydrophone channel is 4 and 1 for the three seismometer components (Geissler et al, 2010).
Figure 2.16 Location of the broad band stations used in this study, ocean Bottom
Seismometers (OBS) blue triangles and Portugal Land stations green triangles.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
5.2 Data format
The most and basic challenge to solve in seismology is to locate the recorded seismic events.
For that purpose geophysicists need at least 3 stations. We therefore precise an OBS’s
network as being a group of stations working together to listen to what that is happen under
the ground. The OBS’s recorded seismic data are collected in SEED format (Standard for
Exchange of Earthquake Data). SEED is the most common International used waveform
format. There are thus many ways of representing the response information, SEED is the most
complete and best defined but also the most complex. However, the most standard open
format currently available is the MiniSEED format used by Quanterra recorders and a few
others. It is also an agreed upon international format; MiniSEED is similar to SEED format of
data only with heathers; the present the propensity is that newer recorders use MiniSEED, and
can accommodate a huge mass memory capacity. In this project (NEAREST) all events
recorded by the OBS’s and eight Portugal land-stations were extracted for processing and
analysis, in order to determine their hypocenters and local magnitudes. The first work we
have to do is to choose the program for processing our data.
5.3 Seisan analysis software
The early improvement in seismic sensors and experiment provided very large quantity of
data. the source locations and origin time becomes very useful for several studies to cover
different area. The data used in this study come from Ocean Bottom Seismometers digital
record. after that we have to choose a program to use to analyze all recorded events, The
earliest, formal earthquake locations use arrival time information from seismic phases applied
in direct-search procedures such as graphical methods (Milne 1886) or simple grid-searches
(Reid, 1910). Presently, earthquakes location can be adequately computed using modern
location techniques and there are numerous advanced location programs in use and most are
free standing. However, in some cases they are also tightly integrated with general processing
software like HYPOCENTER in SEISAN. and most programs are made for local events.
HYPO71, was first released in 1971, and was program designed to determine hypocenter,
magnitude, and first motion pattern of local earthquakes. HYPO71 is still a quite used
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
program. (Lee and Lahr, 1975). HYPOINVERSE, is among location programs used to
determine earthquake locations and magnitudes from seismic network data like first-arrival P
and S arrival times, amplitudes and coda durations.
HYPOCENTER can use nearly all seismic phases and azimuths of arrivals and calculates
most magnitudes. It is the main program used in SEISAN for earthquake location and
magnitude determination (Lienert and Havskov,.1995). There are also many programs that are
currently used like HYPOELLIPSE, HYPOSAT, PITSA, SAC and SHM analysis software
(Lars, 2009). We have seen all these common processing software listed above; we choose to
analyze our data using the SEISAN program. As well known, the exact location of a source,
radiating seismic energy, is one of the most important tasks in practical seismology and most
seismologists have been from time to time involved in this task. For this purpose and to avoid
the errors in the arrival time picks, we have consecrated a rigorous attention when we are
picking P and S phases. The SEISAN seismic analysis system is a complete set of programs
and a simple database for analyzing earthquakes from analog and digital data. Then we have
chosen the SEISAN processing Program for these reasons. Furthermore, SEISAN has several
advantages. It is mostly designed for local events, and also there are several options integrated
in the program, such as the VELEST routine that can be used via SEISAN.
With SEISAN it is possible for local and global earthquakes to enter phase readings manually
or directly pick them with a cursor on seismograms, events can located, edited, determine
spectral parameters, seismic moment, determining the azimuth of arrival from 3-component
stations and plot epicenters. The system consists of a set of programs tied to the same
database. Using the search programs it is possible to use different criteria to search the
database for particular events and work with this subset without extracting the events
(Havskov et al 2010). One of the reasons why we have chosen SEISAN is that most programs
are made for local events, which is the case our study, we are not interesting in analyzing of
teleseismic events.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
5.4 Seismicity
The OBS Network allowed the detection of a large number of local events that could not
otherwise be detected with the Morocco, Spain or Portugal land stations. These are Local
earthquakes with magnitude
. After collecting the data, we have proceed to location of
the events recorded using both the OBS and Portuguese land stations. It is clear that the
seismic rays travel through a part of the continental crust to reach the land stations. The
influence of the continental crust is taken into account when we are picking the phases of the
first arrivals, to which we have assigned a different weight to promote the localization by
OBS’s for the offshore events. The initial hypocentral locations were performed using the
SEISAN software package (Havskov, J. et al.,1999),
The OBS network revealed that the seismicity in the study area is characterized by a nearlycontinuous activity of low magnitude events. The
that
occurred on January 1st, 2008 earthquake is the largest event recorded by the OBS network
and the Portugal seismic stations used in this observation period. The large number of
earthquakes recorded in this one-year time period shows that the Gulf of Cadiz is a quite
active area and testifies of the great sensitivity of the deployed OBS network, This activity is
largely due to the convergence between the plates and is distributed over a wide area of
deformation. In southern of Iberia, numerous active thrust faults were identified (Zitellini et
al., 2001; Gracia et al., 2003,
Zitellini et al., 2004) and associated with instrumental
seismicity, including the Horseshoe fault, the Marquês de Pombal fault and Sâo Vicente fault,
while the Tagus and the Seine Abyssal Plains are remaind aseismic in this period (Lbadaoui et
al 2012). In general seismicity in the Gulf of Cadiz and south Iberia peninsula, is
accommodated by structures located north of the SWIM Fault Zone. Figure (2.17) shows that
the seismic activity concentrates in three main blocs; namely clusters around the south of the
Gorringe Ridge, the eastern part of the Horseshoe Abyssal Plain; and, north and NW of the
Horseshoe fault, near and along the Marques de Pombal plateau and the Sâo Vicente Fault.
Few earthquakes occurred in the accretionary wedge zone while dispersed seismicity is found
south of the Algarve Bassin and Portimâo Bank Figure (2.17).
The determined local
magnitudes showed that approximately 61 % of the events have local magnitudes less than 2,
and 22 % have magnitudes between 2 and 3. The depth distribution of the hypocenters varies
136
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
from shallow to intermediate and show that the majority of the located events are between 20
and 80 Km depth Figure (2.19), Figure (2.20) , with very few events with depths shallower
than 20 km.
Figure 2.17 Seismicity of the Gulf of Cadiz as recorded between august 2007 and July 2008 as shown
by the black circles; AB: Algarve Bassin, AJB: Alentijo Bassin the inclined blue line represents the
SWIM faults zone (SFZ), and red lines are the possible faults.CP: Coral Pach, GB : Gorringe Bank , GF :
Gorringe fault , HsF : horseshoe fault ,MPF : Marques de Pombal fault , PB: Portimâo Bank, PSF: Pereira
de Sousa fault, RV : Rharb Valley , SVC: Sâo vicente Canyon, SVF: Sâo vicente fault,
137
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Also, a NW-SE trend of seismicity that crosses the horseshoe fault can be observed to the
south of our study area Figure (2.19) and seismic profile 2 on Figure (2.20). This seismic
alignment seems to occur rather in segments of seismicity, with depths that vary between 35
and 80 km. This trend of seismicity is nearly along and parallel to the SFZ and may thus, be
correlated to the SWIM faults, which were mapped on the surface, mainly based on
bathymetric scanning. These profiles further reveal the thrusting geometries in this area,
evidenced by active faulting and associated processes.
Figure 2.18 Seismic profiles shown in the map Figure (2.9), showing events that occurred
within 40 km distance from the profile. This profile indicates a more continuous pattern of
seismicity. HsF: Horseshoe fault, GB: Gorring Bank.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
The first seismicity cross section Figure (2.19) shows three main blocs of earthquakes along
the Gorringe bank, Sâo Vicente fault and the accretionary wedge, while the second profile
shows that seismicity concentrates in two areas, in the Gorringe bank and horseshoe fault. On
profile 2, we can identify a pattern that seems to indicate a thrusting shape of the horseshoe
fault, with a slope of about 50 km in length.
