Geophysical Journal International

12 Pages·2014·1.23 MB·English

Geophysical Journal International
Geophys.J.Int.(2015)200,390–401 doi:10.1093/gji/ggu398
GJISeismology
Focusing of elastic waves for microseismic imaging
Johannes Douma and Roel Snieder
CenterforWavePhenomena,ColoradoSchoolofMines,Golden,CO80401,USA.E-mail:[email protected]
Accepted2014October13.Received2014October10;inoriginalform2013December18
SUMMARY
Microseismiceventsgeneratecompressivewavesandshearwaves,whichcanberecordedat
receivers.WepresentatheorythatshowshowelasticPandSwavesseparatelybackpropagate
totheoriginalsourcelocation.TheserefocusedPandSwavefieldsarefreeofsingularities.We D
o
alsodemonstrateatechniquethatenhancestheabilitytoimagethespatialfocusforeachwave w
n
typeusingelasticwaves.Theimprovedspatialfocusobtainedisachievedinavelocitymodel loa
d
forwhich theinterfaceboundaries areapproximate butwherethe mean slownessiscorrect. ed
Deconvolutiondesignsasignaltoberebroadcastedfromthereceivers,usingonlythewaves fro
m
recordedateachreceiver,suchthatthewavefieldhasanoptimaltemporalfocusatthesource h
location. We demonstrate theoretically and numerically that improved temporal focusing of ttp://g
elasticwavesleadstoimprovedspatialfocusingforeachwavetype.Thisproposedtechnique ji.o
onlyinvolvesasimplepre-processingsteptotherecordeddataanditscostishencenegligible x
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comparedtothetotalcostofmicroseismicimaging. rd
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Keywords: Imageprocessing;Fourieranalysis;Earthquakesourceobservations;Computa- rn
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tionalseismology;Fracturesandfaults. ls.o
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the wavefield is sampled at only a limited number of locations, o
INTRODUCTION S
thenitisnotobviousthattimereversalistheoptimalwaytofocus ch
o
Duetohydraulicfracturingbecomingacommonpracticeforuncon- energyontheoriginalsource.Muchresearchhasbeencarriedout o
l o
ventionalgasandoilfields,therehasbeenanincreasedinterestinto on focusing sparsely sampled wavefields (Parvulescu 1961; Fink f M
thestudyofmicroseismicevents.Clustersofmicroseismicevents 1997; Roux & Fink 2000; Tanter et al. 2000, 2001; Aubry et al. in
e
delineate faults and the formation of fractures, and can indicate 2001; Bertaix et al. 2004; Jonsson et al. 2004; Montaldo et al. s
o
neworreactivatingregionsoffailure.Thesemicroseismicevents 2004;Vignonetal.2006;Larmatetal.2010;Gallotetal.2011). n N
can be generated naturally or as a result of hydraulic stimulation Inthispaper,weexploreasimpleextensiontotimereversal,based o
v
e
(Duncan2005;Kendalletal.2011).Therefore,thepetroleumin- ondeconvolution,aspreviouslyderivedbyAndersonetal.(2014). m
b
dustrydesirestodevelopmoreaccuratewaysoflocating,andmoni- Wehaveshownearlierthatdeconvolutionimprovesthelocatingof er 2
toringmicroseismiceventstopotentiallyimprovetheirrelationship microseismiceventsinanacousticmedium(Doumaetal.2013).We 4
toproductionandcompletiondata(Foulger&Julian2012). nowdemonstratedeconvolution’sabilitytoimprovetheimagingof , 20
1
Acommonprocessingmethodtolocatemicroseismiceventsor amicroseismiceventinanelasticmedium.Thismethodisarobust, 4
earthquakes is based on picking arrival times of the acoustic and thoughsimplified,versionoftheinversefilter(Tanteretal.2000,
shearwaves.Thisprocess,however,isdifficulttodoaccuratelywhen 2001; Gallot et al. 2011). It calculates a signal to be rebroadcast
significantnoiseispresentinthedata(Boseetal.2009;Bancroft fromthereceiversuchthattheoutputatthefocallocationbecomes
etal.2010;Kummerow2010;Songetal.2010;Haylesetal.2011). anapproximatedeltafunctionδ(t)andusesonlytherecordedsignals
An alternative approach, which requires less user interaction and ateachreceiver.
allowsformoreaccuracy,isusingtimereversaltoimagethefocus Aswithallimagingmethods,reversetimeimagingisunableto
of the microseismic events or earthquakes at the source location locatethemicroseismiceventtoapointlocationwhenthevelocity
(Larmatetal.2006,2010;Lu2008;Steineretal.2008;Lu&Willis modelusedforthebackpropagationdiffersfromthetruevelocity
2008;Artmanetal.2010).Inthisimagingapproach,oneusestime model,orwhentheapertureislimited;itcausesthespatialimageto
reversaltofocustherecordedsignalatthesourcelocationinboth defocus.Inthenumericalexampleusedforthispaper,theaperture
timeandspace.Theadvantageoftimereversalisthatitdoesnot usedisnotperfect,thus,thewavefieldisnotknownateverypoint
require picking of arrival times which is important when dealing in both time and space. Additionally, we complicate our model
withnoisydata. by adding noise and backpropagating our wavefields not through
Ifonewouldtimereversethewavesateverypointinspace,the thecorrectvelocitymodelbutthroughasmoothedversionofthe
wavefieldwillfocusontotheoriginalsourcelocation.If,however, velocitymodel.
390 (cid:2)C TheAuthors2014.PublishedbyOxfordUniversityPressonbehalfofTheRoyalAstronomicalSociety.
