Kallweit and Wood 1982 Geophysics

GEOPHYSICS, VOL. 47, NO. 7 (JULY 1982); P. 1035-1046, 14 FIGS

The limits of resolution of zero-phase wavelets

R. S. Kallweit* and L. C. Wood*

ABSTRACT

This investigation deals with resolving reflections from thin beds rather than the detection of events that may or may not be resolved. Resolution is approached by considering a thinning bed and how accurately measured times on a seismic trace represent actual, vertical two-way traveltimes through the bed. Theoretical developments are in terms of frequency and time rather than wavelength and thickness because the latter two variables require knowledge of interval velocities. These results are compared with similar studies by Rayleigh, Ricker (19.53), and Widess (1973, 1980). We show that the temporal resolution of a broadband wavelet with a white spectrum is con- trolled by its highest terminal frequency f,,, and the resolution limit approximates I/ ( I .5 fU), provided the wavelet’s band ratio exceeds two octaves. The practical limit of resolution, however, occurs at a one-quarter wavelength condition and approximates I /( 1.4 fJ. The resolving power of zero-phase wavelets can be compared quanti- tatively once a wavelet is known in the time domain.

INTRODUCTION

Current efforts to increase the resolving power of the seismic method make it appropriate to examine closely our fundamental concepts of resolvability. Most reservoirs are small in the vertical dimension, and even those that are not thin out to zero thickness at the edges. A problem of major importance to an explorationist is defining thicknesses in a vertical sense and determining horizontal dimensions of reservoirs. This paper deals exclusively with vertical resolution of seismic reflection data and develops simple formulas for estimating the limits of resolution. Recent studies of wavelet resolution (Koefoed, 1981; Widess, 1980) pointed out the complexity of the problem. Koefoed proposed that there can be no single definitive measure of wavelet resolving power, only trade-off considerations between central lobe breadth, ratio of side lobe amplitude to maximum amplitude, and length of side- tail oscillations. Widess (1981), however, by defining resolving power as the maximum amplitude squared divided by the energy of the wavelet, ignored these trade-off considerations among wavelets having a common energy value. We approach the resolu-

tion problem by considering a thinning bed and how accurately measured times on synthetic traces represent actual, vertical two-way traveltimes through the bed.

Identifying individual reflections from the top and bottom of a thin bed differs from the problem of detecting the presence of the bed. It is necessary to identify two separate concepts: detection and resolution. Detection deals with recording a composite reflection with a sufficiently large signal-to-noise (S/N) ratio, regardless of whether the composite reflection can be resolved into the separate wavelets which compose it. Thus an event that is detectable may or may not be resolvable. Resolution is pri- marily a problem of frequency band, whereas detection (Farr, 1976) is principally one of acquisition. The discussion of resolution is restricted here to noise-free models of isolated thin beds, and the formulas derived here may be considered upper bounds on what can be achieved in practice.

HISTORICAL DEFINITIONS OF RESOLUTION

Let us begin by briefly examining Rayleigh’s criterion of resolution, considering next a criterion developed by Ricker (1953), and finally reviewing the work of Widess (1973). In each case we relate theoretical limits of resolution to parameters that can be measured on the wavelet that is convolved with a reflectivity sequence.

Rayleigh’s criterion

In the optical analogy, a point source is similar to a reflection spike, the optical instrument analogous to the earth, and the diffraction pattern plays the role of the band-limited wavelet. The resolving power of an optical instrument is its ability to produce separate images of objects lying close to each other. Diffraction patterns set an upper limit to resolution in a manner analogous to a propagating seismic wavelet (Figure I). Thus, if the laws of geometrical optics were to apply perfectly, optical instruments could focus parallel light to a point image. This situation, however. is analogous to having a broadband Dirac-delta function (spike) for a seismic wavelet. Huygen’s principle and the physical nature of light waves produce instead a diffraction pattern from the optical instrument, wherein the intensity of the central maximum has a finite width inversely proportional to aperture width, It becomes clear that images of two objects will coalesce into a single image and therefore not be resolved when their separation

Presented at the 47th Annual International SEG Meeting September 21, 1977, in Calgary. Manuscript received by the Editor April 2, 1979; revised manuscript received June 30, 1981. *Amoco Production Company, P.O. Box 591, Tulsa, OK 74102. SNORPAC Exploration Services, Inc., 6160 S. Syracuse Way, Englewood, CO 8011 I. 0016-8033182/0701--1035$03.00. 0 1982 Society of Exploration Geophysicists. All rights reserved.