GB
SVF
AW
Figure 2.19 Seismic profiles shown in the map Figure (2.26), showing events that occurred
within 40 km distance from the profile, this shows three separate clusters of seismicity. AW:
Accretionary wedge, GB: Gorring Bank , SVF: Sao Vicente fault.
GB
HsF
Figure 2.20 Seismic profiles shown in the map Figure (2.26), showing events that occurred
within 40 km distance from the profile, this shows three separate clusters of seismicity. AW:
Accretionary wedge, GB: Gorring Bank , HsF: horsechoe Fault.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
5.5 Inversion method and procedure
A tomographic algorithm, LOTOS-10 (Local Tomography system of program) is designed for
simultaneous inversion for P and S velocity structures and source coordinates. The LOTOS10 algorithm can be directly applied to very different data sets without complicated tuning of
parameters. It has a quite wide range of possibilities for performing different test and is quite
easy to operate.
Figure 2.21 Chart showing the General structure of the LOTOS code
working process.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
LOTOS code is easy to use, it consists of many of folders and files well organized, the main
folder contains eight folders and four files:
COMMON folder: contains subfolder for the control of the display.
DATA folder contains:
Model_01 folder contains two files, this folder contains :
1) MAJOR_PARAM : contain the parameters for source location and inversion.
2) ref_start : contain the starting velocity model using for inversion.
MAJOR_PARAM is a file containing most of the settings parameters (e.g. sources location
and conversion settings), the file control all subroutines written in FORTRAN language,
through specified the keys introduced in the file. In this file we can specify the study area
(geographical coordinates) in
AREA_CENTER rows, it allow to set the choice of the
inversion of P and S waves or
, and if is it synthetic or the actual data. In this file we can
set all the parameters needed for optimization of the initial model. To set the parameter of
damping and noise effect, you have to perform a several series of synthetic test, in this work a
selective settings of the data have been made, we don’t have a huge quantity of data, but we
have an excellent coverage area Figure (2.22), Figure (2.23), our results comes from a set in
which the residue should not exceed 1, and the maximum number of outliers must not exceed
30%, if it is not the case the event is rejected. For the 3D setting parameter, we have chosen
the parameterization with nodes rather sells grid construction , and performing the process of
inversion in multi-orientation direction, this orientation deals with our seismicity trend, we set
the minimum spacing between nodes at 5 km, the nodes are Imbricates on vertical lines
distributed regularly in the study area previously chosen; the nodes are installed according to
the density of the ray existing and first computed using the bending ray tracing theory (see
bending tracing).the velocity distribution is approximated linearly between two adjacent
nodes. The file ref_start is a file with extension .dat, below the reference model used in this
inversion process.
1.79
0.000
5.500
10.000
Ratio
3.10
5.58
6.75
vp/vs
1.73
3.11
3.77
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
24.000
31.000
90.000
500.000
7.50
8.13
8.38
10.00
4.19
4.54
4.68
5.58
This file consists of three columns, the first one contain de values of the chosen depth , and
the second one contain the velocities in km/s and the third one contain the ratio
.
Figure 2.22 Ray paths in the map view at depth of 20, 40, 60 and 80 Km showing the
coverage paths, purple point are the stations.
Figure 2.23 Ray paths in the map view in a vertical cross section shown by grey dots.
Blue triangles are the stations.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ

Inidata folder when we have to put the two files ray.dat and stat-ft.dat in appropriate
file format.
EXAMPLE OF LOTOS INPUT FILE FORMAT ( ray.dat )
-10.84300
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
2
1
2
1
2
2
1
2
1
2
2
1
2
-9.699000
1
2
1
1
2
1
2
1
1
1
1
2
35.88600
26
26
21
21
45
45
51
51
41
41
31
31
18
18
22
22
15
15
40
40
16
12
12
47
47
23
11
11
13
13
19
43
43
36.68900
14
14
18
13
13
15
15
22
23
21
17
17
42.10000
5.988281
11.06836
7.439453
14.17969
7.410156
14.16992
9.208984
15.97852
9.660156
16.83008
11.95898
20.46875
12.59961
22.70898
12.81836
23.11914
14.92969
26.10937
17.86914
30.69922
30.56836
18.59961
31.86914
19.83984
35.70898
36.09961
20.41016
36.54883
21.13867
36.54883
51.69922
29.39844
50.66016
15.00000
7.101562
12.64062
10.75000
10.67969
19.74219
12.40234
22.23047
12.55078
12.94922
15.97266
16.49219
29.26953
143
33
26
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
1
2
1
2
1
1
1
2
1
2
1
2
1
2
28
28
27
27
29
26
30
20
11
11
31
31
32
32
17.19922
31.30078
17.76953
31.69922
19.08984
21.69141
21.57031
38.62109
22.50000
39.92969
22.06250
39.74219
24.83203
45.50000
The second file in the folder inidata is stat_ft.dat, this file contains the geographical
coordinates of the stations.
EXAMPLE OF LOTOS INPUT FILE FORMAT ( stat_ft.dat )
-07.03900
-07.93117
-07.86633
-11.44995
-10.73435
-10.22993
-9.700130
-8.800130
-8.249570
-10.93892
-10.34030
-9.750400
-9.100180
-8.599950
-10.40025
-9.285130
-9.282530
-9.573120
38.17450
37.24300
38.02633
37.05038
37.02558
37.10048
36.94997
35.99980
35.94983
35.77972
35.70988
35.62993
35.59978
35.64973
35.35015
35.11682
36.53183
36.36078
-0.20500
-0.47100
-0.27000
4.800000
2.269000
3.935000
1.980000
3.360000
2.061000
4.764000
4.605000
4.394000
3.442000
2.575000
4.101000
4.745000
1.234000
1.234000
PBAR
PBDV
PBEJ
n0701
n0702
n0703
n0704
NRT15
NRT16
NRT17
NRT18
NRT19
NRT20
NRT21
NRT22
NRT23
NRT24
NRT25
Here above we show an example of an input file, each event is composed of one line (head)
and three columns, the line contains the coordinates of the epicenters, earthquake depth and
number of phases used (33 for the first event and 26 for the second), the first column contains
two possible index (1) for the
phase and (2) for , the second column contains the number
of stations used in locating the events, and finally the third column contains the travel time,
the second is the file containing the station coordinates (Latitudes , Longitudes and
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Elevations). It is necessary to generate this file in this exact format to be accepted and red by
the LOTOS code, in our case, and for those who use SEISAN (Havskov and Ottemöller,
1999) for locating seismic events, input file ray.dat is automatically generated by SEISANLOTOS format conversion given in appendix.
Map folder : contains two files :
Coastal_line file to allow determination of coastal limits in maps.
Polit_bound file to allow determination of politic boundaries between different countries.
FIG_files folder : with subfolders contains GRD, BLN and DAT file which represent the
intermediary and final results. They can be used directly in SURFER for visualization.
PICS folder : with subfolders contains PNG bitmap files with previewing the results
PROG_1D_MODEL folder : folder with programs for preliminary location of sources and
1D velocity model optimization.
PROGRAMS folder : contains all the source code Fortran.
Subr folder: folder which contains all the subroutines. It is necessary only if re-compiling of
the programs will be performed.
Tmp folder: contains temporary files which are used only for current calculations.
all_areas file : file which defines areas and models to be processed for inversion.
model file : file with information about currently processed model.
preview_key file : with a key for previewing should be also defined.
START file: command to start.