Elasticwavesformicroseismicimaging 391
Inthispaper,wefirstderivearelationshipbetweenthetemporal where c is the appropriate wave velocity (α or β depending on
focusandtheincomingwaveforanelasticmedium.Thetheoryis thewavetype).Itisunderstoodthatthetotalwavefieldfollowsby
used to show that improved temporal focusing leads to improved multiplyingwiththeappropriatevectorsphericalharmonicasgiven
spatial focusing for each wave type due to different sources (ex- inexpressions(2)–(4).Theradialdependenceofthewavefieldin
plosive,pointforceanddoublecouple).Wethenshowanumerical thefrequencydomain,therefore,isgivenby
exampleinwhichahorizontalpointforceexciteselasticwaves.
U(r,ω)= A(ω)j(kr), (8)
l l l
THEORY whUerseinAgl(tωhe)FisoaurFieorucrioenrvceonetfifioncife(nt)t.=(cid:8)F(ω)e−iωtdω,thewavefield
In this section, we show that improved temporal focusing leads inthetimedomainisgivenby
(cid:9)
to improved spatial focusing for both P and S waves. Just as in
the theory for the excitation of elastic waves (Aki & Richards u(r,t)= A(ω)j(kr)e−iωtdω. (9)
l l l
2002), we treat the medium as locally homogeneous near the fo-
cus.Thecomplexityofthewavefieldduetoheterogeneityiscap- Thewavesimpingingonthefocusaredeterminedbytheincoming
tured in the wave that is incident on the focus. We thus consider waves.Forlargeradiusr,theincomingwavescanbewrittenas
a locally homogeneous elastic medium where either P, SV or SH
f(t+r/c)
wavesareincidentonafocalpoint.Accordingtoexpression(8.13) uinc(r,t)= l as r →∞. (10)
l r D
of Aki & Richards (2002), the elastic wavefield can locally be o
w
expressedas We show in the Appendix that the total wavefield is given for all n
(cid:2)(cid:3) valuesofrby, loa
d
u(r,θ,ϕ)= lm Ul(r)Rlm(θ,(cid:4)ϕ)+Vl(r)Slm(θ,ϕ) ul(r,t)=(−i)l 2c (cid:9) (−iω)Fl(ω)jl(kr)e−iωtdω, (11) ed from
+W(r)Tm(θ,ϕ) . (1) h
TRhlme(θv,ecϕt)o=rspYhlme(riθlc,aϕl)hrˆalrm(Ponwicasviens)e,q.(1)aregivenby (2) iwnhtBeegreercaaFtuilo(snωes),oiwusrethdeemeFrpiovluoartyiioetrhnetriafnonvlsolfolovwremisngorefnpofelt(aatt)te.iodn:differentiations and ttp://gji.oxfo
Slm(θ,ϕ)= (cid:5)l(l1+1)(cid:6)∂Ylm∂(θθ,ϕ)θˆ+ sin1θ ∂Ylm∂(ϕθ,ϕ)ϕˆ(cid:7) fl(n)(t)≡ dndftln(t). (12) rdjourna
(cid:6) (SV waves)(cid:7), (3) tFiomrens.egInattihveefvreaqluueesncoyfdno,mthaiisn,ntohteatnioontatiimonpl(i1e2s)intrtaengsrlaattiensginflt(ot) n als.org/
Tlm(θ,ϕ)= (cid:5)l(l1+1) sin1θ ∂Ylm∂(ϕθ,ϕ)θˆ− ∂Ylm∂(θθ,ϕ)ϕˆ Fl(n)(ω)≡(−iω)nFl(ω). (13) t Colora
Next,werelatethetotalwavefieldu(r,t)tothesphericalBessel d
(SH waves), (4) functionoforder0whichmakesitposlsibletoevaluatetheFourier o Sc
h
whereY (θ,ϕ)denotethesphericalharmonics.Theradialfunctions integral in eq. (11) analytically. This derivation, found in the oo
UHla(nrk),elVfllmu(rnc)tiaonndsWwlh(ri)charseatsispfhyeeriqc.a(l8B.6e)sosfelAfkuin&ctiRonicshaorrdssp(h2e0r0ic2a)l. Appendix,give(cid:6)s1: d (cid:7)l(cid:10) f(−l)(t+r/c)− f(−l)(t−r/c)(cid:11) l of Min
Westressthatthevectorsphericalharmonicsofexpressions(2)–(4) u(r,t)=clrl l l . es
l r dr r o
donotdenotethenormalmodesofafinitesystem,insteadtheyare n
N
representationsofthewavefieldinalocallyhomogeneousmedium. (14) o
v
Weconsiderthecaseofanincomingwavethat,atalargedistance em
fromthefocalpointr=0,isgivenbyf(t+r/c)/r.Westudythe This expression shows an explicit relationship between the be
properties of this incoming wave at thel focus for every angular total wavefield ul(r, t) for all values of r, and the incident wave, r 24
degree l separately. For a perfect aperture, the angular degree l fl(t+r/c)/ratadistance.Notethatusingthenotationofeqs(12) , 20
describesanexplosivesourcewhenl=0,apointforcewhenl=1 and (13), the incoming wave is integrated l times in the factor 14
andadouble-couplesourcewhenl=2.Sincethereisnosourceat fl(−l)(t±r/c),andthatthedifferentialoperator(1/r)(d/dr)actsl
thelocationr=0,thesolutionisfiniteandisthereforegivenby times as well. The term fl(−l)(t+r/c) is the wave that converges
sphericalHankelfunctionsj: on the focal point, while f(−l)(t−r/c) gives the outgoing wave
l l
thatradiatesfromthefocusaftertheincomingwaveshavepassed
Ul(r)∝ jl(kr) where k =ω/α, (5) throughthatpoint.Theincidentwavef(t+r/c)andtheoutgoing
l
withαrepresentingtheP-wavevelocity,and wavefl(t−r/c)haveoppositesignbecausethefocusisacausticin
twoangulardirections,hencetheMaslowindexincreasesbytwo,
Vl(r) and Wl(r)∝ jl(kr) with k =ω/β, (6) which corresponds to a sign change (Chapman 2004). Note that
where β is the S-wave velocity. Thus, the radial variation of the apointsourceis,bydefinition,locatedatacausticoftheexcited
orbackpropagatedwavefield.Thetheorypresentedtakesthisfully
wavefield is proportional to j(kr), with k the wavenumber of
l intoaccount.
