Maximum energy traveltimes calculated in the seismic frequency band (ps.gz 740K) (pdf 678K) (src 5132K)
**Nichols D.**

Prestack Kirchhoff migration using first arrival traveltimes has been
shown to fail in areas of complex structure. I propose a new method
for calculating traveltimes that estimates the traveltime of the
maximum energy arrival, rather than the first arrival.
The method estimates a traveltime that is valid in the seismic
frequency band, not the usual high frequency approximation. Instead
of solving the eikonal equation for the traveltime, I solve the
Helmholtz equation to estimate the wavefield for a few frequencies. I
then perform a parametric fit to the wavefield to estimate a
traveltime, amplitude, and phase.
The images created by using these parameters are shown to be superior
to those created by using first arrival traveltimes, or those created
using maximum amplitude traveltimes calculated by paraxial ray
tracing.

Band-limited Green's functions in 3-D (ps.gz 108K) (pdf 119K) (src 698K)
**Nichols D. and Palacharla G.**

This document was generated using the
**LaTeX**2`HTML` translator Version 97.1 (release) (July 13th, 1997)
Copyright © 1993, 1994, 1995, 1996, 1997,
Nikos Drakos,
Computer Based Learning Unit, University of Leeds.
The command line arguments were:

**latex2html** `-dir /net/sepwww/public/docs/sep80/gopal1/paper_html index.ltx`.
...

Review of traveltime computation methods (ps.gz 34K) (pdf 53K) (src 68K)
**Audebert F., Bevc D., Biondi B., Lumley D., Nichols D., Palacharla G., Rekdal T., and Urdaneta H.**

We overview a wide range of traveltime computation methods,
most of them novelties from the last five years or so. We first
classify the methods, then we give a standardized description. This
unified form allows a convenient comparison of these methods. Each
description first outlines the basics principles of the given
method. It then identifies its main features: assumptions made on
the velocity model, strong points, and weak points. Finally, we attempt
a quantitative rating of all the methods. The rating
is based on geophysical and efficiency criteria.

Marmousi traveltime computation and imaging comparisons (ps.gz 1158K) (pdf 1010K) (src 8698K)
**Audebert F., Biondi B., Lumley D., Nichols D., Rekdal T., and Urdaneta H.**

We compare several migration traveltime computation methods in
the complex Marmousi velocity model. The methods considered include:
Band-limited Green's functions, paraxial ray tracing, ``NORSAR''
wavefront construction, Zhang's local ray tracing, van Trier - Symes'
upwind finite-difference scheme, and Podvin's eikonal solver. Each of
the methods was tested by overlaying traveltime arrivals on a
finite-difference wave-equation shot gather from a source buried in
depth at the target zone. Based on these results, a few methods were
selected for a full Kirchhoff prestack depth migration test on the
Marmousi dataset. We find that if reasonable estimates of the most
energetic arrival traveltime and phase are used, Kirchhoff migration
can provide images of the Marmousi model that are of similar quality
to finite-difference migration images.

Ray methods in rough models (ps.gz 1105K) (pdf 988K) (src 6695K)
**Rekdal T. and Biondi B.**

First arrival traveltime methods have proven to fail to image
deeper structure, when most of the reflected energy is transported by
later arrivals.
The family of ray tracing methods is one of the few available alternatives
to generate Green's functions with multiple arrivals. Dynamic ray methods
also compute amplitudes and phase-shifts.
Even in rough structures, ray methods can be used to
compute useful Green's functions, if the model is properly smoothed.
We discuss ways to obtain a good image with Green's functions
computed with the paraxial ray tracing method in the 2-D Marmousi model.
Our paraxial ray tracing implementation uses the paraxial and
regularity conditions to limit the region where the paraxial solution for
a specific ray is useful.
Smoothing the model, does not only make the model more ``ray valid'',
but also makes the high frequency wavefield more like the band-limited
solution. We apply a merging of phases technique
to reduce two nearby arrivals into one. This technique reduces the high
frequency noise
in the image significantly at a similar cost as using just the most
energetic arrival. Smoothing over a length proportional
to the local wavelength of the center-frequency, results in a better
image than just smoothing over a constant length. Using the two most
energetic arrivals, instead of just the most energetic, improves the image
of the ``target'' zone.

