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At!'INVESTIGATION OFCONSTANTVELOCITY. : GRADIENT, EFFECTSINSEISM IC ANALYSIS

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BV.

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-"0 ""- .". 'Athesis sutnftted to the

Scho~l

^ofG";aduate

._Studies in pary-ialfulfillmentof..the:"

requirementsfor,the'degr ee nt"""

Mastet.ofEngineering:

Fa cul tyofEngineed rJ9and,AppliedScience Mt5IlorialUniver,si t y nfNewfoundla nd

Augus.t1983

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St.John.'s· Newfoundl and

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Canada

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TOMY.FRIEND Y~WATH I [.HUNASIHGHE.

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. . ' . .

ANINVESTIGATION OF.CONSTANT VELOCITY GRADIENT,EFFECTS~NSEISMIC ANALYSIS.

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ABSTRACT

. Inf onnatfon~oncernin9acoust icvelocHfe s'pla ys,animportantrole

. . '

.

^{; .}

,

inseismic"analysis. Acoustic.v'l;!l o'c'ttyofthemediumcanbe a pprox - jmatedtiy

a,~\P.Ol.ynom.ial fu~~t1on o~ep~. The ,conc~Pt

^thatvelocHy

ver tes l1nearl YwitH depthhasbeen constder ed.b); manyresearchers' .

_

: i:'~S

^an,

,appropr·ia~.e !S~IIIlPt!O~.

^for'

devel0P.lng ~~'he ·~~locttY. ^{p rof' tle'} fO~·.

^;

• ,_I •.'."':amedf~:' ~'~ • •" . ,' ." ;"." .... ?-

.'In . ,_..

;tu~y:

_. ^{the lin}_{. ,}^ear'

~el0cf~Y'

_{.. .}

P;0f.11~ appr~~~h

_. ^has_.

~e~

_{. '}

\u

_\

seti

_'.

~o

_r"^._,

.es~fmatetne.paraeet er sof themed,iU~.. The~onnalmoveo~~._r~\1ation-1 shipf,or a singlezer0-e:tfppfng 'reflector has-be~nderived;' ,A metho -\ dologyforobtainingthep~rameter~suchas the.reflectorgecrnetry and t. ,heconstant

. V~loc~lt"/grad~

^{. '}

ent

^{of the}

. me~fu~

^{.f rcrn}

till~

^{. surface}

cbserveble x-tda~~'ispresented.~

Fur t her , the lfnea'r velocityproftlemodelfor,aslopingreflec~9r^.has _ ____ _ _~ been~~rf.Ved.ba sedon the'nomlalrayanal ysis : Based onthe knowledg e :of ttle,ve.locityprofile.t.heleastsqueres-techntwehasbee'nusedto

. " ' " . ~ . ' " ",

identifyth~refl ect orge~etry ~rOOlret.lecttondata. ,Thex- t da'ta

~has beenobta'inedby shifting the shot/rece'1verpositionProgressively... ' A'comparativeanalys is,hasbee~ 'mad~toobta~ntheeffectof constant.

~el o.c1tygradientin setsntcan!llysis.

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ACKN OWlEDG EMENT

"1 wfsh.toexpres slIy gra titudeto

Qr:

W.J .Ve; t er,_: rof.essor·in

El ec tric alEngineering ,t'l8llod al~n 1v~rsity'ofN~Wfo~ridland~~anada, forprov1d'1~^metheoppor.~n.ltY'^to'do're~earch ~nd~~^{hisglJYl ance·..}

• '.He.hasbeen.ins~l"\JI11ental.in'the~~Hia.i10n,·.c.~d~·ct'andcOO1pl~tion

'''of,

^thisr.~sea.r.ch ",,?rk.. · ^.

.

. '

. :My'thanksaredue toDr..acss.Peter s .Deanof

Eng1neerf~g

and Dr' T; R.

.

C~a~l,ASS;cia~e' De.in of.;,.,[ng-I~~e~l~g

^{'; or}

p~OY f~ing

^the

necess:~:ry

^fac-

HHiesforIconductin g,

'~his re~ear~h

^work."

It.is

~y d~ty

^to

tha'~kDl' .

F.'Aldr1ch, DeanofGraduat eStudt es for. makf ng1tpc sstblefor meto per-suestu~ iesat.ManorialUn1y.ers1ty of Newfoundland.

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TABlEOFCONTENTS

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v,

.TITLE.

ACKlIOWLEIXi EMENT

.1.1 1.2 CHAPTER It

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vii

1

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2₁₂

.

'<, 14

is 15 16 20 22 25 26

·29

" .

3.

39

40

4 1

42

CONCLUSION Am> FURTHERAREASOFRESEARCH

·,A~~~~~~~tO~nt~~a;~:e~~m:f..constant Velocity Analysisof the'Eff ect" of COnstantVel'ocHy

nrecfe nt-cnRaypath" - A"nalyslsof the Effect ofConsta ntVeloc 1ty

Gradfent.onthe Devfation Ccmpared.loConsta nt

~V el oc. i tyAsslM1pt10n

Analysisofthe £ff'ects'ofCons~~'tVeloc.1ty, . Gradfent.on. Est 1matten-of-the--Raflector,~eanetry Impl 1cat1onso{F1ndfngs

