Wavelet-Tuning Cand.Real. Knut Sørsdal University of Oslo

(Article is under construction)

This article is based on a chapter in the book “Basic Theory of Exploration Seismology” of Costain and Coruh and on a chapter in the book “Reflection Seismology” by Kenneth Waters. I have expanded Costain and Coruh’s discussion of Ricker Wavelet – put in some thoughts from Waters, and done some computations in Mathematica.

The phenomenon of wavelet tuning is related to the interaction of the wavelets reflecting from the top and bottom of an interval. Tuning is controlled by the length of the wavelet with respect to the two-way time thickness of the interval. From this definition it is obvious that there will be no interaction when the thickness of the interval is greater than the length of the wavelet; however, when the thickness of the interval is smaller than the length of the wavelet, a partial interaction between the two wavelets will result in a composite waveform with a different shape as the response of the "thin bed." This has to do with the problem of resolution and diffraction that has many applications in physics and mathematics.

There are excellent references in the literature on the thin-layer, wavelet tuning, and resolution.
We have:
M.B.Widess: How thin is a thin bed? Geophys. Prosp.38:1176:1180,1973
Kallweit, R.S and Wood,L.C. The Limits of resolution of zero-phase wavelets. Geophysics 476:1035-1046, 1982 [92]
A.J.Berkhout. Section 1. seismic exploration. In Seismic resolution . Handbook of Geophysical Exploration 1984 [18]
Kallweit and Wood [92] reported on the limits of resolution of reflections from thin beds and Berkhout [18, this Handbook Series, Volume 12, p. 48] on the resolution and detect ability of thin beds. The zero-phase wavelet is required for resolution as well as detectability [18].

1.1 Resolution and Diffractions

In optics, an instrument is always qualified by a number which describes, to the initiated, the limit of fineness of detail that can be seen when the instrument is used. In the case of a telescope, this is the angular separation between two points of light that can just be resolved, that is, distinguished as separate. In light microscopes and electron microscopes, the qualification is similar but is sometimes translated into a linear measure. For example, it may be possible to resolve two points (say) 0.001 mm apart. Films used in cameras have a limit of resolution of (say) 200 lines per millimeter. That is to say, if black lines on the film were closer together than this, the area would appear to be a uniform gray color rather than showing a separation between the lines.

It must be remembered that, in optics, we are dealing with visible light having wavelengths from about 4000 to 7000 A (4 to 7 x 10 "⁷ m), usually very small compared with the detail of the objects being investigated, and the eye is sensitive only to variations in light intensity, not to variations in phase. Now it is known that, even with the most perfect optical system, free of all forms of aberration, the image of a point source is not a point but illumination over a central finite area, followed in a radial direction by successive dark and light rings. These are the famous Fraunhofer diffraction patterns (Born and Wolf, 1959, p. 391), the intensity of which is given, as a function of radial distance from the center, in Figure 1. Although this distribution is not quite a [(sin K\)/Kx]² distribution, it approaches it, and in fact a small rectangular aperture does give rise to such a distribution in both directions parallel to the sides of the rectangle.

In Fig.1. two Fraunhofer diffraction curves have been added together with a separation such that the peak of one curve lies at the first minimum of the other-such a separation occurs at an angle equal to 0.61 (/I/a), where A is the wavelength of the light being used and a is diameter of the telescope or microscope aperture being used.

There is no need for further detail here, because we have already noted that the eye is sensitive to intensity (amplitude) only, the radiation being used is monochromatic, and there is no possibility of waveform variation. Although optical theory suggests a resolution of two neighboring point sources, which is dependent on the wavelength of the light being used, there are few other analogies we can use.

Fig.1.1. Fraunhofer diffraction at a circular aperture. The function is y = [2 J(x)/x]². The sum of two Fraunhofer diffraction curves at separation 0.61 λ/a (Rayleigh criterion).

1.2 SEISMIC RESOLUTION

The question can be asked, What do we wish to distinguish from what ? In a certain (physical) sense, we do not know which part of a seismic cross section or which part of a geological section we wish to examine in detail. Until a specific objective has been stated from other evidence, it can be argued that, in most cases, it is the greater part of the geological section that may need to be examined in detail. The aim of all seismic prospecting is to learn as much as possible about a geological section—the manner in which the rocks are laid down, folded, fractured, and faulted, their mineral constitution, and the amount and type of fluid contained in the pores.