the wave type under consideration. In the following, we study
We now demonstrate how we can use the expression for the
wavefieldswithradialdependencej(kr)anddenotethewavenum-
l wavefieldnearthefocalpointtodemonstratethatimprovedtemporal
beras
focusingleadstoimprovedspatialfocusing.ThespatialfocusR(r)
l
k =ω/c, (7) isdefinedasthewavefieldattimet=0.Itfollowsfromexpression
392 J.DoumaandR.Snieder
(14)thatthespatialfocusisgivenby: range of time values −t < t < t, where t is the half width of
f f f
(cid:6) (cid:7)(cid:10) (cid:11) thetemporalfocus.Expression(19)impliesthatthespatialfocus
Rl(r)≡ul(r,t =0)=clrl r1ddr l fl(−l)(r/c)−rfl(−l)(−r/c) . d(riaffdeirussaipsparlewciaaybslypofrsoitmivez)e.roAfogroovadluteemsopforratlhfaotcsuastis(fsym0al≤l tr)<thcutsf
f
impliesagoodspatialfocus.
(15)
Thespatialandtemporalfocus,andtheirrelationshipdefinedby
This expression gives the spatial focus in terms of the incoming eqs (15), (17) and (19) all depend on the order l. In this section,
wave. weshowtheexplicitformsoftheseexpressionsforthecasel=0,
Onemightbetemptedtodefinethetemporalfocusasu(r=0, l=1andl =2.Thesecasesarerelevantforanexplosivesource
l
t).Thisfield,accordingtoexpression(9),isproportionaltoj(kr) (l=0),pointforce(l=1)anddouble-couplesource(l=2)ifthe
l
inthefrequencydomain.Thezeroth-orderBesselfunctionj (kr)is aperturewouldbeperfect.Forl=0,eqs(15),(17)and(19)become,
0
non-zeroforr=0,butj(kr=0)=0forl ≥1(Arfken&Weber respectively,
l
2001).Thismeansthatforl≥1,thewavefieldvanishesatthefocal (cid:6) (cid:7)
f (r/c)− f (−r/c)
point.Physically,thisisduetothefactthatforl≥1,thefocalpoint R (r)= 0 0 , (20)
(r=0)islocatedattheintersectionofnodallines.SinceU(r=0,t) 0 r
l
vanishesatr=0forl≥1,thisquantityisnotausefuldiagnostic 2
ofthetemporalfocus.Toremedythis,wedefinethetemporalfocus T0(t)= c f0(1)(t) and (21)
insteadas D
(cid:10) (cid:11) o
dl c T(−1)(r/c)−T(−1)(−r/c) wn
T(t)≡ u(r =0,t). (16) R (r)= 0 0 . (22) lo
l drl l 0 2 r ad
e
d
nAosns-hzoerwonfoinrrth=e0A.pWpeenddeirxiv,ethinetlh-tehAdpeprievnadtiivxethoeffjol(lklro)wiisnfignreitleatainodn Eqs(20)–(22)arethesameasshowninthepreviousderivationof from
Andersonetal.(2014),whichdealtwithanexplosivesourceinan h
betweenthetemporalfocusandtheincomingwaves: acousticmediumwherethetemporalfocusisthetimederivativeof ttp
Tl(t)= c2lb+l1 fl(l+1)(t), (17) theWineconmowingcownativneu.e the derivation for R(r) due to a point force ://gji.ox
(l=1)anddouble-couple(l=2).Forl=1,eqs(15),(17)and(19) fo
whereblisgivenby become,respectively, rdjo
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b = 2l(l!)2 . (18) 1 rna
l (2l+1)! R1(r)= r (f1((cid:12)r/c)+ f1(−r/c)) (cid:13) ls.org
Sbpe=cifi1,callyb, =1/3, b =2/15. −rc2 f1(−1)(r/c)− f1(−1)(−r/c) , (23) at Co/
0 1 2 lo
ra
Accordingtoeq.(17),thetemporalfocusthusisproportionaltothe 3 d
(l+1)-thtimederivativeoftheincomingwave.Ofthese(l+1) T1(t)= 2c2 f1(2)(t) and (24) o Sc
h
derivatives,givenbyeq.(17),lareduetothederivativesindefinition o
ra(cli→1nom6dm0).tbfaliOk(ntinan+tegiotrtnihm/e(cfe)ll(i−dtrme+irtfilvrr(/at→ct−i)v−e0r/,ifscgl()idtvu=−eesrrlti→/omc0t)h)2/e(rr.f/aUccr)stifntl(cid:8)h(gat)ta(T=1a4y2c)locfrlo(cid:8)e(ntxt)ap.iannssitohne IRt1m(ra)y=ap−2p3ce22a3rcr13th(cid:12)r1aT2t1((cid:12)e−Tq2).1((−(r23/)3c(r))/+isc)sT−i1n(−gT2u)1((l−a−3r)r(a/−tcrr)(cid:13)/=c)(cid:13)0..Eventhoughe(a2c5h) ol of Mines on Nove
m
Thisexplainsanadditionaltimederivativeinexpression(17). ofthetwotermsinthisexpressiondivergeasr→0,thesingular- b
e
Iitniteesgcraatnincegl.thTishiosnccaengibveesvefr1(i−fi1e)(dr/bcy)w=riati0n(rg/fc1)(r+/cO)(=r2)a.0In+seOrt(inr)g. r 24, 20
thisintoeq.(23)gives 1
4
SPATIAL AND TEMPORAL FOCUS FOR 1 c (cid:3) (cid:4)
EACH ANGULAR COMPONENT R1(r)= r (2a0+O(r))−r2 2a0(r/c)+O(r2) . (26)
Thespatialandtemporalfocusdefinedbyexpressions(15)and(17)
Thetermsproportionaltoa ,whichcausedeachoftheindividual
0
bothrelatetotheincomingwavesandcanbecombinedtoexplicitly
termsinexpression(23)tobesingular,cancelout.Theremainder
relatethespatialandtemporalfocus: ofeq.(26)isfiniteasr→0.
