backus paper

8/6/2019 Backus Paper

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GEOPHYSICS, VOL. XXIV, NO. 2 (APRIL, 19593, PP. 233-261, 20 FIGS.

WATER REVERBE RATIONS-THEIR NATURE AKD ELIMINATION*MILO M. BACKUSt

ABSTRACTIn offshore shooting the validity of previously recorded seismic data has been severely limited by

multiple reflections within the water layer. The magnitude of this problem is dependent on the thick-ness and the nature of the boundaries of the water layer.The effect of the water layer is treated as a linear filtering mechanism, and it is suggested thatmost apparent water reverberation records probably contain some approximate subsurface structuralinformation, even in their present form.The use of inverse filtering techniques for the removal or attenuation of the water reverberationefiect is discussed. Examples show the application of the technique to conventional magnetically re-corded offshore data. It has been found that the effectiveness of the method is strongly dependenton the instrumental parameters used n the recordingof the original data.

ITTRODUCTIOI;

In marine seismic operations, the wate r-air interface is a flat, strong reflector,with a reflection coefficient close to - 1. In many areas the water-bottom inter-face is also a strong reflector. W e then have an energy trap-a non-attenuatingimedium bounded bye two strong reflecti-ng i-nterfaces. A pulse ge~nerated n thetrap, or entering the trap from below, will be successively reflected between thetwo interfaces, with a t ime interval equal to the two-way travel t ime and anamplitude deca y dependent on the reflection coefficients. As a result, valid pri-mary reflections from depth are obscured by previously established reverbera-tions.This water reverberation problem was first recognized on the basis of theapparent periodicity of a suite of seismogram s from the Persian Gulf and wastreated for the one-layer case by w ave-guide theory (Burg et al., 19 51). Sincethen, singing reco rds have been recognized as a serious and widesprea d limitationto the acquisition of valid subsurface structural information, particularly in thePersian Gulf and in Lake Jlaracaibo. The problem h as also been recognized inthe Gulf of Mexico and off the coast of California, and it is present to somedegree in any marine operation.

The wa ter reverberation problem h as been examined experimentally in modelstudies by Sarrafian (1956). The equivalent problem for a dipping bottom wasstudied and utilized in the interpretation by Poulter (1950) in his Antarcticstudies.

In the first part o f this paper, the water reverberation problem is examinedapproximately as a linear filtering problem. The predicted effect of the waterlayer on seismic data is discussed on the b asis of this analysis. In the secon d

* Paper read at the 28th Annual International Meeting of the Society at San dntonio on October16, 1958. Manuscript received by the Editor December 3, 1958.

t Geophysical Service Inc., Dallas, Texas.233


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234 MILO M. BACKUSsection, experimental data on the nature of water reverberation records is pre-sented. Finally, the use of inverse filtering to eliminate or reduce the wate r re-verberation effect is discussed and some examples are presented.

MARINE SHOOTI&G In- TERMS OF LINEAR YILTERIXGIt is useful to regard the reflection seismic method as an attempt to obtain ap-

proximately the impulse respon se of the subsurface to Vertically travelling energy.The amplitude peaks in the impulse response are then interpreted in terms ofsubsurface layering. Filtering of the impulse resp onse in the instruments, neces-sary for a satisfactory signal-to-noise ratio, results in a degradation of the data

FIG. 1. Summary of marine-reflectinn technique in terms of linear filtering.

as does filtering in the ground due to frequency dependent attenuation. Thewater layer in marine work may be regarded a s an additional undesirable andextremely sharp filter which is acting on the data. A picture of the factors actingon the data in marine work which may be considered approximately as filteringeffects is provided in Figure 1. The actual order of occurrence is shifted for clarityin developing the final form of the seismic signal.

The marine seismic source j typically 15 to 25 lb of high velocity dynamitedetonated 4 to 6 ft below the surface, produces a disturbance which may, on theseismic time scale, be regarded essentially as a positive pressure impulse (Cole,1948). The reflection from the water-air interface is presumably small due to theln~ oft energy in cavitation. The pressure pulse reverberates within the waterlayer, producing a series of pulses. The signal is further shaped according tothe position of the transducer with respect to the free surface, and this effect willbe treated as a separate filtering effect. The ringing signal is also transmitted intothe subsurface and simply reflected back into the water layer. This is the portion


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WATER REVERBERATIONS 23.5of the filtering process which provides the desired data. Upon their re-entry intothe water layer, arrivals from the subsurface ring again. Primary reflections thusessentially pass through tw o sections of the water-laye r filter. Similarly, sectionmultiples may essentially pass through a number of sections of the water-layerfilter. The data then passes through the instrumental filters and conventionally-is nonlinearly treated by automatic gain control.

