GEOPHYSICS, VOL. 70, NO. 2 (MARCH-APRIL 2005); P. D19–D28, 6 FIGS.
10.1190/1.1897028
Refraction across an angular unconformity between nonparallel TI media
John Hodgkinson1 and R. James Brown1,2
phase angles and phase velocities of the incident and refracted waves obey Snell’s law across the interface. We
also demonstrate, using auxiliary angles, that the description of refraction between elliptically anisotropic media by
stretching the media to make them isotropic, then applying
isotropic refraction, is also valid for our general angularunconformity case. However, both stretching (1D) and either scaling (2D) or shearing must be applied correctly and
separately to the two media.
The refraction algorithm developed from this theory and
another developed by Byun in terms of phase-velocity theory are currently the only published noniterative algorithms known to us for refraction across an angular unconformity where the axes of anisotropy are parallel neither to
each other nor to the interface.
Based on this theory, we have developed a demonstration program, AUXDEMOC, that computes the refractedray angles for any combination of parameters by the two
equivalent methods: (1) anisotropic refraction and (2)
stretching plus isotropic refraction. This program can be
downloaded from http://www.crewes.org/ under Free Software.
ABSTRACT
Assuming elliptical wavefronts, we reformulate refraction theory for transversely isotropic (TI) media based on
the use of the auxiliary angle, α, which is intermediate between the phase angle, θ , and the group angle, φ. When
considering the application of stretching to transform elliptically anisotropic media into isotropic media, the auxiliary
angle is a natural one to use because both θ and φ → α
under such stretching. Our present formulation for TI media makes the assumption of elliptical anisotropy, which is
valid generally for SH-waves but only as a special case for
P-and SV-waves, where, in the SV case, the only possible
ellipses are circles. Nevertheless, the theory has useful applications for P-waves over limited ranges of propagation
direction (e.g., in the short-spread approximation).
Our formulation provides explicit results for all angles
of incidence and for what we term an angular unconformity between two TI media, that is, for all orientations
of the axes of symmetry for each of the media, and for
all orientations of the interface, assuming these two axes
and the interface normal to lie in the same vertical plane.
Our conclusions have been verified by showing that the
INTRODUCTION
For example, Vander Stoep (1966) discussed the quasianisotropy resulting from the presence of unrecognized thin
high- and low-velocity layers, and applied an elliptical Pwave approximation in a limited range of angles about the
vertical. Hodgkinson (1970) demonstrated that for P-waves
with angles of incidence in the zone between 0◦ (vertical)
and 45◦ , quasi-anisotropy resulting from the unrecognized interbedding of dolomite (∼6700 m/s) and anhydrite (∼4600
m/s), could be represented by elliptical anisotropy to within
0.5%. And Helbig (1983) stated that even though P- and SVwavefronts “can never be ellipsoids if anisotropy is the result of lamellation, pieces of the wavefront can be represented
with sufficient accuracy by an ellipsoid.” Various others have
assumed P-wave elliptical anisotropy with this in mind (e.g.
Elliptical anisotropy
As shown by Daley and Hron (1979b) and noted by
Thomsen (1986), for the case of an SH-wave in a transversely
isotropic (TI) medium with a vertical symmetry axis (VTI),
the wavefronts in the 2D representation are always elliptical — for both weak and strong TI. Although this has long
been known to be untrue in general for P- and SV-waves in
TI media (e.g., Postma, 1955), the special case of elliptical Pwavefronts in TI media was often assumed in early studies in
order to simplify the mathematics (e.g. Postma, 1955; Uhrig
and Van Melle, 1955; Kleyn, 1956; Dunoyer de Segonzac and
Laherrere, 1959).
Manuscript received by the Editor December 26, 2001; revised manuscript received November 30, 2003; published online March 22, 2005.
1
University of Calgary, Department of Geology and Geophysics, Calgary, Alberta, Canada T2N 1N4. E-mail: jonhod800@hotmail.com.
2
Presently Faculty of Science and Technology, University of the Faroe Islands, P.O. Box 2109, FO-165 Argir, Faroe Islands. E-mail:
rjbrown@setur.fo; rjbrown@ucalgary.ca.
c 2005 Society of Exploration Geophysicists. All rights reserved.
D19
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D20
Hodgkinson and Brown
Daley and Hron, 1979a; Byun, 1982; Tsvankin and Thomsen,
1994; Slawinski et al., 2000, 2003).
The algebraic simplicity of elliptical anisotropy allows a logical extension of its theory to cover SH-waves refracted at an
angular unconformity or in other similar scenarios, using only
explicit expressions and ray (group) velocities. An evaluation
of the behavior of SH-waves in such a situation should lead to
a clearer understanding of the behavior of P-waves in a similar situation, in particular where the surface offset distance
is less than the target/reflector depth (see, e.g., Tsvankin and
Thomsen, 1994).
