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0.092

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Seismic 23

in terms of W idesss 1982 ) criteria. Model studiesconfirm

that resolution has a linear relationship o signal bandw idth

but is affectedby noise n a nonlinear way. Under conditions

of high noise, more benefit s gained rom an imp rovemen t n

the signal-to-noiseSIN) ratio than from an increase n signal

bandwidth.A squareshaped requency spectrumgives bet-

ter resolution han a taperedspectrum.Resolution s shown

to be a function of the effective bandw idth. One H z of

bandwidthat low frequ ency is just as important as I Hz at

high frequency.

Reflection polarity affects the peak amplitude of the

composite waveform of closely spaced reflections. For

noise-free mod els, polarity is critical in the prediction of

boundary spacing based on waveform amplitude. Polarity

has little significance when noise is introdu ced into th e

model.

FIG. 5

combined with appropriate processingallow an approxi-

mately correc t interval velocity beneath the bit. From the

interval velocity, dep th may be co mpu ted as a function of

time and velocity. Dep th and interval velocity provide for

calculations o r pore pressureand porosity to be made by

standard ormulas. Pore pressurecalculation depen ds pri-

marily on depthand interval transit times. Reliability is then

a function of th e accuracyof the inversion. Porositypredic-

tion however requires more knowledge or assumptionsof

rock properties.Tests made on appropriateVSP data result

in reasonablevalues of pore pressure nd porosity. Estima-

tion o f rock prope rties has sufficient merit to continue

researchand developmentof the concept.

Seismic 24dnterpretation:

Theory

Limit of Resolution as a Function of Noise, S24.1

Frequency Spectrum and Reflection Polarity

Harry R. Espey, Grotran, Inc.

A first priority of seism ic nterpretation s the recognition

of valid reflection events. Noise-free mod el studieshave in

the past shown he resolutionof closely spaced eflections o

be limited by th e signal bandw idth, bandsh ape,and phase

spectrum . Such reso lution has little meaning if reflections

cannot be recognized n the presenceof typical noise. The

ability to identify a distinct event is limited by th e amplitude

and frequency spectrumof the noise relative to the signal.

The bandwidthand bandshape f the signal spectrumaffect

the distinctness f individual events as well a s the resolution

of closely spaced eflections.

The resolvingpower of different waveformsare evaluated

Resolving ower

Widess quantified resolving power in terms of the p eak

signal amplitude a,) relative to the total wav eform energy

0.

P,, = c,,,,E.

Base d on the form ula, a spike reflection containing all

frequencies) has infinite resolving pow er, and an infinite

time duration event reflection containing

1

frequency) has

zero resolving power. The formula provides a convenient

means of evaluating the resolvingpower of different wave-

forms.

Wides s defined resolutionas the reciprocalof the resolv-

ing power and suggested formula for resolution under

noise-freeconditions.

T,- = l/ 2 x bandw idth).

Kallweit definedresolution n terms of the upper frequency

of the spectrum f,,).

The two formulasgive very similar resultswhen the wavelet

band ratio excee ds wo octaves , but W idesss definition is

more general since t is not restrictedby bandwidth.

Spectrum hape

The zero-phasewavelets shown in Figure I both have a

bandwidthof lo-40 Hz, but the spectrumof wavelet A hasa

box shape and B has a sin ramp at the lower and upper

boundaries.Wavelet A a ppears o h ave superiorquality as a

function of side lobe energy, but the resolving pow er,

calculated by Widess s form ula, is g reater for wavelet B.

Wavelet B also looks more like a spike function. The

resolving pow er calculations indicate that clo sely space d

reflectionboundariescan be resolvedbetter with wavelet B

than with A. This is shown to be true in model studies.

The resolvingpower of reflectionwavele ts s a function of

the effective bandwidth of the sp ectrum. A wavelet with

specifiedbandwidth and box-shapedspectrumgives better

resolution than a wavelet with a sin ramp function at the

spectrum dges.A linear ramp function prod uces ess reso-

lution than a sin ramp function. The total area under the

frequency spectrumcurve controls the resolution. A spec-

trum with a linear ramp will give the same esolutionas a box

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Seismic 24

605

FIG. 1. Effect of ban d shape on wedge resolution.

spectrum over a lesserbandwidthwhen the area under each

spectrumcurve is equal.