The w ater-layer filter not only constitutes a sharply ringing filter but onewith characteristics which vary along the line. The records we obtain are thensimilar (except for a compounding effect on section multiples) to what we couldobtain by using a very ringing filter on land records, varying the filter fromprofile to profile.The Water-Lay er Filter

d, = pressure transducer depth.d,= water depth.7, = 2d,/l-, = two-way travel time between seis and water surface.7,=2dw/T,= two-way travel time between water surface and water bottom.

1/1p1- V&uR=----------= reflection coefficient of wate r-rock interface.FlPlf T/U&U~1= 2dJVl= two-way travel t ime between ocean bottom and subsurface

reflection.

RI=1/2p2- VIP1

= reflection coefficient at subsurface interface~l,Pz+~PlConsider an upwa rd travelling impulsive plane wave, entering the water layer

from below, arriving at Z=d, at 1=0 (Figure 2). The transducer output wouldrepresent the successive reflections between the two interfaces and would be ofthe form,

g(t) = S(l) - 6(1 - 7,) - R6(t - 7,,,)+ RS(t - Tc - 7,)+ R%(t - 27,,) -

= ,go (-~YRW ~ - m,j - 2 (-1)nR7L6(1 - UT,, - 7.)71 = j(t) - f(l - Ts), (11

wheref(lj = 5 (- l)RQ (t - ~7~). (21n=O

For an impulsive input to the water layer, g(t) is measured as the output. Equa-tion (1) thus represents the impulse response of the filter equivalent to theeffect of the water layer and the measurem ent position. This filter may be broken


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236 MILO M. BACKUS

-rr R, = -I

FIG. 2. Schematicdiagram of a water layer (VW,p,J boundedabove by a perfect reflector RF - 1,and below by a hard bottom of reflectioncoefficientR.up into two parts: (a) that which is independent of transducer dep th [equation(2)], and (b) the filtering effect due to the relationship between seismom eterand the free surface. iYe shall consider that part of the filter which is independentof the means of recording as the water-layer filter.

The equivalent frequency response may be obtained from (2).

S* ccF(w) = --m C,(- l)RQ (t - nr,)e -+dt (3 )CEwhich is the binomial expansion for

1j+) = ..__ ~ ,1 + Re-iu1 F(o) 1 = (1 + R?+ 2R cos [(JT~])-~/~, (4a)

@(CO ) tan-- R sin (~7~)1 + R cos wT~J (4b)In the limiting case of a perfectly rigid bo ttom, R = 1 and equation (4a) be-

comes

which blows up at the resonant frequencies,,AL?$5 (6)w

correspo nding to equation (2) in the article by Burg et al. (1951).Equation (4a) is plotted in Figure 3 for a water de pth, d,=lOO ft, for two

cases:~R-l, and R=O .5 Then water lawyer s equivalent to a sharply peakedcomb filter with resonance at a fundamental frequency of 123 cps and at theodd harmonics 37 cps, 62 cps, 87 cps, etc. If we recorded data which h ad been


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WATER REVERBERATIONS 237passed through this filter, using a pass-band from 2G55 cps, we would essentiallyobtain a 37-cps sine wave. Often the energy returning from the subsurface is en-riched in certain frequenc ies, and the filtering due to seismom eter dep th is gentlypeaked . Thus we would expect often to obtain a nearly pure sine wave even ona wide-band recording.For different water depths the frequency scale in Figure 3 is merely expandedor compressed. As R , the ocean-bottom reelection coefficient, changes, the degreeof peaking at resonance changes. For an incident plane wave travelling at anangle i with the vertical, the effective filtering of the wate r layer is given byreplacing r, in equation (4) by rU cos i.

If the ocean bottom is acous tically s oft, that is, if R


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238 MILO M. BACKUS

In addition to a resonance at f= 0, we have resonance at a fundamental and allharmonic frequencies. For the case illustrated in Figure 3, resonance occurs at0, 25, 50, 75, 100 cps, etc. The nature of one example of a thin, acoustically softlayer on a lake bottom has been discussed by Jo nes et al. (195 8).