1035

1036 Kallweit and Wood

LIGHT BEAM

DIFFRACTION

SIN * X

X2

PATTERN

lNTENSiTY m + = z-Tt;- TROUGH

+4-

RESOLVED RAYLEIGH ‘S LIMIT UNRESOLVED OF RESOLUTION

DECREASING IMAGE SEPARATION -

FIG. I, Rayleigh’s criterion states that the limit of an optical instrument to distinguish separate images of objects lying close together occurs when the two diffraction images are separated by a distance equal to the peak-to-trough distance of the instrument’s diffraction pattern.

INFLECTION POINTS -

WAVELET b = WAVELET BREADTH

b/Z = PEAK - TO - TROUGH time

2Tt-,= FIRST ZERO CROSSING INTERVAL

RESOLVED RAYLEIGH’S RICKER’S UNRE!iiOLVED CRITERION CRITERION

DECRE/XNG IMAGE SEPARATION -

FIG. 2. Rayleigh’s limit of resolution occurs when images are separated by the peak-to-trough time interval, whereas Ricker’s limit occurs when they are separated by a time interval equal to the separation between inflection points.

Resolution Zero-Phase Wavelets

becomes somewhat less than the width of the central diffraction maximum.

The criterion established by Rayleigh (Jenkins and White, 1957. p. 300) is to define the peak-to-trough separation (b/2). that is. the central-maximum to adjacent-minimum time interval of a diffraction pattern, as the limit of resolution. In other words, two point-source objects are said to be resolved when their separation is equal to or exceeds the peak-to-trough separation of the diffraction pattern of the optical instrument used for observation. Similarly. objects are said to be unresolved when their separation is less than the peak-to-trough separation as shown in Figure 1. Note also in Figure I that diffraction wavelet breadth or trough interval (b) and diffraction wavelet width or first zero-crossing interval (2Ta) are identical. This is not true of seismic wavelets containing negative side lobe energy, as seen in Figure 2. We will show that whereas wavelet breadth appears to be directly related to seismic resolution, wavelet width, for wavelets with white spectrums, does not appear to be. Thus Rayleigh’s limit of resolvability relates to the first derivative (i.e., trough time) of the wavelet. Rayleigh recognized, however, that smaller separations could be detected with sensitive intensity measuring instruments such as microphotometers because the composite of the two diffraction patterns continues to exhibit two distinct central peaks at separations less than the peak-to-trough interval (Figure 2).

Ricker’s criterion

Ricker ( 19.53) studied the composite waveform as a function of separation and observed, as did Rayleigh, that central maxima exhibit two lesser peaks as separations decrease, merging finally into single major peaks with no subsidiary maxima. Ricker chose the limit of resolvability to be that separation where the composite waveform has a curvature of zero at its central maximum. i.e., a “flat spot” (Figure 2). Ricker (1953. p. 774) showed that the resolution limit can be determined by differentiating the wavelet twice. More simply, a flat spot or zero-curvature condition occurs when two spikes are separated by an interval equal to the separation between the inflection points on the central maximum (lobe) of the convolving wavelet.

DEPTH

FIG. 3. The basis for two-term reflectivity spike models consists of a wedge of material bounded above and below by dissimilar material.

Ricker (1953) first discovered this elegant property and applied it to the case of equal-amplitude spikes with equal polarities. Widess (1973) devoted his study exclusively to the case of equal amplitudes and opposite polarities, which we discuss next.

Widess’ criterion

Widess (1973) described a study of the composite waveform obtained by convolving a zero-phase wavelet with two spikes of equal amplitudes and opposite polarities. A fundamental observation is that as spike separations decrease, the effect of convolving a wavelet with two spikes of opposite polarities is one of differentiation.

Widess observed that as separations decrease, there is a point where the composite wavelet stabilizes into a good replica of the derivative waveform such that, for all practical purposes, there is no change in the peak-to-trough time but rather only a change in amplitude of the composite waveform. Widess concluded by

HORIZONTAL DISTANCE

(Jo

3 o- --Y I- r

SAME POLARITY OPPOSITE POLARITY

FIG. 4. The time-domain response of a wedge model (Figure 3) consists of two-term reflectivity sequences with equal or unequal spike amplitudes having either the same or opposite polarities.

