Reflection Seismology. Theory, Data Processing and Interpretation

255 Pages·2014·3.409 MB·English

by  Yang Wencai (Auth.)

REFLECTION
SEISMOLOGY:
THEORY, DATA
PROCESSING AND
INTERPRETATION
W Y
ENCAI ANG
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Preface
ReflectionseismologybelongstoasmallbranchofsolidEarthphysics.
Doesithaveitsowntheoreticalsystems?ThisisthequestionthatIkeep
considering.Ifonecopieswordbywordtheformulafromelasticorcon-
tinuum mechanics to form the theory of reflection seismology, one looks
upon reflection seismology as a branch of applied technology, but not
asabranchofappliedscience.Eventoday,manygeoscientistsstillregard
reflection seismology as engineering technology that surveys mineral
exploration.Asreflectionseismologists,wecannotblametheirignorance,
but can only blame ourselves for not having built the theoretical system
perfectly andcompletely.
It is true that reflection seismology was an oil/gas exploration tech-
nique at the beginning. An oil field was found according to the seismic
reflection data in the continent of Oklahoma in the USA in 1926, which
provesthatseismicreflectionisreallyagoodpracticaltechniquehaving
enormous economic worth. After the 1930s, based on decades of effort,
especiallyafter the invention of computers, thetheory ofreflection seis-
mology has been developed rapidly. Some monographs on reflection
seismology have been published in the 1970s. However, because the
renewal speed of seismic exploration methods has been too fast, these
monographs cannot include enough newly developed content on the
theory of reflection seismology. On the other hand, textbooks used in
universities mostly quote elastic mechanics formulas, and have not
absorbed enough newly developed theories on reflection seismology.
From 1983 to 2013, I have written a few monographs on geophysical
inversion theory and methods that have been my main research area.
As the foundation of performing seismic inversion is built on solving
the forward problems in reflection seismology, I have devoted my
time for summarizing the theory of reflection seismology, merging for-
ward and inverse problems into a textbook, and bringing out the best
in each other.
This book is designed as a textbook for professional graduate stu-
dentsmajoringinappliedgeophysics.Itisimpossibletocoveralldevel-
opments in reflection seismology theories. I have outlined just the
equationsystemsofreflectionseismologybasedonthetheoryofcontin-
uum mechanics. The equation family constructs the theoretical frame-
work of reflection seismology. I do not want to emphasize the abstract
beautyoftheseequations,butinthisbook,Itrytoputupsystematically
ix
x
PREFACE
a skeleton that is formed with equations and mathematical formulae,
showing the spirit and values of classical physics. Equations and
mathematical formulae are the essential language for communication
between human beings and computers, applicable to all. One can be
an excellent geologist who may not be good at mastering this kind of
language, but one cannot be an excellent seismologist who is not
good at mastering this kind of language.
The corresponding teaching of how to use this textbook takes about
50h.Asteachingmaterials,theauthorsmustconsiderthefollowingthree
points and guard against three misleading factors, namely first, write
teachingmaterialsasresearchreports,discussalotofdetails,andmislead
studentstoignorethebackbone.Second,writethetextbookasacollection
of practical techniques that will be updated soon, and do not probe into
their theoretical bases deeply. Students tend to listen excitedly, think
thatalotofusefulknowledgewouldbeobtained,butignoretheacademic
ability and marrow. Third, write the teaching materials as an encyclope-
diathatincludeseveryaspectofthesubject,butnotfullysystematically.
Books become very voluminous when they provide too much of knowl-
edgeforstudents,butbecomeunfavorablefortraininggraduatestudents
whohave asystematic thinkingability.Toavoid misleadingthereaders,
thisbookadherestothemainidea,anddoesdiscussthemainbranchesin
detail; it only explains the theory and not already existing market-based
technology. I am not afraid of writing little, but only afraid of writing
too much. The backbone must explain clearly, the minor branches may
only be clicked. It not only lets students assimilate information but also
confers the ability to combine bits of knowledge to solve practical prob-
lems in application.
