The rock physics handbook, second edition

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The Rock Physics Handbook, Second Edition
Tools for Seismic Analysis of Porous Media

The science of rock physics addresses the relationships between geophysical
observations and the underlying physical properties of rocks, such as composition,
porosity, and pore fluid content. The Rock Physics Handbook distills a vast quantity
of background theory and laboratory results into a series of concise, self-contained
chapters, which can be quickly accessed by those seeking practical solutions to
problems in geophysical data interpretation.
In addition to the wide range of topics presented in the First Edition (including
wave propagation, effective media, elasticity, electrical properties, and pore fluid
flow and diffusion), this Second Edition also presents major new chapters on granular
material and velocity–porosity–clay models for clastic sediments. Other new and
expanded topics include anisotropic seismic signatures, nonlinear elasticity, wave
propagation in thin layers, borehole waves, models for fractured media, poroelastic
models, attenuation models, and cross-property relations between seismic and electrical
parameters. This new edition also provides an enhanced set of appendices with key
empirical results, data tables, and an atlas of reservoir rock properties expanded to
include carbonates, clays, and gas hydrates.
Supported by a website hosting MATLAB routines for implementing the various
rock physics formulas presented in the book, the Second Edition of The Rock Physics
Handbook is a vital resource for advanced students and university faculty, as well as
in-house geophysicists and engineers working in the petroleum industry. It will also
be of interest to practitioners of environmental geophysics, geomechanics, and energy
resources engineering interested in quantitative subsurface characterization and
modeling of sediment properties.
Gary Mavko received his Ph.D. in Geophysics from Stanford University in 1977

where he is now Professor (Research) of Geophysics. Professor Mavko co-directs the
Stanford Rock Physics and Borehole Geophysics Project (SRB), a group of approximately 25 researchers working on problems related to wave propagation in earth
materials. Professor Mavko is also a co-author of Quantitative Seismic Interpretation
(Cambridge University Press, 2005), and has been an invited instructor for numerous
industry courses on rock physics for seismic reservoir characterization. He received
the Honorary Membership award from the Society of Exploration Geophysicists
(SEG) in 2001, and was the SEG Distinguished Lecturer in 2006.
Tapan Mukerji received his Ph.D. in Geophysics from Stanford University in 1995 and

is now an Associate Professor (Research) in Energy Resources Engineering and


a member of the Stanford Rock Physics Project at Stanford University. Professor
Mukerji co-directs the Stanford Center for Reservoir Forecasting (SCRF) focusing
on problems related to uncertainty and data integration for reservoir modeling. His
research interests include wave propagation and statistical rock physics, and he
specializes in applied rock physics and geostatistical methods for seismic reservoir
characterization, fracture detection, time-lapse monitoring, and shallow subsurface
environmental applications. Professor Mukerji is also a co-author of Quantitative
Seismic Interpretation, and has taught numerous industry courses. He received
the Karcher award from the Society of Exploration Geophysicists in 2000.
Jack Dvorkin received his Ph.D. in Continuum Mechanics in 1980 from Moscow

University in the USSR. He has worked in the Petroleum Industry in the USSR and
USA, and is currently a Senior Research Scientist with the Stanford Rock Physics
Project at Stanford University. Dr Dvorkin has been an invited instructor for numerous industry courses throughout the world, on rock physics and quantitative seismic
interpretation. He is a member of American Geophysical Union, Society of Exploration Geophysicists, American Association of Petroleum Geologists, and the Society
of Petroleum Engineers.

The Rock Physics
Handbook, Second Edition
Tools for Seismic Analysis of Porous Media

Gary Mavko
Stanford University, USA

Tapan Mukerji
Stanford University, USA

Jack Dvorkin
Stanford University, USA


CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,
São Paulo, Delhi, Dubai, Tokyo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521861366
© G. Mavko, T. Mukerji, and J. Dvorkin 2009
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2009
ISBN-13

978-0-511-65062-8

eBook (NetLibrary)

ISBN-13

978-0-521-86136-6

Hardback

Cambridge University Press has no responsibility for the persistence or accuracy
of urls for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.