c2l+1 (cid:6)1 d (cid:7)l(cid:10)T(−2l−1)(r/c)−T(−2l−1)(−r/c)(cid:11) Forl=2,eqs(15),(17)and(19)become,respectively,
R(r)= rl l l . (cid:12) (cid:13)
l 2bl r dr r R (r)=1(f (r/c)− f (−r/c))−3c f(−1)(r/c)+ f(−1)(−r/c)
2 r 2 2 r2 2 2
(19)
(cid:12) (cid:13)
Eq.(19)isthemainresultofthetheory.Althoughtherelation(19) +3c2 f(−2)(r/c)− f(−2)(−r/c) , (27)
r3 2 2
betweenthespatialandtemporalfocusiscomplicated,itdoesshow
that good temporal focusing implies good spatial focusing. Good
temporalfocusingimpliesthatTl(t)isstronglypeakedneart=0, T (t)= 4 f(3)(t) and (28)
that is, that T(t) only differs appreciably from zero for a small 2 15c3 2
l
Elasticwavesformicroseismicimaging 393
R2(r)= 154c3r1(cid:12)T2(−3)(r/c)−T2(−3)(−r/c)(cid:13) oonntehaechanwgalevseatnydpeo.nTshpearceefoarned,etiamche.wavehasitsowndependence
If one were to use one component of the motion, such as the
−454c4r12 (cid:12)Tl(−4)(r/c)+Tl(−4)(−r/c)(cid:13) xa-rceosmuppeornpeonste,dthoenseoaucrhceothperor.pSeirnticeesffoorrathfiexeddifsfoeurerncet wmaevcehatnyipsems
the radiation pattern of P and S waves are different, the focused
+454c5r13 (cid:12)Tl(−5)(r/c)−Tl(−5)(−r/c)(cid:13). (29) wanaivsemfi.eIlndosrddoerntootapvrooivdidmeicxlienagroinffPo-rmanadtiSo-nraadbioatuetdthweasvoeus,rcoenemmecuhs-t
decomposethewavefieldusingthedivergenceandcurlinorderto
Thewavefieldscomputedarefiniteatthefocalpointr=0when investigatethefocusofeachwavetype(PandS)separately.
onerefocuseseitherPorSwaves.Incontrast,whenPandSwaves Ifonedoesnothaveaperfectaperture,ablurredfocuswilloccur,
areexcitedbyapointforce,theP-wavecomponentandtheS-wave andthefocuscannotbecharacterizedbyoneangulardegreelbut
componentbehaveas1/r3asr→0,hencethePwaveandSwave bythesuperpositionofdifferentangulardegreesl.Thisappliesto
separately have a non-integrable singularity at r = 0, while their thespatialfocusachievedinournumericalmodelling.
sum has an 1/r singularity Wu (1985), which is integrable. The
refocused wavefields do not display this behaviour because these
fieldsaresource-freeatr=0,andthereforethewavefieldisfinite.
NUMERICAL EXAMPLE
Therefore,thePandSwavescanberefocusedseparatelywithout D
o
causingsingularities,andthetreatmentgivenhereisapplicableto Weillustratethetheorywithanumericalexample.Weusetheve- w
n
P,SVandSHwavesseparately. locitymodelshowninthetoppanelofFig.1topropagatethesource lo
a
d
Theexpressionsabovemustbemultipliedwiththeappropriate wavefieldtothereceivers.Themodelconsistsofhorizontallycon- e
d
vectorsphericalharmonic(2)–(4)toobtainthefullfocusedwave- tinuouslayerswhoseP-wavevelocitiesrangefromapproximately fro
field.Foreachwavetype,adifferentsphericalharmonicmustbe 4.7kms−1to5.9kms−1andS-wavevelocitiesrangefromapproxi- m
multipliedtocharacterizethewavefield.Additionally,ineqs(2)–(4), mately2.3kms−1to2.9kms−1.Inpractice,onedoesnotknowthe http
thevectorsphericalharmonicsaresummedovertheangularorder truevelocitymodel.Forthisreason,weusedthesmoothedvelocity ://g
landdegreemwhichcapturestheimprintofthesourceproperties model,showninthebottompanelofFig.1,forthebackpropagation. ji.o
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, 2
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1
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Figure1. P-wavevelocitymodelsofthenumericalexperimentwithunitsofkms−1.Toppanelindicatesthecorrectvelocitymodelandrepresentsthevelocity
modelusedtopropagatethesourcewavefieldthroughthemedium.Bottompanelindicatesthesmoothedvelocitymodelwithcorrectmeanslowness.This
modelisusedforbackpropagationofthetime-reversedsignalandoptimizedinversesignal.Theplussymbolsrepresentthereceivers,thecirculardotrepresents
thesource.TheS-wavevelocitywasthesamebuthadvelocitiesvaluesequaltohalfoftheP-wavevelocity.
