Appendix ASpherical Harmonic Functions and Wigner 3-jSymbols

Spherical Harmonic Functions

According to the Rodrigues formula, the Legendre polynomial of order l is given by

Pl.z/ D 1

2l lŠ

d l

d zl.z2 � 1/l for jzj < 1: (A.1)

By using Legendre polynomials, the associate Legendre function of order .l;m/ isdefined by

Pml .z/ D �

1 � z2�m2dm

d zmPl .z/ for l = m = 0: (A.2)

Here we define function

�lm .�/ D

8ˆˆ<

ˆˆ:

.�1/m i ls2l C 1

2

.l �m/Š

.l Cm/ŠPml .cos �/ for m = 0;

i l

s2l C 1

2

.l � jmj/Š

.l C jmj/Š Pjmjl .cos �/ for m < 0;

(A.3)

where the phase factor i l is introduced (after Landau and Lifshitz 2003, p. 91).This phase factor is the most natural choice from the viewpoint of the theory of theaddition of angular momenta. By using�lm.�/ and

˚m .'/ D 1p2�eim'; (A.4)

we define the normalized spherical harmonic function as

Ylm .�; '/ D �lm .�/˚m .'/ ; (A.5)

H. Sato et al., Seismic Wave Propagation and Scattering in the Heterogeneous Earth:Second Edition, DOI 10.1007/978-3-642-23029-5,© Springer-Verlag Berlin Heidelberg 2012

457

458 A Spherical Harmonic Functions and Wigner 3-j Symbols

where we note that

Y �l;m .�; '/ D .�1/l�m Yl;�m .�; '/ : (A.6)

Normalized spherical harmonic functions satisfy the orthonormal relation:

I

Y �l1m1

.�; '/ Yl2m2 .�; '/ d˝ .�; '/ D ıl1l2ım1m2; (A.7)

where d˝ .�; '/ D sin �d�d' is an infinitesimal solid angle element. The sphericalharmonic closure relation (Arfken and Weber 1995) gives the delta function for thesolid angle as

ı˝�xI x0� D ı˝

��; 'I � 0; ' 0� D

1X

lD0

lX

mD�lYlm .�; '/ Y

�lm

�� 0; ' 0�

D1X

lD0

lX

mD�lY �lm .�; '/ Ylm

�� 0; ' 0� ;

(A.8)

where x D .r; �; '/ and x0 D .r 0; � 0; ' 0/ in spherical coordinates. We note that

I

ı˝��; 'I � 0; ' 0� d˝

�� 0; ' 0� D 1;

I

f�� 0; ' 0�ı˝

��; 'I � 0; ' 0� d˝

�� 0; ' 0� D f .�; '/ ;

(A.9)

for any function f .�; '/.

Addition Theorem and Expansion Formulas

The addition theorem holds for Legendre polynomials:

Pl .cos / D 4�

2l C 1

lX

mD�lYlm .�; '/ Y

�lm

�� 0; ' 0�

D 4�

2l C 1

lX

mD�lY �lm .�; '/ Ylm

�� 0; ' 0� ; (A.10)

where D cos � cos � 0 C sin � sin � 0 cos .' � ' 0/.There is an expansion formula by using spherical harmonic functions (see

Gradshteyn and Ryzhik 2007, p. 941):

A Spherical Harmonic Functions and Wigner 3-j Symbols 459

eikr cos D1X

lD0i l .2l C 1/jl .kr/Pl .cos /

D 4�

1X

lD0i ljl .kr/

lX

mD�lYlm .�; '/ Y

�lm

�� 0; ' 0� ; (A.11)

where jl .z/ D p�=.2z/ JlC1=2 .z/ is the spherical Bessel function. We note that

jl .�z/ D .�1/l jl .z/.Suppose that r1, r2 and r3 are the sides of a triangle that the angle between the

sides r1 and r2 is equal to , where > 0 and r3 Dpr21 C r22 � 2r1r2 cos . There

is another expansion formula (see Gradshteyn and Ryzhik 2007, p. 940):

eikr3

r3D �i

2pr1r2

1X

lD0.2l C 1/ JlC 1

2.kr1/H

.1/

lC 12

.kr2/Pl .cos /

D i

1X

lD0.2l C 1/ jl .kr1/ h

.1/

l .kr2/Pl .cos / for r1 > r2; (A.12)

where h.1/l .z/ D p�=.2z/H.1/

lC1=2 .z/ is the spherical Hankel function of the firstkind. It’s imaginary part is

sin kr3r3

D1X

lD0.2l C 1/ jl .kr1/ jl .kr2/ Pl .cos / (A.13)

since h.1/l .z/ D jl .z/C i nl .z/.

