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CWP -- TABLE OF CONTENTS
Nonhyperbolic reflection moveout of -waves:
An overview and comparison of reasons [pdf 376K]
Sergey Fomel and Vladimir Grechka
The familiar hyperbolic approximation of -wave reflection
moveout is exact for homogeneous isotropic or elliptically anisotropic
media above a planar reflector.
Any realistic combination of heterogeneity, reflector curvature, and nonelliptic
anisotropy will cause departures from hyperbolic moveout at large
offsets. Here, we analyze the similarities and differences in
the influence of those three factors on -wave reflection traveltimes.
Using the weak-anisotropy approximation
for velocities in transversely isotropic media with a vertical symmetry axis
(VTI model), we show that although the nonhyperbolic moveout due to both
heterogeneity and reflector curvature
can be interpreted in terms of effective anisotropy, such anisotropy
is inherently different from that of a generic homogeneous VTI model.
Time-shift imaging condition in seismic migration [pdf 2.2M]
Paul Sava and Sergey Fomel
Seismic imaging based on single-scattering approximation
is based on analysis of the match between the
source and receiver wavefields at every image location.
Wavefields at depth are functions of space and time and are
reconstructed from surface data either by integral
methods (Kirchhoff migration) or by differential methods
(reverse-time or wavefield extrapolation migration).
Different methods can be used to analyze wavefield matching,
of which cross-correlation is a popular option.
a simple imaging condition requires time cross-correlation
of source and receiver wavefields, followed by extraction
of the zero time lag.
A generalized imaging condition operates by cross-correlation
in both space and time, followed by image extraction at
zero time lag.
Images at different spatial cross-correlation lags are indicators of
imaging accuracy and are also used for image angle-decomposition.
In this paper, we introduce an alternative prestack imaging condition
in which we preserve multiple lags of the time cross-correlation.
Prestack images are described as functions of
time-shifts as opposed to space-shifts
between source and receiver wavefields.
This imaging condition is applicable to migration by
Kirchhoff, wavefield extrapolation or reverse-time techniques.
The transformation allows construction of
common-image gathers presented as function of either
time-shift or reflection angle at every location in space.
Inaccurate migration velocity is revealed by angle-domain
common-image gathers with non-flat events.
Computational experiments using a synthetic dataset
from a complex salt model
demonstrate the main features of the method.
Imaging overturning reflections by
Riemannian Wavefield Extrapolation [pdf 2.1M]
Correctly propagating waves from overhanging reflectors is crucial for imaging in complex geology. This type of reflections are difficult or impossible to use in imaging using one-way downward continuation, because they violate an intrinsic assumption of this imaging method, i.e. vertical upward propagation of reflection data.
Riemannian wavefield extrapolation is one of the techniques developed to address the limitations of one-way wavefield extrapolation in Cartesian coordinates. This method generalizes one-way wavefield extrapolation to general Riemannian coordinate system. Such coordinate systems can be constructed in different ways, one possibility being construction using ray tracing in a smooth velocity model from a starting plane in the imaged volume. This approach incorporates partially the propagation path into the coordinate system and leaves the balance for the one-way wavefield extrapolation operator. Thus, wavefield extrapolation follows overturning wave paths and extrapolated waves using low-order operators, which makes the extrapolation operation fast and robust.
Stereographic imaging condition for wave-equation migration [pdf 840K]
Imaging under the single-scattering approximation consists of two
steps: wavefield reconstruction of source and receiver wavefields from
simulated and recorded data, respectively, and imaging from the
extrapolated wavefields of the locations where reflectors occur.
Conventionally, the imaging condition indicates the presence of
reflectors when propagation times of reflections in the source and
receiver wavefields match. The main drawback of conventional
cross-correlation imaging condition is that it ignores the local
spatial coherence of reflection events and relies only on their
propagation time. This leads to interference between unrelated events
that occur at the same time. Sources of cross-talk include seismic
events corresponding to different seismic experiments, or different
propagation paths, or different types of reflections (primary or
multiple) or different wave modes (P or S). An alternative imaging
condition operates on the same extrapolated wavefields, but
cross-correlation takes place in a higher-dimensional domain where
seismic events are separated based on their local space-time
slope. Events are matched based on two parameters (time and local
slope), thus justifying the name ``stereographic'' for this imaging
condition. Stereographic imaging attenuates wavefield cross-talk and
reduces imaging artifacts compared with conventional imaging.
Applications of the stereographic imaging condition include
simultaneous imaging of multiple seismic experiments, multiple
attenuation in the imaging condition, and attenuation of cross-talk
between multiple wavefield branches or between multiple wave modes.
Numeric implementation of wave-equation migration velocity analysis operators [pdf 916K]
Paul Sava and Ioan Vlad
Wave-equation migration velocity analysis (MVA) is a technique similar
to wave-equation tomography because it is designed to update velocity
models using information derived from full seismic wavefields. On the
other hand, wave-equation MVA is similar to conventional,
traveltime-based MVA because it derives the information used for model
updates from properties of migrated images, e.g. focusing and moveout.
