Wave propagation and sampling theory--Part II: Sampling ...€¦ · Wave propagation and sampling...

GEOPHYSICS, VOL. 41, NO. 2 (FEBRUARY 1982); P. 222-236, 15 FIGS

Wave propagation and sampling theory-Part II: Sampling theory and complex waves

J. Mot-let*, G. Arensz, E. Fourgeau*, and D. Giard$

ABSTRACT

Morlet et al (1982, this issue) showed the advantages of using complex values for both waves and characteristics of the media. We simulated the theoretical tools we present here, using the Goupillaud-Kunetz algorithm.

Now we present sampling methods for complex signals or traces corresponding to received waves, and sampling methods for complex characterization of multilayered or heterogeneous media.

Regarding the complex signals, we present a two- dimecsional(2-D) method of sampling in the time-frequency domain using a special or “extended” Gabor expansion on a set of basic wavelets adapted to phase preservation. Such a 2-D expansion permits us to handle in a proper manner instantaneous frequency spectra. We show the differences between “wavelet resolution” and “sampling grid resolution.” We also show the importance of phase preservation in high-resolution seismic.

Regarding the media, we show how analytical studies of wave propagation in periodic structured layers could help when trying to characterize the physical properties of the layers and their large scale granularity as a result of complex deconvolution. Analytical studies of wave propagation in periodic structures are well known in solid state physics, and lead to the so-called “Bloch waves.”

The introduction of complex waves leads to replacing the classical wave equation by a Schriidinger equation.

Finally, we show that complex wave equations, Gabor expansion, and Bloch waves are three different ways of ‘introducing the tools of quantum mechanics in high- resolution seismic (Gabor, 1946; Kittel, 1976, Morlet, 1975). And conversely, the Goupillaud-Kunetz algorithm and an extended Gabor expansion may be of some use in solid state physics.

GABOR EXPANSION AND SAMPLING THEORY OF COMPLEX WAVES

We develop the following studies to obtain better methods of quantification of quasi-periodic signals received on punctual re- ceivers. This applies to any type of signals carried by any type

of waves. Furthermore, we quantify the information received by an analogic sensor or transmitter with maximum fidelity.

The received signal is a time function, and the standard methods of A/D conversion involve (1) a “sample and hold” operation at constant time intervals; (2) a “measurement” operation of the analog amplitudes of the registered samples by comparison with unit voltages; and (3) a digital recording of the result of the measurements.

These methods are perfectly valid, if and only if the following environment is present. To increase the fidelity, we must increase simultaneously the sampling rate, i.e., the number of samples, and the accuracy, i.e., the number of bits per sample. This leads to a serious economic problem, even in the new tech- nological environment, since it implies: high transmission rate for the digitized information (bitsisec), very large numbers of recorded information to be recorded and processed (samples and bits).

On the contrary, the alternate sampling method we present here possesses the following advantages. The elementary samples give a direct measure of the elementary received energy (by its square root), thus “quantifying the information” arriving to the sensor. The high frequencies, which carry high-resolution information, are sampled at higher sampling rate than the lower ones that are very poor in resolution. The traces are digitized as a complex function of both time and frequency. Therefore, in such a method of quantification, the phase is recorded with an accuracy which is frequency independent. This condition is needed to carry out in a proper way any processing method based on signal enhancement by interferences.

Complex wave function

D. Gabor (1946, 195 1) introduced the method of 2-D sampling in a time-frequency domain to combine the advantages of the two standard methods of sampling as time-domain sampling and frequency-domain sampling. Such a method leads to mathematical models which fit better to wave propagation than either of the two alternate standard methods. It is easy to introduce this method, as given below.

Monochromatic waves.-A real monochromatic signal can be represented as:

s(t) = a cos it + b sin ot,

Presented at the 50th Annual International SEG Meeting November 19, 1980 in Houston as “Signal filtering and velocity dispersion through multilayered media.” 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. Napolkon Bonaparte, 92500 Rueil Malmaison, France. *ELF Aquitaine Company, S.N.E. A. (P), Tour G&&ale, Cedex 22, 92088 Paris La defense France. 0016-8033/82/0201-222$03.00. 0 1982 Society of Exploration Geophysicists. All rights reserved.

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Sampling Theory and Complex Waves-Part II 223

cos wt = Re(e’“‘), sin wt = -Re(ie’“‘),

where Re stands for the real part of a complex quantity; then

s(t) = a(eiw’ + Ciw’)/2 + b(elw’ - e-‘“‘)/2i,

and

s(t) = a(P’ + eCiwr)/2 - ib(eiw’ - eCimr)/2,

or more simply,

s(t) = 1/2[(a - ib)e’“’ + (a + ib)e-‘“‘I.

In this last expression, the first term represents the positive frequencies and the second, the negative frequencies.

To make these expressions simpler, we may write

u - ib = A eiQ,

and then,

s(t) = 1/2A[e i(wr++) + e-i(wl++)],

Note that the second term of the last expression is the conjugate of the first. The signal s(t) is thus fully defined by the first term only. In the angular frequency domain, the Fourier transform of s(t) is S(w) = a + ib = A e-IQ. The monochromatic signal s(t) may be presented as a function of two variables, in a time-frequency domain: S(t, w) = A ercwr ‘*‘. A and $ here are two parameters which may have any value.

Broadband waves.-In a time-frequency representation, we consider the waves as built up from interferences, i.e., complex summation, of a large or infinite number of monochromatic waves. This leads to the formulation known as the inverse Fourier transform:

s(t) = I

m ]/2A(0)[e’I”‘++‘W’I + ,-~(“J~++‘““]&,

0

or using angular frequencies,

I

Cz s(t) = 1/2?T 1/2A(4[e ~lwt+$(w)l + e-iIw~+6(w)l]d,,

0

Here we will use only positive frequencies w E [0, ~1. Let us define

$(t) = 1/27r /~A(w)e”““~‘w’ldw. 0

Then

s(r) = 1/2[4J(t) + +*(t)].

9(t) is called the complex signal corresponding to s(t). NOW, going back to the Fourier transform, we can introduce

the negative frequencies for a real signal s(t), implying

i

A(w) = A(o),

4(-w) = -d?(o).

