Blocky models via the L1/L2 hybrid norm [pdf 112K][source] Jon Claerbout This paper seeks to define robust,
efficient solvers of regressions of nature with two goals:
(1) straightforward parameterization,
and (2) ``blocky'' solutions.
It uses an hybrid norm characterized
by a residual of transition between and
for data fitting and another for model styling.
Both the steepest descent and conjugate direction methods
The 1-D blind deconvolution problem is formulated
in a manner intended to lead to both
a blocky impedance function
and a source waveform.
No results are given.
Generalized-norm conjugate direction solver [pdf 220K][source] Mohammad Maysami and Nader Moussa In optimization problems, the norm outperforms the norm in presence of noise and when a blocky or sparse solution is appropriate.
These applications call for a solver that can redefine the optimum criteria for a particular problem.
We have implemented a generalized norm solver that is useful for a wide range of problems. Our solver modularizes the norm function so that it can easily be interchanged to experiment with different schemes on any particular geophysical problem. We implement , , and two additional norms: Huber and Hybrid . These are useful for problems that seek the benefits of both the and norms.
Dix inversion constrained by L1-norm optimization [pdf 308K][source] Yunyue (Elita) Li and Mohammad Maysami To accurately invert for velocity in a model with a blocky
interval velocity inversion using Dix
inversion, we set up our optimization objective function using
criterion. In this study, we analyze and test an improved version of
the Iterative Reweighted Least Squares (IRLS) solver, a hybrid
solver and a conjugate direction solver. We use a 1-D
synthetic velocity data set and a 1-D field RMS velocity data
set as test cases. The results of the inversion are promising for
applications on realistic geophysical problems.
Applications of the generalized norm solver [pdf 288K][source] Mandy Wong and Nader Moussa and Mohammad Maysami The application of a L1/L2 regression solver, termed the generalized norm solver, to two test cases, shows that it is potentially an efficient method for L1 inversion and is easy to parameterize. The generalized norm solver iterates with conjugate direction. Our first test case, the line fitting problem, shows that the generalized solver is capable of removing outliers in data. Our second test case, the 1D Galilee problem, shows that the generalized solver can produce a satisfactory ``blocky" solution. In terms of parameters, a low threshold value, if giving convergent solution, gives the best result. Experience shows the optimal number of inner loop iterations is one.
Alternatives to conjugate direction optimization for sparse solutions to geophysical problems [pdf 140K][source] Nader Moussa Throughout much of this summer, we experimented with extensions to the conjugate direction method to find optimal solutions to sparse geophysical problems. However, this category of techniques is not unique in its ability to optimize L1-styled fitting goals. We also investigated a variety of other techniques, including a pure L1 solution via the weighted median; a steepest-descent algorithm using the signum-function as a gradient of the true L1 norm; and a totally different approach using the Simplex Algorithm, by mapping our objective function into a linear programming form. Categorically, the approaches that relied on the true L1 method failed due to what we believe is a theoretical shortcoming of the direct application of the pure L1 norm to geophysical optimization problems. The use of linear programming turned out to be quite successful. This could be an interesting option for future research in geophysical optimization.
Velocity model building
Measuring velocity from zero-offset data by image focusing analysis [pdf 1.5M][source] Biondo Biondi Migration velocity can be estimated from zero-offset data
by analyzing focusing and defocusing of residual-migrated images.
The accuracy of these velocity estimates is limited
by the inherent ambiguity between velocity and reflector curvature.
However, velocity resolution improves
when reflectors with
different curvatures are present,
as demonstrated by simple synthetic examples.
The application of the proposed method to
zero-offset field data recorded in the New York harbor
yields a velocity function that is consistent with available
geologic information and clearly improves the
focusing of the reflectors.
Gradient of image-space wave-equation tomography by the adjoint-state method [pdf 3.0M][source] Claudio Guerra Optimization with gradient-descent techniques requires computing the gradient of the objective function. The gradient can be determined by using the Frech
t derivatives, but, for practical problems, this can be very expensive. The gradient can be more efficiently computed by the adjoint-state method, which does not require the use of the Frech
t derivatives. Here, I derive the gradient of the image-space wave-equation tomography using the adjoint-state method. I also show its application with a numerical example using image-space phase-encoded gathers.
Geophysical data integration and its application to seismic tomography [pdf 584K][source] Mohammad Maysami For oil exploration and reservoir monitoring purposes, we probe the earth's subsurface with a variety of geophysical methods, generating data with different natures, scales and frequency content. This diversity represents a large problem when trying to integrate all the gathered information.
The concept of a shared earth by all these geophysical surveys suggests the presence of structural similarities in different data sets. For that purpose, it is necessary to work with geophysical properties that are scale-independent and not physical properties in individual layers.
