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A generalized form of Snell's law in anisotropic media

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We have reformulated the law governing the refraction of rays at a planar interface separating two anisotropic media in terms of slowness surfaces. Equations connecting ray directions and phase-slowness angles are derived using geometrical properties of the gradient operator in slowness space. A numerical example shows that, even in weakly anisotropic media, the ray trajectory governed by the anisotropic Snell’s law is significantly different from that obtained using the isotropic form. This could have important implications for such considerations as imaging (e.g., migration) and lithology analysis (e.g., amplitude variation with offset). Expressions are shown specifically for compressional (qP) waves but they can easily be extended to SH waves by equating the anisotropic parameters (i.e., ε=δ⇒γ ) and to qSV and converted waves by similar means. The analytic expressions presented are more complicated than the standard form of Snell’s law. To facilitate practical application, we include our Mathematica code.
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GEOPHYSICS, VOL. 65, NO. 2 (MARCH-APRIL 2000); P. 632637, 4 FIGS., 1 TABLE.
A generalized form of Snell’s law in anisotropic media
Michael A. Slawinski
, Rapha¨el A. Slawinski
, R. James Brown
,
and John M. Parkin
∗∗
ABSTRACT
We have reformulated the law governing the refrac-
tion of rays at a planar interface separating two aniso-
tropic media in terms of slowness surfaces. Equations
connecting ray directions and phase-slowness angles are
derived using geometrical properties of the gradient op-
erator in slowness space. A numerical example shows
that, even in weakly anisotropic media, the ray trajec-
tory governed by the anisotropic Snell’s law is signifi-
cantly different from that obtained using the isotropic
form. This could have important implications for such
considerations as imaging (e.g., migration) and lithology
analysis (e.g., amplitude variation with offset).
Expressions are shown specifically for compressional
(qP) waves but they can easily be extended to SH waves
by equating the anisotropic parameters (i.e., ε = δ γ )
and to qSV and converted waves by similar means.
The analytic expressions presented are more compli-
cated than the standard form of Snell’s law. To facilitate
practical application, we include our Mathematica code.
INTRODUCTION
The laws governing ray bending have been investigated since
the time of Ptolemy (2nd century A.D.). Prior to the work of
Snell (1591–1626), ray bending had been described by equating
the ratio of angles, and not their sines, to the ratio of veloci-
ties, quite accurate nevertheless for small angles. Snell’s law
extended the validity to all angles of incidence. However, it
assumes both media to be isotropic. In the present paper, we
extend Snell’s expression to encompass anisotropic media.
For simplicity, we assume horizontally layered media and a
dependence of velocity on angle of incidence that is the same
Presented at the 67th Annual International Meeting, Society of Exploration Geophysics. Manuscript received by the Editor March 30, 1998; revised
manuscript received September 27, 1999.
The University of Calgary, Dept. of Mechanical Engineering, Calgary, Alberta T2N 1N4, Canada. E-mail: geomech@telusplanet.net.
The University of Calgary, Dept. of Geology and Geophysics, Calgary, Alberta T2N 1N4, Canada. E-mail: slawinski@geo. ucalgary.ca; jbrown@geo.
ucalgary.ca.
∗∗
Formerly Baker Atlas International, Inc., Calgary, Alberta, Canada; presently PanCanadian Petroleum Ltd., 150–9th Ave. SW, Calgary, Alberta,
Canada. E-mail: john parkin@pcp.ca.
c°
2000 Society of Exploration Geophysicists. All rights reserved.
for all azimuths, that is, the media considered exhibit transverse
isotropy (TI) with a vertical symmetry axis (TIV). Although
this case has 2-D geometry, the general methodology is equally
applicable to three dimensions.
We proceed by two different routes: (1) by applying the con-
tinuity of k
x
, the component of the wave number, k, tangential
to the interface between two media, and (2) via Fermat’s princi-
ple, or the stationarity of traveltime between the source located
in one medium and receiver located in the other medium. Both
routes (the continuity conditions and the stationary traveltime
concept) yield the same expression for the refraction law, i.e.,
Snell’s law.
Thus, Snell’s law in horizontally layered media and for a
given frequency can conveniently be restated as the require-
ment that k
x
be continuous across the boundary. This property
holds for both isotropic and anisotropic media regardless of
the type of body wave generated at the boundary (longitudinal
or transverse) and forms the basis for our strategy of calculat-
ing reflected and transmitted angles. The approach following
Fermat’s principle can be formulated through the calculus of
variations (e.g., Slawinski and Webster, 1999).
Present exploration methods of data acquisition and pro-
cessing offer the potential to investigate certain subtle charac-
teristics of the subsurface. Consequently, rigorous and widely
applicable tools are needed. A refraction law for anisotropic
media contributes to more refined and efficient exploration
techniques.
METHOD AND RESULTS
The formulation presented in the next two sections (geomet-
rical and mathematical) states equations for a general velocity
anisotropy expressed in terms of 3-D Cartesian coordinates.