Kallweit and Wood

TWO - WAY LAYER THICKNESS (MILLISECONDS)

0

50

100

150

200 A

TEMPORAL RESOLUTION (TR)

FIG. 5. The seismic response of a wavelet convolved with two spikes of equal amplitude and equal polarity is a composite wavelet that varies as a function of spike separation, i.e.. layer thickness.

inspection that the limiting separation for wavelet stabilization occurs when the bed thickness (i.e., spike separation) is equal to l/8 of a wavelength of the predominant frequency of the propagating wavelet. He remarked (1973, p. 1180) that beds thinner than 118 of a wavelength can be resolved in principle by measuring changes in amplitude of the composite reflection. Use of amplitudes for this purpose, however, is difficult in actual practice because one must normalize or calibrate amplitudes to a known beds thickness with a known zero-phase wavelet. (See discussions by Lindsey et al, 1976; Meckel and Nath, 1977; Neidell and Poggiagliolmi, 1977; Nath et al, 1977; Sheriff, 1977; Schramm et al, 1977; and Clement, 1976.) For Widess the l/8- wavelength separation establishes the limit of resolvability of a thin bed because further decreases in spike separation do not appear to have corresponding changes in peak-to-trough times on a visual basis.

The main objective of this paper is to develop concepts of resolution which unify Rayleigh, Ricker, and Widess’ viewpoints, thereby removing polarity considerations from resolvability.

THE BASIC MODEL

To assess the relative merits of the above resolution criteria in providing a measure of the ability to produce a seismic trace on which the reflections from the top and bottom of a thin bed could be picked visually for a meaningful estimate of bed thickness, we performed simple model studies of an isolated thinning bed.

The fundamental model consists of a wedge of material bounded above and below by dissimilar materials (Figure 3). The two- term reflectivity sequence associated with the wedge is shown in Figure 4 for the two cases of equal and opposite polarity reflections

where it is assumed that the amplitudes of these reflections are equal. The corresponding zero-phase band-limited events are shown in Figures 5 and 7. The use of zero-phase wavelets over their nonzero-phase counterparts simplifies further developments because reflection arrival times correspond to peaks and troughs (Schoenberger, 1974; Berkhout, 1974).

Identical polarity

Observe in Figure 5 as Ray!eigh did (Jenkins and White, 1957, p. 300; Ricker, 1953) that the central maxima exhibit two distinct peaks as separations decrease, merging finally into a single major peak. The limit of resolution, which we shall call “temporal resolution” (TR), occurs when the true separation decreases to an interval where a flat spot or zero-curvature region appears on the central maximum of the composite waveform (Ricker, 1953). Ricker (1953) showed this limit can be determined by equating the second derivative of the convolving wavelet to zero as discussed in Appendix A. Temporal resolution is thus equivalent to the time separation between the inflection points on the central lobe of the convolving wavelet. Tuning thickness (Rayleigh’s criterion), expressed in time (b/2). is the point where apparent thickness (peak-to-peak time) is precisely the same as true thickness and can be determined by equating the first derivative of the convolving wavelet to zero as shown in Appendix B. It is equivalent to the wavelet’s central peak-to- adjacent trough time

Figure 6 shows apparent thickness (i.e., peak-to-peak time) and maximum absolute amplitude of the composite waveform as a function of true bed thickness for a 25-Hz Ricker wavelet. It can be seen that above tuning thickness, peak-to-peak time mea-

Resolution Zero-Phase Wavelets 1039

FIG. 6. Resolution and detection graphs for two spikes of equal amplitude and equal polarity convolved with a 2%Hz Ricker wavelet show apparent thickness (peak-to-peak time) and maximum absolute amplitude of the composite wavelet as a function of true thickness (spike separation).

surements are good approximations of bed thicknesses, whereas below tuning thickness. amplitude information must be used. This is shown by observing that the thickness curve crosses the 45 degree line at tuning thickness and then rapidly approaches a limiting value.

The waveforms in Figure 5 show that maximum composite waveform amplitudes decrease to a minimum and then increase for smaller separations, The minimum occurs at tuning thickness as seen in Figure 6; it is the difference between the peak absolute amplitude of the central lobe and the peak absolute amplitude of the adjacent side lobe of the wavelet. The amplitude curve then increases nonlinearly and finally doubles its value at small separations in the limit of zero thickness.

Opposite polarity

In Figure 7 there appears to be a minimum separation point where the composite wavelet stabilizes into a good replica of the derivative waveform beyond which there are no further signi- ficant changes in the peak-to-trough times. However, it can be seen from Figure 8, for a 25-Hz Ricker wavelet, that there is no point where the wavelet stabilizes into a good replica of the derivative waveform in the noise-free case except at the limiting value of zero thickness. Therefore, apparent wavelet peak-to- trough time stabilization is a poor criterion for determining true

TWO - WAY LAYER THICKNESS (MILLISECONDS)

26 24 22 20 18 16 14 12 10 8 6 4 2

BASIC WAVELET DERIVATIVE OF BASIC WAVELET

FIG. 7. Widess (1973) observed that a wavelet convolved with two spikes of equal amplitude and opposite polarity converges to the derivative of the convolving wavelet as spike separation decreases.