The same results may be obtained in a straightforward manner by- the useof the wave equation and the appropriate boundary conditions.The Filterilzg EJect of Seismom eter Depth

The complete alteration of the signal in equation (1) w as broken up into twoparts. \Ie shall now examine the filtering which is associated with the transducerposition relative to the free surface. As shown in equation (l), the effect of pres-sure-seismometer depth is equivalent to a filter with an impulse response,

j(t) = b(1) - cqt Ts). (9 )This m erely expresse s the fact that for an uptravelling wave, the pressure trans-ducer in a homog eneous half-space sees the direct arrival and the inverted re-flection from the free interface. The frequency response of this filter is,

1 ;,(W) j = / 2 sin (7) j

(10)

(lOa)

1Ob)Similarly, for a velocity seismom eter at de pth d,, the filtering effect would be ofthe form, F,(w) = 1 + e-iwra. (11)

The amplitude and phase response [equation (10) ] is illustrated in Figure 5for seismometer dep ths of 10 and 25 ft. Phas e lag is positive. The effect of thisfilter on seismic reflection c haracter is also illustrated in Figure 4, wh ich sho wsvertical reflection energy recorded from a vertical pressure spread, using auto-matic gain control. The predicted increase in low frequency response with in-creasing seismom eter depth is apparent. It may also be seen that small variationsin seismom eter depth are less significant when th e peak of the seismometer re-sponse is som ewha t to the right of the signal spectrum. For this reason, andbecause of the general shape of the filter, a pressure seismom eter depth of 10-Eft is most commonly used in marine work.ICffect of the IT7ater-L ayer Filler OILSubsurjace Rejlectious

iYe now shall consider the effect of the water layer on the seismic reflection


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WATER REVERBERATIONS 239

FEET26IO14162226303436

ILLlJSTRIITlcW Of EFFECT Ct- SEISAPWETE RDEPTH CW IifFLECTM CHCRACTER

Upward Traveling lmpulsivrPIone Wove f(t) q 6(1)

FIG. 4. Filtering effect of seismometer depth for a pressure seismometer.

process in the simple case in Figure 5 in which there is a single subsurfac-e re-flector. Consider the injection into the system of a dow nward travelling impulsiveplane pressure wave, at t = 0, z = d,. In addition to the signal trapp ed in the wa terlayer [of the form in equation (l)], we would meas ure energy reflected from thesubsurface interface. The initial arrival will be of the form

f&) = (1 - P )R& - r1 - T,, + TsJ, (12)whe re (1 --R) is the two-way transmission coefficient at the water-rock interface.

At t= [~l+(n+l)~,+~,], (n+l) in phase arrivals, reflected up from the sub-surface once and reflected upward from the water-bottom ?z times, will be de-


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240 MILO M. BACKUS

FIG. 5. Schematic diagram of the same conditions as shown in Figure 2 with theaddition of a single subsurface reflector of reflection coefficient RI.

tected . Th e press ure signal resulting from all events singly reflected from thesubsurface interface will thus be of the form

g(t) = f(t) - f(t - 7, )f(t) = (1 - R2 )Ri 5 (-l)+ + ljm[t - T1 (S + ijJw+ Ts]. (13)

?Z=O

Similarly, by considering all possible ray-path segm ents with asso ciated re-flection and transmission coefficients, and summing in-phase arrivals, we canformulate the complete expression for the signal picked up by the transducer.

g(O = SO) + f(t) - f0 - TJ,j(t) = 2 (- l)fRV@ +zc k=j-1- ~7, + TV) + c Rlj c (- l)(l - R2)(i--k)RX

n= , j=l k=O (14)

. 2 (pzj- k,___-n= (j - K)!n! i(-1, ~+-k+W[t - $2 - (92+ j - k)Tw + 7,q.

In equation (14) thejth term represents all energy which has been reflected up-ward from the deep interface j times. The Kth term for a particular j representsall energy wh ich ha s been reflected upward j times from the deep reflector andhas suffered K downw ard reflections from the wa ter-rock interface. The 92th termin the inner sum represents the energy for a givenj and K which has been multiplyreflected n times in the water-laye r trap. Th e multiplier,

(n +j - K)!ij - k) 1 II !

represents the number of different permutations of the different ray-path seg-


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FIG. 6. Ray paths corresponding to particular terms in equation (14). This illustrates the basisfor the initial increase in amplitude of the water reverberations due to an initial arrival from depthwhen the water-bottom coefficient is greater than 0.5. Looking at the upper example, the initialarrival from depth would have an amplitude of RI (i-R2). The ratio of the second reverberation tothe initial arrival would then be 3RZ, since there are three independent permutations of the ray-pathsegments which constitute simultaneous in-phase arrivals. Thus if R2 is greater than +, the secondwater multiple from a primary reflection will have higher amplitude than the initial arrival from theprimary reflection.

ments for a given j, K, n. In Figure 6 the ray pa ths (disto rted to a point sourc efor clarity) corresponding to several particular terms in equation (14) a re illus-trated.

The only term in equation (14) which represents a desired signal is the(j= 1, K=O, n=O) term of the second sum. The o ther terms all represent ringingdue to the water-laye r filter and section multiples.