Geometrical formulation
Snell’s law can be illustrated using phase-slowness surfaces
for both incident and transmitted media (e.g., Auld, 1973;
632
Snell’s Law in Anisotropic Media 633
Helbig, 1994). The geometrical construction is facilitated by the
fact that the phase-slowness vectors of the incident, reflected,
and refracted waves are coplanar. Their being coplanar follows
from the kinematic requirement that boundary conditions be
satisfied at all times and at every point of the interface. For
TI therefore, and without any loss of generality, it is possi-
ble to choose a Cartesian coordinate system such that all the
phase-slowness vectors lie in the xz-plane. The familiar case of
isotropic media is considered in Figure 1.
Mathematical formulation
The geometrical approach illustrated in Figure 1 for the
isotropic case (i.e., spherical slowness surfaces) is extended
to include more general scenarios where the slowness sur-
face is an arbitrary surface in slowness space. Consider two
anisotropic media separated by a planar, horizontal interface.
Let the phase-slowness surface in the upper medium be given
by the level surface of a function f (x, y, z) in slowness space
FIG. 1. The geometrical construction yielding reflection and
transmission angles of slowness vectors in an isotropic medium
using the phase-slowness curve. The curves represent phase
slowness in the medium of incidence and transmission, whereas
the Cartesian axes x and z correspond to the horizontal and
vertical slownesses, respectively. The same concept applies in
an anisotropic medium except that the xz-plane cross-section
of the phase-slowness surface does not, in general, form a
sphere. The thin lines within the circles (radii) are collinear
with the phase-slowness vectors; the thick lines, normal to
the phase-slowness surface correspond to the group-slowness
vectors. The symbols ϑ
i
r
and ϑ
t
are the angles between
phase-slowness vectors for incident, reflected, and transmit-
ted waves and the normal to the interface; θ
i
r
and θ
t
are the
angles between group-slowness vectors for incident, reflected,
and transmitted waves and the normal to the interface (i.e., ray
angles). In the isotropic case, ϑ
i
= θ
i
r
=θ
r
,and ϑ
t
= θ
t
.
spanned by Cartesian coordinates x, y, and z:
f (x, y, z) = a. (1)
Similarly, let the phase-slowness surface in the lower medium
be given by the level surface of a function g(x, y, z) in slowness
space:
g(x, y, z) = b. (2)
In slowness space, x, y, and z have dimensions of slowness, as
does r, which we use below as a general symbol for slowness.
Consider a ray incident on the boundary from above. All
phase-slowness vectors (for incident, reflected, and transmitted
waves) must be coplanar so, without loss of generality, we can
take them to lie in the xz-plane (Figure 2). Denoting the phase-
slowness vector as m, the continuity conditions require that
m
i
·
¯
x = m
r
·
¯
x = m
t
·
¯
x, (3)
where
¯
x is a unit vector in the x-direction and subscripts
i, r, and t refer to incident, reflected, and transmitted waves,
respectively.
The group(ray)-slowness vector, w, is normal to the phase-
slowness surface. Using the property that the gradient points
FIG. 2. The geometrical construction yielding reflection and
transmission angles of slowness vectors in an anisotropic
medium using the phase-slowness curve. The curves represent
phase slowness in the medium of incidence and transmission,
whereas the Cartesian axes x and z correspond to the hor-
izontal and vertical slownesses, respectively. This is an illus-
tration of ray angles for incident, reflected, and transmitted
rays in anisotropic media separated by a horizontal, planar in-
terface using phase-slowness surfaces described by functions
f and g.Themvectors correspond to phase-slowness and
w vectors to group slowness; θ and ϑ correspond to ray an-
gle and phase angle, respectively for incident, reflected, and
transmitted waves. Note that ϑ
i
6= θ
i
r
6= θ
r
, and ϑ
t
6= θ
t
(cf. Figure 1).
634 Slawinski et al.
in the direction in which f increases most rapidly and that it is
normal to any surface of constant f gives
w
i
k∇ f(x,y,z)|
(x
i
,y
i
,z
i
)
, (4)
i.e., the ray vector, w, is parallel to the gradient. Normalizing,
and choosing the function f to have a minimum at the origin
O(0,0,0) and to be monotonically increasing outwards yields
¯
w
i
=−
f(x,y,z)|
(x
i
,y
i
,z
i
)
|∇ f (x, y, z)k
(x
i
,y
i
,z
i
)
, (5)
where the negative sign ensures that the incident unit ray vec-
tor,
¯
w
i
, points towards the boundary. From the definition of
dot product, the cosine of the angle of incidence (the angle be-
tween the ray vector and the normal to the interface) is given
by
cos θ
i
= (
¯
z) ·
¯
w
i
=
¯
z ·∇f(x,y,z)|
(x
i
,y
i
,z
i
)
|∇ f (x, y, z)k
(x
i
,y
i
,z
i
)
, (6)
where
¯
z is a unit vector in the z-direction.
The normalized transmitted ray vector is given by
¯
w
t
=
g(x, y, z)|
(x
t
,y
t
,z
t
)
|∇g(x, y, z)k
(x
t
,y
t
,z
t
)
, (7)
and thus the cosine of the angle of transmission (again with
respect to the interface normal) is given by
cos θ
t
= (
¯
z) ·
¯
w
t
=−
¯
z·∇g(x,y,z)|
(x
t
,y
t
,z
t
)
|∇g(x, y, z)k
(x
t
,y
t
,z
t
)
. (8)
In evaluating the above expression, one uses the facts that
the horizontal phase-slowness components are equal (i.e., x
t
=
x
i
), and that y
i
and y
t
are zero by the choice of the coordinate
system. Hence, z can be found by substituting x and y into
equations (1) or (2). Note that physical solutions must have
¯
w
t
· (
¯
z) 0.