a b/2 TWO WAY TRUE THICKNESS

IMILLISECONOSI

FIG. 8. Resolution and detection graphs for two spikes of equal amplitude and opposite polarity convolved with a 25-Hz Ricker wavelet show apparent thickness (peak-to-trough time) and maximum absolute amplitude of the composite wavelet as a function of true thickness (spike separation).

bed thickness. Observe the thickness curve as true thickness decreases from 50 msec to zero. At about 33 msec, the curve deviates downward from the 4%degree line, crosses back over the 45-degree line at precisely tuning thickness, and then asymptotically approaches a limiting value at zero true thickness equal to the peak-to-trough time of the derivative wavelet. Since this time can be obtained by differentiating the convolving wavelet twice with respect to time and equating to zero, this value is the same as the separation between the inflection points on the central lobe of the convolving wavelet or temporal resolution. The apparent stability of peak-to-trough composite waveform time below tuning thickness sets a minimum apparent bed thickness that varies in value between tuning thickness and wavelet temporal resolution which is a property inherent in the convolving waveform, and it is unrelated to true bed thickness.

The amplitude curve in Figure 8 shows that at tuning thickness, composite waveform amplitudes reach a maximum equal to the sum of the maximum absolute amplitudes of the central peak and adjacent side lobe of the convolving wavelet and then decrease nonlinearly to zero. We also note that at the true thickness separation of h,,/8.5, the peak absolute amplitude of the composite waveform equals the peak absolute amplitude of the convolving Ricker wavelet.

Thus Ricker’s (1953) zero-curvature criterion for temporal resolution can be applied to the two-term reflectivity series of equal strength and opposite polarity as well as to the case of equal amplitude and equal polarity. Ricker’s criterion (temporal resolution) is thus generalized to apply without polarity constraints.

TEMPORAL RESOLUTION OF SEISMIC DATA

The above discussion shows that Rayleigh’s limit is a wavelet’s peak-to-trough time which corresponds with tuning thickness. whereas Ricker’s limit, which is the time separation between the wavelet’s inflection points, corresponds to temporal resolution.

A wavelet frequency band determines tuning thickness as well as temporal resolution. Appendices B and C derive simple formulas relating both tuning thickness and temporal resolution with spectral frequency band for Ricker and sine wavelets, which are discussed in turn below.

Ricker wavelets

The Ricker ( 1953) wavelet (velocity type) is a useful wavelet to analyze for its resolving power because of its widespread use in seismic modeling and interpretation. Temporal resolution can be expressed either in terms of peak frequency or in terms of predominant frequency. Peak frequency f,, is defined as the frequency component with the largest amplitude, whereas predominant frequency J, is the reciprocal of the trough-to-trough time or breadth about the central lobe. It is shown in Appendix B that temporal resolution for a Ricker wavelet is

and tuning thickness is given by

b I _=- 2 2.6 fp

(2)

Sine wavelets

A sine wavelet has all frequencies present with equal amplitudes (i.e.. a white spectrum) between a lower terminal frequency fc and an upper terminal frequency fi,. We refer to this frequency range as the wavelet’s frequency band cfc, fi,). Other terms used are bandwidth (f,, - fp) and band ratio (f,,/fc ). Examples of seismic data that might have a basic wavelet corresponding to a sine wavelet are Vibroseis” recordings, wavelet processed records, and signature deconvolutions.

It is shown in Appendix C that temporal resolution (TR) and tuning thickness (b/2) for a sine wavelet can be expressed as a function of the upper terminal frequency as

I TR = ~

((.2/2)J,'

and

b I _=- 2 (‘Ifi,

(4)

Parameters cl and cz are functions of upper and lower terminal frequencies and therefore frequency band. The graph in Figure 9 shows parameters cI and c2/2 versus band ratio, and we see that they converge to the asymptotic values 1.398 and 1.509, respectively, as band ratio increases. The rapid flattening of the curves beginning at around two octaves demonstrates that wavelets with flat amplitude spectra whose terminal frequencies em- brace two or more octaves have inflection points and peak-to- trough times that depend mainly on f,,. Consequently, temporal resolution and tuning thickness may be expressed with excellent approximation as

and

b I _=- 2- l.df,,’

(6)

c Trademark of Conoco, Inc.


1.3 1 I I I I I I 1 Vf, - 1 2 3 4 5 6 7 8 9

OCTAVES- 1 2 3

BANDRATIO FIG. 9. Resolution parameters for tuning thickness (~1) and temporal resolution (~2/2) for sine wavelets versus band ratio.

