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Fundamentals of Seismic Wave Propagation

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�, Φ [0] ray phase term (5.4.28)
v(0) [M−1T2] Green particle amplitude coefficient (5.4.28)
P(0), t(0)j [L−2T] Green stress amplitude coefficient (5.4.28)
f (�)(ω) [T(1−�)/2] source spectral term (5.2.64)
MS – source excitation functions (8.0.12)
HHHR – receiver conversion coefficients (8.0.12)
0.2 Symbols xxi
Table 0.1. concluded. The error terms, EEEN and EEEHj , are for three dimensions.
They have an extra unit [L] in two dimensions.
Symbols Units Description Equation
lˆ, mˆ, nˆ [0] interface basis vectors (6.0.1)
w, W – velocity-traction vectors (6.1.1)
Ti j , TTT , R, T [0] reflection/transmission coefficients (6.3.4)
h [M−1/2LT1/2] interface polarization conversion (6.6.3)
� [L−2T2] q2β − p2 (6.3.54)
� – coefficient denominator (6.3.61)
Y [M−1L2T] admittance (9.1.28)
A – differential systems (6.3.1)
Q [0] interface ray discontinuity (6.3.22)
C [0] coupling matrix (6.7.6)
[γ ] [0] perturbation coefficients (6.7.22)
γ [L−1] differential coefficients (7.2.96)
r [0] component vector (7.2.7)
FFF – transformed source matrix (7.1.7)
F – plane-wave fundamental matrix (7.2.2)
P – plane-wave propagator matrix (7.2.4)
L – Langer matrix (7.2.132)
B [L−1T] Langer differential matrix (7.2.134)
AAA – Langer propagator (7.2.144)
X – phase propagator (7.2.202)
� [L] head-wave length (9.1.41)
γ [LT−1] Rayleigh wave velocity (9.1.63)
BJ K [L2T−2] ray perturbation terms (10.2.8)
EEEN [L−3] error in equation of motion (10.3.32)
EEEHj [M−1L−1T2] error in constitutive equation (10.3.33)
DDD(�) [M−1L4−2�T�] scattering dyadic (10.3.53)
�E [L−1] scalar Born error kernel (10.3.53)
�B [L−1T] scalar Born perturbation kernel (10.3.71)
�K [0] scalar Kirchhoff kernel (10.4.11)
xxii Preliminaries
0.3 Special functions
Special functions, both standard and non-standard, are very useful in the mathe-
matics of wave propagation. Many special functions – trigonometrical, hyperbolic,
Bessel, Hankel, Airy, etc. – are widely used and the definitions and notation are
(almost) standardized. We follow the definitions given in Abramowitz and Stegun
(1965). Other functions are not as widely used and some are unique to this book.
Table 0.2 lists these and where they are first used or defined.
Table 0.2. Special functions, description and equations where used first.
Function Description Equation
[. . .] saltus function (2.4.1)
unit(. . .) = [. . .] unit function Section 0.1.3
[. . . , . . .] commutator (6.7.12)
{. . .} second-order minors (7.2.205)
Re(. . .) real part (3.1.10)
Im(. . .) imaginary part (3.1.13)
sgn(. . .) sign (scalar) (2.1.9)
normalized (vector) (0.1.2)
signature (matrix) (10.1.44)
δ(t) Dirac delta function (3.1.19)
�(t) analytic delta function (3.1.21)
δ(m)(t) m-th integral of delta function (5.1.2)
H(t) Heaviside step function (4.5.71)
B(t) boxcar function (8.4.13)
λ(t) lambda function H(t) t−1/2 (4.5.85)
�(t) analytic lambda function (5.2.82)
µ(t) mu function H(t) t1/2 (9.2.30)
Aj(x), Bj(x) generalized Airy functions (7.2.144)
J (t) two-dimensional Green function (8.1.37)
C(t, x) Airy caustic function (9.2.46)
Fi(t) Fresnel time function (9.2.60)
F(t, y) Fresnel shadow function (9.2.61)
Fr(ω) Fresnel spectral function (9.3.60)
Tun(ω, ψ) tunnelling spectral function (9.3.41)
Tun(t, ψ) tunnelling time function (9.3.45)
Sh(m)(t) deep shadow funtion (9.3.101)
0.4 Canonical signals
The main objective of this book is to outline the mathematics and results nec-
essary to describe the canonical signals generated by an impulsive source in an
elastic, heterogeneous medium. Table 0.3 lists the first-motion approximations for
0.4 Canonical signals xxiii
Table 0.3. Signal types with their first-motion approximation and approximate
spectra for an impulsive point source.
