Appendix BMath review

This appendix is not intended to teach anyone vector calculus or complex number theory,but simply to list some of the important definitions to assist those whose skills may havebecome rusty. Many of the equations are expressed in both standard vector notation and theindex notation used in this book.

B.1 Vector calculus

Consider a Cartesian coordinate system with x, y, and z axes. The length, or magnitude, ofa vector u is written as u. A vector may be expressed in terms of its components as

u = uxx + uyy + uzz, (B.1)where x, y, and z are unit length vectors in the x, y, and z directions. The dot product oftwo vectors is a scalar (a single number) and is defined as

= u v = uv cos (B.2)= uxvx + uyvy + uzvz, (B.3)

where is the angle between the two vectors. It follows that u v = 0 when u and v areorthogonal and that for unit vectors x x = 1. Note that for a unit vector u, the dot productu v gives the length of the orthogonal projection of v onto u.

The cross-product between two vectors is a third vector that points in a direction per-pendicular to both (according to the right-hand rule). The cross-product can be expressedin component form as

u v = (uyvz uzvy)x + (uzvx uxvz)y + (uxvy uyvx)z, (B.4)and the length of this vector may be expressed as

u v = uv sin . (B.5)

353

354 A P P E N D I X B . M A T H R E V I E W

v

u

v u = v cos()

Figure B.1 The dot product ofa vector with a unit vector isthe length of the projectiononto the unit vector.

Note that the dot product is commutative, but not the cross-product, that is

u v = v u, (B.6)u v = (v u). (B.7)

A second-order tensor, U, is a linear operator that produces one vector from another,that is,

u = Uv, (B.8)ui = Uijvj (sum over j = 1, 2, 3).

Here we introduce the use of index notation; i and j are assumed to take on the values 1,2, and 3 for the x, y, and z components, respectively. Notice that in a Cartesian coordinatesystem, the second-order tensor U has the form of a 3 3 matrix. We also begin usingthe summation convention; repeated indices in a product are assumed to be summed overvalues from 1 to 3.

The projection property of the dot product can be used to express a vector in a different(i.e., rotated) Cartesian coordinate system. If the new coordinate axes are defined by theorthogonal unit vectors x, y, and z (expressed in the original x,y,z coordinates), then thex coordinate of a vector v is given by x v. In this way the vector in the new coordinatesystem is given by

v = x1 x2 x3y1 y2 y3z1 z2 z3

v Av, (B.9)

where A is the transformation tensor with components equal to the cosines of the an-gles between the primed and unprimed axes. We can express the same equation in indexnotation as

vi = Aijvj. (B.10)

B.1 V E C T O R C A L C U L U S 355

Because the rows of A are orthogonal unit vectors it follows that

ATA = 1 0 00 1 0

0 0 1

= I, (B.11)

where I is the identity matrix.We often will also want a way to transform a Cartesian tensor to a new coordinate system.

This can be obtained by applying the transformation tensor A to both sides of (B.8)

u = Uv (B.12)Au = AUv (B.13)Au = AU(ATA)v (B.14)Au = AUAT(Av) (B.15)u = AUATv (B.16)

and we see that the tensor operator that produces u from v in the primed coordinate systemis given by

U = AUAT (B.17)

which we can use to convert U to U. In Chapter 2, we use the eigenvector matrix N to rotatethe stress tensor into its principal axes coordinate system. The definition of N is similar toA except that the unit vectors are set to the columns rather than the rows. Thus NT = Aand in this case the transformation equation is

U = NTUN (B.18)

Useful matrix identities include

A(B + C) = AB + AC (B.19)(A + B)T = AT + BT (B.20)(AB)T = BTAT (B.21)(AB)1 = B1A1 (B.22)(A1)T = (AT)1 (B.23)

where for the last two we assume the existence of inverses of A and B.Functions that vary with position are termed fields; we can have scalar fields, vector

fields, and tensor fields. In this case we may define spatial derivatives, such as the gradient,divergence, Laplacian, and curl.

356 A P P E N D I X B . M A T H R E V I E W

The gradient of a scalar field, written , is a vector field, defined by the partial deriva-tives of the scalar in x, y, and z directions:

u = = x

x + y

y + z

z, (B.24)

ui = i,where i is shorthand notation for /x, /y, and /z for i = 1, 2, and 3 respectively. Thegradient vector, , is normal to surfaces of constant .

The gradient of a vector field is a tensor field:

U = u, (B.25)Uij = iuj.

The divergence of a vector field, written u, is a scalar field:

= u = uxx

+ uyy

+ uzz

(B.26)

= iui (sum over i = 1, 2, 3).The divergence of a second-order tensor field is a vector field:

u = U, (B.27)uj = iUij (sum over i = 1, 2, 3).

The Laplacian of a scalar field, written 2, is a scalar field:

= 2 = = 2

x2+

2

y2+

2

z2(B.28)

= jj (sum over j = 1, 2, 3).The Laplacian of a vector field is a vector field:

u = 2v = v, (B.29)ui = jjvi (sum over j = 1, 2, 3).

