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Seismic Reflection Moveout for Azimuthally Anisotropic Mediaby Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on {Date} in partial fulfillment of the requirements for the degree of Doctor of Philosophy ABSTRACT
Anisotropy can exist in the subsurface as an intrinsic property, or as an
induced property, or as a combination of both. Induced anisotropy can result
due to preferred orientation of grains, thin layering, and/or the presence
of fractures. In this thesis, we study the reflection moveout for Compressional
and Shear waves in horizontally stratified azimuthally anisotropic media.
Reflection moveout in common-midpoint (CMP) gathers is generally treated
by an azimuthally-isotropic hyperbolic equation. Recently, special treatment
for the reflection moveout in azimuthally anisotropic media has been introduced
due to the fact that the moveout in such cases is azimuthally dependent.
Here, the normal moveout (NMO) equation parameterized by the exact normal-moveout
(NMO) velocity is studied. We verify numerically (through synthetic data)
the accuracy of the exact (i.e., analytic) azimuthal description of the
NMO velocity. We show that the azimuthal variation of the NMO velocity
has, in general, a relatively simple elliptical form. We also show that
the NMO equation is sufficiently accurate for $P$- and Shear-wave propagation
on conventional spreadlengths (e.g., lenghts close to the reflector depth).
The influence of anisotropy causes the deviation of the moveout curve
from a hyperbola, even in a homogeneous anisotropic layer with a horizontal
interface. Hence, reflection moveout in azimuthally anisotropic media
is not only azimuthally dependent but it is also nonhyperbolic. To account
for the nonhyperbolic moveout, we have derived an exact expression for
the azimuthally dependent quartic coefficient of the Taylor series expansion
for the two-way traveltimes [$t^2(x^2)$] that is valid for any pure mode
of wave propagation. As a result, we introduce an analytic representation
for the quartic coefficient for pure mode reflection in anisotropic medium
with an arbitrary strength of anisotropy. In addition, we present an analytic
expression for large-offset, nonhyperbolic reflection moveout (NHMO).
Special attention is given in this study toward $P$-wave propagation
in orthorhombic (ORT) and horizontal tranverse isotropic (HTI) medium
with horizontal interfaces. The presence of azimuthal anisotropy causes
shear waves to split into fast and slow shear waves. As a result, the
quartic coefficient and the reflection moveout for shear wave propagation
are briefly addressed in this thesis.
The quartic coefficient has a relatively simple form, especially for
shear waves. The quartic moveout coefficient and the normal-moveout velocity
are substituted into the nonhyperbolic moveout equation, which is originally
designed for vertical transverse isotropic (VTI) media. Synthetic examples
show that this equation accurately describes the azimuthally dependent
$P$-wave reflection traveltimes, even for spreadlengths (offsets) twice
as large as the reflector depth. The reflection moveout for shear-wave
propagation (i.e., fast and slow), on the other hand, is purely hyperbolic
in the direction normal to the polarization. In addition, the nonhyperbolic
portion of the moveout for shear-wave propagation reaches its maximum
along the polarization direction, and it decreases rapidly away from the
direction of polarization. Hence, the anisotropy-induced nonhyperbolic
reflection moveout for shear-wave propagation is only significant in the
vicinity of the polarization directions (e.g., $\pm 30^\circ$) and for
large offset-to-depth ratios.
In multilayered azimuthally anisotropic media, the NMO velocity and
the quartic moveout coefficient can be calculated with good accuracy using
Dix-type averaging (e.g., the known averaging equations for VTI media).
The interval NMO velocities and the interval quartic coefficients, however,
are azimuthally dependent. This allows us to extend the nonhyperbolic
moveout (NHMO) equation, originally designed for VTI media, to more general
horizontally stratified azimuthally anisotropic media. Numerical examples,
from reflection moveout in orthorhombic and HTI media, show that this
NHMO equation accurately describes the azimuthally dependent $P$-wave
reflection traveltimes, even on spreadlengths twice as large as the reflector
depth.
In 3-D (or, 2-D) land seismic data acquisitions (also in water-bottom
cable surveys), we have relatively full control on offset and azimuthal
coverage. In conventional marine surveys, however, the azimuthal coverage
is quite limited. Except in limited cases, current seismic data acquisition
surveys do not utilize the azimuthal variation of the reflection moveout
and the NMO velocity to obtain better medium parameter-estimation or imaging.
Here, we address the inverse problem of using the azimuthal dependence
of the NMO velocity in azimuthally anisotropic media. The error and sensitivity
analysis provide insight into the inverse problem and how it relates to
seismic data acquisition. We explore the limitations involved in using
the azimuthal variation in the NMO velocity to invert for the medium parameters.
Moreover, we address issues such as the effect of sectorization in 3-D
seismic data surveys on such inversion. Finally, we study how the resolution
of the inverse problem is influenced by the number of NMO velocities that
enter the inversion process.