While the phase-slowness vectors, m, are coplanar for the
incident, reflected, and transmitted waves, the ray vectors, w,
need not lie in the same plane. Their directions are deter-
mined by the normals to the phase-slowness surfaces. They will,
however, remain in the same plane if the phase-slowness sur-
faces are rotationally symmetric about the z-axis. If the phase-
slowness surface does not possess rotational symmetry, the in-
cident, reflected and transmitted group vectors need not be
coplanar. In such a case the angle of deviation, χ, from the
sagittal plane, assumed to coincide with the xz-plane and con-
taining all phase-slowness vectors, m, can be found by consid-
ering the projection, w
xz
, of the ray vector, w, on this plane:
w
xz
= [w · x,0,w · z]. (9)
From the definition of scalar (dot) product, it follows that
cos χ =
w · w
xz
|wkw
xz
|
. (10)
The geometrical formulation presented above can be
adapted for incident, reflected, or transmitted rays. Other con-
cepts, such as total internal reflection, also emerge naturally
from this formulation.
QUASI-COMPRESSIONAL WAVE
To illustrate the approach presented above, we derive a
refraction law for quasi-compressional (qP) waves. In equa-
tion (11), qP-wave phase velocity is expressed in terms of two
anisotropic parameters, δ and ε (Thomsen, 1986). In the mathe-
matical formulation, we have expressed angles of incidence and
transmission as a function of the ray parameter, m
x0
or x
0
, com-
mon to both media.
The phase velocity, v
qP
,ofaqP-wave in a weakly anisotropic
medium is given by Thomsen (1986) as
v
qP
(ξ) = α(1 + δ sin
2
ξ cos
2
ξ + ε cos
4
ξ), (11)
where α is the velocity for propagation perpendicular to the
interface, and δ and ε are anisotropic parameters. Here, the
phase angle, ξ, is the phase latitude, the complement of
the phase colatitude, ϑ, used by Thomsen (1986).
The slowness curve, m r, in the medium of incidence can
be expressed as
r(ξ) =
1
v
qP
(ξ)
=
1
α(1 + δ sin
2
ξ cos
2
ξ + ε cos
4
ξ)
. (12)
In the 2-D TIV case, using a standard expression for the
normal to a curve expressed in polar coordinates, we use an
equation relating ray angle of incidence, θ, and the phase lati-
tude, ξ:
θ = arctan
dr
dξ
r tan ξ
dr
dξ
tan ξ + r
, (13)
(e.g., Anton, 1984).
Knowing the characteristics of the medium of incidence
(α
i
i
i
) (using the subscript i or t to refer to the medium
of incidence or transmission, respectively), and given the ray
angle of incidence, θ
i
, equation (13) still can not, in general, be
solved explicitly for the corresponding phase angle, ξ
i
. It can,
however, be solved numerically (see Appendix A). Once the
phase angle, ξ
i
, is found, the corresponding ray parameter, x
0
,
can be calculated as
x
0
=
cos ξ
i
v
qP
(ξ
i
)
=
cos ξ
i
α
i
¡
1 + δ
i
sin
2
ξ
i
cos
2
ξ
i
+ ε
i
cos
4
ξ
i
¢
.
(14)
Having found the ray parameter, x
0
, which is continuous
across the interface, we calculate the phase latitude in the
medium of transmission, ξ
t
, from
x
0
=
cos ξ
i
v
qP
(ξ
i
)
=
cos ξ
t
v
qP
(ξ
t
)
=
cos ξ
t
α
t
¡
1 + δ
t
sin
2
ξ
t
cos
2
ξ
t
+ ε
t
cos
4
ξ
t
¢
. (15)
Equation (15) can be rewritten as a quartic in cos ξ
t
and solved
for the phase angle, ξ
t
;
α
t
x
0
(ε
t
δ
t
) cos
4
ξ
t
+ α
t
x
0
δ
t
cos
2
ξ
t
cos ξ
t
+ α
t
x
0
= 0,
(16)
where physically acceptable values of cos ξ
t
must be real and
in the range [0, 1].
Having found the phase latitude in the medium of transmis-
sion, ξ
t
, and knowing that the transmitted ray is normal to the
slowness curve, we can calculate the ray angle of transmission,
Snell’s Law in Anisotropic Media 635
θ
t
, explicitly from equation (13). The above method was per-
formed based on the weak-anisotropy assumption, given the
properties of the media of incidence and transmission.