This book is divided into 7 chapters as follows: The first chapter
reviews the basic wave theory of classical physics, and the second sum-
marizeselasticwaveequationsthatbuildupthefoundationofreflection
seismology. Particle dynamics used to be improperly applied to explain
vibration and wave propagation in the past, and I have tried my best to
correctimproperconceptsbyusingcontinuummechanics.Chapterthree
summarizesseismicwavepropagationinthesolidEarth,especiallysome
special characteristics different from elastic waves in common solid
media. Chapter four deals with wave equation changes along with seis-
micdataprocessing,togetherwithchangesontherelevantboundarycon-
ditions.Chapterfivediscussestheintegralsolutionsofwavepropagation
problems and Green’s function methods. Chapter six discusses the
decomposition and continuation of seismic wave fields, with emphasis
on the properties of wave equations with variable coefficients and the
operatorexpansionmethodfortheproblemsunderstudy.Thelastchap-
ter briefly introduces inverse problems involved in seismic exploration
and typical solving numericalmethods.
xi
PREFACE
IwouldliketothankprofessorsFuChengyiandW.Telfordforguid-
ingmeonsolidEarthgeophysics.IalsoappreciatethehelpofDrs.Yang
Wuyang, Li Lin, Xie Chunhui, Wang Enli, Wang Wanli, and Zhu
Xiaoshan in the translation of this book to English. The author is much
indebtedtoMohanapriyanRajendranforimprovingtheEnglishpresen-
tationofthisbookandgratefultoallcolleaguesandreaderswhoinform
errors left in this book.
Wencai Yang
National Lab on Tectonics and Dynamics,
Institute of Geology, CAGS.PRC
C H A P T E R
1
Introduction to the
Wave Theory
O U T L I N E
1.1. Wave Motion inContinuous Media 2
1.2. Vibration 5
1.3. Propagation and Diffusion 6
1.4. AcousticWave Equation 8
1.5. AcousticWave Equationwith ComplexCoefficients 11
1.5.1. ComplexElastic Modulus and the Complex Wave Velocity 11
1.5.2. DampingWave Equations inViscoelastic Media 13
1.5.3. Viscoelastic Models 14
1.6. AcousticWave Equationwith VariantDensity orVelocity 16
1.7. Summary 18
Physics is the study of the motion of matter. It originates from New-
tonian mechanics theory. In classical mechanics, which is established by
Newtonintheseventeenthcentury,particledynamicswasfirstemployed
to describe the motion of macroscopic objects. However, we must deal
withcontinuouslyconstructedmediasuchastheearth;theapplicationof
particle dynamics has some limitations. Thus, scientists developed
continuum mechanics in the early twentieth century, including fluid
mechanicsandsolidmechanics.Quantummechanicshasbeendeveloped
in the same period to describe the motion of microscopic particles. The
theoryinthisbookisbasedoncontinuummechanicsandthemethodsof
mathematical physics.
ReflectionSeismology 1 (cid:1)2014PetroleumIndustryPress.Publishedby
http://dx.doi.org/10.1016/B978-0-12-409538-0.00001-4 ElsevierInc.Allrightsreserved.
2
1. INTRODUCTIONTOTHEWAVETHEORY
Themotionofmacroscopicobjectsingeneralcanbedividedintothree
types:Thefirstisdisplacement,suchaslinearmovement,rotation,flight,
and flow. The second is vibration and wave motion, such as periodic
motion,waterwave,acousticwave,andseismicwave.Thethirdischaotic
movement,suchasintermittentmotion,turbulence,andnonlinearwave
motion.Thisbookonlydiscussesclassicalvibrationandwavetheory.Asa
kind of physical movement, vibration can be described using an initial
value problem of ordinary differential equations for a closed system, in
whichenergyandinformationdonotgetexchangedbetweenthesystem
and the outside world. For an open system, in which energy and infor-
mation get exchanged between the system and the outside world, the
initialandboundaryvalueproblemsofpartialdifferentialequationsmust
beapplied.Thischapterdiscussesthebasicwavetheory,focusingonthe
acoustic wave equation andrelatedwave behaviors.