Wavefront construction using waverays (ps.gz 482K) (pdf 484K) (src 1858K)
**Urdaneta H.**

A method for computing first arrival traveltimes and amplitudes in
a general two-dimensional (2-D) velocity model is
presented. The method is the result of merging two
recently published ray tracing methods. The product is a
very robust algorithm that is able to produce broadband wave phenomena,
such as dispersion and wavelength dependent scattering.
Its ability to produce broadband wave phenomena, is achieved
by performing a wavelength-dependent smoothing of the velocity model
across wavefronts. In the limit of high frequency, the method
reduces to geometrical ray theory.
The method is able to illuminate areas of large geometrical
spreading where conventional ray tracing methods may give no
arrivals. The method is tested on synthetic complex
velocity models.

Constraints on minimum velocity variance in seismic traveltime tomography (ps.gz 27K) (pdf 38K) (src 14K)
**Berryman J. G.**

Traveltime data together with known spacing between sources and receivers
for seismic transmission tomography can be used to determine rigorous
constraints on the minimum and maximum wave speeds in the propagating medium.
These constraints lead to a new *minimum variance criterion* on inversion
algorithms. These results do not improve the velocity reconstructions
directly, but provide rigorous and easily computed figures of merit to
help evaluate the difficulty of the reconstruction problem and the performance
of tomographic inversion codes. Specifically, these criteria may be used
during preprocessing to decide whether linear or nonlinear traveltime
tomography methods are required for analyzing a given data set, or during
postprocessing to determine whether regularization methods used to constrain
the maximum model variance were overly restrictive.

3-D prestack migration of common-azimuth data (ps.gz 40K) (pdf 55K) (src 267K)
**Biondi B. and Palacharla G.**

In principle, downward continuation of 3-D prestack data
should be carried out in a 5-D computational space,
even when the dataset to be migrated is only 3-D or 4-D.
Unless this efficiency issue is solved,
3-D prestack migration methods based on the solution
of the one-way wave equation are
uncompetitive with Kirchhoff methods.
We present a method for downward continuing common-azimuth data
in the frequency-wavenumber domain in the original 4-D space
of the common-azimuth data geometry.
The method is based on a stationary-phase
approximation of the one-way wave equation and
can be applied to both phase-shift and Stolt migrations.
The proposed migration methods are exact for constant velocity,
and approximate for velocity varying with depth.
However, results of some numerical experiments
on synthetic data show that the approximation is good
even in presence of strong vertical velocity gradients.

Transformation of 3-D prestack data by Azimuth Moveout (ps.gz 175K) (pdf 171K) (src 1730K)
**Biondi B. and Chemingui N.**

We introduce a new partial-migration operator, named Azimuth Moveout (AMO),
that rotates the azimuth and modifies the offset of 3-D prestack data.
AMO can be effectively applied to
improve the accuracy and to reduce the computational
cost of 3-D prestack imaging.
For example, a 3-D prestack dataset can be drastically reduced in size
by coherent partial-stacking after AMO.
The reduced dataset can be then imaged by prestack depth migration,
a process that would have been too expensive to apply to
the original dataset.
AMO can also be effectively used for regularizing data geometries
(e.g. correct for cable feather) and for interpolating
unevenly sampled data.
AMO is defined as the cascade of DMO and inverse DMO at
different offsets and azimuths.
We derive the time-space domain formulation of the AMO
operator by first deriving its Fourier domain representation,
and then analytically evaluating the stationary-phase approximation.
The impulse response of AMO is a surface in the time-midpoint space;
the shape of the surface is a skewed saddle, and its
spatial extent is determined by the amount
of azimuth rotation and offset continuation to be applied to the data.
When the azimuth rotation is small (),
the AMO operator is compact and inexpensive to apply in the
time-space domain.
We successfully tested AMO by coherently stacking traces
with similar offsets and azimuths from a synthetic land survey.