Helbig (1983), Meadows (1985), Blair and Korringa (1987),
Dellinger and Muir (1988) and Dellinger (1991) have demonstrated that elliptically anisotropic layers may be stretched in
such a manner as to transform wavefront ellipses into circles,
representing isotropy. Rays may then be refracted through
the resulting transformed image using isotropic Snell refraction across each interface. Helbig (1983) considered only cases
for which an axis of symmetry in the second (refracting)
medium is normal to the overlying interface. Dellinger and
Muir (1988) claimed that the same stretching principle also
extends to multiple layers, provided the stretching process is
“consistent across layer boundaries.” They gave a rough outline of a procedure for doing this and stated without proof
that their procedure could be applied for arbitrarily dipping
symmetry axes and interfaces. Below, we demonstrate the
validity of this stretching process analytically, in particular
for the case of an arbitrarily dipping interface and lowermedium symmetry axis, that is, for our angular-unconformity
model.
In this paper, we consider the 2D problem in which the vertical plane of propagation contains the symmetry axes and the
interface normal — therefore, also the dip direction — and we
extend the refraction analysis to cases of angular unconformities or other zones of vertically changing dips.
SNELL’S LAW FOR TRANSVERSELY
ISOTROPIC MEDIA
Thomsen’s (1986) γ , δ, and ε have become the parameters
of choice in describing a TI medium, but in discussing elliptical anisotropy, we follow Dunoyer de Segonzac and Laherrere
(1959) in defining the anisotropy, factor, k, as the ratio of maximum to minimum velocities. It is related to Thomsen’s (1986)
SH anisotropy parameter, γ , as follows. Using Thomsen’s notation,
γ =
2
(π/2) − β02
vSH
,
2β02
(1)
where vSH (π/2) is the horizontal SH velocity and β0 ≡ vSH (0)
is the vertical SH velocity. In the notation used in this paper,
this equation is written as:
γ =
k2V 2 − V 2
k2 − 1
,
=
2V 2
2
(2)
therefore,
k 2 = 1 + 2γ .
(3)
Figure 1 shows an elliptical wave propagating in a VTI
medium. The velocity of vertical propagation, V, is represented by the semiminor axis of the group-velocity ellipse, and
the horizontal velocity, kV, by the semimajor axis. The radius
vector AE specifies the group velocity, V(φ), at E; and the corresponding group or ray angle, φ, specifies the direction along
which the energy propagates. The phase velocity, v(θ ), at E
is the speed of propagation of the plane wave
that is tangent to the wavefront ellipse at
E, in the direction normal to the wavefront
specified by the angle θ. The angle AE K is
the auxiliary angle, α, as defined by Dunoyer
de Segonzac and Laherrere (1959). This angle proves particularly useful in the application of stretching in order to transform
elliptically anisotropic media into isotropic
media.
Appendix A uses Figure 1 to show that relationships among the angles φ, α, and θ are
tan φ = k tan α = k 2 tan θ.
(4)
The group (ray) velocity in the direction φ is
shown to be
V (φ) = V cos α/cos φ,
(5)
and the phase (wavefront) velocity in the direction θ is
v(θ ) = V (φ) cos(φ − θ ),
(6)
where v(θ) is the phase velocity corresponding to the group velocity V(φ). In terms of
the phase angle, θ, and the vertical velocity,
V, the phase velocity is
Figure 1. Relationship between group angle and phase angle for propagation in
an elliptically anisotropic medium.
v(θ ) = V
cos2 θ + k 2 sin2 θ .
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(7)
Refraction, Angular Unconformity, TI Media
Figure 2 shows an incident (upper) medium (medium 1) of
anisotropy factor k1 and vertical velocity V1 , separated from a
second medium (k2 , V2 ) by a horizontal plane interface that is
parallel to the major axes of the wavefront (group-velocity) ellipses in both the upper and lower media. The line CEN (Figure 2) would represent the advancing wavefront if medium 1
extended below the interface, whereas the line CM represents
the actual wavefront in medium 2 at the same instant. The
distance AC may be represented by the parameters of either
medium, allowing establishment of a relationship between α1
and α2 , and thus between φ1 and φ2 .
From equation A-9 of Appendix A, we
can write:
D21
A ray in medium 1 is incident on the interface at A at an
angle φ1 to the y-axis. The extended ray meets the primary
ellipse (characterized by k1 and V1 ) at E, and the wavefront
tangent at E represents the hypothetical plane wavefront at
time t had no interface existed. This plane-wave component
intersects the x-axis at C at an angle θ1 and meets the interface
at R. The tangent from R to the secondary ellipse (k2 , V2 ) represents the refracted plane wavefront in medium 2 after time t.
It meets the secondary ellipse at M and makes an angle θ2 with
the x -axis. The line QM is normal to the x -axis.
AC = V1 /sin α1 = V2 /sin α2 , (8)
where tan φ = k tan α for each of the two
media.