Then the inverse Fourier transform becomes

I

+aC s(t) = l/27r l/2 A (w)e’~w’+~(“)ldo.

-@z

CornPare the above expression, representing the real signal

s(f) to the formula giving the complex signal. The integration

domain is twice as large, but the amplitude A is reduced by half. Regarding the Fourier transform, we may consider we obtain

it by complex crosscorrelation of the signal s(t) with the set of complex exponential functions (cosine and sine) representing the elementary monochromatic waves which are the set of basic waves for the decomposition. Using positive frequencies only, we may read the sum of these complex crosscorrelations as follows:

+CC i-X

S(0) = I

s(-t) . cos wtdt + i I

s(-t) sin ordf, -zC --m

or

I

+a0 S(w) = s(--t)e’w’df,

-Cz

or, permuting t and -t,

S*(w) = lirn s(t)e-iwrdf --x

and introducing the complex signal

I

+m S(0) = l/2 $*(t) e”“‘dt,

-m

which represents the complex crosscorrelation of the signal +(t) with the complex monochromatic wave erwt,

Using the negative frequencies, we may read

S(0) = l/2 j’ o s(-t)eiw’dt.

Utility of a 2-D expansion of the waves

In physics, modeling wave propagation requires the use of pseudoperiodic transient signals. Both time representation and frequency representation are poorly adapted to model wave propagation. On the other hand, modeling phenomena involving interferences require representing the signals using complex functions. Using only the real part of signals in modeling may induce wrong results.

The time-frequency representation of the signals is

S(t, w) = II

A(?, w)F(t’ - I, CO’ - CO) I

elwl(r’-I)+~(l,W)ldo’dt’,

where F(t’ -- t, w’ - 0) represents a local sampling function. The integration domain I is a 2-D domain: time and frequency.

Regarding the signals as information carriers, we may call this 2-D domain, after D. Gabor, the “information plane.” This domain is defined by

t E {-=, +a}, Cd E {--5. +m}.

Following R. Balian, to obtain a correct modeling of wave propagation, all of the functions and variables involved in the wave equation must be taken as having complex values, Further- more, we can derive a good approximation of the wave field, especially for diffraction phenomena, by interferences of the elementary waves related to any elementary possible trajectory in the complex space-time domain. We will limit ourselves in the following developments to complex functions of the two real variables-time and frequency. The wave equation then becomes similar to a SchrBdinger equation.

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224 Morlet et al

The wave functions representing the solutions of this equation have the physical dimension of the square root of the energy. They could be compared to the solutions of the Schrodinger equation, hence to probability amplitudes (Encyclopedia Britannica, 1967).

Gabor expansion

Sampling theory in the time-frequency domain.-Using a time-frequency representation, we define “instantaneous frequency spectra,” and thereby obtain a better mathematical modeling of the real signals than with the “analytical signal,” leading to a single instantaneous frequency and a corresponding instantaneous phase.

Representing S(r, o) is a practical 2-D sampling problem involving r and o. Gabor showed the way, using a regular sampling grid leading to a regular pavement of the information plane with rectangular cells of constant size, the sampling rate being At for the time scale and A.w for the frequency scale. In each cell C(t,, w,) or C(i, j) of this grid, the elementary sample of the function s(t, o) is defined by its amplitude A and its phase +:

A([,, w,) = A(i,j), +(r,, 0,) = +(i,j).

A sample of the function S is not to be considered as a scalar value and not even as a complex number, but as a result of the weighting of the function S in the cell C (i, j); a set of elementary signals is taken as a base for the decomposition of the function S.

We are familiar with two particular cases of such a sampling method:

(a) The time-domain sampling.-In this particular case, the elementary signals taken as the basic wavelets are Dirdc functions, infinitely narrow in the time domain, but with an infinite amplitude; then At = 0 and Aw = 30.

In practice we use as basic wavelets signals which we can physically and numerically represent. In the sampling grid, the cells corresponding to such basic wavelets have dimensions Ar = E (very small value), and Aw = k/E (very large value).

(b) The frequency domain sampling.-ln this case, the elementary basic wavelets are complex monochromatic exponential waves which read e’“‘. The cells of the corresponding sampling grid have as dimensions Ar = 3~ and Aw = 0.

In practice, we must use as basic wavelets signals which may be physically and numerically represented. The corresponding sampling grid has dimensions A r = k/~ (very large), and A w = E

(very small).

Sampling theory and resolution.-The sampling methods for signals were developed by physicists as an extrapolation starting from the experimental measurement methods of scalar physical quantities by weighting. This measurement of scnlur yuunriries, corresponding to specific characteristics of different material objects, is achieved by comparison to unit weights.

The main characteristics of this deterministic weighting are (1) The results obtained from a sequence of measures achieved on a set of objects do not depend upon the order of the objects in the sequence of the elementary measurement operations. (2) Each elementary measure corresponds to a specific object and is therefore fully independent of the measures achieved on the other objects. (3) The measure is a linear operation, where the addition operation must be correctly defined. (4) If there is some instru- mental random noise, we can enhance the measurement accuracy for any specific object by using statistical methods. We may simply repeat the weighting operation, then average the results of this.

We may transpose this type of measurement method to the sampling of pseudoperiodic signals or waves, but we must first remember the following points. (1) The sampling of signals is achieved by a sequence of successive weighting operations, the order of which is imposed. (2) Two successive elementary measurements may not correspond to totally independent information or events. It is therefore very difficult, if not impossible, to distinguish and separate perfectly two elementary independent signals (this differs fundamentally from the measurement operations in particle counters). This leads to the notion of resolving power or resolution, which represents the ability for the measurement method to separate two elementary signals corresponding to two independent information or events.

(3) The measured quantity is no longer scalar, but rather of a complex and periodic type. The unit weights in such a measurement operation must be complex elementary signals. On the other hand, at the end of the quantification into sampled values, the result of the measurement operation is a physical quantity whose physical dimension is the square root of energy. There- fore, the linearity of the measurement operations is observed for complex signals.

(4) If the signals are affected by random or deterministic noise, we may enhance the accuracy of the measure of the signals, thereby the information they carry. We must repeat the measurement operations in such a way to decouple signals and noise.