Part II: Attaining accuracy with depth-variable velocity dip moveout (ps.gz 143K) (pdf 181K) (src 2655K)
**Blondel P.**

The oil industry's need for high-quality images of the
Earth's interior has become more and more pressing in last ten
years. In practice, the quality of seismic imaging increases with
the amount of data acquired. Therefore, seismic surveys often
include three-dimensional and high-coverage acquisitions that
result in a huge quantity of data to process. For timely results
each step of the processing sequence must be as fast as possible.
A time-consuming step in a standard seismic imaging
flow is dip moveout correction. Therefore, enhancing the speed of
this step without sacrificing precision is essential to improve
the state of seismic imaging.
The rapid processing of a huge amount of data requires us to simplify
complex algorithms. Under the constant velocity assumption, the
formulation of the dip moveout correction reduces to the equation
for an ellipse. Because its expression remains simple in three dimensions,
the process is computationally efficient. However, in the case of an
irregular data acquisition geometry, the chaotic spatial spreading
forbids a trace-parallel implementation. This problem is solved by
a time-parallel implementation that allows a fast processing for
any azimuthal distribution in the data.
There are two ways to improve precision in amplitude balancing
and focusing of the seismic images with the dip moveout process.
The first is to apply a proper weight along the operator,
which is achieved at almost no extra computational cost. The
second way is to consider depth-variable velocity. The dip moveout
correction then becomes computationally expensive. However, an
approximation valid for gently dipping reflectors allows the
variable-velocity process to be almost as fast as the
constant-velocity process while improving the focusing of
seismic events. This method is easily applicable in three
dimensions as a first-order approximation of the theoretic
``saddle'' operator.

3-D migration velocity analysis with kinematic Kirchhoff migration (ps.gz 36K) (pdf 52K) (src 28K)
**Audebert F. and Teng L.**

Migration velocity analysis includes both depth-focusing
analysis and residual curvature analysis of coherency panels.
In either method, it is widely used for 2-D velocity analysis
in regions of complex geological structures. Unfortunately, since complex
structures tend to be 3-D structures, 3-D prestack depth migration
is desirable. The most commonly proposed candidate is kinematic
Kirchhoff migration. This paper describes how we can use it as a
datuming-migrating tool, to produce migration velocity analysis panels
in 3-D seismics.

3-D prestack depth migration: Why Kirchhoff? (ps.gz 39K) (pdf 54K) (src 30K)
**Audebert F. c.**

3-D prestack depth migration is emerging as an industrial imaging tool.
Many competing algorithms have been proposed by various authors. Nevertheless,
the real bottleneck in 3-D prestack processes is the sheer amount of input
and output data. This paper reviews the principal migration methods
on the basis of their capacity for data manipulation,
without evaluating the algorithms themselves.
It concludes that Kirchhoff methods are the best first candidates,
while other methods dealing with subsets of input data may have
to be taken into account in the future.

Seismic monitoring of oil production: A feasibility study (ps.gz 49K) (pdf 98K) (src 4629K)
**Lumley D., Nur A., Strandenes S., Dvorkin J., and Packwood J.**

We perform a feasibility study on the likelihood of seismically
detecting and interpreting the time-varying changes in a North Sea reservoir
during solution-gas-drive oil production from a horizontal well.
This study integrates reservoir engineering fluid-flow simulations,
rock physics measurements and transformations,
and prestack seismic modeling and migration on a real but anonymous
North Sea reservoir model.
We calculate spatial distributions of reservoir rock properties from the
fluid-flow simulation data, and map the associated
seismic responses at three production-time snapshots:
prior to any oil production (Base Survey), after 56 days (Monitor 1),
and after 113 days (Monitor 2) of oil production.
Multi-offset seismic surveys are simulated for each of these
three production times. Using realistic seismic acquisition
parameters, we are able to successfully
detect and monitor dynamic gascap expansion in the reservoir
during the fluid-flow simulation of the oil production process.
Evidence of gas coning is clearly visible in the
prestack-migrated difference sections at realistic seismic
noise levels and frequency bandwidth.