The most direct form of measurement uses the various forms of logs obtained in holes drilled through formations, and logs that contain information capable of being determined from seismic measurements are:

1. P-wave velocity logs.

2. S-wave velocity logs.

3. Density logs.

A perfectly logged single hole gives information applicable to a small volume around the hole, so that many holes are necessary to give some of the geological information sought. A seismograph would approach perfection if it could, in a vertical sense, measure the same quantities as the three logs listed, to the same degree of detail.

One answer to the general question about resolution is that we want to maximize the detail with which the vertical variation of the two seismic velocities and the density are obtained. In other words, we would like to be able to determine, as closely as one can using well logs, the depths at which the lithology and connate fluids change.

In another sense, however, the seismic method has been used to acquire information on the lateral variation in some or all of the quantities listed earlier, becoming a cheaper alternative to the drilling of many wells. We must expect therefore to be able to detect horizontal changes in the same elementary parameters—not only to detect them but to place them correctly in space. One can ask, How closely can this be done? And this is another form of resolution about which information must be forthcoming.

Finally, with only three parameters, the bulk and shear moduli of elasticity and the density, that can physically affect the seismic waves, even in the perfect case, how much knowledge of the economic factors associated with hydrocarbon production can be obtained ? And what is the precision to be expected ?

These are complex questions which do not have easy answers.

8.3 VERTICAL SEISMIC RESOLUTION

The characteristics of reflections, as related to the velocity and density logs connected to a relation between laminar velocity changes in the earth and reflection coefficients :

dR = ½ d(ln V) (1.1)

and this can be extended, if the density also varies, to

dR = ½ d(ln Z) (1.2)

where Z is the acoustic impedance of the rock— the product of the proper velocity and the density— which is a function of the depth h or the two-way reflection time t. It is possible to define a piecewise continuous function called the reflectivity:

r(t)=lim d/dt R = ½ d/dt(lnZ) (1.3)

It may be advisable to point out that the reflection coefficients we have been dealing with have been obtained at constant sampling rates. They really represent the product of the reflectivity and the sampling rate, although this has not been explicitly stated. Going back now to (1.2), we note that it is not a linear function of the acoustic impedance (a small change in impedance when the average impedance is low has a higher reflectivity than the same change when the average impedance is high). The reflectivity, leaving out some complications for the moment, is the quantity that gives rise to the seismic reflection record, but it is not as easy to interpret, geologically, as would be the actual rock property—the impedance. Thus we transform (1.2) by two steps:

(1.4)

and

(1.5)

This gives a relationship which allows calculation of Z(t) as a function of t and an assumed, or known, impedance at the beginning t₀ of the log. It does, however, assume a knowledge of reflection coefficients. There are some difficulties.

1. It is the impulse response (including all multiples and transmission losses) that is actually responsible for the seismic reflection trace.

2. This impulse response is very wide-band in frequency and, to obtain the seismic record, it has to be filtered with the effective bandwidth pulse received by the seismic system.

3. It is assumed that it is possible to produce a seismic trace as though it has been generated by plane waves (i.e., the spherical divergence has been removed exactly).

4. It is assumed that the exact values of the reflection coefficients are known. The process is nonlinear and responds in a different manner to large reflections than to small ones. Thus a knowledge of the reflection coefficient scaled by some unknown constant does not allow the exact acoustic impedance log to be determined.

The effect of all these restrictions is to limit the fidelity with which the acoustic impedance log can be displayed. It is a seismic approximate impedance log (SAIL).

Fig.1.2. The impulse response of isolated thin beds

These restrictions are now examined in more detail. It is interesting, since we are concerned mostly with questions involving thin layers, to examine the case of an isolated thin layer, that is, thin compared with the wavelengths of the seismic pulse. The impulse response, as shown in fig.2, consists of a series of rapidly diminishing pulses, equally spaced in time by the two-way transit time for the layer 2d/V, where d is the thickness and V is the relevant velocity. The signs of the separate pulses are either:

1. A first impulse from the upper surface, followed by a second of the same sign and then later ones of alternate sign (for a layer of intermediate acoustic impedance between two extremes^ or

2. A first impulse from the upper surface, followed by decreasing pulses all of the opposite sign (for a layer whose impedance is either less than, or greater than, the impedance on either side).