Signal enhancement by phase coherence is achieved by constructive interference of complex amplitudes, to be compared to interference of probability amplitudes.

time resolution, frequency resolution.-As seen earlier for the particular cases of time-domain sampling and frequency- domain sampling, the dimensions (diameter or bandwidth) of the signals in the two conjugate domains (using the terminology of quantum mechanics) are related by the relationship At Aw 2 k

with k = a numeric constant. This inequality, called the Schwartz inequality, is similar to the uncertainty principle of Heisenberg. Choosing a sampling grid in the information plane adapted to transient or pseudoperiodic signals like seismic traces, we must compromise between the values Ar and Aw when defining the elementary signals chosen for the time-frequency representation of the traces. The association of the grid and the basic wavelets leads to a quantitative definition of the 2-D resolution (time and frequency).

Basic wavelets with Gaussian envelopes: Gahor wavelets. -We could choose in either domain (time or frequency) rectangular envelopes for the basic signals. But then, in the conjugate domain, its transform is a sin x/x function. Using such signals involves disturbing effects in computing (Gibbs phenomena). Furthermore, they are not real physical signals. There- fore, we prefer to use Gaussian envelopes for the basic wavelets, where the same type of function is its transform in the conjugate domain.

Such signals are closer to physical signals and their envelope corresponds to the square root of the intensity in diffraction spots, which are directly related to the notion of resolution.

Such complex wavelets were introduced by Gabor (Gabor, 1946). In the time domain, they are represented by the product of a complex sine function by a Gaussian envelope.

The real part of such wavelets (or cosine wavelet) is a zero- phase wavelet, symmetric in the time domain. The imaginary part (or sine wavelet) is antisymmetric, and in quadrature with the corresponding cosine wavelet.



In the particular expansion used by Gabor, the information plane is paved using a regular grid, the cells of which have their dimensions related by At A w = k. Therefore, in the time domain, At is constant and the envelopes of every basic wavelet are identical; in the frequency domain, the sampling rate is constant, having the value Aw.

Gabor also noticed that such a grid leads to a decomposition which is not orthogonal.

Energy carried by a wave and Poynting’s vector

Before introducing computing methods for the energy carried by a wavelet, we first recall some fundamentals of wave propagation. For monochromatic plane waves propagating in homogeneous media, we can compute the energy which flows through a unit area, normal to the direction of the wave propagation. In the medium, the energy density, or total energy per unit volume, is the sum of the kinetic and potential energies related to the wave. The kinetic energy is related to the particle velocity in the medium. The potential energy is related to the work of the internal stresses.

In plane-wave propagation theory, the same 1-D wave equation applies to elastic waves, the two fundamental functions being P = excess pressure or stress and U = Particle velocity.

MEAN PERIOD T= 20 MS. MEAN -FREQUENCY_ fz50Hz. DIAMETER At q 40MS.

_ , .-50 0 50

FIG. I. Example of Gabor complex wavelet used as a basic wavelet in the Gabor expansion.

Therefore, for elastic waves as for EM waves, the energy flow across a unit area is equal to the flux of Poynting’s vector through the surface of unit area.

For elastic waves, Poynting’s vector is (Dieulesaint and Royer, 1974) P,. = PU.

For a plane wave propagating downward through a homogeneous medium, we saw in Morlet et al (1982) P = ZU. Hence Poynt- ing’s vector is:

P, = PZJ = zu* = pvu”.

In the following theoretical development, we will use time average of the functions, according to the notation:

I

'2

(X) = Mean (X) = I/(rz - tl) Xdt.

II

Thus the mean flux of Poynting’s vector across a surface element ds, normal to the propagation direction, is

(F) = (P,)dS = p(U*)VdS.

That is the mean energy flow of the plane wave (Officer, 1958). In the propagating wave, we may introduce the notions of

kinetic energy, related to the real part of the complex wave function, and of potential energy, related to the imaginary part of the complex wave function. Since both of them are in quadrature, the total energy carried by a monochromatic wave is thus constant versus time and space coordinates in a homogeneous medium.

Utility of a logarithmic scale in the frequency domain

In physics, there is no real advantage in handling negative frequencies. On the other hand, using complex signals leads us to use only the positive frequencies. However, this introduces a disturbing discontinuity at zero frequency in our mathematical models. Using a logarithmic scale for the frequencies (octaves), we can avoid this problem, as is well known in seismic processing.

There are other advantages to using a logarithmic scale in the frequency domain, as we will show later. This use leads us to introduce a new type of expansion, i.e., an extended Gabor expansion.

Main characteristics of the basic wavelets.-Taking zero for the time origin and normalizing the maximum amplitude to 1, we can write the complex function representing Gabor wavelets:

s(t) = e-‘2”A”21”2emy,t

with the two parameters w0 = mean angular frequency, At = duration (or diameter) defined as the time interval separating the two points on the envelope where the modulus drops to the value l/2.

The modulus of the Fourier transform of g(t) is found to be

G(@) = l/2 (n/in 2)1~*Ate-“/1”*‘lA”“-wo”412,

If we write Aw = bandwidth, defined as the frequency interval separating the two points where the modulus drops to half the maximum, we obtain

Atho = 8 In 2 = 5.5452

In the frequency domain, we then have

ArAf = (4 In ~)/II = 0.8825 ,

similar to the uncertainty principle for Gabor wavelets.

The standard formulation of the uncertainty principle for a Gabor wavelet is

AtAo = l/2.


Morlet et al

FREQUENCY (Hz. 1 12.5 25 50 100

FIG. 2. Set of Gabor basic wavelets used in the Gabor expansion (practical example)

Energy carried by a Gabor wavelet.-The amplitude carried by a monochromatic wave is (for a complex monochromatic wave function)

s(r) = ” erw(r-riv)

where U = maximum particle velocity in the propagation medium, p = density of the medium, the complex wave function writes x = space coordinate, and V = celerity of the wave.

Then the total carried energy density (kinetic + potential) is Wr = U’p. The mean kinetic energy density (computed on one period) carried by the wave would be half of the above value, i.e., (W,.) = 1/2u’p.

For a time-limited signal, defined as a truncated sine wave of duration dt, the total energy carried by a plane wave beam of unit section is WT = U’pdt.

