Ao—A 051 Leo UNIVERSITY OF SOUTHERN CALIFORNIA LOS ANGELES DEPT 0——ETC F~~ 8/ItRECURSIVE DERIVATION OF REFLECTION COEFFICIENTS FROM NOISY SEIS——ETC (U)1977 N E NAHI. .J H MENDEL. L H SILVERMAN APOSR 75 2797

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Presented at 1978 IEEE InternationalCon ference on Acous ti ncs, Speech , and SignalProcessing, Tu l sa , Oklahoma , April 1 978

RECURSIVE DERIVA TION OF REFLECTION COEFFICIENTSFROM NOISY SEISMIC DA TA

N. E. Nahi , J. M. Mendel , and L. M. Silverman

Departmeut of Electrical EngineeringUn ivers i ty of Southern CaliforniaLea A ng eles , Californ ia 90007

A B STRACT discretj zed and decoavolved observed data ,We conside r p lane-wave motion at norm al m ci- (55) The normal equations of the preced ing step

have the Toeplitz st ructure which makes it possiblede uce in a hor iz onta l ly laye red system. The sys- ~ utilize the vary efficient Levinson aLgorithm tote rn is assumed lossless , and on ly the coznpres- rec~irsivel y solve for the ref lection coefficients.sional waves are treated. A procedure is introduced

for determining the ref lection coeff icients of the In this paper the method of solution to the inverselayered system when the observed seismic data problem stated above is fundamentally modified tomay contain random noise. No decouvo lut ion of the cope with the existence of the noise in the measure-measured seismic data is required b y the proce- ment data , ofte n without need for any deco nvotut ion.du re when the input is a narrow wavelet. More specificall y, althoug h again a unifo rm laye r ed1. INTRODUCTIO N sy stem is assumed , th e choice of number of layers

can now be made independen t of the sampling rate:~ recent years much at tent ion has been given to r equirement of the data (step (S2) above) often re-the problem of determining reflection coefficients suit ing in the need for far fewer layers. No decon-for a layered media from the observed sei.m~c vol ution is n ece ssa ry ( st ep (S 1) ) for wavelets of dur-dat a [1_4J . In line with the customary a ssumpt ion s ation of the order of twice the layer travel times.and rest r ictio ns , we also limit our attention to a The exact deconvolution of step( Si ) is ei ther nothor izonta l l y strat if ied nonabsorptive earth with possible in pr actice or , at th e least , will Lur the rag .ve rt ical ly t rav e l ing plane compression a l waves. gravate the harmfu l effects of the noise in the ob-Such a system is completel y described by a set of se rvation [5J . Furthermore , the deco nv olut iuu is areflection coefficients and travel times within lay- time consuming operation. Finally, the pr ocedu r e isers. ve ry simple to derive and does not need the con-

A fundamental procedure described in detail in cepis of z-tr ansiorms , minimum phase, fo rwa rd andbackward pol ynomials , spectral fact orization , etc.the above references for deriving values of the re- The results reduce to the exist ing solution of the in-flectio n coefficients can be summarized b y the fol- verse problem in the absence of noise and with alowing assumptions and steps. spike input signa l (wavelet) [1).

Standard Assumptions:2. STATEMENT OF THE PROBLEM

(Al) The input wavelet is assumed known. We are considering a uniform K layered system(A Z) The data is assumed noise free, and normal incident compre ssional waves. Figure 1(A 3) The layered system is assumed to have uni-

form travel times between layers where a number re pr esents suc h a system whe re d , ( t ) is the down-of the layers are h ypothetical , i .e. , they may not going wave at the bottom of the j il laye r and u (t) iscorrespond to an actual interface of the subs t ruc- the ~ip -going wave at the top of the la yer. The re-

flection , downward t ransmission and upward traus-ture and are associated with zero re flection and mission coefficient , associated with the in terface atunity t ransmission coeff icients, the bottom of j t i layer are denoted r ,, t , and t~ re-

