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Abstract

The only restriction on the values of the elasticity parameters is the stability condition. Within this condition, we examine Christoffel equation for nondetached $qP$ slowness surfaces in transversely isotropic media. If the $qP$ slowness surface is detached, each root of the solubility condition corresponds to a distinct smooth wavefront. If the $qP$ slowness surface is nondetached, the roots are elliptical but do not correspond to distinct wavefronts; also, the $qP$ and $qSV$ slowness surfaces are not smooth.
On Christoffel roots for nondetached slowness surfaces
Len Bos
, Michael A. Slawinski
, Theodore Stanoev
Abstract
The only restriction on the values of the elasticity parameters is the stability condition.
Within this condition, we examine Christoffel equation for nondetached qP slowness surfaces in
transversely isotropic media. If the qP slowness surface is detached, each root of the solubility
condition corresponds to a distinct smooth wavefront. If the qP slowness surface is nondetached,
the roots are elliptical but do not correspond to distinct wavefronts; also, the qP and qSV
slowness surfaces are not smooth.
1 Introduction
Since the studies of Rudzki (1911)1, characterizing shapes of wavefronts in anisotropic media has
been of interest to seismologists. Postma (1955) derived a condition for elliptical velocity dependence
in homogeneous transversely isotropic media that is equivalent to alternating isotropic layers. This
condition was generalized by Berryman (1979) for “any horizontally stratified, homogeneous material
whose constituent layers are isotropic.” The proof for nonexistence of ellipticity of qP wavefronts
in media resulting from lamellation came from Helbig (1979), in response to Levin (1978). Shortly
thereafter, Helbig (1983, p. 826) stated the following. (1) The wavefront of qP waves is never an
ellipsoid; (2) the wavefront of qSV waves is never an ellipsoid; (3) the wavefront of SH waves is
always an oblate ellipsoid. Lamellation, which is described by Helbig (1979, 1983) as fine layering
on a scale small compared with the wavelength, is tantamount to using the Backus (1962) average;
throughout this paper, we use the methodology of the latter.
We consider the three roots of the solubility condition of the Christoffel equation to which we refer
as Christoffel roots. These roots correspond to the wavefront-slowness surfaces of the three waves
that propagate in an anisotropic Hookean solid. Herein, we examine transversely isotropic media
that results from the Backus average of isotropic layers. We derive the conditions under which the
spherical-coordinate plots of the three roots are ellipsoidal; we refer to such roots as elliptical. In
accordance with polar reciprocity, the ellipticity of wavefront slownesses is equivalent to ellipticity
of wavefronts.
As it turns out, a necessary condition for the ellipticity of roots is the nondetachment of the qP slow-
ness surface. Although the Hookean solids that represent most materials encountered in seismology
exhibit a detached qP slowness surface, the existence of both detached and nondetached slowness
surfaces is, indeed, permissible within the stability condition of the elasticity tensor (Bucataru and
Slawinski, 2009). Mathematically, this condition is the positive definiteness of the elasticity tensor.
Dipartimento di Informatica, Universit`a di Verona, Italy, leonardpeter.bos@univr.it
Department of Earth Sciences, Memorial University of Newfoundland, mslawins@mac.com
Department of Earth Sciences, Memorial University of Newfoundland, theodore.stanoev@gmail.com
1This publication, which was presented to the Academy of Sciences at Cracow in 1911, has been translated with
comments by Klaus Helbig and Michael A. Slawinski; it appears as Rudzki (2003).
1
arXiv:1903.02514v1 [physics.geo-ph] 6 Mar 2019
2 Christoffel equation in Backus media
The existence of waves in anisotropic media is governed by the Christoffel equation; its solubility
condition is (e.g., Slawinski, 2015, Section 7.3)
det
3
X
j=1
3
X
`=1
cijk` pjp`δik
= 0 , i, k = 1 ,2,3,
where cijk` is a density-scaled elasticity tensor and pis the wavefront-slowness vector. The three
roots of this bicubic equation can be stated as the expressions for the wavefront speeds of the qP ,
qSV and SH waves.