The main motivation for using wave-equation MVA is derived from its
consistency with the corresponding wave-equation migration, which
makes this technique robust and capable of handling multipathing
characterizing media with large and sharp velocity contrasts.
The wave-equation MVA operators are constructed using linearizations
of conventional wavefield extrapolation operators, assuming small
perturbations relative to the background velocity model. Similarly to
typical wavefield extrapolation operators, the wave-equation MVA
operators can be implemented in the mixed space-wavenumber domain
using approximations of different orders of accuracy.
As for wave-equation migration, wave-equation MVA can be formulated in
different imaging frameworks, depending on the type of data used and
image optimization criteria. Examples of imaging frameworks correspond
to zero-offset migration (designed for imaging based on focusing
properties of the image), survey-sinking migration (designed for
imaging based on moveout analysis using narrow-azimuth data) and
shot-record migration (also designed for imaging based on moveout
analysis, but using wide-azimuth data).
The wave-equation MVA operators formulated for the various imaging
frameworks are similar because they share common elements derived from
linearizations of the single square-root equation. Such operators
represent the core of iterative velocity estimation based on
diffraction focusing or semblance analysis, and their applicability in
practice requires efficient and accurate implementation. This tutorial
concentrates strictly on the numeric implementation of those operators
and not on their use for iterative migration velocity analysis.
Interferometric imaging condition for wave-equation migration [pdf 6.5M]
Paul Sava and Oleg Poliannikov
The fidelity of depth seismic imaging depends on the accuracy of the
velocity models used for wavefield reconstruction. Models can be
decomposed in two components corresponding to large scale and small
scale variations. In practice, the large scale velocity model
component can be estimated with high accuracy using repeated
migration/tomography cycles, but the small scale component
cannot. When the Earth has significant small-scale velocity
components, wavefield reconstruction does not completely describe the
recorded data and migrated images are perturbed by artifacts.
There are two possible ways to address this problem: improve wavefield
reconstruction by estimating more accurate velocity models and image
using conventional techniques (e.g. wavefield cross-correlation), or
reconstruct wavefields with conventional methods using the known
background velocity model, but improve the imaging condition to
alleviate the artifacts caused by the imprecise reconstruction, which
is what we suggest in this paper.
We describe the unknown component of the velocity model as a random
function with local spatial correlations. Imaging data perturbed by
such random variations is characterized by statistical instability,
i.e. various wavefield components image at wrong locations that depend
on the actual realization of the random model. Statistical stability
can be achieved by pre-processing the reconstructed wavefields prior
to the imaging condition. We employ Wigner distribution functions to
attenuate the random noise present in the reconstructed wavefields,
parametrized as a function of image coordinates. Wavefield filtering
using Wigner distribution functions and conventional imaging can be
lumped-together into a new form of imaging condition which we call an
``interferometric imaging condition'' due to its similarity to
concepts from recent work on interferometry. The interferometric
imaging condition can be formulated both for zero-offset and for
multi-offset data, leading to robust and efficient imaging procedures
that are effective in attenuating imaging artifacts due to unknown
Isotropic angle-domain elastic reverse-time migration [pdf 1.5M]
Jia Yan and Paul Sava
Multicomponent data are not usually processed with specifically
designed procedures, but with procedures analogous to the ones used
for single-component data. In isotropic media, the vertical
and horizontal components of the data are commonly taken as
proxies for the P- and S-wave modes which are imaged
independently with acoustic wave
equations. This procedure works only if the vertical and
horizontal component accurately represent P- and S-wave modes, which
is not true in general. Therefore, multicomponent images constructed
with this procedure exhibit artifacts caused by the incorrect wave
mode separation at the surface.
An alternative procedure for elastic imaging uses the full
vector fields for wavefield reconstruction and imaging. The wavefields
are reconstructed using the multicomponent data as a boundary
condition for a numerical solution to the elastic wave equation. The
key component for wavefield migration is the imaging condition that
evaluates the match between wavefields reconstructed from sources and
receivers. For vector wavefields, a simple component-by-component
cross-correlation between two wavefields leads to artifacts caused by
crosstalk between the unseparated wave modes. An alternative method is
to separate elastic wavefields after reconstruction in the subsurface
and implement the imaging condition as cross-correlation of pure wave
modes instead of the Cartesian components of the displacement
wavefield. This approach leads to images that are easier to interpret,
since they describe reflectivity of specified wave modes at interfaces
of physical properties.
As for imaging with acoustic wavefields, the elastic imaging condition
can be formulated conventionally (cross-correlation with zero lag in
space and time), as well as extended to non-zero space and time
lags. The elastic images produced by an extended imaging condition can
be used for angle decomposition of primary (PP or SS) and converted
(PS or SP) reflectivity. Angle gathers constructed with this procedure
have applications for migration velocity analysis and amplitude versus
Elastic wave-mode separation for TTI media [pdf 3.1M]
Jia Yan and Paul Sava
Seismic waves propagate through the earth as a superposition of
Seismic imaging in areas characterized by complex geology requires
techniques based on accurate reconstruction of the seismic wavefields.