The sequence of impulses decays rapidly and in practice is important only for very large velocity and density contrasts which occur, for example, with thin coal beds in a siltstone matrix or for a gas sand in a shale-water-sand environment. The values plotted in this and some following diagrams are approximately 0.3—which is not a common reflection coefficient in the earth.

Next, we have to deal with the situation that each pulse in the impulse response has been filtered by the combination equipment-earth filter. It is assumed that the equipment phase has not been removed by an inverse filter such as deconvolution or a special filter designed for the purpose. In either case, each constituent pulse is a symmetrical (zero-phase) pulse. On fig.1.3. we have a Rickerpuls as initial pulse and the thin beds from fig.1.2.

Fig.1.3

We first look at the effect of integrating the wide-band reflection coefficient case. A perfect integrator can be regarded as a process or circuit by which a delta function is transformed into a positive step function. This is easily seen since, if the delta function occurs at a time τ, integration up to that time yields zero, by the definition of the delta function. At time x, the integral yields a unit value, which is retained for all times thereafter. Mathematically,

(1.6)

= 1 (otherwise)

If such an integrator is applied to a sinusoidal wave train,

(1.7)

which shows that the integrated output has an amplitude that decreases inversely as the frequency and is retarded in phase by n/2 radians. Thus, for constant-amplitude input, doubling the frequency (an increase of one octave) reduces the amplitude of the integrated output to one-half.

Figure 1.4 .shows the effect of approximating the delta function by a narrow rectangular pulse to obtain an approximate step function—with a steep ramp replacing the step because of the finite width of digital sampling. The step function can also be regarded as the impedance change between the upper and lower layer of the reflection. The amplitude is in arbitrary units because, not knowing the size of the reflection coefficient, we cannot evaluate the exponential function. In Figure 8.3b the integral of a nominal 5- to 100-Hz, flat-spectrum, autocorrelation pulse is shown. Actually, the spectrum had a taper over 4 Hz at each end of the spectrum centered over the nominal end points. Tapering a spectrum of constant bandwidth at the two ends has the following effects :

1.The high-frequency oscillations tend to die out more rapidly with time, the more slowly the tapering is done at the high-frequency end.

2.The low-frequency oscillations on either side tend to die out more rapidly with time, the more slowly the tapering is done at the low-frequency end.
3.The central breadth of the autocorrelation function tends to increase, the more tapering is done.

Various forms of taper have been used, but in practice the linear taper is almost always employed.

Fig.1.4 Integration of an approximation to a delta function

Figure 1.3 shows that, while the integration of a reflection impulse (whose spectrum contains frequencies from zero to some high value connected with the thickness of the pulse) gives an acoustic impedance change over a small interval of time and the change remains after the impulse has passed, it is obvious that the integration of the band-limited zero-phase pulse results in a fast change in apparent impedance (controlled by the high-frequency end of the spectrum) preceded, and followed, by slow decays controlled by the low-frequency end of the spectrum. We are thus forcibly made aware of the need—in order to achieve our goal of acoustic impedance fidelity— of a very wide band of frequencies. Not only are the high frequencies important, but also the lows. This statement is reinforced by several examples later.

We now go one stage further in the process of deriving the practical SAIL trace from the reflection trace, since we know that the absolute values of the reflection coefficients are not likely to be known (exceptions are sometimes possible in certain geological situations in which isolated interfaces between very thick constant-velocity rocks exist).

Instead of the true impedance Z(t), we are now dealing with an approximate impedance function Z(t), and the reflection coefficient trace must be replaced by a special processed record trace S(t). Since the absolute scale of S(t) is not known, the 2 in front of the integral sign has to be replaced by a more general constant A. Thus we have

(1.8)

Provided that is always small which, in practice appears to be nearly true, the exponential term on the right can be expanded, retaining only the constant and first-order terms:

(1.9)

Z(0) (1.10)

Thus the fractional change in impedance (as indicated by the SAIL trace) is simply the seismic trace integrated and displayed on an arbitrary scale.

In the remainder of this section, the question of resolution is examined from the point of view of the accuracy with which the variations in the actual acoustic impedance can be forecast (from the surface) by the integration of properly processed seismic traces.