Wigner 3-j Symbols

The integral of three spherical harmonic functions is written by using the Wigner3-j symbols (Landau and Lifshitz 2003, p. 444):

.Ylm/l 0m0

l 00m00 �I

Y �l 0m0 .�; '/ Ylm .�; '/ Yl 00m00 .�; '/ d˝ .�; '/

D .�1/m0

i l�l 0Cl 00l 0 l l 00

�m0 m m00�

l 0 l l 000 0 0

�

�p

14�

�2l 0 C 1

�.2l C 1/

�2l 00 C 1

�:

(A.14)

460 A Spherical Harmonic Functions and Wigner 3-j Symbols

The above integral vanishes except whenm0 D mCm00 and jl � l 00j � l 0 � jl C l 00j(the triangular condition) according to the selection rules corresponding to theaddition of angular momenta. Then, lCl 0 Cl 00 is even. Wigner 3-j symbols are purereal. There are published programs for the numerical computation of the Wigner 3-jsymbols (Shriner and Thompson 1993; Wolfram 1991).

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Index

Acoustic impedance, 141Active seismic experiment, 276, 329Active volcano, 8, 38, 79, 278Addition theorem, 296, 435, 436, 458African plume, 118Alaska, 98Ambient noise, 3, 59, 401, 413Angular brackets, 11Angular correlation function, 330Angular spectrum function (ASF), 346Apparent duration, 5, 202, 377Approximate solution, 257Array analysis, 3, 45Asama volcano, 276ASO in Kanto, Japan, 51, 370Attenuation measurement, 99Autocorrelation function (ACF), 19, 20, 39,

118, 131, 189, 401

Back-arc side, 52, 106, 375Back scattering, 69, 142, 176, 307Backscattering coefficient, 7, 67, 133, 134,

198S-to-S backscattering coefficient, 147, 175

Basalt, 159Basic scattering pattern, 142, 173

body to body wave scattering, 151Rayleigh to body waves, 150

Basin and Range Province, 154Binary interaction approximation, 215Biot, 158Birch’s law, 16, 141, 172, 189Body-wave reflection, 203Body waves, 446Boltzmann equation, 222, 230Boltzmann lattice gas approach, 77

Born approximation, 125, 160, 161, 270, 370,430

Born scattering coefficients, 9, 315Boundary integral method, 77, 182Bounded 1-D medium, 425Bourret approximation, 217Branch cut, 254, 271, 273Brownian particle motion, 246b value, 110

California, 37, 85, 154, 329, 369Cartesian coordinates, 127, 137, 242, 347, 417Cavity, 77, 180Cell, 222Center-of-mass coordinates, 324Central Africa, 118Characteristic time, 156, 343, 350, 353, 354ChISS, 98Coda, 2, 41

attenuation, 67, 80, 81, 316deconvolution, 116, 117interferometry, 114magnitude residuals, 108normalization method, 92, 123, 154, 260Q, 67spectral ratio, 116

Coherence radius, 338Coherent wave, 336

energy, 236Collision zone, 57, 395Confluent hypergeometric function, 326Conservation of total energy, 72, 241, 253, 368Continuity equation, 129Conventional Born approximation, 161, 220Conversion between body and Rayleigh waves,

208

H. Sato et al., Seismic Wave Propagation and Scattering in the Heterogeneous Earth:Second Edition, DOI 10.1007/978-3-642-23029-5,© Springer-Verlag Berlin Heidelberg 2012