6.5.1 Using the LOTOS code
To use the code, you must first have the seismic data. input files for our calculations include a
list of the stations with coordinates and elevations, and a catalog of the arrival times from
local earthquakes, in our case, the work was done by exploiting a huge database of seismic
data recorded by OBS’s in the Gulf of Cadiz, the data are collected in the SEED format, then
converted to SEISAN format for processing (Havskov, J. et al.,1999). After that all events
occurring in the period between 2007 and 2008 in Gulf of Cadiz are extracted and grouped
using SEISEI from different seismic networks (Ocean bottom seismometers and Portuguese
land stations). SEISAN is the software that provides a complete set of programs written
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
mainly in Fortran and a simple database to analyze digital data. The program allows a perfect
management of the database of seismic events, and accurate analysis especially if the data are
of good quality. All seismic data recorded by 24 stations OBS has been analyzed by manual
picking arrival times of
and phases.
More than 600 events was analyzed, we used the VELEST program for data inversion
(Kissling, 1994) to derive velocity models in one dimension as the starting velocity model, as
well known, achieving good results depends on carefully choosing the initial model and we
have defined all free parameters such as amplitude, damping and smoothing, by checking a
series of synthetic tests (see appendix). Then calculations begin with preliminary earthquake
locations in a 1-D velocity model. We used a starting 1-D velocity model obtained when the
database was relocated using the VELEST inversion algorithm (Kissling et al,. 1994) we
further tested several other initial velocity models (cf. to section V below). At this stage, the
source coordinates and origin times are determined using a grid search method (Koulakov et
al., 2006).
Several causes can produce errors when trying to locate sources, for example the outliers in
the initial data set, incorrect identification of phases, or an excessive residuals related to
wrong picking of phases, for the purpose of good source location performing and avoiding
these problems, the method allows the optimization of a Goal Function (Koulakov and
Sobolev, 2006) that reflects the probability of the source location in the 3-Dimensional space.
Model travel-times are computed using tabulated values calculated in the 1-D velocity model
at a preliminary stage, and matrix inversion for P and S velocities and source parameters are
performed using a least-squares method (Paige, C et al,.1982),( Van der Sluis, A et al,. 1987).
5.5.2 One dimensional velocity optimization and preliminary source location algorithm
The inversion of seismic data requires a credible reference model, data are selected for
optimization. From the entire events, only data distributed as uniformly with depth are
selected. This is done by selecting for each depth interval the events with the maximum
number of recorded phases. Then a travel time table in a current 1D model is calculated. the
model is defined manually according to priori previous research study. Travel times between
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
sources at different depths to the receivers at different epicentral distances are computed in a
1D model using this analytical formula
Where
is the ray parameter (Nolet, 1981). The algorithm allows the incidence angles of the
rays to be defined in order to achieve similar distances between rays at the surface.
When the travel times are defined they are corrected for elevations of stations and then the
sources are located based on calculating a goal function (Koulakov, Sobolev, 2006). This
location algorithm is very stable. For example, it can find the correct source coordinates even
if it is located at a distance of 400-500 km from the initial searching point. After that, the
calculation of the first derivative matrix along the rays is performed. Each element of the
matrix
the
is equal to the time deviation along the
ray caused by a unit velocity variation at
depth level. The depth levels are defined uniformly and the velocity between the levels
is approximated as linear. Matrix inversion is performed simultaneously for the P and S data
using the matrix previously computed. In addition to the velocity parameters, the matrix
contains the elements to correct the source parameters (
). The data vector
contains the residuals. Regularization is performed by adding a special smoothing block. Each
line of this block contains two equal non-zero elements with opposite signs that correspond to
neighboring depth levels. The data vector in this block is zero; increasing the weight of this
block smoothes the solution. If there is a-priori information about existence of interfaces (e.g.
Moho), it can be included in the inversion. In this case, the link between the pair of nodes just
above and below the interface would be skipped. Optimum values for free parameters
(Smoothing coefficients and weights for the source parameters) are evaluated on the basis of
synthetic modeling. The inversion of this sparse matrix is performed using the LSQR method
(Page, Saunders, 1982, Van der Sluis, Van der Vorst, 1987). A sum of the obtained velocity
variations and the current reference model is used as a reference model for the next iteration.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
5.6 Starting velocity model
In general case; nonlinear parameter estimation problems can often be solved by choosing a
starting model and then iteratively improving it until a good solution is obtained. In this work
we start with the model proposed in the 2008 NEAREST-cruise and was improved
subsequently (Matias,, et al., 2009). Afterwards, all recorded events are localized with the
resulting appropriate initial velocity model obtained using the VELEST inversion algorithm
(Kissling, et al., 1994). The data set thus collected, consists of more than 600 local events
Figure (2.24). The quality of the initial earthquake locations is rather high, with an RMS
travel time residual less than 0.8 s. These events allowed us to obtain a total of 9194 arrival
times, consisting of 2968 P-wave and 6226-S wave arrival times.
Figure 2.24 Earthquakes location of more than 600
events recorded during NEAREST cruise survey.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
One dimensional initial velocity model optimization
The inversion process requires an initial 1-D velocity model; we used different starting
velocity models in order to select one which gives optimal locations. Thus, we tested seven
initial velocity models, including the one yielded by the VELEST algorithm Figure (2.25, a).
(Kissling et al., 1994), in addition to several variants of these initial models
GULF OF CADIZ VELOCITY MODELS
10
0
-10
-20
-30
-40
DEPTH (km)
-50
-60
-70
-80
-90
-100
-110
Legend Title
-120
OBS model
-130
OBS + Portugal mod
-140
Portugal Model
-150
5.50
6.0
6.50
7.0
7.50
8.0
8.50
9.0
P-wave velocity (km/sec)
Figure 2.25.a Velocity model obtained by the VELEST algorithm , the black line
plot represent the model given by OBS location , the red line represent the model
given using both OBS’s and Portugal land station and the blue model is given using
only the Portugal land stations.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.25.b Different starting P-velocity models used for optimization of
the initial velocity model; model 1 is the velocity model proposed in the
NEAREST-2008 cruise report, model 3 is the model derived using
VELEST, while models 2,4,5,6 and 7 are initial-velocity models with slight
modifications of the previous ones.
Based on the analysis of the resulting velocity distributions and the RMS of residuals, we find
that the 1-D velocity model which gives the least RMS residuals in the Gulf of Cadiz region
corresponds to model 3 presented in Table (2.1) and Figure (2.25.b), which is the model found
using the VELEST inversion algorithm (Kissling et al 1994),. This model consists of a crust
with three layers with an interface at ~6 km depth, and the Moho at 30 km depth. Using the
LOTOS software, Model 3 was then optimized and yielded the model given in Table (2.1).
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TAB 2.1 P and S velocities in the reference 1-d model after
Optimization by the lotos software.
Vp (Km/s)
Vs (Km/s)
0
2.713239
0.919177
10
6.093362
3.755473
20
6.839791
3.845567
30
7.766315
4.629175
40
8.243161
4.554774
50
7.922880
4.675756
60
7.808963
4.645917
70
7.722842
4.563402
80
7.685386
4.518101
90
7.939282
4.506800
100
8.103004
4.549830
Depth (Km)
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
6. Synthetic tests
A rather common way of assessing the reliability of tomographic models consists of so-called
synthetic tests, where the ability to image a known input model often a regular pattern of
alternating positive velocity variations is tested, such test mainly addressed the resolution of
the inversion scheme. It is a reliable technique that helps examine the resolution of the used
data, before inversion, we performed several series of synthetic tests in order to get the
optimum parameterization parameters, the average amplitude of noise was defined at 0.1 s
and 0.1 s for P and S waves, respectively. Figure (2.26) and Figure (2.27) show the result of
our synthetic test and checkerboard test respectively , in map view at depths between 25 and
50 km and in two vertical sections AA’ and BB’, it can be seen that in the upper section the
periodic anomalies are reconstructed in most parts of the study area, the Figures bellow show
the results for this tests, the resolution is much higher in area bounded by, the Gorringe ridge
in the west, Saô Vicente Cap in the north, Portimaô Bank in the East and the Coral patch ridge
in the south, especially for S velocity model, We lost reliability of model S-W of the
horseshoe abyssal plain and in the S-E of Coral patch ridge, however, the largest part of the
adequately sampled inversion volume exhibits good resolution of this structure.
