GEOPHYSICS,
VOL.
41, NO.
2 (FEBRUARY
1982); P. 222-236,
15 FIGS
Wave propagation and sampling theory-Part
Sampling theory and complex waves
II:
J. Mot-let*, G. Arensz, E. Fourgeau*, and D. Giard$
ABSTRACT
Morlet et al (1982, this issue) showed the advantagesof
using complex values for both waves and characteristicsof
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
heterogeneousmedia.
Regarding the complex signals, we present a twodimecsional(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 highresolution 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 receivers. 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 standardmethods
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 technological 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 possessesthe 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 advantagesof 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
represented as:
real monochromatic signal can be
s(t) = a cos it
+ b sin ot,
Presentedat the 50th AnnualInternationalSEG MeetingNovember 19, 1980 in Houstonas “Signal filtering and velocity dispersion throughmultilayered
media.” Manuscriptreceivedby the EditorJuly 11, 1980; revised manuscript received May 8, 1981.
*ELF AquitaineCompany,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
cos wt = Re(e’“‘),
sin wt = -Re(ie’“‘),
where Re standsfor 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
223
II
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 crosscorrelationof 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:
i-X
+CC
s(-t) sin ordf,
s(-t) . cos wtdt + i
S(0) =
I --m
I -zC
or
S(w) =
+a0
s(--t)e’w’df,
I -Cz
or, permuting t and
-t,
u - ib = A eiQ,
S*(w)
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.
= lirn
--x
s(t)e-iwrdf
and introducing the complex signal
+m
S(0) = l/2
I -m
$*(t)
e”“‘dt,
which represents the complex crosscorrelation of the signal
+(t) with the complex monochromatic wave erwt,
Using the negative frequencies, we may read
S(0) = l/2 j’
Broadband waves.-In
a time-frequency representation, we
consider the waves as built up from interferences, i.e., complex
summation, of a large or infinite number of monochromatic
waves. This leads to the formulation known as the inverse
Fourier transform:
m
]/2A(0)[e’I”‘++‘W’I
+ ,-~(“J~++‘““]&,
s(t) =
I0
or using angular frequencies,
Cz
s(t) = 1/2?T
I 0 1/2A(4[e
~lwt+$(w)l+ e-iIw~+6(w)l]d,,
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
A(w)
i 4(-w)
= A(o),
= -d?(o).
Then the inverse Fourier transform becomes
s(t) = l/27r
+aC
l/2 A (w)e’~w’+~(“)ldo.
I -@z
CornPare the above expression, representing the real signal
to the formula giving the complex signal. The integration
s(f)
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) =
A(?, w)F(t’ - I, CO’- CO)
II
I
elwl(r’-I)+~(l,W)ldo’dt’,
where F(t’ -- t, w’ - 0) representsa 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},
CdE {--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, Furthermore, 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 SchrBdingerequation.
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Morlet et al
224
The wave functions representing the solutions of this equation
have the physical dimension of the squareroot 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 instantaneousfrequency 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:
this particular case,
(a) The time-domain sampling.-In
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
samplinggrid 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 sequenceof the elementary measurementoperations. (2) Each
elementary measure correspondsto 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 instrumental 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 successiveelementary measurementsmay 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 representsthe 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. Therefore, 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 measureof 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 frequencydomain 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. Therefore, 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 representedby the product of
a complex sine function by a Gaussian envelope.
The real part of such wavelets (or cosine wavelet) is a zerophase wavelet, symmetric in the time domain. The imaginary part
(or sine wavelet) is antisymmetric, and in quadrature with the
corresponding cosine wavelet.
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Sampling Theory and Complex Waves-Part
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.
II
225
Therefore, for elastic waves as for EM waves, the energy flow
acrossa 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 propagatingdownward througha homogeneous
medium, we saw in Morlet et al (1982) P = ZU. Hence Poynting’s vector is:
P, = PZJ = zu*
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.
= pvu”.
In the following theoretical development, we will use time average
of the functions, according to the notation:
'2
(X) = Mean (X) = I/(rz
-
tl)
I
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 constantversustime and spacecoordinatesin a homogeneousmedium.
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.
MEAN PERIOD
T= 20 MS.
MEAN -FREQUENCY_ fz50Hz.
