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3-D Seismic

Common-azimuth prestack depth migration of a North Sea data set (ps 1058K) (src 5914K)
Biondi B.
Common-azimuth migration accurately imaged in depth a marine 3-D prestack data set recorded in the North Sea. The comparison of the results of common-azimuth migration with the results of multiple 2-D in-line migrations show that common-azimuth migration correctly focused the reflections along the cross-line direction, as well as the in-line direction. Because common-azimuth migration takes into account the 3-D features of the velocity model it focuses sub-salt reflections better than 2-D migration. Common-azimuth migration can be also used to extend the depth-focusing analysis methodology to 3-D. I hope that refinements in the velocity model achieved by depth-focusing analysis can even further improve the migration results presented in this report.
Azimuth moveout for 3-D prestack imaging (ps 1299K) (src 2763K)
Biondi B., Fomel S., and Chemingui N.
We introduce a new partial prestack-migration operator, named Azimuth MoveOut (AMO), that rotates the azimuth and modifies the offset of 3-D prestack data. AMO can improve the accuracy and reduce the computational cost of 3-D prestack imaging. We have successfully applied AMO to the partial stacking of a 3-D marine data set over a range of offsets and azimuths. Our results show that when AMO is included in the processing flow, the high-frequency steeply-dipping energy is better preserved during partial stacking than when conventional partial-stacking methodologies are used. Because the test data set requires 3-D prestack depth migration to handle strong lateral variations in velocity, the results of our tests support the applicability of AMO to prestack depth-imaging problems. AMO is defined as the cascade of a 3-D prestack imaging operator with the corresponding 3-D prestack modeling. To derive analytical expressions for the AMO impulse response, we used both constant-velocity DMO and its inverse, as well as constant-velocity prestack migration and modeling. Because 3-D prestack data is typically irregularly sampled in the surface coordinates, AMO is naturally applied as an integral operator in the time-space domain. The AMO impulse response is a skewed saddle surface in the time-midpoint space. Its shape depends on the amount of azimuth rotation and offset continuation to be applied to the data, but it is velocity independent. The AMO spatial aperture, on the contrary, does depend on the minimum velocity. When the azimuth rotation is small ($20^{}$), the AMO impulse response is compact and its application as an integral operator is inexpensive. Implementing AMO as an integral operator is not straightforward because the AMO saddle may have a strong curvature when it is expressed in the usual midpoint coordinates. To regularize the AMO saddle, we introduce an appropriate transformation of the midpoint axes that leads to an effective implementation.
Antialiasing 3D prestack Kirchhoff datuming (ps 73K) (src 204K)
Crawley S.
Irregular spatial sampling complicates antialiasing. 3D seismic surveys are designed to have even sampling in midpoints, but often have irregular sampling in offset. This presents the difficulty that prestack integral operators may have quite different bandwidth along different axes. 3D prestack Kirchoff datuming can be performed in midpoint and offset coordinates, where the data's dip range can be asssumed to be limited relative to its dip range in shot and receiver coordinates. Operator bandwidth can be preserved for a limited range of dips, even when sampling is quite sparse. A midpoint and offset formulation of Kirchoff datuming is more expensive, but may be necessary to make the operator useful for many 3D geometries.
Normalization of Kirchhoff imaging for variable fold (ps 424K) (src 1705K)
Chemingui N. and Biondi B.
For multichannel recording, abundance of data traces represents a key element in the processing stage and the quality of the final image. Therefore, seismic acquisition geometries are often designed to optimize the fold coverage in the survey. However, this often neglects a crucial processing requirement: a good distribution of offset and azimuth in the CMP bins. The processing of seismic data for amplitude inversion requires both accuracy of algorithms and proper handling of irregular geometry. Sampling irregularities in the form of varying fold may introduce noise, cause amplitude distortions, and even structural distortions when wave equation processes such as DMO, AMO, and prestack migration are applied to the data. In the context of amplitude-preserving processing, we present a new development in our true-amplitude sequence for processing wide-azimuth 3D surveys. We extend the concept of multiplicity to wave equation processes and explore a normalization procedure to correct for the variations in fold distribution. Given a common-offset common-azimuth 3D subset, we correct for the fold variations by normalizing each input trace according to the local fold of its corresponding bin. The normalization corrects for data trace redundancy but does not account for missing traces or for the spatial distribution within local bins. Results of the application of integral migration and integral AMO as summation over the preconditioned data have proved to correct and equalize the imaging processes for the effects of fold variations.
Three dimensional dynamic ray tracing in complex geological structures (ps 428K) (src 2663K)
Sun Y., Clapp R. G., and Biondi B.
We implement a robust three dimensional dynamic ray tracing algorithm, that can be applied to complex geological structures. We use a Runge-Kutta solver to solve the dynamic ray tracing system. This solver has the ability to adapt its step length in accordance with the local gradient of the slowness field. We applied the ray tracing method to three synthetic models. The results were accurate and robust when the model was smoothed with a 3-D triangle filter to make the models more ``ray valid''.
Visualization of irregularly sampled seismic data with AVS (ps 604K) (src 563K)
Mora C. B., Clapp R. G., and Biondi B.
We present a visualizer for multi-dimensional irregularly sampled data sets using SEPlib90 (SEP's library to handle irregularly sampled data) data access routines and AVS's (Application Visualization System) visualization environment. We developed two AVS modules that read, stack and regularize irregularly sampled data sets, and two modules that extract 2-D slices from n-dimensional data and generate the corresponding surface in 3-D space. These modules can be combined with standard AVS modules to display the data. Information on the reliability of interpolated data is also generated and displayed simultaneously with the seismic data for improved interpretation.


