Wave propagation and sampling theory--Part 1: Complex ...€¦ · For layered media, as found in...

GEOPHYSICS, VOL. 47, NO. 2 (FEBRUARY 1982); P. 203-221, 33 FIGS.

Wave propagation and sampling theory-Part I: Complex signal and scattering in multilayered media

J. Mot-let*, G. ArensS, E. Fourgeau*, and cj. Giard$

r ABSTRACT

From experimental studies in digital processing of seismic reflection data, geophysicists know that a seismic signal does vary in amplitude, shape, frequency and phase, versus propagation time To enhance the resolution of the seismic reflection method, we must investigate these variations in more detail.

We present quantitative results of theoretical studies on propagation of plane waves for normal incidence, through perfectly elastic multilayered media.

As wavelet shapes, we use zero-phase cosine wavelets modulated by a Gaussian envelope and the corresponding complex wavelets. A finite set of such wavelets, for an appropriate sampling of the frequency domain, may be taken as the basic wavelets for a Gabor expansion of any signal or trace in a two-dimensional (2-D) domain (time and frequency). We can then compute the wave propagation using complex functions and thereby obtain quantitative results including energy and phase of the propagating signals. These results appear as complex 2-D functions of time and frequency, i.e., as “instantaneous frequency spectra. ’ ’

Choosing a constant sampling rate on the logarithmic scale in the frequency domain leads to an appropriate sampling method for phase preservation of the complex signals or traces. For this purpose, we developed a Gabor expansion involving basic wavelets with a constant timeduration/mean period ratio.

For layered media, as found in sedimentary basins, we can distinguish two main types of series: (1) progressive series, and (2) cyclic or quasi-cyclic series. The second type is of high interest in hydrocarbon exploration.

Progressive series do not involve noticeable distortions of the seismic signal. We studied, therefore, the wave propagation in cyclic series and, first, simple models made up of two components (binary media). Such periodic structures have a spatial period. We present synthetic traces computed in the time domain using the Goupillaud- Kunetz model of propagation for one-dimensional (1-D) synthetic seismograms.

Three different cases appear for signal scattering, depending upon the value of the ratio wavelength of the signal/spatial period of the medium.

(1) Large wavelengths

The composite medium is fully transparent, but phase delaying. It acts like an homogeneous medium, with an “effective velocity” and an “effective impedance.”

(2) Short wavelengths

For wavelengths close to twice the spatial period of the medium, the composite medium strongly attenuates the transmission, and superreflectivity occurs as counterpart.

(3) Intermediate wavelengths

For intermediate values of the frequency, velocity dispersion versus frequency appears.

All these phenomena are studied in the frequency domain, by analytic formulation of the transfer functions of the composite media for transmission and reflection. Such phenomena are similar to Bloch waves in crystal lattices as studied in solid state physics, with only a difference in scale, and we checked their conformity with laboratory measure- ments.

Such models give us an easy way to introduce the use of effeciive velocities and impedances which are frequency dependent, i.e., complex. They will be helpful for further developments of “complex deconvolution.”

The above results can be extended to quasi-cyclic media made up of a random distribution of double layers. For signal transmission, quasi-cyclic series act as a high cutfilter with possible time delay, velocity dispersion, and “constant Q” type of law for attenuation. For signal reflection they act as a low cut filter, with possible superreflections.

These studies could be extended to three-dimensional (3-D) binary models (grains and pores in a porous reservoir), in agreement with well-known acoustic properties of gas reservoirs (theory of bright spots).

We present some applications to real well data. Velocity dispersion may explain: mistying between sonic logs and

Presented at the 50th Annual International SEG Meeting November 19, 1980 in Houston as “Signal filtering and velocity dispersion through multilayered tiedia.” Manuscript received by the Editor July 11, 1980; revised manuscript received May 8, 1981. *ELF Aquitaine Company, O.R.I.C. Lab, 370 bis Av. NapolCon Bonaparte, 92500 Rueil Malmaison, France. SELF Aquitaine Company, S.N.E.A. (P). Tour GCnkrale, Cedex 22, 92088 Paris La defense France. 0016-8033/82/0201-203$03.00. 0 1982 Society of Exploration Geophysicists. All rights reserved.

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204 Morlet et al

velocity surveys; mistying between synthetic seismograms and seismic sections.

Effective velocities lower than mean velocities may explain: very low velocities for P- and S-waves, especially in weathered shallow layers; anomalies on the values of the ratio VA/VP ; mistying between P and S seismic sections.

Finally, the Gabor expansion provides a tool to obtain sampled instantaneous frequency spectra, and to carry out a suitable recording and processing method in high- resolution seismic, especially to preserve the phase information. Such a processing will involve complex signals, complex traces, complex velocities and complex impedances.

For practical purpose, this paper comprises two separate parts. Here, we present some interesting features on the scattering of seismic signals obtained by a simulation method using simple models. We show the usefulness of the notions of complex signals, complex velocities, and complex impedances, and overall of the Gabor expansion, by simulation on very simple models.

Morlet et al, (1982, this issue) will be concerned with the development of fundamental notions useful to handle seismic data from the sampling method of recording to processing methods. There we give theoretical and prac-. tical tools to sample and handle these data in the time- frequency domain, using complex functions.

INTRODUCTION

A fundamental objective of all research geophysicists is to increase the resolving power of the seismic reflection method. But a disturbing fact appears: It is very difficult, and often impossible, to resolve the seismic records into elementary interfaces especially because of the large ratio of signal wavelength/layer thickness, which is sometimes much greater than unity.

However, since our problem is to resolve layers rather than to separate interfaces, we can use the models of quantum mechanics, more specifically those of wave propagation in crystal lattices.

Referring to SchrBdinger, let us note that “Quantum Mechanics stands in the same relation to ordinary Classical Mechanics that physical optics does to geometrical optics” (Cohen-Tannoudji et al, 1977; Morlet, 1975; Encyclopedia Britannica, 1967).

From this viewpoint, how does the geologic series appear to a geophysicist? In sedimentary basins, the geologic layers are cyclic or quasi-cyclic rather than progressive (especially series contain- ing fluid hydrocarbons or coal). Fortunately, this is helpful for the seismic reflection method, because information quantity and reflectivity are related.

We will therefore choose the following specifications for our models: (1) For the media, we will model the geologic series as a random superposition of cyclic or quasi-cyclic series, rather than as a random distribution of layers. In such a distribution, an elementary series is made up of two layers or more; there are therefore no individual interfaces but a sequence of “interface clusters,” and sometimes reflectivity can appear at the boundary between two successive cyclic series. Starting with binary periodic media, we will show the physical meaning of effective velocities and impedances and their frequency dependence.

(2) For rhe signal, we will use models with the following properties: simple but realistic; suitable for energy and frequency quantification; suitable for attenuation and resolution studies. More specifically, we will use a set of Gabor complex wavelets, for various frequencies, simulating therefore a time-frequency sampling method for the wave propagation studies.

We will demonstrate the interest of a time-frequency method of sampling, leading to a specific type of Gabor expansion of the traces into basic wavelets.

(3) For the algorithm, we will use the I-D wave propagation model known as the Goupillaud-Kunetz algorithm for synthetic seismograms. We will show thereafter what type of assumptions are necessary to extend the results to 3-D models of rocks (especially porous unconsolidated reservoirs).

Finally, we will give examples of practical applications.

WAVE PROPAGATION IN PERIODIC MEDIA

First, we make the following assumptions for simplicity: plane waves; normal incidence; 1-D perfectly elastic layered media; simple models of layering, i.e., cyclic series made of two components, and simple signals. Later, extensions could be made as follows: more complex media (quasi-cyclic l-D, then 2-D and 3-D); oblique incidences, more complex signals. and quasi-plane waves.