I

-0.04 set -7

Bond (Hz) Bondratio Bandwidth

‘I * f” (Octaves) (Hz) 2 To (msl

24,27

17,34

IO,41

6,45

3.48

6,48

12,48

24.48

44,48

3 19.6 19.4 19.4

17 19.6 18.9 18.3

31 19.6 17.1 16.0

39 19.6 15.8 14.7

45 19.6 14.9 13.8

42 18.5 14.9 13.8

36 16.7 14.6

24

4

13.9

10.9

13.4

10.9

13.7

13.0

10.9

b/2 (ms)

TR (ms)

FIG. 10. A suite of wavelets shows the relationships between frequency band (f; , J;,), band ratio (f,,/J;). bandwidth (f,, - ft). primary lobe zero-crossing interval [2To = I /(J;, + .fc )I, tuning thickness (h/2), and temporal resolution (TR).


for

Appendix C derives these relationships theoretically. For much of the data currently available with modern data processing and acquisition, these formulas are very useful interpretive tools.

There are other parameters in addition to temporal resolution and tuning thickness in the time domain, and terminal frequencies and band ratio in the frequency domain, that might be related to sine wavelet resolution. These are central lobe zero-crossing interval (27’“), where 2To = 1 /cff + f,,), midfrequency (f; +

.f;,)/2, and bandwidth (f;, - fc). These parameters are compared below for their relevance to resolution.

Figure IO shows a series of sine wavelets. Wavelets A through E have bandwidths that increase about a common midfrequency. Observe that 2To remains constant while tuning thickness (b/2) and temporal resolution (T,) decrease with increasing bandwidth. This suggests that either resolution remains unchanged or increases with increasing bandwidth.

Next compare wavelets E through 1 where fi, is held constant and bandwidth is decreased. Now both 2To, b/2, and TR decrease with decreasing bandwidth, suggesting (contrary to the above) increasing resolution with decreasing bandwidth. Clearly, a

TWO- WAY VERTICAL SPIKE SEPARATION (MILLISECONDS)

26 24 22 20 I8 16 I4 I2 IO 8 6 4 2 BAND

BANDRATIO

BANDWIDTH

-0

b/2

TR

(3,481 Hz

16 (4-OCTAVES)

45 Hz

19.6 ms

14.9 ms

13.8 ms

BAND

BANDRATIO

BANDWIDTH

2 To

b/2

TR

(12,471 Hz

4 (L-OCTAVES)

35 Hz

16.9 ms

14.9 ms

I4 .o

BAND (22,43) Hz BANDRATIO 2 (I-OCTAVE 1

BANDWIDTH 21 Hz

2 To 15.4 ms

b/2 14.9 ms

TR 14.5 ms

FIG. I I. The opposite polarity wedge model convolved with sine wavelets having identical tuning thicknesses (b/2) show relationships of wavelet parameters.


I- . . .’ 1 -- (0.50) Hz LOIPOII c

I ---- (12.5.501 Hz 2-Octave f ; : $

0 II I I I I 1 0 0 0 10 b/2 20 30 40 50

TWO-WAY TRUE THICKNESS (MILLISECONDS1

FIG. 12. Resolution and detection graphs for the opposite polarity case compare a low-pass sine wavelet to a 2-octave sine wavelet having identical upper terminal frequencies.

comparison of sine wavelets per se yields little insight into resolution.

Returning to the basic wedge model, Figure I I displays the sine wavelets (3, 48) Hz, (12. 47) Hz, and (22. 43) Hz convolved with the opposite polarity case (Figure 7 wedge model). These wavelets have identical tuning thicknesses (b/2) and band ratios of 4 octaves, 2 octaves, and I octave, respectively. Observe that reflection peak-to-trough times between the middle and upper seismogram are virtually identical and vary slightly between the middle and lower models. Therefore wavelet resolving power, measured as a function of how accurately reflection peak-to- trough times on the traces represent actual two-way traveltimes through the thinning bed, is virtually the same for the 4-octave and 2-octave wavelets and close for the 2-octave and l-octave responses.

In comparing the 2-octave wavelet to the 4-octave wavelet, we find that the only frequency-related parameter that does not change significantly is J,. This suggests that sine wavelet resolution is directly related to the upper terminal frequency and in- sensitive to band ratio provided band ratios are 2 octaves and greater. Figure I2 confirms this. A 2-octave wavelet and a low- pass wavelet with a common f,, are compared. The thickness graphs closely track for all relevant bed thicknesses and converge at temporal resolution (TR).

