Full seismic waveform modelling and inversion

354 Pages·2011·9.759 MB·English

by  Andreas Fichtner (auth.)

Andreas Fichtner
Full Seismic Waveform
Modelling and Inversion
With contributions by Florian Bleibinhaus
and Yann Capdeville
123
AndreasFichtner
UtrechtUniversity
DepartmentofEarthSciences
Budapestlaan4
3584CEUtrecht
TheNetherlands
fi[email protected]
AdvancesinGeophysicalandEnvironmentalMechanicsandMathematics
ISSN1866-8348 e-ISSN1866-8356
ISBN978-3-642-15806-3 e-ISBN978-3-642-15807-0
DOI10.1007/978-3-642-15807-0
SpringerHeidelbergDordrechtLondonNewYork
(cid:2)c Springer-VerlagBerlinHeidelberg2011
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TomyfamilyandCarolin

Foreword
Our planet is permanently vibrating, excited by oceans, atmosphere, earthquakes,
orman-madesources.Luckily,Earth’sphysicalpropertiesaresuchthatthesevibra-
tions–elasticwavestobemorespecific–oftenpropagatetolargedistancescarrying
informationonthemediumtheyencounteralongtheway.Theproblemofmaking
aneducatedguessatthesubsurfacestructurefromobservationsofgroundmotions
isasoldasinstrumentalseismologyitself(sonotthatold,maybeacenturyorso).
Let us call the problem seismic tomography akin to CT scanning in medicine, a
field seismologists have always envied. Because of limitations in illuminating the
Earthwithsufficientcoveragewehaveneverobtainedthesharpanddetailedinternal
structuressofamiliarfrommedicalimaging.
Uptonowwehavemostlycutcornersinthewaywemodelandfitourseismic
data.Wetypicallyreducelong,wigglyseismogramstoafewbytesofinformation
(e.g. travel times, phase velocities) and try to explain these data with approximate
theories.Thishasbeenforagoodreason.Ourcomputersweresimplynotfastand
bigenoughtoallowthecalculationofcompletewavefieldsthrough3DEarthstruc-
tures.Frequentlythedatajustdonotwarranttheuseofsophisticatedphysics.
Thesituationregardingcomputationalpowerinconnectionwith3Dwaveprop-
agationhasdramaticallychangedinthepastfewyears.Evenonaglobalscalethe
calculationofwavefieldsacrossthecompleteobservedfrequencyrangeisinsight.
Onsmallerscales(continents,basins,volcanoes,reservoirs)wearealreadywitness-
ingtheemergenceof3Dwavepropagationasthetoolfordatamodelling,inversion
andparameterstudies.
This was Albert Tarantola’s (and others) dream 25 years ago: just simulate the
physicscorrectlyandletthedata(i.e.themisfittoatheoreticalseismogram)decide
whethertheEarthmodelisgoodornot.Inhisworld(theprobabilisticapproach)this
shouldbedoneusingaMonteCarlo-typeapproach:calculatezillionsofmodelsand
usealltheresultstoestimateparametersanduncertainties.Unfortunately,wearenot
there yet. We still need to resort to linearisations around (hopefully good) starting
modelsandemployadjoint-typetechniquestoupdateourEarthmodels.Fortunately,
inmanysituations,goodstartingmodelscanbefound,makingiterativewaveform
inversion the preferred tool to improve our Earth models using most or all of the
observeddata.
vii
viii Foreword
Thebookinyourhandsisthefirsttoprovideabroadoverviewonhowtosolve
the forward problem to calculate accurate seismograms in 3D Earth models, to
compare theory with observations and to iteratively update Earth models until a
satisfactoryfittoobservationsisachieved.Manyquestionsremain,inparticularin
connectionwithproperlyassessinguncertainties.Yet,thetoolsdescribedherewill
beessentialinthecomingyearstomoveforwardinreconstructingthestructureof
Earth’sinterioronallscales.Inthatsensetheywillhaveastrongimpactonmany
related fields in Earth sciences such as geodynamics, earthquake physics, explo-
rationgeophysics,tectonics,volcanology.Thebookshouldbeessentialreadingfor
everyoneinterestedingettingthebestoutofseismicobservationsintermsofEarth’s
physicalproperties.
Munich,Germany HeinerIgel
October2010
Preface
Full waveform inversion is a novel variant of seismic tomography that is charac-
terisedbythenumericalsolutionoftheequationsofmotion,theexploitationoffull
waveforminformationandtheiterativeimprovementofthetomographicimagesthat
accounts for non-linearity in the relation between model parameters and synthetic
data.Thenumericalsolutionsensuretheaccuratemodellingofseismicwavepropa-
gationthroughrealisticallyheterogeneousEarthmodels,thusmakingfullwaveform
inversionthetomographicmethodofchoicewhenthemediumpropertiesarehighly
variable.Thankstothecombinationofnumericalsolutionsandadjointtechniques,
any type of waveform can be exploited for the benefit of improved tomographic
resolution–withouttheneedtoidentifyparticularwaveformsintermsofclassical
seismicphasessuchasPorS.
Thepurposeofthisbookistopresentthenecessaryingredientsforafullwave-
forminversionappliedtorealdata.Itisintendedtoserveasanaccessibleintroduc-
tion to the topic for advanced students and professionals. The level is such that it
could be used as a source of advanced teaching material for specialised seminars,