3-D seismic monitoring of an active steamflood (ps.gz 16K) (pdf 23K) (src 121K)
**E. Lumley D.**

I present an idealized physical model of steamflood fluid-flow
and make rock physics predictions of seismic impedance changes that might
be observable in 3-D surface seismic monitoring data.
Four distinct fluid-flow phases are considered. Closest to the
steam injection well, a small hot steam zone should be very visible
in seismic data due to a predicted 30% decrease in P-wave velocity (Vp)
compared to
pre-steam reservoir conditions. With increasing radial distance from the
injector, an annulus of hot water is likely to be seismically transparent
due to a weak (5%) decrease in Vp, but a larger radial annulus of
hot oil may be visible due to a predicted 10-15% decrease in Vp.
A high-pressure cold oil front is predicted to propagate away from the
injector one order of magnitude faster than the thermal fronts.
If the increased pore pressure due to steam injection forces the
high-pressure cold oil to cross the bubble point, a large *increase*
in Vp of at least 15% may be visible in seismic monitor data.
This model is currently being tested on a 3-D field data set.
Preliminary results are exciting and will be published later this year.

Measuring and modeling attenuation in rocks (ps.gz 15K) (pdf 17K) (src 4K)
**Muir F.**

This document was generated using the
**LaTeX**2`HTML` translator Version 97.1 (release) (July 13th, 1997)
Copyright © 1993, 1994, 1995, 1996, 1997,
Nikos Drakos,
Computer Based Learning Unit, University of Leeds.
The command line arguments were:

**latex2html** `-dir /net/sepwww/public/docs/sep80/francis2/paper_html index.ltx`.
...

Seismic AVO analysis of methane hydrate structures (ps.gz 446K) (pdf 1146K) (src 3287K)
**Ecker C. and Lumley D. E.**

Marine seismic data from the Blake Outer Ridge offshore Florida show strong
``bottom simulating reflections'' (BSR) associated with methane hydrate
occurence in deep marine sediments. We use a detailed amplitude versus
offset (AVO) analysis of these data to explore the validity of models which
might explain the origin of the bottom simulating reflector. After careful
preprocessing steps, we determine a BSR model which can successfully reproduce
the observed AVO responses. The P- and S-velocity behavior predicted by the
forward modeling is further investigated by estimating the P- and S-impedance
contrasts at all subsurface positions. Our results indicate that the Blake Outer
Ridge BSR is compatible with a model of methane hydrate in sediment, overlaying
a layer of free methane gas-saturated sediment. The hydrate-bearing
sediments seem to be characterized by a high P-wave velocity of
approximately 2.5 km/s, an anomalously low S-wave velocity of
approximately 0.5 km/s, and a thickness of around 190 meters.
The underlaying gas-saturated sediments have a P-wave velocity of 1.6 km/s, an
S-wave velocity of 1.1 km/s, and a thickness of approximately 250 meters.

Methane hydrate rock physics models for the Blake Outer Ridge (ps.gz 195K) (pdf 220K) (src 563K)
**Ecker C.**

Seismic analyses of methane hydrate data from the Blake Outer Ridge
indicate high P-wave velocity and anomalously low S-wave velocity in
sediments containing methane hydrates. In an attempt to explain this observed
P-wave and S-wave velocity structure
at the transition from gas to hydrates,
the effect of different hydrate models on elastic moduli
and velocities are explored. After construction
of an initial gas model, the properties of the hydrates are estimated
using the bound averaging method together with the Voigt and Reuss bounds
for elastic moduli.
The result suggests that the hydrates becomes part of the rock matrix and
softens the pores by fracturing. The additional formation of ice is required
to obtain the desired P- to S-wave velocity
ratio in the hydrate bearing sediments,
indicating temperature conditions around the freezing point of water.