Returning to the isolated thin-layer case, as a suitable starting point, we note that each of the impulses in the impulse response has to be convolved with an integrated symmetrical autocorrelation pulse. In the examples that follow, two pulses are used, having the same low-frequency spectrum (7 Hz, tapered by 4 Hz, centered on 7 Hz) but, to show the effect of the higher frequencies, in one case the upper frequency is 43 Hz and in the other 90 Hz, with a 20 Hz taper, centering on the nominal frequency. These pulses, their amplitude spectra, and the integrated pulses resulting from them are shown in Figure 1.4. Note that, in the case of the 7-431 pulse, the rise time is near 0.020 sec, whereas for the 7-901 pulse, the rise time is approximately 0.012 sec.

As far as the spectra are concerned, we know that integration results in a 6-dB increase in power (double amplitude) for every doubling of the frequency (octave). This is well shown in these figures. One way of integrating the pulse is in fact to make this change in the amplitude spectrum, change the phase by 90°, and Fourier-synthesize the pulse.

In Figures 1.5 and 1.6 are shown the results of a model computer program which computes the reflection response from a generalized layered system. Models corresponding to 10, 20, 40, 60, 80, and 200 ft are computed, the two-way times being 0.002,0.004, 0.006, 0.008, and 0.020 sec, respectively. The velocities outside and inside the layer were 5000 and 10,000 ft/sec, respectively, giving a reflection coefficient at the layer boundary equal to 0.33.

In these plots, the velocity log is first displayed, showing the different layer thicknesses and the velocities just mentioned. The first column then shows reflection coefficients, positive for downgoing waves to the right. In this display, all vertical logs, except the velocity log, are given their time amplitude relative to one another, although for each layer the maximum excursion is limited to 1 in. Thus we can use the first reflection coefficient trace for scaling purposes.

Some features to be noted are:

1. All beds up to the 60-ft bed are thin beds as far as both of the pulses are concerned. The amplitudes continue to grow with bed thickness (although not linearly).

2. For all these thin beds the higher-frequency pulse output is larger than the lower-frequency output, but the shape does not change. Here we have the first difference between the seismic and the light case. For thin beds, amplitude is a measure, in some sense, of the reflecting layer thickness. Resolution, in the light case, does not change with intensity.

3. We do not see any appreciable separation in the Rayleigh criterion sense between the two sides of the SAIL trace until the layer thickness is greater than 80 ft, but it is visible at 200 ft.

4. Between 60 and 80 ft, the higher-frequency pulse output decreases in size, whereas the lower-frequency pulse output increases.

5. The output pulses from the thin layers are very close to the original autocorrelation pulses, before integration. This is because a positive pulse, followed closely by an equal and opposite pulse, effectively differentiates the input pulse. Hence the integration has been countered by the differentiation performed by the impulse response.

6. Both pulses effectively resolve the separate edges of the 200-ft layer, producing a almost square-topped SAIL trace, although it is evident that the layer indication starts from a base very much lower than the reference line. The lack of low frequencies and zero frequency make their effect felt.

7. The impulse response shows that the predicted multiple reflections within the layer cause a slight tail, but their effect is barely noticeable.

In general, a single frequency is reflected from the lower side of a thin layer of this type with a phase change of 180°. Therefore, in order to emerge in phase with the reflection from the upper surface, a phase change of an additional 180° must occur because of the travel path. This happens when the layer has a thickness of one-quarter wavelength for the frequency and velocities concerned. We would expect the maximum amplitude reflection to occur when the layer is one-quarter of the wavelength of the average frequency in a pulse. For the 7-901 pulse, the average frequency must be near 48 Hz, and this, in a medium of 10,000 ft/sec velocity, has a wavelength of 204 ft, and the optimum reflection should be obtained with a layer about 51 ft thick. This corresponds closely to the example given, even though this too simple analysis has been given for a single arithmetic average frequency.

An output almost one-half of the maximum is, however, given by a layer only one-sixth of the optimum thickness, and it is evident that smaller reflection indications can be picked when the signal/noise ratio is good, but are these small events real ?

One other example has been presented (Figure 1.7) in which three different thickness layers are included at different depths in the section. There is no significance to be attached to the actual bed thicknesses chosen, nor to their relative depths, Although the outputs from the three layers appear very similar to those computed earlier (in value, so to speak), the impulse response trace can be used as a clue to see that, as the waves pass through each layer, the downgoing energy pulse becomes more and more complex, resulting in changed reflection outputs from the lowest layer and a series of low-frequency events of greater than expected amplitude trailing the last layer output. These can easily be interpreted as thin layers (of alternating velocity contrasts) in their own right. The presence of only a few thin, high-contrast layers can make the SAIL indications ambiguous in meaning.