At q 40MS.
DIAMETER
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) =
_,
.-50
0
50
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.
FIG. I. Example of Gabor complex wavelet used as a basic wavelet
in the Gabor expansion.
The standard formulation of the uncertainty principle for a
Gabor wavelet is
AtAo
= l/2.
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Morlet et al
FREQUENCY(Hz. 1
12.5
25
100
50
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)
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,
s(r) = ” erw(r-riv)
where U = maximum particle velocity in the propagationmedium,
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
(2riA~)~ In 212 dt
wT=
P
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 representsa particular wavelet of the subsetdefined 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.
-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 subsetof wavelets, as for photons,
the carried energy is proportional to the frequency.
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
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Sampling Theory and Complex Waves-Part
, Frequency
2 samples
pr cell:
N+4
1
C(t,f):cosine
N+3
( S(t,f):sins
CONSTANT
N+2
AREA
PER CELL
At.Af=k
N+l
I
I
I
I
N
I‘
I
(when
I
I
I
0
f in Hz)
II
227
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 nonorthogonal 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,
I
j) = 8 In 2,
time
but the value of Aw(i, j) is frequency dependent
FIG. 3. Time-frequency sampling grid used for the Gabor expansion (practical example for four-octave bandwidth).
Aw(i, j) = l/k’(8
In 2)/27rw,,
thus, for the wavelets of Figures 1 and 2, for k’ = 2,
Ao,/w,
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
squaredamplitude.
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 timefrequency 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
measurementsin 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 representsin signal sampling theory the
mathematicalorthogonality condition for the elementary functions.
‘We may note here that obtaining independent sampled values
for adjacent sampling cells in the grid is analogous to deconvolving. 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
= (2 In 2)/n
with
Awj = Ao(i,
j).
The sampling rate of the grid in the frequency domain is then
constantwhen 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
+=
s(ti,
w,)
=
I -cc
sWg*(f
-
f,,
o,)dt
or
+2
S(r,. 0,) =
I --a
s(-t)g(t
- ti, w,)dt.
This summation is performed by discretization of the time
functions, 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 representsa 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
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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 appearsto 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 standarddeviations, the
uncertainty principle becomes exactly At Aw = l/2.
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
accuracyon 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:rcell.
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.
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
representationof the constant Q laws for attenuation.
Concluding remarks
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*edthe utihty of logarithmic sidles to the quantih<atlon of
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 extrapolating 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 trammission. 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 standardsampling methoda (based on a constant
sampling rate in the time domain).
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 enhancing 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 multichdnnel processing method.
ANALYllCAL STUDIES OF PROPAGATION
IN PERIODIC MEDlA
These studies were developed by numerous authors for various
branchesof applrcatronsin 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 assistantsE. 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
1
I I
= A(w) x efWZ’”
+ B(w) x r rw.-‘V
1/z
where
P:
uv=
L2;
I
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,
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Sampling Theory and Complex Waves-Part
II
229
R = (ZJ - z, )/(Z,
and
w = 2 x -r/T.
The propagation time is t = z/V, and the phase, C$= 2 x 7~ x
r/T. We introduce the following vectors:
+ z, ).
Mathematical developments become simpler if we detine one
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
cos2S
M,. =
- R’cos2A
__._-_
I -- R2
2iZ, (cos Z t R cos A) (sin 2: - R sin A)
I-R’
(COSX - R cos A)(sin 2: + R sin A)
~
I -R2
-2i
IZ,
cos 22: - R’ cos 2A
I -R’
I
Therefore
X(z) = A(w) X eim X X(+)
+ B(w) X e-‘+
X X(-)
This matrix can be written in the simple form
introducing the matrices M and M _ ’ ,
M=
; Mm’ = 112 x 1 ,iz
_l’,z
1
iZ sin 6
cos 4
M, =
(i/Z)
sin C$
cos C$l ’
where C$is defined by the implicit equation
cos I$ = t (cos 2Z - R’cos
and the vector Y (z) = M X X(z), we have
2A)/(l
- R’)
and where
Z’ = Zi’ x
(cos 2; t R cos A)(sin Z; - R sin A)
(COST-RcosA)(sin2;+RsinA)’
Thus, from depth zOto depth z, we can write Y(z) = A X Y (au).