489

490 Index

Conversion scattering, 141, 148, 200, 235, 268,391

Convolution type reciprocity, 420Corner frequency, 97, 188Correction terms, 171Correlation between Q�1

c and the b-value, 111Correlation distance, 20, 53, 76, 131, 204, 246,

398Correlation of log-amplitude fluctuation, 323Correlation type reciprocity, 410, 418, 421,

436Coupled diffusion equation, 240, 2781979 Coyote Lake earthquake, 114Crack, 15, 77, 180Cross-correlation function (CCF), 59, 401, 443

of ambient noise, 443of random waves, 423of singly-scattered coda waves, 440

Crustal inhomogeneity, 205, 291Cutoff scattering angle, 167, 178Cylindrical wavelet, 362

Difference coordinates, 324Differential operator, 212, 443Differential scattering cross section, 65, 129,

144Diffraction

effect, 179, 324tomography, 37, 105

Diffusionapproximation, 237, 270, 278equation, 72, 239model, 8, 72, 73, 240, 276process, 316solution, 72, 132, 248, 256, 280

Diffusivity, 72, 73, 239, 280Directional distribution of mean energy

density, 230, 232, 233, 294Directional Green’s function, 295Direct simulation Monte Carlo method, 310Dislocation, 159Dispersion relation, 148, 211, 216, 220Double seismic zone, 119Doublet analysis, 113Duration magnitude, 42, 84Dyson equation, 216

Earthquakefault, 290swarm, 447

Eigenfunctions of Rayleigh wave, 148Einstein summation convention, 10

Elastic vector waves, 135, 315Elastic wave, 50, 143, 161, 185, 221, 270, 418Elastic wave case, 235, 240Elliptic theta function, 346, 351Emperor Seamount Chain, 120Energy density, 41, 84, 130, 191, 221, 270Energy partition, 281Energy-flux density, 65, 130, 188, 230Energy-flux model, 73Ensemble, 19, 162, 189, 211, 322, 381, 388

average, 27, 131, 189, 193, 214, 381of random media, 20of random noise, 403

Envelopebroadening, 6, 50, 52, 369correlation method, 122

Equipartitionof radiated energy, 422state, 241, 270, 281, 405

Equivalent body force, 136, 170, 172Exchanged wavenumber, 130, 139, 172Excitation of the orthogonal component, 55Expansion formula, 296Exponential ACF, 23, 132, 146, 162, 167, 175,

217

f -k analysis, 46, 89F-net, 90F-P duration time, 42, 45Far-field, 137, 422

displacement vector, 186Fast (micro) variables, 222Fenton Hill, 38, 57, 268FFT, 75, 274, 354Finite difference, 73, 75, 143, 179, 362, 390First Fresnel zone, 325First-order perturbation method, 125First-order smoothing method, 215Fluctuations of elastic coefficients, 170Fore-arc side, 6, 53, 106, 372Forward scattering, 52, 179, 328Fourier transform, 11, 20, 45, 128, 139, 142,

184, 222, 249, 383Fourier-Laplace

domain, 247transform, 253, 279

Four single-scattering modes, 192, 199Fractional fluctuation, 20, 77, 126, 139, 163,

189, 223, 390Fractional velocity fluctuation, 8Free surface, 56, 78, 148, 200, 203, 243, 312,

392Frequency-independent Q�1, 156

Index 491

Fresnel zone, 37, 339

Gauss hypergeometric function, 284Gaussian ACF, 22, 23, 26, 27, 325, 334, 339,

343, 350, 361, 362, 383, 389Gaussian source time function, 277General ACFs, 220General source and receiver locations, 68Geological evidence, 13Geometrical spreading factor, 153, 203, 275Geothermal area, 447Grain boundary mechanism, 159Grain size, 27, 160Granite, 27, 158Green’s function, 31, 127, 137, 149, 166, 212,

213, 233, 237, 268, 332, 402, 419,429

Green’s function retrieval, 61, 402Guided wave, 119Gutenberg-Richter formula, 84

Half-space, 179, 204, 369Harvard CMT, 293Helioseismology, 401Higher mode Rayleigh wave, 91, 303, 309Hilbert transform, 41, 413Hi-net, 60, 84, 96, 263, 396, 446Honshu, Japan, 38, 96, 159, 290, 372, 373, 446Horizontal-component seismograms, 93Hot dry rock, 38