Contents

Preface

page xi

1

Basic tools

1.1
1.2
1.3

1.4

The Fourier transform
The Hilbert transform and analytic signal
Statistics and probability
Coordinate transformations

1
6
9
18

2

Elasticity and Hooke’s law

21

2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8

21
23
35

39
40
43
47

2.9
2.10
2.11
2.12

Elastic moduli: isotropic form of Hooke’s law
Anisotropic form of Hooke’s law
Thomsen’s notation for weak elastic anisotropy
Tsvankin’s extended Thomsen parameters for orthorhombic media
Third-order nonlinear elasticity
Effective stress properties of rocks
Stress-induced anisotropy in rocks
Strain components and equations of motion in cylindrical and
spherical coordinate systems
Deformation of inclusions and cavities in elastic solids
Deformation of a circular hole: borehole stresses
Mohr’s circles
Static and dynamic moduli

3

Seismic wave propagation

81

3.1
3.2

Seismic velocities
Phase, group, and energy velocities

81
83

v

1

54
56
68
74
76


vi

Contents

3.3
3.4
3.5
3.6
3.7
3.8

3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16

NMO in isotropic and anisotropic media
Impedance, reflectivity, and transmissivity
Reflectivity and amplitude variations with offset (AVO) in isotropic media
Plane-wave reflectivity in anisotropic media
Elastic impedance
Viscoelasticity and Q
Kramers–Kronig relations between velocity dispersion and Q
Waves in layered media: full-waveform synthetic seismograms
Waves in layered media: stratigraphic filtering and velocity dispersion
Waves in layered media: frequency-dependent anisotropy, dispersion,
and attenuation
Scale-dependent seismic velocities in heterogeneous media
Scattering attenuation
Waves in cylindrical rods: the resonant bar
Waves in boreholes

138
146
150
155
160

4

Effective elastic media: bounds and mixing laws

169

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17

Hashin–Shtrikman–Walpole bounds
Voigt and Reuss bounds
Wood’s formula
Voigt–Reuss–Hill average moduli estimate
Composite with uniform shear modulus

Rock and pore compressibilities and some pitfalls
Kuster and Tokso¨z formulation for effective moduli
Self-consistent approximations of effective moduli
Differential effective medium model
Hudson’s model for cracked media
Eshelby–Cheng model for cracked anisotropic media
T-matrix inclusion models for effective moduli
Elastic constants in finely layered media: Backus average
Elastic constants in finely layered media: general layer anisotropy
Poroelastic Backus average
Seismic response to fractures
Bound-filling models

169
174
175
177
178
179
183
185
190
194
203
205
210
215
216
219
224

5

Granular media

229

5.1
5.2

Packing and sorting of spheres
Thomas–Stieber model for sand–shale systems

229
237

86
93
96
105
115
121
127
129
134


vii

Contents

5.3
5.4
5.5

Particle size and sorting
Random spherical grain packings: contact models and
effective moduli
Ordered spherical grain packings: effective moduli

245
264

6

Fluid effects on wave propagation

266

6.1
6.2
6.3
6.4

266
272
273

6.20
6.21

Biot’s velocity relations
Geertsma–Smit approximations of Biot’s relations
Gassmann’s relations: isotropic form
Brown and Korringa’s generalized Gassmann equations for
mixed mineralogy
Fluid substitution in anisotropic rocks
Generalized Gassmann’s equations for composite porous media
Generalized Gassmann equations for solid pore-filling material
Fluid substitution in thinly laminated reservoirs
BAM: Marion’s bounding average method
Mavko–Jizba squirt relations
Extension of Mavko–Jizba squirt relations for all frequencies
Biot–squirt model
Chapman et al. squirt model
Anisotropic squirt
Common features of fluid-related velocity dispersion mechanisms
Dvorkin–Mavko attenuation model
Partial and multiphase saturations
Partial saturation: White and Dutta–Ode´ model for velocity
dispersion and attenuation
Velocity dispersion, attenuation, and dynamic permeability in
heterogeneous poroelastic media
Waves in a pure viscous fluid
Physical properties of gases and fluids

331
338
339

7

Empirical relations

347

7.1

Velocity–porosity models: critical porosity and Nur’s modified
Voigt average
Velocity–porosity models: Geertsma’s empirical relations for
compressibility
Velocity–porosity models: Wyllie’s time-average equation
Velocity–porosity models: Raymer–Hunt–Gardner relations

6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
6.17
6.18
6.19

7.2
7.3
7.4

242

282
284
287
290
292
295
297
298
302
304
306
310
315
320
326

347
350
350
353


viii

Contents

7.5

7.8
7.9
7.10
7.11
7.12
7.13
7.14

Velocity–porosity–clay models: Han’s empirical relations
for shaley sandstones
Velocity–porosity–clay models: Tosaya’s empirical relations for
shaley sandstones
Velocity–porosity–clay models: Castagna’s empirical relations
for velocities
VP–VS–density models: Brocher’s compilation
VP–VS relations
Velocity–density relations
Eaton and Bowers pore-pressure relations
Kan and Swan pore-pressure relations
Attenuation and quality factor relations
Velocity–porosity–strength relations

