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Equivalent Media

What is Equivalent in an ``Equivalent Medium? (ps 41K) (src 3K)
Nichols D. and Karrenbach M.
The static behaviour of a medium consisting of infinite horizontal layers can be described by an equivalent homogeneous medium (Schoenberg and Muir, 1989). In this article we give a simplified notation for the calculation of the equivalent medium and discuss the question of how an equivalent medium should be defined. If stress is applied on the top boundary of a stack of layers and the homogeneous equivalent medium, both media show identical changes in elastic deformation energy within the medium and the same exterior displacements and average forces on the boundaries.

A general theory for equivalent media (ps 50K) (src 6K)
Karrenbach M.
Conservation of deformation energy, volume and mass lead to a general formulation for finding the static homogeneous equivalent of an arbitrary heterogeneous anisotropic medium. The homogeneous equivalent has identical integral properties and on a large scale, shows the same average behavior in displacements and stresses. The medium is not restricted to a special geometry.
Modeling vector fields in regions with irregular boundaries (ps 117K) (src 273K)
Muir F., Dellinger J., Etgen J., and Nichols D.
Geologists often see the Earth as homogeneous blocks separated by smooth curved boundaries. In contrast, computer modeling algorithms based on finite-difference schemes need to have the elastic constants of the Earth specified at the points of a regular grid. A common way of gridding a geological model is to lay the computer model grid down and use whatever elastic constants lie beneath each grid point. This naive method may result in artifacts; for example, a gently sloping interface will be modeled as a coarse staircase, which generates unwanted diffractions at the stair edges. ...

Cross-hole data processing

Cross-hole seismology at SEP (ps 34K) (src 2K)
Muir F.
Although surface-to-surface geometry is still pre-eminent in seismic exploration, the increasing use of the seismic method in oil field exploitation applications such as reservoir delineation has led to the development of new downhole tools, and with these tools the need for new data processing techniques. To this end, the SEP conducted a weekly seminar/workshop throughout the Summer Quarter on cross-hole data processing.

Rotation and wavelet estimation using crosshole data (ps 326K) (src 581K)Biondi:
Cole S. and Karrenbach M.
A first task in processing our three-component crosshole dataset is to separate P and S wavetypes. Given the arrival direction of incident energy, one can rotate and combine the three components to produce P, SV, and SH wave sections. We developed an automatic scheme to perform this separation. A window of data is rotated over all possible arrival directions and the direction that maximizes the power of a given wave type is chosen. The entire trace is then rotated according to this direction to produce ...
Deconvolution of cross-hole data (ps 28K) (src 1K)
Vanyan L.
One of the works of the cross-hole SEP subseminar during the summer quarter, 1990 was processing 3-component data obtained using an experimental borehole vibrosource. In (Cole and Karrenbach,1990) the ``rotation" of the data is described. Technically the rotation is transformation of the coordinate system and respectively the components of the data; new axes were chosen the following way: 1) along the direction source-receiver, 2) normal to the first axis in the vertical plane, 3) perpendicular to the vertical plane. The purpose of such processing is separating the waves with ...
Traveltime inversion of a cross-well dataset for elliptically anisotropic media (ps 75K) (src 39K)
Filho C. A. C.
Using the paraxial elliptical approximation for the dispersion relation around a horizontal axis, I applied a homogeneous and a layered traveltime inversion scheme to a common-receiver, multi-component, cross-well dataset. Each wave-type was inverted independently for the two parameters that describe the elastic model for media with elliptical anisotropy. The layered inversion scheme decomposes the model into a set of symmetric and anti-symmetric square functions. Results of the homogeneous inversion for the P and SV waves show a considerable degree of elliptical anisotropy. Even with the absence of SH information we were able to obtain four of the five transverse isotropic elastic parameters. Although the insufficient coverage of this specific dataset compromises the uniqueness of the solution, the first iterations of the layered scheme represent at least one resolution-step ahead of the homogeneous results.
A modified anisotropic system (ps 32K) (src 1K)
Muir F.
Some years back, Muir & Dellinger (1985), in an attempt to simplify the processing of multi-component seismic data, introduced a system of equations that unified ray and wave theory in an approximate manner. This allowed, for example, time/offset relations to be converted to dispersion relations which could then form the basis for anisotropic imaging schemes. This paper removes one drawback from that scheme-the use of an anelliptic factor that had no counterpart in normal experience-and ...

Velocity analysis and migration

Extending a CMP gather by optimization in velocity space (ps 100K) (src 134K)
Claerbout J. F.
A test of linear inversion theory yields weak and unconvincing extrapolations of a CMP gather.
An ambient noise dataset from a producing field (ps 539K) (src 932K)
Cole S. and Vanyan L.
We have recently begun looking at an interesting dataset where ambient seismic noise was recorded using a 480 channel array of buried geophones in a producing field. Pumps provide strong, broadband coherent energy. Using techniques that we have applied to other ambient noise datasets, we hope to locate sources of noise on the surface and at depth, and infer some details about the subsurface.
A Generalized Phase-Shift Method (ps 58K) (src 5K)
Ji J.
A basic part of seismic migration is downward continuation of surface data into the subsurface. Gazdag (1978) introduced the phase-shift method which yields complete accuracy for laterally uniform structures. Kosloff and Kessler (1987) showed how the phase-shift method can be generalized for an arbitrary velocity structure in the space-frequency domain. The generalized phase-shift method by Kosloff and Kessler is instructive to understand how the phase-shift method works and can be implemented in both space-frequency domain and wavenumber-frequency domain.