Signal First-Motion Spectrum
Direct ray δ(t − R/c)/R eiωR/c/R
(homogeneous) (4.5.75)
Direct ray
(stratified) (5.7.17) (9.1.5) δ(t − T )/(dX/dp)1/2 eiωT /(dX/dp)1/2
Direct ray
(inhomogeneous) (5.4.27) δ(t − T )/R eiωT /R
Partial reflection T δ(t − T )/(dX/dp)1/2 T eiωT /(dX/dp)1/2
(interface) (9.1.5)
Partial reflection γ (p, ζ )|t=T,x=X
(stratified) (9.1.21)
Total reflection Re(T �(t − T ))/(dX/dp)1/2 T eiωT /(dX/dp)1/2
(interface) (9.1.55)
Turning ray – forward δ(t − T )/( − dX/dp)1/2 eiωT /( − dX/dp)1/2
branch (9.2.5)
Turning ray – reversed δ¯(t − T )/(dX/dp)1/2 i−sgn(ω)eiωT /(dX/dp)1/2
branch (9.2.9)
General ray (5.4.38) (6.8.3) Re (i−σT �(t − T )) /R i−sgn(ω)σT eiωT /R
Head wave (9.1.52) H(t − Tn)/�3/2n x1/2 − eiωTn /iω�3/2n x1/2
Interface wave Re
x−1/2�(t − Tpole) ∂p∂T −1
(9.1.75) (9.1.89)
Tunnelling wave (9.1.136) Re (GGG�(t − T˜ ))
Airy caustic ddt C
t − T˜ A, 2�a/(9X ′′A)1/3
ω1/6eiωT˜ A−iπ/4
(9.2.46) (9.3.53) × Ai (−(3ω�T˜ A/2)2/3)
Fresnel shadow ddt F
(t − T )/(T˜ 1 − T ), ± 1
(9.2.61) (9.3.60)
Deep shadow Sh(3)
3(t − T˜ 1)/(V1�1)3
(9.3.97) (9.3.101)
the various signals described in this text. It summarizes their salient functional
features, and cross-references the equations in the text where fuller results can be
found. The time domain and spectral results are given for the particle displace-
ment assuming an impulsive (delta function), point force in three dimensions.
More complete expressions without the first-motion approximation are given in
the text which remain valid more generally. The expressions in the table indicate
the type of signals expected. For brevity, only the most important factors from the
results are included, i.e. the terms that vary most rapidly. The complete expressions
can be found in the text in the equations given.
Numerical simulations of the propagation of elastic waves in realistic Earth mod-
els can now be calculated routinely and used as an aid to survey design, interpre-
tation and inversion of data. The theory of elastodynamics is complicated enough,
and models depend on enough multiple parameters, that computers are almost
essential to evaluate final results numerically. Nevertheless a wide variety of meth-
ods have been developed ranging from exact analytical results (in homogeneous
media and in homogeneous layered media, e.g. the Cagniard method), through
approximations (asymptotic or iterative, e.g. ray theory and the WKBJ method),
transform methods in stratified media (propagator matrix methods, e.g. the reflec-
tivity method), to purely numerical methods (e.g. finite-difference, finite-element
or spectral-element methods), in one, two and three-dimensional models. Recent
extensions of approximate methods, e.g. the Maslov method, quasi-isotropic ray
theory, and Born scattering theory and the Kirchhoff surface integral method ap-
plied to anisotropic, complex media have extended the range of application and/or
validity of the basic methods.
Although the purely numerical methods can now be used routinely in mod-
elling and interpretation, the analytic, asymptotic and approximate methods are
still useful. There are three main reasons why the simpler, approximate but less
expensive methods are useful and worth studying (and developing further). First,
complete numerical calculations in realistic Earth models are as complicated to
interpret as real data. Interpretation normally requires different parts of the sig-
nal to be identified and used in interpretation. Signals that are easy to interpret
are usually well modelled with approximate, inexpensive theories, e.g. geomet-
rical ray theory. Our intuitive understanding of wave propagation normally cor-
responds to these theories. As no complete, robust, non-linear inverse theory has
been developed, we must use simple modelling theories to interpret real data and
understand (and check) numerical calculations. Secondly, the analytic and approxi-
mate modelling methods allow the properties of different parts of the signals to

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