Determination of near seafloor properties from ocean-bottom recordings (ps 185K) (src 131K)
Ecker C. and Sayers C.
We compare two methods of determining the near seafloor parameters (density, P-wave and S-wave velocity) from data recorded by ocean-bottom seismometers (OBS) and ocean-bottom hydrophones (OBH). The first method is based on AVO from ocean-bottom pressure and vertical particle velocity, and the second on AVO from ocean-bottom vertical and radial particle velocity. Using simple synthetic seismograms, we evaluate the parameter estimation for a simple water over half-space model and explore how the results are influenced by layering and poroelastic effects. The method using two particle velocities seems to be more robust and more sensitive to parameter changes over a broader range of angles. Introducing either a 50 m or 100 m thick layer underlaying the seafloor, this method still yields P-wave velocity estimates with errors less than 3% and S-wave velocities with errors less than 7%. The use of pressure and vertical particle velocity does not result in good estimation of P-wave velocity, S-wave velocity or density. In the case of poroelasticity, the pressure-vertical particle velocity method produces reasonably good results for the P-wave velocity ($$ 1% error) and S-wave velocity ($$ 2% error), while introducing a significant error into the density (20% error). Furthermore, poroelasticity has a considerable effect on the radial particle velocity, thus causing the inversion based on the vertical and radial particle velocity components to yield errors of more than 30% for the S-wave velocity estimation.
Iterative methods of optimization with application to crosswell tomography (ps 75K) (src 44K)
Berryman J. G. and Fomel S.
We review the theory of iterative optimization, revealing the common origin of different optimization methods and reformulating the pseudoinverse, model resolution, and data resolution operators in terms of effective iterative estimates. Examples from crosswell tomography illustrate the theory and suggest efficient methods of its implementation.
A prospect for super resolution (ps 48K) (src 44K)
Claerbout J.
Wouldn't it be great if I could take signals of 10-30 Hz bandwidth from 100 different offsets and construct a zero-offset trace with 5-100 Hz bandwidth? This would not violate Shannon's sampling theorem which theoretically allows us to have a transform from 100 signals of 20 Hz bandwidth to one signal at 2000 Hz bandwidth. The trouble is that simple NMO is not such a transformation. ...