Another consideration in wavelet resolution is interference effects of side lobe amplitudes. It is evident that side lobe amplitudes increase as band ratio decreases to become indistinguish- able from the primary lobe at a monofrequency. The ability of the wedge model seismogram to reflect true bed thickness deteri- orates rapidly at less than about a I .5-octave band ratio, making resolution considerations in this range meaningless.

Comparison of sine and Ricker wavelets

Consider the responses to the reflectivity series models with a (12.5. 50) Hz sine wavelet. Figure I3 shows the time-amplitude

FIG. 13. Resolution and detection curves for a (12.5, 50) Hz sine wavelet show tuning thickness (b/2) and temporal resolution (Ta) limits.

plots for the two cases of equal top and bottom reflection co- efficients with equal and unequal polarities. In the opposite polarity case, peak-to-trough time and normalized amplitudes are plotted against true thickness, whereas peak-to-peak measurements are made for the equal polarity case. It is apparent that the time-separation plot for both cases oscillates around the 45. degree line, and this deviation decreases as the true thickness increases. Figures 6 and 8 show similar plots for a 2%Hz Ricker wavelet. Note that the deviation of the time plot from the 45.degree line starts at a considerably lesser bed thickness. Since both of these sine and Ricker wavelets have the same temporal resolution, this deviation is due to the side lobe effect (tuning) of the wavelet on the peak-to-trough or peak-to-peak measurements. By comparing synthetic traces generated with different convolving wavelets having identical temporal resolution, the effects of side lobe tuning on resolution and detection may be studied. This concept is of considerable importance since it establishes seismic events to be differentiated from artifacts created by wavelet tuning effects in a seismic section.

DISCUSSION

There is considerable vagueness in the current literature deal- ing with resolution concepts because of ambiguity in the detini- tions of the terms “frequency” and “wavelength.” Predominant frequency corresponds to predominant wavelength, peak frequency relates to peak wavelength, while maximum frequency corresponds to minimum wavelength and minimum frequency to maximum wavelength. In applying the above resolution formulas to reservoir thickness calculations. it is imperative to distinguish clearly between these various types of frequencies. The term peak frequency used in conjunction with Ricker wavelets desig- nates the frequency component having the largest value in the Fourier amplitude spectrum. Terminal frequencies j’,, and f; define maximum and minimum frequencies for band-limited transients such as sine wavelets. Widess (1973) defined predominant fre-


quency to be that frequency obtained by computing the reciprocal of the time interval between the wavelet’s two central side lobes; in other words, predominant frequency is obtained by recipro- cating the wavelet’s breadth time h. Referring to Figure 2, predominant frequency equals l/b. With these definitions in mind some of the confusion in establishing thin-bed thickness limits can be removed. Consider, for example, the model studied by Widess (1973) using a Ricker wavelet similar to that illustrated in Figure 7.

As shown in Appendix B, temporal resolution is given by

T, zz I 3.0 f, ’

where Jb is the peak frequency, and tuning thickness is given by

b 1 _=- 2 2.6 f, ’

Parameter b/2 may be used to relate peak frequency f,, to predominant frequency &

I fp= i= 1.3f,,. (9)

Temporal resolution and tuning thickness for a Ricker wavelet can now be expressed in terms of predominant frequency with temporal resolution given by

TR = -!- 2.31 f, ’

(10)

and tuning thickness by

15_OCT\ I-3-62-70

pG-\ 7-g-62-70

1 /-LG\ 16-17-62-70

21-23-62-70

HZ

b I

2 2 f;I

These correspond to limiting bed thicknesses of

LIZR = &

(1 I)

(12)

at temporal resolution and

in the case of tuning thickness where A, = V/f,, is the predominant wavelength through a bed of interval velocity V. Similar relationships can be established Ibr sine wavelets.

Practical applications

We have shown that the parameters TR and b/2 are directly related to resolution and may be equated to terminal frequencies and band ratio in the frequency domain. In particular, we have shown that provided a wavelet band ratio of at least 4 (2 octaves) is maintained, the resolving power of a sine wavelet is directly related to fi,, but largely independent of band ratio beyond 2 octaves.