Migration, tomography, seismic inversion theory, and how they relate to each other (ps 75K) (src 11K)
Kneib G.
The Kirchhoff integral can only be used to propagate wavefields forward. Backward propagation is based on the Porter-Bojarski integral. Both integrals simplify if data is registered on a plane. Transmission and reflection data should be combined to increase spatial resolution. Angles and frequencies should vary as much as possible to cover the largest area possible of the k-space. Migration is identical to diffraction tomography for varying frequencies if the weak scatterer assumption is valid in a constant background velocity medium. Holography is backpropagation and imaging of a monofrequent wavefield. Linear and non-linear inversions improve the migration and tomography results because they consider the data as imperfect.
Finite-difference calculation of Residual-migration operators (ps 251K) (src 713K)
Zhang L.
Using a kinematic approach, I derive two sets of partial differential equations whose solutions define the kinematics of residual-migration operators. These operators can transform an image migrated with one velocity model to an image migrated with another velocity model. Under the assumption of knowing the partial derivatives of traveltimes with respect to the coordinates of velocity models on a regular grid, these two sets of partial differential equations can then be solved with standard finite-difference techniques. This algorithm can efficiently calculate the residual-migration operators for common shot gathers or constant offset sections migrated with general velocity models. Examples with residual profile migration prove that the accuracy of the algorithm is sufficient for seismic applications.

Deconvolution and picking techniques

Correlation-picking deconvolution (ps 173K) (src 210K)
Filho C. A. C.
A scaled correlation between the residuals and the wavelet is used as the basis of a new deconvolution approach, which resolves first the strongest events and then, as the number of iteration increases, the weaker events of the reflectivity sequence. An important property of this method is that the retrieved series has the same frequency content as the original sequence. The results obtained with the application of this algorithm to real data are equivalent to a sharper version of the results of predictive deconvolution.
Automatic dip-picking by non-linear optimization (ps 201K) (src 803K)
Zhang L. and Claerbout J.
We have developed a new automatic dip-picking method that is superior to the dip-scan method in several aspects. In our method, we estimate the dip of an event through the relative time-shift between neighboring traces. The optimal time-shift is defined to be the minimizer of a non-quadratic objective function that measures the discrepancies between the neighboring traces after these traces are shifted relative to one another. To eliminate aliasing effects, data-dependent weighting functions are included in the objective function. This non-linear optimization problem is solved by searching. Once a preliminary solution is obtained, the objective function is approximately reduced to a quadratic form and the residual time-shift is then estimated by solving a linear equation. In the end, the time-shift is converted into the dip. Examples with synthetic and field data show that the combination of the linear and non-linear optimizations enables our algorithm to have the properties of antialiasing, high resolution and high accuracy. The applications of the algorithm include event-picking, moveout corrections, local dip-filtering and missing data interpolation.

Active documents and computer graphics

Active documents and reproducible results (ps.gz 15K) (pdf 37K)
Claerbout J.
A revolution in education and technology transfer will follow from the marriage of word processing and software command scripts. In this marriage, here being called an active document (a-doc), an author attaches to every figure caption a pushbutton or a name tag usable to recalculate the figure from from all its data, parameters, and programs. An a-doc provides a concrete definition of reproducibility in computational oriented research. Given suitable interactive software, an active document is easily converted into an interactive document (i-doc). I have two textbooks undergoing conversion to i-books.

Why Active documents need cake (ps.gz 8K) (pdf 22K)
Claerbout J. and Nichols D.
An active document (a-doc) is software that reproduces a document including its plots. When plot files are absent the a-doc software should regenerate them. One reason to author active documents with the freely-available utility cake instead of the UNIX utility make is because cake handles an environment where intermediate files are missing.

Progress towards the interactive book (ps 400K) (src 550K)
Karrenbach M. and Nichols D.
We have been able to use recent advances in workstation technology and software to make progress towards Jon's goal of an ``interactive book''. By using the ``xtex'' previewer we can preview documents written in the typesetting language TEX. Furthermore by using special TEX commands we can add ``hot spots'' to the previewed document, when the user clicks on these areas a sequence of commands is executed on the workstation. This ability makes the document interactive, the text can describe an idea and then clicking on a button will start a program that shows the idea in action. Clicking on the button can also regenerate a figure within the text, either the same figure or a version made with new parameters.
Xvpanel: interactive programs in one line or less (ps 48K) (src 8K)
Cole S.
In SEP-65, I described an interactive data processing environment that I developed called IPE. Briefly, IPE allows you to use existing batch programs in a pseudo-interactive manner, complete with buttons and sliders that can be used to change parameter values. No programming is required; all that is needed is an ASCII description of the parameters for each program. While IPE is fine for many purposes, in other cases it is overkill. This became clear during the development of interactive documents at ...
X3D (extensible 3-D visualization) update (ps 86K) (src 273K)
Ottolini R.
The X3D graphics system has been upgraded to XView user interface with the additional graphics functionality of illuminated surfaces and transformable raster images.


A geometrical approach to seismic processing: the method of discontinuities (ps 191K) (src 46K)
Goldin S.
What do we expect from a theory? Efficiency, prediction, and explanation. Certainly the first aim seems to be the most important. Many specialists apply a theory mainly with the purpose to construct effective algorithms. But I believe the prediction and explanation by no means are less important. Though we do use computers, we ourselves are not computers at all. And an investigator who possesses algorithms but does not understand why and how it works cannot estimate proper results of processing and realize all ...

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Stanford Exploration Project