chapter 1 time and frequency

8/13/2019 Chapter 1 Time and Frequency

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Chapter 1 Time and Frequency

Bohr maintains that there are two complementary aspects of reality. These are

like two distinct planes which we cannot exactly focus simultaneously.

Louis de Broglie (18921987)

One standard way of generating a time-frequency response builds on the Fourier

transform (FT). We are going to develop concepts related to the continuous wavelet

transform (CWT) and related somewhat to the short-time Fourier transform (STFT).

Begin with the idea of an experiment that yields a time series. It could be a seismicexperiment in the field or in a laboratory, an audio recording, an electrocardiogram of

the electrical activity of the heart, or any other measurement that has time-variable

output. The data can have various properties. A time function can be periodic, such as a

single sine or cosine wave, a voice perfectly holding one musical note, or the regular

resting heartbeat. Ideally, the first two contain only one frequency, whereas the last has

many frequencies but is still periodic. For such cases, the FT is ideal. If a heartbeat is

recorded for, say, 1000 s and we do a Fourier transform, the amplitude spectrum will

precisely identify the frequencies contained in the signal. That is possible because the

frequency content is unchanging; the heart beats in a regular and repeatable pattern for

the entire measurement period. The signal is stationary and unchanging, and the FT is

the ideal tool for analysis.However, we also have to deal with nonperiodic signals. One example would be a

heartbeat under stress such as while taking a university examination, when the rate

would go up or down according to the difficulty of questions, growing time constraints,

exhilaration, despair, and so forth. If we measured three or four minutes of data in such

a case and did an FT of the time series, the amplitude spectrum would indicate some

kind of overall average frequency content. It would not be possible, however, to tell

whether the heart was beating fast or slow early in the recording or late or in the mid-

dle. The Fourier transform in this case tells only some of the story. The wavelet trans-

form is one example of a time-frequency method that tells more of the story.

In such cases, we are interested in local spectral properties. What is the spectrum

in the vicinity of 1 s or 5 s in our data? Those are questions that the Fourier transformcannot answer.

There is a long history of the Fourier transform in seismology. If a time series is

involved, the transform variable is frequency. For a space series (different locations at

the same time), the transform variable is the wavenumber. Much of data processing can

be done in the Fourier domain, including various kinds of filtering, migration, and

noise removal. The Fourier transformer is well understood, fast, and invertible the

gold standard of transforms.

Distinguished Instructor Short Course 1


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Fourier transform

We will use the convention that a time function,g(t), and the Fourier transform

of that function,g(), are in the time or frequency domain as indicated by the argu-

ment list rather than by some variation on the function symbol.The forward FT is defined as usual,

g g t e dti t( ) ( ) , =

(1)

where scaling constants have been omitted, i = -1, and angular frequency (in radians

per second) is related to linear frequency ( fin cycles per second or Hertz) by =2f.

The functiong() is complex and can be expressed in polar formg() =Aeito reveal

the amplitude spectrum,A(), and phase spectrum, (). The inverse FT is given by

g t g e di t( ) ( ) .=

(2)

Briefly, we will look at some simple examples. Figure 1a shows the real-valued

time-domain input data, with the vertical axis being time. All physical experiments will

yield real values in the measured

time series. Figure 1b and 1c

shows the FT of the time-domain

output in this case, in which the

vertical axis is frequency. In the

Fourier domain, we have a fre-

quency series in which each valueis typically complex. The plots

show the real (Figure 1b) and

imaginary (Figure 1c) parts.

From these, the amplitude and

phase at each frequency can be

calculated.

Figure 2 shows the time series

again, along with the amplitude

and phase spectra. In this exam-

ple, the amplitude spectrum shows

that the signal is relatively strongnear 40 Hz, although a notch

appears at exactly 40 Hz. Never-

theless, there is generally a peak in

the amplitude spectrum near

40 Hz and another at about 65 Hz.

That is valuable information, but

Elements of Seismic Dispersion

2 Society of Exploration Geophysicists

0a) b) c)

0.2

0.4

0.6

0.8

1.0

Time(s)

0

Trace

0

20

40

60

80

Fre

quency(Hz)

0

FT real

0

20

40

60

80

0

FT imaginary

Figure 1.(a) Time series along with the (b) real and (c)

imaginary parts of its Fourier transform.

0

0.2

0.4

0.6

0.8

1.0

Tim

e(s)

0a) b) c)

Trace

0

20

40

60

80

Freque

ncy(Hz)

FT amplitude

0

20

40

60

80

180 0 180

FT phase

Figure 2.(a) The time series of Figure 1 with its (b)

Fourier amplitude and (c) phase spectra.


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with no indication whether the 65-Hz energy

is distributed evenly throughout the time

series or resides in only part of it.

A signal generated by summing four

cosines can appear to be quite irregular(Figure 3), but the FT amplitude spectrum

precisely identifies the constituent frequen-

height, indicating the relative time-domain

amplitude of each cosine that went into

building the time series. Therefore, the spec-

tral amplitude has information; the locations

of peaks in the amplitude spectrum tell us

the frequencies involved. We can think of

the complicated time series as having been

generated by four spikes in the Fourierdomain. Conversely, four spikes in the time domain would yield a complicated Fourier

amplitude and phase spectrum resulting from interference. That exemplifies the well-

known Fourier time-frequency symmetry or duality.

Delta function input

It is useful to consider the ultimate nonstationary time series, an impulse (or

delta function) that exists only at a single time point (Liner, 2010). Because the Four-

ier transform decomposes a transient time signal into independent harmonic com-

ponents, the functiong() would seem to provide exact frequency information but

no time localization. In other words, the FT can precisely detect which frequenciesreside in the data but yields no information about the time position of signal fea tures.

We can investigate that claim by considering the FT of a delta function.

Our first approach is graphical. Consider a single spike in the time domain slightly

after zero time (Figure 4a). The FT results (Figure 4b through 4d) show real and imagi-

nary, amplitude, and phase, respectively. The amplitude spectrum (Figure 4c) is flat and

contains no information about time location of the spike. The key is the phase spec-

trum (Figure 4d), which is seen to have a constant slope; the time-localization informa-

tion resides in that slope, the derivative of the phase with respect to the frequency.