358
359
363

380
383
383
384
386

8

Flow and diffusion

389

8.1
8.2
8.3
8.4
8.5
8.6
8.7

Darcy’s law
Viscous flow
Capillary forces
Kozeny–Carman relation for flow
Permeability relations with Swi
Permeability of fractured formations
Diffusion and filtration: special cases

389
394

396
401
407
410
411

9

Electrical properties

414

9.1
9.2
9.3
9.4
9.5

Bounds and effective medium models
Velocity dispersion and attenuation
Empirical relations
Electrical conductivity in porous rocks
Cross-property bounds and relations between elastic and
electrical parameters

414
418
421
424
429

Appendices

437

Typical rock properties
Conversions
Physical constants
Moduli and density of common minerals
Velocities and moduli of ice and methane hydrate

437
452
456
457
457

7.6
7.7

A.1
A.2
A.3
A.4
A.5

355
357

ix

Contents

A.6
A.7
A.8

Physical properties of common gases
Velocity, moduli, and density of carbon dioxide
Standard temperature and pressure

468
474
474

References
Index

479
503



Preface to the Second Edition

In the decade since publication of the Rock Physics Handbook, research and use of
rock physics has thrived. We hope that the First Edition has played a useful role in
this era by making the scattered and eclectic mass of rock physics knowledge more
accessible to experts and nonexperts, alike.

While preparing this Second Edition, our objective was still to summarize in a
convenient form many of the commonly needed theoretical and empirical relations of
rock physics. Our approach was to present results, with a few of the key assumptions
and limitations, and almost never any derivations. Our intention was to create a quick
reference and not a textbook. Hence, we chose to encapsulate a broad range of topics
rather than to give in-depth coverage of a few. Even so, there are many topics that we
have not addressed. While we have summarized the assumptions and limitations of
each result, we hope that the brevity of our discussions does not give the impression
that application of any rock physics result to real rocks is free of pitfalls. We assume
that the reader will be generally aware of the various topics, and, if not, we provide a
few references to the more complete descriptions in books and journals.
The handbook contains 101 sections on basic mathematical tools, elasticity theory,
wave propagation, effective media, elasticity and poroelasticity, granular media, and
pore-fluid flow and diffusion, plus overviews of dispersion mechanisms, fluid substitution, and VP–VS relations. The book also presents empirical results derived from
reservoir rocks, sediments, and granular media, as well as tables of mineral data and
an atlas of reservoir rock properties. The emphasis still focuses on elastic and seismic
topics, though the discussion of electrical and cross seismic-electrical relations has
grown. An associated website ( offers MATLAB codes
for many of the models and results described in the Second Edition.
In this Second Edition, Chapter 2 has been expanded to include new discussions on
elastic anisotropy including the Kelvin notation and eigenvalues for stiffnesses,
effective stress behavior of rocks, and stress-induced elasticity anisotropy. Chapter 3
includes new material on anisotropic normal moveout (NMO) and reflectivity,
amplitude variation with offset (AVO) relations, plus a new section on elastic
impedance (including anisotropic forms), and updates on wave propagation in stratified media, and borehole waves. Chapter 4 includes updates of inclusion-based
effective media models, thinly layered media, and fractured rocks. Chapter 5 contains
xi