394 J.DoumaandR.Snieder
P S
D
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ttp
P, TR S, TR ://g
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P, DC S, DC f M
in
e
s
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Figure2. DecomposedPwavefieldattimeoffocusduetohorizontalpoint Figure3. DecomposedSwavefieldattimeoffocusduetohorizontalpoint n N
force.ToppanelisPwavejustafterthehorizontalpointforceisemitted. force.ToppanelisSwavejustafterthehorizontalpointforceisemitted. o
v
Middlepanelindicatestheresultofinjectingthetimereversedsignalback Middlepanelindicatestheresultofinjectingthetime-reversedsignalback em
intothesmoothedvelocitymodelfromthereceiverlocations.Bottompanel intothesmoothedvelocitymodelfromthereceiverlocations.Bottompanel be
showsresultofinjectingtheinversesignalcalculatedusingdeconvolution showsresultofinjectingtheinversesignalcalculatedusingdeconvolution r 2
4
backintothesmoothedvelocitymodelfromthereceiverlocations. backintothesmoothedvelocitymodelfromthereceiverlocations. , 2
0
1
4
Thevelocitymodelissmoothedbyusinga2-Dtrianglesmoothing calculated the energy ratio of the signal to noise, defined as the
of the slowness with a smoothing radius of .185 km in the x and ratio of the sum of the signal and noise squared, to be equal to
z directions(Fomel2007).Thissmoothedvelocitymodelhasthe 0.89.
samemeanslownessasthecorrectvelocitymodel. It is important to note that all the numerical work is done in-
Wefirstuseahorizontalpointforcelocatedat(x,z)=(0.51km, plane. The source wavefield propagates through our model. The
2.68km).ThesourceischaracterizedbyRickerwaveletwithdom- vertical and horizontal displacement, which are recorded at the
inantfrequencyof100Hz.Thereare56receiversdistributedover receiverlocations,areorientedin-plane.WhenwesolvefortheS
twoverticalboreholesinourmodel.Thex-locationsofthereceiver componentbycalculatingthecurlofthewavefieldusingthevertical
boreholes are 0.74 km and 0.88 km, respectively. The receivers and horizontal displacement, we imply that we are investigating
rangefromadepthof2.36kmto2.86kmwithaspacingof18.5m. the SV component. Therefore, we do not model the out of plane
In order to make the numerical simulation more realistic, we SH component. In this numerical example, sensors are deployed
addedbandpasslimitednoisetoourrecordeddataatthereceivers in two vertical boreholes. The theory presented here is, however,
beforeapplyingtimereversalordeconvolution.Theadditivenoise equallyvalidfora2-Dsensornetworkplacedon,ornear,theEarth’s
only contains frequencies within the bandwidth of the data. We surface.
Elasticwavesformicroseismicimaging 395
P, TR P, DC
D
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lo
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e
d
fro
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S, TR S, DC ttp://g
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Figure4. Temporalfocusedimagesduetoahorizontalpointforceproducedbybackpropagatingthecalculatedtime-reversedsignalsusingtimereversaland rn
a
deconvolutionforverticalboreholearray.Parts(a)and(b)aretemporalfocusofthePwaveduetotimereversalanddeconvolution,respectively.Parts(c)and ls
.o
(d)aretemporalfocusoftheSwaveduetotimereversalanddeconvolution,respectively. rg
a/
Horizontalpointforce Thederivationandexplanationofthesetwomethodsarediscussed t Co
Thissectiondescribesthenumericalmodellingthatdemonstrates inmoredetailbyDoumaetal.(2013). lora
Afterbackpropagation,thewavefieldisdecomposedintoPand do
thatimprovedtemporalfocusingleadstoimprovedspatialfocusing S
Scomponentsforacrucialreason.Wedemonstratedinthe‘The- c
foreachwavetype.Wefirstmodelthewavefieldduetoahorizontal h
ory’ section that improved temporal focusing leads to improved oo
pveorintitcaflordciespelaxcceitmateionntsaotftthheeswoauvrceefiellodcaatrieonth.eTnhreechoorrdizeodnatatleaancdh spatialfocusingforeachwavetype.Wedonotconsiderthefocus l of M
fortheverticalorhorizontaldisplacements.Rather,weusethedis-
receiver. Afterwards, we apply either time reversal or deconvolu- in
tion to the recorded signals to generate the wavefields, which are placement components to calculate the P and S wavefields using es o
divergenceandcurl,respectively.ThisallowsustoretrievetheP n
backpropagated. N
andSwavesthathavebackpropagatedfromthesources.Foreach o
Weusethetime-reversedordeconvolvedsignalstoexcitewaves v
wavetype,ourtheorypredictsthatanimprovedtemporalfocusing em
thatbackpropagatethroughthesmoothedvelocitymodel,usingthe leadstoimprovedspatialfocusing. be
followingforcesactingateachofthereceiverlocations: Wefirstmodelthewavefieldduetoahorizontalpointforceex- r 24
F(cid:10)TimeReverse=(Ux(−t),Uz(−t)), (30) citationatthesourcelocation.ThetoppanelofFigs2and3show , 201
the P and S wavefields’ radiation pattern just after the horizontal 4
(cid:3) (cid:4) pointforceisemittedandrepresentapureangulardegreel=1.In
F(cid:10) = UInverse(t),Uinverse(t) . (31) perfectsourceimaging,wewouldreconstructtheseradiationpat-
Deconv x z
terns.However,ouraperatureisnotperfectandwebackpropagate
(cid:10) (cid:10)
Here, FTimeReverse and FDeconv arethesourcefunctionsfortimere- throughasmootherversionofthevelocitymodel.Thus,wedonot
versalanddeconvolution,respectively,andUx(t)andUz(t)arethe expecttobeabletoreconstructtheseradiationpatternsperfectly.