Conceptually, the value of the slope is related to the time location of the spike. If a time

series were to consist of a single spike at any time, we could find out where it lives by

measuring the slope of the phase.To quantify that concept, it is convenient to introduce the Dirac delta function (or

unit impulse) heuristically as

t tt t

t t-( )=

=

00

0

1

0,

(3)



0a) b)

0.2

0.4

0.6

0.8

1.0

Time

(s)

0

Trace

0

20

40

60

80

Frequen

cy(Hz)

FT amplitude

Figure 3.(a) Time series composed of

summing four cosine waves of constantfrequency for all time. (b) The amplitudespectrum easily identifies the input fre-

quencies.

Delta-function input


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where the time point t0is termed the support of the delta function, i.e., the delta func-

tion is zero except at the infinitesimal time point t0. A key feature of a deltafunction is

the sifting property it exhibits under the action of integration,

t t g t dt g t -( ) ( ) = ( )-

0 0 .

(4)

To find the Fourier transform of a delta function, letg(t) =(t t0), write the FT integral,

and apply the sifting property to yield

g t t e dt ei t i t ( )= -( ) = 0 0 . (5)

Recognizing that the result is already in polar formAei, where (A, ) are amplitude

and phase, it is clear that the amplitude spectrum,A=1, contains no information about

the time location of the spike. The phase, however,

= =t ft0 02 , (6)

is linear, and the spike location, t0, is encoded in the phase slope. For this elementary

case, the spike location can be recovered exactly by

t

ddf0

1

2=

.

(7)



0.0 0.1 0.2 0.3 0.41.0

0.5

0.0

0.5

1.0

Time (s)

Amplitude

Time function

0 10 20 30 40 50 600.10

0.05

0.00

0.05

0.10

Frequency (Hz)

Amplitude

Fourier real and imaginary

0 10 20 30 40 50 600.0

0.5

1.0

1.5

2.0

Frequency (Hz)

Amp

litude

Amplitude spectrum

0 10 20 30 40 50 60180

90

0

90

180

Frequency (Hz)

Degrees

Phase spectrum

a) c)

b) d)

Figure 4.Fourier transform of a delta function not at zero time. (a) Time series formed by a singleimpulse slightly away from zero time. (b) Real and imaginary parts of the Fourtier transform. (c) FTamplitude spectrum, which is constant for all frequencies. (d) Phase spectrum, whose slope encodes

the spike-time information.


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Thus, time-localization information is found in the Fourier transform, but it is

encoded in the phase. The temptation is to say, Great, now I can nd my spike, and

everything is done. Not so quick!

When the signal consists of two spikes (Figure 5), the phase spectrum (Figure 5d)

becomes more complicated. Careful examination of this example reveals that the phaseslope is the same as in the single-spike case, but that is because the two spikes are cen-

tered on the time of the original spike. That fact leads to the conclusion that for two

spikes of equal amplitude, the phase slope tells us the center-point time between them.

In addition, the amplitude spectrum now provides some information. Specifically, it has

developed a notch (or zero value). We will see later that a relationship exists between

notch frequency and time delay between spikes.

In summary, for a signal with two spikes of equal amplitude, phase slope can iden-

tify the center time of the spikes, and the amplitude notch tells us the time between

spikes. Here, the time-frequency information resides in a combined analysis of ampli-

tude and phase spectra.

There is an interesting aside here. The Fourier transform is linear, meaning that ifwe add two time signals and then do the transform, we obtain the same result as if we

had done the transforms separately and then added results in the frequency domain.

However, think about the two-spike case. Each spike as a separate time series will give

an amplitude spectrum that is flat, and adding the flat spectra obviously yields a flat

spectrum. Nevertheless, we have seen that the amplitude spectrum of a two-spike

signal is not flat but has a notch. Is there a paradox here? No. Linearity of the Fourier



0.0 0.1 0.2 0.3 0.41.0

0.5

0.0

0.5

1.0

Time (s)

Amplitude

Time function

0 10 20 30 40 50 600.2

0.1

0.0

0.1

0.2

Frequency (Hz)

Am

plitude

Fourier real and imaginary

0 10 20 30 40 50 600.0

0.5

1.0

1.5

2.0

Frequency (Hz)

Amplitude

Amplitude spectrum

0 10 20 30 40 50 60180

90

0

90

180

Frequency (Hz)

De

gress

Phase spectrum

a) c)

b) d)

Figure 5.Fourier transform of two spikes. (a) Time series formed by two impulses centered on thetime of the spike shown in Figure 4a. (b) Real and imaginary parts of the Fourier transform. (c) FTamplitude spectrum with notch. (d) Discontinuous phase spectrum.


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transform applies to the real and imaginary parts of the frequency-domain representa-

tion. Once we convert to amplitude and phase, linearity is lost, i.e., summing two

time series will give a different amplitude spectrum than summing the individual

amplitude spectra.

The notch phenomenon for a time-domain spike pair is considered mathematicallyin Appendix A. The spikes considered here have the same sign and amplitude, so from

equation A-9 of Appendix A, we can write

f

nt

nn = +

D =

1 2

2 0 1 2, , , ,...,

(8)

where fnis the nth notch frequency. That relationship says that from observation of the

frequency associated with, say, the first spectral notch, f0, the time separation of the

spikes can be calculated from

Dt t t f= - =2 1

0

1

2.

(9)

That accomplishment is a muted victory because we do not find the absolute spike times,

only the delay between them.

Can this approach be extended to more complicated signals? First, recognize that

the notch in Figure 5 is a hard zero only because the spike amplitudes are equal. Other-

wise, a local minimum but not an actual zero would occur in the amplitude spectrum

(see Appendix A). Perhaps this minimum location, along with phase slope, would still

let us solve the problem. But what about three spikes or spikes with opposite polarities

or differing amplitudes? What about 20 spikes? Clearly, this path of investigation is

limited. I want to make it clear that time-localization information indeed does exist inthe Fourier transform, but it is extremely hard to unravel. Going to more complicated

signals, this approach becomes unworkable.