xii

Preface

extensive new sections on granular media, including packing, particle size, sorting,
sand–clay mixture models, and elastic effective medium models for granular materials. Chapter 6 expands the discussion of fluid effects on elastic properties, including
fluid substitution in laminated media, and models for fluid-related velocity dispersion
in heterogeneous poroelastic media. Chapter 7 contains new sections on empirical
velocity–porosity–mineralogy relations, VP –VS relations, pore-pressure relations,
static and dynamic moduli, and velocity–strength relations. Chapter 8 has new
discussions on capillary effects, irreducible water saturation, permeability, and flow
in fractures. Chapter 9 includes new relations between electrical and seismic properties. The Appendices has new tables of physical constants and properties for common
gases, ice, and methane hydrate.
This Handbook is complementary to a number of other excellent books. For indepth discussions of specific rock physics topics, we recommend Fundamentals of
Rock Mechanics, 4th Edition, by Jaeger, Cook, and Zimmerman; Compressibility
of Sandstones, by Zimmerman; Physical Properties of Rocks: Fundamentals and
Principles of Petrophysics, by Schon; Acoustics of Porous Media, by Bourbie´, Coussy,
and Zinszner; Introduction to the Physics of Rocks, by Gue´guen and Palciauskas;
A Geoscientist’s Guide to Petrophysics, by Zinszner and Pellerin; Theory of Linear
Poroelasticity, by Wang; Underground Sound, by White; Mechanics of Composite
Materials, by Christensen; The Theory of Composites, by Milton; Random Heterogeneous Materials, by Torquato; Rock Physics and Phase Relations, edited by Ahrens;
and Offset Dependent Reflectivity – Theory and Practice of AVO Analysis, edited by
Castagna and Backus. For excellent collections and discussions of classic rock physics
papers we recommend Seismic and Acoustic Velocities in Reservoir Rocks, Volumes 1,
2 and 3, edited by Wang and Nur; Elastic Properties and Equations of State, edited
by Shankland and Bass; Seismic Wave Attenuation, by Tokso¨z and Johnston; and
Classics of Elastic Wave Theory, edited by Pelissier et al.
We wish to thank the students, scientific staff, and industrial affiliates of the
Stanford Rock Physics and Borehole Geophysics (SRB) project for many valuable
comments and insights. While preparing the Second Edition we found discussions with
Tiziana Vanorio, Kaushik Bandyopadhyay, Ezequiel Gonzalez, Youngseuk Keehm,

Robert Zimmermann, Boris Gurevich, Juan-Mauricio Florez, Anyela Marcote-Rios,
Mike Payne, Mike Batzle, Jim Berryman, Pratap Sahay, and Tor Arne Johansen, to be
extremely helpful. Li Teng contributed to the chapter on anisotropic AVOZ, and
Ran Bachrach contributed to the chapter on dielectric properties. Dawn Burgess
helped tremendously with editing, graphics, and content. We also wish to thank the
readers of the First Edition who helped us to track down and fix errata.
And as always, we are indebted to Amos Nur, whose work, past and present, has
helped to make the field of rock physics what it is today.
Gary Mavko, Tapan Mukerji, and Jack Dvorkin.


1

Basic tools

1.1

The Fourier transform
Synopsis
The Fourier transform of f(x) is defined as
Z 1
FðsÞ ¼
f ðxÞeÀi2pxs dx
À1

The inverse Fourier transform is given by
Z 1
f ðxÞ ¼
FðsÞeþi2pxs ds
À1

Evenness and oddness
A function E(x) is even if E(x) ¼ E(–x). A function O(x) is odd if O(x) ¼ –O(–x).
The Fourier transform has the following properties for even and odd functions:
 Even functions. The Fourier transform of an even function is even. A real even
function transforms to a real even function. An imaginary even function transforms
to an imaginary even function.
 Odd functions. The Fourier transform of an odd function is odd. A real odd function
transforms to an imaginary odd function. An imaginary odd function transforms to
a real odd function (i.e., the “realness” flips when the Fourier transform of an odd
function is taken).
real even (RE) ! real even (RE)
imaginary even (IE) ! imaginary even (IE)
real odd (RO) ! imaginary odd (IO)
imaginary odd (IO) ! real odd (RO)
Any function can be expressed in terms of its even and odd parts:
f ðxÞ ¼ EðxÞ þ OðxÞ
where
EðxÞ ¼ 12½ f ðxÞ þ f ðÀxފ
OðxÞ ¼ 12½ f ðxÞ À f ðÀxފ
1


2

Basic tools

Then, for an arbitrary complex function we can summarize these relations as
(Bracewell, 1965)
f ðxÞ ¼ reðxÞ þ i ieðxÞ þ roðxÞ þ i ioðxÞ

FðxÞ ¼ REðsÞ þ i IEðsÞ þ ROðsÞ þ i IOðsÞ
As a consequence, a real function f(x) has a Fourier transform that is hermitian,
F(s) ¼ F*(–s), where * refers to the complex conjugate.
For a more general complex function, f(x), we can tabulate some additional
properties (Bracewell, 1965):
f ðxÞ , FðsÞ
f à ðxÞ , Fà ðÀsÞ
f à ðÀxÞ , Fà ðsÞ
f ðÀxÞ , FðÀsÞ
2 Re f ðxÞ , FðsÞ þ Fà ðÀsÞ
2 Im f ðxÞ , FðsÞ À Fà ðÀsÞ
f ðxÞ þ f à ðÀxÞ , 2 ReFðsÞ
f ðxÞ À f à ðÀxÞ , 2 ImFðsÞ
The convolution of two functions f(x) and g(x) is
Z þ1
Z þ1
f ðzÞ gðx À zÞ dz ¼
f ðx À zÞ gðzÞ dz
f ðxÞ Ã gðxÞ ¼
À1