Equation 8 is clearly in the form of
Snell’s law for isotropic media of velocities
V1 = k1 V1 and V2 = k2 V2 for angles of incidence α1 and α2 , respectively. In isotropic
media, the angle of incidence is identical
with both the group and phase angles, φ and
θ. From equation 8, both are clearly equal
to theauxiliary angle as well (i.e., α = φ =
θ ). This could also be termed a generalized
Snell’s law for elliptically anisotropic media,
as demonstrated by Dunoyer de Segonzac
and Laherrere (1959). It is restricted to the
case in which both media have an axis of
symmetry parallel to the interface.
REFRACTION BETWEEN
NONPARALLEL TI MEDIA
A more general case is shown in Figure
3, that of two TI media separated by a refracting interface where, in general, the axes
of anisotropy are not parallel to each other,
nor is either axis parallel to the interface.
In this case, the primary ellipse (medium
1) is shown in black and represents elliptical anisotropy in the incident medium. Its
x- and y-axes lie along AX and AY, respectively. The secondary ellipse (medium
2) is shown in red and represents elliptical anisotropy in the refracting medium. Its
axes are rotated counterclockwise through
an angle λ and lie along AX and AY , the
x - and y -axes.
The interface AR between media 1 and 2
(Figure 3) is at an angle ω to the x-axis and,
in general, ω = λ. Positive angles are measured counterclockwise from their reference
axes. When dealing with refraction at a planar interface, the ray angle of incidence is
the angle between the normal to the interface and the ray. Here, the ray angle of incidence in medium 1 is φ1 − ω. Similarly, the
ray angle of refraction is φ2 + λ − ω.
Figure 2. Refraction across an interface between two TI media where the interface normal and the axes of anisotropy are all parallel.
Figure 3. Refraction across an interface between two TI media where the axes of
anisotropy are parallel neither to each other nor to the interface normal.
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D22
Hodgkinson and Brown
The line AM represents the refracted ray and forms an angle φ2 with the y -axis. The angle between the refracted ray and
the normal to the interface, considered to be the ray angle of
refraction, is φ2 + λ − ω.
As with the simpler anisotropic model of Figure 2, our objective here is to determine the distance AR along the refracting interface in terms of the parameters of both media, 1 and
2. In Appendix B, we derive an expression for sin α2 , equation
B-15:
sin α2 =
ac ±
b2 (a 2 + b2 − c2 )
,
a 2 + b2
(9)
(10)
The choice of sign of the square root in equation 9 is determined by the phase angle, θ1 . If the absolute value of (θ1 −
ω) exceeds π , then use plus before the square root; otherwise,
use minus.
Equations 9 and 10 relate α2 to α1 (through c) in terms of
k1 , k2 , V1 , V2 , λ, and ω, all of which are constants for a particular model, allowing α2 to be obtained directly from α1 . Then,
using equation 4, φ2 may be obtained from φ1 .
Alternatively, sin α1 may be obtained directly as detailed in
equation B-20:
sin α1 =
eg ±
f 2 (e2 + f 2 − g 2 )
,
e2 + f 2
(11)
where e, f, and g are defined in Appendix B.
Evaluation of equation 9
The value of c (equations 10 and B-13) takes on a range of
values as α1 changes from its critical value in one direction to
its critical value in the other. The square root in equation 9 is
zero when either b2 = 0 or c2 = a 2 + b2 , that is, for the two
following cases.
Case 1: b = k2 sin (λ − ω) = 0. — In this case, a = 1, sin α2 =
±c, λ = ω, and the interface lies along the x -axis for all models
in which the values of λ and ω are equal. Critical values of α1
occur when sin α2 = ± 1, so that the refracted ray lies along the
x -axis, which in this case coincides with the interface. Thus,
when b = 0, the ray lies along the positive x -axis (c = +1) or
along the negative x -axis (c = −1).
The condition cited by Helbig (1983) in his lemma referring
to the stretching of elliptically isotropic media is an example
of this case 1. When λ = ω, equation B-13 reduces to
sin α2 =
k2 V2
(sin α1 cos ω − k1 cos α1 sin ω).
k1 V1
sin α2 =
V2
sin α1 ,
V1
(13)
equivalent to equation 8, or Snell’s law in its simplest form.
Below, we also show that the restriction by Helbig (1983) to
cases where λ = ω (i.e., where the symmetry axis in medium
2 is parallel to the interface normal) is unnecessary. For this
general case, equation B-13 without the parameter substitutions becomes
k2 V2
sin α2 cos(λ − ω) + k2 cos α2 sin(λ − ω)
=
. (14)
sin α1 cos ω − k1 cos α1 sin ω
k1 V1
where a and b are defined in Appendix B and
k2 V2
c=
(sin α1 cos ω − k1 cos α1 sin ω).
k1 V1
In the case λ = ω = 0 (i.e., both symmetry axes are parallel
to the interface normal), equation 12 reduces to
(12)
Below, we show that this is a form of Helbig’s (1983) relationship for refraction after stretching the media for the case when
both axes of symmetry are normal to the interface (Figure 2).
Case 2: b =/ 0 and c2 = a2 + b2 . — In this case, sin α2 = ±a/c.