For a Gabor wavelet with a Gaussian envelope, the total energy carried is obtained by integrating the previous formula, for a modulus of the complex wave varying as a Gaussian function of time

wT= P (2riA~)~ In 212 dt

-n

= k U’pht,

with k = 1/2[11/(2 In 2)]lJ2 = 0.7526 For a monochromatic wave, the particle velocity U is related

to the particle displacement A by the relation

U = Aw.

Hence, we may write

WI = kA2w2pAr.

And if ohr = constant. as is the case for a set of wavelets with constant shape ratio, we have WT = KA’pw.

Therefore, for this particular subset of wavelets, as for photons, the carried energy is proportional to the frequency.

Gabor wavelets with constant shape ratio.-Consider the subset of the Gabor wavelets defined by the following relations:

At = k’2-rr/oo = k’T,

where T = mean period of the wavelet, and k’ = numerical constant.

Such wavelets may be deduced from each other by merely changing the time scale. They have a constant shape ratio.

Taking into account the relation

AtAw = 8 In 2,

we find

Ao = (8 In 2)/At = (l/k’)w,(8 In 2)/2x;

thus Aw = k,oo. where k, is a numerical constant. In other words, in the frequency domain their bandwidth is proportional to their mean frequency.

Figure 1 represents a particular wavelet of the subset defined by k’ = 2. Figure 2 represents in the frequency domain (with logarithmic scales) a set of wavelets from this family. In the logarithmic representation, the wavelets of the family defined by k’ = 2 derive from each other by a translation in the octave domain.

Sampling grid for Gabor wavelets with constant shape ratio. (a) Energy distribution in the information plane.-Once

a 2-D signal (or trace) sampling in the time-frequency domain is performed, we need an accurate representation of its energy and phase distribution, i.e., of its complex amplitude as a 2-D function of time and frequency.

We know that for monochromatic waves, the energy carried is proportional to U’.

Since neither pure monochromatic waves nor Dirac pulses exist in practice, we may define for a time interval d t, for a frequency interval do, the total energy carried by a sampling cell C having dimensions at and a o.

If (U) represents the mean density of the distribution of U in the cell C, the energy quantity in the cell is



, Frequency

1 2 samples pr cell:

( C(t,f):cosine

S(t,f):sins

N+2 CONSTANT AREA

PER CELL

At.Af=k N+l

I I I ‘I I I (when f in Hz)

N I I I I

0 time

N+4

N+3

FIG. 3. Time-frequency sampling grid used for the Gabor expansion (practical example for four-octave bandwidth).

W,(C) = (U)2pxraw = kp(U)2

Therefore, for a given propagation medium and cells of constant area k, the elementary energy per cell is proportional to the squared amplitude.

Finally, since U is a complex amplitude, we represent it in each cell by a complex value given by its amplitude A (t. o) and its phase $(t, w). The energy carried is therefore proportional to A2.

(b) Grid resolution and wavelet resolution.-Sampling theory implies an ambiguity about the notion of resolution. In fact, we must distinguish two sets of symbols in the time- frequency domain: At and A o for the wavelet dimensions, at and aw for the cell dimensions of the sampling grid. In fact, we perform signal sampling by decomposition on a base made up of a set of basic wavelets used as unit wavelets in the weighting operation of the signal.

In contrast to our experiment of the simple mass measurement, for example, two alternate but mutually exclusive solutions are possible.

(1) If we need independent elementary results of the individual measurements in the adjacent cells, A r = a t and A o = a w represent simultaneously the dimensions of the cells in the grid, and also the dimensions of the wavelets in the time-frequency domain. This restrictive condition represents in signal sampling theory the mathematical orthogonality condition for the elementary functions.

‘We may note here that obtaining independent sampled values for adjacent sampling cells in the grid is analogous to decon- volving. There is, therefore, a strong resemblance between complex deconvolution and decomposition on a base made up of a set of orthogonal functions.

(2) If we need to preserve the information carried by the signal with the maximum of fidelity, we must be able to predict the value of the signal in any intermediate point between sampled values on the grid and the values at and aw, defining the grid resolution, must be simultaneously small enough to permit this prediction by interpolation between the sampled values on the grid. This interpolation may be performed using linear operators. In the last case, the fact that we assume possible the interpolation operation implies that the adjacent sampled values are not independent.

(c) Sampling grid in the time-octave domain.-The main advantage of the Gabor wavelets with constant shape ratio is

that, in spite of their nonorthogonality to each other, they still achieve good resolution in the time domain, due to a good phase resolution. A set of such wavelets may be used as a base of non- orthogonal functions for the decomposition of the Gabor expansion. First adjusting the cell dimensions to the wavelet dimensions, we obtain a sampling grid more complicated than the first grid proposed by Gabor. In this new expansion, the area of each elementary cell remains constant. The area for the cell C(t,, w,) would be:

At(i, j)Ao(i, j) = 8 In 2,

but the value of Aw(i, j) is frequency dependent

Aw(i, j) = l/k’(8 In 2)/27rw,,

thus, for the wavelets of Figures 1 and 2, for k’ = 2,

Ao,/w, = (2 In 2)/n with Awj = Ao(i, j).

The sampling rate of the grid in the frequency domain is then constant when we use a logarithmic scale for the frequencies, and in such a representation, the wavelets of the base are obtained by translation from the first (see Figure 2). The area for each cell C(i, j) being constant, this involves,

At(i, j) = At, = (8 In 2)/Awj = 4n/w,.

On Figure 3, the sampling rate is four wavelets per octave in the frequency domain. We will see later the practical values for the time-frequency sampling of seismic traces.

Computing method for the direct Gabor expansion

As in the Fourier transform case, the value of the complex number representing an elementary sample corresponding to a cell C(i, j) of the grid is obtained by complex crosscorrelation of the signal s(t) with the two Gabor wavelets (cosine and sine) corresponding to this particular cell. We can thus write

I

+=

s(ti, w,) = sWg*(f - f,, o,)dt

-cc

or

+2

S(r,. 0,) = I

s(-t)g(t - ti, w,)dt. --a

This summation is performed by discretization of the timefunctions, followed by a dot product. The same operations must be performed for all cells of the time-frequency sampling grid.