Standard Steus: spectivel y whe r e t~~1+r ,, t~= l - r~, T h e one wa y trav-e l time between laye r s ~s de noted by 1’.( Si) The observed seismic data is decocivolved us-

ing the input waveform to produce the system re- The input to the system , d0(t) . is assumed knownspouse to a unit spike input . (the wavelet) and the output may be e i ther u~(t) (i n1S2) The number of layers is chosen hig h enough to the marine environment) or u0(t ) . Th. measuredresult in travel times short compared with the in- seismic data, y(t) , consist, of the output and an addi-verse of the b&ndwidth of the observed seismic tive noise component n(t). The source of this noisedata. may be the instrument measurement noise , th e un-( S3) The deconvolved data is sampled with samp- c e r t a i n t y in the knowled ge of the input wavelet orli ne interva l equal to the chosen one-wa y t rave l response to unwanted inputs (ambient noise). It isti me between layers. 5 desired to process y( t ) , t g O and derive values for( S4) The system s t ruc ture is used to arr ive at a the reflection coefficients r , j 1 K (r , may beset of normal equations ( l inear simultaneous aqua- assumed known in cases such as the marine euvi-tto~ s) in terms of reflection coefficients and the ronmeat).

course the deconvolutlon may be performed in 3. STATE EQUATIONSdisc rete time I sin g the same sampl ing int erva~. U sing the notat ion of Fig...l ,fo r a genera l j t~ laye r

APPrO V Pt~bi1e re j~~~~~dlStrj bt~~~1~~lt~~d

2

we have ( 6 , 7]. U~(c) ~ r + u 3(t )u ,( t +c)dt . (12 )t~u~, , ( t ) + r~d (t ) ( 1)

Integrating both sides of Eq.(1 O) from -~~ to +., andd .,1(t + ..) = — r :u~,,t (t) 4- t~d (t ) . (2) using Eqs. ( 11) and (12) , we fi nd thatThese equations are valid for j = 1 K — i . The D~.~( c ) — U ~.1( c) = t~/ t~(D~( c ) — U ~( £)] (13)They shouLd be augmented at the surface with

where j =0 , 1, 2 K. This is a general izat ion ofu,(t) = t~u 1( t) l -r ~d0 (t ) (3) the weLl— know n[1] energy t ransfe r (Kunet z ) relation.d, (t +r ) = -r0u .(t ) + t 0d,(t) (4) Note that in our der ivat ion ,inputd ,(t) is not assumed

to be an impulse acid the seismic data is not dis-acid at the basement with cretized.u5(t +r ) = t~u~~1(t) + r 1d5(t ) r,d~(t) (5) Iterat ing( 13) , starting with j L and ending withd~.1(t ) — r ,_ 1u~~1(t) +;d,(t ) = t,d,(t ) . (6) j K , we obtai n

Equatio ns (3 , 4) and (5 , ó) can be derived from ( 1) and(2) ( lett ing j = 0 , 1 K ] b y noting that u0(t ) is take n D141(c) =

~~[DL

(~

) -UL( c) ] (14)

at the bottom of layer 0 and d551(t ) represents the where L can take on the values of 0, 1 or K ,down-going wave Leaving the last interface and is ,~,not reflected by any other interface : hence u~, .( t ) .0. ~ ~ t, and fl’ ~ t’~ t (. In the marine case thi s rela-0 0These equatio ns , called causal func tio na l, are not tionship is used with L 1 . In the non-marine casedif ference equatio ns since t is the continuous time it is used with L 0 .variable [6].

5. APPLICATIO N TO MARINE ENVIRONMENTUsing the state equations given above , it ca n beshow ci [Appeci. B, 9] tha t the function d541 satisfies In this section we will direct our attention to thethe equation marine case and shal l express D~ .~( c ) in terms of

measured signals. To do this , we set r0 = 1 and wed11[t +K—) +a .d K..~[t +(K-2)r ]+ . . . +a~_~d~.1t t _ ( K _ 2) r] see from (14) tha t we must express D~( ~) -U ,( c) interms of the measured signals. The f i r s t layer can+r or~d1.1[t_ Kr 1 = ~i t 1d~(t) ( 7) be depicted as in Fig. 2. Observe that (4) becomes

No te tha t the coefficien t of the h ighes t te rm of the d1(t+ T) = _u~(t ) +Zd. 0(t) . (15)

left band side is un ity and tha t of the L owest te rm isr0r,. The precise form of the other coefficients is From (11 , 12. 13), we can evaluate the diffe r ence 5not important. Onl y the s t ructural fo rm of (7) wilt termbe utilized in the sequel. In this equation , the Un- D1( ~) -U1( ~) ~~ P( e) = r [(2d 0 ( t )_ u 1(t ) )kniowrt s are the reflections coefficients which are -.