We conclude that although the Fourier transform contains some time-localization

information, extracting it becomes unreasonable quickly, for even simple impulsive

time functions. Something better is needed, and that something is extension from the

frequency domain to the time-frequency domain.

Continuous wavelet transform

We will discuss primarily the continuous wavelet transform while recognizing that

it is only one of a broad class of transform methods that deliver a time-frequency (TF)spectrum. The short-time Fourier transform was the basis of early work in seismic spec-

tral decomposition (Gridley and Partyka, 1997), in which broadband seismic data are

decomposed into narrow or single-frequency bands for display and interpretation, a

method particularly suited to modern 3D seismic data. The Gabor transform, similar to

the STFT, is widely available for seismic use as part of the Seismic Un*x open software

package (Cohen and Stockwell, 2010). Matched-pursuit decomposition (MPD) also has

been used in seismic applications (Chakraborty and Okaya, 1994).




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Short-time Fourier transform (STFT)

The short-time Fourier transform is particularly useful, so let us explain how it

works. First, it might seem like a good idea to chop up the input time series into short

signals and do a Fourier transform of each one. The frequency sampling rate in a digital

Fourier transform is given by 1/(2tmax), where tmaxis the duration of the signal. For

example, if the time series is chopped into 100-ms pieces, the frequency sample interval

will be 5 Hz. Clearly, that approach will give poor frequency resolution.

A better approach is described by Chakraborty and Okaya (1994). The input time

series is time-segmented by application of a bank of Gaussian lters,

g t g e e dc t i( , ) ( ) ,( ) = - -

-

2

(10)

where the argument list again is used to denote the domain forg, defines the center

time of the Gaussian window, and cdetermines the Gaussian width (c= for conven-tional STFT). Choice of ceffectively controls the time-frequency resolution of the STFT.

For a given lter, the Gaussian window preserves data values at the center and

tapers away on either side. The signal is still the same length as the original but is

nearly zero away from the current filter location. Taking the FT of this filtered signal

creates a spectrum that is associated with the central time of the window, thus forming

one row in a time-frequency space. Progressing through a series of filters generates the

complete TF space with frequency sampling given by the original signal length. A trade-

off between filter length and frequency resolution remains, but it is not as dramatic as

simply chopping up the time series.

The STFT is theoretically invertible for the case c=, with reconstruction of the

time series by summation over frequency

g t g t e di t( ) ( , ) .= -

-

(11)

Introduction to the continuous wavelet transform (CWT)

The continuous wavelet transform, using a complex Morlet-analyzing wavelet, has

a close connection to the Fourier transform and is a powerful analysis tool for decom-

posing broadband wavefield data. Since the 1980s, wavelet transforms have been applied

to a wide range of seismic problems.The CWT is one method of investigating the time-frequency details of data whose

spectral content varies with time (nonstationary time series). Motivation for the CWT,

along with a discussion of its relationship to the Fourier and Gabor transforms, can be

found in Goupillaud et al. (1984a).

As a brief overview, we note that French geophysicist Jean Morlet worked with

nonstationary time series in the late 1970s to find an alternative to the short-time

Fourier transform. The STFT was known to have poor localization in both time and




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frequency, although it was a first step beyond the standard Fourier transform in the

analysis of such data. Morlets original wavelet-transform idea was developed in collabo-

ration with theoretical physicist Alex Grossmann, whose contributions included an

exact inversion formula. A series of fundamental papers flowed from this collaboration

(Grossmann and Morlet, 1984; Goupillaud et al., 1984a, 1984b). Connections soon wererecognized between Morlets wavelet transform and earlier methods. For further details,

the interested reader is referred to Daubechies account of the early history of the wave-

let transform (Daubechies, 1996).

Here we describe implementation details of the CWT as applied to wavefield mea-

surements, using reflection-seismic data as a specific example. A classic paper in the

general field of time-frequency decomposition of reflection-seismic data is Chakraborty

and Okaya (1995), which discusses the short-time Fourier transform, continuous wavelet

transform, discrete wavelet transform, and matched-pursuit decomposition. However,

application to field seismic data is limited to the matched-pursuit method.

The CWT implementation described here uses a complex Morlet analyzing wavelet.

As a time-domain Gaussian-tapered complex exponential, this particular wavelet has anatural and compelling connection to the venerable Fourier transform. An inventory

of several other analyzing wavelets can be found in the excellent summary paper of

Torrence and Compo (1998).

Passage to the continuous wavelet transform

To move toward the wavelet transform, let us take a different view of the Fourier

transform. Think of the FT integral as crosscorrelating the input signal against a series

of single-frequency waveforms (real cosines and imaginary sines). For example, at 8 Hz,

the FT integral tells us to correlate (slide, multiply, and sum) the input signal against an

8-Hz sine/cosine wave. If the correlation yields a large number, it implies similarity

between the two, and the signal is said to have a large, 8-Hz content. Evaluation of theFourier integral is in effect a pattern-recognition process the more the input signal

resembles the analyzing wave-

form, the larger the pattern-indi-

cator output (Fourier coefficient).

An important aspect of the

digital FT is linear frequency

sampling. In the frequency

domain, the values are spaced

df=1/(2tmax) apart. Morlet et al.

(1982a, 1982b) found advantages

to frequency sampling based on

powers of 2, termed dyadic sam-

pling. This has a natural connec-

tion to octaves in music.

Figure 6a shows a time series

consisting of a single spike, and

Figure 6b shows a 2D time-fre-

quency panel. How is this panel



0a) b)

0.2

0.4

0.6

0.8

Time(s)

0

Data

0

0.2

0.4

0.6

0.8

0 10 20 30 40 50 60 70 80

Frequency (Hz)

FT integrand (real)

Figure 6.Fourier-transform time-frequency impulse

response. (a) Time series consisting of a spike at 0.5 s.(b) Real part of the FT integrand, the time-frequencyfunction g(t)eit.


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made? It is simply a plot of the quantity inside the FT integral (the integrand), represent-

ing the input signal multiplied by the FT kernel

g t g t ei t, ( ) = ( ) .

(12)

As equation 12 shows, this is a complex function of time and frequency (only the real

part is shown in the figure). Any given frequency, say, 40 Hz, is a perfect cosine wave.