A crucial component of the methods in this category, collectively
known as wave-equation migration, is the imaging condition
which extracts information about the discontinuities of physical
properties from the reconstructed wavefields at every location in
Conventional acoustic migration techniques image a scalar wavefield
representing the P wave-mode, in contrast with elastic migration
techniques which image a vector wavefield representing both the P and
For elastic imaging, it is desirable that the reconstructed vector
fields are decomposed in pure wave-modes, such that the imaging
condition produces interpretable images, characterizing for example PP
or PS reflectivity.
In anisotropic media, wave-mode separation can be achieved by
projection of the reconstructed vector fields on the polarization
vectors characterizing various wave modes. For heterogeneous media,
the polarization directions change with position, therefore wave-mode
separation needs to be implemented using space-domain filters.
For transversely isotropic media with a tilted symmetry axis (TTI),
the polarization vectors depend on the elastic material parameters,
including the tilt angles. Using these parameters, I separate
the wave-modes by constructing nine filters corresponding to the nine
Cartesian components of the three polarization directions at every
Since the S polarization vectors in TI media are not defined in the
singular directions, e.g. along the symmetry axes, I construct these
vectors by exploiting the orthogonality between the SV and SH
polarization vectors, as well as their orthogonality with the P
polarization vector. This procedure allows one to separate S wave-modes
which are only kinematically correct.
Realistic synthetic examples show that this wave-mode separation is
effective for both 2D and 3D models with high heterogeneity and strong
Elastic wave-mode separation for VTI media [pdf 3.8M]
Jia Yan and Paul Sava
Elastic wave propagation in anisotropic media is well represented by
elastic wave equations. Modeling based on elastic wave equations
characterizes both kinematics and dynamics correctly. However, because
P and S modes are both propagated using elastic wave equations, there
is a need to separate P and S modes to obtain clean elastic images.
The separation of wave modes to P and S from isotropic elastic
wavefields is typically done using Helmholtz decomposition. However,
Helmholtz decomposition using conventional divergence and curl
operators in anisotropic media does not give satisfactory results and
leaves the different wave modes only partially separated. The
separation of anisotropic wavefields requires the use of more
sophisticated operators which depend on local material parameters.
Anisotropic wavefield separation operators are constructed using the
polarization vectors evaluated by solving
the Christoffel equation at each point of the medium. These
polarization vectors can be represented in the space domain as
localized filtering operators, which resemble conventional derivative
The spatially-variable ``pseudo'' derivative operators perform well in
heterogeneous VTI media even at places of rapid velocity/density
Synthetic results indicate that the operators can be used to separate
wavefields for VTI media with an arbitrary degree of anisotropy.
Wide-azimuth angle gathers for wave-equation migration [pdf 1.9M]
Paul Sava and Ioan Vlad
Extended common-image-point-gathers (CIP) contain all the necessary
information for decomposition of reflectivity as a function of the
reflection and azimuth angles at selected locations in the
subsurface. This decomposition operates after the imaging condition
applied to wavefields reconstructed by any type of wide-azimuth
migration method, e.g. using downward continuation or time reversal.
The reflection and azimuth angles are derived from the extended
images using analytic relations between the space-lag and time-lag
extensions. The transformation amounts to a linear Radon transform
applied to the CIPs obtained after the application of the extended
imaging condition. If information about the reflector dip is
available at the CIP locations, then only two components of the
space-lag vectors are required, thus reducing computational cost and
increasing the affordability of the method.
Applications of this method include the study of subsurface
illumination in areas of complex geology where ray-based methods are
not usable, and the study of amplitude variation with reflection and
azimuth angles if the subsurface subsurface illumination is
sufficiently dense. Migration velocity analysis could also be
implemented in the angle domain, although an equivalent
implementation in the extended domain is cheaper and more effective.
Micro-earthquake monitoring with sparsely-sampled data [pdf 3.2M]
Micro-seismicity can be used to monitor the migration of fluids during
reservoir production and hydro-fracturing operations in brittle
formations or for studies of naturally occurring earthquakes in fault
zones. Micro-earthquake locations can be inferred using wave-equation
imaging under the exploding reflector model, assuming densely sampled
data and known velocity.
Seismicity is usually monitored with sparse networks of seismic
sensors, for example located in boreholes. The sparsity of the sensor
network itself degrades the accuracy of the estimated locations, even
when the velocity model is accurately known. This constraint limits
the resolution at which fluid pathways can be inferred.
Wavefields reconstructed in known velocity using data recorded with
sparse arrays can be described as having a random character due to the
incomplete interference of wave components. Similarly, wavefields
reconstructed in unknown velocity using data recorded with dense
arrays can be described as having a random character due to the
inconsistent interference of wave components. In both cases, the
fluctuations obstruct focusing that occurs at source
This situation can be improved using interferometry in the imaging
process. Reverse-time imaging with an interferometric imaging
condition attenuates random fluctuations, thus producing crisper
images which support the process of robust automatic micro-earthquake
The similarity of random wavefield fluctuations due to model
fluctuations and sparse acquisition are illustrated in this paper with a
realistic synthetic example.
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