Illumination by noise source, 405Inada granite, 58Incident plane

P-wave, 136, 171S-wave, 137wave, 65, 126, 136, 162

Infinitesimal shift of the pole, 128Integral equation, 181, 232Integral form, 232, 234, 246, 336Integral-differential equation, 232, 236Intensity, 53, 127, 340Intensity spectral density (ISD), 340, 381Intermediate scale, 337Intrinsic absorption, 159Intrinsic attenuation, 5, 49, 54, 73, 105, 153,

154, 258, 370Inversion for energy radiation, 290Irregular surface topography, 77Island arc, 57, 395Isochronal scattering curve, 305

Isochronal scattering shell, 69, 195, 204, 300Isotropic scattering, 7, 49, 72, 132, 246, 258,

282, 432, 433Isotropic scattering with conversion, 267Isotropic source radiation, 246, 294Iwate volcano, 114Izu Peninsula, 49, 372

Japan, 276, 291arc, 6Sea, 60

Jordan’s lemma, 128, 219, 254, 272, 427JST, 121

� value, 142Kamchatka, 108, 379Kanto, 27, 42, 47, 49, 51, 56, 93, 264Kanto–Tokai region, Japan, 52, 95, 259, 3721995 Kobe earthquake, 115KTB deep well, 27Kurikoma volcano, 376Kyushu, Japan, 18, 84, 155, 264, 447

Lame coefficients, 135, 379, 419Laplace transform, 11, 233, 255, 269, 272Lapse time, 1, 41, 84, 92, 193, 211, 261, 360,

425dependence, 86

Large Aperture Seismic Array (LASA), 118,329

Large cylindrical shell, 410Large spherical shell, 405, 433Laser Doppler vibrometer, 57Late coda, 309Lg-coda, 85Lg-wave, 97, 317Line-of-sight propagation, 320Lithospheric heterogeneity, 6, 54, 103, 370Local Born approximation, 214, 225Local earthquake, 56, 105, 175, 201, 243, 245,

413Localized velocity inhomogeneity, 125Local magnitude, 42, 96, 189, 260, 370Log-amplitude fluctuation, 321, 326, 328, 3291989 Loma Prieta earthquake, 111Long period band, 89, 303Los Angeles, 445Lower crust, 32, 85, 275Lower mantle, 8, 107, 315, 370Low-frequency events, 121Lunar rock, 159

492 Index

Lunar seismograms, 79

Major arc, 304Mantle, 54Markov approximation, 52, 315, 322, 336, 371Mass operator, 215Mean energy density, 229Mean free path, 6, 66, 316Mean Green’s function, 216Mean square (MS)

envelope, 41, 86, 185, 267, 391fractional fluctuation, 20, 26, 163, 214of the phase fluctuation, 324trace, 41

Mean wave, 214, 336Merapi volcano, 114Microearthquakes, 198Microseisms, 59, 63, 401, 413Mid-crustal reflector, 322004 Mid-Niigata earthquake, 447Minor arc, 304Modified Bessel function, 24, 248, 250, 352Moho, 1, 29, 32, 33, 77, 316, 371, 446Monte Carlo simulation, 259, 309, 356Moving window cross-correlation analysis,

114Mt. Dodaira, 93Mt. St. Helens volcano, 109Multiple isotropic scattering model, 8, 277,

308Multiple Lapse Time Window Analysis

(MLTWA), 258, 266, 267, 293Multiple scattering, 87, 132, 182, 202, 237,

245, 255, 278, 368, 440Multi-scaling, 222Mutual coherence function (MCF), 338

Nevada, 15, 99NIED, 2, 35, 44, 116Nikko area, northern Kanto, Japan, 205Nikko–Shirane volcano, 32Noise body forces, 422Noise source, 403Nonisotropic ACF, 26Nonisotropic random elastic media, 397Nonisotropic random media, 53, 244Nonisotropic scattering, 142, 185, 294Nonlinear site effect, 116, 447Nonspherical source radiation, 282Nonvolcanic deep low-frequency tremor, 121Non-volcanic tremor, 447NORSAR, 118, 329