The important concept here is complete isolation of frequencies in the FT kernel, a

property that will be lost in the continuous wavelet transform.

Does this plot really contain any time-frequency information? Visually, a pattern is

seen centered on the time location of the spike. However, what we would like to see on

the TF plot is a display that is zero away from the time location of the spike, not a col-

lection of infinite cosines filling the space. Therefore, information about the spike loca-

tion is present but is hidden in the phase, as discussed earlier.

Nonlinear frequency sampling relates to the concept of scales. Nonlinear (or dy-

adic) sampling is described as a certain number of octaves, each containing a certainnumber of voices. The octave and voice indices are used to compute nonlinear scale

values as well as an integer-scale number index. Mathematical details are given later.

This concept can be used to remap the frequency axis of the TF plot in Figure 6b

from frequency to scale number, as shown in Figure 7. There is no new information

here, but the change to dyadic

frequency sampling alters the look

of things. Because it is an inverse

relationship between scale index

and frequency, low frequency now

shows up on the right. There is a

unique relationship between scalenumber and frequency for the

Fourier integrand. For example,

scale 20 equates to about 33 Hz;

an input cosine with that single

frequency would show up on a

single scale.

How do we make the passage

to the continuous wavelet trans-

form? The FT has sine- and cosine-

analyzing waveforms. The CWT

uses some other kind of waveform

as a basis for analysis. The key is

that for a CWT, the analyzing

waveform does not extend indefi-

nitely but has only a short life.

Many types of wavelets have been

studied; we will concentrate on one

called a complex Morlet wavelet.



0a) b)

0.2

0.4

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Time(s)

0

Data

0

0.2

0.4

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0.8

10 20 30 40 50

FT integrand (real)

c)

0

40

80

120Frequenc

y(Hz)

10 20 30 40 50

Scale number

Scale-frequency plot

Figure 7.Fourier-transform time-frequency impulseresponse. (a) Time series consisting of a spike at 0.5 s.

(b) Real part of the FT integrand, the time-frequencyfunction g(t)eit, now plotted with dyadic frequencysampling. (c) Crossplot of frequency and scale number.


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Another important point with CWT is the dyadic frequency sampling. For a time

series several thousand points long, the Fourier transform contains an equal number of

frequencies, whereas dyadic sampling of the CWT typically will contain only a few

dozen scales. This sparse representation in the transform domain is possible and ade-

quate because each CWT scale contains a band of frequencies, and the bands overlapfrom one scale to the next. Because the bands overlap, there is no such thing as a 20-Hz

wavelet. The term X Hz for a wavelet transform implies a band of frequencies on

either side of X.

Figure 8 will aid us in making the transition from Fourier to wavelet transforms.

Figure 8a shows a 30-Hz complex exponential (solid line =real; dashed line =imaginary)

time function. This Fourier wavelet, if we can call it that, is given by

t ei f t( ) = 2 0 ,

(13)

where f0is now a parameter and tis time. Taking a Fourier transform of this input signal

(Figure 8b) results in one frequency-domain spike at 30 Hz. Figure 8c shows the same

situation with a complex Morelet wavelet that consists of the Fourier wavelet multiplied

by a Gaussian envelope,

t e et c i f t( ) = -( )/ .

2

02

(14)



0.04 0.02 0.00 0.02 0.041.0

0.5

0.0

0.5

1.0

Time (s)

A

mplitude

30-Hz Fourier wavelet

0 20 40 60 80 100 120

0.0

0.2

0.4

0.6

0.8

1.0

Frequency (Hz)

Amplitude

Amplitude spectrum

0.04 0.02 0.00 0.02 0.041.0

0.5

0.0

0.5

1.0

Time (s)

A

mplitude

30-Hz Morlet wavelet

0 20 40 60 80 100 120

0.0

0.2

0.4

0.6

0.8

1.0

Frequency (Hz)

Amplitude

Amplitude spectrum

a)

b)

c)

d)

Figure 8.Fourier and Morlet wavelet comparison. (a) Time-domain Fourier wavelet consisting of a30-Hz real cosine (solid line) and imaginary sine (dashed line). (b) Fourier amplitude spectrumof the time series shown in part (a). Note that only one frequency (30 Hz) is present. (c) Morlet

wavelet (real and imaginary) with 30-Hz center frequency. (d) The amplitude spectrum of part (c)shows that the Morlet wavelet consists of a Gaussian band of frequencies centered on 30 Hz.


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See the difference? The Morlet wavelet does not live forever; it has compact time

support. The only difference, as seen from equation 14, is a Gaussian damping envelope

defined by a new parameter, c. That parameter determines the width, in the time

domain, of the Gaussian damping envelope that multiplies the cosine and sine waves

inside it. If the damping factor cincreases, there are more oscillations in the wavelet.Thus, time resolution is decreased, and frequency resolution is increased. That is a vital

point to understand. When the Gaussian envelope tightens up, time resolution is

increased. That action also lowers frequency resolution because fewer wavelet oscilla-

tions can be used to determine frequencies in the signal.

Conversely, when the envelope widens, time resolution is lowered, and frequency

resolution is increased. In practice, cis a tuning parameter that lets us customize the

CWT, depending on the application.

Notice in the amplitude spectrum that when we mention a 30-Hz Morlet wavelet,

30 refers to the peak or central frequency, but the wavelet contains significant energy at

lower and higher frequencies. Specically, the Morlet wavelet spectrum is Gaussian

because the Fourier transform of a Gaussian function (the Morlet time envelope) is alsoGaussian (Bracewell, 1965). That means that, say, a 31-Hz Morlet wavelet would have

significant overlap with a 30-Hz wavelet. Thus, the dyadic sampling scheme is used

because dense linear frequency sampling is unnecessary to characterize or reconstruct

the signal.

Figure 9a shows a 33.5-Hz time signal that represents a dyadic scale index of 20.

