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Introduction to Seismic llYg111rn l'lethods brlanKussel

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9.1 Introduct ion

Inthissect ionwevl i ] l lookatatypeofinversionthatgoesunoerseveralnames,incluoingtravelt imeinversion,raypat,hinversion,andseismictomography.Thelasttermtendstobeoverusedatthemoment 'soit isimport 'anttousethetermcorrect ly. Insect iong.3weshallshowanexamplewhichmaybeconsideredasseismictomography.Asal lof theothernamessuggest 'however,seismictravelt i rneinversionusesasetoftravelt imemeasurements o infer the structure oi t 'he ear th ' The parameters

'which are

extractedarevelocit iesanddepths,andthereforeagrossmodelofearthstructurecanbeder ived.. Ini t ial ly , thiswascons. idered[heult imategoal 'but i thasbecomeobviousthatthisaccuratesetofveiocityversusdepthneasur.ementscan be used effect ively to constrain other types

of inversion'Forexample,. thevelocit iescouldbeused,asthelowfrequencycomponent. in

.recursive inversion, or as the velocity control for a depth migrat ion '

The way in which travelt ime inversion is carr iedset, of t irnes from a dataset' These pick's lnay cofitetypes of seismic datasets:

out is to f irst Pick afrom anY of three bas ic

( 1) Surf ace se' ismi f i leasurements- shots anc lgeophones n the surface'

(2) VSPmeasurements- shots on surface, geop' l" ronesn wel l ' and

(3) Cross-hole measurements- shots and geoPhones oth in wel l '

once the times have been picked, they must be rnade o fit a model of thesubsurface. In the next sect ion, we wir l look at sorr* straighforward

examplesof using travelt ime picks in order to resolve the earth's ve' locity

and deptnstructure -

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Intro

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Introduct ion to Seismic Inversion Hethods - --+&*rb*****+*****r Brian Russell,r*ffi *******iii;'i;*il;*;;; iiii; *t *rr *****rerrrc*****rr*****t*****r**#s

1.2 NumericalExamplesf Travel i re-Inversi on

Consider the simp' lest poss ibl e case'9.1 shows the t:^avel paths that wou' ldconf i gurat ions given a square area oftravelt imes in Figure 9'1 woult l s imply be:

(1) Surfaceseismic:(Z) VSP:(3) Cross-hole:

a constant veloci tY. resu' l t

f rom thedi nrensi n L bY L

earth. Figurethree geometry

Note that the

Lpt ={Tr-pt=L p

p = t / ? L,

p = t lFL, anop = t / L,

or

where. P=l/U.contain the same nformat ion'Obviously, al l three sets of measurements

However, ' i f the veloci ty (or slowness p) anO the depth are both unknown'nei ther ' .one can be determine

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Introduction to Seismic Inversion l.lethods Brian Russel ldffi***********rc***litl-t********ffirt***t.*********************************

Source Receiver

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Figure 9.3 Separation of the earth into smallfor sei smic travel t inre i nversion.

Reflecting

constani velocity blocks(BorOinget a1, 1986)

boundaries

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br lan RUSSe IIntroducti r .g:?H*r-Tftl:l-T::Hii**********r.*rr**rr'r***trrr****t*********Jrk*t****l

Letus.pPlYthestraightrayapproximat ioninthesimp.|ecaseofhavingsirnplytwoolocksofdi f ferentveloci ty. In i 'h iscase,Wghavecoupiedtogetherbot 'hsurfaceandvsPmeasurements.TwopossiblerecordingarrangementsareshowninFiguresg.4andg.5.Trresi tuat ionsi l lustratedareobviously oversimpri f ied since we have assumed

straight ray approximat ion n

both boxes' That' s' there is no refract ionat the veloci ty discont inuity '

andtheref lect ionpoint isdirect lyat thecenterof thetwoboxes.However'i fweassumethattheve]ocit iesvaryonlysl ight ly, th isapproximat. ionlSreasonabl '

Let us srart with tne situat ion i l lustrated byFigure g-4' In this case'

t 'hereisasingleshot'withgeophonesbothonthesurfaceandinaboreholeatthe base of tne layer. If we assume that the

sicres of trre boxes are unity inlength (1 cm or m or km ) ' the t ' ravelt ime equations

are

(1) For the raYPath from.,= nl7e1*

S to Rt:8,, where Pl= 1/veloci tY in box Iol= l /veloci tY in box Z,L(2) For the raypath from S to Rtfl

.^=11 o.' + 't3 Pz.'z z 'L z - 'Thus, the total probiem can be expressed in matr ix

form as:

l ; ' lu1 [" l= [" . ] " . AP='lg [ L l= I ' l -lz L) ( ,. .E Y J Lo'J L" l 'The solut ion to the previous equation is then

1p=A-tUnfortunately,aquickt ' ryatso]vingtheaboveequat ionwil lshowthat

the determinant of A is 0' vrhich Ineans thatthe inverse is nonreal izable '

Physical ly, th is ist .e l l inguSthatthetwotravelpathsspendequalproport ions of their paths i t t each box'

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Inrroouct, lon o,lg :Tic lnversion Methods grian Russe.l l****t*************i**********************************************************

VSP aypaths for a single shot.

R2

Figune 9.5 Surface and VSpraypaths for two separate shots.

:

Figure 9.4 Surface and

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