Stress-induced transverse isotropy in rocks (ps.gz 42K) (pdf 55K) (src 524K)
**Schwartz L. M., Murphy W. F., III, and Berryman J. G.**

The application of uniaxial pressure can induce elastic anisotropy
in otherwise isotropic rocks. We consider models based on two very
different rock classes, granites and weakly consolidated granular
systems. We show (1) that these models share common underlying assumptions,
(2) that they lead to similar qualitative behavior, and (3) that both provide
a microscopic basis for *elliptical anisotropy*. In contrast,
a finely layered transversely isotropic medium always shows anelliptical
anisotropy. In the granular case,
we make experimentally verifiable predictions regarding the horizontally
propagating modes based on the measured behavior of the vertical modes.

Angle-dependent reflectivity by profile imaging (ps.gz 19K) (pdf 26K) (src 3262K)
**Ji J. and Palacharla G.**

Among the suite of prestack imaging methods available, profile imaging
method is the most suited for providing wave-theoretical
angle-dependent reflectivity estimates.
In this study, we tested profile imaging method for
its ability to preserve angle-dependent reflectivity
on the image.
A synthetic shot gather was generated
by an elastic finite-difference modeling code (Karrenbach, 1992)
for a shale and gas-sand model (Ostrander, 1982).
The profile imaging was performed with
phase-shift and finite-difference approaches.

An AVO analysis project (ps.gz 993K) (pdf 872K) (src 8859K)
**Ecker C., Lumley D., Levin S., Rekdal T., Berlioux A., Clapp R., Wang Y., and Ji J.**

We present an AVO data set consisting of
raw prestack seismic data, petrophysical information and well-logs.
These data are the focus of an AVO workshop sponsored by Mobil.
SEP's AVO project goals include: true-amplitude preprocessing including
multiple suppression, AVO analysis and impedance inversion,
and developing lithologic/hydrocarbon indicators based on rock physics
properties and seismic attributes. We discuss preliminary results
in model building, synthetic seismogram generation and preprocessing,
and briefly outline our planned research.

Amplitude balancing for AVO analysis (ps.gz 19K) (pdf 27K) (src 1507K)
**Berlioux A. and Lumley D.**

Source and receiver amplitude variations can distort AVO analysis of
prestack seismic reflection data. We therefore perform an amplitude
balancing of seismic traces from a marine data set. We address this
problem by computing the total energy of each trace in the seismic
survey, and remove the global low-wavenumber amplitude trend from the
resulting 2-D trace energy map. We then estimate an amplitude
coefficient for each physical hydrophone position in the recording
cable, and each shot position along the survey line. We apply these
coefficients to the original seismic traces and successfully remove
most of the source and receiver amplitude variation.

Near-surface velocity estimation and layer replacement (ps.gz 638K) (pdf 666K) (src 10452K)
**Bevc D.**

In this paper I describe progress towards developing a method of
inverting for complex near-surface velocity structure by combining
two inversion techniques in a two-step method.
The first step is based on the qualitative observation that trace power
calculated from seismic data and plotted in midpoint-offset coordinates
results in distinctive trajectories which depend on the position and
depth of near-surface velocity anomalies.
By summing along these trajectories it is possible to create an image which
positions the anomalies in terms of lateral position and depth.
In the second step, this image will be input to a wavefield
extrapolation inversion routine as a preliminary velocity model.
Once the velocity structure is
estimated, the seismic data can be numerically propagated through it to remove
distortions caused by focusing and kinematics effects.
I present examples of downward continuing synthetic data through near-surface
velocity models using both Kirchhoff and finite-difference wavefield
extrapolation methods.

Matrix formulation of adjoint Kirchhoff datuming (ps.gz 26K) (pdf 33K) (src 6K)
**Bevc D.**

I present the matrix formulation of Kirchhoff wave
equation datuming and show that upward and downward continuation are
adjoint to each other. By expressing the datuming operator explicitly
in matrix form, it
is possible to examine the nature of the Hessian and to show
that Kirchhoff datuming is not idempotent.