The Fourier wavelet time-frequency plot (Figure 9b) is basically zero everywhere except

at 33.5 Hz, showing that this wavelet has perfect frequency resolution. The Morlet CWT

result for the same case (Figure 9c) shows a frequency band centered around scale index

20 (33.5 Hz). It could be argued that this is not an improvement over the FT result

because the information about frequency of the input signal is smeared across the

Morlet-wavelet Gaussian frequency response. That is true, but we are thinking ahead tosituations in which the frequency content changes with time. In that case, as we have



0

0.2

0.4

0.6

0.8

1.0

10 20 30 40 50Scale index

Fourier (real)

0

a) b) c)

0.2

0.4

0.6

0.8

1.0

Time(s)

0

Data

0

0.2

0.4

0.6

0.8

1.0

10 20 30 40 50Scale index

Morlet (real)

Figure 9.Comparison of Fourier and Morlet wavelets. (a) 33.5-Hz time signal. (b) Real part ofFourier time-frequency representation with dyadic frequency sampling. (c) Real part of Morlet

wavelet transform showing band of frequencies associated with one scale.


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seen previously, the FT will

give us no time-dependent

information.

In Figure 10, we again

see an input signal com-posed of a single spike and,

in the middle, the FT inte-

grand (real part) in dyadic

frequency sampling. As

discussed earlier, each scale

uniquely relates to a fre-

quency value, so each

column in this plot is an

infinite cosine of fixed

frequency. Figure 10c

shows the CWT response tothe same input. Notice that

the CWT result is localized

in time and centered on

the spike. Moving across

the image horizontally at

the spike time confirms

that it contains all frequen-

cies.

As the Morlet-wavelet

damping parameter cis

increased (Figure 11), thetime-domain Gaussian

envelope widens, thus

decreasing time localiza-

tion. In addition, by allow-

ing more oscillations to

exist in the wavelet, the

frequency resolution

thereby is increased, as

mentioned earlier.

The top graphs in

Figure 11b and 11c showthe maximum amplitude

at each scale and indicate

that the amplitudes are

constant. In the CWT

definition, there is typi-

cally a scale factor of the



0

0.2

0.4

0.6

0.8

10 20 30 40 50

Scale index

Fourier (real)

0

a) b) c)

0.2

0.4

0.6

0.8

Time(s)

0

Data

0

0.2

0.4

0.6

0.8

10 20 30 40 50

Scale index

Morlet (real)

Figure 10.Fourier and Morlet-wavelet impulse response. (a)Input time signal that has a single impulse. (b) Real part of Fou-

rier time-frequency representation with dyadic frequency sam-

pling. (c) Real part of Morlet-wavelet transform showing timelocation and spectrum of the impulse.

0

a) b) c)

0.2

0.4

0.6

0.8

Time(s)

0

Data

0

0.2

0.4

0.6

0.8

10 20 30 40 50

Scale index

CWT real (c= .004)

10 20 30 40 500

1

2

Maximum amplitude

0

0.2

0.4

0.6

0.8

10 20 30 40 50

Scale index

CWT real (c= .008)

10 20 30 40 500

1

2

Maximum amplitude

Figure 11.Impulse response of Morlet-wavelet CWT. (a) Inputdata consisting of zero values and one unit-amplitude spike.

There are 250 time samples, and the time-sample interval isdt=.004 s. (b) Real part of the CWT of the data using c=0.004

and p=0. This is a five-octave, 10-voice CWT with maximumfrequency of 125 Hz (Nyquist). The scale axis is associated withcentral frequency of the Morlet wavelet through Figure 14 below.

(c) Real part of the impulse response using c=.008 and p=0.In each case, the upper plot is maximum value at each scale

index, showing constant peak amplitude.


13/22

form ap, where ais the scale value andpis a constant (see equation 15 below). Choos-

ingp=0 will result in the flat impulse response across scales, as shown. That would be

appropriate for some kinds of spectral decomposition in which the goal is to visualize

all frequency components equally.

Clearly, however, this choice ofpdoes not conserve energy or power between theinput and TF response, i.e., the energy (in a squared-amplitude sense) in the TF plane is

much greater than the energy in the spike. Reconstructing the signal with this choice of

p-value would introduce much more low-frequency energy than existed in the original.

Energy (and reconstruction) can be conserved in this sense by choosingp=, as is

done in quantitative applications such as singularity analysis. A full discussion of CWT

normalization can be found in Torrence and Compo (1998).

A more complex case, in Figure 12, shows an input signal that begins at 10 Hz

and ends 1 s later at 60 Hz with a short taper on each end. This is a linear sweep signal

such as is used in vibroseis (although in that case, the signal would be more like 10 s

long). A Fourier transform would show only that the signal has energy between 10

and 60 Hz (scales 35 and 10). The Fourier spectrum would show nothing about howthose frequencies are distributed over time, meaning that there is no time-localization

information. The CWT result clearly shows the change of frequency content over

time. By adjusting the damping parameter c, we can fine-tune this plot, as shown in

Figure 12c.

CWT normalization and reconstruction

For readers who are less interested in mathematics, this section can be skipped. For

others, let us give a bit of detail. The continuous wavelet transform can be defined in a



0

0.2

0.4

0.6

0.8

10 20 30 40 50

Scale index

CWT (c= .004)

0

a) b) c)

0.2

0.4

0.6

0.8

Time(s)

0

Data

0

0.2

0.4

0.6

0.8

10 20 30 40 50

Scale index

CWT (c= .012)

Figure 12.CWT Morlet comparison for a linear sweep. (a) Input 1-s time signal with linear fre-quency change from 10 to 60 Hz (scale index 35 to 10). (b) Amplitude spectrum of Morlet wavelet

(c=.004). This spectrum contains both time and frequency resolution. The TF signal is linear butappears to be nonlinear because of dyadic sampling on the scale axis. (c) A damping value of

c=.012 gives better TF resolution.


14/22

variety of ways that differ with respect to normalization constants and conjugation. We

define the transform as

g a b a g t t b

a

dtp, ,( )= ( ) -

-

(15)

where (t) is the analyzing wavelet, bis a timelike translation variable, ais a dimension-

less frequency-scale variable, andpis a real normalization parameter. As with the FT,

the argument indicates the physical domain,g(t), or the transform domain,g(a,b). This

convention is used routinely in geophysics when multidimensional transforms often are

encountered (Claerbout, 1985; Liner, 2004).