This formulation become s cumbersom e in more complex multilayered cases,hut it is quite convenient for examining the amplitude relationships for thedifferent kinds of arrivals. Amplitude as a function of time for the arrivals cor-responding to j= 0, 1, 2, 3 is plotted for R= 0.3, 0.5, and 0.7 respectively inFigures 7-9. The cases plotted are for R1=O.l, T~=.ST~.oweve r, the results fordifferent values of RI and 71 may be conveniently obtained merely by shiftingthe curves. For exam ple, to obtain the results for rl= 10.67,, each curve isshifted 5.6j units to the right. To obtain results for ir=.5~~, R1= 0.2, all curvesare shifted 6j decibels upward.

The curve forj= 0, which represents the energy which never leaves the waterlayer, s how s simple exponential decay,

j&R = Rl/ zz e --h ;,


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242 MILO M. BACKUSActually, the curve for j= 1, which represents energy reflected upward only

once from the subsurface, initially increases with time for R>O.5. The rate ofdecay for j = 1 is always less than that in equation (15)) approaching (15) as k-+ CCIThus we would expect that subsurface reflected energy will always eventuallybecome dominant over water confined energy, no matter how high R is, except

.00001

.000001

FIG. 7. Relative amplitudes and signs for termsj=O, 1, 2, 3, Case 1.Calculated for a plane wave, subsurface attenuation neglected.


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FIG. 8. Relative amplitudes and signs for termsj=O, 1, 2, 3, Case 2.Calculated for a plane urave, subsurface attenuation neglected.

when R = 1. Furtherm ore, an inspection of Figures 7-9, considering reasonablevalues fo r reflection coefficients, suggests that, in general, subsurface reflectedenergy should becom e dominant early on most marine rec ords.

Section multiple reflection energy, illustrated forj= 2 and j= 3, sho ws a morepronounced increase with time The initial peculiar behavior is due to inter-


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ference between the reverberations corresponding to different values of k. How-ever, the energy for k=O always rapidly becom es dominant, and it is the onlysignificant energy involved at the point of maximum amp litude. Thus the domi-nant portion of the section multiple energy essentially passes through the water-layer filterj times. The mere presence of a perfect reflector at the air-water inter-

FIG. 9. Relative amplitudes and signs for termsj=O, 1, 2, 3, Case 3.Calculated for a plane wave, subsurface attenuation neglected.


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WATER REVERBERATIONS 245face increases the expectation of section multiples over that encountered on land.How ever, the enhancement or compounding of section multiples by the water-layer filter is an even more significant fa ctor. Section multiples involving mo rethan one subsurface interface would be even more emphasized on marine records,due again to the existence of multiple in-phase ray paths. W e sho uld, therefore,expect section multiple energy to be very important on mo st marine singingrecords.

By regarding the amplitude coefficient for a particular j, k as a continuousfunction of n, we can predic t th e delay be tween the initial arrival and the pointof maximum energy arrival. For example, for j= 1, the maximum amplitude,31,~~ ccurs at

jZ=%,,+I--~].

The maximum amplitude is then,(17)

In Figure 9, the initial arrival from the subsurface is concealed by the water-confined reverberations. In this particular case, we would not pick the energyfrom the primary reflection until a time 2rW to 37, after th e initial arrival. Ingeneral, later initial arrivals of primary reflections would be buried under previ-ously established reverberations, and we would only read the data after it hadundergone a number of reverberations within the water layer. This delay couldeasily be on the ord er of one second. ,4 very small change in any of the reflectioncoefficients would change considerably the point on the record where we wouldread the data conn ected with any particular primary reflection.Slructural Implicatiom of a Conoen tioplal Singing R ecord Interpretation

\Ve can thus infer tha t a conventional interpretation of reco rds in a singingarea should give the following results.

(1)

(2 )

(3 )

(4 )

Very shallow picks on the record might represent water-confined energyand hence give a simply exagge rated picture of the ocean-bottom struc-ture. In general, howev er, subsurface energy should become dominantearly on the record.When data connected with a given primary reflector are picked, theywould often be considerably delayed in time and would be structurallybiased by the water-bottom structure.The records should lack continuity, and the dominant energy at anyparticular time would be connected with different. subsurface reflectorsas one moves laterally across a section.Deep er structural indications would, in general, represent water rever-


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246 MZLO M. BACKUSberations excited by section multiples; hence they w ould sho w exagg eratedshallow structure biased by exaggerated water-bottom structure.

(5) True deep unconformable structure w ould be mask ed by reverberationsconnected with shallow reflectors.