Figure 2 shows noise-free models for a geologicwedge

which varies in thickness rom .004 to .042 sec. The mo dels

were generatedby convolving zero-ph asewaveletswith the

reflectivity function. Mode l A wavele t has a bo x spectrum

ranging rom lo-40 Hz and B has the same bandwidth, but

shapedwith a sin ramp function. Model C wavelet has a sin

ramp spectrumwith a bandwidth of 6.6 to 43.3 Hz giving it

the same effective bandwidth as model A. A .OlO-set

wedgeseparationproducesa compositewaveform suggest-

ing two reflection events for m odel A, but a se parationof

.012 set is necessary or distinctness n model B. Model C

wavelet has an absolute bandwidth greater than model A,

but the resolution s essentially h e same. The wavelet used

in model C was designed o have an effective bandwidth

equal to that of mo del As wavele t.

The results shown n Figures

1

and 2 indicate that better

resolutionmay be obtained by using a bo x-shaped pectrum

for b and-pass iltering of seismicdata. This is contrary to the

usual practiceof ram ping he filter function to minimize the

ringing effect. Th e resultsalso ndicate hat the amplitude

of a vibrator sw eep signal should not be ram ped at the

beginning and end of the sweep. The sweep is comm only

ramped o minimize side lobe energy. but at the sacrificeof

total energy radiated from the base plate. The resolving

FIG. 2. Effect of band shape on w edge resolution.

powerof the nonrampedsweep s greater,and the maximum

energy of the vibrator system s utilized,

Bandwidth and center frequency

Wides ss formula sugg estshat resolution s a function of

the spectrumbandwidthalone. Model studies onfirmed hat

a zero-phasewavelet with a box spectrum anging rom I5 to

30 Hz gives half the resolutionof a w avelet with a spectrum

of 10 o 40 Hz. A wavelet with a spectrumof IO to 70 Hz has

twice the resolutionof the 10 to 40 Hz wavelet.

The center frequency of the spectrumhas little effect of

resolution. A wavelet with 30 H z bandwidth has the same

resolutionwhether he spectrum asa center requencyof 35

or 135Hz. This is illustrated n Figure 3 usinga wedgemodel

convolvedwith two waveletshaving he samebandwidthbut

different central frequency. The wavelet with the higher

center frequency has a cyclic appearancewith a short time

betwen cycle peaks, but the time duration of the complete

wavelet is essentially he same as the wavelet with a lower

frequencyspectrum.Since the time duration of the wavelet

is a function of the bandw idth, no adva ntage s gained by

shifting o a higher requency.

The results of the m odel study in Figure 3 suggest hat

seismicdata acquisitionshouldbe carried out in a way that

preserves he maximum bandwidth. Filtering out low fre-

quency energy, during acquisition or process ing, ives the

misleading impression of high resolution due to closely

spacedcycle peaks n the signal waveform. The time dura-

tion of the wavelet can be reduced by the addition of low

frequency energy, thereby giving better resolution. Low

frequencyenergy propagates ith the least amount of atten-

uation and is therefor e available with g reater S/N strength

than high frequency energy. One Hz of b andwidthat a low

frequency s just as important as I Hz at a high frequency.

Noise

Recognition of a single reflection event as well as the

distinctionof two closely spaced vents depends n the level

of noise relative to signal amplitude. Resolution.defined by

noise-freemodels,has ittle meaningwhen a reflectionevent

cannot be reco gnized n real data containingnoise.The most

5-35 HZ

120-150 HZ

FIG. 3.

Effect of central frequencyon resolution.

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Seismic 24

critical noise occupies he same requency spectrumas the

desired signal. Noise lying ou tside the signal spectrumcan

be filtered out without loss of resolution.

Wides s ncludes noise in a definition of resolving pow er,

where r is the S/N ratio. The resolution T,) w as previously

shown to be equal to the inverse of resolving power P,).

Therefore,

T,. = 1 + l/r/2 x bandwidth.