We can also proceed using a more general mathematical for-
mulation. TIV media (i.e., 2-D velocity anisotropy) can be char-
acterized by a slowness curve. Consequently, for such cases,
the former approach is sufficient. For potentially more compli-
cated media (i.e., exhibiting a 3-D velocity anisotropy), a more
general method, involving slowness surfaces (as opposed to
slowness curves) is necessary. To illustrate its use based on the
TI-media case, let g(r) be a function in slowness space de-
fined by
g(r)=
1
r
α
t
¡
δ
t
sin
2
ξ cos
2
ξ + ε
t
cos
4
ξ
¢
, (17)
where r is the radius of the slowness surface (i.e., the magni-
tude of the slowness). The slowness curve is given by the level
surface of g(r), i.e., by the set of points (r) for which
g(r)= α
t
. (18)
The gradient can be expressed as
g(r)=r
g
r
+4
1
r
g
∂ξ
, (19)
where the angle is measured with respect to the x-axis, r is
the radial unit vector, and 4 is the azimuthal unit vector (i.e.,
perpendicular to the radius).
The propagation (ray, group) vector is always perpendicular
to the slowness surface (i.e., its direction is parallel to the gra-
dient of g). For g given by expression (17), the gradient can be
written as follows:
g(r)= r
µ
1
r
2
+4
Ã
α
t
sin
¡
2ξ
¢£
δ
t
cos(2ξ) 2ε
t
cos
2
ξ
¤
r
!
.
(20)
In polar coordinates, the Cartesian unit vector, z, can be ex-
pressed as
z = r sin ξ + 4 cos ξ. (21)
This form is used in the desired dot product, i.e., equation (8).
The transmitted group angle, θ
t
, which the group slowness vec-
tor makes with the normal to the interface, can be expressed in
terms of the given transmitted phase angle, ξ
t
, calculated using
equation (16), as
cos θ
t
=
α
t
cos ξ
t
sin(2ξ
t
)
¡
δ
t
cos(2ξ
t
) 2ε
t
cos
2
ξ
t
¢
sin ξ
t
r(ξ
t
)
s
1
£
r(ξ
t
)]
2
+ α
t
sin(2ξ
t
)
£
δ
t
cos(2ξ
t
) 2ε
t
cos
2
ξ
t
¤
2
,
(22)
where,
r(ξ
t
) =
1
α
t
¡
1 + δ
t
sin
2
ξ
t
cos
2
ξ + ε
t
cos
4
ξ
t
¢
. (23)
Numerical example
Consider a planar horizontal interface between two anisotropic
media. In the upper medium, the vertical wave speed is
α
1
= 3000 m/s, and the anisotropic parameters are ε
1
=−0.2
and δ
1
= 0.1. In the lower medium, the vertical wave speed is
α
2
= 4000 m/s, and anisotropic parameters are ε
2
= 0.15 and
δ
2
=−0.2.The ray strikes the interface from above (Figure 3)
at an incidence angle of θ
i
= 30
. The calculations (Appendix
A) yield the results in Table I.
Notice that based on vertical wave speeds, as would be the
case for α
1
and α
2
obtained from “zero-offset” vertical seismic
profiles (VSP), an incidence angle of 30
would yield (by the
standard form of Snell’s law) a refraction angle of 41.81
. This is
significantly different from the result of 64.01
obtained based
on the weak-anisotropy assumption.
Particularly complicated phenomena can be observed in the
case of converted waves (qP to qSV). For instance, at a flat
horizontal interface, a ray can bend towards or away from the
normal, depending on the angle of incidence (e.g., Slawinski,
1996). In particular cases, such bending leads to nonuniqueness
of raypath, i.e., given a source and receiver, there are several
stationary points of the traveltime function and consequently
several ray trajectories.
There are several ways to confirm the correctness of the solu-
tion (i.e., to verify that the results obtained using an algorithm
Table 1. Group and phase angles and velocities of incidence
and transmission.
Incidence Transmission
Group angles 30.00
64.01
Phase angles 35.57
51.53
Group velocities 3013 m/s 4133 m/s
Phase velocities 2998 m/s 4036 m/s
FIG. 3. Quasi-compressional (qP) wave slowness curves (i.e.,
cross-sections of corresponding slowness surfaces in a ver-
tical plane). Parameters are α
1
= 3000 m/s, α
2
= 4000 m/s,
ε
1
=−0.2
2
=0.15
1
=0.1
2
=−0.2. The outer curve corre-
sponds to the “slower” medium of incidence. The inner curve
corresponds to the “faster” medium of transmission.
636 Slawinski et al.
are in agreement with certain fundamental requirements). The
fulfillment of those requirements constitutes necessary condi-
tions for the validity of the method. First, the phase and group
angles, ϑ, and θ, as well as the magnitudes of phase and group
velocities, v and V, must satisfy the following equation in either
medium:
cos(θ ϑ) =
¯
¯
¯
¯
v(ϑ)
V(θ )
¯
¯
¯
¯
. (24)
This can be confirmed graphically with the help of Figure 4,
from which it is clear that
cos ζ = v/V. (25)
Second, the phase angles and phase velocities must satisfy the
following equation across the interface:
sin(ϑ
1
)
v
1
(ϑ
1
)
=
sin(ϑ
2
)
v
2
(ϑ
2
)
, (26)
where the subscripts 1 and 2 correspond to the upper and lower
media, respectively. Expression (26) is the standard form of
Snell’s law, which is always valid for phase angles and phase
velocities. Third, Fermat’s principle of stationary time must be
satisfied (e.g., Helbig, 1994). This might not be obvious from a
quick inspection, and it might require a reformulation in terms
of the Euler-Lagrange equation (e.g., Slawinski and Webster,
1999).