Thus, we start with deterministic models and then introduce statistical models involving various distributions of cyclic series.

The 1-D wave equation

We introduce here the notations we will use, and recall briefly the fundamentals of wave propagation.

Propagation in a homogeneous medium.-We assume (1) perfect elasticity (Hooke’s law), (2) propagation in an un- bounded medium, (3) compressional waves. The elastic wave equation is then

a2F/az2 = (i/v$(a2F/ar*),

which is valid for F being particle displacement, cubical dilatation, excess pressure or stress, and particle velocity. Two fundamental characteristics for the wave are P = excess pressure or stress (related to potential energy), U = particle velocity (related to kinetic energy).

The modulus of a complex linear combination of these last two quantities is proportional to the square root of the total energy carried by the wave (see Morlet et al, 1982, this issue).

Two fundamental parameters for the medium are V = velocity of compressional waves, and Z = impedance, with the following relations:

v = (E/p)“‘, z = (p/J”2

where E = Young’s modulus, and p = density.

The general solution of the wave equation is

= A(o) x eiwz’” 1

I I 1/Z

with w = angular frequency, and z = depth. Thus, for a progressive downgoing wave B(o) = 0 and P = Z x U, in conformity with the principle of the equipartition of the total energy in potential and kinetic energies (the total energy being thus represented by a complex function). For a progressive upgoing wave.A(o)=OandP= -ZXU.

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Complex Signal and Scattering-Part I 205

HOMOGENEOUS UPPER MEDIUM

Pe

Ze = PeVe

TRAVELING time : I,;1

VELOCITY : VI IMPEDANCE : 21 = f’1 Vl

PERIODICAL SERIES

HOMOGENEOUS SUBSTRATUM

FIG. 1. Periodic model for the propagation medium

Propagation through a plane interface.-We assume here (1) plane waves, (2) normal incidence, and (3) two semiinfinite media. Writing first the two fundamental equations for continuity of the wave function at the interface, for particle velocities, and conservation of the energy applied to the energy flow through the interface, for kinetic energies of incident, reflected, and transmitted waves, then combining these two relations, permits us to introduce the reflection coefficient at the interface, for particle velocity U:

R = (Z, - Z,)/(Z, + Z2)>

where Zt = impedance of the incident medium, and Z2 = impedance of the transmission medium.

For the excess pressure P, the sign of R is opposite, and

R = (Z, - Z,)/(Zz + Z,).

Note that similar equations hold for shear waves, and more gen- erally for electromagnetic (EM) waves and electric currents.

Characteristics of the media

For simplicity, we start with a 1-D model made of a finite periodic series as shown in Figure I. For each elementary homogeneous component medium, the four characteristic parameters are: T = one way traveltime, p = density, V = velocity (or celerity) of the wave, Z = p X V = impedance (redundant parameter).

For a binary periodic series made up of two component media, the parameters are (1) for each elementary structure or “motif,” made up of two layers: there are four parameters for medium I (subscript 1), and four parameters for medium 2 (subscript 2); (2) there is a total N of elementary structures or motifs, and (3) the characteristic parameters are for upper medium or “entry medium” (subscript e), substratum (subscript s).

We will study a very large range of parameter values. including very low densities, as observed in the case of gas. The

MEAN PERIOD T= 2DMS. MEAN FREQUENCY- fz50Hz. DIAMETER At q 4DMS.

AMPLITUDE

I

FIG. 2. Example of Gabor complex wavelet used as signal model.


206 Morlet et al

investigations for very strong impedance contrasts will be used when extrapolating the results to 3-D models of aerated rocks or gas reservoirs.

Characteristics of the signals

To study the energy and dispersion, we need to measure amplitudes on envelopes, phases, times and delays. Therefore we need short signals, easy to define in both time and frequency domains.

Such complex wavelets were studied by D. Gabor and lead to a time-frequency decomposition of the signals or seismic traces (Gabor, 1946, 1951). They are made of a pure sine wave modulated by a Gaussian function as shown in Figure 2. The cosine and sine parts of the complex wavelet are in quadrature. In the frequency domain, their amplitude response is a Gaussian function, as in the time domain.

Such wavelets are defined by the following parameters:

T = mean period, f = 1 /T = mean frequency,

A0 = maximum amplitude of wavelet, At = diameter or duration, Af = bandwidth.

At and Af give physical limits of the wavelet in the time and the frequency domain, respectively.

This definition being somewhat arbitrary, we can choose the practical limits of the wavelets as the points where the amplitude of the Gaussian envelope drops to half of the maximum value (Claerbout, 1976).

From the point of view of information theory, At measures the ability of the wavelet to distinguish or discretize successive informations in the time domain, Af measures the ability of the wavelet to distinguish or discretize successive informations in the frequency domain. Therefore we may introduce the following definitions: At = resolution in time domain, Af = resolution in frequency domain. These two quantities are related by

AtxAf=k

a relation similar to the “uncertainty principle” of quantum mechanics, the strict equality being valid for symmetric “zero phase’ ’ wavelets.

Computing k, we find (Morlet et al, 1982), k = (4 X In 2)/ 7~ = 0.8825 (that is close to 1).

In standard seismic processing, we can neglect the velocity dispersion, while trying to minimize At, and thus maximizing Af,’ that is, use a broadband spectrum. But when dispersion occurs, the wave velocity is frequency dependent. Thus we must process information in both the time and the frequency domains, using a 2-D sampling. For that reason, instantaneous frequency spectra are interesting functions.

We have to choose an adequate compromise between resolu- tions in the time and the frequency domains. Following the concepts of D. Gabor, we use a simple type of time-frequency decomposition to analyze the information (Gabor, 1951).

Instead of the Gabors’ decomposition, we use basic wavelets of constant shaw ratio: A t/T = constant. Such wavelets have a constant shape in the frequency domain: Af/f = constant, and they possess a remarkable property. The total energy carried by those wavelets is proportional to their mean frequency. This very simple result (Morlet et al, 1982) may be compared to the fundamental relationship W = h X f, giving the energy carried by a photon in the case of EM waves (where h = Planck’s constant).

FREQUENCY SPECTRA

FREOUENCVIH~.~

FIG. 3. Set of Gabor basic wavelets (logarithmic scale in the frequency domain).

Using a logarithmic scale in the frequency domain, the amplitude spectra of such wavelets are of constant shape as shown in Figure 3. In most cases, we use complex wavelets with A t/T = 2 which leads to a bandwidth close to l/2 octave (Figures 2 and 3). In spite of some theoretical difficulties with the Gabor expansion due to the nonorthogonality of the basic wavelets (Morlet et al, 1982), simulation shows that it is an interesting tool for signal analysis.

First model of propagation: The synthetic seismogram

As the propagation model, we use the Goupillaud-Kunetz algorithm (Baranov and Kunetz, 1960; Berryman et al, 1958) described in Figure 4. This is the classical algorithm for calculat- ing a synthetic seismogram from a set of constant traveltime layers defined by their impedances.

We first compute the reflection coefficients at all interfaces from the impedances of the adjacent layers. We then compute the impulse response step by step for increasing i, then j (Baranov and Kunetz, 1960).

For each interface k, we can output as an elementary trace a synthetic impulse response. We thus obtain a set of traces representing a synthetic vertical seismic profile (VSP). To avoid spurious reflections, we assume that the reflection coefficient at the surface is zero.

INTERI :ACE NUMBER)

U (i, j) = IJ (i, i,- 1) x [l + R (k)] + D (i - 1, i)x R (k)

U(i,j) = U(i,)-1) x [-R(k)] + D(i-l,j)x[l-R(k)]

R (k) = Reflection Coefficient at the kth interface

FIG. 4. I-D propagation diagram (Goupillaud-Kunetz algorithm).