À1

Convolution theorem
If f(x) has the Fourier transform F(s), and g(x) has the Fourier transform G(s), then the
Fourier transform of the convolution f(x) * g(x) is the product F(s) G(s).
The cross-correlation of two functions f(x) and g(x) is
Z þ1
Z þ1

Ã
Ã
f ðxÞ ? gðxÞ ¼
f ðz À xÞ gðzÞ dz ¼
f à ðzÞ gðz þ xÞ dz
À1

À1

where f* refers to the complex conjugate of f. When the two functions are the same,
f*(x) ★ f(x) is called the autocorrelation of f(x).

Energy spectrum
The modulus squared of the Fourier transform jF(s)j2 ¼ F(s) F*(s) is sometimes
called the energy spectrum or simply the spectrum.
If f(x) has the Fourier transform F(s), then the autocorrelation of f(x) has the
Fourier transform jF(s)j2.


3

1.1 The Fourier transform

Phase spectrum
The Fourier transform F(s) is most generally a complex function, which can be
written as
FðsÞ ¼ jFjei’ ¼ Re FðsÞ þ i Im FðsÞ
where jFj is the modulus and ’ is the phase, given by
’ ¼ tanÀ1 ½Im FðsÞ=Re Fðsފ
The function ’(s) is sometimes also called the phase spectrum.

Obviously, both the modulus and phase must be known to completely specify the
Fourier transform F(s) or its transform pair in the other domain, f(x). Consequently, an
infinite number of functions f(x) , F(s) are consistent with a given spectrum jF(s)j2.
The zero-phase equivalent function (or zero-phase equivalent wavelet) corresponding to a given spectrum is
FðsÞ ¼ jFðsÞj
Z 1
f ðxÞ ¼
jFðsÞj eþi2pxs ds
À1

which implies that F(s) is real and f(x) is hermitian. In the case of zero-phase real
wavelets, then, both F(s) and f(x) are real even functions.
The minimum-phase equivalent function or wavelet corresponding to a spectrum is the
unique one that is both causal and invertible. A simple way to compute the minimumphase equivalent of a spectrum jF(s)j2 is to perform the following steps (Claerbout, 1992):
(1) Take the logarithm, B(s) ¼ ln jF(s)j.
(2) Take the Fourier transform, B(s) ) b(x).
(3) Multiply b(x) by zero for x < 0 and by 2 for x > 0. If done numerically, leave the
values of b at zero and the Nyquist frequency unchanged.
(4) Transform back, giving B(s) + i’(s), where ’ is the desired phase spectrum.
(5) Take the complex exponential to yield the minimum-phase function: Fmp(s) =
exp[B(s) þ i’(s)] ¼ jF(s)jei’(s).
(6) The causal minimum-phase wavelet is the Fourier transform of Fmp(s) ) fmp(x).
Another way of saying this is that the phase spectrum of the minimum-phase
equivalent function is the Hilbert transform (see Section 1.2 on the Hilbert transform)
of the log of the energy spectrum.

Sampling theorem
A function f(x) is said to be band limited if its Fourier transform is nonzero only
within a finite range of frequencies, jsj < sc, where sc is sometimes called the cut-off
frequency. The function f(x) is fully specified if sampled at equal spacing not

exceeding Dx ¼ 1/(2sc). Equivalently, a time series sampled at interval Dt adequately
describes the frequency components out to the Nyquist frequency fN ¼ 1/(2Dt).


Basic tools

2

2
Π(s)
Boxcar(s)

sinc(x)

Sinc(x)

4

0

−2
−3

−2

−1

0
x

1

2

3

0

−2

−2

−1

0
s

1

2

Figure 1.1.1 Plots of the function sinc(x) and its Fourier transform Å(s).