Deep SeismicExploration Sha)lowSe1sm fcAppl1cjt.1ons

~~~~llMoveout,'RelatfonShiP

^{forthe .}

^{'; 1}

linear.Veloci ty Prof il e Model :,'

I

Es timat ion'of the Paramete rsof theMedium ., . From...."'.rfl ceHeasur enents USfl'l9." NO)"ll1dl I'.

. HovequtRela ti on" ' ,".. • ' _ . _ THEONUiAYTRAVELTIME'FOR

f;O~Al INCIDENCE

..Iilo...

. .. . . ·•.1

LINEAR VELOCITYPROFILE MODEL.FOR DIE'PING

REFLECTORS t.·.· '''',

General ,

Nonnal'R4YAnalysisonADippingReflector US,1ngt.tnearVel OC,ityProfi,leMod~l

N~~~~1~.iia~f~~oaCh .. .

^to

th~~~le~t~':

ANALysts OF ERRORSDUETO COKSTflNTVELOCITY ASSUMPTION IFTHE HEDIltlHAS A'LItlEAR VELOCITYPROFILE

pAGE NO;'

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46

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:.48 50 51

"

'.58

61 62 63

64

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^~

72 '76

71

77

7.

79 .0 .0

.,

az'

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VelocHy Spectra

n

CoherencyHea.sur Em e nt ~ Uses ofthe VelocitySpectra LINEAR VELOCITY

P~'oFtlE M'~DllJM

I ;

.-,,'

I ·

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a.a

3~J.1 3.4 111-8 3.5.0 3.5.-'

! 3.5.2

APPEND'i~

^3.5

.'CHAPT~IV

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4.1

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_{.~AG E}NO.

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I

REFERENCES"

.S I

BiBliOGRAPHY 90

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. :(-: ^{. .... , - .} _-j.-

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.,

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.;

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Tabl e .1

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LISTOF.TABLES· .

'--"

Pag!!

'.' .

I

',"c

5.3 "Errors,Due.,toConstantVe lOCli · AssUmptlon If,the HediulI,t!a, '.

.Con§,tant.Veloc 1tY,Gradi ent " .0.'.5

. ~. " .. . : -r ' .

129

)

'

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, ..,

95, -

112 108 109"

'11,0.

:.11\' ':"page

P3

'to 9"4."

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".': .

: :;',-

..-.~,

, .'

.1;' . .

, ll ST.Of"ILwd'RATIONS

• .. , ....

.vll1·

Figure

;--

Oblique' Inctdenteaten-Interrace

I . ...._. '." ' . ;. _ ..:" .

3.3.1 CoherenCi,esfO~_·.Va.r tqus .Traje~t9ri,~.~.

\ ,",,'3,3.

'. 2- V~l~ct~ Sp~'t;~ra Showing Ref1ec t'~n

^loss

cau~ed

^•

\ByS~Y I • '.

<:»:

'J)j.O

S:~~~.tw1~e",_.c:om;~~nt. 'a~d, ~f ~.~:~!~eli~_!_,t;:. pt~~ne5~

j.S;,·;.,

Canln.o~·Dep~h:,_P~fnt.^DataGat~rf n9

'.~

'~''-

- ,

12~

~-125

126

127.

I

12&

.c:> ,

Page

117

i

118

119

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II

12Q I

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'21

^{~. b}

I

I

122

l'

":'123'

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i· tx

.. '

.",

5.4.4 5.4.1

5'.3.11 DepthZ~vs"=Error,(Zo-~)^40rl'~^1., Corllspondence..Slope11'10Deg-r.ees

"' 1 ' , ..

Dept h2₀vsError(Io •;)Fp.r·1~1 Corr es pondenc e,.4-20Degree s "

ErrOr~·ti,u! to''const~n't velocftY·As'si..Ptl~n ·

^.'

Zo" '500•Meter s ' .

Err ors DueTo'Co~stant^velocity AsSumpt io n'.'.•,

. \.. Zo ~,2000.~et~!rs ..

5.4.3 .:~ro,:~,Du~o~;i;;~ nt,VelocftYA~5lmlptfon;

0, .~ . ' .

Corr esPondingShUt'of.Ppfnts.fro:lTrueReflector ..du.!'toCon'StantV.elo~ty.AS_slnp~fon

". 5.4.2 .Figur e

5:\.5 Raypaths.

f~t

^linear,-OelodtrPrott le.Model,

• a "

20Deg:~e6S. -~o ~ 500~~lers"

" :t.'~~ ·'_:,··

^{5.2-.6 Raypaths}

_a".

20 Degrees'.^For'linear.^VelocityZ,'!'1000^{Prof tle}Meter s^MoeI'^&)., '

,•.~~ , .0_ ' ' ••

;'..",5.3.7· 'Depth

~ y~ .Err.?r(X~ ~

^XN)"

Due,T~~nstant

Vel-oc1tyASSU·l!I.~tionlt~.;10 Degrees

~· 5.3.8.

^DeP:h

Z~ v~

^Error(Ip

:x~') Du~" !o .c·onst~nt·

.Veloc ity Assumption.6-20'Degrees ....

5',3.9 De.;th

Z~ y'~'~~ro,.·.(z,i ~

^Zp·l.Dueto

CO;'lS~ nt:.

Veloctty~ssump t1on ;:6-IO'oegrees I

S.3 .1 0",·D,eJU:h Z

6

^{'VsError,(IK- Zp):p ueTo Constant"}

....V"el oclty:As~ump.tlon.61120'[Tegree~ .

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.

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I~

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~.

OVERVIEW . "

•We

pr~sent her~

^abriefsurrrnaryof the

t~riten~

^{theSUbsequl ntchapters.}

Detll'iledref erences~h~tnere turee-enoe qtvenhereasthey"are. IN;,esen t'.edattheappropr iatepoints,i n 'the subsequentma te~ia l .

' -.

. . , ... . ^.

j .

In:hap~I'.th~:sei.snlc,~bjectlvesarebrieflyd,!SCUSSedinorderto. facilitate the u,nder sta ndfng-ofthe probl em. Datagathering,processing, migrationand

i riterPt~tatf on

^are.

the

maln poi nt s ccnstdered.

9bje<:tive~

of the-thesisere alsoprGsented.

In Chap terII;the fundillTle nta ls'of acoust ic'wavepropagat1 ona nd~thei r rela tions tothep~ra~eter~oftheA).edfaaredisc,lls sed."-i~differen t raytheor et icalapproa~hesto the _waveequat i onar epresented; Th~

behav.,our'ofther~ypath.s inavertlc~l1yinhomog eneous mediumare

.

^'

. .

'consid.er~., The an:'l.lys1sperta iril ng'~th:ra~paths' "in a-con.stant

,ve.l~CitY.lIlediunT^{andin an}·one.(!~ll1e.ns10~ally continuous:l~:~^velocity.

gradi entmedium~spr.esent~.. .

:

I~

the':conclud ing par t oA thh

chap~.er

^the

a;a,~~sh

^of

ref1~cti.on

^,

,phenomenaaten interfaceis considered. Thisanalysisgives sene

U'nd~r,~t~~.ding ~~t ~he

^{.refl,ec,tion}

i;?~f1ci entand

^the

~cou,stlc 1mp~ance .'

contrastre 1atio ns h i p.a~medtenboundari es . ItshOws theeffect of lnc:ide nce'ang.le.on

r,~flecd~n..!nd ~r~i~smhsion coe;~~iCfents. "

'. •,' . I .

Chapt er

nr

is.d fV f~ ed~ntt?'.twopar~~^MAuK!"B~,In'p~rt,~'thes~ent- ,"wfse constant ve\ocf t ypr~f i 1multl1aye~ed'II~1um.1scons1de'red~ Als o.

the

,~e~~,od.S

^of

:e~tirilati lig

^{"t he'refiec,tor}

ge"'."~trY"f,~crn re~\l ection

data ""

been.presented . Part BIsdevoted totheana,lysis pertai ning.to the'linear'

,~.

I.

I

l ~

.I. :

.

~

.

:'

'I

velocityprof ilemodel. A zero-dlppl ngref lector15considered. The.. • n~mal.oveoutrelat ion ship15 derivedinthefom~IChtssufubltfO!

obta ini ngthe par4llet er sof

the:

lIedt.... Thel~asts~.arestechntqJe^h' usedt9obtaintheparallet er s.,Thesolati onre sults ar-e-pres ented. .InChapt erIV,the-ttneer.velocityp~ofilemodelfora dippingref l ect or

hasbeencons;dered.b,s"edonthenonnal ray

a~~lYS1s. ·

^ASlll table:' th- enat'ical'model'has'been

f;nnuht~ ""for

^{obta ining}

t~e

re: ' ector,9eoo1etr y, fr~ref'7ctfondata. Amethod

oi

obtaining.reflectfon~~at~

is

·present,ed•.

. , .

,

Chapte rV15devoted toanerr oranalysl ~ofusingt~econ stantvelocity prof l1emodeltoest.lmatetherefl ectorgeanetry In

a

linear"velocity pro f1'Jemedtlol!l. Alinearve locU -!proftl emedl U/lto hcon ~ideredinthe , forward.ce- puta ti on,.t.e..ino~tainingthedata. Int~~ever sete-lPU- tatten,t.e.Inestimating theparameters~f t~elIIedlUllan ISsumpti on,is lIade that eevelocity of,t he.edt,C.isa~onstint,~alto~heaverage .velocityofthe dat agener llttng'systl!ll, l.heresul ~ n terrorsofthe

'.<t'm,' " ar-e

" to,.';

fer'hm

~" ":t;" "...

^I,ti"

·d'~~ .~

^{•'. .}

InC~PterVt,the~a 1 n'~onc'usio~sand furtherareasofres~a~chare present ed"

,

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CHAPTER. I OBJECTIVESOF-TtiE~PORATI~l'SElSM ICS

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2,

1.1 SEISM IC

OBJE~I'~ES

^{.\ •}

Set Siltc encJP.asses tne brGad

ra~e