To make mathematical formulas clear, we use bold English letters for
vectorsor matrices, and normal Greek letters for scalars in thisbook.
1.1. WAVE MOTION IN CONTINUOUS MEDIA
Wave motion refers to the propagation of vibration in continuous
media, which can bedescribed by using the following formulations:
Wave motion¼vibrationþpropagation (in continuousmedia);
Vibration¼periodicmotion that anobjectmovesaround its
equilibrium point;
Propagation¼interaction between thevibration andadjoiningmass
grains and diffusionof the vibrationenergy.
The above concepts are limited to the situation that the vibration
equilibrium point of mass grains is fixed and is stationary during wave
propagation; theyareaccuratefor theelastic waves propagatinginsolid
media. Generalized waves involve situations in which the vibration
equilibriumpointisnotfixedandmovable.Forexample,waterwavesare
akindofgeneralizedwaveswithmovingequilibriumpoints.Wedonot
discuss generalized waves in thisbook.
Three assumptions are usually accepted in the study of continuum
mechanics and are as follows (Fung,1977; Spencer, 1980; Du Xun, 1985):
1. Mass motion follows the law of conservation of mass, that isthe first
derivative of mass M with respectto time tiszero:
dM
¼0 (1.1)
dt
2. Theequilibriumpointofmassegrainvibrationinawavefieldisfixed
and stable.
3
1.1. WAVEMOTIONINCONTINUOUSMEDIA
3. Vibrationoccurs in continuous media,and the interaction between
adjacentmassgrainsfollowsthecontinuityequation,whichdefinesthe
continuous media.
What kind of media can be defined as continuous media? In other
words,whatkindofmechanicallawsarefollowedbycontinuousmedia?
Definition: for a small deformation, continuous media are defined by
the continuity equation as follows:
r¼r0ð1(cid:2)divuÞ (1.2)
Inthisequation,rdenotesthedensityofamassgraininthemediawith
respecttotimet;r0denotesthedensityofthemassgrainwithrespecttoan
initial time t0; u denotes the displacement vector of the mass grain; divu
indicatesthedivergenceofthedisplacementvector.Equation(1.2)couldbe
derived from Eqn (1.1), which describes the conservation of mass. The
productofthegrainvolumeanddensityiscalledthemasselement.
SetVasthevolumeofcontinuousmediaandthemassinEqn(1.1)can
be expressedas follows:
Z
M¼ rdV
V
In Cartesian coordinates, according to the law of the conservation of
mass,masscan beexpressedas follows:
Z Z
M¼ rdV ¼ r0dV (1.3)
V V
wherer0representsthedensityattheinitialtimet0.Becausedv¼dxdydz
anddvo ¼dxo dyo dzo, denotinga determinant as
(cid:2) (cid:2)
(cid:2) (cid:2)
(cid:2) vx vx vx(cid:2)
(cid:2)vx0 vy0 vz0(cid:2)
(cid:2) (cid:2)
J ¼(cid:2)(cid:2) vy vy vy(cid:2)(cid:2) (1.4)
(cid:2)vx0 vy0 vz0(cid:2)
(cid:2) (cid:2)
(cid:2) vz vz vz (cid:2)
vx0 vy0 vz0
Wehave
dV ¼JdV0
Substituting Eqn (1.4) into Eqn (1.3) yields
Z Z
rJdV ¼ r dV (1.5)
o o o
Vo Vo
4
1. INTRODUCTIONTOTHEWAVETHEORY
or
Z
ðrJ(cid:2)r ÞdV ¼0 (1.6)
o o
Vo
Hence,theintegral kernel shouldbeequal to zeroas
r0 ¼rJ (1.7)
Suppose a small deformation is being generated during the wave
motion.Then,
(cid:2) (cid:2) (cid:2) (cid:2)
(cid:2) (cid:2) (cid:2) (cid:2)
(cid:2)(cid:2)vux(cid:2)(cid:2)(cid:3)1 and (cid:2)(cid:2)vux(cid:2)(cid:2)(cid:3)1; etc:
vy vz
Ifoneignoresthesecond-orderterms,Eqn(1.3)canbetransformedas
J ¼1þvuxþvuyþvuz ¼1þdivu (1.8)
vx vy vz
IfwesubstituteEqn(1.8)intoEqn(1.7),thecontinuousEqn(1.2)canbe
obtained.