The numerical process to recover the intermediate points between samples is to
convolve with the sinc function:
2sc sincð2sc xÞ ¼ 2sc sinðp2sc xÞ=p2sc x
where
sincðxÞ 

sinðpxÞ

px

which has the properties:
'
sincð0Þ ¼ 1
n ¼ nonzero integer
sincðnÞ ¼ 0
The Fourier transform of sinc(x) is the boxcar function Å(s):
8
1
>
< 0 jsj > 2
ÅðsÞ ¼

>
:

12

/

jsj ¼ 12

1

jsj < 12

Plots of the function sinc(x) and its Fourier transform Å(s) are shown in Figure 1.1.1.
One can see from the convolution and similarity theorems below that convolving
with 2sc sinc(2scx) is equivalent to multiplying by Å(s/2sc) in the frequency domain

(i.e., zeroing out all frequencies jsj > sc and passing all frequencies jsj < sc.

Numerical details
Consider a band-limited function g(t) sampled at N points at equal intervals: g(0),
g(Dt), g(2Dt), . . . , g((N – 1)Dt). A typical fast Fourier transform (FFT) routine will
yield N equally spaced values of the Fourier transform, G( f ), often arranged as




N
N
þ1
þ2
Á Á Á ðN À 1Þ
N
1
2
3
ÁÁÁ
2
2
Gð0Þ GðÁf Þ Gð2Áf Þ Á Á Á GðÆ fN Þ GðÀfN þ Áf Þ Á Á Á GðÀ2Áf Þ GðÀÁf Þ


5

1.1 The Fourier transform

time domain sample rate Dt

Nyquist frequency fN ¼ 1/(2Dt)
frequency domain sample rate Df ¼ 1/(NDt)
Note that, because of “wraparound,” the sample at (N/2 þ 1) represents both ÆfN.

Spectral estimation and windowing
It is often desirable in rock physics and seismic analysis to estimate the spectrum of
a wavelet or seismic trace. The most common, easiest, and, in some ways, the worst
way is simply to chop out a piece of the data, take the Fourier transform, and find
its magnitude. The problem is related to sample length. If the true data function is f(t),
a small sample of the data can be thought of as
&
f ðtÞ;
a t b
fsample ðtÞ ¼
0;
elsewhere
or
 1

t À 2ða þ bÞ
fsample ðtÞ ¼ f ðtÞ Å
bÀa
where Å(t) is the boxcar function discussed above. Taking the Fourier transform of
the data sample gives
Fsample ðsÞ ¼ FðsÞ Ã ½jb À aj sincððb À aÞsÞeÀipðaþbÞs Š
More generally, we can “window” the sample with some other function o(t):
fsample ðtÞ ¼ f ðtÞ oðtÞ
yielding
Fsample ðsÞ ¼ FðsÞ Ã WðsÞ
Thus, the estimated spectrum can be highly contaminated by the Fourier transform

of the window, often with the effect of smoothing and distorting the spectrum due to
the convolution with the window spectrum W(s). This can be particularly severe in
the analysis of ultrasonic waveforms in the laboratory, where often only the first 1 to
112 cycles are included in the window. The solution to the problem is not easy, and
there is an extensive literature (e.g., Jenkins and Watts, 1968; Marple, 1987) on
spectral estimation. Our advice is to be aware of the artifacts of windowing and to
experiment to determine the sensitivity of the results, such as the spectral ratio or the
phase velocity, to the choice of window size and shape.

Fourier transform theorems
Tables 1.1.1 and 1.1.2 summarize some useful theorems (Bracewell, 1965). If f(x) has
the Fourier transform F(s), and g(x) has the Fourier transform G(s), then the Fourier


6

Basic tools

Table 1.1.1

Fourier transform theorems.

Theorem

x-domain

s-domain

Similarity

f(ax)

,

1 s
F
jaj a

Addition

f(x) þ g(x)

,

F(s) þ G(s)

Shift

f(x – a)

,

Modulation

f(x) cos ox

,

Convolution

f(x) * g(x)

,

e–i2pasF(s)
1 
! 1 
!
F sÀ
þ F sþ
2
2p
2
2p
F(s) G(s)

Autocorrelation

f(x) * f (–x)

,

jF(s)j2

Derivative

f 0 (x)

,

i2psF(s)

Table 1.1.2

*

Some additional theorems.

Derivative of convolution
Rayleigh
Power
( f and g real)

d
½ f ðxÞ Ã gðxފ ¼ f 0 ðxÞ Ã gðxÞ ¼ f ðxÞ Ã g0 ðxÞ
dx
Z 1
Z 1
j f ðxÞj2 dx ¼
jFðsÞj2 ds
À1
À1
Z 1
Z 1
f ðxÞ gà ðxÞ dx ¼
FðsÞ GÃ ðsÞ ds
À1
À1
Z 1
Z 1

f ðxÞ gðÀxÞ dx ¼
FðsÞ GðsÞ ds
À1

À1

transform pairs in the x-domain and the s-domain are as shown in the tables.
Table 1.1.3 lists some useful Fourier transform pairs.