recordedsignals.Theinversesignalofatime-seriesg(t)isdefined Inordertoshowthatdeconvolutiongeneratesanimprovedspatial
as focus,wefirstdemonstratethatdeconvolutionenhancesthetempo-
ralfocus.Thus,wecalculatethetemporalfocusingforthePand
ginverse(t)(cid:9)g(t)=δ(t), (32)
Scomponentasaresultofdeconvolution(eq.31)comparedwith
where (cid:9) denotes convolution. In order to solve for ginverse(t) and timereversal(eq.30).Wedefinedthetemporalfocusingineq.(16)
avoidinstabilityforginverse(ω)wheng(ω)=0,wehavetoapplya asthel-thderivativeoftheincomingwavefield.Thisisnecessary
waterlevelregularization.Thus, because the wavefield is zero at our source location due of nodal
lines.Inordertodemonstrateimprovedtemporalfocusingforahor-
1 g∗(ω)
ginverse(ω)= ⇒ . (33) izontalpointforce,wetakethederivativeofthePwavefieldinthe
g(ω) |g∗(ω)|2+(cid:10) x-directionandthederivativeoftheSwavefieldinthez-direction
396 J.DoumaandR.Snieder
P, TR P, DC
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Figure5. Spatialfocusedimagesduetoahorizontalpointforceproducedbybackpropagatingthecalculatedtime-reversedsignalsusingtimereversaland als
deconvolution.Parts(a)and(b)are1-DslicesofFig.2throughdepth2.68km.Parts(c)and(d)are1-DslicesofFig.3throughxlocation0.51km.Notethe .o
rg
differentscalesusedforthecross-sectioninthexandzdirection. a/
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because these derivatives are the radial derivatives perpendicular timereversal(middlepanel)anddeconvolution(bottompanel)for lora
tothenodallinesforeachwavetype.Wechangethedirectionof theScomponent.Thiswasexpectedduetodeconvolutionandtime do
S
thederivativebecausetheradiationpatternofthePwaveduetoa reversalproducingsimilartemporalfocusesfortheSwave. c
h
horizontalpointforceisadipoleinthex-directionwhiletheS-wave Theaperture,overwhichwerecordthedatathatwebackprop- oo
radiation has a dipole pattern oriented in the z-direction (Aki & agate,isnotperfect.Thiscausesthespatialfocuses,createdusing l of M
Richards 2002). This is visible in the top panel of Figs 2 and 3, timereversalanddeconvolutionshowninFigs2and3,tonotbe
in
which show the radiation patterns of the P and S wavefields just confined to one angular degree l because the spatial focuses are es
o
afterthesourcehasacted.Wecalculatethederivativesasdefined blurredinthez-direction.Aperfectspatialfocuswouldconsistof n
earlier to show the temporal focus for the P and S wave at the onlythel=1component.Figs5(a,b)and(c,d)showcross-sections No
v
sourcelocation.ComparingFigs4(a)to(b),onecanclearlynote ofthebackpropagatedwavefieldsinFigs2and3inthex-andz- em
thatdeconvolutionhassignificantlyimprovedthetemporalfocus- directions,respectively,sothatitiseasiertoassestheimprovements be
ingcomparedtotimereversalforthePwave.Incontrast,Figs4(c) andcomparisonsbetweenthetwomethods.Notethatthescalesof r 24
and(d)showthatbothtimereversalanddeconvolutionproducea thehorizontalaxisforFigs5(a)and(b)aredifferentfromFigs5(c) , 2
0
1
similartemporalfocusfortheSwave.Becauseimprovedtemporal and(d).Fig.5(a)demonstratesthattimereversalisnotabletocreate 4
focusing implies better spatial focusing, see eq. (26), one would awell-defineddipolefocusinthex-direction,whichrepresentsthe
expecttoseeanimprovedspatialfocusforthePwaveusingdecon- radiationpatternofaPwaveduetoahorizontalpointforce.Fig.5(b)
volutioncomparedtousingtimereversal.Additionally,wedonot showsthatdeconvolutionisabletoreconstructthedipoleradiation
expecttheSwave’sspatialfocustoimproveusingdeconvolution patternofthePwaveduetoahorizontalpointforce.Figs5(c)and
becausethetemporalfocusdidnotimprove. (d)demonstratethatinthisexamplethereseemstobenosignifi-
Afterhavingdemonstratedthatdeconvolutionimprovedthetem- cantdifferencebetweentimereversal(c)anddeconvolution(d)to
poralfocusingforthePwave,wecomparethespatialfocusgen- reconstructtheSwave’sfocus.