^of-phenonenaInvo l vi ngnatunl ordel1berlteexc itatio n,ofthe eart horalocalsecti onof.theearth's su"",,,ce.the.ttendantwayepropa~atton,anddtver ses1gn.latNt medhlllinteracti onsinvolv ed.Ins~ch:pht"n(JIena..SeismicPhe~an~' 1nvol~eabrtl. d rangeofphysic alpdnc'lple s.and,canprehensionof thepllenOOlenatpvcr ves an1nt erd iscl pUMryappro.tllwithaspectsfr ?D.- geology,physicalandeechantc etproperti es ofsubst ances ,rod.

mechanlc.s,engineering,oceanogr aphyandsediment ology ,dataacqu1si'tlon and.proce ssing,mathEmaticsandsignal th.,ory" and o,thers . Onegene rall y It.!:.esadh t fncti o nbetweenseismology and.I!xplorat~~n se i.snits.,Sehmology15 abranch

o f

geophYSics.which

is

concerned wi th deep-eertn phen(lllena,str onggroundMotion andrelatedearth-~

quakehazard,volcanoesandrelated .an-causeddistur bances,etc.

. .

Exploration s.elSllicsisidentified aspertainsgenera lly tothe search fdrhydroca rbonandMineralresources,andinyo1vil'l9verydeli- berat eprobingsignal

.

and ext ensivesensinganc!inter pr etationof^.the lIedidresponses. Other appl1cat1o.nsin explora tionsetsefcspertain tostudyofgeol ogic almedia ingener al,study·of deepa~Shall ow ocean.sed illtntswithi ntheframework ofoceanography, andgeot echnical and engineerj,1'I9s~dies,ofmedia.

t•

The

brofJ

object tve of~xplora tory'seismic analysis15 toextra,ctfrom theresponses, estilllatesofmedill11geaJletry.cooposit ion,and,p.aramet ers whic" are lnfluent lalintheintetact io nof signi l Indmedia . seismic analysis isassocia ted withgeophysicalMode11ng (mode lwilding is•

)

systematiccoordinationof theoretical andemptrtcalelements'of the knowledqe int o ajoi nt construct). Inseism1cs ,therea l earth is approx imatedbya modelincertainsignificdl'lt respect s.

Our study ismai nl y concer nedwith the subsur-facestructureofthe seabedand of deep mediawith hydrocarbon patentia l. Se1!imicmethods have becomeindi spensabl e"in the searchof all~nd.gas. Theyare ut il 1zed for theex ~' orat1on ofnew reservo l rs and a 1so~he eva l ­

.~a t l onofdisCOverie$deX15"t.~rIdfie lds . Dr11l1~gact-fvH1e$,fOr ,

_the search ofhydro car bOn aregenera~lYgUidedby ins ightfromthe

deep seismi cdata andfr om bore-hole-tnecm art c n. Ttie engineering criteria for desi gn

.

"andsiting

.

of offshore structuresinvolv ed in

^.

the dri llingandextrecttcn-ecttvtttesare based on the shallow.seismic data.

The seismic~eflectionmethodIs an

.

acousticimaging tec

^.

hnique.

^.

The mainobjectivei.s to collect1nf.onnationfromthe earth'ssubsurface bymeasuri~gandanaly~ingtheresponsetoseismicexcitationsat the earth'ssurface. Hostof the analysisdealswith~pressionJlwav~es

::' ::' ,:;:~t::,::;::'::, ::::':;;,~,:::: : ::',::::h::'::'/:::::~

s-veves }. .

1/ j ,'

, In explorattcn sefsntcs •.the distur ba.ncecreated byaset

I

tcenergy sourc epropagatesthrough the ear t handisrefl ect edfront.emedium discontinuities. Thereflect ed signal consists of the pr ar yref1ect~ns as wellas multiplerefl ections. The arrival time of the primary '"f1'~tl'''^at

th',

surrececontetns

'of,",,,,,, . - I ^{;. ,.t1,"} ',"00'"

- ,

c'

\ ,

of the subsurface strata. Refl ectionstrengthscantain infonnJtion

"'boutcontr astin

.

charact eris ticimpedanc es at

th~ I

^{di sco ntir..,1ties.}

=;::~:~i~:

^bed

^tv

^{tded . .. , . ,}

^{"' I n} I'"~ ".ss"m,e

^tfcns,

•tneerpret et.tcn.

.

^'

•

In~p' orat~onsefsmics,arraysof tr,a.nSduc~rslare us,e4 to coll ect the

.data•. Ther earetourprjncipa~typesof traCelgath~rs .namely , cm:non- J

s~urc eg".t her.coornon-rec~ fv ergat her ,comnon~rffsetgat her, and~lJTJJ1on-

!

depth-pointgather. Tnetype~a the.rto be

rSe<!

depends onth~Objec.- ttves. Usually;a totalof24or 48sensors. areusedforeach gather .

;hesesensorsignalsare fir s t mul t i plex,ed

t~ether

^andthenreco rded on m.;,et" top" . ln,jewof the'bo.ve,thef.e

1

^{0rdedrespo.nseshavetobe}

danu i~iplex edbef oreprocess ing; SUbsequent 'r' sO¥.timecor r-ec'ttons -forver -tous'phenomenaarecar ri ed out . The timecorrecti~n^{comprises • •}

. . I

stati ccorr e cti onand dynamic corre ct ion. T~OU9 hthest atic correctton one att8'llpts to remove tilevari ations from a alouscond itionsof the earth,

speC1f~Cally.

^{theelevat ion}eff &tsa'•.eff ect s

of. g~eatly ".:~

,dlff eri,"gsurface layerveloc H i:sr The stait'corr ect 'tc nIsn~t req~ir~

for setwtcrecordingatsea. The dynamicc rr e ctlonisthe ti mecor-r-ae- tlon appli edfor thepat hdiff erence s. It ependsongeanetryof spread andref lector dept h:

Thesignal- to-no1se ratio(SNR ) of ael9:n1c espcnses can beenha nced i

!

I

,

greatly throog hsuperpositionor stacking of agath er. Stacking

s~u,~\~e

done 'af ter nornel -move-cut . correct ton(f,e . the

dyn~ic

.c orrec tio~l ' Inadditio ntothe inher ent.enhancsnentofthe signa.l-:" . tc-nctse ratio, sucking reduce s'tileeffectofmultiplereflections.

.

..

Am~ng.the segat hers,t~ecemnOn-dep:h~po..tntga,tller .1,5.g,.neral 1Y1l.s e

J ,)

for cons tructionofthe' velocityspec traplotused·fo.~obta i ningth~.

estimates of velociti es . Itcane'tscbeusedto obtai n the inte rval' vetectet es_a~the datumveloc ity. Inadditio n,_YelOci~:pectraar-e usedinapplica tions likecheckin g'thepresl!nceCof '"Siia'febody or 011

." .

.

and gas.r eservc tr s. Fur ther , theamount ofmult iplespres en tinthe seismicgat hercan alsobeobtained usingtheve'locttyspectra.

Da t a~roC eSSi n\' 1scarri ed.ou.t by ,using amodel;wh~ chisdevel oped, . based on"the"Ph}siC~l inSight.ofthe phenone na.

.Inexcto - at tco,s ei smlcs,the refl ect i on amp' itudes ,through their .{, dependen ce oncereaet er sforthenedteonthetwo siles ofa~int er fa ce- canc~ntrfbu t~.,t o the interpr et a ti on ofgeol ogkal deta il. In erecu ce..·l arg eampl itu deerr ectsknowna~bri ght spotsmay indica t e anint er face' betweenaporous gas- bearingmediumandanott-erwa t er · beari ngmediumori.strong ly refl ect i ngcaprock. Such a'pher umencn can,sane tlme~beseenon refl ec tion sehTlogralts and~n.veioc ity'~pectra displays . '.,. ....

If 1twerep~ss i bl etoprobethemediumwithan ideal impuls e, the reflec tion responsewouldbetheimpulseresponseof the~i~m,a~d wouldcO~h i nimpulses fran themedi umdlscontlnu~tie s ,.~pract ice, th~^{probing pu'lseisat bestimpulse-like}and the pulse shape~ay

frequentlybe.u nknoen. Inorderto enhancetheprimary.reflecti?ns',.

, , ~'j'lt"

whic h co n ta i n theinforma ti on ofintere staboutthe medium.one needs to reaove thenon-idealputseshapeeffec.ts-. as N,ellasany'stron~' revebere ttons,ghost andot her multi ple ref l ec ti ons', Themet~of predi ctivedeconvolu~lpn1.ntrOd.uceliby Robinson(1967)~a.sbeen successfu llyus ed forthesetasks. 'A fundamentalassumJtionfor this-method"fsthatthe

ref1ect1v it~

^a't'

.t~

interface ,arestatistically

.

^'"

-

"randOOIso.that theearth'impulsere5~on5e^can.be consideredtobe a

randcxnslgn,ll. Wtth the-addi tj~'nalassutnptionsthatthesourcewavelet is..mi nimumphas eand thatthela1fr ed'earth

f~'

^{a 'Hne er}

s~ten.

^1t

1 s

then possible to effect a deccnpestticnof thesource'wavele tand the(randlXil)tnncvattcnsi9nalrepresenttna.the medium impulse response.

The"ef f ect1vehe ss of deconvol utionfllteringon'ac~aldata depends on the extent.towhiCh the inherentassumptio nsap~lY~nthe actua,l • si tu atio n. Inerectteetheseassu~ ptionsmaynot~_upheld. Thewid e·