Discussions of resolution usually are based on the seismic pulse itself, and some additional distinctions are made with respect to resolution and definition, the former being the ability of an object to give a high-frequency output—which may, however, be ringy if it is derived from a spectrum that is too narrow (the light case can be regarded as an example in which only the envelope of a narrow-band frequency wave is effective for resolution; see Figure 1.8—while the latter is a measure of discreteness of he pulse itself in which the waveform is useful in denning two closely spaced objects. The discussions run somewhat parallel, but the direct explanation in terms of the integrated waveform not only involves both these definitions implicitly but also serves the direct interpretation of seismic data problem much more effectively. So far, these discussions of resolutions have been given in the vacuum of a noiseee environment. It is obvious that changes in amplitude of reflections from thin beds—or indications of layer velocity changes—are interpretable only when random .;ianges of the same nature due to geological or other noise are inappreciable.

Wavelet shaping in thin beds

Now we go more into the concept of thin beds and we will differ between differentiating and integrating.

It is important to understand two facts that was illustrated on fig. 1.1:

1.A thin bed acts as a differentiator when the top and bottom of the interval are represented by reflection coefficient of opposite polarity.

2.A thin interval acts as an integrator when the polarity of the reflection coefficients is the same.

This can be visualized by considering a single frequency like fig.1

When we differentiate this frequency we get:

Integration gives this result.

Fig.1.1.One cycle of a windowed cosine wave.

We examine a zero phase Ricker wavelet defined by:

1.1

F_d is the dominant frequency of the Ricker Wavelet and t is two-way travel time.

The breadth of the Ricker wavelet in seconds is given by:

and is about 16 ms.

A Ricker wavelet with a 50-Hertz dominant frequency and its amplitude spectrum are shown in Figure 2 .a

It is obvious that the Ricker wavelets reflected from the top and bottom of an interval with opposite polarities will interact constructively when the two-way thickness of the interval is equal to the one half of the wavelet breadth. Similarly, the wavelets reflected from the top and bottom of an interval with the same polarity will interact destructively when the two-way thickness of the interval is equal to the one half of the wavelet breadth. Interaction of the wavelets with opposite polarity will result in differentiation where the effect of the differentiation is controlled by the ratio between the two-way layer thickness and one-half of the wavelet breadth.

A differentiated Ricker wavelet

If the reflection coefficients from the top and bottom of thin-layer have opposite polarity, then the wavelet interaction acts as a differentiation. We will regard this case first.

An analytically differentiated Ricker wavelet is given by:

tf_d² e ^–(πfdt)2 (389.636364 (f_d t)² – 59.2176264) 1.2

and is shown in Figure 3.a.

Variation of the peak-to-trough amplitude of the composite wavelet formed by the interaction is shown in Figure 3.b.

For a reflection coefficient of 0.25, the peak-trough amplitude of the composite wavelet quickly increases from 0 (no layer) to 0.44 while the ratio between the layer thickness and one-half breadth of the wavelet increases from 0 to 0.9. Then the peak-trough amplitude of the composite wavelet slowly decreases and asymptotically reaches the peak-trough amplitude (0.36)of the Ricker wavelet that is scaled by the reflection coefficient (0.25). The maximum tuned peak-trough amplitude of the composite wavelet corresponds to a 23% increase over the peak-trough amplitude of the interaction-free reflected wavelet. Obviously, this percentage gain is related to the type and shape of the wavelet. The increase in amplitude caused by thin-layer tuning will be greater with the relative increase in the side lobes of the wavelet.

Three snapshots that simulates wavelet interaction as the layer thickness changes are shown in Figure 3.b. The curves, in order from low- to high-amplitude, correspond to ratios of 0.142,1.0, and 1.57 between the two-way layer thickness and half-breadth of the wavelet. The change in the amplitude and shape of the composite wavelet demonstrate the importance of the effect of the thin-layer and tuning. It can easily lead to misinterpretation such as false amplitude anomalies if the effect of the thin-layer and tuning is not recognized.

