Plane-wave migration in tilted coordinates (ps.gz 22388K) (pdf 3588K) (src 47349K)
**Shan G. and Biondi B.**

Plane-wave migration in tilted coordinates is powerful to image steeply dipping reflectors,
although the one-way wave-equation operator is used. In plane-wave migration,
the recorded surface data are transferred to plane-wave data by slant-stack processing.
Both the source plane-wave and the corresponding slant-stacking data
are extrapolated into the subsurface, and images are generated by cross-correlating these two
wavefields. For each plane-wave source, we assign tilted coordinates whose direction depends on
the propagation direction of the plane-wave. For isotropic media, the one-way wave-equation
operator does not change. For vertical transversely isotropic (VTI) media,
one-way wave-equation operators for tilted transversely isotropic (TI) media are
required because of the rotation of the coordinates. I apply this method to the BP 2004 velocity benchmark,
to a synthetic dataset for VTI media, and to a real anisotropic dataset. The numerical examples show
that plane-wave migration in tilted coordinates can image steeply dipping reflectors
and overturned waves, and it is a good tool for salt delineation.

Generalized Riemannian wavefield extrapolation (ps.gz 1172K) (pdf 615K) (src 29949K)
**Shragge J.**

This paper extends wavefield extrapolation to generalized
Riemannian spaces. The key component is the development of a
dispersion relationship appropriate for propagating wavefield on
generalized non-orthogonal meshes. This wavenumber contains a number
of mixed-domain fields in addition to velocity that represent
coordinate system geometry. An extended split-step Fourier
approximation of the extrapolation wavenumber is developed, which
provides accurate results when multiple reference parameters sets
are used. Three examples are presented that demonstrate the
validity of the theory. An important consequence is that greater
emphasis can be placed on generating smoother computational meshes
rather than satisfying restrictive semi-orthogonal criteria. This
result should lead to more accurate and efficient generalized
Riemannian wavefield extrapolation.

Mesh generation using differential methods (ps.gz 733K) (pdf 812K) (src 15598K)
**Shragge J.**

This paper examines a differential gridding method for generating
computational meshes appropriate for
solving partial differential equations. Differential methods pose
mesh generation as an elliptical boundary value problem within a
framework of differential geometry. Generalized Laplacian operators
are used to propagate the known coordinate values on the boundary
points into the interior in a smooth manner. The methodology allows
for the specification of monitor functions that provide mesh
regularization and prevent grid clustering. Examples are provided
for two seismic imaging applications: wave-equation Green's
function generation and wave-equation migration from topography. In
both cases, the resulting regularized meshes have minimal convexity
and are conformal to the the prescribed boundaries.

Prestack exploding-reflectors modeling for migration velocity analysis (ps.gz 1030K) (pdf 451K) (src 450864K)
**Biondi B.**

I introduce a prestack generalization of exploding-reflectors modeling
that has the potential to drastically reduce the computational
cost of Migration Velocity Analysis (MVA) based on wavefield-continuation
migration and modeling.
The method aims at synthesizing,
starting form a prestack migrated image,
a limited number of independent experiments that contain all the velocity
information needed for MVA.
The basic element of the method is the modeling
of one isolated Subsurface-Offset Domain Common Image Gather (SODCIG)
by using the prestack image as an initial condition.
Taking advantage of the limited range of
the subsurface offsets in the migrated image,
we can combine many SODCIGs in the same modeling
experiment without diminishing the velocity information
contained in the data.
The number of independent experiments required depends
on how close we are to the correct migration velocity.
Theoretically, this number is the same as the number of
subsurface offsets needed to represent the prestack image
from which we started the modeling.
Several numerical examples illustrate the basic concepts of the
proposed method, and demonstrate its potential usefulness
for MVA.