embedded in the coefficients i , . , . , a5,.1, r0r 5 and ‘ [Zd 0(t+ c)_ u 1(t + c ) ] — u ~( t) u~(t + c) l d t (16)~~~~ The input ;(t) is assumed known . Equation (7) o ra.is the s tart ing point for our inverse procedure: P( c) = ‘~~~4d,( t)d ,(t + ~ )dt- ~ :2d o (t ) u 1(t ÷c ) d thowever, it is in terms of a signal which , in general ,is not measurable. In the following two sections we

- i~~2uit d

~(t+ e)dt : ( 17)r elate d5.1t t) t o measured seismic data. We do this

so th at we will be able to extract the reflection co- hence , P( c) can be evaluated from a knowled ge ofeL’icients from measured data. d~(t) and u ,( t ) for any desi red e. Observe , also ,4. A GENERALIZED ENERGY TRANSFER that( 14) with 1 1 can be wri t ten in terms of P( €) ,

(KUN ET Z) RELATIO N using ( 16) , as

Consider c to be a non-negative continuous or D,+~(c) = ~‘ Plc) . (18)disc rete variable wi thd in i ensionof t ime . Equations( 1)and ( 2) where j 0 , 1 K are multiplied by We should point Out at this s tage tha t the quan.u , ( t +-+c) and d~,,1(t+ r+ c) respectively resul t ing in tity u 1( t) needed in ( 17) is onl y available th r ough th e