The amplitude spectra of the composite wavelets given in Figure 3.c along with the spectrum of the original Ricker wavelet, which is represented by the curve that has its peak at 50 Hertz, are shown in Figure 3.d. The curve with a peak about 60 Hertz corresponds to the ratio of 0.142 between the two-way layer thickness and half-breadth of the wavelet while the curve with a peak around 40 Hertz represents the ratio of 1.57. The third curve with a peak at about 50 Hertz represents the ratio of 1.0. The general interpretation is that a thinning layer in the thinning direction will first shift the frequency band to lower frequencies and then shift it to higher frequencies. Therefore, recognition of the location of tuning via this change in the amplitude spectra might lead to the estimation of the layer thickness.

Fig.4.a-h shows the plots of the wavelet in single plots. It is the same plots that are combined on fig. 3. For the same layerthickness/one-half wavelet breadth one lot at a time:

Fig.4 shows the same as fig.3.b but with a higher frequency with a constant breadth. We can see that the breadth of the Ricker wavelet in the thin bed is bigger than on fig.3. Fig.5.b shows the specter and this has a lower frequency.

Fig.1.3 Wavelet interaction as a differentiation. Peak to through amplitude variation that results from the differentiation of Ricker Wavelets as a function of two-way layer thickness and one-half of the breadth of the wavelet.

Fig.4Differentiation of amplityde spectrum. Curves identified with their peak from left to right represent ratio 1.57,1.0, and 0.142 between layer thickness and half breadth of the wavelet.

An integrated Ricker wavelet

If the reflection coefficients from the top and bottom of a thin-layer have the same polarity, then the wavelet interaction acts as an integration. The variation in the peak-trough amplitude of the composite wavelet formed by such an interaction is shown in Figure 5.a

Again a reflection coefficient of 0.25 was considered at the top and bottom of the layer. The peak-trough amplitude of the composite (on fig.6.a) wavelet reaches its maximum at the smallest thickness and decreases as the ratio between the two-way layer thickness and half-breadth of the wavelet increases. The peak-trough amplitude has its minimum near the ratio 1.0 and asymptotically reaches the peak-trough amplitude (0.36) of the Ricker wavelet that is scaled by the reflection coefficient (0.25). The change in the peak-trough amplitude between the minimum and maximum corresponds to a variation of about 158%. The change in the maximum peak-trough amplitude over the peak-trough amplitude of the Ricker wavelet scaled by the reflection coefficient (0.25) is about 75%. Obviously, these percentage gains are determined by the type and shape of the wavelet.

The change in the peak-trough amplitudes will be greater with the relative increase in the amplitudes of the side lobes of the wavelet. Three snapshots that simulates wavelet interaction as the layer thickness changes are shown in Figure 6.. The curves, in the order from shorter length to longer length of the composite wavelet, correspond to the ratio of 0.142, 1.0, and 1.57 between the two-way layer thickness and half-breadth of the wavelet. The change in the amplitude and shape of the composite wavelet exhibit the importance of the effect of the thin layer and tuning as an integration process. As in the case of wavelet differentiation, wavelet integration can easily lead to misinterpretation such as false amplitude anomalies if the effect of the thin layer and tuning is not recognized.

Figure .a6: Wavelet interaction as an integration. The thinnest layer thickness causes the highest peak-trough amplitude with relatively shorter length of the composite wavelet. The smallest peak-trough amplitude of the composite wavelet occurs around the ratio of 1.0 between the two-way layer thickness and one-half of the breadth of the wavelet.

The amplitude spectra of the composite wavelets given in Figure 5.24 along with the spectrum of the original flicker wavelet, which is represented by the curve that has its peak at 50 Hertz, are shown in Figure 5.25. The curve with a peak about 60 Hertz corresponds to the ratio of 0.142 between the two-way layer thickness and half-breadth of the wavelet, while the curve with a peak around 40 Hertz represents the ratio of 1.57.

Figure 6.b: Effect of wavelet integration on amplitude Spectrum. The curves identified with their peak from left to right represent the ratio of 1.57, 1.0, and 0.142 between the two-way layer thickness and half-breadth of the wavelet. The fourth curve with the peak at 50 Hertz represents the original Ricker wavelet.