It is important to note, however, that in the case of a reasonably conformablesubsurface, a flat water bottom, and large simple structures, sufficient structuralinformation is included in the ringing reco rds as they stand, to locate thesestructures approximately. The effect of the water layer on primary reflectionenergy is equivalent to the use of very tight ringing filters on land, e xcept thatthe characteristics of the filter are changed from profile to profile.

EXPERIMEh-TAL DtlTA OS THE SATURE OF WATER REVERBERATIOiS RECORDS

In Figure 10 a textbook-type example of the water reverberation problem inthe Persian Gulf is provided. %ater depth is 195 ft according to fathometer data.Successive arrivals with a constant time interval are very apparent. However,the time interval indicates a trap depth of 210 ft. Note that successive reflectionsare inverted due to the inversion at the free surface and lack o f inversion at thewater bottom. Velocities comp uted from normal moveout and the delta-T due todip across the reco rd are shown.Ll ormal Moreout Data

The results of a conventional normal movement analysis of 15 profiles fromthe Persian Gulf (Figure 19) are shown in Figure 11. Average velocities werecom puted by using straight ray-path formu lae, and the vertical lines in Figure 11show the standard deviation. The co mputed velocity ranges from about 6 000 to8000 ft per sec. There is no evidence for average velocity values of 5000 ft perset which should be present if purely water-confined energy were dominant onany part of the record. The actual velocity as a function of depth is not well estab-lished for this area! but it is expected to be of the order of 7000 ft per set to adepth of 1000 to 1500 ft, and interval velocity below that depth should increasewith depth from an initial value of about 10,000 ft p er sec. On the basis of Figure11, it may be inferred that energy associated with the very shallow section(reverberations set up by primaries and section multiplesj is dominant over thefirst 1.0 to 1.5 set, with deeper penetration dominating in the interval from 2 to3 sec.

An additional suite of more than 20 0 profiles from an entirely different wate r-reverberation area were similarly analyzed with similar results. The m easurednormal-movement velocities were consistently intermediate between expectedvalues for subsurface primaries and purely water-confined energy.

It is concluded on the basis of normal moveout that the dominant data onnear!y all of the water-reverber ation records we have studied represents waterreverberations connected with reflections from the subsurface and that the


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24 8 MILO M. BACK US

FIG. 11. .\verage velocit)- versus center trace time from a normalmoveout anal>-& of the profiles shown in Figure 19.

travel path is made up of the ordinary subsurface travel path plus a path repre-senting a large number of successive reflections within the w ater layer .

A more striking example of the normal moveout criterion is supplied inFigure 12 in which the singing is not nearly as obvious as in the prev ious exam-ples. This reco rd was dynamically corrected for the expected normal moveoutfor primary events. Up to about 0.5 set the events have quite excessive normalmoveout, in this case actually close to that for a SOO O-ft-per-seeaverage velocity.At about 0.6 set, an event ap pears with nearly the proper normal moveout, andsuccessive reverberations connected with this subsurface reflection may be seenfollowing. This s ort of behavior is very common in areas where a moderatesinging record problem is present.

The utility of dynamically corrected record sections is immediately apparentfor this type of work where normal moveout constitutes a significant criterion inthe identification of events.Frequency _,I zalysis

The Fourier transform of four segments of the fourth trace of the profileillustrated in Figure 10 was comp uted. The amplitude as a function of frequencyis displayed in Figure 13. Amplitude coefficients were com puted at l-cps intervalsover the range 45 cps to 63 cps, and a t 3-cps intervals over the rest of the rangefrom 6 to 12 0 cps. The pass band through which the data had been filteredprevious to analysis was about 20 to 12 0 cps including the pressure transducer


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FLG.13. Frequencyanalysisof four segmentsof Trace 4, Figure 10.response. Seismometer depth was 2.5 ft, and the effective response of this filter isshown in Figure 4. Water depth was 195 ft. Accepting 42 cps and 5 4 cps as reso-nant frequencies of the water-layer filter, we predict resonances at 6 cps and allodd harmonics p lotted as open circles in Figure 13. The sharply p eaked response ,particularly at 42 and 54 cps, is quite apparent on all segments of the record.The data fits the concept of regarding the w ater layer as a filter if we accept atrap depth different from that indicated by the fathometer. This difference hasbeen found generally to be quite comm on in the Persian Gulf and has been at-tributed to the presence of a thin, soft layer overlying the hard bottom. The highfrequency fathometer sees the top of this thin layer a s the first dominant re-flector, while the low er seismic frequencies are most affected by the hard bottominterface.