In this paper, I overview two methods for extracting structural information from data and using it as a constraint to the seismic tomography problem to compare different techniques and their effectiveness.
Reverse Time Migration
Selecting the right hardware for Reverse Time Migration [pdf 1.5M][source] Robert G. Clapp and Haohuan Fu and Olav Lindtjorn The optimal computational platform for Reverse Time Migration (RTM) has recently become a topic of
significant debate, with proponents of the
Central Processing Unit (CPU), General Purpose Graphics Processing Unit (GPGPU), and Field
Programmable Gate Arrays (FPGA) all claiming superiority.
The difficulty of comparing these three platforms for RTM performance is that the underlying
leads to significantly different algorithmic approaches.
The flexibility of the CPU allows for significant algorithmic changes, which
can lead to more than an order of magnitude improvement in performance.
The GPGPU's large number of computational threads and overall memory bandwidth provide
a significant uplift but require a simpler algorithmic approach, requiring more
computation for the same size problem.
The FPGA's streaming programming model results in an attractive but different cost metric.
The current lack of a standardized high-level language is problematic.
Reverse Time Migration of up and down going signal for ocean bottom data [pdf 3.5M][source] Mandy Wong and Biondo L. Biondi and Shuki Ronen We present the results of reverse time migration (RTM) on ocean-bottom data as a precursor to applying reverse time migration and inversion of multi-component ocean-bottom data using the two-way acoustic wave-equation. We propose a joint-inversion scheme that constructively combines up- and down-going migration results and removes spurious artifacts in the final image. Reverse time migration of up-going data gives stronger reflector amplitude and weaker artifacts than migration of down-going data; however, due to mirror-imaging, the area of sub-surface illumination is narrower when using up-going energy. In addition, we observe that a new class of artifacts is present due to RTM injection of receiver wavefields with the ocean bottom geometry.
Theory and practice of interpolation in the pyramid domain [pdf 25M][source] Antoine Guitton and Jon Claerbout With the pyramid transform, 2-D dip spectra can be characterized by
1-D prediction-error filters (pefs) and 3-D dip spectra by 2-D
pefs. This transform takes data from -space to data in
-space using a simple mapping procedure that leaves
empty locations in the pyramid domain. Missing data in -space
create even more empty bins in -space.
We propose a multi-stage least-squares approach where both
unknown pefs and missing data are estimated. This approach is tested
on synthetic and field data examples where aliasing and irregular
spacing are present.
Schoenberg's angle on fractures and anisotropy: A study in orthotropy [pdf 96K][source] James G. Berryman For vertical-fracture sets at arbitrary orientation angles to each other
- but not perfectly randomly oriented, I present a detailed model
in which the resulting anisotropic fractured medium generally has
orthorhombic symmetry overall. Analysis methods of Schoenberg are emphasized,
together with their connections to other similarly motivated and conceptually
related methods by Sayers and Kachanov, among others. Examples show how
parallel vertical fracture sets having HTI symmetry turn into orthotropic
fractured media if some subsets of the vertical fractures are misaligned
with the others, and then the fractured system can have VTI symmetry
if all the fractures are aligned either randomly, or half parallel and
half perpendicular to a given vertical plane. Another orthotropic case
of vertical fractures in an otherwise VTI earth system treated previously
by Schoenberg and Helbig is compared to, and contrasted with, other
examples treated here.
Poroelastic measurements resulting in complete data sets for granular
and other anisotropic porous media [pdf 184K][source] James G. Berryman Poroelastic analysis usually progresses from assumed knowledge of
dry or drained porous media to the predicted behavior of fluid-saturated and
undrained porous media. Unfortunately, the experimental situation is often incompatible
with these assumptions, especially when field data (from hydrological or oil/gas reservoirs)
are involved. The present work considers several different experimental scenarios
typified by one in which a set of undrained poroelastic (stiffness) constants
has been measured using either ultrasound or seismic wave analysis,
while some or all of the dry or drained constants are
normally unknown. Drained constants for such a poroelastic system can be deduced
for isotropic systems from available data if a complete set of undrained compliance data
for the principal stresses is available, together with a few other commonly measured
quantities such as porosity, fluid bulk modulus, and grain bulk modulus.
Similar results are also developed here for anisotropic systems having up to orthotropic symmetry
if the system is granular (i.e., composed of solid grains assembled into
a solid matrix, either by a cementation process or by applied stress) and the grains are
known to be elastically homogeneous.
Finally, the analysis is also fully developed for anisotropic systems with nonhomogeneous
(more than one mineral type), but still isotropic, grains -
as well as for uniform collections of anisotropic grains as long as their axes of symmetry
are either perfectly aligned or perfectly random.