I 1

FIG. 5. Synthetic VSP in a binary periodic medium (1) low- frequency wavelet, nonadapted embedding impedances.

To obtain the response to a given input signal, we can either introduce it in the algorithm or convolve the signal with the impulse response. The algorithm being linear, both methods give the same results. Even for very strong contrasts, with an impedance ratio Zr /Z, up to 1000, this algorithm is not affected by computing errors.

For the reflection coefficients, we use the classical convention R = (Z, - Z,)/(Z, + Z,). This sign convention is valid for the excess pressure wave P. With this convention, the impulse response for the reflection seismogram gives the excess pressure P as the sum of the upgoing (u) and downgoing (d) waves:

P=d+u.

The same equation gives the pressure for the transmission seismogram, and thus for the VSP, at any layer of the multilayered medium. For the particle velocity U, the corresponding equation would be:

U = (d - u) x l/Z,,

where Z, = impedance of the layer m. Therefore, it is easier to utilize pressures, particularly for the VSP, because the formula giving P is simpler.

Qualitative study usiug interpretation of synthetic traces

Fit we show the displays of VSPs when the signals are the cosine part of Gabor wavelets. To point out the phenomena clearly, and to make the presentation easier, we chose the following characteristics for the binary periodic medium:

Equal traveltimes in both media: T, = TV; Large number of layers: N = I28 elementary motifs, hence

256 layers; Strong impedance ratio: Z2/Z1 = IO.

Although this impedance contrast may appear very strong, it is not unrealistic and could exist in series including gas reservoirs or aerated sands or shales (as we will see later).

In the VSPs presented here, the propagating waves are sampled using one trace per layer, for layers of constant traveltime (in conformity with the computing algorithm).

In the following figures, the time scale corresponds to the vertical axis, and the depth to the horizontal axis. The horizontal scale is not exactly depth, but the k subscript, i.e., one way traveltime for the direct wave. It would represent depth only in the case of constant velocities for all the layers, which means the density contrasts determine the impedance contrasts. There- fore depth must be understood as a simplification for one way traveltime.

Regarding the impedances of the embedding media, two different cases are studied. (I) Upper medium, substratum and medium no. 2 are of same impedance Z, = Z, = Zz. (2) The impedance of the embedding media are adapted to the effective impedance of the composite periodic medium (this being strictly true only

FIG. 6. Synthetic VSP in a binary periodic medium (2) low- frequency wavelet, adapted embedding impedances.

FIG. 7. Synthetic VSP in a binary periodic medium (3) high- frequency wavelet, nonadapted embedding impedances.


208 Morlet et al

for low frequencies). For T, = 72, adapted impedances involve Z, = Z, = (Z, X Z2)“2 as we shall see in the analytical studies developed in Morlet et al (1982).

Regarding the wavelengths, we investigated the three different cases of low, high, and intermediate frequencies using varying values of the ratio period of the signal/traveltime in one “motif *’ = T/r, where T = mean period of the Gabor wavelet, and 7 = 7, + 72 = one way traveltime in an elementary motif of the binary periodic medium. This ratio is the fundamental factor discriminating the propagation mode in a periodic medium, as will be shown in the following developments.

Low frequencies.-The propagation of a low-frequency Gabor wavelet in the binary periodic medium defined earlier is shown in the VSPs presented in Figure 5 for nonadapted embedding impedances and in Figure 6 for adapted embedding impedances. In this example, we used a ratio T/T = 16.

Notice. the following points. (1) In spite of the strong impedance ratio from one layer to the next one (Z,/Z, = IO), which of course gives very large alternating reflection coefficients (with an absolute value higher than 0.8), the wavelet crosses the 256 successive interfaces and the periodic medium is fully transparent. This transparency is clearly shown in Figure 6, for the case of adapted impedances in the embedding media. For nonadapted impedances in the embedding media, the upper and lower interfaces would give, respectively, negative and positive reflections, but no reflection appears within the binary periodic medium (Figure 5). (2) The propagation of the direct wave in these VSPs involves constant slope in the figures for depth = one way traveltime and time = two way traveltime. This slope

FIG. 9. Synthetic VSP in a binary periodic medium (5) intermediate frequency wavelet, nonadapted embedding impedances.

is obvious in the upper and lower embedding homogeneous media. But in the intermediate periodic medium, the slope is higher; thus, the velocity of the actually traveling or effective wavelet is lower than that of the direct wave, which is vanishing or evanescent. (3) The incident wavelet crosses the periodic medium without any noticeable distortion in shape.

In summary, for wavelets of low frequency the periodic medium appears as a homogeneous medium, with a specific effective velocity and a specific effective impedance. The direct wave disappears and the effective wave is a delayed wave. In the time domain, this delayed wave is made up of constructive interferences of a large number of multiple reflections inside the multilayered medium, producing a coherent wavelet. In the frequency domain, it involves phase rotations for monochromatic frequencies, generating a wave packet when the phase is stationary (i.e., quasi-constant in a bounded area of the space-time domain).

We may notice here that the delayed wavelet or wave packet, carrying the energy of the wave, may be compared to a “quasi- particle.” Furthermore, there are some analogies between the structure of the Goupillaud-Kunetz algorithm and that of the Feynman diagrams, which are known to be useful tools in quantum

FE. 8. Synthetic VSP in a binary periodic medium (4) high- FIG. 10. Synthetic VSP in a binary periodic medium (6) inter- frequency wavelet, adapted embedding impedances. mediate frequency wavelet, adapted embedding impedances.



NE 5

T/T i32 -I=1 MS.

FIG. 11. Synthetic seismic section: Pinchout in a periodic medium.

field theory (Mattuck, 1976). This phenomenon may be compared to the propagation of a beam of light in a crystal lattice, which involves multiple scattering of the light beam in a 3-D lattice made up of a regular ionic distribution. This multiple scattering leads to a coherent propagating wave, implying a slower velocity of the light in the crystal and a velocity dispersion versus frequency.

In this way, we may explain the good stability of the phenomenon of propagation of the wave packet when we introduce anomalies in the thicknesses of some layers inside the periodic medium (this was verified by our simulation method using the Goupillaud- Kunetz algorithm).

High frequencies.-For T/T = 2, there is no transmitted signal because the multiple reflections destroy themselves by interference. There is a reflected signal built by constructive interference. This phenomenon persists for higher values of T/T. It appears in the VSPs presented in Figure 7 (nonadapted impedances) and in Figure 8 (adapted impedances). For T/T = 2.83, no energy penetrates the periodic medium, and we ob- serve total reflection of the incident wavelet. For the two above values of T/T, the effective reflection coefficient is therefore close to - 1. For T/T = 5.66, some part of the energy penetrates the periodic medium, with a strong velocity dispersion, but the main part of it is reflected.

Thus, briefly, for high frequencies (wavelengths close to twice the motif thickness), the periodic medium works as a super-

rejkror. These phenomena remain the same even for much

REAL WAVELET (COSINE)

IMAGINARY WAVELET

(SINE)

ENVELOPE WAVELET

(MODULUS1

0 time DOMAIN

0 frequency OOMAIN

MEAN FREQUENCY]

FIG. 12. Main parameters used for quantitative studies of wavelet propagation.

weaker impedance contrasts. This property of superreflectivity is well known and used in thin layer optics. In the 3-D extension, it will be related to bright spots.

Intermediate frequencies.-The VSPs presented in Figure 9 (nonadapted impedances) and Figure 10 (adapted impedances) show an intermediate case. We used the value T/T = 8.

The phenomenon of velocity dispersion versus frequency clearly appears in the periodic medium. Inside the output signal, propagating in the substratum, the frequencies are selected by their location in the space-time domain. The low frequencies stay in the head of the wave packet and the higher frequencies in the later events in the packet. The effective velocity is therefore frequency dependent.