Let us consider a homogeneous transversely isotropic medium, whose elasticity parameters are
cTI =
cTI
1111 cTI
1122 cTI
1133 0 0 0
cTI
1122 cTI
1111 cTI
1133 0 0 0
cTI
1133 cTI
1133 cTI
3333 0 0 0
0 0 0 2 cTI
2323 0 0
0 0 0 0 2cTI
2323 0
0 0 0 0 0 cTI
1111 cTI
1122
.(1)
Herein, superscript TI indicates transverse isotropy resulting from the Backus (1962) average. Using
n3= sin2ϑto express the wavefront orientation, we parameterize the expressions of the three
roots (e.g., Slawinski, 2015, equation (9.2.19), (9.2.20)) as
vqP,qS V =v
u
u
tcTI
3333 cTI
1111(1 n3) + cTI
1111 +cTI
2323 ±
2ρ(2)
and
vSH =scTI
1212 n3+cTI
2323 (1 n3)
ρ,(3)
where ∆ = a(n3)2+b n3+c, with
a=cTI
1111 + 2 cTI
1133 +cTI
3333cTI
1111 2cTI
1133 4cTI
2323 +cTI
3333,(4a)
b= 2 cTI
1111 cTI
2323 2cTI
1111 cTI
3333 + 4 cTI
11332
+ 8 cTI
1133 cTI
2323 + 6 cTI
2323 cTI
3333 + 2 cTI
33332
,(4b)
c=cTI
2323 cTI
33332
.(4c)
The reciprocal of root (3) is elliptical. The reciprocals of roots (2) are elliptical if and only if ∆
is a perfect square; in other words, if and only if the expression is pd+fsin2ϑ, where dand f
are nonzero real constants. This happens if and only if the discriminant of ∆ , which we denote by
Disc (∆) , is zero. In view of expressions (4a)–(4c),
Disc (∆) := 16 cTI
1133 +cTI
23232cTI
1111 cTI
2323 cTI
3333+cTI
11332
+ 2 cTI
1133 cTI
2323 +cTI
2323 cTI
3333= 0 .
2
The solutions are
cTI
2323 =cTI
1133 ,(5a)
cTI
2323 =cTI
1133 and cTI
1111 =cTI
2323 ,(5b)
cTI
3333 =cTI
11332
+cTI
1111 cTI
2323 + 2 cTI
1133 cTI
2323
cTI
1111 cTI
2323
; (5c)
solution (5b) is a special case of solution (5a) but we keep both for convenient referencing below.
For the Backus average, solution (5b) cannot be satisfied within the stability condition; it would
require Pn
i=1 (1/(c1111)i) = Pn
i=1 (1/(c2323)i),which is not allowed. Solution (5c) can be satisfied
if and only if c2323 is the same for all layers, which results in an isotropic average (Backus, 1962,
Section 6).
If we consider a stack of nisotropic layers, whose elasticity parameters are
c1111 ={(c1111)1, . . . , (c1111)n}and c2323 ={(c2323)1, . . . , (c2323)n},
the stability condition for each layer is (e.g., Slawinski, 2018, Exercise 5.3)
(c1111)i>4
3(c2323)i>0, i = 1 , . . . , n .
The Backus-average elasticity parameters are
cTI
1111 =12
nY2
n W 1+4
n(UZ),(6a)
cTI
1122 =12
nY2
n W 1+2
n(U2Z),(6b)
cTI
3333 =n W 1,(6c)
cTI
1133 =12
nYn W 1,(6d)
cTI
2323 =n V 1,(6e)
cTI
1212 =n1U , (6f)
where
U:=
n
X
i=1
(c2323)i, V :=
n
X
i=1
1
(c2323)i
, W :=
n
X
i=1
1
(c1111)i
, Y :=
n
X
i=1
(c2323)i
(c1111)i
, Z :=
n
X
i=1
((c2323)i)2
(c1111)i
.
(7)
A standard form of these parameters is given by, for example, Slawinski (2018, Section 4.2.2); the
expressions, therein, and those of parameterizations (6), are equivalent to A,B,C,F,L,M
of Backus (1962, equations (13)), respectively. The stability of the Backus average is inherited from
the stability of the layers (Slawinski, 2018, Proposition 4.1); in other words, if the layers are stable,
so is the average.
3 Christoffel roots
Solutions (5a)–(5c) can be written in terms of parameterizations (6) as
64 (n(V+W)2V Y )2n22n Y +U(WV) + V Z W Z +Y2
V3W3= 0 ,
3
whose solutions are
Y=n
21 + W V 1,(8a)
W=Vand Y=n , (8b)
Z=n2+U V U W + 2 n Y Y2
VW; (8c)
again, solution (8b) is a special case of solution (8a) but we keep both for convenient referencing
below. We proceed to prove the existence of solution (8a) for n>4 layers, followed by a numerical
example to illustrate the result.