North America, 118Northern California, 95North of India, 118NRCDP, 51Numerical simulation, 118, 134, 159, 198, 398,

450

Oceanic slab, 53, 106, 398Omega-square model, 188One-dimensional case, 246One-fiction approximation, 217Optical theorem, 184, 440, 450Orthogonality, 287, 296, 297, 434Oshima granite, 58

Pacific Ocean, 60Pacific plate, 38, 52, 53, 106Parabolic approximation, 52, 180, 319, 328Parabolic equation, 321, 347, 380, 386Parabolic wave equation, 339Parkfield, 37, 38, 113, 4462004 Parkfield earthquake, 447Particle motion trajectory, 55P-coda, 47, 49, 54, 77, 199, 267, 395Pennsylvania, 369Perturbed velocity, 20Phase fluctuation, 326, 329Phase screen method, 336Philippine Sea plate, 121, 446Phononic lattice solid method, 77Piezoelectric transducer (PZT), 57PKIKP, 117, 118PKiKP, 118, 119PKiKP precursor, 118PKP precursor, 118Plane P-wavelet, 380Plane S-wavelet, 386Plane wave, 45, 63, 130, 181, 192, 349Point scatterer, 432Point shear-dislocation source, 185, 189, 203,

288, 360Polarization, 58, 65, 137, 145, 193, 242, 315,

386of S waves, 235vector, 148

Porosity, 15Power law decay, 87Power spectral density function (PSDF), 8, 19,

20, 131, 146, 189, 191, 194PREM model, 8, 79, 309Principal value, 228Process of stationary increments, 21

Index 493

Prolate spheroidal coordinates, 69, 194, 300,414, 415, 438

Prolate spheroidal shell, 69P-to-P scatterer distribution, 103P-wave, 9, 54, 84, 136, 170, 185, 242, 278, 422

Q�1c , 3, 80, 86, 307

in China, 82in Japan, 83in the Eurasian continent, 83in United States, 82temporal changes, 109

IQ�1S , 258

ScQ�1S , 258

Q�1 factor of the mean wave, 220Q�1P , 102, 153, 181

for the lithosphere, 156Q�1P =Q�1

S for the lithosphere, 157Q�1S , 102, 153, 181, 258

for the lithosphere, 155Quasi-monochromatic wave, 341, 348Quaternary volcano, 53, 54, 376

Radiation angle, 69Radiation boundary condition, 128Radiation pattern, 49, 97, 186, 232, 282, 360Radiative transfer equation, 211, 222, 230,

246, 257, 294, 368Radon transform, 36Random angles, 359Random fluctuation, 13, 174, 189, 194, 322Random inhomogeneity, 18Random media, 1, 21, 64, 76, 131, 164, 185,

211, 315, 322Rayleigh wave, 6, 61, 79, 148, 208, 303, 444,

446Rayleigh wave coda, 88Ray-tracing, 36Realizations of random media, 21, 75, 328Receiver function, 34Reciprocity, 409, 420Reduction of independent parameters, 141Reflected S-phase, 106Reflection surveys, 30Refraction surveys, 29Regional earthquake, 370Residue, 219, 271, 345Resolvent formula, 418, 422RMS

envelope, 40, 120, 205, 276, 354velocity amplitude, 200

Rytov method, 321

Salt body, 32Salton Sea Trough, 445San Joaquin valley, 4452003 San Simeon earthquake, 446Scalar wave, 30, 75, 125, 153, 211, 335, 402,

413case, 232, 238

Scattered wave, 4, 67, 126, 129, 136, 162, 185,268, 361, 413

Scattergram, 18Scattering amplitude, 129, 130, 138, 192, 317,

370, 433Scattering angle, 69, 193Scattering attenuation, 4, 48, 66, 153, 160, 198,

220Scattering coefficient, 3, 65, 130, 145, 146,

160, 208, 231, 245, 328Scattering loss, 49, 173Scattering medium, 4, 64, 246, 401, 450Scattering strength parameter, 324S-coda, 3, 42, 49, 92, 93, 147, 175, 185, 258

attenuation, 3wave excitation, 64, 197

ScS Envelopes, 396Seismic albedo, 5, 258Seismic moment, 97, 188, 290Seismic-moment time function, 186Seismogram envelope, 3, 40, 49, 121, 185,