(Koulakov, 2009) has used a specific technique to estimate the capacity in terms of
contribution of the noise present in the real data used in the conversion process, this technique
consist in performing the inversion separating the data into two categories, those with odd
index in the list of data, and those with an even index, and perform the process of inversion
for each list separately. In addition to checkerboard test, it is very useful to perform this test to
give more credibility to the tomographic results previously found. The test can give a subtle
interpretation of the results; it also allows you to check the lucidity of obtained anomalies.
The checkerboard test is often sufficient to check the robustness of the tomographic results,
however, the odd-even test events test can provide much additional important information that
can often help to estimate the effect of the use of half of the data, and to compare it with the
results obtained using all data, Figure (2.30, 2.31) show the result obtained using odd and
even events inversion process for P and S waves, the parameterization and grid reconstruction
used in this test is the same used for the real data.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.26 Checkerboard test performed for P and S waves in horizontals sections.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
8
Figure 2.27 Synthetic test performed for P and S
waves in horizontals sections at 35 Km depth.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.28 Checkerboard test performed for P waves in
vertical sections, AA’ and BB’ shown in Figure below.
Figure 2.29 Checkerboard test performed for S waves in
vertical sections, AA’ and BB’ shown in Figure below.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.30 Anomalies of P velocities distribution, test with inversion
of two independent data subsets (with odd/even numbers of events).
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.31 Anomalies of P velocities distribution, test with inversion
of two independent data subsets (with odd/even numbers of events).
From the two last Figures, we can see that this additional test exhibit a good correlation
results in most working area.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
7. Results and discussion
Using the relocated hypocenters in the optimized initial model, the inversion scheme
described before allowed us to invert for the three-dimensional structures beneath the Gulf of
Cadiz, high-quality events were used for the inversion of local earthquake data. Thus, Figure
(2.32-36) show the obtained P-velocity tomograms at selected depths; namely at 10 km, 15
km, 25 km, 35 km and 50 km depths, while Figure (2.37-41) show the S-velocity tomograms
at the same depths. These depths were selected for their importance in terms of velocity
distribution and also because the tomograms have higher resolution at these depths. For
shallower depths and for depths greater than 50 km, the synthetic tests show that the results
are rather not reliable (Appendix).
The main general observation from the tomograms of Figure (2.32-36) and Figure (2.37-41),
is that the limits reveled between the different high and low velocity anomalies in this study
area, strike either in a NE-SW or in NW-SE directions. We further notice that the tomograms
of P and S show more or less similar velocity distributions and features. In addition, we
observe that the S-waves tomograms show a better resolution as indicated by the synthetic
tests (see Appendix). This can be explained by the fact that there are about twice as many Sarrivals than P-arrivals in our earthquake hypocentral phases. At depths 35 km and shallower,
P and S velocity distrubutions show a large negative anomaly trending SE of the study area,
approximately delimited into latitudes 35.1°N and 37°N and between 8°W and 10°W
longitudes, this anomaly coincides with the inverted part of the accretionary wedge and
extends all the way to the Portugal continental margin, at shallow depths, this anomaly can be
attributed to the existence of an area with a large concentration of sediments (the accretionary
wedge), while at greater depths, this anomaly could be interpreted as reflecting, that this part
of the Gulf of Cadiz is rather made out of continental crust and thus, its northern part
delineates more or less the limit between continental and oceanic crust. However, the many
alternances between high and low velocity anomalies in these tomographic images do not
seem to clearly delineate a continent-ocean boundary such as previously reported by Stich, et
al,. (2005).
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.32 P-velocity distribution at 10 km. Solid lines show the
existing faults and dotted lines show possible strike-slip faults, and black
dots show the epicenters location given by tomography program and
inclined gray line represent the SWIM fault zone (SFZ).
Figure 2.33 P-velocity distribution at 15 km. Solid lines show the
existing faults and dotted lines show possible strike-slip faults, and black
dots show the epicenters location given by tomography program and
inclined gray line represent the SWIM fault zone (SFZ).
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.34 P-velocity distribution at 25 km. Solid lines show the existing
faults and dotted lines show possible strike-slip faults, and black dots show
the epicenters location given by tomography program and inclined gray line
represent the SWIM fault zone (SFZ).
Figure 2.35 P-velocity distribution at 35 km. Solid lines show the existing
faults and dotted lines show possible strike-slip faults, and black dots show
the epicenters location given by tomography program and inclined gray line
represent the SWIM fault zone (SFZ).
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.36 P-velocity distribution at 50 km. Solid lines show the
existing faults and dotted lines show possible strike-slip faults, and black
dots show the epicenters location given by tomography program and
inclined gray line represent the SWIM fault zone (SFZ).
Figure 2.37 S-velocity distribution anomalies at 10 km. Solid lines show
the existing faults and dotted lines show possible strike-slip faults, and
black dots show the Epicenters location given by tomography program and
inclined gray line represent the SWIM fault zone (SFZ).
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.38 S-velocity distribution anomalies at 15 km. Solid lines show
the existing faults and dotted lines show possible strike-slip faults, and black
dots show the Epicenters location given by tomography program and
inclined gray line represent the SWIM fault zone (SFZ).
Figure 2.39 S-velocity distribution anomalies at 25 km. Solid lines show
the existing faults and dotted lines show possible strike-slip faults, and black
dots show the Epicenters location given by tomography program and
inclined gray line represent the SWIM fault zone (SFZ).
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.40 S-velocity distribution anomalies at 35 km. Solid lines show
the existing faults and dotted lines show possible strike-slip faults, and black
dots show the Epicenters location given by tomography program and inclined
gray line represent the SWIM fault zone (SFZ).
Figure 2.41 S-velocity distribution anomalies at 50 km. Solid lines show
the existing faults and dotted lines show possible strike-slip faults, and black
dots show the Epicenters location given by tomography program and inclined
gray line represent the SWIM fault zone (SFZ).
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.42 Horizontal sections of P-velocity
anomalies at 15,25,35,50 km depth. (perspective
view)
Figure 2.43 Horizontal sections of P-velocity
anomalies at 15,25,35,50 km depth. (perspective
view)
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
7.1 Swim lineaments (SFZ)
Throughout the south of the study area, the tomographic images show a NW-SE belt of high
velocity anomaly distribution extending from the Gorringe Bank to the Accretionary wedge
crossing the horseshoe abyssal plain Figure (2.44). This anomaly approximately follows and
coincides with the SWIM lineaments (SFZ), which consist of a set of strike-slip faults and
proposed to be a zone plate boundary between Eurasia and Nubia plates (Zitellini et al.,
2009).
The SWIM lineament (SFZ) is trending WNW-ESE and occurs from the western part of the
Gorring Bank, cutting across the Horseshoe Abyssal Plain, and accretionary wedge towards
the Morocco margin Figure (2.44), extends over a discontinuous length of about 600 km
(Terrinha et al. 2009). This SWIM lineament was suggested by several authors to be the
boundary between the Nubia and the Eurasia plates, as taken from GPS measurements
(Nocquet and Calais, 2004).
Recently several studies of high-resolution bathymetry of Gulf of Cadiz area was made,
helped to improve the identification of this strike-slip faults lineament (Zitellini et al., 2009).