Computing the inverse Gabor expansion

The retrieving operation of the signal is performed as for the inverse Fourier transform, and we can directly obtain the complex signal through

+(t) = j- do 1, S(t’, w)g(r’ - t, o)F[w(r’ - r)]dr’, w

where F represents a local sampling or weighting function related to each cell of the sampling grid, made to scale the amplitude of the result in spite of the nonorthogonality of the basic wavelets. Again, this operation is performed using the cell distribution of the grid.

Amplitude resolution and phase resolution

In practice, a correct sampling of a complex function implies the possibility of retrieving, by interpolation between the sampled values, any value of the function. This leads in fact to a sampling


228 Morlet et al

method preserving the phase information with an accuracy finer than one cycle, hence to a sampling grid whose resolution makes any cycle skipping impossible at any frequency withm the useful band.

It follows that the time sampling rate of the grid may not be higher than one period, which is the cabe for each of the wavelets corresponding to the different periods. This is the antialiasing condition for the extended Gabor expansion we just described. Furthermore, to detect and to correct offset phenomena due to crossfeed from one frequency band to the others, it suffices to double the sampling rate in the time domain. then taking two samples per period. In the frequency domain, a bamphng rate of four wavelets per octave appears to be suitable for phase preservation

We finally obtain as practical value for the area of each cell of the extended Gabor expansion:

araw = 1/2T27~/T(2”~ - 0.5”‘) = 0 545 ,

which leads to the following consideration. When defining the dimensions of the wavelets in the time-frequency domain, replacing both diameter and bandwidth by standard deviations, the uncertainty principle becomes exactly At Aw = l/2.

the information (in both amplitude and frequency domains), which leads to the binary coding of the information. He also showed that, to increase the information transfer rate, one must increase the signal frequency. Finally, Shannon, when extra- polating his work on discrete information to continuous signals, noticed that the theoretical problem is much more complicated and that such extrapolation is only made possible with some restrictive assumptions. Rather than describing these assumptions, he gave the following practical examples: voice and music tram- mission. In both of these particular cases, the elementary units, i.e., phonemes or notes, arc carried by a signal involving a large number of periods, and the knowledge of the phase is not needed. The problems of short-pulse transmission and of dispersion in communication lines are not included in his works. Therefore, when representing the information carried by complex signals or waves, it is possible to attain a better use of a limited number of bits than in the standard sampling methoda (based on a constant sampling rate in the time domain).

The above conditions insure the validity of the addition operation for the complex amplitudes of the waves, a necessity in any interferential processing method. More specifically. the basic wavelets with constant shape ratio are well adapted to phase preservation, since their duration is proportional to their period At = KT.

The objective of high-resolution methods is to attain a better accuracy on the location of the high-trequency information on the time axis. Correct recording of the phase information is then needed. Furthermore, amplitude preservation is impossible without phese preservation. An accuracy of l/16 cycle for any frequency in the extended Gabor expansion. I e.. 4 bits pe:r cell. is probably enough for amplitude recovery by interfereuce of complex amplitudes (as demonstrated by results obtained in modeling wave propagation). We could therefore neglect the amplitude information as in the sign bit seismic method whenever a high level of information compression is needed.

Finally, as in attenuation or dispersion studies (which are of the greatest interest in high-resolution seismic), when processing a particular narrow frequency band, we must consider every other frequency band as noise. We must then preserve a sufficient partial dynamic range for each of the narrow-frequency bands sampling the frequency domain. If information compression is needed, phase preservation is the fundamental condition for en- hancing amplitude preservation. If the amplitude is not recorded in the extended Gabor expansion we just described, it will be recovered by interferences of probability amplitudes in any multi- chdnnel processing method.

This must be compared, as noticed by R. Balian, with the WKB approximation used in quantum mechanics applied to physical optics. Such an approximation leads to neglecting, in the wave equation, the terms mvolving the moduh of the complex amplitudes using only those involving complex exponentials, i.e., the phases. Finally, it appears that the extended Gabor expansion described earlier should be an interesting recording and processing tool in high-resolution seismic

Practical remarks

It is possible to obtain a regular grid, with a constant sampling rate m both dimensions, taking for coordinates the cycles (or phases) and the octaves. This representation leads to easier methods for computing, interpolating, and more generally handling the data in a computer. It also gives surprisingly simple representation of the constant Q laws for attenuation.

These studies were developed by numerous authors for various branches of applrcatrons in physics (AbCl&s, 1946: Born and Wolf, 1959; Brekhovskikh, 1960; Brillouin, 1946, Dieulesaint and Royer, 1974; Elachi, 1976: d’Erceville and Kunetz, 1963; Rytov, 1956). Most of them worked in the frequency domain, as we will do now. We present here the synthetic approach made by G. Bonnet and his assistants E. de Bazelaire and J. F. Cavassilas; this work led to a thesis by J. P. Dolla, who worked for us on this subject in the G.E. S.S. Y. at Toulon University (Bonnet, 1980; Dolla, 1980). Such a model of wave propagation is called Bloch waves in solid state physics (Kittel. 1976). The main interest in studying wave propagation in periodic media is because of the simple way they give us of introducing complex velocities and complex impedances. On the other hand, we may notice here that physical materials are granular and homogeneous media are only mathematical objects.

Propagation matrix in a homogeneous medium

The vectorized solution of the wave equation is

Concluding remarks where

The sampling method we just presented could appear as contradictory to the works of C. E. Shannon (Shannon. lY4Y). In fact there is no contradiction. First of all. Shannon studied the transmission of discrete information in communication lines. and quite a few of the sampling methods of analog data He show*ed the utihty of logarithmic sidles to the quantih<atlon of

P: u- v= L- 2;

I

ANALYllCAL STUDIES OF PROPAGATION IN PERIODIC MEDlA

= A(w) x efWZ’” 1 I I 1/z

+ B(w) x r rw.-‘V

excess pressure UT stress (related to potential energy), particle velocity (related to kinetic energy), velocity of the waves, impedance of the medium. depth, pcrlod of the WdVC,


and

w = 2 x -r/T.


R = (ZJ - z, )/(Z, + z, ).