Multichannel Processing

The effects of lateral velocity variations and ambient noise source location on seismic imaging by cross-correlation (ps 347K) (src 746K)
Rickett J.
Finite-difference modeling is used to investigate the conjecture that, by cross-correlating noise traces recorded at the surface, we can construct what would be recorded at one of locations if there was a source at the other. Synthetic experiments are conducted on a variety of Earth models with lateral variations in velocity. For each model, conventional seismograms are compared with seismograms constructed by cross-correlating noise traces. Sources of the ambient noise are first taken to be at infinity (planar wavefronts), and then within the zone of interest (curved wavefronts). The kinematics of the reflection events in all the cross-correlation seismograms are consistent with the conventional seismograms, suggesting the conjecture is robust to moderate lateral velocity variations and the location of the source of the background noise. In cases where most of the incoming energy has one angle of incidence, it is shown that by applying a weighting function in the slant-stack domain, the coherency of hyperbolic reflection events is increased.
Suppression of water bottom multiple energy in beam stacked data (ps 488K) (src 3961K)
Holden T. C. and Biondi B.
Water-bottom related multiple energy can be a significant problem for processing and interpretation of seismic data. The standard method of NMO stacking CMP gathers to eliminate multiples is often inadequate and does not address the problem of preserving interesting features on prestack data such as AVO and non-hyperbolic moveout. In this paper we examine primary and multiple energy in beam stacked data and a method of isolating the primary energy in this space. We decompose the data into the beam stacked space by means of an iterative least squares inversion of the beam stack operator. We then apply a masking function to the beam stacked data. The primary energy is then forward-modeled simply using the beam stacking operator. We apply our method to real data and obtain encouraging results.
The pyramid transform and its application to signal/noise separation (ps 734K) (src 4307K)
Sun Y. and Ronen S.
The ``pyramid transform'' is spatial-resampling of data in the frequency-space domain with frequency-dependent grids (FDG), in which the spatial-sampling interval is inversely proportional to the frequency. A cube in the F-X-Y domain is thus transformed to a pyramid. Wavefields do not contain high wavenumbers in low frequencies. Therefore, the pyramid transform is invertible when applied to wavefields. After pyramid transformation, the size of data has shunk dramatically. This feature can save many resources. However, the main benefit is that the low frequencies are not over-parameterized which makes frequency-dependent grids more suitable for inversion and interpolation. For example, spatial prediction filters become independent of the temporal frequency in the pyramid domain. This feature has a great potential in signal/noise separation and trace interpolation. In this paper, we investigate the possibility of applying the pyramid transform to noise suppression. Our results show the prediction filter estimated in the pyramid domain can remove the low temporal frequency random noise, which can not be handled by the prediction filter estimated from the common frequency-independent grids (FIG).


Range of the P-wave anisotropy parameter for finely layered VTI media (ps 73K) (src 16K)
Berryman J. G.
Since the work of Postma (1955) and Backus (1962), much is known about elastic constants in vertical transversely isotropic (VTI) media when the anisotropy is due to fine layering of isotropic elastic materials. I review earlier work and then show that the P-wave anisotropy parameter c11/c33 lies in the range ${14} c_{11}/c_{33} <+2<1/(+2)$,when the layers are themselves composed of isotropic elastic materials with Lamé constants $$ and $$ and the vertical average of the layers is symbolized by $<$. The upper bound is true for all finely layered media of this type, while the lower bound is the best possible for fine layering with two constituents. This lower bound corrects an error in Postma's (1955) paper on two-component VTI media. The method used here shows in general terms that the P-wave anisotropy parameter is smallest when the variation in the layer $$ Lamé parameter is large, independent of the variation in the shear modulus $$. This result is therefore important for applications to porous layers where the effects of fluids influence only the $$ Lamé constants, not $$.Thus, the P-wave anisotropy parameter may be a useful hydrocarbon indicator in some situations.
Wave-equation migration: The Kirchhoff approach (ps 1496K) (src 1445K)
Mallia-Zarb E.
I developed an algorithm that performs 2-D Kirchhoff migration. For testing purposes I generated synthetic data and tested the algorithm using different migration parameters. Thus my results confirmed that, given the correct parameters, Kirchhoff migration successfully converts the data space back to the image space.

Computing at SEP

SEP documents and software on the web (ps 94K) (src 53K)
Schwab M. and Schroeder J.
SEP offers the following free software packages on its web page. Each item is documented in more detail on an HTML page at the SEP web site.

Design of the SEP web pages (ps 237K) (src 189K)
Schwab M. and Schroeder J.
Ideally, the SEP web pages should be rich in content, free of clutter, and quick to download. We suggest two basic types of web pages: table-of-contents pages that contain lists of annotated links, and content pages that contain mostly text and figures. The author of an SEP web page should attract search engines, maintain a consistent layout for related pages, and include a simple, universal navigation button that returns the reader to the SEP home page. Finally, to maintain the web page contents, the corresponding SEP source directory should be equipped with a simple makefile target www that exports the software to the web site.
SEP Computing environment (ps 57K) (src 9K)
Clapp R. G. and Crawley S.
One of the more mysterious things when you begin at SEP is the computing environment. After a year or two you know how things work, but in the meantime you have probably spent many hours rewriting pre-existing code or struggling with fairly pedestrian software which is documented either cryptically, not at all, or in a location unknown to all but the documenter. This paper is an effort to minimize time sinks by giving a basic overview of the SEP computing environment, stating some of the `unwritten rules', and providing the answers or directions to the answers to common questions of new students.