The approach involving exact formulas
In principle, it is possible to carry out all the derivations de-
scribed in this paper using exact equations. For qP-wave phase
velocity, the form equivalent to equation (11) is (Thomsen,
1986)
F(r) =
1
r
α
q
1+εcos
2
ξ + D(ξ) = 0, (27)
where D(ξ) is given by
D(ξ )
1
2
µ
1
β
2
α
2
×
v
u
u
u
u
u
u
t
1 +
4(2δ ε) sin
2
ξ cos
2
ξ
1
β
2
α
2
+
4ε
µ
1
β
2
α
2
+ ε
cos
4
ξ
µ
1
β
2
α
2
2
1
. (28)
DISCUSSION AND CONCLUSIONS
In this paper, we have illustrated the derivation and applica-
tion of the general Snell’s law using compressional (qP) waves.
The same approach, however, can be applied in an analogous
manner to both shear and converted waves (Slawinski, 1996).
For SH-waves, whose velocity dependence can be described
in terms of a single parameter, γ , one can quite conveniently
use an exact expression without the weak-anisotropy approxi-
mation. Notably, the exact expression for a refraction law with
elliptical velocity dependence (equivalent to SH-waves in TI
media) was presented as a result of a VSP study in the Sahara
desert (Dunoyer de Segonzac and Laherrere, 1959). Also, one
FIG. 4. The phase-velocity vector, v, is orthogonal to the wave-
front. The group-velocity vector, V, in an anisotropic yet ho-
mogeneous medium is collinear with the ray under considera-
tion. For an infinitesimal time increment, greatly exaggerated
here (l
0
to l
1
), V is the hypotenuse of a right triangle, and the
enclosed angle ζ is equal to θ ϑ .
can easily reduce expressions provided in this paper to the
weak-anisotropy SH case by equating parameters ε and δ.
ACKNOWLEDGMENTS
The authors acknowledge the considerable support of
the University of Calgary and the CREWES Project. Also,
the authors acknowledge the critical sponsorship of The
Geomechanics Project, namely, Baker Atlas, Integra Scott
Pickford, PanCanadian Petroleum, Petro-Canada Oil and Gas,
and Talisman Energy. In addition, we thank Don Lawton and
an anonymous reviewer for their critical review and cogent
suggestions, and Larry Lines for his stewardship of the review
process.
REFERENCES
Anton, H., 1984, Calculus with analytic geometry: John Wiley & Sons,
Inc.
Auld, B. A., 1973, Acoustic fields and waves in solids, volumes I and
II: John Wiley & Sons, Inc.
Snell’s Law in Anisotropic Media 637
Dunoyer de Segonzac, Ph., and Laherrere, J., 1959, Application of
the continuous velocity log to anisotropic measurements in north-
ern Sahara; results and consequences: Geophys. Prosp., 7, 202–
217.
Helbig, K., 1994, Foundations of anisotropy for exploration seismics:
Pergamon Press.
Slawinski, M. A., 1996, On elastic-wave propagation in anisotropic me-
dia: Reflection/refraction laws, raytracing and traveltime inversion;
Ph.D. thesis, Univ. of Calgary.
Slawinski, M. A., and Webster, P. S., 1999, On generalized ray param-
eters for vertically inhomogeneous and anisotropic media: Can. J.
Explor. Geophys., in press.
Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954–
1966.
Wolfram, S., 1991, Mathematica, a system for doing mathematics by
computer, 2nd edition: Addison-Wesley Publishing Co.
APPENDIX A
MATHEMATICA CODE FOR qP WAVES
Mathematica software can be used to investigate ray bend-
ing at the interface between two (TI) anisotropic media. Given
angle of incidence and parameters of the medium of inci-
dence and the medium of transmission, it provides the angle
of transmission, s. Note that Mathematica software is case-
sensitive.
Qi = angle of incidence in radians
VJ = vertical speed in the medium of incidence
VD = vertical speed in the medium of transmission
EJ = anisotropic parameter in the medium of incidence, ε
DJ = anisotropic parameter in the medium of incidence, ε
ED = anisotropic parameter in the medium of transmission, δ
DD = anisotropic parameter in the medium of transmission, δ
R = Qi (initial guess for the numerical solution)
the next step expresses the slowness curve in the medium of in-
cidence:
ri = 1/(VJ*(1+DJ*Sin[zi]
2*Cos[zi]
2+EJ*Cos[zi]
4))
the next step calculates the derivative w.r.t. phase angle in slow-
ness curve in the medium of incidence:
dri = Sin[2*zi]*(EJDJ*Cos[2*zi]+EJ*Cos[2*zi])/
(VJ*(1+EJ*Cos[zi]
4+DJ*Cos[zi]
2*Sin[zi]
2)
2)
next steps calculate the ray parameter, x:
FindRoot[Cot[Qi]==
(driri*Tan[zi])/(dri*Tan[zi]+ri), {zi,R,Pi,Pi}]
zif = Abs[zi/.%]
x = N[Abs[Cos[zif]/
(VJ*(1+DJ*Sin[zif]
2*Cos[zif]
2+EJ*Cos[zif]
4))]]
the next step calculates the phase latitude in the medium of trans-
mission, z:
FindRoot[VD*x*(EDDD)*C
4+VD*x*DD*C
2
C+VD*x == 0, {C, 0.5, 0, 1.5}]
z = ArcCos[C/.%]
dz = 2*z
r = 1/(VD*(1+DD*Sin[z]
2*Cos[z]
2+ED*Cos[z]
4))
t = VD*Cos[z]*Sin[dz]*(Abs[DD*Cos[dz]2*ED*Cos[z]
2])
Sin[z]/r
b = Sqrt[1/r
2+(VD*Sin[dz]*(DD*Cos[dz]
2*ED*Cos[z]
2))
2]
the next step calculates the angle of transmission,s
s=ArcCos[Abs[t/b]]
... illustrates the complex refractive indices and extinction coefficients of IOH, perovskite (CH3NH3PbI3), and ZnO. The perovskite material system and the metal oxide contact layers have a comparable refractiveindex. Hence, the reflection at the perovskite/metal oxide interface is low, and the entire layer stack exhibits a comparable refractive index.(Slawinski, Slawinski, Brown, & Parkin, 2000) ...