spread epoueet tcns of predictivedeco nvolut i on~veproved its abi,lity for,pri ma r y reflectionenhencenent , In a typicaldat~process ing sequence, thedeconvol ut ionprocedurefoll ows II numberofadditional dig italfilter.appqcationsto ccapresathesourcepulseand to provide agreaterenphaststothedeeper,reflections. This 15 achieved by Wi enershap~ngfilter s. the variation insourcepulse shapewithtra vel time can be accountedforby windowingthe traceandapplyi ng the

I Wien er filter [Robinsonetal (1980)) .

Hl?"lOOlorphicfilt f;!r1ngis agener alization of linearf.llt er1ngfor certainnon.lin~a~filteringproblBns. Itcan-beapplied for the

deconvolut ion. Theresults are satisf'actory under high~'i9nal -to ­

nOi ~

^ratios

I T~fbolet 0979P~metho~

^{canbeusedi.fthe}

fr equencyrange s ofthe'!Pl~iO n~nd

\he

s~stffilresponse ar e significant lydif f erent. Thistechnique has beenconsideredas an alternativeapproach forsei smic deJonvolutio·n. The results are generallyunsatisfactor y,st ncetheise t satc s1gna]shaveusuallylow sfgnal -to- nois e rat fos(SNR). \

.

. --. ^\

^.

The milll.imum entropymethod.\M~l- : beendevelcped,[ Burg(1967~1 for spectral anal ysis..Thistecnntcee produces a powerspectra l

. . I · .

-estimat ecorrespondl,rIg to themostrandomand least predictable

. . i .

.timeserfe$.~Thisest1matlontechn1q~eeffects the minimiz ationof the prediction.er ror~ndof thehi ndsight orr~trospectionerror. For alarge number'_ofda~poin t sthe resultsby thistechnique are very simi lllrto the're sul t sobta.1ned byth~^Wie ner-Le vin son method. The maximum entropyrep'resentation of the'observed dat aIsan.ill.ltor~re ssi'J e process (AR) (Vanden80s(1.971)J . Th'1stechni que15applicableto theob~_e:Veddata to ;heex~entthat ti.,ese sat isf y theautoreg ressive' model hypoth esis. This techniqueis in gener a1super iorto the more

•conventio nalspectr al...nalysts methods![Lacos s'(l971)'.'Burg(1970). , Ulryc h(1972).,'U l~YCh^{(1975»). M}ost O;f the,usualmethod~of spectral analysis haveassociatedwindowfunctio'nswhichare independentof the dat a or of the prop\rt ie$of the randall process . T'hel1ax1inlJ'll entro py

"!ethod(MEM )

a~

maxilllUffi likelihood

~(~rCll Pon

(1969)., 'Lacoss (1971)]do notha~efb ed'wi rdowfunctio ns. Ithas beenshown tha t the reciprocal'ofthemaximum'1H:el1hood spectr umis equa ltotheaverage

,

-/

"of the reetpr-ecns ofthe llax1nl.lnentnl py spect ra.[lkJrg(1,972 ). The

uxflll,lllentropy -ethodliasnotbeen widel y.pp1iedin ellplorati on set setessinc eIn th15lI,el hodtbe

d.u

Is ...tched;lIllthanautoregress ive (AR)process whereas thel'Ied11t1re spons.eisIIOr e appropriatelylIOdelled as anautoreg ressive ..:ovi ngaverage(ARMA)

~rocess "

^The"[H,"technique Ilaybewt tebte forearthquakese tsate analysiswherethedat a are reasonablycons~ste~twithanautoregress heproce~' ThaHEM15of \.

con~id erable·irripor~anceinsl ~at 1onseher-e~horttlllil seri esare encount ered.

-A

,pri nClpal d1ffl'CUHyInapplylngM'~\.lsth~cbotce er.ccereterlength.'This technlquehas'foundma"ny'applications outsfdethesetse tcarea,

Tilestate spaceapproachhas beensuggested !or setsatcsignal anal ysisIH:~ndel(1978)••Baylessetai(1910)•• 8erkhouteta1"'<1916):.

CNnp (1974)••Otteta1 (lg72l Theaut~r!9res.Slveandaovi nglYerag~

(ARMA)IlOdel c:.ln1ft!'represe~tedItnstate.spacefOl"lllKendel(1971J••

.Silviaet al (1919)). In such I\r epresentatlon.sincethe syst9l.

•atrtces are unknown.the taskofestl.ating theIRedI.... 4.p'lse resporis e

·~anesa steteeeepar. et eresti"mationproblem.,Itseeasthat Ljung's corrf!.Cted&t endedKalmanfflt er[Ljung.(l 919IJcanbeusedas a stat e andparameter estimatortn setsetcapplica tions: TheMinfl1~Vadance . (Mendel(1981)1ancPHa~1l1umlikeli hoOd (Konnylo et al.(l983h'technfques

V

^can

a1s~

^{beused fnseismic}appli ca tions

b~sed

^{onstatespacemodeL··}

Themetn

adva ~tage

^tnthestate Space

ap·proa ~1I

^{isthatthe assumpt ions}

such~sstatfonari ~yofthenoise.the lIlin fmtnde lay concept,.ll ne~. a~tyandtime1~~artanceneed.notbe~lde .asinotherdeccovclut lcn

I )

,.

tec h n f~u.

\Inpl"'Oc~n i ngof .ari ne seiSilicdata,atypical seweeceafpr ocedllr~s

15asfoll ows:IRob tnsonetal (l9801I.

·dewul t i l'lexlng

:'rriormat~.. . .

· sort t ng

f~lat1ve aIlP1ibJd~

^scaling .'bandpassfiltering

•predict 1ndeccnveluttcn ..wi'iln!!rfi1,i~rfn9

· CllP sorttng

·Y~l ocityanalys15 .• ItIOcorrection

·;CllP sta cking .,'iIlener fll ter tng

•aodel t ng

·.lgrat1011 '.In t~rp retiltton.

.1

i

I

./tow, turningourattl!fltt onto thelIigratto n.one of the basfcproblans inexplorat t o nsei smology 15toobtafnthecllOrd tnatU ofthesubsurfa c~

structur~s.

^{nereflectionseismQ9 r.}

~oes no~ g h~·the

fnfonniltfon . about-

r:

^true

re~ l

!etto" petnt, 'The setsmogrilm

sho~

^{as ifthe r,efl}

e~.

tfon-occur s direct beneath theCOP~ofntwhenfnrealfty,unl e ssth~

.:I

reflector.is.horizontal ,therefl ection-would'be·' ocated·elsewhere. To I)btatnthetru ere.f1e~.wrpointfrOllltheref1e~ tt on se.1~ram ,a .knowledgeof tiltvel ocity.promeisnee~ssarl. If the-veloct typi-~ft1e

IsknoWl ft tsposs t b~eto,trl.cet.heraypathand hence,todet el'lifne

1-

1-

10

thetruereflect ionpoi nt. InOUT st udy,wenavemadethee ssusptton thatthe veloci t yofthenedlumisYarYill!l'lin~arJYtc'.~nableu~^to' es.tteetethetru e rer j ecttenpoint. The true reflect or geometry identifica t ionfrom the reflec tiondataISknownasmtqr atton..The ....:;igrati on.can

b~" d!V lded

into two; .1in

ca,tegOrieS:' ~·~ely.,

gelllletrica.l

migrationand~ve'equ.a tionmigration. Ther e arediffe~'ent te(hnfqu~s underthese,e,at eg ories:;. The~.teebnfquesar e'di scu~ sefIn.thefol lo,wing:.~

paragraphs .

/ .

^{. ' .}

Th:re are.tWogeomet~~allligratlontec hniques.namely,eaxiJ1!U1icon-.. vexit y,mi g ration andwavefrontmigrati on, In,pdndpte ,~bot ~the migrati on,tecbntqoesar ethe's~efor agivenrecord.Q~ction . T~e max imooconvexity migr ationtakesthe valuesofth«: recordsection, aiongahyperbo1ic~ar'candplJ-t stheir sumatits~pex'lRobfn~~n (1982)1-.

Wavefrontmigration takes tileval ueof ther-ecord.sectrena'-t'a point and'~tstheva}u eevenlyalongthe_c ir cularar cthat.has:t hi s pa1n t .asitsdee pest'paint

[Hagel1~Oorn .

^{(]9S.4),',fkJbra"1}

(19~7 )" 'RObl~~n,

^et

al (1980)"McQuillinet al(1979).. Rob inson (1982) ]. "

Thesave equat i on

.JIIigra ~io'l1

is

cl~sely

^{related'to,.the}

p~obl ~

^0'(

det eminingthe wave fieldtha teXi~ts in the~ropag attngmedia. In ,wa~eequ at1oni!'i gr atl onItisassueed thatthe.s~urcesare:po s i~loned

al'ongthereflectorsur facew1 th streng th proPbr tlona lto-,t he'reflection coeffl cientsandallthesourc e s areactivat ed

~t

^{time t}

.:0 'iB'erlcho~t

(1980lJ.. Therecefve-sare on'tile sur face'of t~^{'gr o und,}_Th~"W'ig!"a~ion pr oblen,1sconsider ed as a depropagatio n fran t1Il1et."tJsur face',of thegroond)to'time t• 0 (surface of the ref l ector) In reve r se

,

. /

.j

. .

r '

. . ^{: ;" "}

^.;

^.

· I'''''

II

\ 0 . \ . ' , ( ~

'd i r ect ion. There~arethreedifferentwe veequa t ion.migration tech-

.~

.

"

.

niQues,namely, Foarl e"rtransfonnwavee~~atlonmtgra~~ on"ffnfte 'differenceappr oa chand Kfrchho! f!lIfgra~n...-Th~/:ur fertransf oryn

tec hnique doesnotallow~ny14 t e ralva,rfatio,1'lof velocHy·a l?ng 'th l'!. • in ti resec c tcn,stnce it,15a nonrecur s 1Ye)tec hnique. The nonrecur s lve technf~es^{cannot be usedwhhJ:he}lat~ral^{'veloci ty}var-tat .tcns."But" . •

t}l~

^{ver"t1ca ;}

V~10cHi vlJr{ailons~-n t:e

handled by

re~rslve

,app; 1ca t i on"

·_'of"th~.Four1er_{". ~}tr"ansfon"_{". .} technique [5t;1

. ^"

t;(l97B) .

^,

:Ro·b1

.

^,',~son.^(1980)J.',

•

,