Statics prediction from cross-correlations in orthogonal directions (ps.gz 77K) (pdf 134K) (src 342K)
**Abma R.**

Traditional reflection statics programs use model traces built from
stacks of nearby traces.
Using cross-correlations between nearby
prestack traces avoids the problems associated
with the model building process.
Cross-correlations between trace pairs positioned in orthogonal directions
with respect to structure, residual NMO, and the shot and receiver terms
allow the full statics problem to be separated into three sets of
problems.
Cross-correlations are performed in gathers that are
independent of, or orthogonal to, one of three parts of the full problem.
These simpler problems may be solved individually with greater ease and
perhaps with more accuracy than traditional approaches.
Since only one term is being predicted within each subset of the problem,
this may be considered a signal-processing
problem rather than a matrix-inversion problem.
The cross-correlations of neighboring traces
may be supplemented with
the creation of cross-correlations between traces that are widely separated.
These extra cross-correlations may
stabilize the problem.
Preliminary results on synthetic data appear promising.

Techniques for determining the structure and properties of permafrost (ps.gz 74K) (pdf 77K) (src 162K)
**Abma R.**

Several methods for predicting the relationship between the
velocity and the liquid-water-to-water-ice ratio in
permafrost are derived and examined,
including a modification of the Voigt average
for three materials
based on critical
porosity.
Field seismic techniques for detecting anomalous velocity or attenuation
changes are reviewed in view of the unique wave propagation problems encountered
in permafrost areas.

The ingredients for a simple auto-statics program (ps.gz 11K) (pdf 12K) (src 3K)
**Muir F.**

This document was generated using the
**LaTeX**2`HTML` translator Version 97.1 (release) (July 13th, 1997)
Copyright © 1993, 1994, 1995, 1996, 1997,
Nikos Drakos,
Computer Based Learning Unit, University of Leeds.
The command line arguments were:

**latex2html** `-dir /net/sepwww/public/docs/sep80/francis1/paper_html index.ltx`.
...

Chapter 1: From prestack migration to migration to zero offset (ps.gz 59K) (pdf 78K) (src 31K)
**Popovici A. M.**

As a condition for further generalization of the migration
to zero-offset in variable velocity media,
I develop the theory for 2-D migration to zero offset (MZO)
in constant velocity media,
starting from prestack migration in midpoint-offset coordinates.
At the end of this chapter I arrive at an integral formulation
for the MZO operator, analytically derived from the double square
root (DSR) prestack migration equation. The integral formulation for
the MZO is very similar in form to the DSR equation, suggesting
a generalization to variable velocity media via a phase-shift
algorithm. Further chapters deal with offset separation and
the depth variable *v*(*z*) and laterally variable *v*(*x*,*z*) velocity
media.

Anti-aliased Kirchhoff 3-D migration (ps.gz 32K) (pdf 52K) (src 21855K)
**E. Lumley D., F. Claerbout J., and Bevc D.**

A significant degradation in the quality of Kirchhoff 3-D migration images
often arises because the migration operator summation trajectory is too steep
for the input seismic trace spacing and frequency content.
We present an operator anti-aliasing method that suppresses this problem,
based on local triangle filtering. The *N*-point anti-alias triangles
are efficiently applied as 3-point filters after causal and anticausal
integration of the seismic trace data. We implement our method on
a massively parallel CM-5 in a memory and floating-point efficient
algorithm, and compare our anti-aliasing method to a standard
Kirchhoff migration using a 3-D salt intrusion dataset from the Gulf of
Mexico. Our results indicate that our anti-aliasing method greatly
enhances the 3-D resolution of steep salt-sediment interfaces and faults,
and suppresses false reflections caused by conventional
Kirchhoff-migration aliasing artifacts.

When is anti-aliasing needed in Kirchhoff migration? (ps.gz 1177K) (pdf 1019K) (src 5770K)
**Bevc D. and Lumley D. E.**

We present criteria to determine when numerical integration
of seismic data will incur operator aliasing. Although there are
many ways to handle operator aliasing, they add expense to the
computational task. This is especially true in three dimensions.
A two-dimensional Kirchhoff migration example illustrates that
the image zone of interest may not always require anti-aliasing and
that considerable cost may be spared by not incorporating it.