Thesis
Full-text available
Metal-halide perovskites are considered as one of the most exciting material systems due to their excellent optoelectronic properties. Notably, the multi-bandgap properties of perovskites have opened an emerging prospect for highly efficient tandem solar cell and color vision applications. So far, only perovskite-based tandem solar cells allow reaching energy conversion efficiencies exceeding 30% at low manufacturing cost. In this thesis, efficient solar cells and color sensors are studied based on metal-halide perovskite materials. Charge transport/contact layers have a significant impact on the electrical and optical properties of perovskite solar cells. Particularly, the front contact, which is a part of the junction of the solar cell, has to be efficient for realizing high energy conversion efficiency. The front contact must provide a lateral charge transport to the terminals and should allow efficient light incoupling while maintaining low optical losses. Hence, In the first part of the thesis, metal-oxides, such as titanium oxide (TiO2), nickel-oxide (NiO), zinc oxide (ZnO), etc., are investigated as potential front contacts for realizing efficient perovskite solar cells. High-quality metal oxide films are prepared by spray pyrolysis deposition (SPD), electron-beam physical vapor deposition (EBPVD), metalorganic chemical vapor deposition (MOCVD), and atomic layer deposition (ALD) techniques. As a first step, the study is carried out to investigate the planar perovskite solar cell performance with different front contacts, which is also used as a reference device structure for future investigations. Subsequently, the study is progressed to the textured perovskite solar cells, which combines the benefit of reaching high shortcircuit current densities and energy conversion efficiencies due to efficient photon management. Efficient photon management allows enhancing photon absorptions in perovskite solar cells through light incoupling and/or light trapping. Herein, light incoupling and light trapping are investigated with the integration of surface textures (e.g. moth-eye, pyramid, optical metasurfaces, etc.) on top of planar perovskite solar cells. A non-resonant optical metasurface is additionally studied as an alternative light-trapping structure for realizing efficient perovskite solar cells, where an array of ZnO nanowires is realized by the templated electrodeposition through a mask of resist. The complex requirements of perovskite solar front contacts and the effect of the front contact on the optics of perovskite solar cells are described in this part of the study. The optics of solar cells is investigated by 3D finite-difference time-domain (FDTD) optical simulations and the electrical effects of solar cells are inspected by the 3D finite element method (FEM). Detailed discussions for the realization of metal oxide films and the influence of photon management on the photovoltaic performance are provided. The second part of this thesis deals with detailed balance calculations and photon management of perovskite-based tandem solar cells. An extended Shockley– Queisser model is used to identify fundamental loss mechanisms and link the losses to the optics of solar cells. The influence of free-carrier absorption of metal oxide films on the optics of low bandgap and/or tandem solar cells is investigated. Herein, an optimized design is proposed for the perovskite/silicon tandem solar cell, which has the potential to reach energy conversion efficiency beyond 30% with a short-circuit current density exceeding 20 mA cm−2 while using realistic device geometry. A hybrid approach is used to investigate the optics of perovskite/silicon tandem solar cells by combining 3D finite-difference time-domain simulations with experimental measurements. Furthermore, multi-bandgap perovskites are employed as absorbers for investigating high-efficiency perovskite/perovskite tandem solar cells at low cost. Details on the nanophotonic design of perovskite-based tandem solar cells are provided. In the final part of this thesis, multi-bandgap perovskite materials are considered for the realization of efficient vertically stacked color sensors. The vertically stacked color sensor consists of three different energy bandgap perovskite diodes (channels), which allows exhibiting excellent color separation without having any color aliasing or color moiré error. The complex material properties of multi-bandgap perovskites are determined by the energy shift modeling. The quantum efficiency of the proposed vertically stacked color sensor is 3 times higher than the conventional filter-based color sensors. The current study focuses on the perovskite color sensor for achieving the quantum efficiency approaching 100%. The quantum efficiency of the investigated sensor is calculated by 3D finite-difference time-domain simulations. The study is further advanced to the realization of the multi-channel color sensor for detecting multispectral imaging, where six individual perovskite diodes are used for the sensor construction. The six-channel sensor outperforms all other characterized sensors. It enables the reconstruction of incident spectra that can be applied to a wide range of areas, such as health, communications, safety, and securities. The colorimetric characterization is performed based on the calculated spectral responsivities of the investigated color sensors. Details on the used materials, the device design, and the colorimetric analysis are provided.