3.1 Nondetachment
Within the constraints of the stability, both detachment and nondetachment are permitted. The
nondetachment of the qP slowness surface occurs if and only if cTI
2323 =cTI
1133 . From solution (5a),
and its parameterization (8a), it follows that roots (2) are elliptical. Hence, we have the following
lemma.
Lemma 1. There exists a Backus average of at least four isotropic layers for which the Christoffel
roots are elliptical.
Proof. We fix a > 4
3and let x(0 ,1] so that
c1111 =na
x,a
x,a
x,a
xoand c2323 =x , 1
x,1
x,1
x.(9)
Following defintions (7), variables Y , V , and Wof solution (8a) become
Y(x) = x2
1
a+
n
X
i=2
(c2323)i
(c1111)i
=x2
a+
n
X
i=2
1/x
a/x =x2
a+n1
a,
V(x) = 1
x+
n
X
i=2
1
(c2323)i
=1
x+
n
X
i=2
1
1/x =1+(n1) x2
x,
W(x) =
n
X
i=1
1
(c1111)i
=
n
X
i=1
1
a/x =n x
a.
We define X(x) := n
21 + W(x)V(x)1, which results in
X(x) = n
2 1 + n x
a1+(n1) x2
x1!=n
21 + 1
an x2
1+(n1) x2.
As x0+, we have
Y(0+)n1
a3
4(n1) , X(0+)n
2,
and, hence, Y(0+)> X(0+) for aclose to 4
3. Furthermore, as x1,
Y(1)1
a+n1
aand X(1)n
21 + 1
a.
Thus, for n>4 , and aclose to, but greater than, 4
3,Y(1)< X(1) .
It follows form the Intermediate Value Theorem that there exists an x(0,1) for which Y(x) =
X(x),which completes the proof.
4
3.2 Numerical example
Let us consider a numerical example, where
x= 0.229852282578204 , a = 1.344000000000000 and Y=X= 2.271452434379770 ; (10)
indeed, there exists a Backus average of at least four isotropic layers, whose Christoffel roots are
elliptical.
Using results (10) with parameters (9), we obtain
c1111 ={5.8472 ,5.8472 ,5.8472 ,5.8472}and c2323 ={0.2299 ,4.3506 ,4.3506 ,4.3506};
the Backus-average parameters, following expressions (6), are
cTI
1111 = 3.6692 , cTI
1122 =2.9717 , cTI
3333 = 5.8472 ,
cTI
1133 =0.7936 , cTI
2323 = 0.7936 , cTI
1212 = 3.3204 .
(11)
The eigenvalues of tensor (1) with values (11) are λ1=λ2= 6.6409 , λ3= 6.0812 , λ4=λ5= 1.5873 ,
λ6= 0.4635 , which belong to a transversely isotropic tensor (B´ona et al., 2007a); since they are
positive, the stability condition of the average are satisfied. Also, Disc (∆) = 1.9883 ×1012 , which
can be considered zero, as required. Consequently, equation (2) becomes
vqP,qS V =v
u
u
t4.4628 + 2.1781 (1 n3)±q62.8718 (0.6373 n3)2
2ρ.(12)
Recalling that n3= sin2θ, and since ρmust be a positive scalar quantity, we see that equation (12)
is pd+fsin2ϑ, as required. Letting ρ= 1 , the three roots are
vqP =v
u
u
tcTI
3333 cTI
1111(1 n3) + cTI
1111 +cTI
2323 +
2ρ=p5.8472 + 5.0536 sin2ϑ(13a)
vqSV =v
u
u
tcTI
3333 cTI
1111(1 n3) + cTI
1111 +cTI
2323
2ρ=p0.7936 2.8756 sin2ϑ , (13b)
vSH =scTI
1212 n3+cTI
2323 (1 n3)
ρ=p0.7936 + 2.5268 sin2ϑ . (13c)
The reciprocals of expressions (13) are ellipses, illustrated in Figure 1. Therein, the grey curve
represents 1/vSH ; the black curves represent 1/vqP and 1/vqSV .