Example of practical application.-Before giving the quantitative results, we present in Figure 11 a synthetic section of reflection seismograms showing how cyclic series could appear in a seismic section.

The model is made up of a binary periodic medium embedded within two homogeneous media, with a constant value of traveltime in each elementary structure or motif: T = 1 msec. The number of motifs increases from the left to the right in the section.

The composite medium is the same we described earlier, except that we replaced Z2/ZI = IO by Zt/Zz = 10, thus reversing the signs of the reflection coefficients (the top reflection is positive and the bottom reflection negative). In such a series, only top and bottom reflections are visible, even though there are strong


210 Morlet et al

Zj/Z2=10 T=2ms II/T?=1 N=20 ZIP2

1 10

0 10 20 30 TRANSlTTlME [ms I R

i

+l

- 0

-1

1 2 3 ------_ MISSION TIME/OIRECT time v)

z 5l= 4’ 3?! :z O=:

0 TRANSMISSION OELAV I msl 100

REFLECTION --

0” REFLECTION time[msl 100 0’ REELECTION timelmsl 100

FIG. 13. Wavelet propagation versus frequency: reflection and FIG. 14. Wavelet propagation versus frequency: reflection and transmission (standard wavelets). transmission (broadband wavelets).

impedance contrasts from one layer to the next. No reflections would appear after deconvolution on a real section even if frequencies up to 1000 Hz or more were recorded. Indeed, they would be reflected by the top layers of the series.

The third reflection, in the lower part of the section, corresponds to the first internal multiple between upper and lower interfaces of the periodic medium.

time delays for the bottom reflections are very large. For example, with 100 motifs, in spite of a two-way time of 200 msec for the direct wave (evanescent), the negative bottom effective reflection is picked at 360 msec. Such delays are much longer than those observed in standard seismic, but for cyclic series with layer thicknesses from one to a few meters, time differences of 10 percent are possible when comparing (1) measured times on sonic logs, using a spacing smaller than the layer thickness, hence not affected by time delays due to intrabed multiples, and (2) measured times on velocity survey data in a well or on seismic recorded data, affected by time delays due to intrabed multiples. In the latter case, sonic times are shorter than seismic times.

Opposite results may be observed, due to velocity dispersion.

for cyclic series involving very thin layering, with layer thicknesses close to the wavelength of the sonic log signal.

Z1/22=10 T=Pms T1/T2=1 N=2l_l 11/l?

0 TRANSMISSION DELAY [msl

TlMtImsl -20 0 20

Quantitative studies of synthetic traces

Fundamentals parameters.-We will now present a method to carry out a quantitative interpretation of seismic traces and sections in terms of time frequency, amplitude, energy, resolution (diameter and bandwidth), and phase.

Figure 12 shows how these parameters can be extracted from synthetic traces computed using Gabor wavelets as signals. For real seismic traces, such a method would require prior filtering of the traces by a zero-phase filter modeling the corresponding Gabor wavelet.

Simulation of a time-frequency decomposition.--In order to develop a method of automatic analysis of the above parameters on seismic traces, we simulate a time-frequency decomposition by taking as the basic wavelets a set of Gabor wavelets with constant shape ratio: At/T = constant; thus, Af/f = constant.

Taking a logarithmic scale in the frequency domain, we use a set of Gabor basic wavelets from which the mean frequency f = 1 /T is sampled with sampling rates of I /2 octave (Figures 13 and 14) and l/4 octave (Figures I5 and 16). In these four figures, only the cosine part of the complex wavelet is used, and two values of the shape ratio are presented: (I) A t/T = 2



Zr/Zz=lO T=2ms T,/Tz=l N=20

0 10 20 30 TRANSIT time tmsl II

111111111111111111111 i

s 3

TR,tJNSMlSSlON

z1/12 10

5

0

II +1

0

-1

0 TRANSMISSION OELAV~sl 100 __

TIMElms -2u 0 uo TIMElms -20 o 20

2 1

FIG. 15. Wavelet propagation versus frequency: transmission FTC. 16. Wavelet propagation versus frequency: transmission (standard wavelets). (broadband wavelets).

in Figures 13 and 15 (standard value in our studies), and (2) At/T = 1 in Figures 14 and 16 (for comparison).

For an automatic study of amplitudes, we picked times on the maxima of envelopes of the output responses (for reflection and transmission). To compute these envelopes, we need to use both sine and cosine parts of the complex wavelets. thus to compute complex synthetic seismograms. Figures 13 and 14 show the responses for transmission and reflection with various signals. The periodic medium is defined in the upper part of the figures as follows: (1) characteristic parameters, (2) impedance log, and (3) reflection coefficients, with locations of the source 5, the reflection pressure receiver R, and the transmission pressure receiver T.

The signals appear in the lower part of the figures as follows: no. 0 = Dirac pulse, nos. 1 to 5 = set of Gabor wavelets of decreasing periods. The corresponding traces for transmission and reflection responses are referenced with the same numbers.

The sign convention used for the reflection response leads to plotting a positive reflection as a trough on the seismogram.

Figures 13 and 14 illustrate the important remark in O’Doherty and Anstey (1971): “More up, less down,” but the energy distribution is strongly frequency-dependent in periodic series. Low frequencies are transmitted and high frequencies are reflected. Furthermore, intermediate frequencies (signal no. 3) are dispersed.

Zl/Z2=10 T=Sms T1/T2=1 N=20 21/12

IJ

0 10 20 30 TRANSIT time tmsl _ n

-4 +l

0

-I -1

0 TRANSMISSION OELAY [ msl 100

SIGNALS

For the same periodic medium, Figures 15 and 16 show the transmission responses with a sampling rate of l/4 octave for the mean frequency of the wavelets. For low frequencies (signal no. l), no shape distortion appears for the signal. However, the wavelet is strongly delayed.

We can thus define two different times: (1) the theoretical timeof the direct wave (without intrabed multiples), and (2) the effective time picked at the maximum of the envelope of the effective transmitted wavelet.

For a one-way transmission time of 40 msec, the transmission delay is 30 msec. Thus the ratio “effective transmission time/ direct wave time” equals 1.75. Therefore, as we can define an effective velocity of the periodic medium (cf., Figure 6) the ratio effective velocity/mean velocity equals 0.57.

Also evident from Figures 15 and 16, high frequencies do not cross the medium (signal nos. 8 and 9), intermediate frequencies are dispersed (signal nos. 5 to 7), and transmission delay increases with frequency (signal nos. 1 to 5).

Figures 13-16 illustrate a practical example of the utility of the time-frequency Gabor expansion applied to seismic traces. Using this decomposition, the concept of the “instantaneous frequency spectrum” is introduced as a powerful extension of the “instantaneous frequency and phase” concept used in the analytical signal (Ville, 1948) and complex trace (Taner et al, 1977).


212 Morlet et al

21/22 = 10 T= 2 ms TI/% = 1 N 46

OUTPUT AMPLITUDE

1 5 10 50 100 FREQUENCY (Hz)

FIG. 17. Filtering effect of a binary periodic medium for wave propagation.

When signal dispersion is prominent (signal nos. 5 and 6), picking effective time becomes impossible and the wavelet loses its identity. On real seismograms, we could interpret such a phenomenon as the disappearance of some frequencies of the signal, probably by absorption, and an increase of the noise level.

From a theoretical point of view, the uncertainty principle becomes At X Af + I, because of the increase of At by dispersion, implying time delays and strong phase rotations.

Using information theory, we can say that the information carried by the signal being similar to “neg-entropy,” velocity dispersion increases the entropy of the signal (Brillouin, 1956). In practical applications, depending upon what is known and unknown, dispersed signals may be deconvolved (as in Vibroseis processing) to zero-phase wavelets, or they may be considered as noise.