... The above relation holds true for all angles of incidence with an assumption of both media being isotropic. Slawinski [60] discussed iso-and aniso-tropic media and extended Snell's law to encompass anisotropic media. In our research, we limit our discussion to the isotropic media only. ...
Thesis
Full-text available
The traditional RF communication is not effective at high-speeds. Radio blackout is commonly associated with the interference from plasma created around high-speed aero-space and re-entry vehicles. Communication using lasers at optical frequencies has emerged as a promising counter to the problem. High-speed flows are characterized by drastic changes in the thermodynamic properties across the shock wave in a flow. The refractive index of a medium, such as air, governs the angular shift in the trajectory of an optical signal. For a fluid, the refractive index is a function of the thermodynamic states. The flow gradients in the shock layer of a three-dimensional geometry, such as cone, are responsible for refractive index gradient which result in continuous refraction of the optical signal. These angular shifts by the shock layer and the shock wave are further summed with the angular shift by atmospheric stratification. The variation of refractive index within shock layer, across shock wave and atmosphere, induces angular shift. An analysis of the horizontal shift in the signal's destination on reaching the surface, for air-to-ground communication, is crucial. The curvature of the Earth is an important factor to determine range of communication angle, horizontal shift, and ensure interception of signal on the surface. The analysis can be extended to the laser weapons for targeting. This thesis presents the development of an analytical framework to investigate the aero-optical and atmospheric effects on an optical signal when targeted from an aero-space vehicle for surface communication. The framework excludes the effects of - turbulence, boundary layer, and chemical reactions. The atmospheric refraction has been validated with the Cassini's model. The entire formulation has been verified computationally for wedge and cone in high-speed flows that capture effects from the ow gradients. Computational implementation of the framework is available as an open-source application software, referred AbRuAn after my family members name (Abhinav, Ruchi, Anil), under the MIT License.
... Rays travel through the model considering the flexible step lengths (Wang 2013(Wang , 2014. Slawinski et al. (2000) and Sadri and Riahi (2010) presented the robust approach to finding the trajectory of ray path in the VTI media. Since the ray parameter p stays constant through the ray path, the phase angle for new points would be computed by checking the Snell's law through the ray trajectory by solving the 4th order equation as: ...
Article
For addressing the behavior of a reservoir with different fluid types, the Biot-Gassmann equation often is the base of the practical simulation. Despite the prevalence application of this equation with the isotropy conditions, the unforeseen errors always expose in simulation results because of the anisotropy state in reality. We investigated the anisotropy model with the integration of two analytical strategies using a three-component VSP data set. We obtained an initial anisotropy model in the region of acquired walkaway VSP using slowness polarization inversion, and updated the anisotropy model with the application of anisotropy ray-tracing and tomography. We applied a layer-stripping approach to the anisotropy model during raytracing to optimize the inversion. Given a computed geomechanical model and extracted rock properties of a carbonate reservoir, we developed the anisotropy Biot-Gassmann model, for finding the elastic moduli. We used the substitution strategy to generate the dynamic model of elastic moduli. We showed how the compressional modulus and rigidity change with the anisotropy model in different fluid content. We found that integration of slowness polarization and raytracing tomography increases the maneuverability to control the predicted anisotropy model and intensifies the convergence rate of the inverse problem. We observed that the isotropy assumption in modeling the elastic parameters makes around 8–10 % drift value in compressional modulus relevant to the reality, whereas rigidity showed reluctant behavior to fluid.
... Yan et al. (2016Yan et al. ( , 2019 analyze the uncertainties in laboratory anisotropy measurement, and they explain why some data sets have better quality control in estimating δ or c 13 than others do. After the VTI model is parameterized, anisotropic ray tracing is performed to calculate the theoretical reflection traveltimes using Snell's law in anisotropic media (Slawinski et al., 2000): ...
Article
Full-text available
Compared with isotropic media, at least two extra parameters are involved in common P-wave seismic data processing and interpretation for transversely isotropic media. Previous synthetic model testing has shown that it is challenging to estimate anisotropy parameters even using extremely low noise level seismic data from a simple geologic setting. Although theoretically independent, anisotropy parameters are not free variables for organic-rich mudrocks whose elastic properties are often approximated by transverse isotropy. One potential approach to improve the accuracy in the estimated anisotropy parameters is to consider the physical relationships between them during the inversion process. To test this proposition, we first modify a commonly used nonhyperbolic reflection moveout equation as a function of the interval anisotropy velocities so that rock-physics constraints could be effectively applied to each layer. The rock-physics constraints are established from data analysis of selected laboratory anisotropy measurement data. The laboratory data are then used to parameterize hundreds of 15-layer transverse isotropy models using a Monte Carlo simulation. The synthetic model testing indicates that the accuracy of the estimated anisotropy parameters can be improved if the relationships between the anisotropy parameters are considered during the inversion process.
... This formula is consistent with Refs. [31,32] . ...