This observation has immediate applications in field acquisition and processing. Since the upper terminal frequency appears to be the controlling factor in resolution. every effort must be made to recover the highest frequencies commensurate with resolution objectives. Expanding band ratio beyond 2 octaves by lowering 1; is a worthy consideration if factors other than thin-bed resolu-

4 200ms I- TR = IOms

FIG. 14. A suite of synthetic seismograms produced by convolving a reflectivity sequence derived from a well log with Ormsby wavelets wherein the upper terminal frequency is kept constant while varying the lower terminal frequency.


tion are important. These other factors might include trace inver- sion studies and reduced side lobe amplitude tuning considerations. Figure 14 demonstrates these conclusions through a set of synthetic seismograms where the low-frequency sides of the convolving Ormsby wavelets (approximations to sine wavelets) are varied while the high sides are kept constant. Observe that peak and trough times and number of reflections remain es- sentially invariant as band ratio is decreased. Reflection amplitudes increase slightly from 5 octaves to 2 octaves and more so from 2 octaves to I .5 octaves. However, even at I .5 octaves the increased amplitudes of events due to sidelobe tuning have not severely compromised one’s ability to differentiate these events.

Reservoir thickness estimates involving clearly defined seismic events such as bright spots may benefit from these concepts. Con- sider a deconvolved section that has been zero-phased and filtered to a (10, 65) Hz band-pass so that tuning thickness becomes

b 1 I _=-= ~ = 11 .O msec 2 1.4 fu (1.4) (65)

and temporal resolution is

TR= ’ 1 - = ~ = 10.3 msec 1.5 fu (1.5) (65)

Consider the thickness estimation of a gas-filled sand having an interval velocity of 8000 ft/sec. Tuning thickness is then (11 .O) (8) / 2 = 44 ft. Thickness estimates for noise-free models may be considered reliable for thicknesses greater than 44 ft and un- reliable for thicknesses less than 44 ft. This is because beds thinner than 44 ft cannot be distinguished from each other on the basis of interval times alone. The maximum theoretical difference between the trough-to-peak times of a 44-ft bed and an infinitely thin one is less than 1 msec. Therefore, in this example no attempt should be made to estimate bed thicknesses on the basis of interval times alone for any beds having trough-to-peak times between 10 and 11 msec.

CONCLUSIONS

Resolution is an important aspect in the interpretation of seismic traces once the data have received proper handling from data acquisition through processing. Interpreting the convolutional model for stratigraphic purposes requires a zero-phase basic wavelet in order for peak-trough times to be valid measurements for estimating bed thicknesses to use in defining reservoir dimensions. The formulas and concepts presented here are imme- diately useful in defining limits of resolvability once signal frequency band has been established. These formulas represent upper bounds because they are based upon noise-free models.

Below the tuning thickness limit, amplitude information en- codes thickness variations provided the entire amplitude variation is caused by tuning effects, and amplitude calibration then per- mits net-pay thickness calculations for arbitrarily thin beds. The study, on the other hand, establishes in a quantitative manner the absolute limits of resolution when amplitude information cannot be used.

The literature contains diverse statements as to the limits of resolvability. The concepts developed above clarify and quantify the limits by showing the practical limit actually corresponds to Rayleigh’s peak-to-trough time separation. This time interval in turn becomes the one-quarter wavelength value when the predominant frequency is used to convert to wavelengths. The most surprising result is that both Rayleigh’s peak-to-trough limit and Kicker’s inflection point criterion seem to depend mainly on the

highest frequency present, within a high degree of accuracy. in the case of a band-limited. zero-phase wavelet with a flat amplitude spectrum extending 2 octaves or more in band ratio.

ACKNOWLEDGMENTS

The authors are indebted to J. R. Moffitt for development of computer programs and technical assistance, and to R. Hastings- James for invaluable aid in revising the manuscript. We also wish to thank Amoco Production Company for permission to publish this study.

REFERENCES

Berkhout, A. .I., 1974, Related properties of minimum phase and zero- phase timefunctions: Geophys. Prosp., v. 22, p. 6X3-703.

Clement, W. A., 1976, A case history of geoseismic modeling of basal Morrow-Springer sands, Watonga-Chickasha trend, Geary, Oklahoma T13N. RIOW: AAPG Special Memoir 26, p. 451-476.

Farr, J. B., 1976, How high is high resolution‘?: Presented at the 46th Annual International SEC Meeting October 26, in Houston.

Jenkins. F. A.. and White. H. E.. 1957. Fundamentals of ootics: New York, McGraw Hill Publishing Co.,, 637 p.

Koefoed, O., 1981, Aspects of vertical seismic resolution: Geophys. Prosp.. v. 29, p, 21-30.

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APPENDIX A A DISCUSSION OF RICKER’S RESOLUTION CRITERION

Consider a zero-phase wavelet convolved with two positive Dirac-delta functions separated by a time interval of 2T. The wavelet complex s(t, T) is

s(r, T) = w(t + T) + w(t - T). (A-1)

A flat spot at the center of the wavelet complex corresponds to setting the second derivative with respect to time equal to zero and evaluating it at the origin of the coordinate system. i.e.,

w”(T) + w"-T) = 0 (A-2)

where double primes denote differentiating twice with respect to the wavelet’s argument. Since zero-phase wavelets are symmetrical in time with symmetrical second derivatives, we obtain

w"(T) = 0.

