


 Nonhyperbolic reflection moveout of waves:
An overview and comparison of reasons  

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In transversely isotropic media, velocities of seismic waves depend
on the direction of propagation measured from the symmetry
axis. Thomsen (1986) introduced a notation for VTI
media by replacing the elastic stiffness coefficients
with the  and wave velocities along the symmetry
axis and three dimensionless anisotropic parameters. As shown by Tsvankin
(Tsvankin, 1996), the wave seismic signatures in
VTI media can be conveniently expressed in terms of
Thomsen's parameters and .
Deviations of these parameters from zero characterize the relative
strength of anisotropy. For small values of these parameters,the
weakanisotropy approximation (Thomsen, 1986; Tsvankin and Thomsen, 1994)
reduces to simple linearization.
The squared group velocity
of waves in weakly anisotropic VTI media can be expressed
as a function of the group angle measured from the vertical
symmetry axis as follows:

(1) 
where is the wave vertical velocity, and and
are Thomsen's dimensionless anisotropic parameters, which
are assumed to be small quantities:

(2) 
Both parameters are equal to zero in isotropic media.
Equation (1) is accurate up to the secondorder terms in
and . We retain this level of accuracy throughout
the paper. As follows from equation
(1), the velocity corresponding to ray
propagation in the horizontal direction is

(3) 
Equation (3) is actually exact, valid
for any strength of anisotropy. Another important quantity is the
normalmoveout (NMO) velocity, , that determines the smalloffset
wave reflection moveout in homogeneous VTI media above a
horizontal reflector. Its exact expression is (Thomsen, 1986)

(4) 
If as, for example, in the ANNIE model proposed by
Schoenberg et al. (1996), the normalmoveout velocity
is equal to the vertical velocity.
It is convenient to rewrite equation (1) in the form

(5) 
where

(6) 
Equation (6) is the weakanisotropy approximation for the
anellipticity coefficient introduced by
Alkhalifah and Tsvankin (1995). For the elliptic anisotropy,
and
. To see why the groupvelocity function becomes elliptic in
this case, note that for small

(7) 
Seismic data often indicate that
, so the anellipticity
coefficient is usually positive.
An equivalent form of equation (1) can be obtained in terms of
the three characteristic velocities , , and :

(8) 
From equation (8), in the linear
approximation the anelliptic behavior of velocity is controlled by
the difference between the normal moveout and horizontal velocities
or, equivalently, by the difference between anisotropic
coefficients and .
We illustrate different types of the group velocities (wavefronts)
in Figure 1.
The wavefront, circular in the isotropic case (Figure
1a), becomes elliptical when
(Figure 1b). In the ANNIE model, the vertical and NMO
velocities are equal (Figure 1c). If and
, the three characteristic velocities satisfy the
inequality
(Figure 1d).


nmofro
Figure 1. Wavefronts in
isotropic medium,
(a),
elliptically anisotropic medium,
(b),
ANNIE model, , (c), and
anisotropic medium with , (d).
Solid curves represent the wavefronts.
Dashed lines correspond to isotropic wavefronts for the vertical and
horizontal velocities.







 Nonhyperbolic reflection moveout of waves:
An overview and comparison of reasons  

Next: HORIZONTAL REFLECTOR BENEATH A
Up: Fomel & Grechka: Nonhyperbolic
Previous: Introduction
20140127