Article
This study aims to extend the radiative energy transfer method (RETM) to anisotropic plates loaded by transverse point force at high frequencies. The flexural wave field of a medium is represented by energy density and intensity. The energy variables are superposed by incoherent rays emitted by actual source in the analyzed domain and fictitious sources on the domain boundary. By applying Fermat’s principle, we theoretically prove that ray paths are straight lines for rays traveling in unbounded homogeneous anisotropic media. The kernel functions of the energy transfer in the domain are derived by considering the energy balance at the loading point. To represent the radiative intensity in a free field, we define a directivity function, which can be determined by the stationary phase approximation of the far field solution. The diffuse reflection of rays at free boundaries leads to a Fredholm integral equation, from which the fictitious sources are determined. Numerical examples show that the energy flow fields and energy distributions of typical anisotropic plates are well predicted by the proposed approach.
... Вектора медленности отраженных и преломленных волн находятся из закона Снеллиуса для анизотропных сред [Slawinski et al., 2000], который в данном случае заключается в равенстве проекций векторов медленности на ось X падающей, отраженных и преломленных волн. Вектора поляризации находятся путем нахождения собственных векторов соответствующей матрицы Кристоффеля. ...
Article
Full-text available
The paper considers an algorithm for calculating reflection coefficients from boundary between two HTI media. Analysis of the presence of anisotropy above and below the target boundary, as well as variations in the parameters of HTI media, was done. Interpretation of reflection data from the boundary between two HTI media with neglect of anisotropy above or below potentially leads to significant errors in estimation of symmetry axes directions, and hence fracturing orientation. Overestimation/underestimation of an elastic parameter in the overlying HTI medium could lead to a corresponding overestimation/underestimation of similar parameter in the underlying target layer in the result of AVAZ inversion. Furthermore, among the anisotropy parameters Thomsen parameter γ has most significant influence on the reflection coefficients dependences. Thus, the parameter γ could be used foremost as a result of the AVAZ inversion.
... The perovskite material system and the metal oxide contact layers have a comparable refractive index. Hence the reflection at the perovskite/metal oxide interface is low and the entire layer stack exhibits a comparable refractive index [90]. ...
Article
Full-text available
Energy conversion efficiency losses and limits of perovskite/silicon tandem solar cells are investigated by detailed balance calculations and photon management. An extended Shockley–Queisser model is used to identify fundamental loss mechanisms and link the losses to the optics of solar cells. Photon management is used to minimize losses and maximize the energy conversion efficiency. The influence of photon management on the solar cell parameters of a perovskite single-junction solar cell and a perovskite/silicon solar cell is discussed in greater details. An optimized solar cell design of a perovskite/silicon tandem solar cell is presented, which allows for the realization of solar cells with energy conversion efficiencies exceeding 32%.
Preprint
In this paper we address two different related problems. We first study the problem of finding a simple shortest path in a $d$-dimensional real space subdivided in several polyhedra endowed with different $\ell_p$-norms. This problem is a variant of the weighted region problem, a classical path problem in computational geometry introduced in Mitchell and Papadimitriou (JACM 38(1):18-73, 1991). As done in the literature for other geodesic path problems, we relate its local optimality condition with Snell's law and provide an extension of this law in our framework space. We propose a solution scheme based on the representation of the problem as a mixed-integer second order cone problem (MISOCP) using the $\ell_p$-norm modeling procedure given in Blanco et al. (Comput Optim Appl 58(3):563-595, 2014). We derive two different MISOCPs formulations, theoretically compare the lower bounds provided by their continuous relaxations, and propose a preprocessing scheme to improve their performance. The usefulness of this approach is validated through computational experiments. The formulations provided are flexible since some extensions of the problem can be handled by transforming the input data in a simple way. The second problem that we consider is the Weber problem that results in this subdivision of $\ell_p$-normed polyhedra. To solve it, we adapt the solution scheme that we developed for the shortest path problem and validate our methodology with extensive computational experiments.
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Anisotropy measurements were carried out in two wells 300 km apart in the Northern Sahara, in order to improve the interpretation of seismic refraction surveys. These measurements were based on; the shortening of experimental oblique travel-times with respect to theoretical, travel-times computed by disregarding anisotropy. A perfect knowledge of the velocity distribution is required in order to eliminate all influence of refraction; therefore a continuous velocity log is indispensable. The results in the two wells are in agreement, and show that anisotropy is essentially a function of lithology. The following values were obtained: Consequences pertaining to seismic refraction are as follows: — Depths computed without anisotropy are too small by 8 % to 9 %, i.e. about 300 meters. — The error on offsets reaches 70 %. — Depth computations carried out by assigning anisotropy factors at the sight of lithology gave very satisfactory results. — High velocity, highly anisotropic formations such as anhydrite play a very important part in the propagation of refracted waves and in interpretation. Consequences pertaining to seismic reflection are as follows: — Velocity profiles yield velocities closer to horizontal velocities than to vertical velocities. — Depth and migration computations are significantly influenced by high velocity, highly anisotropic formations such as anhydrite.
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On elastic-wave propagation in anisotropic media: Reflection/refraction laws, raytracing and traveltime inversion
  • M A Slawinski
Slawinski, M. A., 1996, On elastic-wave propagation in anisotropic media: Reflection/refraction laws, raytracing and traveltime inversion;