Stacking and migration of Oseberg data (ps.gz 1116K) (pdf 962K) (src 4025K)
**Teng L.**

This paper describes methods of stacking and post-stack migration
that I used with a data set from Oseberg field.
CMP(common midpoint) bins are used for brute stack to simplify
the stacking procedure. Both
two-pass 3-D post stack migration and Ristow's
four-way splitting method are applied to migrate the stacked data.
The result of four-pass migration doesn't show significant improvement over
the two-pass result
because brute stack, linear interpolation and other approximation.

Angle-dependent reflectivity by wavefront synthesis imaging (ps.gz 29K) (pdf 31K) (src 9732K)
**Ji J.**

Elsewhere in this report, Ji and Palacharla (1994) show
the difficulty of obtaining angle-dependent reflectivities
by conventional prestack methods such as profile imaging.
Imaging by wavefront synthesis is a good alternative to
achieve such a goal.
One of the wavefront synthesis methods is controlled illumination
(Rietveld and Berkhout, 1992; Berkhout, 1992).
This method consists of two steps, areal shot record synthesis
and areal shot record migration.
Angle-dependent reflectivity images can be obtained by
combining the field shot records in such a way
that the related areal source wave field has a
predefined shape, like a constant ray parameter
or a constant incidence angle at the target depth level.
I tested the algorithm with the Marmousi data.

Toward an exact adjoint: semicircle versus hyperbola (ps.gz 31K) (pdf 37K) (src 8229K)
**Ji J.**

The correctness of an adjoint operator entirely depends
on the correctness of the corresponding forward modeling operator.
In migration, the input-oriented algorithm implies
more correct forward modeling than the output-oriented algorithm does.
One big difference between the two algorithms
is the domain of interpolation,
and this is one of the major sources of migration artifacts.
Whereas the input-oriented algorithm
requires interpolation in the model space,
the output-oriented algorithm
requires interpolation in the data space.
For Kirchhoff migration, the method of semicircle superposition
and the method of hyperbola summation correspond to
the input-oriented and output-oriented algorithms, respectively.
The same analogy can be applied to Stolt migration.
Levin (1994) suggested the input-oriented algorithm
that is the counterpart to the output-oriented algorithm
suggested by the SEP ``in-time" group
(Popovici et al., 1993; Lin et al.,1993; Blondel and Muir, 1993)

Stolt without artifacts? -- dropping the Jacobian (ps.gz 87K) (pdf 146K) (src 209K)
**Levin S. A.**

Two-term approximation to the transmission response through a set of thin layers (ps.gz 54K) (pdf 67K) (src 601K)
**Chemingui N., Clapp R., and Muir F.**

We present a new, far-field solution for approximating the transmission
response through a set of thin layers.
The method relies
on a modification to the O'Doherty & Anstey solution
to the transmissivity
through a layered medium. We tested the new method on a real log
and compared the results to the ODA solution. We show that the approximate
solution is quite accurate and thus can be used to provide connections with
the equivalent medium
theory.

Reciprocity of tensor wavefields (ps.gz 624K) (pdf 615K) (src 2335K)
**Karrenbach M.**

If reciprocity principles are invoked in seismic data processing
algorithms, one has to make sure that those processes maintain reciprocity.
I outline briefly the derivation of reciprocity principles for elastic wave
equations and show that discretized or approximated wave equations
can lose symmetry properties and thus reciprocities. This effect is
important if one is concernd about true-amplitude processes
and if one needs to use
reciprocity arguments. I show an example of an elastic
finite-difference approximation to the wave equation
that is not reciprocal, but can be made reciprocal
by symmetrizing the Green's function kernel properly.