For assumed known parameters, functions, or models, one can use deterministic processing. But for unknown parameters, functions, or models, one must use statistical processing, assuming random distributions for the unknowns.

Morlet et al (1982) develops the theory behind the particular Gabor expansion we use here.

Quantitative study of the filtering effects of a periodic medium.-Figure 17 shows an example of the amplitude responses of a periodic medium for reflection and transmission of Gabor wavelets, obtained by automatic picking of the output seismograms. The composite medium is the same as earlier, but of smaller total thickness: N = 16. Logarithmic scale is used for frequencies.

Notice the following features. (1) For adapted impedances of the embedding media, the periodic medium works as a selective filter reflecting high frequencies and transmitting low fre-

quencies. and the cut-off frequency is close to IO0 Hz in this particular case, in agreement with Figures l3- 16. (2) For nonadapted impedances of the embedding media, the filter is more complex due to the interference of multiple reflections at the lower and upper interfaces of the periodic medium. The filter acts as a thick layer made up of a homogeneous medium charac- terized by its effective impedance and effective velocity.

Using the effective velocity, equal to 0.57 times the velocity of the direct wave (as seen above earlier), it is easy to explain the maximum on the transmission curve for a frequency close to 9 Hz, by constructive interference between the effective wave and its multiple reflections at the lower and upper interfaces of the periodic medium. This reinforcement is maximum when the total thickness of the medium equals one-half wavelength of the signal. For similar reasons, the transmission curve has a minimum for a frequency close to 4.5 Hz when the total thickness of the medium equals one-quarter wavelength of the signal (from de- structive interferences between the effective wave and its multiple reflections). (3) The principle-“More up, less down” -expresses the obvious complementary nature of the amplitude curves, in conformity with energy conservation (energy being proportional to squared amplitudes).

Effective velocity of a periodic medium for low frequencies. -As shown above, we can define (Figure 6) and measure (Figure 15) effective velocity of a periodic medium for low-frequency wavelets.

Figure 18 displays a chart of effective “slowness” (the inverse of velocity). To minimize the number of parameters. we plotted the ratio effective slowness/direct wave slowness which equals the ratio of effective time/direct time Logarithmic scales are used for both coordinates.


Complex Signal and Scattering-Part I

EFFECTIVE TIME/DIRECT time z,/zz q 2 T,/T* q 1

213

Tl’tz2.83

- 1 2 3 45 10 20 30 40 50 100 200

RATIO 21 /ZZ

FIG. 18. Effective slowness of low-frequency wavelets in binary periodic media versus impedance contrast.

A large range of values for the ratio Zr /Z, will be useful later in gas reservoirs or aerated media. Each curve corresponds to a constant value of the ratio T, /TV, which is related to the thickness ratio and thus to the volumic ratio for the two components of the binary medium. This chart was obtained by automatic picking of effective times for low-frequency wavelets.

Notice the symmetry when 7, and r2 permute. This operation changes only the sign of the reflection coefficients, thus not affecting the transmitted signal. But the slowness is maximum forrr = r2.

Attenuation for transmission of high frequencies.--Using automatic picking, we can study the maximum amplitude of the transmitted waves. For high frequencies (ratio T/T close to 2) which are strongly reflected, this amplitude decreases rapidly in the first few layers of the periodic medium. In Figure 19, the amplitude of the transmitted wave is plotted versus the number of motifs.

The impedance ratio Z1/Zz equals 2, representing an unusual but realistic value. The phenomenon shown in Figure 19 can be interpreted as an exponential attenuation through the binary medium. This type of attenuation involves only selective reflectivity or a varying effective velocity versus frequency, re- sulting in an increase of the entropy of the signal without energy losses.

All the phenomena described earlier may be referred to as I-D scattering. and like the anology between the effective wave packet and the quasi-particle, and the Bloch waves, it provides the fundamental basis for further extension to quasi-cyclic series and to 3-D structured media.

Laboratory experiment

We checked the validity of the above theoretical studies with the following laboratory experiment, by M. Lefebvre, in the G.E.S.S.Y. in Toulon. A periodic medium was made of alternating thin plates of steel (thickness, 3 mm; velocity, 6300 m/set; density, 7.8), and synthetic resin (thickness, 1 mm;

N=._:D 40 80 120

Ra. 19. Exponential attenuation of a high-frequency wavelet in a periodic medium.

velocity, 2400 m/set; density, I .2), involving an impedance ratio (Z, /Zz) equal to 16. The mean frequency of the ultrasonic short pulse used was 70 kHz.

We measured the effective velocity by picking effective times for three different values of N(N = IO, 20, 30), thus three different thicknesses. The measured value of the ratio effective time/direct time was found equal to I .8, in agreement with the expected value on the theoretical corresponding model (i.e., 2.1), taking into account the accuracy of the experiment (20 percent).

Analytical studies of propagation in periodic media

Numerous authors developed such studies in the frequency domain for various applications in physics (Abeles, 1946; Born and Wolf, 1959; Brekhovskikh, 1960; Brillouin. 1946; Dieulesaint and Royer, 1974; Elachi, 1976; d’Erceville and Kunetz, 1963: Rytov, 1956). They agree perfectly with our simulations in the time domain as we will show. Morlet et al (1982) presents a synthetic approach developed by Prof. G. Bonnet and his assistants E. de Bazelaire and .1. F. Cavassilas, which led to a thesis by J. P. Dolia, who worked with us on this subject in the G.E.S.S.Y. at Toulon University (Bonnet, 1980; Dolla, 1980). We will use some of the results of these analytical studies in our next investigations.

EXTRAPOLATION TO (#JAM-CYCLIC SERIES

Qualitative studies

We noted earlier the stability of the transmission response for the low-frequency approximation, i.e., from the zero frequency to the cut-off frequency, in the low-frequency passband. From a geologic point of view, in sedimentary basins, especially when they contain fluid or solid hydrocarbons, cyclic and quasi-cyclic series are very common. This important phenomenon has been noticed by numerous authors, particularly O’Doherty and Anstey (1971).


Morlet et al 214

0

100

50

10

5

1

PICKED ON AMPLITUDES OF TRANSMITTEO WAVELETS

-I-

1 5 10 Zl/ZP 50

FIG. 20. Q factor for attenuation of high-frequency wavelets in binary periodic media (versus impedance contrasts).

The structural characteristics of natural objects remain somewhat the same for various observation scales (Mandelbrolt, 1975). Simple physical explanations of the causes of such structural periodicities have become common knowledge (Nicolis and Prigogine, 1978; Haken, 1978).

As geophysicists, we may no longer define the sedimentary series as a random distribution of interfaces or of single layers, but as a random distribution of quasi-cyclic series. With no loss of generality, we can resolve a sedimentary series into a set of basic elements, each element being a binary medium, as follows. The minimum number of layers per element is then 2. In quasi- cyclic series, lower and higher impedances alternate. In progressive series, this is no longer true, but it is nevertheless possible to use the results of the analytical studies giving the transfer function for a double layer, in a progressive embedding media environment, and thus to replace the binary medium by a single equivalent layer. In this last case, the filtering effect for transmission is small enough to make the low-frequency approximation (i.e., the formulas giving effective velocity and effective impedance) valid for all frequencies.

Practically, we can always replace, with a good approximation for transmission studies, any sedimentary layering by a sequence of alternate lower-higher impedance layers.

Quantitative studies

We can therefore conduct statistical studies of wave propagation in sedimentary series as follows. The randomized characteristic parameters for such series will be, for each binary medium (taken as basic element), impedance ratio, volumic ratio, motif thickness, effective velocity (or mean velocity), and total thickness (or number of motifs).