The curl of a vector field is a vector field:

u = v =(vz

y vyz

)x

+(vx

z vzx

)y

+(vy

x vxy

)z. (B.30)

B.1 V E C T O R C A L C U L U S 357

The operator is distributive, that is,

(+ ) = + , (B.31) (u + v) = u + v, (B.32)

(u + v) = u + v. (B.33)

A vector field defined as the gradient of a scalar field is curl free, that is,

() = 0. (B.34)

A vector field defined as the curl of another vector field is divergence free, that is,

( u) = 0. (B.35)

The following identities are often useful:

u = u + u , (B.36) u = u + u, (B.37)

( u) = u 2u. (B.38)

The identity matrix I can be written in index notation as ij where

ij ={

1 for i = j,0 for i = j. (B.39)

When ij appears as part of a product in equations, it can be used to switch the indicesof other terms, that is,

iijuk = juk. (B.40)

Of great importance in continuum mechanics is Gausss theorem, which equates thevolume integral of a vector field to the surface integral of the orthogonal component of thevector field:

V

u dV =S

u n dS, (B.41)

where n is the outward normal vector to the surface.

358 A P P E N D I X B . M A T H R E V I E W

Re(z)

Im(z)

x

yr

z

Figure B.2 The complex number zcan be represented as a point in thecomplex plane.

B.2 Complex numbers

We use complex numbers in this book mostly as a shorthand way to keep track of the phaseand amplitude of harmonic waves. The imaginary number i is defined as

i2 = 1. (B.42)

It follows that

1 = i and 1/i = i. (B.43)

A complex number can be written as

z = x+ iy, (B.44)

where x = Re(z) is the real part of z and y = Im(z) is the imaginary part of z (notethat y itself is a real number). Complex numbers obey the commutative, associative, anddistributive rules of arithmetic. The complex conjugate of z is defined as

z = x iy. (B.45)

Complex numbers may be represented as points on the complex plane (see Fig. B.2),either in Cartesian coordinates by x and y, or in polar coordinates by their phase, , andtheir magnitude, r = |z|. These forms are related by

z = rei = r(cos + i sin ) = x+ yi. (B.46)

The magnitude |z| is also sometimes referred to as the absolute value of z. Note that

y/x = tan (B.47)

and that

zz = (x+ iy)(x iy) = x2 + y2 = |z|2. (B.48)

Now let us illustrate the convenience of complex numbers for describing wave motion. Aharmonic wave of angular frequency is defined by its amplitude a and phase delay

B.2 C O M P L E X N U M B E R S 359

(Fig. B.3), that is,

f(t) = a cos(t ). (B.49)

Using a trigonometric identity for cos(t ), this can be rewritten

f(t) = a cos cost + a sin sint (B.50)= a1 cost + a2 sint, (B.51)

where a1 a cos and a2 a sin . This is a more convenient form because it is a linearfunction of the coefficients a1 and a2. A harmonic wave of arbitrary phase can alwaysbe expressed as a weighted sum of a sine and a cosine function. Two waves of the samefrequency can be summed by adding their sine and cosine coefficients. Note that a and may be recovered from the new coefficients using

a2 = a21 + a22 and = tan1(a2/a1). (B.52)

We can obtain the same relationships using a single complex coefficient A by writingthe function f(t) as a complex exponential function

f(t) = Re [Aeit] . (B.53)Expanding this, we have

f(t) = Re [A (cos(t)+ i sin(t))]= Re [A(cost i sint)] . (B.54)

Now consider the real and imaginary parts of A = x+ iy:

f(t) = Re [(x+ iy)(cost i sint)] . (B.55)

a

t

Figure B.3 The amplitude a and phase delay of a cosine function.

360 A P P E N D I X B . M A T H R E V I E W

The real terms give

f(t) = x cost + y sint. (B.56)This is identical to (B.51) if we assume

a1 = x = Re(A), (B.57)a2 = y = Im(A). (B.58)

In this way a single complex number can keep track of both the amplitude and phase ofharmonic waves. For convenience, equations such as (B.53) usually do not explicitly includethe Re function; in these cases the reader should keep in mind that the real part must alwaysbe taken before the equation has a physical meaning. This applies, for example, to equation(3.36) in Chapter 3.