u3 ( t+ r)u :( t+.+ c) = ~~~~~~~~~~~~~~~~~~~~~~~~~~~ observation

~~~~~~~~~~~~~~~~ +u ~51(t +c)d~(t ) ] ( 8) y(t) = u~(t) + n(t) (19)where nit) is the additive noise. Consequently, P(e)

d35~( t+ r)d .51( t+r+ c) r~u~.,1(t) u 1~1( t+ c) +t~d~( t )d a( t+ C) is not physicatty available; however , we can deficie— r~t2 (u~,.1( t )d~(t+e) +u~,.1(t+c)d 1( t ) i . P( c) by rep lacing yj t) for u 1(t) in(1 7) ,

MultipL ying( S) by t / t~ and adding the resul t ing cx- ~ ( c ) = r ~~ 4do(t) d~( t + c ) d t _ J ~:zd o(t )y( t + e ) dt

pression to (9) yields — .— Z y( t)d 5(t +c)dt (20)

(t ,/ t~)d 1( t)d ,(t +c)+u .51(t)u ~.1(t+ c) . (10) 5Because of the range of integration ic i ( 11) , we ca nLet us define the following correlat ion-type

functions aLso express D~( c) as D:( c) =

d~(t) d :(t+ ~ )dt ( 11) We use this form o f ( l 1 ) i n our development ofD,( ~) — U,( c).

-~~~~

3

which can aLso be writt eci as Note tha t in general the n~ are functions of d,51(t) , asigna l which is not determinable; however , we wilt

~~( c) = P( c) + N( c) • (21) show in the following section tha t when the inputwhere wavelet is narrow enough(ciot necessaril y a spike),

(28) has a unique solut ion for the reflection coeffi-N(c) = -s:za o(t )n(t+c)d t_ s: :zn(t)do(t+c) dt. (22) cients in terms of observable data.The statistics of noise term N(s) can be determined • 7. SPECIA L CASE OF NARROW WAVELETin terms of those of 0(1), Using ~ (e) i~~( 18) yields Let us now consider the case where d0(t) does not

(23) extecid* beyond 2,’, i. e,,

d.0(t) 0 1< 0 , t > Z ~~. (29)where ~ (c) is a known quantity. Equat toa(23) is a Since the time of ar r ival at the Kt ~ inte rface is K,~fundamenta l relationship which will be used in the and the time of arr ival of the f i rs t reflections isderivation of the inverse procedure. (K+2) r,ó. DERIVA TION OF THE NO R MA L EQUATIONS d5+~(t) sO 1< K r (30a)

Equation (7) is the main relation which will beused to derive the reflection coefficients. Dividing = 2~~t j d0(t -K ’r) K r < t ~ (K+2) ’r (3Gb)both sides by tit, acid identifying the resulting coef- more comp licaeed terxn s t > ( K + Z ) r . (30c)ficie cits by 5,, 5,,, we get Froni(27, 30a, 3Gb , and 29) we see that

5,,d , 1[t+K r]+ 91d5,1[t +( K— 2) ~] +‘. .

+ 5,, ~d5,,1[t _ ( K— 2) ‘r)+$5 d,,~1[t -K ,.] = d,,(t) . (24) r:d,,(t )~~.1(t+K~ )dt = an t~ j 2”d ( t )d t . (31)acid thatWe compute the coefficients of this equation by

means of the following ‘Least squares” criterion:i=1 , . . ., K. (32)

mon (%d ,,50[t+K r J + . . . +Note now that ( 28) will have precisely K+1 unknowns. ~‘

- —

-d , , (t ) 3’dt . (25) I~C of them in the vector multipl ying the Toep litzma.-The result of this minimization is equivalent to trix and one on the rig ht- hand side, n1~multipl ying both sides o f ( 7 ) b y d55jt+ ( K_ 2i) i .J and in.. FinaUy, Normal Equation(28) can be written in ategrating from -. to for i 0 , 1,... , K. By either compact matrix form. asapproach , we obtain K+I simultaneous equations ,

which, ustng(l1), become ~~, , x~~~j c (33)

rD~.1(O) ~~.~(zr ) ... D~

(zK~~[11 K ‘ ‘~i.~’1 where

~ c is a (K+1) x (K+1) Toeplitz matrix with thef i rs t row beingt [P(0), P( 2e) P (2Kr) 1; a5 is aD, , 1( 2 r ) D,,,,1(0) ‘.‘ . a 1( = 2 l’1t1 e~ K+1 column vector with f irst and last elements 10acid r,, respectively; and,~~ =cot(~5,O ,O, .. . , O) acid

= 2 ~~(1-r~) ~~ d (1)dt.(26) K

wh er e we have substitu ted t,,=l +r O =Z to represent The Normal Equation (33) can be solved for ~~~~. Thisthe marine ecivironmecit acid onl y produces one of the K reflection coefficients ,

namel y r5. We wilt show, in the foltowing,that iu the~ t . ~~~~~(t)d 5.0(t4K~ _ 2i,’)dt, i=O , 1, 2 K • case ~ the marine environment , nested within (33)

(27) are a set of normal equations , the solutions of whichSubstituting for D5 1 (~) from (23), we find that produce each one of the reflection coefficients. The(26) reduces to absenc e of this usefu l property in the non-marinecase renders the procedure of this paper inapplic-able in tha t case.

Let us now hypothesize a j- layer system (i. e. .P 2 e P0 :

Kr P(O) ~ (2,’) ... ~ (2K~~[1]

the basement layer is the j ’~) consisting of the top(28)

L 0 i-layers of the above K-layer system (Kg j).Ctear-~ (ZK i’) : , , •

~ (0) r5 ly, from (33), we have

__________ ( 34)Note that the (FC+t) s (K+i) matrix on the left has the

To.plltz structure. The terms N(O), N(2~’),... 51f th is condition is ciot satisfied, we can always de-

which appear in P (0), P( 2 r ) . .. ., are random van - convolve the data to achieve thu . Since the re-abLes with known statistics; they wilt be zero if the quirement here is not to deconvolve down to anseismic data is noise free (i. e. • nit) sO). Observe, impulse function (onl y (29) has to be sat isfied) , thisalso tha t the f irs t and last elements of the vector results in a more practical soluttoci.on the Left-hand side of ( Z S) are unity and r5, me-

~For a narrow wavelet and c a 2, the calculation ofspectively, by virtue of the property which we P( s) simplifies since (20) reduces tostated for the d,,51 causal functional equation.

Z.rEquatIon (28) provides the second point ~ci our ~~(c) = -2~ d,,(t ) y(t+c)dt .proced ure f~ r Identif ying the reflection coefficients.

-

4

where a . wilt again have 1 and r , as f i r s t and last tion e r ror variance. As seen , the r e su l t s are cx-elemen~~ . We shal l now show thi t , in the case of act for zero noise variance (as expected) and arethe marine environment , P. is a (j+ 1) x (j+ 1) Toepli tz quite good for var iances of 0.1 and 1. For noisematrix composed of the to~ left co rn er of P~; ~. e., variance of 10, aLthoug h the average is not poor , theits first row is given by [P (o), P(z, ’) P( Zj—)) . e r ro r var iance indicates tha t the estimates are not

For the moment let us ignore the addit ive noise very reliable.ter m i n ( 2 0 ) . Let us denote b y u~(t ) the response of A comparison between the procedure of th ia pa-the i- l ayer sy st em( i .e . , the te rm u~(t ) in Fig. ~ is pçr and the standard procedures described in[ 1.4 }replaced by u~(t )) . Ln( 17) , due to th e fact that d, , ( t ) 0 is warranted. Let us consider the noise term 0(t)for t > 2”~, the last value of u .(t ) contributing to PIE) to be white. Clearl y , (22) indicates tha t the randomis u ,(Z ’r + c) . In determinat ioci of P., with elements va riables N(. ) have finite variances. For this case

= 0 Zj r , for a j - l ayered’sv st ern the re fo re [ t ~~~~te], had we performed the necessary decon-the last val ue ot u~(t ) co n t r ibu t ing to is u~[2 ( j+ i ) ~.]. volution and sampling required by the classical ap-On the Other ha nd, u~(t ) is the res ponse of the K- proach to the inverse problem , the resu l t i ng N (. )layer system , and random variables would have infinite van iance ,clearly render ing the approach meaningless, Of course ,u .(t) = u ~(t ) 0 g t g Z( j+ l ) s ’ (35 ) “approximate ” deco nvolutio~ will eliminate thissi nce the f i rs t re turn from the in terfaces below the problem but at a great sacrif ice in the informationj t i will not a~~ ear earlier than t Z ( j + i ) r . Hence , the available within the seismic data. It should also beelements oi