In Figure 13, the complete dominance of the harmonics at 42 and 54 cpscannot be explained at all on the basis of instrumental filters and can only b epartially explained on the basis of seismom eter depth. This dominance is at-tributed to the relative enrichment of frequencies from 30 to 60 cps in the energyreturning from the subsurface. The relative complexity of the spectra is additionalevidence indicating the importance of the energy returning from the subsurface.Amplitude Decay Data

In Figure 14, several traces from a constant-gain recording in the PersianGulf, and amplitude-versus-time measuremen ts taken from that recording areshown. A plot of amplitude decay with the effect of spherical divergence approxi-


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m3anllldWv 3AllWl3tl

j_ c


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252 MILO M. BACKUSmately removed is also shown. By comparison with Figures T-9, it may be seenthat Figure 14 cannot be explained on the basis of water-confined energy, but isreasonably consistent with the concept of dominant subsurface excited re-verberations.Structural Evidence

Purely water-confined reverberation energy should show structure corre-sponding approximately to an exaggeration of trap-bottom structure, the extentof the exaggeration increasing with time On a particular profile, apparent dipshould increase regularly with time Reverberation energy excited by subsurfacereflections, on the other hand, should be biased by water-bottom structure butshou ld also contain su bsurface structure. On a particular profile, dip-versus-timeshould be made up of segments showing local change as a function of time con-trolled by water-bo ttom structure, but it should be displaced in absolute valuefrom the dip predicted from water-bottom structure. How ever, a study of thesefactors has not led to a detinitive conclusion based on structural evidence.Structural behavior may be examined in the Persian G ulf record section (Figure10) in which the fathometer and trap de pth da ta are shown.

The existence of the water reverberation problem is characterized by an ap-parent singing or dominant repetition interval on the records, by a sharp lypeaked frequency spectrum, by normal m oveout intermediate between that ex-pected for the true section velocity and that expected for w ater velocity, and byan abnormally slow energy decay. Experimental data confirms the predictionthat the dominant energy is connected with subsurface reflected energy and isconsistent with the concept of regarding the water-layer effect approximatelyas a linear filtering effect.

REMOVAL OF THE WATER REVERXERATIOX EFFECTReferring to Figure 15, the water-reverberation effect can be regarded as a

linear filtering effect rathe r than no ise. The desired primary reflection data passesthrough tw o sections of the water-layer filter, with transfer function

(18)If no non-linear proc esse s are involved in record ing th e signal, by pass ing therecorded data through the inverse of the filter in equation (18), the desiredprimary reflection data may be recovered in its original form. Th e frequency re-sponse of the required inverse filter, H(w ), is defined by


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WATER REVERBERATIONS 253or

H(cd) = (1 + Re--j@Jrw)Z. (19)The difference between this filtering conce pt and the usual seismic filtering

concept requires emphasis. In general, w e record a desired seismic signal, f(t, x),to which is added noise, N(t, x). The characteristics off and N are examined interms of frequency and wave-length. We then u se a filter with a pass band limitedto the region wh ere F(w)/N(w) is large, or we design a multiple-seismometerarray with a pass band where F(k)/N(k) is large. We filter out those frequenciesor wave lengths w here the signal-to-noise ratio is low and accept the resultingdegradation in the signal. An equivalent approa ch to singing records would beto regard the w ater reverberations as noise, and use a narrow band-pass filter

DESIRED signal RECORDED SIGNAL

WATER REVERBERATIONS-AN INVERSE FILTERING PROBLEM

DESIRED SE.ISMlCSIGNAL NOISE

f(t) N(t)

F(w) N(w)

F(k) N(k)

RECORDEDSIGNAL

f(t) + N(t) time

F(w) + NW frequency

F(k) + N(k) *aye length

SEISMIC NOISE - A SELECTIVE FILTERING PROBLEMFIG. 15. Block diagraln contrasting the water reverberation and the seismic noise pblems.


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254 MILO M. BACKUSfitted between two resonant pe aks of the w ater-layer filter, or to use an arbitrarycomb-notching filter to eliminate the reverberation frequencies. In th e case ofthe water-reverberation problem, this sort of approa ch would often result in ex-cessive degradation of the desired signal. By using the inverse filter, on the otherhand, the primary reflection events should be recovered with the wave formwhich would have been obtained if the water layer had been removed.

Section-multiple reflections are not removed by this inverse water-layerfiltering, nor are the reverberations from section multiples completely e liminated.How ever, the reverberating energy from section multiples is considerably at-tenuated , while valid initial arrivals from primary reflections are not attenuated.The compounding effect of the water layer on section multiples is reduced cou-siderably. The effect of the inverse filter in equation (19j on the n, j, K term inequation (14) is a reduction by the ratio,

(j- k)(j-- K - 1)(n + j - k) (PC j - k - 1)

which represents also the increase in the ratio of primary reflection amplitudeto the amplitude of energy connected with section multiples. For exam ple, inFigure 9, the amplitude of the reverbera tions due to the first section multiplewould be attenuated by 22 db at (t!r) = 18, and by 33 db at (t/7) = 25.