where
,a
-60)
A=
0
e-“+
‘
0
-+()I
We can then write
X(i)
z M
‘Y(i)
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:
= M~~‘AMX(;~~)
Introducing then d = i - 20, the propagation matrix T(d, ZJ
defined by
e”
and
X(-)
=
Propagation matrix for a periodic medium made up of N motifs
T(d. Z) = M
‘AM.
tinally leading to
T(d, Z) =
C”.
with the following eigenvectors:
X(T) = T(d, Z)X(z,,).
is therefore
and
cos($ - $0)
iZ sin(+ - c$~)
(i/Z) sin(+ - $0)
cos(4J - 40)
By superpositionof N motifs defined this way, i.e., beginning
and ending by l/2 layer of medium MI, the propagation
matrix in a binary medium equals
M,, = My =
Propagation matrix for one motif made up of two layers
The functions P and U are continuousat the interfaces between
the media. Media M, and Mz are defined by the following
parameters:
M,:?,.Z,,d,.
Ma: T?. Zz. dz.
where dk is the layer thickness of the medium Mk, Zk is the impedanceof the medium M1. and r1 is the one way traveltime in one
elementary layer of Mk.
We define the following:
2: = ~F(T, t rJ/T,
A = ?T(T, - T?)/T.
and we use the following convention:
cos N+
iZ sin N+
(i/Z) sin N+
cos N+
Thus MF($, Z) = M,(N$, Z), which is related to the constant slope in Figure 5 and permits us to define the effective
velocity.
Complex transfer functions for transmission and reflection
The incident wave function in the entry medium (indexed e)
can be written
X,(z) = E,(O)X(f)
and generates the transmitted and retlectcd waves
X,(z) = E,(w)X(t)
in the substratum(indexed s),
X, (2) = E, (w) X (-)
in the entry medium (indexed e).
and
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230
Morlet et al
Then the complex transfer function for transmission is
EFFECTIVE
I
1
h,(w) = E,(wlE,(o),
TIME/DIRECT
I
time
I
and for reflection
h,(w)
= E,(o)lE,(o).
The wave vectors in entry medium and substratumare
X,(z)
X(z)
1
= X,(z) + X,(z),
= X,(z)
Introducing the propagation matrix M,,
we have
X,(Z) = M,%(z).
We can compute h,(w) and h,(w), resolving a linear system
of equations. After a few mathematical developments, and defining the following expressions
2.5
T,, = 2&/(Z,
+ Z,),
R,, = (Z, - Z,)/(Z,
+ Z,),
x = cos + = (cos 22 - R2 cos 2A)/(l
TN(X) = cos (N+),
PN(x) = sin (N$)/sin
6
0:4
0:2
0:8
+ (Chebyshev polynomials),
we finally obtain
h,(o) =
1
- R’),
FIG. 4. Chart of effective slownessfor wave propagation in binary
periodic media for large impedance contrasts (low-frequency
approximation).
T,S
TN(X) - iPPiv(x) ’
and
h,(o) =
R,,T,v(x) - iP’P,v(.r)
TN(X) - iPPN(x)
where
a,(sin 22 - R2 sin 2A) + 2azR sin(2 - A)
P=
1 -R’
and
p’ =
a2(sin 2C - R2 sin 2A) + 2c~,R sin(X - A)
with
rv.
--I
=
z: + z,z,
Ty_
z, (Z,
+ Z,)
=
’ -L
z:- ZJ,
Z,@, + Z,) .
1 -
+2/2
= 1 -
2(2;*
-
A2R2)/(l
- R2)
Coming back from phases to times,
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
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 authorsfor 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),
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Sampling Theory and Complex Waves-Part
EFFECTIVE
time
/ DIRECT
II
231
Zl
time
t
1
=
EFFECTIVE IMPEDANCE
l-
-T-l---T-.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
100
0
0
100
200 FREQUENCY (Hz)
0
100
200 FREQUENCY (Hz)
0
100
200
J
t.
I
200 FREQUENCY (Hz)
0
100
200 FREQUENCY (Hz)
I
01
1
0
J
100
I:
FIG. 7. Transfer functions for transmission in binary periodic
media (obtained by Fourier analysis of synthetic traces). Z I /Z, =
IO, Z,, = Z, = Zz, T = 5 msec, N = 15.