3.3 Interpretation
Although values (11) do not represent typical Hookean solid used in seismology, they satisfy the
stability condition and can appear in computational searches. To gain an insight into the appear-
ance of slowness curves in Figure 1, let us examine an example using the density-scaled elasticity
parameters for Green-River shale (e.g., Slawinski (2015, Exercise 9.3); Thomsen (1986, Table 1)),
cTI
1111 = 13.55 , cTI
1133 = 1.47 , cTI
3333 = 9.74 , cTI
2323 = 2.81 , cTI
1212 = 3.81 ,(14)
5
Figure 1: Three Christoffel roots resulting in three slowness curves
(a) (b) (c)
Figure 2: Slowness curves, 1/vqP , 1/vqSV , 1/vSH , for modified values of elasticity parameters for
Green-River shale. As cTI
1133 → −cTI
2323 , the innermost curve ceases to be detached and smooth.
6
where each parameter is scaled by 106; herein, superscript TI refers to an intrinsically transversely
isotropic medium, as opposed to TI , which is a Backus average.
The curves of 1/vqP , 1/vqSV and 1/vS H with values (14) are illustrated in Figure 2(a); such curves
are typical in seismology. The progression of Figures 2(a)–(c), however, illustrates an important
property. For detached qP slowness surfaces, the expressions for the qP ,qSV and SH wavefront
speeds, indeed, correspond to distinct smooth wavefronts. However, if cTI
1133 =cTI
2323 , the qP and
qSV slowness surfaces lose their smoothness. Also, their expressions—not only their curves—become
connected with one another; neither root corresponds to a distinct slowness curve nor does a given
curve result from a single root.
For each slowness surface, the criterion of belonging to a particular wave on either side of an intersec-
tion is not its belonging to a single root but the orientation of the corresponding eigenvectors, which
are the displacement vectors of a given wave (B´ona et al., 2007b). As can be readily shown, for the
innermost surface, the displacement vector is normal to it along the rotation-symmetry axis and in
the plane perpendicular to it; hence, it corresponds to the qP wave. Figure 2 supports heuristically
a rigorous statement based on the eigendecomposition theorem.
4 Ellipticity condition
According to Thomsen (1986), the ellipticity condition is ε=δ, where
ε=cTI
1111 cTI
3333
2cTI
3333
and δ=cTI
1133 +cTI
23232cTI
3333 cTI
23232
2cTI
3333 cTI
3333 cTI
2323,(15)
for either TI or TI . However, equations (13a)–(13c) lead to ellipsoidal forms, even though, therein,
ε=0.2363 6=0.4445 = δ. To avoid this discrepancy, we state the following proposition with a
qualifier.
Proposition 2. The detached qP slowness surface is ellipsoidal if and only if ε=δ.
Proof. Following expressions (15), ε=δif and only if
cTI
2323 =cTI
1133 and cTI
1111 =cTI
2323 or cTI
3333 =cTI
11332+cTI
1111 cTI
2323 + 2 cTI
1133 cTI
2323
cTI
1111 cTI
2323
,
which are solutions (5b) and (5c) , respectively. Solution (5a), cTI
2323 =cTI
1133 , is—in general—the
condition for nondetachment, and—for the Backus average—is the condition for elliptical roots. For
the Backus average, solution (5b) is not allowed within the stability condition and solution (5c)
results in an isotropic average, hence, circular roots. Thus, the ellipticity condition, ε=δ, is valid
for detached qP slowness surfaces only.
To gain an insight into Proposition 2, we modify parameters for Green-River shale—illustrated in
plot 3(a)—by applying expression (5c) to obtain an ellipsoidal qP slowness surface illustrated in
plot 3(b). Applying subsequently expression (5a), we obtain three elliptical Christoffel roots, where
ε6=δ, as expected in view of expressions (13a) and (13b). The innermost slowness surface is neither
detached nor elliptical, as illustrated in plot 3(c).
Let us apply these expressions in the opposite order. Using expression (5a), we obtain the result
illustrated in plots 2(c) and 4(b). Applying subsequently expression (5c), we obtain three elliptical
slowness surfaces, illustrated in plot 4(c). The stability conditions are satisfied but δis indeterminate.
7
(a) (b) (c)
Figure 3: Slowness curves, 1/vqP , 1/vqSV , 1/vSH , for modified values of elasticity parameters for
Green-River shale. Plot (b) corresponds to expression (5c); plot (c) corresponds to expression (5c)
followed by expression (5a).
(a) (b) (c)
Figure 4: Slowness curves, 1/vqP , 1/vqSV , 1/vSH , for modified values of elasticity parameters for
Green-River shale. Plot (b) corresponds to modifications by expression (5a); plot (c) is modified by
expression (5a) followed by expression (5c).
8
However, if we let cTI
2323 ≈ −cTI
1133 , as opposed to cTI
2323 =cTI
1133 , we obtain δ=ε, as a consequence
of detachment.