Mathematical developments become simpler if we detine one

The propagation time is t = z/V, and the phase, C$ = 2 x 7~ x

r/T. We introduce the following vectors:

elementary motif as follows: (I) l/2 layer of medium MI ; (2) 1 layer of medium M,; (3) I /2 layer of medium M,. The propagation matrix for an elementary motif is then

M, = 7-(d,/2, Z,)T(d., Zz)T(d,/2, zr);

which leads to

M,. =

cos2S - R’cos2A 2iZ, (cos Z t R cos A) (sin 2: - R sin A) __._-_ - I -- R2 I-R’

2i (COS X - R cos A)(sin 2: + R sin A) cos 22: - R’ cos 2A - ~ - IZ, I -R2 I -R’ I

Therefore

X(z) = A(w) X eim X X(+) + B(w) X e-‘+ X X(-)

introducing the matrices M and M _ ’ ,

M= ; Mm’ = 112 x 1 ,iz _l’,z 1

and the vector Y (z) = M X X(z), we have

Thus, from depth zO to depth z, we can write Y(z) = A X Y (au). where

,a -60) 0 A=

0 e-“‘+ -+()I

We can then write

Both 4 and Z have a physical meaning: 4 leads to the definition of an effective time and of an effective phase velocity; Z is the effective impedance of the binary medium. The matrix M, has the following eigenvalues:

X(i) z M ‘Y(i) = M~~‘AMX(;~~)

Introducing then d = i - 20, the propagation matrix T(d, ZJ defined by

e” and C”.

with the following eigenvectors:

is therefore

X(T) = T(d, Z)X(z,,). and X(-) =

T(d. Z) = M ‘AM.

tinally leading to

T(d, Z) = cos($ - $0) iZ sin(+ - c$~)

(i/Z) sin(+ - $0) cos(4J - 40)

Propagation matrix for one motif made up of two layers

The functions P and U are continuous at the interfaces between the media. Media M, and Mz are defined by the following parameters:

M,:?,.Z,,d,. Ma: T?. Zz. dz.

Propagation matrix for a periodic medium made up of N motifs

By superposition of N motifs defined this way, i.e., beginning and ending by l/2 layer of medium MI, the propagation matrix in a binary medium equals

M,, = My = cos N+

(i/Z) sin N+

iZ sin N+

cos N+

Thus MF($, Z) = M,(N$, Z), which stant slope in Figure 5 and permits us velocity.

is related to the con-

to define the effective

where dk is the layer thickness of the medium Mk, Zk is the impedance of the medium M1. and r1 is the one way traveltime in one elementary layer of Mk.

We define the following:

2: = ~F(T, t rJ/T, A = ?T(T, - T?)/T.

Complex transfer functions for transmission and reflection

The incident wave function in the entry medium (indexed e) can be written

X,(z) = E,(O)X(f)

and generates the transmitted and retlectcd waves

X,(z) = E,(w)X(t) in the substratum (indexed s),

and

and we use the following convention: X, (2) = E, (w) X (-) in the entry medium (indexed e).

This matrix can be written in the simple form

M, = cos 4 iZ sin 6

(i/Z) sin C$ cos C$l ’

where C$ is defined by the implicit equation

cos I$ = t (cos 2Z - R’cos 2A)/(l - R’)

and where

Z’ = Zi’ x (cos 2; t R cos A)(sin Z; - R sin A)

(COST-RcosA)(sin2;+RsinA)’


230 Morlet et al

Then the complex transfer function for transmission is

h,(w) = E,(wlE,(o),

and for reflection

h,(w) = E,(o)lE,(o).

The wave vectors in entry medium and substratum are

X,(z) = X,(z) + X,(z), X(z) = X,(z)

Introducing the propagation matrix M,, we have

X,(Z) = M,%(z).

We can compute h,(w) and h,(w), resolving a linear system of equations. After a few mathematical developments, and defining the following expressions

T,, = 2&/(Z, + Z,), R,, = (Z, - Z,)/(Z, + Z,),

x = cos + = (cos 22 - R2 cos 2A)/(l - R’),

TN(X) = cos (N+), PN(x) = sin (N$)/sin + (Chebyshev polynomials),

we finally obtain

h,(o) = T,S

TN(X) - iPPiv(x) ’

and

h,(o) = R,,T,v(x) - iP’P,v(.r)

TN(X) - iPPN(x)

where

EFFECTIVE TIME/DIRECT time

1 I I I

2.5

1

1

6 0:2 0:4 0:8

FIG. 4. Chart of effective slowness for wave propagation in binary periodic media for large impedance contrasts (low-frequency approximation).

P= a,(sin 22 - R2 sin 2A) + 2azR sin(2 - A)

1 -R’

and

p’ = a2(sin 2C - R2 sin 2A) + 2c~,R sin(X - A)

with

z: + z,z, z: - ZJ, rv. = Ty_ = 1 - +2/2 = 1 - 2(2;* - A2R2)/(l - R2) --I z, (Z, + Z,) ’ -L Z,@, + Z,) .

Synthesis of the two approaches to the problem

We will now use and compare the results obtained by (1) the analytical studies (developed in the frequency domain), (2) the developments made in Morlet et al (1982, this issue) using the Goupillaud-Kunetz algorithm for a 1-D synthetic seismogram and a set of Gabor wavelets, therefore simulating the Gabor expansion in the time-frequency domain of the propagating waves.

Approximation for low frequencies.-To compute the effective characteristics of the binary medium, we can study the propagation through one elementary motif. We have

+ = 27~7,/T,

where T, = effective time for one motif. When $ is small, the equation

cos + = (cos 2C - R2 cos 2A)/(l - R2)

developed in Taylor series (first two terms) becomes

Coming back from phases to times,

7; = (7, + T2)2 - (7, - Tz)~

I -R2

Then, using the notation,

Y = (TI - T?)/(T, + 711,

we finally obtain

7,/T = [(I - y2R2)/(1 - R’)]“’

This expression equals the ratio mean velocity/effective velocity for the medium.