AMO inversion to a common azimuth dataset: Field data results (ps.gz 2383K) (pdf 870K) (src 336258K)
**Clapp R. G.**

I cast 3-D data regularization as a least-squares inversion problem.
I form a linear operator
that maps from irregular dataspace 5-D data space to a regular 4-D common
azimuth volume using a cascade of linear interpolation and Azimuth Move-out
(AMO) binning. I regularize the inversion by adding
minimizes the difference between various
() cubes by applying
a filter that acts along offset. AMO is used to
transform the cubes to the same *hx* before applying
the filter.
Further efficiency is gained by inverting each frequency independently.
I apply the methodology on marine dataset and compare
the results with two more conventional approaches.

PS - Azimuth moveout: Real data application (ps.gz 1569K) (pdf 2270K) (src 10691K)
**Rosales D. A. and Clapp R. G.**

We present an application of the PS-AMO operator for partial-stacking 3-D
multicomponent, ocean-bottom seismic data.
The partial-stacking problem is defined as a
regularized least-squares objective function.
To preserve the resolution of
dipping events, the regularization term uses the PS-AMO
operator. Application of this methodology on a portion of the Alba 3-D,
multicomponent, ocean-bottom seismic data set
shows that we can satisfactorily obtain an interpolated data set that
honors the physics of converted waves.

Target-oriented wave-equation inversion with regularization in the subsurface-offset domain (ps.gz 358K) (pdf 273K) (src 2187K)
**Valenciano A. A.**

A complex velocity model can cause shadow zones in an image computed by migration due to poor illumination. These shadow zones may contain weak signals masked by artifacts. To reduce artifacts and recover the real signal, a wave-equation target-oriented inversion scheme can be developed that uses an explicitly computed least squares inversion Hessian. To solve the otherwise ill-conditioned inversion problem a model regularization needs to be added. One choice for regularization is to penalize the energy in the image not focused at zero subsurface-offset. The subsurface-offset Hessian needs to be computed by using the adjoint of migration as the modeling operator. Results on Sigsbee model show encouraging results.

Parallel implementation of image segmentation for tracking 3D salt boundaries (ps.gz 429K) (pdf 230K) (src 23409K)
**Lomask J. and Clapp R. G.**

We distribute the modified normalized cuts image segmentation with random boundaries algorithm on a parallel network to track 3D salt boundaries. We identify two key steps of this algorithm for parallelization. Firstly, we parallelize the calculation of the weight matrix. Secondly, we parallelize the matrix-vector product of the eigenvector calculation. This method is demonstrated to be effective on a 3D seismic cube.

Flattening with geological constraints (ps.gz 29K) (pdf 76K) (src 6026K)
**Lomask J. and Guitton A.**

In areas with faults and poor signal/noise ratio, where reflectors can be
discontinuous from place to place, a dip-based flattening technique might not
be able to appropriately track sedimentary layers. To aid the
flattening algorithms, one or few reflectors can be picked. This
information can be then incorporated in our algorithms as geological
constraints. In a first method, we add a model mask to a time domain
solution using a Gauss-Newton approach that incorporates an initial
solution. In a second method, we set the lower and upper bounds of
a constrained optimization algorithm called limited memory BFGS
with bounds (L-BFGS-B). Having incorporated the geological information, the
flattening algorithms can accurately pick reflectors in 2D and 3D for
noisy field data examples. In addition, preliminary results seem to
indicate that the L-BFGS-B method converges faster than the Gauss-Newton
method.

Quaternion-based Signal Processing (ps.gz 1046K) (pdf 1971K) (src 51574K)
**Witten B. and Shragge J.**

Hypercomlex numbers, which have primarily been used for pattern
recognition, offer many useful applications to geophysics. Image
disparity estimation is a hypercomplex, phase-based technique, using
quaternions, that can find differences between subtlety varying
images. This technique relies on applying a quaternionic Fourier
transform, a quaternionic Gabor filter and exploits the symmetries
inherent in the quaternion. Two applications of hypercomplex image
disparity estimation are time lapse analysis and boundary detection.