In this part of the study region along the SFZ, we find a high velocity anomaly belt (mostly
for S phase) from the surface layers towards to about 35 km depth. This belt of high velocity
anomalies shows a transition zone that exhibited a clear separation of two structures, they can
be interpreted as the result of an oblique compressive deformation of Iberia with respect to the
Nubia plate. Our results thus, seem to support the (Zitellini et al, 2009) suggestion that this
zone constitutes the boundary of these two plates (Eurasia and Africa) within the Gulf of
Cadiz. Although they not extend far enough to the south to definitely confirm this
observation, due to lack of data in this part of the study area.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.44 NW-SE belt of high velocity anomaly (SWIM lineament).
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
7.2 Sâo Vicente Canyon
The Marques de Pombal fault Figure (2.45) is a NNE-SSW trending, fault which was first
recognized by (Zitellini et al., 1999), it was interpreted to be the surface expression of a thrust
fault ( Gracia et al, 2003), and as a part of an active fault system located about 100 Km
westward of Sâo Vicente Cap, with a rather high shear stresses acting on it ( Grevemayer et al
2003). Considered the most likely source for the 1755 earthquake, in spite of the estimated
Mw 8.7, it have been suggested an active lithosphere folding, with the inherent topographic
expression, related with the Marques de Pombal-Horseshoe Fault System (MarquesFigueiredo et al, 2010); our model show a wide positive anomaly extending over 50 Km
depth westward of the Sâo Vicente Cap. This anomaly is rather clear and the S wave velocity
anomaly distribution has almost similar features to the P waves velocities Figure (2.45), Thus,
the Marques de Pombal Plateau is likely to be associated to an anomaly related to deep mantle
uplift as well.
Figure 2.45 Horizontal P-velocity distribution at 25, 35 and 50 km, showing high
velocity anomaly southwest Portuguese Margin.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
To the north, the Sâo Vicente Canyon, is an area where our synthetic tests provide higher
accuracy velocity estimates. This area located roughly at 37°00N, is characterized by rough,
irregular physiography; dominated by massive ridges, large seamounts, and extensive abyssal
plains, where active faults, such as the NE–SW trending Marques de Pombal, and the Sâo
Vicente faults (Zitellini et al., 2001), representing a major morphological feature separating
Alentejo and Algarve margins. It follows a simple NE-SW orientation forming a wide
submarine channel terminating on the Horseshoe Abyssal Plain, where the homogeneous
parallel reflectors are interpreted as comprising shelf-derived sediments deposited at the base
of the slope. In its southern part a pronounced hummocky character is seen, (Alves et al,
2000). The seamounts may represent either horsts formed during the Early Cretaceous rifting
episode or uplifted Eocene structures. The Sâo Vicente Canyon appear to be controlled by the
late Hercynian fracture pattern (Sibuet,et al 1987). All these deductions reveal that this part is
also characterized by a complex geological structure. To help understand the structure beneath
the Sâo Vicente Canyon; we have made vertical cross-sections along the profiles shown on
Figure (2.46). These vertical cross sections show tomographic on profiles Figure (2.47-48-4950) and represent the true-velocity seismic vertical section, rather than velocity perturbations.
Thus, vertical sections 1 and 2 are oriented in a NW-SE direction. Vertical cross-section 1
crosses the seismic cluster along the Gorringe Bank, while vertical section 2 crosses the
seismicity cluster along the Sâo Vicente fault (SVF). The tomographic vertical cross-section 2
Figure (2.49, 2.50) show a mantle uplift at the level of the Sâo Vicente fault coinciding with
the Marques de Pombal Plateau (see Figure 5 of Terrinha, et al,. 2009), where both of these
cross-sections (1 & 2) show that the Moho underneath the Gulf of Cadiz is rather deep, with
an average depth of about 26 km below sea surface. This part of the study area was
interpreted as a rectangular shaped monocline structure limited to the east by the São Vicente
Canyon and has been suggested as potential sources of large magnitude earthquakes and
tsunamis (Vizcaino et al, 2006).
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.46 Positions of the vertical cross-sections velocity profiles 1 and 2
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
GB
HsF
Figure 2.47 P velocity distribution in vertical cross-sections 1, the position of vertical
section is shown in the Figure (2.39)., GB: Gorringe Bank. HsF: Horseshoe Fault
GB
HsF
Figure 2.48 S velocity distribution in vertical cross-sections 1, the position of vertical
section is shown in the Figure (2.39)., GB: Gorringe Bank. HsF: Horseshoe Fault.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
SVF
AW
Figure 2.49 P velocity distribution in vertical cross-sections 2, the position of vertical
section is shown in the Figure (2.39), SVF : Sao Vicente Fault, AW : Accretionary wedge.
SVF
AW
Figure 2.50 S velocity distribution in vertical cross-sections 2, the position of vertical
section is shown in the Figure (2.39), SVF : Sao Vicente Fault, AW : Accretionary wedge.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
7.3 Gorringe Bank
To the NW of the study area, a distinct high velocity anomaly extends indeed from 10 km to
35 km depth and approximately coincides with the uplifted Gorringe Bank Figure (2.51).
Below these depths, this high velocity anomaly continues with depth, but appears rather
attenuated. Figure (2.52) shows the density and temperature depth profiles in the Gorringe
bank and its surroundings, it clearly exhibits three pieces density variations, increasing until
14 km with a rapid increase in density and continues increasing slowly up to 40 km, where no
compositional density contrast exists with the surrounding lithospheric mantle (Jiménez, Munt
et al 2010). Below this depth the density decreases, which explains the appearance of the lowvelocity anomaly beneath the Gorring Bank.
Figure 2.51 Horizontal P-velocity distribution at 25, 35 and 50 km, showing the attenuation of
anomaly A between the Gorringe bank and horsechoe abyssal plain.
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BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.52
Density and temperature depth profiles at four positions
identified in Fig (2.26): Tagus Abyssal Plain (profile a), Gorringe Bank
(profile b), and Horseshoe Abyssal Plain (profiles c and d), (Jiménez, Munt et
al 2010).
The tomographic vertical cross-sections 1 Figure (2.53) clearly shows that the Gorringe bank
is an anomaly that involves mantle uplift and that this topographic feature comes from deep
within the mantle, this feature clearly stands out from horizontals P and S velocity distribution
at depths of 10 km, 15 km and 25 km. Some authors explained that the Gorringe Bank
structure is resulting from about 50 km of thrusting of African oceanic crust upon the
Eurasian plate, (Jiménez Munt et al 2010), Profile 1 Figure (2.54) indicates this possible
movement related to a NW-directed thrust carrying upper mantle rocks, explain as well the
existence of that several seamounts in this region, as the Gorringe Bank, and the Coral Patch
seamounts and Ridge, bordered by the abyssal plains, they are incontestably formed by the
NW–SE compression movement during the Iberia– Nubia rapprochement.
173
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.53 Model results showing the lithosphere structure. The map above show the
position of the profile. White dots are earthquake hypocenters, (Jiménez, Munt et al 2010).
Figure 2.54 P- velocity distribution in vertical cross-sections 1, shows the layers going to
50 km depth , precising the direction in which the African oceanic lithosphere thrust Eurasian
plate.
174
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
7.4 Portimao-Bank
The convergence of Iberia–Nubia plates is probably due to other geodynamic processes, at the
moment of slab rupture when the subduction stops (Billi et al, 2011), while the trending
direction of these seamounts is probably due to the oblique convergence, can then justify the
W-E trending of the Portimao-Bank. where a clear high velocity anomaly is shown in
Portimao-Bank Figure (2.55), source of the largest event of January 1st, 2008 (
recorded by NEAREST network, explain the evidence of (Pliocene-Quaternary) associated
with seismic recent activity (Cunha et al, 2012). It is not presumptuous to say, or to highlight
the existence of an active subduction zone beneath the Gulf of Cadiz, GPS data overall system
(Global Positioning System) showed no differential movement through the Strait of Gibraltar
(Stich et al, 2006; Serpelloni et al, 2007), actually, bellow 50Km depth, our velocity profiles
do not show any indication of subduction zone to the SE of our study area.