Similar results were obtained by numerous authors for periodic multilayered media (Rytov, 1956; Brekhovskikh, 1960; d’Erceville and Kunetz, 1962). They agree perfectly with the chart obtained by picking effective times on the low-frequency wavelets. Figure 4 is a chart of effective slowness obtained from the last equation, for a large range of impedance ratios (logarithmic vertical scale),



EFFECTIVE time / DIRECT time

-T-l---T--

Zl =

1 t EFFECTIVE IMPEDANCE

l-

.666

0.8

FIG. 5. Chart of effective slowness for wave propagation in binary periodic media for intermediate impedance contrasts (low frequency approximation). FIG. 6. Chart of effective impedance for wave propagation in

binary periodic media (low-frequency approximation).

1

0 0 100 200 FREQUENCY (Hz) 0 100 200 FREQUENCY (Hz)

01 I . J I I 0 100 200 FREQUENCY (HZ 0 100 200 FREQUENCY (Hz)

1 J I: J t. I

0 100 200 FREQUENCY (Hz) 0 100 200 FREQUENCY (Hz)

FIG. 7. Transfer functions for transmission in binary periodic FIG. 8. Transfer functions for transmission in binary periodic media (obtained by Fourier analysis of synthetic traces). Z I /Z, = media(obtained by analytic method in frequency domain). Z( /Zz = IO, Z,, = Z, = Zz, T = 5 msec, N = 15. 10. Z,. = Z, = Z?. T = 5 msec. N = 15.


Morlet et al

I L J

0 50 FREQUENCYtHz) 100

01 I 1

0 50 FREQUENCY (Hz) 100

1

3 1 !4

50 FREQUENCY (Hz) 100

FIG. 9. Enlargement in the frequency domain of Figure 7. FIG. 10. Enlargement in the I’rcquency domain of Figure 8. ZI/Zz = 10; Z, = Z, = Zz; r = 5 msec; N = 15. Z,/Z? = 10; Z, = Z, = Z,; r 5 msec; N = 15.

useful for gas reservoirs and aerated rocks, as we shall see later. Figure 5 is a similar chart, for a smaller range of impedance

ratios, directly useful in seismic interpretation. The formula giving y versus T, and 72 explains the symmetry

observed in Figures 4 and 5. For low frequencies, the first term of the development in Taylor series leads to the following value for the effective impedance of the periodic medium:

z = (Z,Z,)“2[(1 - yR)(l + YR)]“*.

Figure 6 is a chart of effective impedance directly usable in seismic interpretation (logarithmic vertical scale).

Passbands and forbidden bands in the frequency domain. -The analytical study of the transfer functions h,(w) and h,(w) is easy in the frequency domain. We present here the results for transmission. Two different cases appear, depending upon whether the propagation matrix M, is a rotation matrix or not.

(1) + is real, thus 1x1 < 1 :M, is a rotation matrix. For frequency bands corresponding to this first case, the periodic medium is transparent.

(2) + is imaginary; thus 1x1 > 1 :M,, is not a rotation matrix. For frequency bands corresponding to this second case, the exponentials become real, leading to exponential attenuation, i.e., for an infinite periodic medium, leading to suppression of the transmission. Such bands are known in the theory of wave propagation in crystals under the name of forbidden bands. They correspond to superreflectivity for reflection. From a physical point of view, this superreflectivity is the result of constructive

d 50 FREQUENCYIHZ) 100

1

0

0 50 FREQUENCY (HZ) 100

d

0 50 FREQUENCY (HZ) 100

interferences of distributed reflcctivities, related to clusters of intelFaces.

Figures 7 and 8 show the amplitude response versus frequency for the transfer functions in transmission, computed for 7 L/T* = 10/90, 50/50, and 90/10, using the following two approaches: (1) synthetic seismogram, then Fourier transform (Figure 7), (2) direct computing from h,(o) (I;igure 8). The results coincide perfectly, the few differences being due to sampling problems in the frequency domain. We may notice the periodicity of the spectra for or = ~2. For practical applications in seismic reflection, the side passbands will unfortunately be useless, because of their instability when small anomalies are introduced in the perioaicity of the medium.

Figures 9 and 10 show an enlargement of the frequency scale leading to a better definition for the first passbands shown on Figures 7 and 8. The narrow peaks appearing in these passbands are due to the reinforcement of the effective wave by its multiple reflections at lower and upper interfaces of the periodic medium, and they thus generalize the phenomenon described in Morlet et al (1982).

A general formula for the location of these peaks is

T = (2/k)N7,(w),

where the successive values of 7’. corresponding to each single peak in the spectrum, are given by k = any integer number from 1 up to N (for the first passband). When Z, = Z, = Zz, the two methods are exact for modeling of the propagation in a multilayered periodic medium. For direct computation of the



1. FOURIER ANALYSIS OF SYNTHETIC TRACES

LAYERSSEQUENCE:M~+~~(MI+M~)+MS

0- 0 loo 200 FREQUENCYlHzl

2. DIRECT COMPUTING

LAYERSSEQUENCE:

1

:

c 3 0.

0 100 200 FREQUENCY (Hz)

FIG. 11. Differences between Fourier analysis of synthetic traces and direct computing, due to the half-layer method used in the analytical developments. Z,/Z, = 10, T = 5 msec, T,/T* = 1, N = 15.

response of this layering model, we took, as first and last layers of the periodic medium, one-half layer of medium 2.

This is no longer possible for the general case regarding the embedding media. For example, Figure 11 gives the transfer function for transmission for adapted impedances of the embedding media. In this case, the layer sequences used in the two methods are different; then for the direct computation, the first and last layers are made up of l/2 layer of medium 1. For the first passband, the differences between the transfer functions are limited to the amplitudes of the narrow peaks (due to the differences in the reflectivity values at the boundary interfaces with the embedding media). But the first side passband of the directly computed response is almost destroyed. This implies that the low-frequency passband is not fundamentally affected by small anomalies in the medium’s periodicity, but that is no longer the case for the side passbands.

This confirms parallel studies we made concerning the stability of the transmission response for a set of synthetic seismograms for periodic media including small anomalies. In these last studies, we introduced in the layering model of a synthetic section a progressive pinchout involving a single elementary layer of the periodic medium. These studies were made for different locations of the pinchout in the periodic medium (upper part, lower part, medium). In such cases, we observed a good stability of the transmitted signal.