Flattening with cosine transforms (ps.gz 238K) (pdf 154K) (src 2453K)
**Lomask J. and Fomel S.**

In the Fourier-domain flattening methods presented previously Lomask et al. (2005); Lomask and Claerbout (2002); Lomask (2003) the data has to be mirrored in order to eliminate Fourier artifacts. This means that the data is replicated and reversed so that the boundaries are periodic. This requires four times the memory in 2D and eight times in 3D. In a world where post-stack data cubes can easily be tens of gigabytes in size, efficient memory usage is extremely important.
Here we apply a discrete cosine transform (DCT) approach developed for 2D phase unwrapping Ghiglia and Pritt (1998); Ghiglia and Romero (1994) to our flattening method. As a result, we the reduce memory requirements of the transforms in 2D by a factor of four and in 3D by a factor of eight. We reap an additional factor of two savings from using real instead of complex numbers. Furthermore, the reduction in size significantly reduces computation time as well. We demonstrate its use on a simple 3D synthetic model.

...

Wave-equation angle-domain common-image gathers for converted waves:
Part 2 (ps.gz 5359K) (pdf 1200K) (src 30283K)
**Rosales D. A., Fomel S., Biondi B. L., and Sava P. C.**

Common-image gathers are very useful for velocity and
amplitude analysis. Wavefield-extrapolation methods produce
Angle-Domain Common-Image Gathers (ADCIGs). For the conventional PP
case, ADCIGs are a function of the opening angle.
However, the ADCIGs for converted-wave data (PS-ADCIGs) are a
function of the half-aperture angle, that is, the incidence
angle plus the reflection angle.
In PS-ADCIGs, both the P-to-S velocity ratio () and
the image dip play a major role in transforming
the subsurface offset into the opening angle.
We introduce a simple methodology to compute PS-ADCIGs.
Our methodology exploits the robustness of computing
PP-ADCIGs, and incorporates the velocity ratio (),
with an image dip field, which is estimated along the prestack
image. Our methodology also transforms the
half-aperture angle in PS-ADCIGs into an independent P-incidence angle to form
P-ADCIGs, and an independent S-reflection angle to form
S-ADCIGs.
Numerical studies show that when the
P-to-S velocity ratio and image midpoint information are not
incorporated, the
error in computing PS-ADCIGs is large enough to introduce
artifacts in the velocity model.
Synthetic results show the accuracy of the transformation
introduced in this paper. Real data results on the
2-D Mahogany field show the practical application
and implications for converted-wave angle-domain common-image
gathers.

Residual moveout of 2D multiples in Angle-Domain Common-Image Gathers (ps.gz 1475K) (pdf 497K) (src 41739K)
**Alvarez G.**

I show that, for specularly-reflected multiples, the constant velocity
straight-ray approximation of the residual moveout in Angle-Domain
Common-Image Gathers (ADCIGs) is only appropriate for small aperture
angles. The
approximation is good for the primaries because the difference between
the migration velocity and the true velocity is likely to be small.
For the multiples, however, this difference may be large and correcting
for ray bending produces a better approximation that leads to better
focusing of the multiples in the Radon domain. This in turn allows a
more accurate muting of the multiples. I show results with two ADCIGs,
one synthetic and one real.

Residual moveout in anisotropic angle-domain common image gathers with dipping reflectors (ps.gz 153K) (pdf 209K) (src 640K)
**Jousselin P. and Biondi B.**

We generalise to dipping reflectors the fundamental concepts for quantitatively
relating perturbations in anisotropic parameters to the corresponding
reflector movements in angle-domain common-image gathers (ADCIGs).
We apply the general methodology to the particular case of residual
moveout (RMO) analysis of reflections from dipping reflectors in a
vertical transverse isotropic (VTI) medium. Synthetic examples show
the accuracy of the RMO curves predicted by our kinematic analysis.