200
J
.
FREQUENCY (Hz)
I
I
FREQUENCY (HZ
FIG. 8. Transfer functions for transmission in binary periodic
media(obtained by analytic method in frequency domain). Z( /Zz =
10. Z,. = Z, = Z?. T = 5 msec. N = 15.
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Morlet et al
I
I
01
d
J
L
0
50
FREQUENCYtHz)
100
50
FREQUENCY
100
50
FREQUENCYIHZ)
100
0
50
FREQUENCY
(HZ)
100
0
50
FREQUENCY
(HZ)
100
1
0
(Hz)
1
1
1
3
!4
50
FREQUENCY
(Hz)
100
d
0
FIG. 9. Enlargement in the frequency domain of Figure 7.
ZI/Zz = 10; Z, = Z, = Zz; r = 5 msec; N = 15.
FIG. 10. Enlargement in the I’rcquency domain of Figure 8.
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:
interferences of distributed reflcctivities, related to clusters of
intelFaces.
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
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 passbandswill 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 passbandsshown 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
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Sampling Theory and Complex Waves-Part
233
II
1. FOURIERANALYSIS OF SYNTHETIC TRACES
LAYERSSEQUENCE:M~+~~(MI+M~)+MS
01
0-0
loo
J
I
J
t.
1.
1
0
100
200
FREQUENCY
(Hz1
0
loo
200 FREQUENCY
(Hz)
100
200 FREQUENCY
(Hz)
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.
"0
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.
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 passbandof 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
PHASEVELOCITY
----
GROUPVELOCITY
-
WAVELETVELOCITY -----I-EFFECTIVE
VELOCITY/MEAN
VELOCITY
1
-I-
0.1
I
PERIOL
l(m
t
B
20
40
80
200
400
800
FIG. 13. Velocity dispersion in a binary periodic medium (intermediate impedancecontrasts). Zi /Z, = 10; T = 2 msec; T I /T? =
1.
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Morlet et al
234
GROUP SLOWNESS
-
PHASE SLOWNESS
----
Tl/T2
i
.l
GROUP SLOWNESS
Tl/T2;1
-’
PHASE SLOWNESS -----
VERSUS WAVE PERIOD AND IMPEDANCE RATIO
EFFECTIVEtime / DIRECT time
VERSUS WAVE PERIOD AND IMPEDANCE RATIO
EFFECTIVEtime / DIRECT time
--
I
.oo
L
PERUoo/T
PERIOD/ T
FIG. 14. Charts of effective slowness in a binary periodic medium
(intermediate impedance contrasts).
FIG. 15. Charts of effective slowness in a binary periodic medium
(low-impedance contrasts).
embedding media. Notice that, due to the small reflectivity at
the boundary interfaces, the thin peak amplitudes are much
smaller than those of the previous figures, especially for the
first passband.
traces computed by the Goupillaud-Kunetz algorithm and simulating the Gabor expansion. Due to the bandwidth of the wavelets used, the curve representing the velocity of the wavelets is
smoother than the other two curves.
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 slownesses, 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.
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+/+
group velocity as
VK = d do/d+
= do/w,
= d(d$/dw)-
defines the
.‘
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
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
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Sampling Theory and Complex Waves-Part
for low frequencies, superreflectivity for high frequencies, and
velocity dispersionfor 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,
thesephenomena are no longer negligible, but they can be handled
using complex models for both wave functions and physical
characteristicsof 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 compressionof 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 quantization 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
studiescould 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 studieswill contribute to enhancethe resolving
power of the seismic reflection method.
ACKNOWLEDGMENTS
The authors wish to thank the Elf Aquitaine Group for permission 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 suggestionsduring
the development of the project.
We are especially indebted to G. Henry, R. Le Moal, B.
Michaux, and K. Titchkosky for helpful discussionsand comments
while preparing this material.
We wish to thank R. Balian, of the Service de physique
theorique, C. E. A. (Saclay), for his stimulating supportconcerning
the developments related to theoretical physics.
Finally, we wish to thank P. Detombes, who typed and
corrected this manuscript with diligence and efficiency.
235
II
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