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The
-
transform is the natural domain for anisotropy parameters
estimation in layered or vertically varying media with horizontal
symmetry planes
(van der Baan and Kendall, 2002; Sil and Sen, 2008; Douma and van der Baan, 2008; Tsvankin et al., 2010). Since the
horizontal slowness is preserved upon propagation, the
-
transform
allows simpler and more accurate traveltime modeling (ray tracing) and
inversion (layer stripping). Moreover, the
-
transform is a
plane-wave decomposition. Therefore, the phase
velocity, rather than the group velocity, is the relevant
velocity. The group velocity controls instead traveltime in the
traditional
-
domain (Tsvankin, 2006). Unfortunately, the exact
expressions for the group velocities in terms of the group angle are
difficult to obtain and cumbersome for practical use. As a result, it
requires either ray tracing for exact
-
modeling in anisotropic
media or the use of multi-parameter traveltime approximations. In
this domain, the most straightforward and widely used approximation
for P-waves reflection moveout comes from the Taylor series expansion
of traveltime or squared traveltime around the
zero offset (Taner and Koehler, 1969; Ursin and Stovas, 2006):
Although it is possible to derive exact formulas for all the series coefficients (Tsvankin, 1995,2006; Al-Dajani and Tsvankin, 1998), equation 1 loses its accuracy with increasing offset to depth ratio. Fomel and Stovas (2010) introduced recently a generalized functional form for approximating reflection moveout at large offsets. While the classic Alkhalifah and Tsvankin (1995) 4th-order Taylor/Padé approximation uses three parameters, the generalized approximation involves five parameters, which can be determined from the zero-offset computation and from tracing one nonzero-offset ray. In a homogeneous quasi-acoustic VTI medium (Alkhalifah, 1998), the generalized approximation of Fomel and Stovas (2010) reduces to the three-term traveltime approximation of Fomel (2004), which is practical and more accurate than other known three-parameter formulas for non-hyperbolic moveout.
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commonray1,commonray2
Figure 1. Comparison between event geometry in ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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The
-
domain provides an attractive alternative to computing
P-wave reflection-moveout curves. The
-
transform stacks the data
gathered in
-
domain along straight lines, whose direction
Equation 3 remains exact as long as we use the
exact expression for the phase velocity
(red solid line in
figure 2a). Exact expressions exist for all types of anisotropic
media with a horizontal symmetry plane. Unfortunately, the exact and
the highly accurate (Stovas and Fomel, 2010) expressions for
-
signatures are not very practical because they depend on multiple
parameters. In practice, one may prefer to employ three-parameters
approximate relations for the phase velocity. Although these
signatures are approximate, they are more reliable then the
-
transformed version of their dual-pair in the
-
domain (figure 2b).
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taup,error-taup
Figure 2. (a) ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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We can extend the result in equation 3 to a stack
of
horizontal homogeneous layers with horizontal symmetry
planes. According to Snell's law, the horizontal slowness
is
preserved upon propagation through each layer. Thus, the total
intercept time
from the bottom of
-th layer is the summation
of each interval intercept time
in the
contributing layers:
The summation in equation 4 can be substituted by a convenient
relation in term of the effective parameters obtained from the Dix
average of interval ones. This result will be used in the
next section to derive a closed-form expression
for P-waves
-
reflection moveout in terms of interval or effective
normal-moveout velocity
and horizontal velocity
(or,
alternatively,
): these two parameters control all time-domain
processing steps in VTI media (Alkhalifah and Tsvankin, 1995).
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