Attenuation of 2D specularly-reflected multiples in image space (ps.gz 374K) (pdf 270K) (src 1724K)
**Alvarez G.**

I propose a new method to attenuate specularly-reflected multiples in the
image space. The method is based on the difference in mapping between
primaries and multiples in Subsurface Offset Domain Common Image Gathers
(SODCIGs). I migrate the data with a velocity slower than that of the
primaries but faster than that of the multiples. The primaries are
therefore undermigrated whereas the multiples are overmigrated. For
positive data offsets, the primaries are mapped to positive
subsurface offsets and the multiples to negative subsurface offsets in
SODCIGs. I then apply a tapered mute to eliminate the primaries and do
adjoint migration on the multiples with the same velocity model to
get an estimate of the multiples in data space. Similarly, a tapered mute
is applied to eliminate the multiples and adjoint migration used to
obtain an estimate of the primaries in data space. The estimate of the
multiples is adaptively matched to the data with the estimate of the
primaries used as a weight function to prevent matching the primaries.
I illustrate the method with a 2D synthetic dataset and
show that the primaries can be well recovered although some residual from
the water bottom multiple remains.

Mapping of specularly-reflected multiples to image space: An example with 3D synthetic data (ps.gz 2231K) (pdf 1566K) (src 3641K)
**Alvarez G. and Biondi B.**

In 2D, specularly-reflected multiples, when migrated with the velocity
of the primaries, map to negative subsurface offsets
in Subsurface-Offset-Domain Common-Image Gathers (SODCIGs). In
Angle-Domain Common-Image Gathers (ADCIGs) they map with curvature towards
increasing depths. Here we show, through a 3D synthetic prestack dataset,
that specularly-reflected multiples in 3D have a similar behavior
with an interesting addition: in 3D ADCIGs, the multiples exhibit an
azimuth rotation proportional to the dip of the reflecting interface
generating the multiple. This attribute may be used to discriminate between
primaries and multiples in 3D ADCIGs and therefore help in the attenuation
of the multiples.

Application of Least-Squares Joint Imaging of Multiples and Primaries on Shallow Water-Bottom Data Sets (ps.gz 2091K) (pdf 652K) (src 15808K)
**Vyas M. and Brown M.**

Data contaminated with strong shallow water-bottom multiples is rife with challenges. Application of Least-Squares Joint Imaging of Multiples and Primaries (LSJIMP) on such data sets yields mixed results. LSJIMP solves both the separation and integration simultaneously, as a global least-squares inverse problem. We point out some limitations of LSJIMP by testing it on synthetic data sets that emulate shallow water-bottom marine environments. Some slight modifications have been made, and we suggest some strategies that might make LSJIMP an effective algorithm.

Interpolation with pseudo-primaries (ps.gz 1672K) (pdf 476K) (src 11032K)
**Curry W.**

Large gaps exist in marine data, particularly at near offsets. I generate pseudo-primaries
by cross-correlating a multiple model with the original data.
These pseudo-primaries are used as training data for a non-stationary prediction-
error filter, which is then used to interpolate the missing near offsets. This
method yields good results, and also provides a quality control measure to judge
the usefulness of the pseudo-primaries.

Interpolating diffracted multiples with prediction-error filters (ps.gz 3660K) (pdf 1356K) (src 21083K)
**Curry W.**

Diffracted multiples are a persistent problem, especially in the cross-line
direction. Methods to remove these multiples typically require dense and
complete sampling. I use non-stationary prediction-error filters to
interpolate extra shots in the in-line direction and extra receivers in the
cross-line direction. The dimensionality of the interpolation does not make a
big difference in the in-line direction, but makes a large difference in the
cross-line direction. This is shown with several tests on a synthetic dataset.