eratedbydeconvolutionandtimereversalforeachwavetype.The Ournumericalresultshaveshownthatdeconvolutionwasableto
backpropagatedwavefieldsatt=0forthetwomethodsareshown improvethetemporalfocusforthePwave,whichledtoanimproved
inthemiddleandbottompanelsofFigs2and3.Themiddlepanel reconstructionofthePwavefield’sradiationpattern.However,de-
ofFig.2representsthespatialfocusofthePwaveusingtimere- convolution was not able to improve the temporal focus for the
versal whereas the bottom panel shows the spatial focus of the P S-wave, due to a horizontal point force, which led to it also not
wave using time-reversal. Fig. 2 shows that deconvolution drasti- improvingthereconstructionoftheSwavefield’sradiationpattern.
callyimprovesthespatialfocuscomparedtodeconvolution.Fig.3 ThiscanbeattributedtothefactthatanodallinefortheSwave-
does not show a clear improvement of spatial focusing between field’sradiationpatternintersectsthereceiverarray.Deconvolution
Elasticwavesformicroseismicimaging 397
P S
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P, DC S, DC f M
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Figure6. DecomposedPwavefieldattimeoffocusduetoverticalpoint Figure7. DecomposedSwavefieldattimeoffocusduetoverticalpoint n N
force. Top panel is P wave just after the vertical point force is emitted. force. Top panel is S wave just after the vertical point force is emitted. o
v
Middlepanelindicatestheresultofinjectingthetime-reversedsignalback Middlepanelindicatestheresultofinjectingthetimereversedsignalback em
intothesmoothedvelocitymodelfromthereceiverlocations.Bottompanel intothesmoothedvelocitymodelfromthereceiverlocations.Bottompanel be
showsresultofinjectingtheinversesignalcalculatedusingdeconvolution showsresultofinjectingtheinversesignalcalculatedusingdeconvolution r 2
4
backintothesmoothedvelocitymodelfromthereceiverlocations. backintothesmoothedvelocitymodelfromthereceiverlocations. , 2
0
1
4
assignsalargerweighttothereceiversnearthenodallineinorder
viouslystated,thesourcewaveletwasfirstinjectedintothecorrect
toincreaseaweakrecordedsignal.Thisisunphysicalbecausethere
velocitymodelatthesourcelocation.Thehorizontalandvertical
isnoinformationtobegainedintheseweakrecordedwaveforms
displacementswerethenrecordedatthereceivers.Oncethesesig-
near the nodal lines. These receivers are supposed to record no
nalswererecorded,timereversalanddeconvolutionmethodswere
informationaboutthesourceandshould,therefore,notpropagate
applied in order to calculate our backpropagating signals. These
anyinformationback.However,itsimultaneouslydemonstratesthe
signalscalculatedusingthetwomethodswouldthenbepropagate
robustnatureofdeconvolution.Fortheradiationpatternwhichhas
backintothesmoothedvelocitymodelfromthereceiverpositions.
anodallineintersectingthereceiverarray,deconvolutiondoesnot
WethendecomposedthewavefieldintothePandScomponentat
generateaninaccuratebutratheracomparablereconstructionofthe
thetimeoffocusbytakingthedivergenceandcurl,respectively,of
radiationpatternastimereversal.
thedisplacements.
ThetoppanelsofFigs6and7showthePandSwavefield’sradi-
ationpatternsaftertheverticalpointforceisemittedandrepresents
Verticalpointforce
thepureangulardegreel=1.Asstatedpreviously,onewouldbe
Weappliedthesameanalysistowardsaverticalpointforceusing abletoreconstructtheradiationpatternsinperfectsourceimaging.
thesamenumericalmodelasforthehorizontalpointforce.Aspre- However, weusethesamenumericalmodelasusedforthehori-
398 J.DoumaandR.Snieder
S, TR S, DC
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P, TR P, DC ttp://g
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Figure8. Temporalfocusedimagesduetoaverticalpointforceproducedbybackpropagatingthecalculatedtime-reversedsignalsusingtimereversaland rna
deconvolutionforverticalboreholearray.Parts(a)and(b)aretemporalfocusoftheSwaveduetotimereversalanddeconvolution,respectively.Parts(c)and ls.o
(d)aretemporalfocusofthePwaveduetotimereversalanddeconvolution,respectively. rg
a/
t C
o
lo
zontalpointforce.Thus,wedonotexpecttobeabletoreconstruct at t = 0 using either time reversal (middle panel) or deconvolu- ra
d
theradiationpatternsperfectly. tion(bottompanel)forthePandScomponent,respectively.Fig.7 o S
c
Wefirstcalculatedthetemporalfocusesachievesusingtimere- shows that deconvolution significantly improves the spatial focus h
o
rvaedrsiaatlioonrdpeacttoenrnvoslouftiothnefvoerrtthiceaPl paonidntSfocrocmepaorenernottsa.teBdec9a0u◦sefrothme aacchhiieevveeddffoorrtthheePSwwaavveea;rFeicgo.m6pdaermabolnesutrsaintegsetihthaterthdeecsopnavtioallutfioocni ol of M
thehorizontalpointforce’s,wetakethederivativeofthePwave- ortimereversal.Theseresultswereasexpectedbecausedeconvo- in
e
fieldinthez-directionandthederivativeoftheSwavefieldinthe lutionimprovedthetemporalfocusfortheSwavewhilegenerating s o
n
x-directioninordertotaketheradialderivativesperpendicularto acomparabletemporalfocusforthePwave. N
the nodal lines for each wave type. Comparing Figs 8(a) to (b), Inordertovisualizethedipolefociachievedusingeithermethod, ov
e
onecanclearlynotethatdeconvolutionhassignificantlyimproved we generate 1-D slices through Figs 6 and 7. Figs 9(a) and (b) m
b
the temporal focusing compared to time reversal for the S wave; and(c)and(d)showscross-sectionsofthebackpropagatedwave- er 2
whereas,inFigs8(c)and(d),timereversalanddeconvolutionpro- fieldsinFigs2and3inthex-andz-directions.Fig.8(a)illustrates 4, 2
duceasimilartemporalfocusforthePwave.Therefore,theresults that time reversal is not able to create a well-defined dipole fo- 0
1
4
fortheverticalpointforcearethereverseofthatforthehorizontal cusinthex-direction,whichrepresentstheradiationpatternofan
pointforce.Thisisexpectedastheradiationpatternsarerotatedby S-wave due to a vertical point force. Fig. 9(b) shows that decon-
90◦,whichcausethenodallinestorotate.Wehaveseentheimpact volution is able to reconstruct the dipole radiation pattern of the
ofthesenodallinesuponthefocusweareabletoachieve.Asstated S wave due to a vertical point force. Figs 9(c) and (d) demon-
previously,becauseimprovedtemporalfocusingimpliesbetterspa- stratethatthereseemstobenosignificantdifferencebetweentime
tial focusing, see eq. (26), one would expect to see an improved reversal(c)anddeconvolution(d)toreconstructthePwave’sfo-
spatialfocusnowfortheSwaveusingdeconvolutioncomparedto cus.Theseresultsareadifferentwaytoillustratewhatwasshown
usingtimereversal.However,wedonotexpecttoseeanyimprove- inFigs6and7.