Thus,
tan α2 = ±a/b =
±1
;
k2 tan(λ − ω)
±1
tan(λ − ω)
φ2 =
(15)
therefore,
tan φ2 =
or
π
− (λ − ω),
2
(16)
and the refracted ray lies along the interface in the positive or
negative direction, making φ1 the critical angle. An exhaustive
analysis of critical angles is given in Appendix D.
The salient features of refraction at an angular unconformity are as follows: (1) the incident ray meets the primary ellipse at E, (2) the tangent to the primary ellipse at E meets the
interface at R, (3) the tangent from R to the secondary ellipse
meets the secondary ellipse at M, and (4) the line from A to
M represents the refracted ray.
Figure 4 shows an example in which λ < ω and the refracted
ray lies beyond the x -axis, in the zone between the x -axis and
the interface where the value of φ2 is greater than π /2. For the
case when c = ± a, sin α2 = ± 1; therefore, α2 = φ2 = ±π/2,
and the refracted ray lies along the x -axis in either a positive or negative sense. As the value of φ2 increases beyond
the value of π /2 (x -axis), the value of c increases beyond the
value of a, as indicated by the numbers in the panel below the
diagram. Similarly, for a case where the value of φ2 decreases
beyond the value of −π /2 (x -axis), the value of c decreases
beyond the value of –a. Thus, in each case, if the absolute
value of c exceeds that of a, the refracted ray lies in the zone
beyond the x -axis, between the x -axis and the interface, and
the absolute value of φ2 is greater than π /2.
The only information available from equation 9 is the value
of sin α2 , from which the distinction between angles greater
than or less than π /2 has been lost. Thus, when extracting a
value of α2 from sin α2 , the supplement of the angle must be
taken whenever the value of α2 exceeds π /2, and this is controlled by whether the absolute value of c exceeds that of a.
The specific conditions may be written: if c > a and b < 0,
then α2 = π − α2 , and if −c > a and b > 0, then α2 = −(π +
α2 ).
The lower panel of Figure 4 gives the parameter values used
for the model shown: φ and θ, as well as group and phase
velocities on each side of the refracting interface, and corresponding values for a, b, and c of equation 9. The phase angles
of the incident and refracted waves, relative to the interface
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Refraction, Angular Unconformity, TI Media
normal, are θ1 − ω and θ2 + λ − ω, respectively; the phase velocities on each side of the interface are v1 and v2 . The last line
in the panel demonstrates that
v(θ2 )
v(θ1 )
=
.
sin(θ1 − ω)
sin(θ2 + λ − ω)
(17)
Equation 17 is the standard form of Snell’s law, shown to be
valid for phase angles and phase velocities, confirming equation 9 (see, e.g., Slawinski et al., 2000).
ANISOTROPIC REFRACTION BY
STRETCHING ELLIPTICAL MEDIA
We consider two media, as in Figure 3. The symmetry axis
in medium 2 makes an angle of λ − ω with the interface normal. Without loss of generality, we specify a vertical axis of
symmetry in medium 1 (upper), being the direction of minimum velocity. The interface between the two media is a plane
dipping at an angle ω, and again we assume a 2D problem.
Medium 1 again has anisotropy factor k1 and minimum (vertical) velocity V1 ; similarly for k2 , and V2 in medium 2.
If the upper medium is stretched vertically (i.e., in one dimension) by a factor k1 , then the ellipse will be stretched to
become a circle and the medium becomes isotropic with a velocity of k1 V1 . The effect on the interface will be to increase its
dip from ω to ω , that is, rotate it counterclockwise by ω − ω
where
tan ω = k1 tan ω.
(18)
D23
We then stretch medium 2 by a factor k2 parallel to its
semiminor axis, a symmetry axis. Just as for medium 1, this
makes the medium isotropic, but because of the prior scaling,
its isotropic velocity, V2 , becomes
V2 = k2 V2
cos ω
.
cos ω
(21)
Just as stretching medium 1 both rotated and stretched the
interface, stretching medium 2 also rotates and stretches the
interface. In analogy with the previous stretch and with angles
as in Figure 3, the interface is now rotated clockwise by (λ −
ω) − (λ − ω) and stretched by a factor cos(λ − ω)/cos(λ −
ω) , where
tan(λ − ω) = k2 tan(λ − ω).
(22)
And just as before, we now have to scale medium 1 by this
factor to prevent distorting it. Thus, the isotropic velocity in
medium 1, V1 , is finally
V1 = k1 V1
cos(λ − ω)
.
cos(λ − ω)
(23)
To apply Snell’s law for refraction, the angle of incidence
must be taken with respect to the normal to the interface. Referring to Figure 3 for definitions of parameters, we can write
sin[α2 + (λ − ω) ]
sin(α1 − ω )
=
,
V2
V1
(24)
The effect of this vertical stretch on the
ray angle of incidence, φ1 (measured with respect to the vertical), will be to decrease it to
φ1 ≡ α1 (Figure 1), where
tan φ1 ≡ tan α1 = (tan φ1 )/k1 ,
(19)
analogous to equation A-4. The effect on
θ1 is not quite as straightforward, but it is
clear from Figure 1 that under the stretching, which takes E to E , θ1 will decrease to
θ1 ≡ α1 , and that
tan θ1 ≡ tan α1 = k1 tan θ1 ,
(20)
analogous to equation A-14. Therefore,
phase, group, and auxiliary angles, θ1 , φ1 , and
α1 , are now all equal.