The third curve with a peak about 50 Hertz represents the ratio of 1.0. The general interpretation is that, as in the case of wavelet differentiation, a thinning layer in the thinning direction will first shift the frequency band to lower frequencies and then will shift it to higher frequencies. Wavelet integration results in a much more noticeable change in the amplitude spectra than does differentiation, as shown in Figure 5.25. Recognition of tuning by a change in amplitude spectra might lead to the estimation of layer thickness.

The Ricker wavelet of 80 Hertz and its analytically differentiated and integrated versions are shown in Figure 5.27 for comparison. The differentiated wavelet has smaller amplitudes than the original wavelet while the integrated wavelet has larger amplitudes than the original wavelet. In terms of shape change, the differentiated wavelet has a phase shift of +π /2 while the integrated wavelet has - π /2 phase shift with respect to the original Ricker wavelet of zero phase shift.

The general conclusion from the above summary is that the shape and amplitude of the composite wavelet are determined by the layer thickness (two-way) and the breadth of the wavelet. Wavelet interaction caused by a thin layer can lead to misinterpretations; however, it can also be used as an interpretation tool.

8.4 VERTICAL RESOLUTION AND RANDOM NOISE

Any seismic trace taken in the real world consists of a mixture of signal and other events which consist of all other energy unrelated to the problem under discussion. On a single seismic trace, there is nothing to indicate which is which, unless some of the noise lies outside the frequency band associated with the reflections. The latter may be falsely generated impulses in the computer processing, but they are usually tracked down vigorously and eliminated. For a single source set of traces, or other genetically related system, identification of false events can often be inferred from their rate of change in arrival time at the trace group—their apparent velocity along the surface and, if these false events have no relation to the reflection process being studied, they can at least be partially eliminated by velocity/-/^ filtering or, in specific instances, by a method soon to be described.

However, in the case of random noise, the shape, the time of occurrence on any one trace, and the pattern of arrival times are all unpredictable. For most of the remainder of this section, the seismic trace is assumed to consist of the time reflection signal and additive noise:

T((a) = S(co) + N(ca) (8.6) The coefficient of coherence (Foster and Guinzy, 1967) is given by

₍₈.7)

where x(t) and y(t) are stationary time series with power spectra f_xx(co) and /„.(«), respectively, and cross spectrum ./_A,.(w). It must be emphasized that estimates of coherence are made, and great care is needed. The coefficient of coherence is related to the signal/noise ratio by the expression

S(co) ₌ N(co) :

Thus, if the coefficient of coherence is estimated, the average noise power can be estimated in terms of the average signal. This provides a tool which we may need in predicting if an event seen on a trace is probably related to geology or whether it is noise. Discussion of the evaluation of this probability and the proper techniques for estimating the coherence are outside the scope of this chapter and should be sought in Foster and Guinzy (1967).

One problem, which occurs very often in all discussions of seismic data, is the proper correction for near-surface corrections. This problem is aggravated if higher-than-normal freq uencies are used, since the accuracy of time is related to the pulse duration. Coherency measures in the frequency domain appear to have been unaffected by small time origin inconsistencies, but unfortunately the estimation of the coherence coefficient requires the establishment of a lag window within which the spectrum is estimated, and the phases of the various frequencies are related to the time origin of the relevant window.

Corrected seismic traces to be composited should have the same geology-related reflections occurring at the same time. They can be added together after being scaled in amplitude by some preselected weighting function. At most, the resultant composited trace has a signal/noise ratio that has been improved by the square root of the number of traces added together. Generally, the weighting constants are equal to unity, but in some cases of high noise level the traces can be weighted inversely as their power. When the noise level is this high, however, it appears inadvisable to pursue the objective of high resolution, that is, the attainment of very high fidelity of the predicted acoustic impedance compared with that attainable by logging.

If a well is available in an area and seismic reflection work is performed in its neighborhood, the coefficient of coherence can be obtained between the synthetic trace resulting from the well log information and the field trace(s) recorded in the vicinity. Since it is evidence of the existence on the field traces of information at a particular depth that is required, the synthetic record made should start from reflection coefficients rather than from the impulse response. The reflection coefficients can be filtered to correspond in power spectrum to the field trace, but Foster and Guinzy (1967) have warned against the dangers with some forms of filter. The minimum requirement for this form of analysis is sufficient correspondence between the synthetic and field traces that a corresponding lag window with consistent starting times can be employed. In some cases, where formation alteration may be a problem, particular attention may have to be paid to obtaining velocity information through the use of a long-offset velocity tool in the hote and correcting the density measurements for any formation alteration.