! Thertntt edtrre- enceappreeenis Ccmlonlyusedtnset setemigration•.

'itJ ~

er-ecu- stve"t ectmfque..

~'h~

^d'o'flward

ex tr~Polat'ioft 'r~1t

^at'

depth levelzi· iAz is<;.cm!Xlt~fromtheprev1o.:s:extrap?latt~.n·

result'at zi .j'· (t'."'"l )

t. i.

Th h

t~ch~q4e can'~

^{used with',h t e.ra l} velocityva ria tions

si~ce.1t

^1srecurSi've , The.error- involvedisttie' /', ina jorpNlblen in

.thls:t, ~c~"iill e,

^The

.error, ~ncreases

^1<ilth

de~~,

^"Fhe,"", .

"?" '"

frequen cy dependent""''fiX ~.extra ~l a,{ionstep.4~ 1; Th~S"

,le a dstoan undesirable, ef fectcall eddisp ers ion,,BY ,SJll ect1ng'afloat- 6,

.- L~tt~:t"::~:'::;:h::: ::;]~'"T::·:::h:::: t::':,::::.[:::"· .

spatlal'I' band,11n1'Hedrecursive

migra~i.o.n, ~.t n .~~lect

^..

ing',~ mi9~at1o~

^{• .}

one canusecine~so~cho icedepe~~ing0".theCQTIP1~~{lf t~e,str.ucture,.

r.:ti,s net possible-,torely enone par ti cular m1gra t-lo."techn iQue, Expensiverecursi v er:etflods should"O't be'used'forsimplesubsurfa ces:

·Simpl,e

no~recurs~~~thO~S ~annot .be u's~

^fer

cO'l\plica'~fd s t~·ctu.ral

^•

~, s:"~atlons ,

^{"'. " allmigration}

iechri ~q~esl'Qne h~S 't,o _sel.~t:

the

~ost

suitable

~echn.lque

^depending

~n

^{the par}ticular Se1gni5..

S1tua~}P:-

^'

:'(.

" , . '

. . ~

.

^.

.

parameters<. ofthe~M iumtrce'tteren ect tcn data.

rurthe~~a [tneerve:t'Ocit,v profilemod~lfordlpping.refl ect or s i;

considered. The~ilII!eprob1'dn•tla,S ccnstce recbyMi chaels1977. However, ...in,our ana11sf s,th~,.~rob1EJ11

is

ccnstdered in a.different~y. Amodel 'Whfc"".r elat es

ihe.mo~:inq'Sh.otirece·1ver d;st~'ie

and tr avel timeto

t~e

p,arametersof

~he,~~i~m:'

^1,S

~:e,ri';"~ . ··IJ,Si(I~I.nonna1 r~y.

^{anefysts.,A.n,}

.~p1tcttfmTllu,lais,t~e__Q'~eri,Y~,forO.bta1n.i~~_>t hey~'e.l'eflector,~fnt.

Fina11Y,:

th'~' aboVe.nl~~el

^:\s-

U~,~(to ~btii!~'

^the

'tesu1ta~t

^{erro r'Queto}

'~h~

^"l;onstant

·.v'e~oc1t~ ;~s~niP~ti;h. i·f'.i~e."~edi~IJl·: ~s'

^:l'

f'~ar veio~i

^{ty ,}

prof~le:, .The; aver:.g~~,v~.~~.~~ ~i~; ^{: ihl!.} ;a~a·:ge{l,era·t1ng 's~st~

^,1s

~Sed

:as' thecon~t~ntveloc ity6f~~be.m~~Il~~.,,. ^I

Sfnc;th1,' study is

mai nl~ 'd~alirig " Wnh''rayp~th

anal ysi s.

it ~lris '

.

i

w1ththe

. r~ypath

^{approac h,}

. ~o

^the

'S:'lUt1~~"~f

:: "- "

't~e

^wave

. equ~t~~~

^{,. ,and"} ~

~

uses this'solu~ionfort~.e'r:ajP!l th.anaiY~1s)ncqn'sta n·t"and\inear velocity' profile'media •.bef or epr OCeed] ng:W:-t~em.alnS.~dY;

. ' .

~

. .r-«.

r I

12

"Thevelocityofthe med.fumcan,beapproximated"bY, apol~t;'ialtunctfon

~f,depth-. The5illlPlest.appr~~:Jinationto'theYel ocHY'pro~'i1eis~he .

constant:'1eloclty·pr of ile.~hesegmentwfse 'constant velocityprofile is~oreapprop~ta,tethan the ccnstent~elocl ty'asSlinPt1o·n. Sincethe

ve~,oc!ty'

^{in.amed'1um'is}

ge~er~llY' lnC~~S1ng

^{'wi t h'deptotl}

~ t

^{1,5 more}

.?ppropriatetoccns tder the~elOC.1 tyas a11 ne~rfunction of~epth.

TAe.n~rmal^{movl!?'ltrela tionfora linear}vel oc ft y'pr ofil emedium'was d1sc~S5ed^{bySlo'tn1ck(1959).} Tha<relat1~n5.h~p^1'$.~ot 1~^{a fonn} s,ultabletoestimatetheparamet ers ofthe-medium.,:Ther efor e,d '-

"different,

'eClu~t1on

^{ltructur efsdeveloped}

' ~O'

^thenonnal,

m~v~'~t

^{" .}

r~~tlP:nshjpJror

^"a's1;.glezer.o-d1pp1ng

·ref1~ct~r:.uSin9' 'th~.·li'~~ar

^'..

'''ve'O?i tY

p.~ofl1~ ass~pdon.

^{Thi s}

'~na'bl!,:s

^the

~sdma;i.on

^of

t~~

^.

".I

14

...

.' Acoosti~ ' PRoP~f~im ~ri ,

^IT'S

.RELATIONsltIP.

To'lH ( ~~ER~

OF THEMEDlutI.

'I:

, ,I

i I 1

I~

I

.):

'._----"-~

/

15

a.t.

GENEP.Al.

Soundpropagat io nts governedbyalinearsecondorder part i al different i al equat i on knownas thewave equation. Thegene r althree dimensionalwaveequat ion-canbe wri ttenas[Offi cer (1958)],

'and

v¹W'"

1

1

·.!:

1/1 .,.y- 3t1 wh~reV2~~s^tilescal ar oper\lto r

VZf·~ ~ + ~

+ !2

ax2 ay2 az2

---

"'- ..., Visthe acousticveloci t y ofthemediu m

2:1.1"

•

RAYTHEORETICAL APPROACHTOTHE WAVE EQUATION

'2.ll.

J _f

There are two appr oaches

I

forobtaininga so-lution for theabovewave equert cnin terns of ray.theor y•

• etkonatequatio n

•..,Fermat's principl e

,

Theetkcnalequationis based on wav

..

esur faces andraytheo ry. The wave surfac es arethe loci ofpoin ts which undergothe Slimesotton in II one-to-oneccrr espe ndenceatII giveninstantor/ti me. The rays are 'lonnal·tot.1leeave surfacesand they givethedirecti on of propagation 'ofenergy throughtheeedtue.i

i~;

^{Fern:at : s}

pr~ple

postu lates ray

path sbetween~opoints inamediumasthe path sof mini mum.t ra vel

*qi:Ys~r~~~~; :p~~~~~::~ ~~~~:~~:~ell'ent,or,velocitypot ent ial ,

16

ti me. By solvi ngtheaboveray equatio nsbet ween twopoin t sfor a ..

given earth model,itis possibleto know thetraveltirne~andray pat hs betweenI

t~e

^sourceand receiver..Theser.ay tlleory4proaches arepar t icularlyusefu lin solvi ng the lnver s@probl efAinreflection setsntc s and earthquakeseismology. Inthenext section.each

9ne /

ofthe~pproi!lchesiscenstdered . 2.1. 1.1 EIKOtlAlEoJnoN

~'. The wave equation can be transformedt~afi.r.storderpar~ldt t- _ fere nti a l equat ionknownas the eikcnel equation(Of f ice r (1958). Lee et111 (1981)]. The solution can beinte ;.pr et ed.fn termsofWd\/etront s and rays . Ingeneral. the three-dimensi onalwaveequati on2. 1

h as

^an

associatedcharacteri sti c equati on givenby,

(2.2) I

~whe re.

V-i s theacoustic veloc U y.

Incases stere V isnotaconst ant,equat ion 2.2doe~^notrepresent

...--t he associatedcharacteristi c equat tunof the wave equation2.1."The

eikonal equationwillbe agoodappr oxilMt 10n to the waveequationif thefr~ct10naichange 1n thevelocityoverawaveleng~.h^i,ssinall.

It canbeshownthat a more general soluti on for equation2.1or 2.2 takes thefom.

i j

I '

-- \~

11

{U}

\II,.funct ion representingthewavefrontsurface Vo^ISconsta ntreferencevelo city.

,..

Bysubstit ut i ng equa,tion 2.3, in equation 2.2,weobtain anequation known as the eikonalequation whichis give nbelow,

}fi}2+

(~)2

^t(*)2=

<1>2"

^1'12

wheren 15theindex of refractionendn«

~

{2.4}

The etjonelequatlonleadsdirectlyto~heconceptof rays. It is particularly use f ul in solving'problems 11'1 a,he t erogeneous medium where the velocityis a~un&t-1onof the spatialco-ord inates .The, eikonal equationisa first order partial differentialequation. It's solution,for a specified..t1met,1~givenby

w{x,y,Z) ,.constant {2.SJ

(

^Thisrepresents a surface. inthree-dimensionalspace. For II given valueof wand atagiveninstant of't i me't ,any variableatthe surface will be in phase,put not necessarilyof~hesame amplitude.

This surface is talled 'wavef ro nrsinceit preserves a oneto one correspondance of ecttcnalongthesurface. Thepropagationcan be d~s.cribedb,Ythe time progressionof thewavef ro nt. ~normaltattle wavefrontat any spatial coordinate defines the direction of propagation,

'8

and the loci tracedovt by thesen~nnaldir e ct i onsareref er r ed to as rays .

Thenormalsto thewav'ef ront evolve inaccor dancewith the incremental pathl eng th re la tfons hfp

ds" dx

UWlm" ^(1,6)

where thedenominatorfactors arethedire ct lo nuaber-s of the ncreel, Thedf'\.ect Jon cosines.areproportio nal to the frec tio n numbers',so~hat

dx ..k aw

as ' ax

(1.7)

'(1,8)

dz k aw

as" .

IT

I _I

I

wher ekisaconstantandds is'an fncrane nt/relementof theraypat h.

Stnce an incremental segment'ds' of acurv etn

three.{j1~enSlona'

^space

sat is fi e s