For reconstruction of the original time series, the inverse CWT is given in principle

by the double integral

g t g a b t b

a

da db( )= ( ) -

-

-

, ,

(16)

where the inverse analyzing wavelet need not be the same as the forward transform

wavelet. As discussed by Torrence and Compo (1998), if the inverse wavelet is chosen

to be a delta function, the bintegral can be done analytically via the sifting property

of the delta function. The inverse then is simplified to the real part of the summation

over scales

g t Re g a t da( )= ( )

-

, .

(17)

Complex Morlet wavelet

As discussed earlier, the Fourier-transform kernel is given by

K eFTi ft

=2

, (18)

where i = -1and fis frequency in Hertz. The kernel in a continuous wavelet transform

is simply the analyzing wavelet,

K t

CWT = ( ) .

(19)

For wavelike data, a good choice for the analyzing wavelet is the complex Morletwavelet

t e et c i f t( ) = -( )/

2

02 ,

(20)

where tis time, f0is a frequency parameter, and cis a damping parameter with units of

time. This equation describes a time-domain function that is the product of a Gaussian

and a complex exponential whose center frequency is f0.




15/22

The frequency-domain representation of the complex Morlet wavelet is

f e f f

cf( ) = -( )-( )2

0* , (21)

where normalization factors have been omitted and * denotes convolution. This is again

a Gaussian function, now centered in the frequency domain on f0.

Any of several choices can be made with respect to wavelet normalization. Our

definition, equation 20 along withp=0 in equation 15, normalizes the time-domain

peak amplitude.

Comparing the leading terms in equations 20 and 21, we see a duality that ex-

presses the characteristic CWT resolution trade-off. The damping parameter, c, con-

trols the rate at which the time-domain wavelet and frequency-domain spectrum are

driven toward zero away from their respective peaks. As time localization increases

(small c), frequency localization decreases, and vice versa. The complex Morlet wavelet

has been written in this particular form and notation to emphasize the central role of

the damping factor. The value of this parameter has a first-order influence on any CWTresult. In the limit as c, e.g., the Morlet wavelet defined here becomes equal to the

Fourier kernel (e.g., no time localization).

Figure 13 illustrates the time-frequency resolution properties of the complex Morlet

wavelet. In this example, a 60-Hz Morlet wavelet (Figure 13a)is shown with the real



0.04 0.02 0.00 0.02 0.041.0

0.5

0.0

0.5

1.0a) c)

d)b)

Time (s)

Amplitude


0 20 40 60 80 100 120

0.2

0.4

0.6

0.8

1.0

Frequency (Hz)

Amplitude

Fourier spectrum

0.04 0.02 0.00 0.02 0.041.0

0.5

0.0

0.5

1.0

Time (s)

Amplitude


0 20 40 60 80 100 1200.0

0.2

0.4

0.6

0.8

1.0

Frequency (Hz)

Amplitude

Fourier spectrum

Figure 13.Complex Morlet wavelet in the time and frequency domains. (a) 60-Hz Morlet waveletshowing real (solid line) and imaginary (dashed line) parts. The damping parameter is c=1120.

(b) Fourier amplitude spectrum of the 60-Hz Morlet wavelet showing characteristic Gaussianshape. (c) A 30-Hz Morlet wavelet (c=160) has longer time duration but the same number ofoscillations. (d) The amplitude spectrum of the 30-Hz wavelet is narrower because the time-

domain function is wider.


16/22

part as a solid line and the imaginary part as a dashed line. The damping parameter is

c= 1120, resulting in a Gaussian-tapered amplitude spectrum (Figure 13b). Lowering

the center frequency to 30 Hz and using c= 160 gives a wider time-domain wavelet

(Figure 13c) and a narrower frequency spectrum (Figure 13d). Those results are not

dependent on the absolute size of the frequencies being considered; similar plots couldbe constructed for 60 and 30 MHz.

Note the relationship of the damping parameter in Figure 13 to center frequency of

the wavelet c=1/(2f0). Using this value will preserve a wavelet time span equal to twice

the wavelet period.

When used to compute the CWT, the argument of the analyzing wavelet (equa-

tion 20) is translated and scaled, t b

a

-

. The effect of scaling is twofold. In the real

exponential, the scale value multiplies the damping parameter, effectively broadening

the time-domain extent of the wavelet. In the complex exponential, the scale divides

the frequency, lowering it precisely enough to maintain the number of time-domain

oscillations.

Scales and frequencies

The continuous wavelet transform represented by equation 15 is related to convolu-

tion, as evidenced by the appearance of (tb) in the integral. Thus, the CWT can be

written as

g a t g t t a, * / ,( ) = ( ) ( ) (22)

showing that the CWT can be implemented as a series of complex-valued time-domain

convolutions. Each convolution involves the input data and a version of the analyzing

wavelet modified by the scaling variable. Strictly speaking, the CWT is an inner prod-uct, or the zero lag of a complex-valued correlation, which can be expressed as a con-

volution under certain symmetry conditions. The CWT as a crosscorrelation is under-

standable because we are matching similarities between the signal and the analyzing

wavelet.

From a computational point of view, complex time-domain convolutions are

highly inefficient. The mathematical form of equation 22 suggests implementation in

the Fourier domain, where the convolution will become multiplication. Recalling the

Fourier-transform scaling property for a general time functiong(t), i.e.,

FT g t a ag af/ ,( ){ } = ( )

(23)

equation 22 can be written in the Fourier domain as the compact and efficient result

g a t FT ag f af, .( ) = ( ) ( ){ }-1

(24)

A fundamental contribution of Morlets early work (Morlet et al., 1982a) was

recognition that the natural sampling of this scale variable is dyadic (i.e., logarithmic,




17/22

base 2). If the wavelet is dilated by a factor of 2, this means that the frequency content

has been shifted by one octave. The analogy with music and singing is clear and is

perpetuated by defining the scale range in terms of octaves and voices. The number of

octaves determines the span of frequencies being analyzed, whereas the number of

voices per octave determines the number of samples (scales) across the span.Specifically, let the number of octaves in a CWT be NOand the number of voices

per octave be NV. The octave and voice indices progress as

iO O1, , , ,= -0 2 1 N (25)

i NV V .= -, , , ,0 1 2 1 (26)

The scale value for a given octave iOand voice iVis given by

a i i N

= +( )

2 O V V/

, (27)

and it follows that the smallest scale value is 1 and the largest is

a N

=2 .O V1/N -( )

(28)

The scales are referenced to an index that progresses as

i N Na =1 2, , , O V , (29)

and that index can be calculated with the scales themselves, before any convolutions

are performed, by the double loop

ia = 1

for io = 0, No 1 {

for iv = 0, Nv 1{

-

-

a(ia) = 2^(io+iv/Nv)

ia = ia + 1

}

}}.