Thus, we change the ray angle of incidence to α1 = φ1 = tan−1 [(tan φ1 )/k1 ] (equation 19) and the interface dip to ω =
tan−1 (k1 tan ω) (equation 18). In thus stretching medium 1, medium 2 is not unaltered.
The interface is not simply rotated in changing ω to ω but also stretched or scaled.
As Helbig (1983) has shown, any interface
segment of length before stretching will
be of length (cosω/cosω ) after stretching
— and the entire lower medium has to be
scaled isotropically (i.e., by the same factor in two dimensions) in order to keep
its ellipticity unchanged. This scaling magnifies but does not distort the medium.
Figure 4. Refraction across an interface where λ < ω, |c| > |a|, and the refracted
ray lies beyond the x -axis in the zone between the x -axis and the interface. Such
a case requires that the supplement of α2 be taken before conversion to θ2 . Angles
are defined in Figure 3; a and c are defined in Appendix B.
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D24
Hodgkinson and Brown
in which (α1 − ω ) and [α2 + (λ − ω) ] are the incidence angles
in the stretched (isotropic) media. Then from equations 21, 23,
and 24,
k2 V2 cos ω cos(λ − ω)
sin[α2 + (λ − ω) ]
=
.
sin(α1 − ω )
k1 V1 cos ω cos(λ − ω)
(25)
Using equations 18 and 22 in equation 25, we get
k2 V2
sin α2 cos(λ − ω) + k2 cos α2 sin(λ − ω)
=
. (26)
sin α1 cos ω − k1 cos α1 sin ω
k1 V1
This is exactly the same as equation 14 determined earlier by
straightforward geometry, demonstrating the correctness of
the stretching and scaling procedure followed here.
We concerned ourselves only with the refraction at the interface here and ignored deformation of the upper surface
of medium 1, on which seismic data might be acquired. In
general, this surface will have undergone stretching and rotation whose parameters can be readily determined in carrying
out the stretching and scaling procedure described above. A
rigid rotation of the entire model and a 2D scaling can then
restore the upper surface to its original attitude and metric.
The description by Dellinger and Muir (1988) of a “stretchand-shear” procedure for a similar angular-unconformity geometry as considered here was sparse on the explicit details
of their method in the multilayer case. We have followed the
totally equivalent stretch-and-scale procedure described by
Helbig (1983). Although Helbig introduced the isotropic (2D)
scaling concept, he considered only the special case of λ = ω.
ods: anisotropic, and stretching plus isotropic. Our conclusions
have been numerically verified by showing that the phase angles and phase velocities of the incident and refracted waves
obey the standard form of Snell’s law across the interface.
Among other things, the program computes all angles, displays the plots frame by frame, and demonstrates behavior
at near-critical angles. It has proved a valuable tool in investigating and implementing other refraction methods, for example, when used as a means of evaluating Thomsen’s anelliptical methods with δ set equal to ε to provide elliptical
waveforms. This program, AUXDEMOC, can be downloaded
from http://www.crewes.org/ under Free Software.
ACKNOWLEDGMENTS
J. Hodgkinson, wishes to acknowledge the guidance of
Peter E. Gretener of the University of Calgary for his suggestion of anisotropy as the topic for his M.Sc. thesis in 1970,
and the contributions of Peter J. Savage and John Pelletier of
PanCanadian Petroleum (now EnCana) for fostering the early
development of software for anisotropic ray-tracing programs.
R. J. Brown acknowledges support in part from the Natural
Sciences and Engineering Research Council of Canada and
the sponsors of the CREWES Project. We also acknowledge
the helpful comments provided by the referees, especially the
knowledgeable suggestions of Joe Dellinger.
APPENDIX A
RELATIONSHIPS AMONG φ, α, AND θ
SUGGESTED APPLICATIONS
This extension of theory allows the refraction analysis to be
extended to cases of angular unconformities or other zones
of nonuniform dip, such as in thrust-and-fold belts, clastic
wedges, deltaic environments, extensional basins, etc.
Our analysis is based on group (ray) velocities, rather than
phase velocities. It will be advantageous to use such a method
in scenarios where group velocities and not phase velocities
are observed; for example, in physical modeling, where knowledge of source and receiver positions yields knowledge of ray
directions (group angles). This will also be true in certain circumstances for field surveys; for example, in borehole surveys
with accurately located sources and receivers. If shot and receiver fold are low, it may not be possible to determine phase
velocities accurately.
CONCLUSIONS
Other authors have suggested methods by which rays may
be refracted across interfaces between TI media in similarly complex structural conditions and have written computer
code to model propagation in anisotropic and inhomogeneous
media. However, with the exception of Byun’s theory in terms
of phase velocities, we are not aware of any other published
explicit equations or algorithms to expedite their application.