Figure 12 shows the transfer functions for T, /TV =

10/90, 50150, and 90110, with adapted impedances of the

01 I J t. J 1. 1 0 100 200 FREQUENCY (Hz1

0 loo 200 FREQUENCY (Hz)

"0 100 200 FREQUENCY (Hz)

FIG. 12. Transfer functions for transmission in binary periodic media for adapted impedances of the embedding media (direct computing). Zi/Zz = 10, T = 5 msec, N = 15.

1

0.1

t

PHASE VELOCITY ----

GROUPVELOCITY -

WAVELET VELOCITY -----I--

EFFECTIVE VELOCITY/MEAN VELOCITY

I PERIOL

B 20 40 80 200 400 800

-I-

l(m

FIG. 13. Velocity dispersion in a binary periodic medium (intermediate impedance contrasts). Zi /Z, = 10; T = 2 msec; T I /T? =

1.


234 Morlet et al

I .oo L

i

GROUP SLOWNESS - Tl/T2 .l

PHASE SLOWNESS ----

VERSUS WAVE PERIOD AND IMPEDANCE RATIO


PERIOD / T PERUoo/T

FIG. 14. Charts of effective slowness in a binary periodic medium FIG. 15. Charts of effective slowness in a binary periodic medium (intermediate impedance contrasts). (low-impedance contrasts).

embedding media. Notice that, due to the small reflectivity at traces computed by the Goupillaud-Kunetz algorithm and simu- the boundary interfaces, the thin peak amplitudes are much lating the Gabor expansion. Due to the bandwidth of the wave- smaller than those of the previous figures, especially for the lets used, the curve representing the velocity of the wavelets is first passband. smoother than the other two curves.

Velocity dispersion.-The dispersion equation expresses the propagation velocity versus frequency. It was developed under the implicit form

cos I$ = (cos 22: - R2 cos 2A)/(l - R’),

where

cfl= 07, I: = m/T, A = ~(7, - T~)/T

Introducing d = thickness of one elementary motif. we can define the phase velocity as

The condition of phase stationarity d+/+ = do/w, defines the group velocity as

VK = d do/d+ = d(d$/dw)- ‘.

Therefore V, and VK are easy to compute, using the dispersion equation.

An example of computed phase and group velocities versus the wave’s period T is plotted on Figure 13. We took as reference the mean velocity V, = d/T in the periodic medium. We plotted on the same diagram the effective wavelet velocity obtained by automatic picking of the envelopes maxima of the

GROUP SLOWNESS Tl/T2;1

-’ PHASE SLOWNESS -----

VERSUS WAVE PERIOD AND IMPEDANCE RATIO


--

Owing to the approximation made in the assumption that the effective velocity of the wavelet’s energy is related to the maximum of the signal envelope (which is only true for symmetric wavelets), this curve only gives approximate velocity and thus differs from the group velocity.

Figures 14 and 15 represent charts of group and phase slow- nesses, taking as reference the direct wave slowness, for different values of the ratio Zi/Zz. Both scales are logarithmic. The horizontal scale gives the ratio T/T. These charts can be used for seismic interpretation.

Extrapolation to media made up of any number of components

The above studies are limited to binary media. Bonnet showed that the analytical study can easily be extended to periodic media made up of any number of components (Bonnet, 1980). For such media, it is then possible to define the two following characteristic parameters: effective velocity and effective impedance. For low frequencies, he showed that these characteristic parameters of the periodic medium do not depend upon the order of the layers in the elementary motif.

CONCLUSIONS

We have showed theoretically and experimentally 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, and velocity dispersion for intermediate frequencies. These phenomena do occur in standard seismic prospecting, although they are not always easily noticeable. They largely 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 be handled using complex models for both wave functions and physical characteristics of the layered media.

Regarding the recording and processing methods, we showed that, to increase the resolution for frequency dependent velocity and attenuation, we must work in the time-frequency domain. An extended Gabor expansion, using as basic wavelets a set of constant shape ratio wavelets, permits us to preserve both phases and amplitudes for a large range of frequencies. In spite of the nonorthogonality of the basic wavelets, it is possible to minimize the number of complex samples needed for the discretization of the Gabor expansion. This expansion gives directly instantaneous frequency spectra of the seismic traces.

If compression of the information is needed, we showed that it is sufficient to preserve the phase information of the Gabor expansion. In such a case, the information lost on the amplitudes can be retrieved by the interferences of the probability amplitudes, i .e . , by complex multichannel processing.

To discretize multitrace records (seismic profiles), we can handle the space dimension as we did the time dimension. We can then introduce a space-spatial frequency domain for record sampling. This finally leads us to substitute the 2-D seismic records (time, space) for 4-D records, implying a better quantiza- tion of the seismic information.

Regarding the models for the multilayered media, especially for complex deconvdlution, we showed, using direct modeling, that it is possible to introduce complex values for velocities and impedances using thin-layered binary periodic media. These studies could be extended (as for Bloch waves) to 3-D heterogeneous media. We could therefore model the wave propagation in such media as in macrocrystalline structures and for any wave incidence. Finally, we may use some other models developed in solid state physics for different scales of gkanularity, as far as some models developed in quantum fields theory.

We hope that our studies will contribute to enhance the resolving power of the seismic reflection method.

ACKNOWLEDGMENTS

The authors wish to thank the Elf Aquitaine Group for per- mission to publish this material, and also B. Delapalme, F. Bernard, J. Berthelot, and the Research Staff of the group and its subsidiaries S.N.E.A.(P). and O.R.I.C. for their effectual support.

We express our gratitude to P. Alba and R. Coeroli of Elf Aquitaine, to Ph. Marchal of N.A.T. and Ch. Hemon of the French Institute of Petroleum for their efficient cooperation (permits starting these studies) and for their pertinent suggestions during the development of the project.

We are especially indebted to G. Henry, R. Le Moal, B. Michaux, and K. Titchkosky for helpful discussions and comments while preparing this material.

We wish to thank R. Balian, of the Service de physique theorique, C. E. A. (Saclay), for his stimulating support concerning the developments related to theoretical physics.

Finally, we wish to thank P. Detombes, who typed and corrected this manuscript with diligence and efficiency.

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Wave propagation and sampling theory--Part II: Sampling ...€¦ · Wave propagation and sampling...

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