The appropriate range of octaves and scales depends on the spectral content of the

data and the highest requested frequency in the CWT. Here, the highest CWT fre-quency always will be taken as Nyquist frequency. That is a safe choice, but it has some

minor inefficiency because data usually are sampled in such a way that Nyquist is well

above the highest frequency expected in the signal.

We now have the components necessary to relate scales and frequencies. Consider

a CWT consisting of five octaves and 10 voices per octave. Figure 14a shows a plot of

scale index versus scale value for this case. The scale values range from 1 (ia=1) to

29.86 (ia=50). To associate a frequency with each scale, it is necessary to specify the




18/22

maximum frequency in the CWT transform, which is also the initial frequency of the

analyzing wavelet. Lower frequencies are generated when the initial frequency isdivided by successive scale factors.

Using a typical seismic example of dt=.004 s, the Nyquist frequency is 125 Hz, and

the scale values map into frequencies, as shown in Figure 14b. This also can be seen

(inverted) in Figure 7b. This scale-frequency relationship will be retained in the exam-

ples that follow. It always must be remembered that a CWT scale does not correspond to

a single Fourier frequency but rather to a Gaussian spectrum peaked at the Morlet cen-

tral frequency.

By choosing five octaves descending from a Nyquist frequency of 125 Hz, the low-

est frequency in the CWT is 125/29.86 =4.2 Hz. Typical acquisition procedures for

petroleum seismic data do not preserve frequencies below 5 Hz, so we conclude that a

five-octave transform should span the spectral content of such data adequately.One last implementation detail is specification of the damping parameter, c, dis-

cussed earlier in relation to the Morlet wavelet definition (equation 20). A version of the

CWT described here is available as succwt(source code succwt.c) in the Seismic Un*x

processing system (Cohen and Stockwell, 2010).

A sense of the broad range of seismic topics amenable to wavelet-transform analysis

can be gained from Table 1, which shows published applications along with authors

names and years. Only the first occurrence of any particular application is reported.

Any such compilation is necessarily limited. Significant workers in the field might

not be represented if they worked in fundamental rather than applied areas of the subject.

In addition, keyword searches are fallible a search for wavelet transformmight not catch

a title containing wavelet-packet transform. Nevertheless, Table 1 represents the range andprogress of wavelet-transform applications to reflection-seismology data fairly well.

Examples

Figure 15 illustrates the kind of result CWT produces from reflection-seismic data.

The data (Figure 15a) consist of a marine 2D seismic section from offshore Thailand with

200 migrated traces. Each trace is 2 s long with a time-sample interval of dt=.004 s. The



0 10 20 30 40 500

5

10

15

20

25

30

a) b)

Scale index

Scale

value

Scale values

0 10 20 30 40 500

20

4060

80

100

120

Scale index

Centralfre

quency

(Hz

)

Frequency values

Figure 14.Relationship between scale and frequency values. (a) Plot of scale index versusscale value for a five-octave, 10-voice continuous wavelet transform. (b) By assuming that the

highest frequency in the transform is Nyquist (125 Hz in this case), each scale can be associatedwith a particular frequency.


19/22



Table 1.Some published applications of the wavelet transform to petroleum seismic data.1Citations are listed in the references section of this book.

Application Author(s) and year

Downward continuation LeBras et al., 1992

Shear-wave discrimination Niitsuma et al., 1993

Seismic data processing Miao and Moon, 1994

Wavefield extrapolation Dessing and Wapenaar, 1994

Wave-equation inversion Yang and Qian, 1994

Data compression Reiter and Heller, 1994

Trace interpolation Wang and Li, 1994

Phase quality control Asfirane et al., 1995

Migration Dessing and Wapenaar, 1995

Tomography Li and Ulrych, 1995

Sonic-velocity characterization Li and Haury, 1995

Compressed migration Wang and Pann, 1996

Hilbert attribute analysis Li and Ulrych, 1996

Singularity analysis Cao, 1996

Signal-to-noise ratio and resolution Li et al., 1996

Deconvolution Marenco and Madisetti, 1996

Edge detection Dessing et al., 1996

Velocity filtering Deighan and Watts, 1997

Reflection tomography Carrion, 1997

Borehole data upscaling Verhelst and Berkout, 1997Spectral decomposition Gridley and Partyka, 1997

Amplitude variation with offset Wapenaar, 1998

Reflector characterization Goudswaard and Wapenaar, 1998

3D seismic sequence analysis Yin et al., 1998

Thin-bed analysis Zhu and Li, 1999

Complex media characterization Pivot et al., 1999

Direct hydrocarbon detection Sun et al., 2002

Reservoir characterization Osrio et al., 2003

Time-frequency attribute Sinha et al., 2003SPICE attribute Smythe et al., 2004

Group velocity imaging Liner et al., 2009

1Information in this table is derived from the Digital Cumulative Index maintained by the Societyof Exploration Geophysicists (SEG), which includes publications of SEG, the Canadian Society ofExploration Geophysicists, the Australian Society of Exploration Geophysicists, and the EuropeanAssociation of Geoscientists and Engineers, http//library.seg.org/journals/doc/SEGLIB-home/dci/searchDCI.jsp.

-


20/22

seafloor is clearly visible, along with subsurface geologic features including faults and

stratigraphic terminations. A five-octave, 10-voice CWT is computed from the center

trace, and the amplitude spectrum is displayed for c=.004 (Figure 15b) and c=.012 (Fig-

ure 15c). The scale axis corresponds to central frequency in accordance with Figure 14.