~~~, which a re funct ions of u~(t) and ~ noted that, for the narrow wavelets , no deconvolut jon

wtiich a re funêti on s of u~(t ) , are identica t wheni 35l is required b y the procedure outlined in this paper.i s satisfied. Ici other words , the numerical values 9. CONCLUSIONSot P( O) P (2j — ) wilt be ident icaL to those oi theK-layer system for all j g K. Fur thermore , th e ad- We hav e developed a pr ocedure for extractingditive noise term i n ( 2 l ) is independent of the nurn - reflecti on coeff icients from noisy data which we3cr of layers , as is evide nt from (2 2) . feel is a substantial general izat ion of similar pro-

cedures which have been reported in the l i terature.The set of no rmal equati ons give n b y (3 4 ) , for i Associated with these ear l ier procedures are Stan- K , can now be solved for the vectors a., and dard Assumptions and Steps (see Introduction , p. 1)he nce, thei r last element s , r . ,j 1 K. T~~ ma- which include requirements that the data be noisetrix P5 is Toe plitz and cons~ que n tl y, the Levi cison free and tha t the observed seismic data be deco~-algo rithm [i] can be used to solve for the vectors volved. The procedure of our paper avoids thesea . , j 1 K recur s ively, res t r ic t ive requirements, Fur thermore , our proce-Since the m ’s are re fl ect ion coeff ic ients , for the dure total ly avoids the concepts of z -tran sform s ,

solution to thi ’s problem to be acceptable, each r m imum phase, spectral fac tor i z a t ion , for ward andmust be lass than unity in magnitude. It is shown in reverse polynomial manipulations, etc. , which ap-[9] that any solution of(34 ) with , >0 for all j yields pear in the l i t e rature on thi s subject. Finall y, sincea set of r ,’s which sati~ f~’ this co ndition . More- our derivation is so st ra ightfo rward , it sug ges ts aover if P~, is positive definite , a compat ible solu- nu~nber of extensions , including the following, whichtio n with ~~. > 0 is guaranteed, We see therefore , are present l y under stud y: ( 1) nonstandard locationsthat the requi rement tha t r c 1 has nothing to do of source and sensors (e. g. • bot h in the f i r s t layer) ;with a specific method of so ’ution of the norma l (2 ) minimum mean-square estimation in the non-equations (i.e. , the L.evinso n procedure) . This re- marine environment; and (3) optimal pref i tter ing ofsuit is d i f fe ren t from the comparable r esul t in [1- noisy data.3], where one is left with the impression that a ACKNOWLEDGEMENTSspecifi c method of solution leads to r~ <1,