To obtain the impulse response of the required inverse filter, we transform(19) to the time domain, obtaining

h(t) = 6(t) + 2R6(t - 7) + R %(t - 2~). (20)The frequency response of the required filter is shoivn in Figure 16 for a

water depth of about 200 ft and for a bottom coefficient of 1.0 and 0.5. The phaseresponse of the filter is particularly interesting. In applying inverse filtering toa set of records it is necessary to determine whethe r the dominant reverberationproblem is due to an acoustically hard or soft bottom, and it is necessary to deter-mine the effective trap depth and the water-bottom coefficient for each profile.

The seismom eter-depth filtering effect is of a similar form to the required in-verse filter. Thus if one recorded, in the case of an acoustically hard bottom,with velocity seises ocated at the bottom, the filtering effect is of the form

P(W) = (1 + e-+~~) (21)which constitutes a notching filter with notches at the water-reverberationresonant frequencies. The same effect could be achieved in the case of an acous-tically soft bottom by the use of pressure seismom eters at the trap bottom. Thereare serious drawba cks to this approa ch, in addition to the operational difficulties.First, particularly in hard bottom areas , the significant low er trap boundary isoften d eeper than the physical bottom. Thus dragging the seises on the physicalbottom does not provide the desired result. Second, the use of seismom eter depthto attenuate the water reverberations is a noise-filtering approac h, and would


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WATER REVERBERATIONS 25.5

FIG. 16. Frequency response of the two-section inverse filter.

often, particularly in deep water, be expected to produce excessive degradationof the desired signal.Discussion of Apfromima/iom

The examination of the problem has been confined to the plane-wave case.On the later portion of the record, this analysis should nearly apply directly,since the reflected events at that time behave essentially as plane waves. On theearly part of the record, and in the behavior of the water confined energy, theessential deviations of the point-source p roblem from the plane-wave problem arethe following:

(1) time differences due to normal moveout,(2) amplitude variations due to spherical divergence,(3) variations in the effective reflection coefficient due to the sphe rical wavefront.Due to normal moveout, on the outer traces during the early part of the rec-

ord, the interval between successive water reverberations should be less thanthat on the center traces. The interval should then increase with time tendingto the interval observed on the center traces. How ever, normal moveout effectsmay be removed from the outer traces by a number of available dynamic cor-rection devices. If the normal moveout is removed in a continuous manner, thepoint-source record may be changed as far as time relationships are concerned


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256 MZLO M. BACKUSto the approximate equivalent of the plane-wave case, except for some inter-ference effects.

Due to spherica! divergence, the a_m plitude decays as llr, where r is thelength of the travel path. This factor could be approximately comp ensated forin data processing. Also the seismic amplitude suffers non-geometrical attenua-tion in the subsurface, whereas it suffers practically no non-geom etric attenuationin the water. In Figures 7-9 these factors would increase the importance of termswith small j relative to those with large j. Thes e facto rs have a very significanteffect on the results in Figures 7-9, but they do not alter the basic conclusionswhich have been drawn.

For wave lengths large relative to the distance between the source and theocean bottom, the ocean bottom has an effective reflection coefficient differentfrom that of the plane-wave case and varying with frequency. The filtering errorthus introduced would be greatest at low frequencies and in shallow wa ter. Forexample, using a case discussed by P ekeris (1948, p. 49)) the am plitude differencebetween the plane wave and spherical wave for the first reflection from the bot-tom would be about 12 percent at 20 cps for a 200-ft water depth. Becaus e of thisfactor, one of the sections of the w ater-layer filter through which primaryreflections pass deviates from the plane-wave case at low frequencies.

The problem h as been examined for flat, parallel boundaries. Slight dip orstructure in the ocean bottom (which is greatly magnified on the reverberatingrecord) reduc es slightly the effect of the inverse filtering technique. Severe dipor structure in the trap bottom combined w ith a very high bottom-reflection co-efficient results in considerable complications and the problem is not amenable toinverse filtering techniques.

A number of additional deviations of the actual c ase from the theoreticalcases discussed are readily apparent.Examples

An example of the approxim ate application of this inverse filtering techniqueto a textbook-type record is shown in Figure 17. The data w ere recorded withconventional autom atic gain control, which constitutes a non-linear proc ess, andR was set som ewh at arbitrarily to one, using only a single section of the inversewater-layer filter. This reco rd is from the Persian Gulf; water de pth is about 200ft. These are 75-21 cps playbacks of a wide band recording. N o dynamic cor-rections were applied for playback.