The range of variations for the above parameters must be defined. Velocity range could be delined as depending upon depth.

As an example, we studied the expectable attenuation law for such models of multilayered media. We first studied the following deterministic case: attenuation for transmitted Gabor wavelets (with a bandwidth of 112 octave) in a periodic medium for which T = 2 x 7. This corresponds to the maximum attenuation in the “forbidden band” (Morlet et al, 1982).

Figure 20 shows the Q factor (i.e., the relative energy loss per wavelength) versus the ratio Z, /Z,, obtained by automatic picking of the envelopes maxima, thus representing the attenuation due to forbidden bands. Then, to build a statistical model, we can assume that the impedance ratio, the volumic ratio, and the effective velocity are constant, only the thicknesses being randomized. Therefore, for classes of equal motif thickness sampled at constant rate in a logarithmic scale, we assume the same probability for one motif to be in a given class of thickness (flat distribution, in a logarithmic scale of thicknesses). The range of values for T/T being the same in each class of thickness, the value of Q is the same for any class of motif thickness, i.e., Q is constant versus frequenc,y. This involves a constant Q type of law for attenuation.

To give an estimation of the expected range of values for Q in such a series, we can use the range of values found for Q in Figure 20, for pure periodic media. For the small, but very common, values of the impedance ratios (leading to reflection coefficient range from 0.01 to 0.02), the corresponding range of values for Q varies from 20 to 100.

Assuming a distribution of motif thicknesses in a small number of classes (for example 2 to 3 per octave for the Gabor wavelet used here), we obtain practically decoupled forbidden bands from one class to the others. Taking into account that the values of the impedance ratios decrease with the sampling rate in depth, we can limit the number of octaves to a realistic value (close to 4) to fit with the seismic signals. We define therefore 8 to 12 classes of motif thicknesses. This leads to ranges for Q from 160 to 1200.

Hence, we can conclude that, in addition to the absorption of seismic waves in fluid-filled porous rocks, extensively studied by A. Nur and others (Kjartansson, 1978; Mavko and Nur, 1979; Nur and Simmons, 1969; Winkler and Nur, 1978) in the “Rock Physics Project” at Stanford University, wave attenuation could be primarily due to sedimentary layering and intrabed multiples.

EXTRAPOLATION TO 3-D ROCK MODELS

It is natural to extend the results to 3-D structured media. For the theoretical developments, we may refer to solid state physics and wave propagation in crystal lattices. Of particular interest for 3-D wave propagation and scattering, we mention the following works: Balian and Bloch (1970, 1971, 1972), Balian and Duplantier (1977, 1978).

Qualitative studies

The above results represent a first approximation when study- ing wave propagation in 3-D binary periodic media, An intermediate step is to study 2-D propagation of plane waves in periodic fibered media. Such a type of 2-D model is uncommon in sedimentary series. More common are the 3-D models made of granular materials, for example sands or porous rocks (especially in reservoirs).

In such a case, the periodic or quasi-periodic structure is much



1st COMPONENT : Mi

i 1 2 3

2nd COMPONENT : AIR

{

V2 = 330 mis

P2 = 0.0013

EFFECTIVE VELOCITY (M/S)

10000 I I I .B

6000. I

4000.

2000. _

1000.

600.

400. ~~

FIG. 21. Effective velocity versus volumic ratio in binary media FIG. 22. Effective impedance versus volumic ratio in binary media for strong impedance contrasts. for strong impedance contrasts.

finer than those presented earlier, compared to the wavelengths used in seismic reflection. Hence, we are in the case of the low- frequency approximation. However, wavelengths used in sonic logging are closer to the size of grains; therefore dispersion phenomena could occur (also known as scattering, backscattering, or diffusion).

Quantitative studies

The problem is much more complex for 3-D media, because of wave conversions on a granular scale from f-waves to S-waves, and conversely. We will assume that such conversions are negligible, and therefore that the composite medium works as a fluid. This could be the case, for example, in unconsolidated sediments, without or with limited friction. The problem is to define effective velocity and effective impedance for such rocks.

For large wavelengths, we can assume that the characteristics of the composite medium can be averaged in a volume small compared to the wavelength, without changing the results on wave propagation. This hypothesis is based on the notion of multiple interactions or energy exchanges between the wave packet and the elementary grains of the rock. involving a “many body problem” (Mattuck, 1976).

We have to choose the characteristic parameters to be averaged. If one uses the compressibility (i.e., 1 /E) and the density, the following formula for the “effective velocity” in aerated sands is (Lester, 1932; Domenico. 1974)

1st COMPONENT : Mi

2nd COMPONENT : AIR (V2=330m/s

EFFECTIVE IMPEDANCE

looool

0.4 1 VOLUMIC RATIO ~___~~_ 0.2 % AIR

, I I I

0 20 40 60 80 100

V, = E/p,

where E is defined by

(XI + X*)/E = X,/E, + XJEz,

and p by

(X, + X,) x p = XI x PI + x2 x p2,

XI and X2 being the volume ratios for air and rock.

Replacing volume ratios by thicknesses in the above formulas. and using the relations between the parameters E, p, Z, V, and T. we finally obtain the formula found for binary layered media (Morlet et al, 1982). Therefore, the two different approaches to this problem lead to the same results.

Note that, in aerated sands and gas reservoirs, very low effective velocities and very low effective impedances are observable.

Figures 21 and 22 show the strong effect, similar to the phenomenon of “doping,” of a very small volumic proportion of gas on both velocity and impedance of the doped rock. This effect is quite independent of the characteristics of the second medium (grains of rocks or liquids). Therefore, we can assign to cyclic or quasi-cyclic series including gas reservoirs or aerated sands a very strong impedance ratio, and the value Z, /Z2 = 10 would not be unrealistic in such series, whose effective velocity could bc very low.


216 Morlet et al

zJ/&= 2 T=4ms T1/T2=1 N=20 z1/12

1

2

1

IO 0 TRIINSIT TIME Imsl 50

I R

IIIIlIIIIIIIIIIIll1l 11111111111111111111

S 1 TRANSMISSION TIME/DIRECTVlME 2

II z

5s

'J= 22 IZ 0~ =

0 v) 3

5E

k

2z AZ

0 REFLECTION TlME[msl 100 r \ I v)

;; SIGNALS E E 5.0- -5g g 7.1 - -42 E lO.O-

c

= 14.1. -z 2 20.0- -1Y

TIME[msl -20 V o V 0

20

z1/22= 2 T=4ms T,/Tz=l N=20 21/12

I 0 50 TRANSIT llME (msl R 0 J+lR

III 1111111111111111 llllllll11lllllllllI 0

S i-1

1 l.5 2

5 4 3

: 0

0 TRANSMISSION OELAVlmsl 100 I ,”

5.0 5.9 7.1 1

A SIGNALS

FIG. 23. Approximation of a real cyclic series by a binary periodic medium: reflection and transmission.

FIG. 24. Approximation of a real cyclic series by a binary periodic medium: transmission.

APPLICATIONS TO SEISMIC REFLECTION

A case history

We present here a real case of propagation in a quasi-cyclic series of large thickness with strong impedance ratios, to show that the phenomena we investigated are really observable.

Figure 23 shows a first approximation of the real quasi-cyclic series by a binary periodic medium. limited here to a total thickness corresponding to 20 motifs, and the propagation of Gabor wavelets in this medium (with a frequency sampling of l/2 octave).

Figure 24 shows the transmission in the same medium, with a finer sampling for the mean frequency of the wavelets (l/4 octave).

Figure 25 shows the frequency response and the cut-off frequency at 100 Hz and Figure 26 indicates the effective velocity and the velocity dispersion, using the mean velocity as reference [see Morlet et al (1982) for more details].