This is an interesting relationshipbecause t shows hat for

all pra ctical purp oses eso lution is controlled by the signal

bandwidthwhen the SIN is greater han 211,but resolution s

significantly degr aded when the SIN drop s below l/l. In-

creasing he bandwidth 2 to 1 improves resolving power by

100 percent. An improvem ent in S/N from .5/l to l/l

improvesresolving pow er by 250 percent, but an impro ve-

ment in S/N from 211 o 411only improves resolving power

by 18 percent.

Figure 4 show s the result of introducing noise into th e

wedgemodel shown previously. An S/N ratio of S/l causes

a level of interference hat makes t difficult f not impossible

to recognize he geologic eature. Viewing Figure 4 from a

low grazingangle by turning the pag eon edge) mproves he

perceptionof the wedg e. It may be possible o predict the

thickness f the wedgebecauseof its linear trend. In the case

of a real geologicwedge,with so mew hat rregular hinning, it

would be very difficult to p redict accurately the thickne ss

with any no ise present. Prediction of b ed thicknes son the

basisof am plitudechangeswould be futile. T he model with

an S/N ratio of l/l showssubstantial mprovem ent n resolu-

tion as mightbe expected rom the resolvingpower ormula.

Noise-free model studies suggest hat thin b eds can be

resolvedby observationof subtle changes n the composite

waveform character.

Such

waveform changes would be

difficult o d etect with confidence n th e models n Figure 4.

SIN - .S11

SIN - 111

FIG.4. Effect

of noise on wedge resolution.

Many techniquesare available in seism icdata acquisition

to attenua te noise, including the use of source-rece iver

arrays, vertical stack, and CDP stack. An increase of

bandwidthcan be achievedby improving he low frequency

responseof the d etector and instrument systems, but an

increase n high frequency respon se s difficult due to the

absorptionpropertiesof the sedimen tary ection.Therefore,

more opportunities re available to improve the SIN than to

increase he bandwidthbut from the ab ove analysis here is

little benefit in increasing he S/N greater han 211.

Polarity

Noise-free model studies show that closely spacedbeds

cause reflection wavelets to add togetherconstructivelyor

destructivelydependingon the bed spacing nd the frequen-

cy spectrum of the wavelets. The peak amplitude of the

compositewaveform is also affectedby the p olarity of the

reflections.Under noise-freeconditions,bed thickness can

be predictedon the basisof the wavelet amplitudeprovided

the polarity is known. M odel studiessho w that a signal-to-

noise atio of better than 411 s necessaryf amplitude s to be

used o predict bed thickness.Even a small amoun t of noise

corrupts h e waveform amplitudeand limits the accuracyof

thicknesspredictions. Under typical noise conditions, the

wave form polarity has ittle significance.

Seismic Properties of Thin Layers

A. .I. Berkhout und D. de Vries, Delft Univ. of

Technology, The Netherlands

S24.2

In the seism ic iterature, thin layers are often treated by

ray theory. In this paper we show that this approach

oversimplifies he problem and m ay lead to erroneouscon-

clusions.The theo ry of thin layers s reconsidere drom first

principles. t is show n h at for a correc tderivation of seism ic

properties, he wave equation for bending waves flexural

waves) shouldbe included n the treatmentof elastic bound-

ary conditions. The proposed heo ry leads to new expres-

sions for the reflection and transmissionpropertiesof thin

layers. One of the interesting conclusionsshows hat thin

layers are fully transparent for longitudinal wave s if the

apparen t waveleng th of the incident wave along the thin

layer equals h e wavelengthof the free bending wave inside

the thin layer. Th e proposed heory can also be successfully

applied to thin surface aye rs. As an exam ple, expressions

are derived for the surface particle velocity of an upw ard

traveling wave field in situations whe re a thin we athere d

layer overlies a c onsolidated verburden.

Introduction

Thin layers occur almost everywhere in the subsurface.

Sedimentarysequences re always deposited n thin layers.

How ever, in many situations the difference of the seismic

velocitiesbetween he individual ayers s smalland an entire

packageof thin layers may be consideredas a single thick

one. In sp ecialsituations, hin layers occur n the subsurface

with significantly different elastic prope rties. We mention

low-impe dance oal layers, thin high-velocity imestone ay-

ers, and thin low-velocity surface ayers. In thesesituations

the seismic behavior of thin layers should be considered

separately. Particularly with respect to coal layers, much

wor k has been done in this field.