courses and schools. Throughout the text we assume basic knowledge of elastic
wavepropagationinaseismologicalcontext,asitcanbefound,forinstance,inthe
worksofBåth(1979),Bullen&Bolt(1985),Lay&Wallace(1995),Udías(1999),
Kennett(2001)orAki&Richards(2002).
Thisbookisorganisedinthreepartsthatrepresentthemajorstepstobetakenin
ordertosolveafullwaveforminversionproblem.
PartIisdedicatedtothenumericalsolutionoftheelasticwaveequation.Inthe
course of the past few decades many numerical methods for the discretisation of
the wave equation have been developed – each being particularly well suited for
specificapplications.Thechoicetocoveronlyfinite-differenceandspectral-element
methodsindetailhasthereforebeenparticularlydifficult.Inadditiontobeingmost
frequently used in full waveform inversion, finite-difference and spectral-element
methods also offer the opportunity to introduce many of the concepts that are
fundamental in numerical wave propagation. Therefore, I hope that the content of
Chaps. 2, 3, and 4 may serve as a useful basis for the understanding of numerical
methods that I was not able to cover in a book that is focused on the solution of
theinverseproblem.Chapter5isconcernedwiththedescriptionandthenumerical
implementationofvisco-elasticdissipation.Specialattentionisgiventotheconcept
ix
x Preface
ofmemory variables and theconstruction of Q models withaspecified frequency
dependence. Efficient absorbing boundaries are of outstanding importance when
thecomputationaldomaindoesnotcomprisethewholeEarth.Chapter6therefore
describesabsorbingboundaryconditionsandabsorbingboundarylayerapproaches
suchastheGaussiantapermethodandperfectlymatchedlayers(PML).
The subject of Part II is the solution of the full waveform tomographic inverse
problem. Chapter 7 reviews the foundations of non-linear optimisation, including
the concept of local and global minima, uniqueness, convexity, regularisation and
aselectionofgradient-basedminimisationalgorithms.Theadjointmethod,treated
inChaps.8,9and10,isoneofthemethodologicalcornerstonesoffullwaveform
inversionbecauseitallowsustoefficientlycomputethepartialderivativesofseis-
micobservablesthatareneededingradient-basedminimisationschemes.Thegen-
eral operator formulation of the adjoint method, its application to the elastic wave
equation,theextensiontosecondderivatives andthederivationofexplicitexpres-
sionsforFréchetandHessiankernelsarecoveredwithparticulardetail.Ofoutstand-
ingimportanceinfullwaveforminversionisthechoiceofsuitablemisfitfunctionals
that extract as much waveform information as possible while conforming to the
restrictions imposed by the data and the physics of wave propagation. Chapter 11
summarises several misfit functionals that have proven effective in applications to
realdata.Regardlessofanytechnologicaladvances,physicalintuitionremainsthe
mostimportantingredientinthesolutionofanyinverseproblem.Inthecaseoffull
waveforminversion,thisintuitionmostlycomesfromtheinterpretationofFréchet
kernels,acollectionofwhichispresentedinChap.12.
While full waveform inversion is still a comparatively young method, there is
already an important conclusion to be drawn: Its application is highly problem-
dependent! A numerical method that is efficient for the modelling of body waves
on a regional scale may be inefficient for the simulation of global surface wave
propagation. Misfit functionals that are well suited to image sharp reflectors may
not be able to recover the long-wavelength structure of the Earth. Owing to this
extremeproblemdependence,PartIIIpresentsacollectionofcasestudieswherefull
waveforminversionhasbeenappliedsuccessfully.Chapter13isconcernedwitha
continental-scaleproblem,givingspecialemphasistothenumericalmodelling,data
selectionandmisfitquantification.Alocal-scaleproblemisdescribedinChap.14,
writtenbyFlorianBleibinhaus,whousedtheacousticwaveequationtoimagethe
2DvelocitystructureneartheSanAndreasfault.Ahighlyefficientdatareduction
schemeappliedtoasyntheticproblemattheglobalscaleisthetopicofChap.15,
writtenbyYannCapdeville.
Writingabookonfullseismicwaveformmodellingandinversionturnedoutto
bemorechallengingthanIexpected,becauseitrequirestocoveralargevarietyof
differenttopicswithoutcreatingamonsterthathardlyanyonewouldattempttoread
fromthefirsttothelastpage.Ihavechosenthecontentsuchthatithopefullyallows
thereadertodevelopandapplyhisorherownfullwaveformmodellingandinver-
sionmethods.Mywritingprocesswasmarkedbymanytoughdecisionsconcerning
interestingtopicsthatIwasnotabletopresentatallornotindetail.Forinstance,the
large variety of numerical methods for the solution of the wave equation certainly
Preface xi
deserve at least one separate book. The same is true for the ample fields of non-
linearoptimisationandabsorbingboundarymethodsthatIhadtopresentinarather
condensedform.
Utrecht,TheNetherlands AndreasFichtner
August2010