1.2

The Hilbert transform and analytic signal
Synopsis
The Hilbert transform of f(x) is defined as
Z
1 1 f ðx0 Þ dx0
FHi ðxÞ ¼
p À1 x0 À x
which can be expressed as a convolution of f(x) with (–1/px) by
FHi ¼ À

1
à f ðxÞ
px

The Fourier transform of (–1/px) is (i sgn(s)), that is, þi for positive s and –i for
negative s. Hence, applying the Hilbert transform keeps the Fourier amplitudes or
spectrum the same but changes the phase. Under the Hilbert transform, sin(kx) is
converted to cos(kx), and cos(kx) is converted to –sin(kx). Similarly, the Hilbert
transforms of even functions are odd functions and vice versa.

7

1.2 The Hilbert transform and analytic signal

Table 1.1.3

Some Fourier transform pairs.
sin px




!
i
1
1
d sþ
Àd sÀ
2
2
2

cos px




!

1
1
1
d sþ
þd sÀ
2
2
2

d(x)

1

sinc(x)

Å(s)

sinc2(x)

L(s)

e–px

2

e–ps

2

–1/px

i sgn(s)

x0
x20 þ x2

p exp(–2px0jsj)

e–jxj

jxj–1/2

2
1 þ ð2psÞ2
jsj–1/2

The inverse of the Hilbert transform is itself the Hilbert transform with a change
of sign:
Z
1 1 FHi ðx0 Þ dx0
f ðxÞ ¼ À
p À1 x0 À x
or


1
f ðxÞ ¼ À À
à FHi
px
The analytic signal associated with a real function, f(t), is the complex function

SðtÞ ¼ f ðtÞ À i FHi ðtÞ
As discussed below, the Fourier transform of S(t) is zero for negative frequencies.


8

Basic tools

The instantaneous envelope of the analytic signal is
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
EðtÞ ¼ f 2 ðtÞ þ F2Hi ðtÞ
The instantaneous phase of the analytic signal is
’ðtÞ ¼ tanÀ1 ½ÀFHi ðtÞ=f ðtފ
¼ Im½lnðSðtÞފ
The instantaneous frequency of the analytic signal is
!


d’
d
l dS
o¼
¼ Im lnðSÞ ¼ Im
dt
dt
S dt
Claerbout (1992) has suggested that o can be numerically more stable if the
denominator is rationalized and the functions are locally smoothed, as in the following
equation:
E3

2D
SÃ ðtÞ dSðtÞ
dt
5
" ¼ Im4 Ã
o
hS ðtÞ SðtÞi
where 〈·〉 indicates some form of running average or smoothing.

Causality
The impulse response, I(t), of a real physical system must be causal, that is,
IðtÞ ¼ 0;

for t < 0

The Fourier transform T( f ) of the impulse response of a causal system is sometimes called the transfer function:
Z 1
IðtÞ eÀi2pft dt
Tð f Þ ¼
À1

T( f ) must have the property that the real and imaginary parts are Hilbert transform
pairs, that is, T( f ) will have the form
Tð f Þ ¼ Gð f Þ þ iBð f Þ
where B( f ) is the Hilbert transform of G( f ):
Z

1

Gð f 0 Þ df 0

0
À1 f À f
Z
1 1 Bð f 0 Þ df 0
Gð f Þ ¼ À
p À1 f 0 À f
1
Bð f Þ ¼
p


9

1.3 Statistics and probability

Similarly, if we reverse the domains, an analytic signal of the form
SðtÞ ¼ f ðtÞ À iFHi ðtÞ
must have a Fourier transform that is zero for negative frequencies. In fact, one convenient way to implement the Hilbert transform of a real function is by performing the
following steps:
(1) Take the Fourier transform.
(2) Multiply the Fourier transform by zero for f < 0.
(3) Multiply the Fourier transform by 2 for f > 0.
(4) If done numerically, leave the samples at f ¼ 0 and the Nyquist frequency unchanged.
(5) Take the inverse Fourier transform.
The imaginary part of the result will be the negative Hilbert transform of the real part.