Parameter estimation from the elliptical variations in the normal-moveout
(NMO) velocity in azimuthally anisotropic media is sensitive to the angular
separation between the survey lines in 2D, or equivalently source-to-receiver
azimuth in 3D, and to the set of azimuths used in the inversion procedure.
The accuracy in estimating the orientation of NMO ellipse is also sensitive
to the strength of anisotropy. On the other hand, the accuracy in estimating
the semi-axes of the NMO-velocity ellipse is about the same for any strength
of anisotropy.
To invert for the NMO ellipse parameters at least three NMO-velocity
measurements along distinct azimuth directions are needed. In order to
maximize the accuracy and stability in parameter estimation, it is best
to have the azimuths for the three source-to-receiver directions $60^\circ$
apart. Having more than three distinct source-to-receiver azimuths (e.g.,
full azimuthal coverage) enhances the quality of the estimates.
The azimuthal variation of the NMO velocity in azimuthally anisotropic
media can be utilized to invert for some of the medium parameters which
can be useful in characterizing the zone of interest. In HTI media, for
example, and using the $P$-wave reflection moveout, we can estimate three
key parameters: the vertical velocity $V_{\rm Pvert}$, anisotropy parameter
$\delta^{\rm (V)}$, and the azimuth $\alpha$ of the symmetry-axis. Similarly,
in orthorhombic media, inverting for the semi-axes of the NMO-ellipse
by using $P$--wave normal-moveout (NMO) velocity, allows the computation
of the difference in the anisotropic parameters $\delta^{\rm (1)}$ and
$\delta^{\rm (2)}$. With the nonhyperbolic portion of the reflection moveout,
we can obtain the $\epsilon$ parameters (and $\eta$) which can be used
to estimate or at least constrain the fracture density.
The NMO velocity exhibits similar azimuthal variation for different
azimuthally anisotropic models (e.g., HTI, orthorhombic, and monoclinic).
Hence, it is not possible to distinguish between the different azimuthal
anisotropic models solely on the behavior of the NMO velocity. Additional
information such as well logs and cores can be helpful to classify the
medium. Unlike the NMO velocity, the NHMO coefficient manifests different
azimuthal behavior in different anisotropic models. This distinct azimuthal
variations can lead to distinguishing different types of anisotropic media
by studying the nonhyperbolic portion of the reflection moveout.
To maximize quality in the inversion process, it is recommended that
at the design stage of seismic data acquisition to ensure having small
sector sizes ($\leq 10^\circ$) with adequate fold and offset distribution.
Using three NMO-velocity measurements, $60^\circ$ apart, an azimuthally
anisotropic layer overlain by an azimuthally isotropic overburden (as
might happen for fractured reservoirs) should have a relative thickness
(in time) to the total thickness of at least equal to the ratio of the
error in the NMO (stacking) velocity to the interval anisotropy strength
of the fractured layer. Coverage along more than three azimuths, however,
improves this limitation, which is imposed by Dix differentiation, by
at most 50$\%$ depending on the number of observations (NMO Velocities)
that enter the inversion procedure.
An application of this work to field data is performed using ground penetrating
radar (GPR) surveys in azimuthally anisotropic media. Ground penetrating
radar (GPR) signatures, such as reflection moveout, are sensitive to the
presence of azimuthal anisotropy. Ignoring the effect of anisotropy on
the normal-moveout (NMO) velocity, that characterizes the reflection moveout,
causes erroneous images and time-to-depth conversions. Similar to surface
seismology in azimuthally anisotropic media, the GPR normal moveout (NMO)
velocity, along different orientations of common-midpoint (CMP) gathers,
varies elliptically with azimuth.
Here, we introduce an exact representation for the GPR NMO-velocity in
azimuthally anisotropic media (e.g., HTI, and orthorhombic symmetries).
The NMO-velocity representation is valid for arbitrary strength of anisotropy.
A field data example is presented in which three GPR CMP gathers are acquired
along three different azimuths, $60^\circ$ apart, over a fractured medium.
The data are obtained for the transverse mode of electromagnetic wave
propagation in which the polarization is normal to the incidence plane
of the CMP gathers. Our data analysis demonstrates the azimuthal variation
of the GPR NMO velocity, which is utilized to invert for the local orientation
of the fracture system in the near surface. As a result, this study has
important 3D applications in imaging near surface geologic structures
and in the determination of tectonic-induced fractures in the near surface.
Finally, this thesis provides analytic insight into the behavior of
the reflection moveout, specially the nonhyperbolic moveout and the NMO
velocity, in azimuthally anisotropic media. In addition, this work has
important 3D applications in imaging, modeling, and inversion of reflection
moveout in azimuthally anisotropic media.
Note: This abstract is written in LaTex. Return to Theses Return to ERL Home Updated: May 31, 2002
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