Linux at SEP (ps 46K) (src 6K)
Schwab M. and Hargrove P.
SEP continuously balances its commitment to geophysical research with its maintenance and improvement of its computing environment. We suggest that SEP could save a significant amount of time currently expended on the computing environment by simplifying its system, in particular, by reducing the number of supported computer platforms. We suggest a conceptually different computer environment in which SEP researcher would use a Linux workstation for standard UNIX services and a few special purpose computers for specific tasks such as ...


Geophysics in Object-Oriented Numerics (GOON): An informal conference (ps 254K) (src 205K)
Claerbout J. and Biondi B.
We held an open and informal conference at SEP attempting to facilitate cooperation in introducing Object Oriented computational techniques among workers in Geophysics. Java seems attractive for the next century. For the remainder of this century a wide variety of language combination options were considered and many opinions were expressed. A few who stayed longest at the conference (perhaps the most dedicated to cooperation) developed a linkage of the Gockenbach-Symes HCL (Hilbert Class Library) in C++ to Fortran 90.
Hilbert Class Library: Ideas behind the design (ps 58K) (src 15K)
Schwab M. and Urdaneta H.
The Hilbert Class Library (HCL) is a C++ library for applied, large-scale, numerical optimization. The Hilbert Class Library essentially defines a set of abstract classes which programmers use to derive their application specific data structures, operators, and solvers. The common set of base classes ensures the cooperation of ...
HCL and regular data (ps 68K) (src 26K)
Schroeder J. and Schwab M.
To support traditional SEPlib processing capabilities within the HCL (Hilbert Class Library) framework, we implemented a regular data class, a series of simple operators, and a few of Jon Claerbout's solvers. The multi-dimensional regular data grid of an SEP Data Cube is implemented as an RGF (Regular Grid Function) class. In HCL's linear algebra paradigm, the grid size and its physical units define a vector space. Since the RGF class is derived from HCL's abstract vector ...
An IGF90 tutorial (ps 136K) (src 81K)
Urdaneta H. and Karrenbach M.
We encapsulate SEP's Fortran90 data structure for handling data with an irregular spatial sampling (SEPlib90) in a C++ class representation (IGF90) and provide it with an interface to a C++ library of inversion and optimization algorithms. The IGF90 class provides a mechanism for users to take advantage of the efficient Fortran90 data structures and the object-oriented programming paradigm of C++. We present different examples to illustrate the IGF90 class user interface. We use the IGF90 class and the C++ library of optimization algorithms to solve a missing data problem given a roughening filter. We discuss the design of the C++/Fortran90 interface, and illustrate it with the C++ interface to SEP's Fortran90 data structure and the interface to a Fortran90 operator.
SEPF90 (ps 93K) (src 111K)
Clapp R. G. and Crawley S.
SEPlib90 gives programmers a storage format for irregular 3-D seismic data, and a library of accessor routines for bringing them into memory. However, experience showed early and often that I/O and data handling were cumbersome for SEPlib90 application programmers. Here we present a Fortran 90 library and data structure that overlay SEPlib90. The library simplifies various non-geophysical tasks such as data input and output; and makes windowing, allocation and deallocation of irregular traces, straightforward. In addition, the library serves as the link between SEP's new C++ framework, GOON (Geophysical Object-Oriented Numerics), and SEPlib90.
A generic NMO program (ps 94K) (src 178K)
Fomel S., Crawley S., and Clapp R.
Jon Claerbout's books Processing versus Inversion 1992b and Three-dimensional Filtering 1994 list normal moveout (NMO) among the basic linear operators. Indeed, the NMO transformation plays a kernel role in many applications of geophysical data processing, from simple CMP stacking to prestack migration and velocity analysis. The importance of this role increases with the development ...
Simple linear operators in Fortran 90 (ps 71K) (src 694K)
Fomel S. and Claerbout J.
A linear operator maps an input vector to an output vector. In the adjoint mode, the mapping direction is reverse. The simplest implementation of this idea is a minimal interface operator( adj, add, model, data), where the logical variable adj defines adjoint or forward mode, and variable add defines whether the output of the program should be added to the previous value of the corresponding actual argument. The minimal interface is the ``mathematical'' connection to operators as objects. To provide the ``geophysical'' connection, we need to initialize an operator with the arguments that ...

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Stanford Exploration Project