This paper provides a means whereby ray tracing across such
structural zones may be readily and transparently performed.
To aid in this process, we have developed a demonstration program that computes the refracted-ray angles for any
combination of parameters (velocities, anisotropy parameters,
orientations of the symmetry axes, dip of the interface, and
incident-ray angle) by the two equivalent refraction meth-
Figure 1 shows an ellipse and its auxiliary circle, plotted on
x-y axes with the origin at A. If the ellipse is to represent the
vertical section of a wavefront for which the source is at the
origin, then the semiminor-axis will represent the vertical ray
velocity, V; the semimajor-axis will represent the horizontal
velocity, kV; and the equation of the ellipse will be
x2
k2V 2
+
y2
= 1.
V2
(A-1)
The auxiliary circle has a radius equal to kV. A vertical line
drawn through point E intersects the auxiliary circle at E and
the major axis of the ellipse at K (Figure 1).
If point E has coordinates (x1 , y1 ), then
x12
y12
+
= 1,
k2V 2
V2
or
2
y12 ≡ EK =
k 2 V 2 − x12
. (A-2)
k2
In triangle E AK,
2
E K = k 2 V 2 − x12 ,
therefore
E K
=k
EK
(A-3)
and
tan φ
AK E K
= k,
·
=
tan α
EK AK
or
tan φ = k tan α. (A-4)
The length of the radius vector AE represents Vφ , the group
(ray) velocity in the direction φ with respect to the vertical.
If m is the gradient of the ellipse at E (Figure 1), m has the
value
m=
dy
dx
=
E
−x1
.
k 2 y1
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(A-5)
Refraction, Angular Unconformity, TI Media
ering, as
The equation of the tangent to the ellipse at E is then
y − y1 =
−x1
(x − x1 ),
k 2 y1
D25
vθ = Vφ cos(φ − θ ).
(A-6)
From equations A-15, A-16, and A-17, one can obtain the
phase velocity as
which reduces to
xx1
yy1
+ 2 = 1.
2
2
k V
V
(A-7)
vθ = V
At C, on the x-axis,
y = 0,
(A-17)
x = AC = k 2 V 2 /x1 ,
(A-8)
or, since x1 = AK = AE sin α = kV sin α,
AC =
V
kV
=
,
sin α
sin α
(A-9)
defining V . At N on the y-axis (Figure 1), x = 0 in the equation
for the tangent to the ellipse, therefore
y = AN =
V2
V2
V
=
=
.
y1
cos α
EK
xx1 + yy2 = k V .
2
2
DERIVATION OF EQUATIONS 9 AND 11
Figure B-1 is an enlargement of a portion of Figure 3 showing details of the angles in three triangles (ACR, AHG, and
AHR), to each of which we apply the sine law. In triangle
ACR (in terms of medium 1),
AC
AR
=
.
sin(θ1 − ω)
sin θ1
(A-11)
AR =
AG · sin θ2
sin(θ2 + λ)
k2 V2
sin θ2
·
,
=
sin(θ2 + λ) sin α2
(A-13)
(A-14)
(A-15)
From equations A-3 and A-10, the group velocity, Vφ , in the
direction φ is given by
V cos α
EK
E K
=
=
.
cos φ
k cos φ
cos φ
AG =
k2 V2
.
sin α2
(B-4)
In triangle AHR,
AH · sin(θ2 + λ)
sin(θ2 + λ − ω)
k2 V2
sin θ2
.
=
·
sin α2 sin(θ2 + λ − ω)
AR =
(B-5)
From equations B-2 and B-5,
Then, from equation A-4,
tan φ = k tan α = k 2 tan θ.
(B-3)
since, by analogy with equation A-9,
or
tan α = k tan θ.
(B-2)
AH =
(A-12)
Thus, by equation A-9, the tangents from the ellipse at E, and
from the auxiliary circle at E , meet the x-axis at the same
point, C.
From the geometry of Figure 1, the angle ECK = θ, therefore, from equations A-9 and A-10,
sin α
tan α
AN
V
·
=
,
=
tan θ =
cos α kV
k
AC
k1 V1
sin θ1
.
·
sin α1 sin(θ1 − ω)
In triangle AHG (in terms of medium 2),
When y = 0, this tangent meets the x-axis at a distance x from
the origin, where
kV
k2V 2
k2V 2
=
.
=
x1
kV sin α
sin α
(B-1)
Then, using equation A-9,
(A-10)
By similar reasoning to that used above for the equation
for the tangent to the ellipse at point E, the equation for the
tangent to the auxiliary circle at point E , (x1 , y2 ), is
Vφ ≡ AE =
(A-18)
APPENDIX B
therefore
x=
cos2 θ + k 2 sin2 θ .