Figure 2.55 Horizontal P-velocity distribution at 25, 35 and 50 km, showing high
velocity anomaly in the Porimaô Bank.
175
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
7.5 Horseshoe Abyssal Plain
The part of the horseshoe Abyssal plain we are studying is bounded in its northern side by the
Gorringe Bank and in its southern side by the Coral Patch, characterized by a topographic
pattern that reflects a complex geological structural trend, this is due to the fact that the
Horseshoe Abyssal plain is close to the African plate Eurasia boundary, this area strongly also
influenced by a compression movements, interpreted by some authors as giant submarine
debris flows sourced from the Gulf of Cadiz accretionary wedge, ( Iribarren et al, 2007),
Other authors have suggested the existence of a plate boundary Nubia-Eurasia WNW-ESE
trend; starts from the Gloria fault and finish in Rharb basin; reveal that these deep-sea basins
preserve a record of episodic deposition of turbidites (Gràcia et al 2010), transported towards
the west and coming from the Imbricate accretionary wedge in Gulf of Cadiz that covers the
entire Iberian and Morocco margins, the tomographic images in the northern part of the
horseshoe abyssal plain show a low velocity anomaly separating two blocs of a positive
anomalies along the SWIM lineament Figure (2.56), where earthquake activity is minor,
Undoubtedly this anomaly can be interpreted as the contact zone between Horseshoe Fault
and SWIM lineament Fault Zone, such contact, in addition to the oblique compression
movement that governs this area, can clearly create a tension zone.
176
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Figure 2.56 Horizontal P-velocity distribution at 25, 35 and 50 km, showing
low velocity anomaly in the N-W of the horseshoe Abyssal Plain.
177
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
8. Conclusion
The recorded OBS seismicity data set used in this study has revealed a wealth of new
information. Thus, these data reveal that most of the seismicity in the Gulf of Cadiz occurs at
depths that vary between 20 and 80 km. Very few events occurred at depths shallower than 20
km. Prominent clusters of seismicity are found to be associated with the Gorringe bank, at the
SW branch of the horseshoe fault and along the Marques de Pombal Plateau and the Sâo
Vicente fault. Diffuse seismicity is observed offshore to the south of Portugal and very little
seismicity along the accretionary wedge. A NW-SE band of seismicity is further observed to
the SW of our study area. The hypocenters along this band have depths that vary mostly from
35 to 80 km.
The tomographic inversion of the recorded seismic data yielded results that indicate that
patterns of velocity anomalies within the study area are generally oriented along NE-SW and
NW-SE directions. Both the P- and S-velocity distributions show that a low velocity zone is
found at the SE of our study area. At shallow depths, this LVZ is interpreted as corresponding
to thick sediments associated with the accretionary wedge. The velocity tomograms indicate
that the Moho underneath the Gulf of Cadiz has an average depth of 30 km. On the other
hand, higher velocity anomalies are found to the NW and to the north of our study area. The
NW anomaly coincides with the Gorringe bank, and its prolongation in depth clearly shows
that this bank is not a surficial feature, but is rather connected to a deep mantle uprising.
The tomographic images in the northern part of the horseshoe abyssal plain show a low
velocity anomaly separating two blocs of a positive anomalies along the SWIM lineament,
where earthquake activity is minor, Undoubtedly this anomaly can be interpreted as the
contact zone between Horseshoe Fault and SWIM lineament Fault zone, such contact, in
addition to the oblique compression movement that governs this area, can clearly create a
tension zone. Some authors support the existence of an active subduction zone beneath the
Gulf of Cadiz, GPS data overall system showed no differential movement through the Strait
of Gibraltar (Stich et al, 2006; Serpelloni et al, 2007), actually, bellow 50Km depth, our
velocity profiles do not show any indication of subduction zone to the SE of our study area.
178
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Similarly, our inversion results show that the Marques de Pombal Plateau is related to a deep
mantle anomaly. Both the Gorringe ridge and the Marques de Pombal Plateau anomalies are
associated with high levels of seismicity, thus, indicating that the mantle process behind their
uplift is likely to be still ongoing.
All these results obtained by inversion travel-time of seismic waves are very important,
however, there is still 10 km area that should be studied. In general we use surface waves
tomography technique to investigate the shallow depth rather than technique used this thesis,
this is the technique in terms of the theory, because it deal with another types of waves, called
surface waves (Appedix for more details) . In our next work, we will proceed to investigate
the shallow depth using this technique, in order to image all structure in the Gulf of Cadiz.
179
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
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188
BODY WAVES TOMOGRAPHY OF THE GULF OF CADIZ
Appendix
CHAPTER OUTLINE
1. Least squares method
2. Snell’s law
3. Plane wave
4. Conversion of format Seisan
to Lotos
5. Surface waves Tomography
189
APPENDIX
1. Least squares method
Measurement addresses fundamental responses that arise in many scientific studies (and
everyday activities), Answering several question and making lows for natural behaviors,
involving either direct or indirect measurements, it can be seen also as an integral aspect of
the scientific method, namely testing or proving some hypothesis always requires collecting
data, unfortunately, Even if measuring devices were perfectly accurate, it would not be
humanly possible to read them with perfect precision. In brief, even direct measurement
involves some degree or margin of error, despite that, the scientific method does not
guarantee exact results and definitive proof of a hypothesis, scientists continue to seek ways
to minimize these measurement errors by deferent ways, to construct an increasingly accurate
and thorough understanding of our world.
Nowadays, instruments can generate large amounts of data; yields in the most case over
determined systems of equations that arise in geophysics, to solve these systems, several
algorithms were used to fit the seismological observed data, the most stable and used one is
the least squares method, which is a standard approach to the approximate solution of over
determined systems, and can be used for any real data, there will be some misfit to the
spectrum, which can be measured as error. If the data vector
is in the column space, it will
makes linear combination of a column and it will have the form of linear system
is in the column space ,
, the data solve exactly the problem. However,
be in the column space, but
unknown, the system
, if
has to
is probably not, given probability more equations than
may have no solution and its really problem. In seismology and
in its most cases, thus we have to use a large quantity of data, because we have no reason to
say that this measurement are good or are useless, our purpose is to use all data available to
get the maximum information, for this we often solve
when
is not in the column
space (when there is no solution). So we try to solve for the closest vector in the column space
, where
is the projection on to the column space, we take
instead of
mistakes in measurement and it’s the errors . Now consider the system
188
making a
, in that case
APPENDIX
is square invertible matrix, multiplying two sides in the equation lead to
again instead of , because
,
don’t have the solution.
We can see that geometrically in Figure (3.7.a) below, the tow pieces of
were the projection
and the other part is the error, so
and
Where
is the projection of vector d on to the perpendicular space.
error vector can be expressed as
. The error vector
is the
is perpendicular
to the projection plane so
we get
where
Not that
is the identity matrix when
has an inverse, but it’s not the case we are
looking for.
Figure 3.1.a projection in subspace, p is the projection in column space, d is the data vector and e is the error
vector. b) Illustrate the projection of the data vector in terms of components, showing the error quantities and
is an outlier.
189
APPENDIX
The particular application method of determining
the parameter vector is known as the
method of least squares, consisting of adjustment the parameters of a model function to best
fit a data set let as take the simplest way to illustrate an approximation of a solution to over
determined systems by least squares is to consider the case of tree data points of any measure,
and try to find the best straight line that fit these data points.