mentinspatialfocusingforthePcomponentbecausedeconvolu- Ournumericalresultsfortheverticalpointforcereiteratewhatwe
tionandtimereversalgeneratedcomparabletemporalfociforthe haveseeninthehorizontalpointforceresults.Weconcludethat,for
Pcomponent. anelasticmediawithoutaperfectapertureandtruevelocitymodel,
Onceshownthatdeconvolutionimprovesthetemporalfocusfor improved temporal focusing leads to improved spatial focusing.
theSwaveduetoaverticalpointforce,wecomparethespatialfoci We have shown this both theoretically and numerically to be the
achieved using deconvolution or time reversal for both the P and case.Becausedeconvolutionhastheabilitytoimprovethetemporal
S components. Figs 6 and 7 show the backpropagated wavefields focusing,onecanimprovethespatialfocusing.
Elasticwavesformicroseismicimaging 399
S, TR S, DC
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Figure9. Spatialfocusedimagesduetoaverticalpointforceproducedbybackpropagatingthecalculatedtime-reversedsignalsusingtimereversaland als
deconvolution.Parts(a)and(b)are1-DslicesofFig.2throughdepth2.68km.Parts(c)and(d)are1-DslicesofFig.3throughxlocation0.51km.Notethe .o
rg
differentscalesusedforthecross-sectioninthex-andz-direction. a/
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CONCLUSION Artman,B., Podladtchikov,I.&Witten,B., 2010.Sourcelocationusing S
c
time-reverseimaging,Geophys.Prospect.,58,861–873. ho
We have introduced deconvolution which improves the temporal o
focusing of microseismic events. The workflow for this approach Autbimrya,lJf.o-Fc.u,sTinagntebry,sMpa.,tiGo-etrebmerp,oJr.a,lTfihlotemr.aIsI,.JE.-xLp.e&rimFeinntks,.MAp.,p2li0c0a1ti.onOpto- l of M
is similar to imaging with time-reversed signals. The only differ- focusingthroughabsorbingandreverberatingmedia,J.acoust.Soc.Am., in
enceisthatinsteadofsendingthetime-reversedrecordedsignals 110,48–58. es o
into the medium, we send the inverse signal, as obtained by de- Bancroft, J.C., Wong, J. & Han, L., 2010. Sensitivity measurements for n N
convolution, into the medium. The key result of this work is that locatingmicroseismicevents,CSEGRecorder(February),28–37. o
v
we demonstrate theoretically and numerically that this improved Bertaix,V.,Garson,J.,Quieffin,N.,Catheline,S.,Derosny,J.&Fink,M., em
temporalfocusingleadstoimprovedspatialfocusingforeachwave 2004.Time-reversalbreakingofacousticwavesinacavity,Am.J.Phys., be
typeinanelasticmedium.Thisimprovedspatialfocusingisben- 72(10),1308–1311. r 24
eficialforenhancingthefocusoftheelasticwaves.Thesimplicity Bose, S., Valero, H., Liu, Q., Shenoy, R.G. & Ounadjela, A., 2009. , 20
An automatic procedure to detect microseismic events embedded in 1
androbustnatureofthismethodallowsforasimpleincorporation 4
high noise, SEG Technical Program Expanded Abstracts, pp. 1537–
intoexistingreverse-timeimagingmethods.Additionally,thecost
1541.
ofdeconvolutionisminimalcomparedtorunningthefinitediffer- Chapman, C., 2004. Fundamentals of Seismic Wave Propagation, Cam-
encemodelling.Therefore,itcanbeaddedasapre-processingstep bridgeUniv.Press.
withoutsignificantadditivecost. Douma,J.,Snieder,R.,Fish,A.&Sava,P.,2013.Locatingamicroseismic
eventusingdeconvolution,SEGTechnicalProgramExpandedAbstracts,
pp.2206–2211.
Duncan,P.M.,2005.Isthereafutureforpassiveseismic,FirstBreak,23,
111–115.
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