CoverHalf-titleTitleCopyrightContentsPreface to the First EditionPreface to the Second EditionAcknowledgment1 Introduction1.1 A brief history of seismology1.2 Exercises2 Stress and strain2.1 The stress tensor2.1.1 Example: Computing the traction vector2.1.2 Principal axes of stress2.1.3 Example: Computing the principal axes2.1.4 Deviatoric stress2.1.5 Values for stress2.2 The strain tensor2.2.1 Values for strain2.2.2 Example: Computing strain for a seismic wave2.3 The linear stressstrain relationship2.3.1 Units for elastic moduli2.4 Exercises3 The seismic wave equation3.1 Introduction: The wave equation3.2 The momentum equation3.3 The seismic wave equation3.3.1 Potentials3.4 Plane waves3.4.1 Example: Harmonic plane wave equation3.5 Polarizations of P and S waves3.6 Spherical waves3.7 Methods for computing synthetic seismograms3.8 The future of seismology?3.9 Equations for 2-D isotropic finite differences3.10 Exercises4 Ray theory: Travel times4.1 Snells law4.2 Ray paths for laterally homogeneous models4.2.1 Example: Computing X(p) and T(p)4.2.2 Ray tracing through velocity gradients4.3 Travel time curves and delay times4.3.1 Reduced velocity4.3.2 The tau(p) function4.4 Low-velocity zones4.5 Summary of 1-D ray tracing equations4.6 Spherical-Earth ray tracing4.7 The Earth-flattening transformation4.8 Three-dimensional ray tracing4.9 Ray nomenclature4.9.1 Crustal phases4.9.2 Whole Earth phases4.9.3 PKJKP: The Holy Grail of body wave seismology4.10 Global body-wave observations4.11 Exercises5 Inversion of travel time data5.1 One-dimensional velocity inversion5.2 Straight-line fitting5.2.1 Example: Solving for a layer-cake model5.2.2 Other ways to fit the T(X) curve5.3 Tau(p) Inversion5.3.1 Example: The layer-cake model revisited5.3.2 Obtaining tau(p) constraints5.4 Linear programming and regularization methods5.5 Summary: One-dimensional velocity inversion5.6 Three-dimensional velocity inversion5.6.1 Setting up the tomography problem5.6.2 Solving the tomography problem5.6.3 Tomography complications5.6.4 Finite frequency tomography5.7 Earthquake location5.7.1 Iterative location methods5.7.2 Relative event location methods5.8 Exercises6 Ray theory: Amplitude and phase6.1 Energy in seismic waves6.2 Geometrical spreading in 1-D velocity models6.3 Reflection and transmission coefficients6.3.1 SH-wave reflection and transmission coefficients6.3.2 Example: Computing SH coefficients6.3.3 Vertical incidence coefficients6.3.4 Energy-normalized coefficients6.3.5 Dependence on ray angle6.4 Turning points and Hilbert transforms6.5 Matrix methods for modeling plane waves6.6 Attenuation6.6.1 Example: Computing intrinsic attenuation6.6.2 t* and velocity dispersion6.6.3 The absorption band model6.6.4 The standard linear solid6.6.5 Earths attenuation6.6.6 Observing Q6.6.7 Non-linear attenuation6.6.8 Seismic attenuation and global politics6.7 Exercises7 Reflection seismology7.1 Zero-offset sections7.2 Common midpoint stacking7.3 Sources and deconvolution7.4 Migration7.4.1 Huygens principle7.4.2 Diffraction hyperbolas7.4.3 Migration methods7.5 Velocity analysis7.5.1 Statics corrections7.6 Receiver functions7.7 Kirchhoff theory7.7.1 Kirchhoff applications7.7.2 How to write a Kirchhoff program7.7.3 Kirchhoff migration7.8 Exercises8 Surface waves and normal modes8.1 Love waves8.1.1 Solution for a single layer8.2 Rayleigh waves8.3 Dispersion8.4 Global surface waves8.5 Observing surface waves8.6 Normal modes8.7 Exercises9 Earthquakes and source theory9.1 Greens functions and the moment tensor9.2 Earthquake faults9.2.1 Non-double-couple sources9.3 Radiation patterns and beach balls9.3.1 Example: Plotting a focal mechanism9.4 Far-field pulse shapes9.4.1 Directivity9.4.2 Source spectra9.4.3 Empirical Greens functions9.5 Stress drop9.5.1 Self-similar earthquake scaling9.6 Radiated seismic energy9.6.1 Earthquake energy partitioning9.7 Earthquake magnitude9.7.1 The b value9.7.2 The intensity scale9.8 Finite slip modeling9.9 The heat flow paradox9.10 Exercises10 Earthquake prediction10.1 The earthquake cycle10.2 Earthquake triggering10.3 Searching for precursors10.4 Are earthquakes unpredictable?10.5 Exercises11 Instruments, noise, and anisotropy11.1 Instruments11.1.1 Modern seismographs11.2 Earth noise11.3 Anisotropy11.3.1 Snells law at an interface11.3.2 Weak anisotropy11.3.3 Shear-wave splitting11.3.4 Hexagonal anisotropy11.3.5 Mechanisms for anisotropy11.3.6 Earths anisotropy11.4 ExercisesAppendix A The PREM modelAppendix B Math reviewB.1 Vector calculusB.2 Complex numbersAppendix C The eikonal equationAppendix D Fortran subroutinesAppendix E Time series and Fourier transformsE.1 ConvolutionE.2 Fourier transformE.3 Hilbert transformBibliographyIndex