We have to be careful about least squares, when at least one of the data point is way of, this
measurement change completely the shape of the best fit, this can be called an outlier and its
represented by the fourth data measurement Figure (3.7.b).
The best fit can then be written in this form
or
This is the system representing the equation
that can be written in matrix form as
Where
and
We certainly make tree errors for that example, one error for each equation, the way to find
best fit is to square and to sum all these errors and find the smallest possible sum,
If
has independent columns then
where
, by some function
is invertible for a set of experimental data
that contains
then take the deviations (or residuals)
190
parameters
, we
APPENDIX
We can also form the weighted sum of squares of the deviations or residuals
where the weights
express our confidence in the accuracy of the experimental data. If the
points are equally weighted, the
that
can all be set to 1, we can determine these parameters so
is a minimum
, ……… ,
,
The set of those equations is called the normal equations and serves to determine the
unknown as in
.
Until now we treat the problem as a linear appearance, in the case of a non-linear least squares
problem. Numerical algorithms are used to find the value of model parameters which
minimize the misfit to data. The most algorithms used are the LSQR method (Page, Saunders,
1982, Van Der Luis, Van Der Vorst, 1987). Choosing initial values for the parameters as
initial model, then, the parameters are refined iteratively, that is the values are obtained by
successive approximation.
191
APPENDIX
2. Snell’s law
The Snell’s law describe the behavior of light at the interface of two media with the rectilinear
propagation of light in homogeneous and isotropic medium, these laws are the basis of
geometrical optics, and can also be interpreted by different models: wave model Huygens
(Huygens' principle), model of least action Fermat (Fermat's principle) or model of the
electromagnetic wave of Maxwell. To derive Snell’s law, we consider a plane wave,
propagating in homogenous medium of uniform velocity c, the wave fronts at time
time
are separated by a distance
along the ray path. The ray angle from the vertical,
termed the incidence angle.
Figure 3.2 A plane wave incident on a horizontal surface,
incidence angle,
denote the length of the ray.
We can see from the Figure above
Since
, we have
192
is the
and
is
APPENDIX
Finally we get
, where U is the slowness
, P is the ray
parameter, P can be measured when the arrival’s of the wavefront are known at two different
stations.
Consider now a down going plane wave with horizontal interface between two homogeneous
layers of different velocity and the resulting transmitted plane wave Figure (3.3). If we draw
wave fronts at evenly spaced times along the ray, they will be separated by different distances
in the different layers, and we see that the ray angle at the interface must change to preserve
the timing of the wave fronts across the interface. In the case illustrated the top layer has a
slower velocity
and a correspondingly larger slowness
. The ray parameter
may be expressed in terms of the slowness and ray angle from the vertical within each layer
Figure 3.3 A plane wave crossing a horizontal interface
between two homogeneous half-spaces.
This is basic seismic version of Snell’s law in geometrical optics. Equation (3.9) may also be
obtained from Fermat’s principle, which states that the travel time between two points must
be stationary (usually, but not always, the minimum time) with respect to small variations in
the ray path. For more details see also, (Aki and Richards,2002), (John.A. scales, 1997), (Seth
Stein and Michael Wysession,2003), (Serveny,2000).
193
APPENDIX
3. Plane wave
A plane wave is one of particular waves in which the disturbance is constant over all points of
a plane drawn perpendicular to the direction of propagation. Such a plane is often called a
wavefront, and this wavefront moves perpendicular to itself with the velocity of propagation
c, Figure (3.4).
Figure 3.4 Propagation of plane wave.
Harmonic plane waves can be expressed as
From the Figure above we see that
, hence
Can also be expressed in this form
194
APPENDIX
where
this plane wave represent a simple solutions of wave equations, from the same Figure we
consider the same wave propagating along the x-axis with an apparent velocity , and along
the ray with
, the wavefront reach point P and R at same time
, from the Figure again , we
get
Lead to the formula that gives the relation between the body wave velocity
wave velocity
This formula; can be generated for a homogeneous isotropic
Where the quantity
is the ray parameter.
195
layers.
and surface
APPENDIX
4. Conversion of format Seisan to Lotos
Before reorganization of file, we have to convert our data (Source location and station
coordinates) from the SEISAN format to LOTOS format, the output format obtained is used as
input format for LOTOS processing, and in the following we give the routine used to convert
the files.
Code Fortran for conversion
character*8 re
character*1 pol
character*2 ps
character*5 stac,stacod(200),stbad(200)
real fstat(200),tstat(200)
real tobkr(200),tobkr1(200)
integer istkr(200),ipskr(200)
integer istkr1(200),ipskr1(200)
open(1,file='set.dat')
read(1,'(a8)') re
close(1)
open(1,file='data_in/stations.dat')
open(11,file='data_out/stat_ft.dat')
nst=0
1 continue
read(1,*,end=2)stac,tet,fi,met
!write(*,*)pol,stac,it1,at2,if1,af2,met
if(stac.eq.' ') goto 2
nst=nst+1
stacod(nst)=stac
fstat(nst)=fi
tstat(nst)=tet
z=-met/1000.
write(11,*)fi,tet,z,stac
!write(*,*)stac,fi,tet,z
goto 1
2 close(1)
close(11)
write(*,*)' nst=',nst
open(1,file='data_in/seisan.out')
196
APPENDIX
open(11,file='data_out/rays.dat')
3 read(1,12,end=4)iyr,imt,idy,ihr,imn,sec,tzt,fzt,zzt
12 format(1x,i4,i3,i2,i3,i2,f5.1,4x,f6.3,f8.3,f5.1)
tmzt=ihr*3600.+imn*60.+sec
! write(*,*)' fzt=',fzt,' tzt=',tzt,' zzt=',zzt,' t=',tmzt
! write(*,*)iyr,imt,idy,ihr,imn
34 read(1,'(1x,a4)',end=4)stac
if(stac.ne.'STAT') goto 34
ikr=0
5 read(1,14,err=5) stac,ps,ihr2,imn2,sec2
14 format(1x,a5,3x,a2,7x,2i2,f6.3)
if(ps.eq.' A') goto 5
! write(*,*) stac,pol,ps,ihr2,imn2,sec2
!******************
if(stac.eq.' ') then
if(ikr.eq.0) goto 3
ikr1=ikr
ipskr1=ipskr
istkr1=istkr
tobkr1=tobkr
ikr=0
do i=1,ikr1
ips=ipskr1(i)
ist=istkr1(i)
if(ips.eq.0) cycle
tmin=999999
do i1=1,ikr1
if(istkr1(i1).ne.ist) cycle
if(ipskr1(i1).ne.ips) cycle
if(tobkr1(i1).gt.tmin) cycle
imin=i1
tmin=tobkr1(i1)
end do
do i1=1,ikr1
if(istkr1(i1).ne.ist) cycle
if(ipskr1(i1).ne.ips) cycle
ipskr1(i1)=0
istkr1(i1)=0
end do
ikr=ikr+1
ipskr(ikr)=ips
197
APPENDIX
istkr(ikr)=ist
tobkr(ikr)=tmin
!write(*,*)ikr,ipskr(ikr),istkr(ikr),tobkr(ikr)
end do
if(abs(fzt).lt.0.1) then
!write(*,*)' nzt=',nzt
!write(*,*)fzt,tzt,zzt,ikr
tmin=999999
do i=1,ikr
if(tobkr(i).gt.tmin) cycle
imin=i
tmin=tobkr(i)
end do
fzt=fstat(imin)+0.03
tzt=tstat(imin)+0.03
zzt=5.