Missing Data Interpolation with Gaussian Pyramids (ps.gz 26K) (pdf 78K) (src 2252K)
**Sen S.**

I describe a technique for interpolation of missing data in which local operators of many scales but identical shape serve as basis functions. A data structure known as the Gaussian pyramid is developed to represent image information at different scales. This data structure in essence consists of a series of lowpass filtered versions of the original image stacked up one on top of the other forming a pyramid like structure. I first show how to generate a set of reduced images which stack up to form the Gaussian pyramid structure and then show how we can use this Gaussian pyramid structure to fill in missing data. Several examples of filling in missing data with this algorithm are shown and in most cases the results are comparable with those estimated using a prediction filter approach.

A modified Lloyd algorithm for characterizing vector fields (ps.gz 670K) (pdf 460K) (src 64159K)
**Clapp R. G.**

Lloyd's 1-D method
attempts to identify the major features in a signal's
histogram as
accurately as possible,
with as few points as possible.
The selected points are referred to as the signal's
``codebook''.
The methodology is iterative in nature. It starts
from an initial code book, then repeats
...

Selection of reference anisotropic parameters for wavefield extrapolation by Lloyd's algorithm (ps.gz 3658K) (pdf 786K) (src 15631K)
**Tang Y. and Clapp R. G.**

We propose a method for selecting reference anisotropic parameters in laterally varying anisotropic media
for mixed Fourier-space domain wavefield extrapolation.
We treat the selection problem as a quantization procedure, and use a modified version of the 3D Lloyd's algoritm
for reference-parameter selections. We demonstrate that our method yields a more accurate discription of the
anisotropy model with fewer reference parameters than the uniform sampling approach.
Real data examples illustrate the performance of our method.

Inversion shortcuts by model statistics (ps.gz 1102K) (pdf 936K) (src 40204K)
**Artman B.**

A set of model-space coordinates must be pre-selected for input into
LA according to the amplitude (squared) at all coordinate locations.
Only coordinates with an amplitude greater than a supplied percentage
of the maximum amplitude in the model space are kept. The amplitude
range of model space is then quantized from the minimum threshold to
the maximum (squared) value. Coordinates are repeatedly selected for
input into LA according to the number of quantum levels associated with its
amplitude.
Figure shows graphically how the data input into LA is
...

Optimized implicit finite-difference migration for VTI media (ps.gz 4525K) (pdf 747K) (src 60650K)
**Shan G.**

I develop an implicit finite-difference migration method for vertical transversely isotropic (VTI) media
with laterally varying anisotropy parameters.
I approximate the dispersion relation of VTI media with a rational function series,
the coefficients of which are estimated by least-squares optimization.
These coefficients are functions of Thomsen anisotropy parameters. They
are calculated and stored in a table before the wavefield extrapolation.
The implicit finite-difference scheme for VTI media is almost the same as that of
the isotropic media, except that the coefficients are derived from the pre-calculated table.
In the 3D case, a phase-correction filter is applied after the finite-difference
operator to eliminate the numerical-anisotropy error caused by two-way splitting.
This finite-difference operator for VTI media is accurate to , and
its computational cost is almost the same as the isotropic migration.
I apply this method to a 2D synthetic dataset and a 2D slice of a real 3D dataset to validate
the method.

Seismic waves in rocks with fluids and fractures (ps.gz 2371K) (pdf 331K) (src 5136K)
**Berryman J. G.**

Seismic wave propagation through the earth is often strongly affected
by the presence of fractures. When these fractures are filled with
fluids (oil, gas, water, CO_{2}, etc.), the type and state of the fluid
(liquid or gas) can make a large difference in the response of the
seismic waves. This paper will summarize some early work of the
author on methods of deconstructing the effects of fractures, and
any fluids within these fractures, on seismic wave propagation as
observed in reflection seismic data. Methods to be explored here
include Thomsen's anisotropy parameters for wave moveout (since
fractures often induce elastic anisotropy), and some very convenient
fracture parameters introduced by Sayers and Kachanov that permit a
relatively simple deconstruction of the elastic behavior in terms
of fracture parameters (whenever this is appropriate).

4/6/2006