(~)2; (~)2

⁺

(~)2

^{.. 1,}

we obtai n thefoll owt'ngfrlJllequatio ns 2,4,2.7and 2.8:

••'1'"

19

From equation2,9,

,:! _{" .} ¹ _.

Itis als o possibletorewr ite equat ion2.7asgiven below:

" "

^dx

^,,-

^{aw (2.10 . 1)}

'" _" * _" ^.,-

^aw

. .

^{(2. 10. 2)(Z.l O.3 )}

" "

By.consideringthe derivat ive ~

a t

the equattcn2.10.1 ,alongthe raypath weget ,

d (o M)

,

" ^Os

(1i)

d (n * )

• ( ii-;

^{dx aw}

*

^+.~

^*).

" ^,,- " ^.,-

(2.1l)'

From equations2.10. 1and 2.11weobtain ,

, ( 'x ) an I )

.l. lIS n

as

. "'fi" [_) .2.12.1 \

Slmilaf>ly , it ispossi bleto' get the,.fol1owtngbyconsidering2.10.2

.

"'-~/

and2.10.3.

to.beinvesti gat ed.

20

d

(" ~J

^{an (2.12.2 1}

"

_d

"

(nM) !!!." (2.12.3 )

"

^__3Z.'

Theaboverelat;onsindicatei"atthe refractionindexn governsthe rayevolutionandwavef r ont geometry.

2.1.1.2FER."\AT 'SPRINCIPL E

Theray equationcanbederivedfrom Fennat'sprine.1ple[Officer (1958 ) •.

Car uth ers(1977),Akiet 81(1980)] . Thisis specificallyappealing incas es when theray pathandtraveltime,betweentwoendpoints is

,

rbt sdeviationis basedontheassllTllPt fons thatthe velocityisonly afunct i on of spatial coordinatesand thatthe velocityisconti nuous andhaseont .tnucusfirs t partialderi vatives. The Fermat ' s princ1phi r ' \sta t e sthat thepathwhi charaywill tr acebetweentwo points 15such

• )thatthetraveltimeis an extremum. It actuallymeansthatthe time for a ray to travelbetween twopoints mustbestllt ionll-:'y wi threspect to smallvariationsofthepath(Pilant(1979)].

'~

^{We haveto find}the stat10naryvalueof the.1ntegral I given as

i

I·Vo

f •

dt

(

21

SUbst"itut i ng for dt,we have

(-

B ds

,

1= " I

cA T

B

':E.

I

."

^{ds•..since '( 2.13. 1)}

A V

whereAand B are thetwo endpointsofthetr avel pat h. Ifthelength

•ds'isrepresentedas

a

dUJmlY

\farlabl e~d<1.

^oneoptat ns, dS"~;'^{((* ) 2+}~*)2+(* ) 2]1do

.Equation2.13.1 with2.13~2gives

I .. / B n(,x,f , Z) •[(.~)z+(*")2t<*")2] idQ;

/ BF(x,y,z,

*. ~. ii>

^do

(2.13 .2)

(2.14)

For a stationary vetueoft,Euler 'sequatio ns IlUstbesatisf i ed,namely,

(2.15)

./ \flh1chleadsequation 2.14tothe followi ng expltcltfonn:"'

[(.~l' + (*) ,

^+(,*)2J^{ivn _}

~

^{( • n}

(a~na)

^)..0

[(* )2+'<* )2+<*)2JI

1

It can be,furt her simplifiedto thefollow1ngbyvirt ue oftheequattus 2.15Ind2. 13.2.

d "

os

{n

of)

^(2.16)

)

2.2 .

Theequat io n 2.16 is ident1ca1 toequati on 2. 12.which had been obtaine d throughtheeikonalecuetfcn. Hencefermat ' sprincipl e indic ates that thestationary t1rrepathis theraypathsrvenbythe eikonalequatio n'l

RAVS HI AVERTICA Ll V ItlflOMQI;ENEOUSMEDIUM ..

\.

Proceeding further ,the ver tical ly inhomog~e ousmediumis de- fi nedas -the oneinwhich the lredium parametersvar y as afunct io n of dept hon l ~cancomputethe timenecessary foradis t urbance to p~opaga tefr om apointAtopointBalongaraypath givenbyT(AB) (P1la nt(J 979ll,

T{AB)..

J

B~ ..

J

B [(dz)2+(dx)2)~^.

A V A V •

" J

B [1+~dZ/dX)2)j.^dx A

1fZI,,~,then

Inthi scase ,for travel timeto be stati onary,an,extr emllll necess i t at es that,

,

23

Using-thesta tio naryconditiononeobta i ns.

(2 .17)

<•.

By combini ng~2.1? Il.n~2.18oneobt ai ns,

f(~ .

^{z' ) -}

z" ~, ~ ~

[1...(1 ')2]-1..p wherePisacons tant which isknown as., roay

parll.~te r.

^IfwedesIg- natethe incidentangle at depth zas e,then~ ..

i' ..

coteand

./ sine{z)... p

VIz ) This is the generalhed.l0l"'lll of Snell'slaw.

,

^('.191.

Theray parallet erPcanbe"related to twophysicallyobservable q'uant 1t i es

~tthe sur face,namely inci de nce angle(6

S)andacous't1C velOCity(V s)' The relat1onsh1pis,

P . ~

Y,

- - - - -- -- - - '\ >

-- '--

Vappare nt horizontal veloci ty

dT

iii

.

I ^~.

'"

whereTisthetrave l - t i me to depthcoordinat~.zencounteredat arange coor d i nat e x.

ane

At themaximum depthof penet ration-'ofthe ray,e ..90⁰

I • .

p . _'- V...x

~ = -:-

_{.max .}

" , ,' " ', '

(

Basedon theabove,the apparent hor izontal,surface velocityisequalto"

themedi umv~loc1tyatthe o(j~p.th·of greatest penet rettcn.'

• • c

By differentiatingequation 2.19 wi threspectto arclength ds alongthe

f) . .

ray,oneobtai ns,

- v·

²

sin e

~ ^+,

v·

l N!Se~..O. (2.20 )

8,t ^dV

" ^{~ . '~ '"} *

^COS8

Byslbst i tu t i ngin2.20

. ^.

_{~ .. p~ ~ (2';21)}

-'Thi Sindicat es thatth/c urvL of a ray 1na vertical lyinhomoge neous medi umisdi rect l y proportionalto"the velocity gr:adient.

.

~.

LiL_

~romequat i on 2.l.i.2'nd 2.21,oneobtains, dV

"

^(2.22.1l

·'1

1 !

•:l-;

25 By geometricalrelat ionship ,we know that

.. ^./

^{dsiIi • ctiSil '}

Itisalsopossibletowrite2.19 or 2.21inthef~rm,­

' .

.

I

cose·•

.g; .. ~ .P

The rela ti onshi ps givenb,yequations2.19.. 2.21and2.22willbe -usedti""

', ~ , - . "

ana'lysethebehaviourof theray pa th~ inmedia havi ng'diff er ent veloci ty prot t tes,especia llyconstantvelocity prof il e-andlinear veloc ityprofi le.

2:2.1 RAY PATHS IN A CONSTANTVELOCITY MEDIUM

Ina

,on~tant

^{veloci ty}

~dlum

^V(zl'"Vo:c"onst ant',hence

*

^e.0"and

ther e: orebythe ecuetion2',21,~.^O. Accordi ngl y e{z)":',;,80-consu n.t\...

A

c~nstant

velocity mediumdoesnotla lter\the \rayd1;e.cti on. ,Theraycon- 'serves its,in itial inci dent angle. The mi ni mulll· t imep~th'bet weenanytwo

~i nts

^Aand

^a

^{alongtheraypat h}

i~'\a

^{straigh t line.}

~

.. .

IfA....

~

(xA'YA'

z~l ra;{d

a - {xa,'Ya',

~Bl

thenthe~athlength bet ween

, .

A

::.

and

. . .

a !s'V.!,venby

.

.S=

'r

.(xa -xAl2+ (Ya -'f Al2 .+ (zB'-zA)2jl' the travel tin:et

I s

given by

t • S/V ; and

; )

....

I I _{- - - - - -} ^{"' .}

'.

the dtrecttoncosines for thefayABaregi ven by

Henc e ,"theang.l~fr~ydeparture,at the source ,atsgi'ven by.

'.e

~ 'co~-l

^f.(ZB

~ - ~'A)/S]

..-

Sofar we

ha~e

'been

concerne~ ·Wl~h.thli/~ypa.th$' . ~- a const~~t .·vel0CitY_~

mode lwnicn:isthe simplestmodel. Inthen~x,tsection we \11.11con-

"' t . ' , '.

stder ra)1lil.ths tn a linearvelcc.~profilemedium.

(2.23)

..

might b~,approximatedby.econstantgradientprof ile:

.

^\

The const antgr adie ntveloc1ty

. .

profi l eis given by V(z) ".)-0 +gZ,

' .

.

"

2". 2.2 RAY PATHSIN ONE.OIMENSIONAlLYCONTINUOUSLINEARV~LOCITYPROFILEMEDIA Thesetssrtc've.'oci t ygene,ral1Y.increa~~swith.depthin theea~1l"s '

. .

crus t. Itisvetycenmonto considervelocity,tobe,a

conttnu~s

functi on, ofdepthri:t her than tousethe cons.tantveloci ty assumption

eve\ter

^(J9inl.

~

Greenhalgh.e\al (l98Q".. Telford etal(197,8)••~!Jbraletal(1980)•.•.. Slotnick(1959)]. The next simpi es tv

el~o'ty

profiletoa

~~nst~.nt

^{J .} velocityprofileis one where the velo~1tyisaftneerfuncti~n',?fdepth, Inttl1!.sltuottions wherethe constantvelodty9rad.ientassumptton~is' 1nv~l1d.,the mediulJl can b'eeonsid~,redto consist'ofa number.of~epth.

~ntervals• In each of these depthinter...~ls \thevelo citypr ofil e

I ,

.

,

I

I

I

:,.

j' :t.

.

^{' r-:'.}

27

when!YO..

a~

9if"!

constan~~.· th~

^valuesof

wtItc~ /de~P';In ~he

partt~llhir'~t~lt10~:.:Bytaking.the dedvative withrespect toz of theabov~ ~u.atton".weget,

., :}, ~r;)l ' ~" , ^" ~: ^.

•We . ,

alre~~~ ~war~,t~~t\tw!

;olfow,1 ng

;e1aton'~h;PS ~~ '~~Plicab~'e ^{. ' '\}

;" en ,;,,,",,,u' ·" ""'.,t o f . ,r'~ff;,'·d~.. " "'; " \'" ''

.dx • ds,'.'

;~n '·e(z>. ""t2\4} 1 )

dz.•,ds .;cos'e(d . (2.•

2~.2)

ds, ..

veil

.dt ^(2.2.4,:3)

~re

^e(zl

i~

^{\; ray.ng1e df}

.t·n~\~en~~·

^at