!write(*,*)fzt,tzt,zzt,ikr
!pause
end if
write(11,*)fzt,tzt,zzt,ikr
nzt=nzt+1
!if(mod(nzt,50).eq.0)
write(*,*)iyr,imt,idy,ihr,imn
nray=nray+ikr
do i=1,ikr
write(11,*)ipskr(i),istkr(i),tobkr(i)
!write(*,*)ipskr(i),istkr(i),tobkr(i)
end do
goto 3
end if
!**************************************
tmst=ihr2*3600.+imn2*60.+sec2
time=tmst-tmzt
ips=0
if(ps.eq.' P') ips=1
if(ps.eq.'EP') ips=1
if(ps.eq.'IP') ips=1
if(ps.eq.' S') ips=2
if(ps.eq.'ES') ips=2
if(ps.eq.'IS') ips=2
if(ps.eq.' A') ips=3
if(ips.eq.0) then
198
APPENDIX
write(*,*)' Cannot find phase ps=',ps
!pause
goto 5
end if
do ist=1,nst
if(stac.eq.stacod(ist)) goto 15
end do
!write(*,*)' cannot find station:',stac
if(nbad.ne.0) then
do ibad=1,nbad
if(stac.eq.stbad(ibad)) goto 5
end do
end if
nbad=nbad+1
stbad(nbad)=stac
!write(*,*)nbad,stac
goto 5
15 continue
if(ips.ne.1.and.ips.ne.2) goto 5
ikr=ikr+1
ipskr(ikr)=ips
istkr(ikr)=ist
tobkr(ikr)=time
goto 5
4 close(1)
write(*,*)' n_events=',nzt,' n_rays=',nray
write(*,*)' unknown stations:'
open(11,file='data_out/unknown_sta.dat')
do i=1,nbad
write(*,*)i,stbad(i)
write(11,*)stbad(i)
end do
close(11)
stop
end
199
APPENDIX
5. Surface waves Tomography
When an earthquake is happed, the seismograms are dominated by a large longer period
waves arriving after body waves, this waves are the surface waves, Two types of surface
waves, known as love and Rayleigh waves, shows a large surface wave train arriving on a
seismometer’s, transverse component shows the arrival of Love waves, followed by Rayleigh
waves on a vertical and radial component.
Notice that the love waves resulting from SH waves and Rayleigh are the result of P and SV
interactions, it’s bright and straightforward to show the surface waves on seismograms. It’s
necessary to indicate also that, at large distance from the sources the surface waves are
prominent on seismograms, and the radial component of Rayleigh wave is different from the
vertical one. When the displacement of the vertical component is zero, the radial displacement
is maximum and vice-versa, After a large earthquake, contrary to the body waves whose
energy spreads three dimensionally, the surface waves can circle the globe many times, and
their energy spreads tow dimensionally and it’s concentrated near the earth surface.
Dispersion:
The dispersion phenomenon, known as geometrical dispersion, occurs in the earth when the
velocity along the surface varied with the frequency, The fact that the surface velocities vary
depending on the depth range sampled by each period makes surface wave dispersion
valuable for studying earth structure, Use the observed Rayleigh waves dispersion allow us to
examine the possible correlation between The lateral change in Rayleigh waves dispersion
and geology structure.
The dispersion further can be explored easily if we consider the sum of tow harmonic waves
with slightly different frequencies and wave numbers.
The dispersive waves of different frequencies propagate with different velocities. This can
express these waves according to frequency, using the Fourier transform.
200
APPENDIX
is a complex function can be written as :
The inverse transform of allows
us to express the displacement field.
With
initial phase generated by seismic source), A first order linearization of the wave
(
number in about
than:
So, we can write the displacement field in the form:
iwdw
iwdw
With:
201
APPENDIX
we see that G varies with
where
is a velocity of traveling wave expressed
by G function, G varies much more slowly than
. We can say that G is
the envelope function of the wave packet. v represents the velocity of the envelope of the
wave packet, or of the wave group. We then define the group velocity as:
U(w) can be infered from the dispersion relation:
For an infinitesimal variation of w and k, the dispersion relation can be written as:
If
are also a solution of the dispersion equation, then the group velocity is
given by:
On seismogram, we remark that the wave with long period arrive or appear first on
seismogram. Therefore, those waves are faster. The group velocity is found by dividing the
distance between the sources and receivers by the travel time of the wave group, so for each
period we seek the group velocity and plot group velocity – period curve.
This technique is applied using the Fourier transform of a recorded signal, knowing that a
signal which is periodic and exists for all time has a discrete frequency spectrum, therefore
the Fourier transform is used to isolate the wave group of different periods.
202
APPENDIX
Objectives:
Determination of S-velocity models for shallow structure, using the Rayleigh waveforms
which depend strongly on the shallow velocity structure of the medium.
Method:
Group the seismic events in source zones to get an average dispersion curve for each sourcestation path.
Nb: it’s important to group the seismic events (short path : same elastic properties of the
medium). The coordinate differences for a group of events must be less than or equal to 0.2
degrees in latitude and longitude to consider them grouped in the same source zone. To avoid
the time lag, we have to take the instrument response into account. The figure below shows
the path coverage for the study area.
Apply the digital filtering with a combination of Multiple Filter Technique (MFT) (Badal et
al) .The Multiple Filter Technique (MFT) is a filtering technique (Dziewonski et al., 1969),
which is used to obtain the group-velocity dispersion curve from a pre-processed trace
(instrument corrected). It’s used to study variations of amplitude (or energy) of a signa1 as a
function of velocity (time) and period (frequency), the relative energy contour maps obtained
by multiple filtering of the sample wave trains permit determination of the respective
dispersion curves, as the group velocity of the fundamental mode is that associated with the
observed maximum spectral amplitudes. A simple computer routine is used to locate all these
spectral amplitudes whereby the corresponding group times that give us the observed group
velocity dispersion (Y. Chen et al.2009)
The time in which the envelope of the filtering seismic signal given by
where
is the Gaussian filter
203
APPENDIX
Reaches the maximum, is the group time for the frequency
selected as center of the
Gaussian filter. The group velocity is obtained dividing the epicentral distance by the group
time.
Group-velocity dispersion curve can be computed by means of the procedure shown in the
flow chart presented below:
Figure 3.5 Chart of processing method.
When the MFT has been applied for an interval of centre periods Tn, we get the contour map
of relative energy normalized to 99 decibels, as a function of period and group time. The
white curve denotes the group time inferred from the energy map.
And the group-time curve can be inferred from the energy map. After, the group-velocity
curve is calculated from the group times and the epicentral distance, dividing the epicentral
distance by the group time for each period. Apply Time Variable Filtering signal (TVF). The
Time Variable Filtering (TVF) is a filtering technique that does not give a dispersion curve
but a smooth signal (a signal time-variable filtered), in which all effects of noise, higher
204
APPENDIX
modes and other undesirable perturbations can be removed (Cara, 1973). Nb: combination of
MFT and TVF works better than the application of the MFT alone, because the signal/noise
ratio is highly increased. Apply an inversion theory to get S-wave velocity models.
Initial model requirement.
Regionalization:
Sometimes we have to proceed to the regionalization as is known as a process to obtain
dispersion curves in sub-regions from ray-path velocities crossing two or more different
structures.
Reliability:
1) We have to take just the traces which present a well developed Rayleigh wave train and
discard the others.
2) We than match the:
Theoretical dispersion (Calculated).
Observed dispersion.
The reliability is checked by the comparison between the observed group velocity (considered
as observed data) and the theoretical group velocity (calculated from the actual model by
forward modelling and considered as theoretical data).
Resolution:
Calculate the resolving kernels at various reference depths. This resolution is obtained when
the absolute maxima fall over the reference depths.
205
APPENDIX
Figure 3.5 Resolution Kernel (Badal et al 2003).
The solution for the inversion problem is more reliable when the maxima of these resolving
kernels are narrower.
206