In Figure 18, an example from Lake Maracaibo is provided. N ote that thedominant frequencies on the unprocessed record are 30 cp s and 5 8 cps; this in-dicates that the dominant reverberation problem was due to an acousticallysoft bottom.

Figures 19 and 20 are variable density record sections of a suite of recordsfrom the Persian Gulf, illustrating the approxim ate application of the inversefiltering technique. The suite consists of 1.5 profiles providing continuous sub-surface coverage of about 3.5 miles.


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WATER REVERBERATIOXS 259

FIG. 20. This record section is identical to Figure 19 except that inverse filtering has also beenapproximately applied. Any valid events should show reverse normal-moveout.


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260 MILO M. BACKUSThe data we re recorded on magneD ISCs from linear arrays of flat-response

piezoelectric pressure transducers at a depth of 2.5 ft, using a 15120-cps passband and conventional autom atic gain control. F igure 19 is a 7.521 -cps p laybackcorrected for observednormal moveout. That is, the dynamic corrections appliedwere larger than those for expected section velocity. The process ed section(Figure 20) had the same dynamic corrections applied; hence valid reflectionevents should show reverse normal moveout. The inverse filtering technique wasonly approximately applied in the same sense as indicated for Figure 17.Discussion of Exam ples

An examination of Figures 17-20 show s that ap proximate inverse filteringhas removed the dominant singing effect or periodicity on the records and hasbrought out som e apparent reflection events w hich we re mask ed by the ringingon the unprocessed records. These uncovered events in gen eral have less normalmoveout than the dominant energy appearing at the same record time on theunprocessed records. This is particularly apparent in the reverse normal moveoutof the events in the process ed Persian Gulf record section. It means that theprocessed data at a given time represents deeper subsurface penetration. Some ofthe events on the processed records are probably initial primary reflection ar-rivals, and others are initial section-multiple arrivals. Insufficient velocity controlis available to establish clearly w hich events are primary reflections. The qualityof the reflections on the process ed data is poor, and this is due in p art to thefact that only an appro xima te application of the inverse filtering principal hasbeen made . These comparisons illustrate wha t can be done in the data processingof conventionally recorded marine seismic data.

For the optimum application of this technique special high fidelity recordingis necessary. The non-linear AGC must be eliminated or optimally minimized.The amount of signal attenuation in inverse filtering may be as high as 30 dbin extreme cases, for examp le, in the dead areas on the proc essed record inFigure 17. Thus this data-processing technique truly requires the full dynamicrange available on today s magnetic record ers.

Experimental and theoretical evidence indicates that on the majority ofsinging records the dominant energy represents water reverberations excited byreflections from the subsurface. Singing records in untreated form thus containsome structural information about the subsurface.

The effect of the water layer may be approximately treated as a linear filteringeffect and thus may be removed or attenuated by applying the principle of in-verse filtering during d ata processing . The application of inverse filtering con-stitutes a practical production technique which, under ideal conditions, approx-imately results in the elimination of the dominant apparent ringing on marinerecords, the recovery of primary reflection events with the wave form w hich they


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WATER REVERBERATIONS 261would have had if the water-rock interface had been removed, and a substantialincrease in the ratio of valid p rimary reflection events to section multiple energy.

ACKiXOWLEDGMEiXTS

The author thanks Willis S. Shelton who did much of the computational anddata-processing work involved and offered many valuable comm ents on theproblem. The author thanks G eophysical Service Incorporated for the permissionto publish this paper.

REFERENCESBortfeld, R., 1956, Multiple Reflexionen in Nordwestdeutschland: Geophys. Prosp., v. 4, p. 394-423.Burg, K. E., Ewing, Maurice, Press, Frank, and Stulken, E. J., 1951, X seismic wave guide phenome-

non: Geophysics, v. 16, p. 594-612.Cole, R. H., 1948, Underwater explosions: Princeton, Princeton University Press.Jones, J. L., et al., 1958, rlcoustic characteristics of a lake bottom: Acoustical Society merica Jour.,

v. 30, p. 142-145.Pekeris, C. L., 1948, Theory of propagation of explosive sound in shallow water: Geol. Sot. America

Mem. 27.Poulter, T. C., 1950, Geophysical studies of the Antarctic: Palo Alto, Stanford Research Institute.Sarrafian, G. P., 1956, Marine seismic model: Geophysics, v. 21, p. 32&336.Smith, W. O., 19.58, Recent underwater surveys using low frequency sound to locate shallow bedrock:

Geol. Sot. America Bull., v. 69, p. 69-98.

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