~~~th

^r:'Thetr;s".vel t1lM!

t

ita

~tWHn^{two'pointsA and B:}a1.o~9~the^raypath ca~^berepre~~~ted

-', by.•

. . . .

~.

B t ..

. ^,

I

^.

.d,t

.»I

i i

I

'8

•9y inte gra ti ng, we l,anobt a i,"theraytimerela.t.lons hi p

t(6, 60) ..

~

^{'I n}

[~:~f :~~~))

Siililarlr,x(e)andz(e)canbafound fromequatlo n 2.22.1 and 2.24,whence

(2 . 25)

Z(6, 6 0)

1 e~ecos e.de ·1 [slne-sto oe) (2.26 )

r-

0,_ 0'

1'9

K(9,6

0) 1

e~ine. de ^{1 [cos8}0-COS8] (2.27)

P , 1'9

By elimi nat i ngthe angle efromequati ons 2.26and2.27.theray- pathis obt ainedas afunct io nof xand z, as gi venbelow.

(2.28)

By suita blysubst itut i ngfrOIiequat ion2.19~weget , /

Vo ^{"2 ' , Va Z · -} Vo 2

(x- ~):- + ^(z+

T) ".

(9 51n80) wh~re . ~o•ray,depart.ure angle atthedatum"surface•

. :

~ ,

.

It will.be see:nfrom equatio n 2.29, tha t theraypath 15 circula r ,

' . V • ,. •

V ~V'

havlrlg radius - ' - andcent er (-' -'- • •-.!?)

!Isin90' .9~an^{60'. 9}

~^ispO~.1tfve^fO!"IIregion.where theveloc 1ty is1ncreas1hgwlth depth,so that from equatio'n2. 21,

~

^{h'positive. Hencethe ra;}

willcurv e upwards.

as

Similarly, wecan concludethat for a regionwher e the velocity isdecreas in gwithdepth,a raywillcur vedownwards, towar ds'(J

region of minimum velocity. ;rhe sa«lE!res ultjsal so to be ex- pec t edfroma qual itat ivec~nside r"ationofthemot ion ofthe

• .s everronts • Thepor ti on of thewavefront whichis 1naregi on of

highe rvel ocity w111trave l fasterthanthatina lower velocity .regi o nand the wavefrontw11.1be benttoward the reg1?"oflowe r

velocity.

Inpe t roleum exploratl onwe ereus uaJly dealingwithsor e-or- Tess flat-lyin9.. bedding . As are sultofsl owchanqesindensityan d .,.l ast1cpropert ies ofthebeds thechanges inse i smic velocity

aswenove horiz ontally aresmall.. Thehoriz~nta 'vari at ions are gene rally much less rapid than,thevari ations in theverticaldirec- tion. This maybe due tolithologicalchanges

and

theincreasing pre s surewithdept h . Sincethe hori zontalchanges are gradual\frje

' an

^{take it into}

^ac~unt

by di vi d ing the< ur Veyarea into smalle:

area s-tnsuc haway that horizontalvar ia tionscanbeignored.

Then.thesamevert icalve'locftydist ributi on canbe used withdif- ferentpar ameters•

..

-.i ' _

2.3 REFLECTION- IHTROOUCTION ..

The seismicmethodsexploit theelasticpropert1~sof the material.

10se tsetc

P"'P~"';'

,,, 011 and

g.;

0",

'"~

^censtderthe f",,,ncr cont e ntof thepro b in gsi gnal. Itis~b servedth~tthe hlgher the frequency, the hi gher isthe resolution,but at the same time pene- trationdepth

,

islower dueto larger

.

attenuationcaused

.

by absorption

j i

I

I

i' l-

30

andd1s p; rsTo n._Als o,theconver:se istruef.e.thelower thefre-, quency··thela r ger the penetrationbut,thewea~erthe resol ut ion.

Thereforetheretsa tradeoffbetwee nresoluti onandpenetration.

Low frequenciesrangingfrom0.5 Hz - 60 Hz'are of'practicalint er- ...

est for thepenet ra tion'~ePthofupto10,000mezer.~indeep seismic, expl oration studies. Inshal lowsetsetesandgeot echnica lsetsntcs up to10KHz or even100 KHz ts.csec withpenet~ti onof10m to

100meters. \ /

selsmicvelocitiesareV~ depende nt uponthemed1a .incontrast

toradiowaves. Hence the velotity informat ion playsanimport ant rolefor extractingthepr~perti;sof themedia . Inthe case ofnor - malincide nce ,the modeconve rsionsuchas ccepressicnalwavestocom- ., pressional and shear waves doesnot takepl ace. Butfornon-normal

incidence, thereflectedenergyappearspartlyin thefonn of compres-. sionalwaves andpartlyin the

1 0 ",..

of shear waves. The COlllPress iona l andshearwavesvelocitiesaregover ned by the pro per ti es ofthe lI1edia • . Forfurtherextract i on of mediumpropert iesonehasto consi dernon- nonnal incidences.