8.5 VERTICAL RESOLUTION AND DEPTH OF PENETRATION

At some depths, the random noise level eliminates the chance of obtaining adequate fidelity of the seismic acoustic impedance log compared with the real log. The level of random noise is controlled by such factors as wind noise acting either directly on the geophones or indirectly throttgh shaking the ground, scattering of horizontally traveling waves by random inhomogeneities, and so on. The transmission characteristics of the water layer, for example, are excellent, and irregularities of bottom scatterers within the water (fish, gas bubbles, or man-made artifacts) combine to give incoherent scattered energy which persists up to high frequencies.

However, it has been shown that the penetration of high frequencies directly into the earth and the return of high-frequency energy from the target layers is limited by:

1. Attenuation through solid friction.

2. Loss of energy or change in frequency by reverberations and transmission losses due to many layer boundaries.

To a first approximation, the loss of amplitude follows an exponential law:

A(z) = /loe--a = the attenuation constant combining both effects and can be written as

uflQc

f = frequency

1/2 ⁼ specific dissipation constant (effective for both effects) c = velocity

Then the loss, in decibels, in traveling a distance d through the earth is

nfd

Decibel loss = 8.686 — Qc

the graph of which is a straight line. Note that the loss is proportional to both d and / Figure 8.9 illustrates diagrammatically how either of these factors changes Ik spectrum so that the high-frequency effective output drops below the noise level a< the frequency or depth is increased.

As an example, suppose that we wish to propagate 200 Hz to the layers just below Pierre shale, which has a thickness of 5000 ft, a velocity of 7800 ft/sec, and an effective Q of 20. Suppose also that an effective signal/noise ratio of 10 has been established for a frequency of 30 Hz; we would like to know how much higher the signal input musl be in order to achieve the same signal/noise ratio at 200 Hz.

The decibel loss at 30 Hz from some reference level is The decibel loss at 200 Hz from the same reference level is 8.68671 x 200 x 10,000

20 x 7800

= 349.84 dB

A required gain in signal of 297.36 dB, or about 10¹⁵ in amplitude, would be necessary to accomplish our objective—an impossibility.

Since this section is not meant to be negative toward the accomplishment of high-frequency penetration, it is pointed out that, in a section having an effective Q of 100, the difference in input for 100 Hz and 30 Hz is only about 24.5 dB, an increase in amplitude of only 16.8, an achievable result.

Experiments with shear waves have illustrated how drastic this exponential decay lay can be, and caution is required that an objective is within the realms of possibility before attempting very costly experimentation. At best, the results may be attainable only in some types of lithological sections.

8.6 VERTICAL RESOLUTION AND THE CHARACTERISTICS OF NEAR-SURFACE VELOCITY LAYERING

It was previously stated that near-surface layering, because of the reverberations produced and the effect of these reverberations on the character of the downgoing pulse, can cause loss of high frequencies. This loss is further accentuated by transmission of the reflection pulse upward.

This situation is illustrated by three model studies (Figures 8.10 through 8.12), in which the layering near the surface was chosen to be alternating in characteristics of velocity and the layer thicknesses were chosen randomly between 0 and 100 ft by a suitable computer program. The only difference in input to these three examples is the addition of 10 layers at a time—from 10 to 20 to 30. Reflection coefficients and impulse responses were computed using a zero reflection coefficient at the surface, to avoid extra complications. It can be seen that the additional layers all tend to make the impulse response for the deepest boundary deteriorate as far as high-frequency character is concerned. The impulse response is chiefly a single spike for the 10 layers, although followed by some small consecutive positive values, but the single spike continually diminishes until, at 30 layers, the succession of small consecutive positive spikes becomes dominant.

This impulse response change is reflected in the synthetic reflection records. Whereas for the 10-layer case the 07-901 filter gives a sharper indication of velocity increase at a time of 0.53 sec, by the time the 30-layer case is considered there is no more resolution than with the 07-431 filter. Although this case may seem extreme, it must be remembered that the usual shallow section has many more layers (to represent it properly) than the 30 chosen here, although the reflection coefficients involved may not always be as large.

The synthetic technique of inserting a large, isolated velocity boundary below part of a log is a very useful one for studying attenuation due to just reverberations.