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 | Fractal heterogeneities in sonic logs
and low-frequency scattering attenuation |  |
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A scalar wave
in a weakly inhomogeneous medium
(Karal and Keller, 1964; Chernov, 1960; Tatarski, 1961)
satisfies the Helmholtz wave equation
where
is a small perturbation of the medium from homogeneity, and
.
Phase velocity
is the background velocity.
Assuming a second-order stationary statistical distribution
for fluctuations
and a zero expectation value
, spatial covariance of the velocity variations is
defined by relation 1.
Expectation
of random plane-wave realizations
is calculated (Karal and Keller, 1964)
using a perturbation theory to the second order in
by
![$\displaystyle \left[\Delta+k_0^2(1+\sigma^2)\right]\langle u(\mathbf{x},\omega)...
...mega)
\,\langle u(\mathbf{x}^{\prime},\omega)\rangle~d\mathbf{x}^{\prime}= 0\,,$](img152.png) |
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(17) |
where
is Green's function of the operator
,
and integration is performed over the 3D space.
The dispersion relation for a plane wave propagating in the heterogeneous medium follows as
where
is the relative position, with an absolute value
,
and
is the 3D isotropic free-space Green's function with the outward radiation condition
(Bleistein et al., 2001).
The path of waves should be sufficiently long to significantly sample medium heterogeneities statistically (Gist, 1994).
The Born approximation is present because of Green's function.
Heterogeneities with the isotropic correlation function
produce an isotropic wave vector
.
Combining Green's function in equation 19 with the isotropic integral in equation 5 reduces
the squared dispersion relation of equation 18 to
Second-order expansion
in the solution constrains validity to the domain
, where
is the characteristic length scale of the heterogeneities.
The second-order approximation for the 3D dispersion relation is finally
Quantity
, introduced above, is related to the real and even function
, defined by the isotropic integral of equation 4:
Connection to the O'Doherty-Anstey formula is detailed in Appendix A.
 |
 |
 |
 | Fractal heterogeneities in sonic logs
and low-frequency scattering attenuation |  |
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Next: Attenuation in 3D fractal
Up: Scattering attenuation in 3D
Previous: Scattering attenuation in 3D
2013-07-26