We notice a relative time delay of 6 percent for low frequencies and an additional relative time delay of 6 percent for frequencies close to 60 Hz.

Impedances and reflection coefficients of a real quasi-cyclic series Si appear as logs versus transit time (i.e.. one way traveltime) in Figure 27. The sampling rate is 1 msec, and the total thickness of the series is close to 1000 m.

Figure 28 shows the same logs for the sedimentary substratum Sz of the series Si Compared to S, , the series S2 is noncyclic.

z7/i!2=2 -f,‘blS -6 / r2 =1 N=20

AMPLITUDE 1

3.5

1 = 0 ii% ,

7 I , L

FREQUENC'r IHtl

200 300 400 500

FOG. 25. Approximation of a real cyclic series by a binary periodic medium: filtering effect in the frequency domain.



z1/22=2 T= 4ms T1/T2= 1

PHASEVELOCITY ----

GROUPVELOCITY -

WAVELET VELOCITY -------I

EFFECTIVE VELOCITY / MEAN I

VELOCITY

PERIOD (ms) ii 9

FIG. 26. Approximation of a real cyclic series by a binary periodic medium: velocity dispersion.

IMPEDANCE 4

TRAN’SIT TIME’ TRANSIT time2000 I 1 I * 2000-’ I I I I I I 1 8)

500 1000MS. 1000 1500 MS.

TRANSIT time-1 , w

500 1000MS.

FIG. 27. Impedance and reflection logs of a cyclic series (well data).

FIG. 28. Impedance and reflection logs of a noncyclic series (well data).

Propagation in series S, of Gabor wavelets (with a frequency sampling of l/4 octave) is shown in Figure 29 for both transmission and reflection. The first trace on the left represents the impulse response to a Dirac pulse. The cut-off frequency for transmission appears close to 100 Hz.

Figure 30 shows the transfer functions for transmission through series S, , S2, and S1 + S2, with superimposed smoothed responses obtained by a “method of medians.” The noncyclic series S2 works as an all-pass filter. On the contrary, the quasi- cyclic series S, works as a strong filter for high frequencies. Two cut-off frequencies appear, for different amplitude levels: the first at 50 Hz, and the second at 100 Hz.

Figure 31 shows the strong filtering effect of high frequencies due to intrabed multiples. The reflection seismograms were computed with and without multiples for a set of basic Gabor wavelets. For high frequencies, numerous reflections appear in the seismograms without multiples, but they disappear in the seismograms with multiples. This can be explained by the phenomenon of superreflectivity: the high-frequency energy is reflected at the top of the quasi-cyclic series S, .

To obtain approximate quantitative results for interpretation, we can estimate the transmission delays of effective wavelets versus total traveltime, taking as reference the traveltime of the evanescent direct wavelet.

In a first, rough approximation, we will assume that, for each mean frequency of a wavelet, we can apply the formulas valid for low frequencies. Using a generalized formula (Bonnet, 1980),

IMPEDANCE t I

R

1

TRANSIT time-1 - 1 I I w

1000 1500 MS.


Morlet et al

PERIODS (MS.1 0 5 10 20 40 80 160 320

FIG. 29. Propagation of Gabor wavelets in a cyclic series: reflection and transmission.

0 50 100 150 200 250

C--L -I u 50 i-00 i-5 0 200 150

FREQUENCY (Hr.)

T s2

FIG. 30. Filtering effect for transmission in a cyclic series.

we can write, for each value of the total traveltime (Morlet et al, 1982):

/, m \

c z, c 1 /z, ‘I2 7,/T,j = ( i-l 1=I

- x ~ 1 - \m m /

where m = number of layers of constant transit time taken in the running integration window, T, = effective transmission time in the window, rd = direct transmission time in the window, and Z, = impedance of the ith layer.

Then for each sampled time interval (for example, I msec) we have Ar/rd = (T, - T~)/T~.

Finally we can obtain the total transmission delay by integration versus total traveltime.

Figure 32 shows the transmission delays estimated by this rough method for different values of the window m, thus of frequency. Due to the formula used, the larger the integration window, the longer the time delay. We must note, however, that (1) For narrow windows (high frequencies), the low-frequency approximation is not necessarily valid; (2) for very large integration windows (very low frequencies), the signal may be shorter than the window. Then the formula is no longer valid.

However, for intermediate value\ of the window, the formula gives a good estimate of the time delays. In such a case, we can estimate the transmission delay, restricted to the series S, , as close to 15 msec, in conformity with the delays picked on the envelopes of Gabor wavelets in Figure 29. This involves an effective velocity 3.5 percent lower than the mean velocity for low frequencies.


5

Complex Signal and Scattering-Part I

10 20 40 80

PERIODS (MS.)

160 320

219

FIG. 3 1. High-frequency attenuation due to intrabed multiples in cyclic series

On the other hand, the effective traveltime measured from a velocity survey in the well was 15 msec longer than the traveltime given by the sonic log in the series S, , leading to an effective velocity 7 percent lower. Such a value tallies with the results of the theoretical binary periodic model, if we assume that the sonic log was able, in this particular case, to measure the real velocity in each elementary layer of series St.

TWO WAY DELAY (MS.) ?0 10

FIG. 32. Rough estimation of propagation delays due to intrabed multiples in a cyclic series (versus frequency).

The differences between the values of effective velocities obtained from synthetic seismograms (-3.5 percent) and from well shooting (-7 percent) could be explained by various corrections (for example, on instantaneous velocities of the sonic log) to reach a better fit between the sonic log and the velocity survey in the well.

Similarly, and also in a first approximation, we can estimate the distributed reflectivities versus the frequency, assuming that we can compute the effective impedance using the generalized formula for low frequencies

Z,= $Z,X5 l/Zi ( > 112

i=l i-l

Defining the averaged impedance by

,,I

Z, = C ZilW

we can express the distributed reflectivity for different values of the running integration window m (for different frequencies) as follows:

Rd = (z, - =&)/(z, + z,).

Figure 33 represents such a distributed reflectivity for different values of the running integration window m. High frequencies (on the left) lead to higher distributed reflectivities.

These studies explain why, in a large area surrounding this well, high frequencies disappear under the series St and only very low-frequency reflections are visible in the deeper part of the sections.

Similar studies (Schoenberger and Levin, 1974, 1978) have been presented on this subject in the past.

Finally, we can assume that. when attenuation appears in sedimentary quasi-cyclic series, velocity dispersion should be expected. This remark holds for the weathered shallow layers; a small number of alternate aerated and compact layers, thus


Mot-let et al

INTEGRATION WINDOW

a 16 32 64 128 256 P--I__ L- l-.1 _.__L~_I. TWO WAY (MS.)

FIG. 33. Rough estimation of distributed reflectivity due to intrabed multiples in a cyclic series (versus frequency).

having strong impedance contrasts, involve both high-frequency attenuation and velocity dispersion.

Gas reservoirs in quasi-cyclic series

We tried to obtain quantitative comparisons between models and real data in quasi-cyclic series including gas reservoirs. Un- fortunately, we could only give qualitative estimations of what would be the phenomena of superreflectivity. which may increase the amplitudes of the bright spots.

The reason it was impossible to make quantitative comparison is that, in the areas of interest for detecting such phenomena. no valid sonic logs were recorded because of the very low velocities in the gas reservoirs.

Shear waves in quasi-cyclic series

In quasi-cyclic series made up of alternate hard rocks (lime- stone or sandstone) and thinly layered rocks (shales or clays), we may expect strong impedance contrasts for shear waves. There- fore, effective velocities may be much lower than mean velocities, and, if it is not the case for compressional waves, low values of the ratio V,5/V,, (shear to compressional velocities) should be expected. Such values were observed for VT/V,, = 0.43 in alternate series made up of limestones and shales, compared to V,/V,, = 0.62 in reference series.