It is true that wave motion involves only small deformations, but ex-
plosions involve large deformations. In seismic explorations, explosions
areusedasthevibrationsourcestoproduceseismicwaves.Bothvibration
and movement of mass grains occur around the shots, where one has to
use the theory of explosion that we do not discuss in this book. We will
explainthemovementinthefarfieldarea,wheretheequilibriumpoints
arefixed andstableand small deformationsareaccepted.
ThecontinuityEqn(1.2)impliesthatthedensityofthemasselements
may vary during wave motion and that the variation amplitude is pro-
portional to the divergence ofthe displacement vector.
The movements of the mass grains in nature can be classified into
severalkinds,andvibrationandwavemotionaretwoofthem.Thereare
manyphysicalbranchesthatdescribethemovementsofthemassgrains,
including classical mechanics, continuum mechanics, and nonlinear dy-
namics. Classical mechanics is the study of mass motion in free space,
usuallywithoutadditionalconstraints,andishelpfulforthestudyofgas
motion in a vacuum and the Brownian motion of molecules. Mass
movementfollowsNewton’slawsofmotionanduniversalgravitationin
an enclosed dynamic system and can be expressed by some ordinary
differential equations with initial values. Continuum mechanics studies
constrainedmotionincontinuumspace,itisbasedonNewton’sequation
of motion and specific constitutive equation, and it can be described by
partial differential equations with initial and boundary values. The
constitutive equation in elastic mechanics is called the generalized
Hooke’s law. Continuity Eqn (1.2) is the starting point of continuum
5
1.2. VIBRATION
mechanics, which used to be divided into statics and dynamics. Wave
motion isthe resultof some forces, andbelongsto dynamics.
1.2. VIBRATION
Vibration can be described as the motion of a mass constraint to an
equilibrium point with a limited distance. No matter how the mass
oscillates, it will always return to the equilibrium point, due to the di-
rection of the force acting on the oscillator always pointing to the equi-
libriumpoint.Onlywhenthedirectionofforce(i.e.elasticforceinasolid)
isoppositethatofthemovement,thevibrationcanbealwaysaroundthe
equilibrium point. Therefore, vibration is a kind of mass motion whose
workingforceand displacement arein reversedirections.
Inthecaseofone-dimensional(1D)motion,wedenotetheelasticforce
as F, andthe displacement of movement isindicated as u; then, Hooke’s
law is
F¼(cid:2)ku (1.9)
where k indicates the elastic coefficient, the negative sign indicates that
the direction of the force is opposite to the displacement. Denoting the
mass of the oscillator as m, we can apply the equation using Newton’s
second law as (Budak et al., 1964)
2
d u
m ¼(cid:2)ku¼F (1.10)
2
dt
Settingthecircularfrequencyasu,wesubstituteu2 ¼ k intoEqn(1.10)
m
and obtain
2
d uþu2u¼0 (1.11)
2
dt
Theaboveequationisthevibrationequationwithoutdamping,andits
general solutionis
u¼AcosðutþfÞ (1.12)
whereAindicates the amplitudeand findicatesthevibration phase.
Any movement will be affected by resistance, and a vibration with
resistance is called the damping vibration. Damping comes from the
constraints of surrounding media and creates a resistance f , which is
r
proportionaltothespeedoftheoscillationwhenthespeedisnottoohigh
and their directions are opposite. If we denote g as the proportional
coefficient, then
du
f ¼(cid:2)g (1.13)
r
dt