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Waves propagating in a homogeneous medium can be described and extrapolated
in either the timespace ``group'' or frequencywavenumber ``phase'' domains.
In the standard scalar isotropic case life is simple;
there is one soundspeed characteristic
of the medium, and it is the same whether we are talking about
rays (group velocity) or plane waves (phase velocity).
The scalar isotropic ray (or groupvelocity) equation in polar coordinates
is just
 
(1) 
the corresponding dispersion relation (phasevelocity equation)
is
 
(2) 
With anisotropy the propagation velocity depends on the propagation direction,
and the group and phase velocities and directions are no longer the same.
Convenient closedform ``group'' and ``phase'' equations like
equations () and () are usually
not simultaneously available.
Even worse,
the number of parameters required to describe the medium can
be intractably large.
Scalar elliptical anisotropy does not suffer from these difficulties.
There are only two velocities, horizontal and vertical.
It is true that the ray equation for scalar elliptical anisotropy
appears a bit complicated at first glance:
 
(3) 
where is the grouppropagation (ray) direction
(with being vertical).
If we reexpress equation () in terms of
slowness squared M = 1/V^{2}, however,
it simplifies considerably:
 
(4) 
The dispersion relation for elliptical anisotropy is similarly simple:
 
(5) 
where
W_{z} is the vertical velocity squared
and
W_{x} is the horizontal velocity squared.
Another form of equation () displays the fundamental
nature of elliptical anisotropy more clearly.
If you divide equation () through by
, you get
 
(6) 
where is phasevelocity squared and
is the wavepropagation (phase) direction,
with being vertical.
The identity of form between
equations () and () is not accidental.
An ellipse is merely a stretched circle, and
elliptical anisotropy is merely stretched isotropy.
The Fourier similarity theorem
tells us that a ``stretch'' (linear transformation)
in the (x,z) domain corresponds to
an ``inverse stretch'' (another linear transformation)
in the (k_{x}, k_{z}) domain.
Equations () and ()
look the same because they are both equations for stretched circles,
i.e., ellipses.
Figure shows how elliptical anisotropy can be used as
a firstorder paraxial approximation in both the group and phase domains.
Although the groupvelocity and phaseslowness representations
of this transversely isotropic medium look very different,
the firstorder paraxial approximation is elliptical in both domains.
(The TI medium is Greenhorn Shale from Jones and Wang (1981).)
This paraxial ellipse is also of considerable practical importance.
The familiar isotropic moveout equation is
 
(7) 

T(0)^{2} = h^{2} / V_{z}^{2}
,

(8) 
where T(x) is the traveltime at offset x,
is the moveout velocity,
h is the vertical layer thickness,
and V_{z} is the vertical velocity.
(This is oneway traveltime.)
In the isotropic case we only have one velocity, and set
.Converting from isotropy to elliptical anisotropy with a vertical axis
is as simple as replacing z with a new stretched coordinate
. Because the z coordinate does not occur in
equation () it remains completely unchanged in the new
coordinate system.
The rescaled z coordinate does trivially affect equation
by scaling the numerator and denominator equally,
leaving T(0) unchanged.
(h and V_{z} scale with z, because
they involve the same vertical length units.)
The moveout equation for elliptical anisotropy
thus has the same form as for isotropy; we merely have
to let and V_{z} become independent.
The paraxial moveout velocity , unaffected
by the stretch in z, must be the horizontal velocity V_{x}
of the paraxial ellipse.
Because h and V_{z} can be scaled together with no change in the recorded
traveltimes, the vertical scale in elliptical anisotropy evidently can't be
determined from surface kinematics alone (Dellinger and Muir, 1988).
Next: THE FIRST ANELLIPTIC APPROXIMATION
Up: Dellinger, Muir, & Karrenbach:
Previous: Introduction
Stanford Exploration Project
11/17/1997