For compressional waves, such phenomena occur in weathered shallow layers, especially in aerated shales, involving strong attenuation of high frequencies and velocity dispersion. Due to the lower velocities of shear waves, the ratio wavelength/layer thickness is lower than for compressional waves. Therefore, the phenomena of high-frequency attenuation and velocity dispersion must be more prevalent.

High-resolution seismic

In high-resolution seismic, the frequency band is larger than in standard exploration and may reach four or even five octaves. This method will be much more sensitive to the phenomena of attenuation and velocity dispersion. Such phenomena must be

taken into account. in data processing and even in data recording, to obtain high resolution on the layering. Our studies and the theoretical developments given in Morlet et al (1982) could become an important tool for oil field development using high- resolution seismic.

We showed by simulation on synthetic and real data that the propagation of seismic waves in sedimentary series is frequency- dependent, and involves the following phenomena:

Transparency for low frequencies; Superreflectivity for high frequencies; Velocity dispersion for intermediate frequencies.

These phenomena do exist in standard seismic prospecting, although they are not always easily noticeable. They help to explain the experimental constant Q law for wave attenuation, which is valid when taken from a statistical standpoint.

When high resolution is needed, these phenomena are no longer negligible, but they can still be handled.

These phenomena could be more important for shear waves in quasi-cyclic series including shales. Regarding the recording and processing methods, we showed that, to increase the resolution, for frequency dependent velocity and attenuation, we must work in both the time and the frequency domains.

Gabor wavelets of constant shape ratio (cf., Figure 3) can be taken as the basic wavelets for the sampling of the seismic traces in a special Gabor expansion in the time-frequency domain (Morlet et al, 1982). This expansion gives directly the instantaneous frequency spectra of the \cismic traces.

We hope that this technique will contribute to enhance the re-

solving power of the seismic reflection method.

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Morlet, J., Arens, G.. Fourgeau, E.. and Giard, D., 1982. Wave propagation and sampling theory-Part II: Sampling theory and complex waves: Geophysics, v. 47, this issue, p. 222-236.

Nicolis, G., and Prigogine, I., 1978, Self-organisation in nonequilibrium systems: Wiley Interscience.

Nur, A., and Simmons, G., 1969, Stress induced velocity anisotropy in rock- An experimental study: J. Geophys. Res., v. 74, p. 6667-6674.

O’Doherty, R. F., and Anstey, N. A., 1971, refections on amplitudes: Geophys. Prosp., v. 19, p. 430-458.

Rytov, S. M., 1956, Acoustical properties of thinly laminated medium: Soviet Phys. Acoust., v. 2, p, 68-80.

Schoenberger, M., and Levin, F. K., 1974, Apparent attenuation due to intrabed multiples: Geophysics, v. 39, p. 278-291.

__ 1978, Apparent attenuation due to intrabed multiples, II: Geo- physics, v. 43, p. 730-737.

Taner, M. T., Koehler, F.. Reilly, M. D., and Cobum, K. C.. 1977, Continuous high resolution velocity analysis: Presented at the 47th Annual International SEG Meeting, September 21. in Calgary.

Ville, J., 1948, Theorie et applications de la notion de signal analytique: Rev. Cables et Transmissions, v. 1, p. 61-74.

Winkler, K., and Nur, A., 1978, Attenuation and velocity in dry and water saturated Massilon sandstone: Presented at the 48th Annual International SEG Meeting, November 1, in San Francisco.

REFERENCES FOR GENERAL READING Treitel, S., and Robinson, E. A., 1966, Seismic wave propagation in

layered media in terms of communication theory: Geophysics, v. 31, D. 17-32.

Backus, M. M., 1959, Water reverberations-Their nature and elimi- Widess, M. B., 1973, How thin is a thin bed’?: Geophysics, v. 38, nation: Geophysics, v. 24, p. 2333261. p. 1176-l 180.

Bamberger, A., Chavent, G., and Lailly, P., 1977, Une application de la theorie du controle a un probleme inverse de sismique: Annal Geophys.. Table 33. fast. l/2, p. 183-200.

Bois, P.. Chauveau. J.. Grau, G.. and Lavergne. M.. 1960, Seiamo- grammes synthetiquea-Possibilities techniques de rtalisation et limita- tions: Geophya. Prosp.. v. 8, p. 260-314.

Bois, P., and Hemon, Ch., 1964. Condition que doit remplir une fonction pour rep&enter un sismogramme impulsionnel a reflexions multiples, quel que soit le coefficient de reflexion en surface: Annal. Geuphys.. Table 20. fast. 4. p, 509-51 I.

Bois. P., Hemon, Ch., and Marechal, N., 1965, Influence de la largeur du pas d’echantillonnage du carottage continu de vitesses sur les sismo- grammes synthetiques a multiples: Geophys. Prosp.. v. 13, p. 666104.

Bonnet. G., 1968, Considerations sur la representation et l’analyse harmonique des signaux deterministes ou aleatoires: Annal. Telecomm.. v. 23, p. 62-86.

Bouachache, B., and Flandrin. P., 1978, Representation con)ointe en temps et frequence des signaux d’energie Iinie: I.C.P.I.. 31 Place Beliecour. Lyon.

Bouix, M., and Palewa, E., 1969, Methode theorique de determination de I’epaisseur et des caracteristiques electriques de couches geologiques paralleles: Revue du CETHEDEC, v. 17, p. I-13.

Feynman, R. P., Leighton, R. B., and Sands, M., 1965, Lectures on Physics, v. II, sec. 27-3: New York, Addison Wesley.

Goupillaud, P., 1961, An approach to inverse filtering of near surface layer effects from seismic records: Geophysics, v. 26, p. 754-760.

Grant, F. S., and West. G. F.. 1965, Interpretation theory in applied geophysics: New York, McGraw-Hill.

Ishimaru, A., 1978, Wave propagation and scattering in random media, v. I and II: New York, Academic Press.

Karal, F. C., and Keller, J. B., 1959, Elastic wave propagation in homogeneous and inhomogeneous media: J. Acoust. Sot. Am., v. 31, p. 694-705.

Korvin, G., 1976, Certain problems of seismic and ultrasonic wave propagation in a medium with inhomogeneities of random distribution -Wave attenuation and scattering on random inhomogeneities: Trans. E.L.G.I., p. 3-37.

Morlet, J., and Schwaetzer, T., 1961, Mesures d’amplitude des ondes ultrasoniques dans les sondages: Geophys. Prosp., v. 9, p. 544-567.

~- 1962, Mesures d’amplitude dans les sondages-Le log d’attenuation: Geophys. Prosp , v. IO, p. 539-547.

Neidell, N. S., and Taner, M. T., 1971, Semblance and other coherency measures for multichannel data: Geophysics, v. 36, p. 482-497.

Officer, Ch. B., 1958, Introduction to the theory of sound transmission: New York, McGraw-Hill.

Phona, T. .I., and Tsang, L., 1978, Determination of the average micro- scopic dimension in granular media using ultrasonic pulses-Theory and experiment: Presented at the 48th Annual International SEG Meet- ing, November 1, in San Francisco.

Ricker, N., 1953, The form and laws of propagation of seismic wavelets: Geophysics, v. 18, p. 10-40.

-._ 1953, Wavelet contraction, wavelet expansion, and the control of seismic resolution: Geophysics, v. 18, p. 769-792.

Szaraniec, E., 1976, Fundamental functions for horizontally stratified earth: Geophys. Prosp.. v. 24, p. 5288548.

Tolstoy, I., and Usdin. E., 1953, Dispersive properties of stratified elastic and liquid media-A ray theory: Geophysics, v. 18, p. 844 869.


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