In contrast to these results, the Backus average—for either order of expressions (5a) and (5c)—does
not satisfy the stability condition.
5 Conclusion
The only restriction on the values of the elasticity parameters is the stability condition. Within this
condition, we examine properties of the Christoffel roots for nondetached qP slowness surfaces in
transversely isotropic media. The qP slowness surface is detached if and only if cTI
2323 6=cTI
1133 .
Under such a condition, each root corresponds to a distinct smooth wavefront. The qP slowness
surface is nondetached if and only if cTI
2323 =cTI
1133 . Under such conditions, the roots are elliptical
but do not correspond to distinct wavefronts; also, the qP and qSV slowness surfaces are not smooth.
Acknowledgements
The authors wish to acknowledge Ran Bachrach for fruitful discussions leading them to revisit the
classic theorem and statements of Helbig (1979, 1983), and Elena Patarini for her graphical support.
This research was performed in the context of The Geomechanics Project supported by Husky En-
ergy. Also, this research was partially supported by the Natural Sciences and Engineering Research
Council of Canada, grant 202259.
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10
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"The author dedicates this book to readers who are concerned with finding out the status of concepts, statements and hypotheses, and with clarifying and rearranging them in a logical order. It is thus not intended to teach tools and techniques of the trade, but to discuss the foundations on which seismology — and in a larger sense, the theory of wave propagation in solids — is built. A key question is: why and to what degree can a theory developed for an elastic continuum be used to investigate the propagation of waves in the Earth, which is neither a continuum nor fully elastic. But the scrutiny of the foundations goes much deeper: material symmetry, effective tensors, equivalent media; the influence (or, rather, the lack thereof) of gravitational and thermal effects and the rotation of the Earth, are discussed ab initio. The variational principles of Fermat and Hamilton and their consequences for the propagation of elastic waves, causality, Noether's theorem and its consequences on conservation of energy and conservation of linear momentum are but a few topics that are investigated in the process to establish seismology as a science and to investigate its relation to subjects like realism and empiricism in natural sciences, to the nature of explanations and predictions, and to experimental verification and refutation. A study of this book will help the reader to firmly put seismology in the larger context of mathematical physics and to understand its range of validity. I would not like to miss it from my bookshelf." Klaus Helbig
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A periodic structure consisting of alternating plane, parallel, isotropic, and homogeneous elastic layers can be replaced by a homogeneous, transversely isotropic material as far as its gross‐scale elastic behavior is concerned. The five elastic moduli of the equivalent transversely isotropic medium are accordingly expressed in terms of the elastic properties and the ratio of the thicknesses of the individual isotropic layers. Imposing the condition that the Lamé constants in the isotropic layers are positive, a number of inequalities are derived, showing limitations of the values the five elastic constants of the anisotropic medium can assume. The wave equation is derived from the stress‐strain relations and the equation of motion. It is shown that there are in general three characteristic velocities, all functions of the direction of the propagation. A graphical procedure is given for the derivation of these characteristic velocities from the five elastic moduli and the average density of the medium. A few numerical examples are presented in which the graphical procedure is applied. Examples are given of cases which are likely to be encountered in nature, as well as of cases which emphasize the peculiarities which may occur for a physically possible, but less likely, choice of properties of the constituent isotropic layers. The concept of a wave surface is briefly discussed. It is indicated that one branch of a wave surface may have cusps. Finally, a few remarks are made on the possible application of this theory to actual field problems.