As discussed earlier, Nyquist has been chosen as the highest frequency in the trans-

form. Note the decay in amplitude beyond scale 45, indicating that our choice of fiveoctaves does indeed span the low-frequency content of the data. One immediate obser-

vation is the loss of bandwidth experienced by seismic waves that have traveled farther

through the earth. The seafloor reflection has significant energy from scales 45 to 8 (8

to 75 Hz), and the deepest reflectors are limited to scales 45 to 15 (8 to 50 Hz) because of

loss of high frequencies by scattering and attenuation processes (Liner, 2004). It follows

that the CWT is perhaps a natural tool for estimating subsurface attenuation.

Although loss of bandwidth can be seen, the appearance of this CWT result is influ-

enced strongly by the choice of damping parameter. With a value of c=0.004, the CWT

has maximum time localization but poor frequency resolution. Computing the CWT with

c=.012 allows bandwidth changes to be observed more easily, along with spectral notches

which develop by interference as more oscillations are allowed in the analyzing wavelet.By computing a CWT such as the one shown in Figure 15 for every trace in a 3D

migrated seismic volume, a 4D volume of data would be generated whose coordinates

are time, x-bin,y-bin, and scale. From that, a series of seismic 3D volumes could be

extracted, each corresponding to a unique center frequency. That is the concept of the

spectral-decomposition seismic attribute (Gridley and Partyka, 1997), although other

methods isolate individual frequencies better than the CWT does (Puryear and Cas-

tagna, 2008). In petroleum seismology, attributes are data types computed from primary



0.5

a) b) c)

1.0

1.5

2.0

2.5

Time(s)

20015010050

Trace

Migration data

0.5

1.0

1.5

2.0

2.5

10 20 30 40 50 10 20 30 40 50

Scale index

Trace 100 CWT

0.5

1.0

1.5

2.0

2.5

Scale index

Trace 100 CWT

Figure 15.Continuous wavelet transform of seismic data. (a) Migrated 2D seismic section.(b) CWT amplitude spectrum of center trace (c=0.004). (c) CWT amplitude spectrum of center

trace (c=0.012).


21/22

amplitude data to visually enhance

or isolate features of interest.

Figure 16 shows an example of

CWT spectral decomposition. The

broadband data (Figure 16a) consistof a small subset of the offshore

Thailand data. This section begins

just below the seafloor. From Fourier

analysis, we find that the frequency

content is 8 to 72 Hz, with a domi-

nant frequency of 40 Hz. Scale 15 of

the CWT spectrum in Figure 15

represents a narrow-band, 50-Hz

version of one input trace. Repeating

this for all input traces, a narrow-

band expression of the seismic linecan be generated, as shown in Figure

16b. Note that two intervals between

1.5 and 2.0 s show anomalous behav-

ior in the 50-Hz result. The value of

spectral decomposition is that when

calibrated to well control, such anom-

alies might provide information

about hydrocarbon reservoirs.

One of the great strengths of

the CWT is the vast body of theoreti-

cal work associated with it. It is morethan just another time-frequency

spectrum. Our second application

example, the SPICE attribute (spectral imaging of correlative events), draws on this

theoretical work by computing and displaying the slope of amplitude versus scale

(Smythe et al., 2004; Li and Liner, 2008). This gives an estimate of a quantity called the

Hlder exponent, which represents a way of classifying kinds of discontinuities in digi-

tal data, such as spikes, steps, and various kinds of ramps. In the seismic application,

those discontinuities occur in the acoustic-impedance earth model and are passed

through to the reflection-coefficient series and ultimately to the band-limited seismic

trace itself (Smythe et al., 2004).

To a skilled seismic interpreter, the data in Figure 16a tell a story of faulting,unconformities, and stratigraphy. In the mind of such an interpreter, a subsurface

model would evolve through long and detailed analysis of the seismic image. The

SPICE attribute (Figure 16c) represents an automated step toward that subsurface

model. Whereas the seismic image is a composite of many interfering thin-bed

reection events and amplitude oscillations, the SPICE result is a layered subsurface

model showing stratigraphic relationships that are difficult to infer from seismic-

amplitude data.



1.5

b)

c)

a)

2.0

Tim

e(s)

Distance (km)

Broadband migrated data

Narrow-band data (50 Hz)

SPICE attribute

5 6 7 8 9

1.5

2.0

Time(s)

1.5

2.0

Time(s)

Figure 16.Spectral decomposition and SPICE. (a)

Migrated 2D seismic section. (b) Narrow-band datawith center frequency of 50 Hz showing anomalous

zones (dark areas). (c) SPICE-attribute section com-puted from migrated seismic data shows subsurface

layering better than a conventional seismic section.


22/22

As a final note, consider the progress of wavelet transform and spectral-decom-

position methods as evidenced by SEG Technical Program Expanded Abstracts and

Geophysicspapers on those subjects (Figure 17). A group of early adoption papers is

seen peaking in 1996, followed later by a broader, accelerating body of work represent-

ing growing familiarity with the nature and power of time-frequency transforms.In summary, the continuous wavelet transform using a complex Morlet analyzing

wavelet has a close connection to the Fourier transform and is a powerful analysis tool

for decomposing broadband wavefield data. Care must be taken in choosing the range

of octaves in the transform, and particular attention must be paid to the Gaussian

damping parameter.

Applied to reflection-seismic data, the CWT can form the basis of spectral decom-

position (Gridley and Partyka, 1997) or more exotic attributes such as SPICE (Li and

Liner, 2008). A wide range of seismic-wavelet applications has been reported since the

early 1980s, and there is a trend related to development of advanced interpretation

tools. The rate of innovation likely will quicken as wavelet methods move into more

general use.


0

5

10

15

20

25

30

35

40

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

SEGp

apercount

Year

WT SD

Figure 17.Number of papers on spectral decomposition (SD) and wavelet transform (WT) pub-lished in SEG Technical Program Expanded Abstracts and GEOPHYSICSfrom 1992 through 2010.

chapter 1 time and frequency

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