240, 274, 319Semblance coefficient, 47, 201SH-wave, 77, 179, 203Single backscattering model, 66Single isotropic scattering model, 68, 303Single scattering, 3, 50, 80, 178, 191, 204, 211,

299, 393term, 254, 271

Single station method, 100Site amplification, 93, 206, 293Site factor, 115Slow (macro) variables, 222Slowness, 35, 47, 89, 91, 329Smoothing method, 214Snell’s law, 203, 311SOFAR channel, 119Solid angle element, 10, 65, 229, 295, 406Source duration, 191Source energy spectral density, 188Source radiation, 92, 97, 185, 245, 360Source-receiver reciprocity, 403, 409, 410,

418, 420, 432Southern California, 110, 445Spatial variation, 103Specific intensity, 230Spectral decay method, 154

494 Index

Spectral density, 11Spherical coordinates, 20, 67, 126, 186, 217,

271, 347Spherical earth, 92, 303Spherical harmonic function, 246, 283, 297,

434, 458Spherical harmonics expansion, 283, 296Spherical P-wavelet, 396Spherical wavelet, 347Split-step solution, 357Spurious term, 440Stable continent, 57, 395Stacked RMS envelope, 87Stochastic ray-path method, 359Stress tensor, 135, 170Structure function, 21, 339Superposition of plane waves, 361, 380, 388,

392Surface waves, 444

envelope, 303SV-wave, 151, 179, 203S-wave, 4, 19, 79, 136, 145, 153, 186, 236,

258, 370, 422S-wave envelope, 5, 52, 370S-wave polarization, 147, 192, 243

T90, 3771976 Tangshan earthquake, 108Teleseismic P-wave, 34, 56, 103, 317, 329,

369, 395Temporal change, 107, 199, 239, 257, 367, 446

in site factors, 115Tethys trench, 118tF , 84tF�P , 84Thermal diffusivity, 160Thermoelastic effect, 159Thermoelasticity, 160Three-component RMS envelopes, 200Three-component seismograms, 49, 202, 260Three-dimensional case, 2512003 Tokachi-Oki earthquake, 115Total scattering coefficient, 3, 6, 64, 78, 132,

133, 147, 175, 231, 274, 368Total scattering cross-section, 66Transport scattering coefficient, 7, 73, 132,

240, 369Transverse component, 5, 57, 140, 187, 385Transverse correlation function, 330Transverse plane, 323, 339Travel-time correction, 170

corrected Born approximation, 165, 167,169, 194

corrected PP-scattering amplitude, 171corrected scattering amplitude, 172corrected SS-scattering amplitude, 172corrected wavefield, 166

Travel-time fluctuation, 4, 165, 179, 199, 327Triangular condition, 297, 460Trms , 377Tsukuba, 42Turbidity, coefficient, 328T waves, 119Twelve single-scattering modes, 203Two-dimensional acoustic random media, 178Two-dimensional case, 249Two-frequency mutual coherence function

(TFMCF), 340, 346, 361TYM, 290Type curves, 99

Ultrasonic waves, 57, 77Uniformly distributed noise sources, 437U. S. Pacific Northwest, 981971 Ust–Kamchatsk earthquake, 108

Vector wave, 169, 170, 315, 378scattering, 135

Velocity inhomogeneity spectra, 376Velocity ratio, 17, 138Velocity shift, 331Velocity source spectrum, 189Velocity tomography, 1, 34, 39, 445Ventura, 445Volcanic eruption, 108Volcanic front, 6, 52, 105, 372Volcano, 53, 205, 245, 275von Karman ACF, 23, 147, 176, 352, 364

Wandering effect, 315, 342Wave equation, 3, 37, 126, 166, 216, 379, 402,

419, 425, 430Wave parameter D, 325Well-log data, 18, 27, 116, 1411984 Western Nagano earthquake, 1092000 Western Tottori earthquake, 116Western U.S.A., 15, 99White spectrum, 329, 407, 413, 416, 425, 436,

439Wigner 3-j symbol, 297, 460