Inexplorati onsets nt cswe anatzsethe reflectedsignals whi ch carry informatio'naboutthe

efr~4surface

^•.The refl ect i ons of

se1smi~

wavesoccuratinterf aces.with signifi cant changeof acousti cimpeda~ce . The reflectioncoef f ic i ent ishighly dependent on the'tncteence

angl~

whichisthe anglebetween a raypatll Indthenonnalto the surface~i theinc ident poi nt. Hence there! .1ect edenergydepends ontheincident angle inthecaseof non-normal1nc:..i dences.

',- i.

\ I

I

31

Now,we willatten'4lt to derive the appli ca blerel ati onship s for the

"'fase of acoustic

me~ia

where we studythe ref l ect i onat aninterface in the subseque ntparagraph [Temkin(1981).:Caruthers (1977). , Ki nsl er et al(1962)). The ccrrespcndf nqj-el etfcnshfps forela s t i c medi a are consi derablymore compl ex andvirtually int r actable for

.

. ^.

analytical insi ght. The phenomer'\a ha ve beenextensivelystudie d by Knot t (1899) andZoe ppr i t ! (1919)IC,erv enyandnevtnere (1971), AkiandRi chards (1980)andmany others.

The parametric dependenci es forthesesituatio nsare moreconv; nie ntly stud iedon computer generatedplotsforreflecti on coeffi cients,as a funct io nofin~identangle[Youngand Dra;l e(1976 ) ., Telfor det",1 (1976).,Aki and Richards(1980)J.

2.3.1 REFLECTIONATANINTERFACE.

Considertherefl ec~ionandtr ansmis si on of a pla neacoustic waveat a planeboundary betweentwomediahavin gdif fer ent densitiesand soundspeeds. Referri ngto~igure2,(x,y,z) is a coordi nat esp tern suchthatthe boundary15thez·0 plane. The·x-.axis is parallel.t o lln~sof intersecti onof thewavef r ont s andzE0 plane. Weshallcon- sider two coor dinat esy and z, We,c~si derasinglesl nusoi dalcom- p~lRentPifromthe.integralsumofthe Foorierser ies representa tl on oftheplane wave.

(2.30)

Pi is an inci dentuponthe z•0plane wlthan incident angle8i• This inci dentwave genl;!rat esreflectedand~r.nsm1ttedwavesPrand P

z'

1

1

. 1

I . 6

J2

whichare gfvenresP.ectivelyby [Carothers(1977).,Temkin(198 1)) ,

, Pr 8e~p[J

(!.- .

!. -"r.tlJ (2.31)

P

z

^{Ce ll.p {J(~.}

r.o ,:, p

tll (2,32 )

(2.33.1) .)

(2.3 3.2)

(2.33.3 )

('.34)

(2.3S)

(2.36) (ys1ne~-lcos8r)

J Pl(y. O) •

i

Pz~Y~O)

!, . , .,

r,

!r .

!

., r,

~'•.!:.

.,

V;

The bounda ryconditi ons beeo..e (y;1ne~+Icos8

2)

TheP1and Pzare the.sol ut io nsof the'olav e eqU1ltto n2:',

whereP, • PI.+ P,

BancCmay_.becOIIIplell.to41;countfo_Irphas e shifts: The~'S^{ar e the}

.propagationvect orsf~rthe respect1veweves • The ! . ! . 'sc~ nbe

writtenas,

t,e, lilt · IlIr ^IIlz • ^III

At theint er facetheboundarycondi ti ons are independentof time.

t

r .

Th1$15pos~lbleonlyifthefrequenci es of the tr ansmi ttedand ref lect ed wavesare the same asthos e ofth~inciden twave,

i

_I

! l

Ii I _- - - - -

33

The-equation2.36showsthatt~ enannalpar ti cl e velocit i esinthe two medi abethe~ameat theboundar y ,f.e.8

f"Or

Theequati on-2.35ghe.sthe Snell' s

l!w

relati on

r ~

v, v ,

(2. 3 7)

, / (2.38)

Equat ion2.38toget her with 2.37givesthelawofrefracti on,

Theacoust ic wavesmaybeexpr essed by

';

Aexp[j w(_ _^{ys10e1 + Zcos81} ~Il

V , V,

"

Y51081 ^{Z .COS OJ} til

Bexp[jw (- -

V , V ,

Apply in g the boundaryconditions atz•0 oneobtains

12.3 91

.(2.40)

J4

Wedefinethe"reflectio ncoefficie~"R by

"

A

l'j

Clnd the"t ran s mis sion coef fic i ent"Tby

,

. .Itcanbe eas ll): shown that

and

Ingener~l~andTarecomplex.

Bys~1vingthe aboveequationsfor R; and Toneobta ins ,

(2.42)

(2.4~

-,

(2.44)

(2.45)

,I '.

coso, casa Z

R . ~ 0ZV2

~+~

PlV1 PZV Z

PZVZCOS&1-P,V1COSOZ PZVZcos.ol+P,vlcoseZ

J5

cos

8 ,

T . 2 ..~

cos6, + COS92

~ ii2V2.

Uslng Snel l' sla'll re l a tio nitcan be derived that

R .. PZVl cos

& ,

PI (Y12 - V/51h2

0,]1 PZYZCOSI'll + P,

[Y/ -

^{V/ 51n26}1J¹

Atnormal inci dence9, .. 0,whichreduces 2: 46to

(2.4 6)

(2. 47)

(2.46)

The product of density and eceuset c-vetccttv isdefinedas"acoustic impedance"of themed 1~m.t.e. Z..pVEacoust icimpedance

By subs ti t ut i ngtAtZ.48

,7<",

22 - 1, )

R " •

22 +1, \"

o .'"

/

36 I

IfPZV

z ..

^{plV, the re fl ection}coef~,cientfornanna l incidence beccees zero, wher eas fo robliqueincidence"t herefl ecti on coeffie1 entisnot equalto

..

zer o. Forobliqueincidence ,tileref lect iencoef f ic i ent

.

va nis hes and oneobtai nstotaltransmis sionfor.anangleofinciden~e6,•whi ch isknownastheangle oftntroetsston. Thevalueof6

1 tsobtai ned fromt heT'1!lation

Pl tan61 o~ v~s;n²aj.

1 • tan

B ,

p~^(Vf-V~^stnl0;) (pZV

Z)2.(p,V 1)2

:

p~ ( V~

^-

V~)

Manyinferences can bedr awnfrom theabove equationconsider ingthe com~ara t iv eval ue s ofvel oci t i esanddensities ofthemediaasg;veo beloJ:

Int romis s ionisonlypossibleifeither:

P~V2 .elVl and V, Vz

The pa ra met ercombinationsforana'1,:,- wate r or itwater-afrint erf ace art! suchthatther eisno angleofincidenceforwhtch one can off e r totaltransmi ssion.

..

If V2^> VI ataninte r face.thenthereexists an angle 9t•8cr it'

..IJI"""

such thatthere1Tacted" ray has an angle of 11/2radia ns.t,e•itis

\

~--~--- _... ..

-

•

"

coi nciden twith themediulboundary. FrdnSn ell 's law,this crtttcet angle acis gover ned~y

.', r,

Atcrft1call nci dencethe ref ractedwave does notpenetrate into the second medl00, sotha t T•0and R..1,t.e, totalreflection occurs .

-, Again,ifV2^>V

1,therecanbe_ang~esofincidence8 1great,ertha n"

thecri t icalangl e.'Inther-ay-t hec r-etfcmod elthetransmfssfcn, coefficie ntTta kes"pur elyimagi nary values . TlrIs leadstothe interpretat ionofa lo ss)ess interface,wavepropagat i nginthe second medium along theboundar ywit harapidlydecayingpenetra t i o n

dept h. \..

/

So farwehaveconsidered1ndetail thereflections atint e rfaces . Thisstudyw111 befollowedby:a.wa ~h anal ysis,whi chw~ lfarmthe fundsnentelbas isforourfor thc01l fngdiscuss ionontheest imat ion of ~J acoust ic velociti esfromthescrracemeasurem ent sproposedin the nextcha pter.

: 1'

,J]

r ,

38

QlAPTER·I I I ESTIMAT-I OIl OFACOUSTICIlEDI\IIYt:LOCITlES

F'1lOHSlIfACEI(.ISUREJOTS

!- "

I•

\ - I

,

,"

""---'

^"

"

",

J9

Ill-A. ---...

MULTI LAYEREDMEDIUMWITH SEGMENTWISE CONSTANT VELOCITYPROFILES

e. -"

• I

,j

\

1