The work reported on in this paper was per-8. EXPERIMENTAL RESULTS formed at the Un i v . of Southern Calif. , Los A ng eles ,A seve n layer system was chosen with the fo l- CA , under National Science Foundation Grants NSF

lowi ng reflection coefficients: r,= 1: r , 0. 1; r 2 ENG 74-022 97 .AO 1 and ENG 75-03423 , Air Force Of-0. 15: r 3= -O. 3; r 4 0. 25; r 5=O. 12; r,.’O.’OS; r, 0. 2. fice Of Scientific Research Grant AF OSR75 —2 79 7 ,We used a one-way layer trave l time and data Chevron Oil Field Research Co . Contract-76 andsampling rate of 20 msec and 2 nisec. respectivel y. Teled yne Exploration Co. Contract TEC-76, The

simulations were performed by Mr.Alan Yu,aThe input wavelet was choseci as depicted in graduate etudent .Figure 3. Note tha t this is a non-minimum phase REFERENCESfunction. It was specificall y chosen as such to in-dicate tha t the method introduced in this paper is 1. J. F. Claerbout , Fundamentals of Geophysicalnot limited to minimum phase wavelet s. Figure 4 Data Processinc , McGraw-Hill , N. Y., 1976,Is the synthetic seismogram response of the sy.- , L A Robinson & S. Treitel , “The Spectral Fucic-ta m. Figure 5 is obtained b y adding white Gauss- tion of a Layered System and the Determinationian noise with variance 1 to the sampled ,eismmoo - of the Waveforms at Depth , ’ Geonhy sical Pros-gram. Similar results were obtained for vari

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ances of 0. 1 and 10. These responses were then p~5ct~~g, to be published , 19 77 .• u t i l ized to produce estimates oi reflection coeffi 3. L.A. Robinson , Mult ichannel Time Series Anal~-cie nt values. The results of a Monte-Carlo simu-

~~ ~ i~h Di~ it al Computer Programs , Hoiden-lation for 100 di f fe ren t samples of noise appear in ~av , San Francisco , 1967 (Chapt. 3).Tables I and 2. Table 1 p r e s e n t s the mean value

of th. estimate, and Table 2 presents the esti.ma- If n (t ) is not white , the n the va riance of NI ’ ) may~ot be infinite , but will be very large.

5

4. E. A. Robinson , “D y namic Predic t ive Deconvolu-tio n,”Geophvstcal Prospect ing , v.23, Dec. 1975.

5 . K. B. Theriault ~ A .B . B aggeroer . ”Struc ture -

I~ HiEstimation from Acoust ic Ref le c t ion Measure Iment s , ” presented at unknown conference , ° “

I I~~. N. E. Nahi ~ J, M. Mendel , ‘A Time-Domain A p- ,~~~~,

•proach to Seismogram Synthesis for LayeredMedia ,” prese nted at the 4o ’~ An n. lot ’ I. Mtg , o f ,,,

~ 4

Society of Exploration Geophys ic is ts , Housto n , ITexas,Oct,Z4-28,ig7~ , .,.~~‘7, J, M. Mendel . N. E. Nahj , L. M, Silverman ~ H, D,Washbu r n , ” State Space Model s oi LossIe~ s Lay. ~~ ,,ered Media,” presented at 1977 Joint Aut o. Cont . 0.000 0.100(~~ 0 °•““~~~

I N S ~CConi. • San Francisco , Calif. , June 197, ,S. F.A. Gray bi l l , A n Introduction to Linear S t a t i st i - Figure 3. Source Wavet e tcal Models , VoL. 1, McGraw-Hilt , N. 1., 19o1.9. N. L. Nah i, J, M. Mendel & L, M. Silve rman, “Re- 6.0 1 - . • • • • . •

curs ive Derivat ion of Ref lec t ion Coeff icientsf rom Noisy Seismic Data , t S C E E Repor t 49 ’ . ..oo ~Jul y 1977. Also , sub mitted for publication.

5.80 —- — ~~~. 0 (6010 05100)

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Figure 1. K Layered System

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Figure 2. Firs t Layer in the Marine Case Figure 5. Noisy Sampled Seismogram r~ 1.0)

~sbI. I — • - ‘labl.A .r,g. of .6. Litton.,.. i r~ o~ R.fl .c~~oe co.Lnc~.o. 1*. 000 I4...a

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0.0 0 .0 1.0 0 . 0 0 0 00.1 0. 100 1063 0. 100 8052 0. 1060533

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0 0.0 200236C 0 .23 76 2 10 4623.306• r 6 0.00 0.0498 3276 0.03640646 3. ~b Z 0 2 2 3 r~ 3 0.090 40674 0 . 07 1380 735 .0057r . 0.20 0.2 0 13461 0. 2024367 3. 20~~3440 I - -

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