1.3

Statistics and probability
Synopsis

The sample mean, m, of a set of n data points, xi, is the arithmetic average of the data
values:
n
1X
xi
n i¼1

m¼

The median is the midpoint of the observed values if they are arranged in increasing
order. The sample variance, s2, is the average squared difference of the observed
values from the mean:
2 ¼

n
1X
ðxi À mÞ2
n i¼1

(An unbiased estimate of the population variance is often found by dividing the sum
given above by (n – 1) instead of by n.)
The standard deviation, s, is the square root of the variance, while the coefficient
of variation is s/m. The mean deviation, a, is
a¼

n
1X
jxi À mj
n i¼1

Regression
When trying to determine whether two different data variables, x and y, are related,
we often estimate the correlation coefficient, r, given by (e.g., Young, 1962)
1 Pn
i¼1 ðxi À mx Þðyi À my Þ
;
¼n
x y

where jj

1


10

Basic tools

where sx and sy are the standard deviations of the two distributions and mx and my
are their means. The correlation coefficient gives a measure of how close the points
come to falling along a straight line in a scatter plot of x versus y. jrj ¼ 1 if the points
lie perfectly along a line, and jrj < 1 if there is scatter about the line. The numerator
of this expression is the sample covariance, Cxy, which is defined as
Cxy ¼

n
1X
ðxi À mx Þðyi À my Þ
n i¼1

It is important to remember that the correlation coefficient is a measure of the linear
relation between x and y. If they are related in a nonlinear way, the correlation
coefficient will be misleadingly small.
The simplest recipe for estimating the linear relation between two variables, x and y,
is linear regression, in which we assume a relation of the form:
y ¼ ax þ b
The coefficients that provide the best fit to the measured values of y, in the leastsquares sense, are
y
;
x

a¼

b ¼ my À amx

More explicitly,
a¼

n

P

P
P
xi yi À ð xi Þð yi Þ
;
P
P
n x2i À ð xi Þ2

slope

P ÀP 2 Á
P
P
ð yi Þ
xi À ð xi yi Þð xi Þ
;
b¼
P
P
n x2i À ð xi Þ2

intercept

The scatter or variation of y-values around the regression line can be described by
the sum of the squared errors as
E2 ¼

n
X

ðyi À y^i Þ2

i¼1

where y^i is the value predicted from the regression line. This can be expressed as a
variance around the regression line as
^y2 ¼

n
1X
ðyi À y^i Þ2
n i¼1

The square of the correlation coefficient r is the coefficient of determination, often
denoted by r2, which is a measure of the regression variance relative to the total
variance in the variable y, expressed as


11

1.3 Statistics and probability

variance of y around the linear regression
total variance of y
Pn
^y2
ðyi À y^i Þ2
¼
1
À
¼ 1 À Pni¼1
2
y2
i¼1 ðyi À my Þ

r 2 ¼ 2 ¼ 1 À

The inverse relation is

^y2 ¼ y2 ð1 À r 2 Þ
Often, when doing a linear regression the choice of dependent and independent
variables is arbitrary. The form above treats x as independent and exact and assigns
errors to y. It often makes just as much sense to reverse their roles, and we can find
a regression of the form
x ¼ a0 y þ b0
Generally a 6¼ 1/a0 unless the data are
perfectly correlated. In fact, the correlation
pffiffiffiffiffiffi
coefficient, r, can be written as  ¼ aa0 .
The coefficients of the linear regression among three variables of the form
z ¼ a þ bx þ cy
are given by
Cxz Cyy À Cxy Cyz
2
Cxx Cyy À Cxy
Cxx Cyz À Cxy Cxz
c¼
2
Cxx Cyy À Cxy
b¼

a ¼ mz À mx b À my c
The coefficients of the n-dimensional linear regression of the form
z ¼ c0 þ c1 x1 þ c2 x2 þ Á Á Á þ cn xn
are given by
2 3
2
3
c0

ð1Þ
z
6 c1 7
6 7 À
ð2Þ 7
ÁÀ1 6
6 c2 7
6z 7
6 7 ¼ MT M MT 6 .. 7
6 .. 7
4 . 5
4 . 5
zðkÞ
cn
where the k sets
2
ð1Þ
1 x1
6
6 1 xð2Þ
1
6
M¼6
..
6 ..
4.
.
1

ðkÞ

x1

of independent variables form columns 2:(n þ 1) in the matrix M:
3
ð1 Þ
x2
Á Á Á xðn1Þ
7
ð2 Þ
x2
Á Á Á xðn2Þ 7
7
..
.. 7
7
.
. 5
ðk Þ

x2

ÁÁÁ

xðnkÞ