(A-16)
Several authors (e.g., Musgrave, 1970; Daley and Hron,
1979b) give the phase (wavefront) velocity in the direction θ,
measured from the vertical, that is, from the normal to the lay-
k1 V1 sin α2 sin θ1
sin(θ1 − ω)
=
·
.
sin(θ2 + λ − ω)
k2 V2 sin α1 sin θ2
(B-6)
where the angles (θ1 − ω) and (θ2 + λ − ω) are the phase angles relative to the interface in the lower and upper media,
1 and 2, respectively — here, the angles between the wavefront tangents and the interface (Figure B-1). This is the key
equation relating α1 to α2 in terms of the constants of a given
model: k1 , k2 , V1 , V2 , ω, and λ, and of the phase angles, θ1 and
θ2 , relative to the “fast” symmetry directions (the semimajor
axes).
Using equation A-14 to eliminate α from equation B-6 results in
V1 cos2 θ1 + k12 sin2 θ1
sin(θ1 − ω)
v1
= ,
= sin(θ2 + λ − ω)
v2
V2 cos2 θ2 + k22 sin2 θ2
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(B-7)
D26
Hodgkinson and Brown
which represents Snell’s law for phase velocities and angles
across the interface.
In order to obtain a relationship between α1 and α2 (Figure
3) in terms only of the model constants, we need to eliminate
θ1 and θ2 from equation B-6. We first let
R1 =
k12
+
tan2
α1
and
R2 =
k22
+
tan2
α2 . (B-8)
Then, again using equation A-14,
tan α1
and
R1
tan α1 cos ω − k1 sin ω
sin(θ1 − ω) =
.
R1
sin θ1 =
the solution to which is
sin α2 =
e sin α1 − f cos α1 =
(B-9)
(B-11)
(B-12)
Substituting equations B-11 and B-12 into equation B-6
leads to
k2 V2
· (e sin α1 − f cos α1 ) = c,
k1 V1
(B-13)
defining c. This equation then reduces to a quadratic in sin α2 :
(a 2 + b2 ) sin2 α2 − 2ac sin α2 + (c2 − b2 ) = 0,
k1 V1
(a sin α2 + b cos α2 ).
k2 V2
(B-17)
(B-14)
(B-18)
This reduces to a quadratic in sin α1 :
(e2 + f 2 ) sin2 α1 − 2eg sin α1 + (g 2 − f 2 ) = 0, (B-19)
the solution to which is
sin α1 =
a sin α2 + b cos α2 =
k1 V1
(a sin α2 + b cos α2 ). (B-16)
k2 V2
e sin α1 − f cos α1 = g.
(B-10)
and
e tan α1 − f
sin(θ1 − ω) =
.
R1
(B-15)
Then,
Let a = cos(λ − ω), b = k2 sin(λ − ω), e = cos ω, and f =
k1 sin ω. Then,
a tan α2 + b
,
R2
b2 (a 2 + b2 − c2 )
.
(a 2 + b2 )
Let
g=
sin(θ2 + λ − ω) =
Equation B-15 (equation 9 in the main text) expresses α2 in
terms of α1 (Figure 3) and the constants of the model.
In order to compute critical values of α1 , an expression is
needed for α1 in terms of α2 . Equation B-13 may be rearranged
as
Also,
tan α2
sin θ2 =
and sin(θ2 + λ − ω)
R2
tan α2 cos(λ − ω) + k2 sin(λ − ω)
=
.
R2
ac ±
eg ±
f 2 (e2 + f 2 − g 2 )
.
(e2 + f 2 )
(B-20)
Equation B-20 (equation 11 in the main text) expresses α1
in terms of α2 (Figure 3) and is used for the determination of
critical angles.
APPENDIX C
LOCATION OF INTERFACE RELATIVE TO
INTERSECTION POINTS OF ELLIPSES
In general, the two ellipses representing anisotropy in the
two media will intersect at two pairs of points, joined by two
diagonals. If the refracting interface passes
through one of these pairs of intersections,
it can be shown that this position of the interface represents the limiting case between
those models having critical angles and those
having no critical angles. We first derive an
expression for the angle, σ , between the positive x-axis and the line joining one pair of intersection points (Figure C-1). The equation
of the primary ellipse is
x2
y2
+
= 1.
k12 V12
V12
(C-1)
The equation of the secondary ellipse, referred to the same axes as the first, is
(x cos λ + y sin λ)2
k22 V22
Figure B-1. Enlarged portion of Figure 3, showing details of the angles in triangles
ACR, AHG, and AHR.
+
(y cos λ − x sin λ)2
= 1, (C-2)
V22
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Refraction, Angular Unconformity, TI Media
D27
and the equation of a line through the origin, at an inclination
σ to the x-axis, is
the secondary ellipse lies inside the primary ellipse. Such a
model has no critical angles.
y = x tan σ.
APPENDIX D
(C-3)
Solving these three equations simultaneously for tan σ yields
two values of σ : the angles of inclination with respect to the xaxis of the lines joining the two pairs of diametrically opposite
intersections of the ellipses. We get
tan σ =
sin λ cos λ(k22 − 1) ± (k2 V2 /V1 )
cos2 λ + k22 sin2 λ + Z/k12 − V12 /V22
Z
,
(C-4)
where
Z = sin2 λ + k22 cos2 λ −
k22 V22
.