Wave propagation in the heterogeneous lower crust - Finite Difference calculations (ps.gz 1929K) (pdf 1678K) (src 8981K)
**Karrenbach M., Ritter J., and Fuchs K.**

Wave propagation in heterogeneous media is not only characterized
by reflection, transmission and conversion of seismic energy but also by
effects such as scattering and tunneling and can be observed on many scales.
We investigate elastic wave propagation in
the lower crust of the earth. It is remarkable that distance
and time scales in a deep crustal reflection problem
can be easily transformed into an exploration/production oriented problem.
In that analog, the lower crust corresponds to some fractured medium
or a medium with laminated inter bedding of source rocks,
such as, sand and shale.
We model surface seismic reflection data by positioning the source close to
the surface. Wide-angle refraction data are simulated by placing the source
into the lower crust. Teleseismic data are generated by having
a plane or point source beneath the target zone.
On that scale, a source with a frequency of 1Hz essentially sees an
equivalent homogeneous
medium, while a source with a dominant frequency of 5Hz, sees fine scale
discontinuities as observed in various real data.
Using a finite-difference technique, we employ models with spatially varying
subsurface parameters.
The fine scale heterogeneities are thin reflector segments,
whose length and distance from each other are governed by a Poisson's
probability distribution.
Wave type conversions are surprisingly well confined and can be easily
identified in seismograms as on snapshots.
The ultimate goal of this investigation is to determine whether we can
image those reflector segments and determine their Vp/Vs ratio.

An investigation into eliminating surface multiples (pdf 1678K)
**Wang Y. and Levin S.**

Since the earliest days of the
Stanford Exploration Project, the elimination of free-surface marine multiples
has been a target of processing research. In recent years the plane-wave
methods of Riley and Claerbout (1974) have been superceded by fully two- and
three-dimensional extensions of the method at Delf University
(Vershcuur et al., 1988). Alternative derivations of the newer
approach have also been put fort by Carvalho et al. (1991) based on
scattering theory. Interestingy, there is an apparent disagreement among
the various approaches regarding the correct formulation of the algorithm.

In this paper we first develop our own mathematical derivation in order
to clarify the method. Using Green's identity, the soultion of the
inhomogeneous acoustic wave equation without free surface reflection is
implicitly expressed as a Fredholm integral equation of the second kind.
The associated kernels can be found by up-going and down-going wave
decomposition. We then show by means of a simple but instructive example
that solving the implicit integral equation by a Neumann series is
numerically unwise, despite the fact that the series reduces to a finite
sum, and argue for a treatment of the implicit relation by optimization
methods. Finally we outline our plans to tackle this optimization within
the framework of the C++ linear operators (CLOP) machinery recently
developed at the SEP.

Modeling 3-D anisotropic fractal media (ps.gz 167K) (pdf 179K) (src 398K)
**Chemingui N.**

This paper presents stochastic descriptions of anisotropic fractal media.
Second order statistics are used to represent the continuous random field as a
stationary zero-mean process completely specified by its two-point covariance
function.
In analogy to the two-dimensional Goff and
Jordan model for seafloor morphology, I present the von Karman functions as
a generalization to media with exponential correlation functions.
I also compute a two-state
model by mapping the random field from continuous realizations to a binary
field. The method can find application in modeling impedances from fractal
media and in fluid flow problems.

Building models with GOCAD (ps.gz 28K) (pdf 45K) (src 17K)
**Berlioux A.**

The aim of this paper is to give a quick overview of some of the
different tools available in the GOCAD software package to build 2-D
and 3-D models. With the help of this paper and the GOCAD Manual
(), someone without any knowledge about GOCAD should
be able to build any complex model from its representation on a
sheet of paper.

Iterative velocity model building for 3-D depth migration by integrating GOCAD and AVS (ps.gz 217K) (pdf 163K) (src 1414K)
**Clapp R. G. and Biondi B.**

We have developed a procedure for building a 3-D velocity model
starting from a 2-D geological model in order to image 3-D poststack data.
The 3-D model was built using GOCAD, and it was iteratively refined
by interpreting the result of 3-D poststack depth migrations using AVS.
At the beginning of the iterative process, when only few reflectors
are detectable in the migrated cube, we interpret only one surface.
From the deformations measured on the selected reflector
we compute a displacement field for every point in the model.
The application of the displacement field to the GOCAD
model used for migration leads to an improved
model. The migration of the data with this improved
model shows better focusing of some reflectors and allows
a more complete interpretation of the structure.

The ``unwritten'' computing rules at SEP (ps.gz 12K) (pdf 14K) (src 5K)
**Popovici A. M., Nichols D., and Bevc D.**

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5/15/2001