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Shortly after his appointment to the first geophysical professorship (1895 at the Jagiellonian University of Cracow), Rudzki had published two papers in which he made a strong case for anisotropy of crustal rocks [Beitr. Geophys. 2 (1898); Bull. Acad. Sci. Crac. (1899)]. He had solved the Christoffel equation for transversely isotropic media in terms of what we today would call the ‘‘slowness surface’’. Rudzki regarded this as the representation of the wave surface in line coordinates. The conversion to point coordinates lead to an equation of 12th degree. Rudzki had determined a few points, but this was not sufficient to obtain an impression of the wavefront. Costanzi [Boll. Soc. Sismol. Ital. 7 (1901)] had suggested to simplify the coordinate conversion by expressing the solution of the Christoffel equation in a parameter form. The first part of the current paper describes the implementation of this idea. For the first time, the cusps in the wave surface became visible. The results of this first part have been discussed and expanded by Helbig [Beitr. Geophys. 67 (1958) 177; Bull. Seismol. Soc. Am. 56 (1966) 527; Helbig, K., 1994. Foundations of Anisotropy for Exploration Geophysics. Pergamon] and Khatkevich [Isv. Akad. Nauk. SSSR, Ser. Geofiz. 9 (1964) 788]. In a second part, Rudzki applied the ideas to orthorhombic media. The process is straightforward: the elements of the characteristic determinant are of order 2 in the three line coordinates (the three slowness components), with squares of coordinates in the diagonal elements and products of two coordinates in the off-diagonal elements. The elements are easily manipulated so that they are expressed in terms of squares only. Next, the determinant is expanded in terms of rows. This leads to three (equivalent) expressions. The vanishing of any of the three expressions means that the characteristic determinant vanishes, i.e., it corresponds to a solution of the Christoffel equation. Each of the equations can be used to determine one of the sheets of the line coordinates of the wave surface (point coordinates of the slowness surface). To this end, it is expressed in terms of two parameters, which have been chosen strictly for mathematical convenience. After conversion of the line coordinates to point coordinates (formation of the envelopes), one obtains a parameter expression for the wave surface. Until today, the second part of the Rudzki’s paper has not been closely studied. However, a blind test of the equations showed that they indeed describe the wave surface of orthorhombic media. The final sections discuss a few interesting aspects, among them the stability conditions for orthorhombic media and the condition under which a transversely isotropic medium transmits pure P- and S-waves.
Article
Compressional waves in horizontally layered media exhibit very weak long-wave anisotropy for short offset seismic data within the physically relevant range of parameters. Shear waves have much stronger anisotropic behavior. The author's results generalize the analogous results of Krey and Helbig (1956) in several respects: (1) The inequality (c//1//1-c//4//4)(c//3//3-c//4//4) greater than equivalent to (c//1//3 plus c//4//4)**2 derived by Postma (1955) for periodic isotropic, two-layered media is shown to be valid for any homogeneous, transversely isotropic medium; (2) a general perturbation scheme for analyzing the angular dependence of the phase velocity is formulated and readily yields Krey and Helbig's results in limiting cases; and (3) the effects of relaxing the assumption of constant Poisson ratio are considered. The phase and group velocities for all three modes of elastic wave propagation are illustrated for typical layered media with (1) one-quarter limestone and three-quarters sandstone, (2) half-limestone and half-sandstone, and (3) three-quarters limestone and one-quarter sandstone. It is concluded that anisotropic effects are greatest in areas where the layering is quite thin (1-50 ft), so that the wavelengths of the seismic signal are greater than the layer thickness and the layers are of alternately high- and low-velocity materials.
Article
Levin treats the subject concisely and exhaustively. Nevertheless, I feel a few comments to be indicated. My first point is rather general: of the three surfaces mentioned in the Appendix, the phase velocity surface (or normal surface) is easiest to calculate, since it is nothing but the graphical representation of the plane‐wave solutions for each direction. The wave surface has the greatest intuitive appeal, since it has the shape of the far‐field wavefront generated by an impulsive point source. The slowness surface, though apparently an insignificant transformation of the phase‐velocity surface, has the greatest significance for two reasons: (1) The projection of the slowness vector on a plane (the “component” of the slowness vector) is the apparent slowness, a quantity directly observed in seismic measurement. Continuity of wave‐fronts across an interface—the idea on which Snell’s law is based—is synonymous with continuity of apparent (or trace) slownesses; and (2) the slowness surface is the polar r...
Article
Assuming media having a velocity dependence on angle which is an ellipse, previously reported time-distance relations are confirmed for reflections from single interfaces, for reflections from sections of beds separated by horizontal interfaces, and for refraction arrivals, and the expression for diffractions is added. Expressions for plane-wave reflection and transmission coefficients at an interface separating two transversely isotropic media are also derived. None of the properties differs greatly from those for isotropic media. However, velocities found from seismic surface reflections or refractions are horizontal components. There seems to be no way of obtaining vertical components of velocity from surface measurements alone and hence no way to compute depths from surface data.
Article
Although the wavefronts of P and SV waves can never be ellipsoids if the anisotropy is the result of lamellation, pieces of the wavefront can be represented with sufficient accuracy by an ellipsoid. This representation allows a simple determination of the ratio 'zero-offset limit of stacking velocity/vertical velocity'. Constraints on the parameters of the thin layers that constitute a lamellated medium can be translated into constraints of the above velocity ratio. For P waves this ratio is centered around unity for a wide range of constituent parameters. -from Author