V12
(C-5)
As expected, the values of σ are dependent only upon the values of λ, k, and V. The lines joining the intersection points are
shown as S1 and S2 (Figure C-1).
We next set up the situation in which an incident ray lies
along the interface and the interface lies along the line joining
the intersection points of the two group-velocity ellipses (i.e.,
along S1 or S2). This requires two conditions: (1) g = gc (as
found for the refracted ray lying along the interface), and (2)
g 2 = e2 + f 2 .
In explanation of the second condition;
when g 2 = e2 + f 2 , sin α1 = ± eg/(e2 + f 2 ) =
± e/g, from which it follows, using equation
B-18, that
sin α1 =
±e
,
e sin α1 − f cos α1
CRITICAL ANGLES AND EVALUATION
OF EQUATION 11
Critical angles are controlled by the location of the interface
with respect to the points of intersection of the group-velocity
ellipses representing the two media (Figure C-1). Unless one
ellipse is wholly contained within the other for all orientations
of the two, the ellipses will intersect at two pairs of points lying at opposite ends of two diagonals, S1 and S2 (Figure C-1).
Appendix C describes how to determine the angles (σ1 and σ2 )
made by these diagonals with the x-axis.
Since wavefront (group-velocity) surfaces and phaseslowness surfaces are the polar reciprocal surfaces of each
other (see, e.g., Crampin, 1989; Helbig, 1994), one wavefront
surface wholly containing the second corresponds to the first
phase-slowness surface being wholly contained by the second.
Slowness surfaces have been used to examine anisotropic refraction, Snell’s law, critical angles, etc. (e.g., Helbig, 1994;
Slawinski et al., 2000), and our wavefront-ellipse approach is
wholly analogous.
If the interface lies in the sector between S1 and S2 (Figure
C-1) in which the primary ellipse (medium 1) lies inside the
(C-6)
which reduces to
e
= ± tan α1 .
f
(C-7)
But by definition (Appendix B),
1
e
=
,
f
k1 tan ω
(C-8)
±1
= k1 tan α1 = tan φ1 .
tan ω
(C-9)
or
therefore,
φ1 =
π
∓ ω,
2
(C-10)
and the incident ray lies along the interface in
a positive or negative sense.
Thus, when the incident ray lies along the
interface, and the interface lies along S1 or S2,
there is no refraction of the ray. The condition
e2 + f 2 = gc2 represents the limiting condition
for models having critical angles and those
that do not. If e2 + f 2 < gc2 , the argument of
the square root in equation 11 is less than
zero, and the interface lies in a sector in which
Figure C-1. The lines S1 and S2 through the intersection points of the primary
and secondary ellipses divide the model into four sectors. Since the interface is
located in a sector in which the secondary ellipse lies inside the primary ellipse,
this model has no critical angles.
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D28
Hodgkinson and Brown
secondary ellipse, there will be critical angles on the incident
side of the interface. If the interface lies in the sector in which
the secondary ellipse lies inside the primary ellipse, as in Figure C-1, there will be no critical angles on the incident side of
the interface.
If the primary ellipse lies wholly within the secondary, all
orientations of the interface will give rise to critical angles on
the incident side, because, in all cases, the ray will have traveled from low to high velocity. If the secondary ellipse lies totally within the primary, the reverse is true, and there will be
no critical angles at any orientation of the interface.
Equation 11, for obtaining α1 from α2 , is used in practice
only to compute critical angles. To compute the critical angles
for a specific model, the condition is set for the refracted ray
to lie along the interface, which, from equation 16, is that
φ2 =
π
− (λ − ω),
2
from which
α2 = tan−1
(tan φ2 )
.
k2
(D-1)
The values of e and f are constant for any one model, and
the value of g for this critical value of α2 is the critical value,
gc , where
gc =
k1 V1
k2 V2
[sin α2 cos(λ − ω) + k2 cos α2 sin(λ − ω)].
(D-2)
This value of gc is then used in equation 11 to compute the
critical values α1+ and α1− , from which φ1+ and φ1− are obtained
using equation A-4.
After finding the value of sin α1 from sin α2 using equation 11, recovering the value of α1 from sin α1 may result in
an ambiguity, since sin(π /2+ε) = sin(π /2−ε). A resolution is
possible only if it is known from some other parameter which
ambiguity to accept. The other parameter in this context is
the relationship between e, f, and g: if gc >e and f > 0, then
φ1+ ⇒ π − φ1+ ; if gc > e and f < 0, then φ1− ⇒ −(π + φ1− ),
where φ1+ and φ1− represent the positive and negative critical
angles.
As demonstrated in Appendix C, the condition for a model
to have no critical angles is
e2 + f 2 < gc2 .
(D-3)
It is evident that control of the recovery of α2 from sin α2
and of α1 from sin α1 is essential. However, the criteria involved in the resolution of any ambiguities use values of a, b c,
e, f, and g, and all these values are readily available from the
computations of equations 9 and 11.
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