A value of T corresponding to the inflection point on the main


lobe of the wavelet satisfies the latter equation. In general there may be many values of T or roots satisfying this equation. How- ever, zero curvature is satisfied by a value of T corresponding to the inllection point on the wavelet’s main lobe, and it is the smallest value of T satisfying the zero-curvature criterion.

and, as above, define r as the band ratio ,f;,/fc. Setting the first derivative equal to zero yields

APPENDIX B RESOLUTION OF RICKER WAVELETS

A Ricker wavelet K(t) can be expressed analytically as

K(t) = [I - 2(f,~rr)~] exp[-vf,f)2]. (B-1)

Frequency fp denotes the peak frequency in the wavelet’s spectrum; it should not be confused with predominant frequency which is about 30 percent greater. Differentiating the expression for K(t),

i K(t) = 2(~f,)~t[2(7rf,t)’ - 31 exp[-(njpt)‘]. (B-2)

and equating to zero yield the tuning thickness time as

b I _=_

2.6 .fJI (B-3)

2

The above equation quantifies Rayleigh’s resolution criterion for a Ricker wavelet.

Temporal resolution TR is derived by solving for the separation between inflection points

K”(t) = (2&f,:)‘t4 - 12(7&JZf2 + 3 = 0, (B-4)

yielding

TR = 1 3.0 f"

O-5)

Both Rayleigh’s and Ricker’s criterion of resolution (inflection point separation and peak-to-trough time) depend solely on the peak frequency of the Ricker wavelet.

APPENDIX C RESOLUTION OF SING WAVELETS

A general expression for a sine wavelet M*(I) can be derived by subtracting two low-pass sine wavelets

w(t) = 2f,, sin 27rS,t ?f? sin 2Tf; t

2vf,,t - 27Ff;t I (C-1)

where f;, and fv are upper and lower terminal frequencies, respectively. A change of variables simplifies the solution of first and second derivative normal equations. Without any loss of generality, define a real variable c such that the time variable t can be expressed as

,=I (,fl, ’

(C-2)

2n 2rr 2ri 27l 2V 27F - cos - - sin - - - cos - + sin - = 0, (C-3) <’ <’ <’ (’ t-c rc’

and equating the second derivative to zero provides a similar but more complicated relationship. These two equations are the normal equations of our resolution problem. Peak-to-trough times and inflection points depend upon terminal frequencies

.I;, and .f; , that is, they are a function of both parameters c and r. Parameter r describes the sine wavelet’s frequency band as a ratio, whereas parameter (’ expresses time as a function of the upper terminal frequency only. A normal equation can be solved by first choosing a band ratio r and then solving for values of c satisfying the equation; parameter c is then plotted as a function of r. Let cl denote the solution to the first-derivative normal equation and (‘?, the solution to the second-derivative normal equation. Peak-to-trough time or tuning thickness is given by

b 1 _=- 2 (,lfr, ’

whereas inflection point times are equal to

TR I _=~ 2 (.2./U

(C-4)

(C-5)

Since temporal resolution is the intlection-point time separation, it is equal to twice the inflection point time or

TR rz --!-- k2/2)f,,

Each normal equation has an infinite number of roots. As in Appendix B, the smallest time or largest magnitude root in terms of parameter c corresponds to the central lobe complex, whereas the remaining roots correspond to side lobes. Consequently, the solution of interest is the largest root of c because it describes central lobe trough times and intlection points.

An iterative numerical method solves the normal equations for the largest value of c for any specified band ratio r. Resolution constants c, and c2/2 for the first- and second-derivative normal equations as a function of r are plotted in Figure 9. Both parameters c, and 1’?/2 converge to asymptotic values. Peak-to- trough and inflection-point time> are constant, for all practical purposes, if r exceeds 2 octave\. The peak-to-trough asymptote is cl = 1.398. whereas the inflection point asymptote is c2/2 = 1.509. Two simple equations describe Rayleigh’s and Ricker’s resolution criteria for low-pass sine wavelets and approximate wavelets having band ratios greater than 2 octaves:

and

b 1 _=~ 2 I ,398 j;<

(Rayleigh) (C-7)

TR = 1 I ,509 s,,

(Ricker)

Note that both equations depend only on the upper terminal ,. frequency

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Kallweit and Wood 1982 Geophysics