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SEP-79 (1994)

Visualization & Preprocessing

Visualization of multi-dimensional seismic datasets with CM-AVS

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Biondi B. and Trier J. v.
We have developed a visualizer for large 3-D prestack seismic surveys that takes advantage of the data handling (memory and I/O) and computational capabilities of SEP's CM-5. The visualization application is a collection of AVS and CM/AVS modules and networks. The CM/AVS modules read, window, gain, and slice large multi-dimensional (up to six dimensions) datasets on the CM-5 and then pass the images to a workstation for displaying or 3-D rendering with standard AVS modules. We demonstrate our application by visualizing a 3-D prestack marine survey and a 3-D migrated volume together with a 3-D velocity function.

Lateral prediction for noise attenuation by t-x and f-x techniques

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Abma R. and Claerbout J.
Attenuating random noise with a prediction filter in the time-space domain generally produces results similar to those of predictions done in the frequency-space domain. However, in the presence of moderate- to high-amplitude noise, time-space, or t-x prediction passes less random noise than does frequency-space, or f-x prediction. The f-x prediction may also produce false events in the presence of strong parallel events where t-x prediction does not. These advantages of t-x prediction are the result of its ability to control the length of the prediction filter in time. An f-x prediction produces an effective t-x domain filter that is as long in time as the input data. Gulunay's f-x domain prediction, also referred to as FXDECON, tends to bias the predictions toward the traces nearest the output trace, allowing more noise to be passed, but this bias may be overcome by modifying the system of equations used to calculate the filter. The three-dimensional extension to the two-dimensional t-x and f-x prediction techniques allows improved noise attenuation, because more samples are used in the predictions, and the requirement that events be strictly linear is relaxed.

3-D data infill and extrapolation

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Claerbout J. F.
Here I deal with data given on a uniform 3-D mesh (or 1-D or 2-D). I provide code for recovering missing data in the interior of the cube or abutting one wall of the 3-D data cube by a two-stage linear least squares process. First I fit a 3-D prediction-error filter (PEF) to the extended cube ignoring regression equations that involve missing elements. Then I estimate those missing elements by finding the data that minimizes the power out of the PEF. As in my other work since SEP-73, the volume can be broken into subcubes in which separate problems are solved and pieced together.

Operator design for three-dimensional data extrapolation

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Abma R.
In this paper, extrapolation of data in three dimensions is done in two steps. First, a three-dimensional annihilation filter is calculated from the available data. Next, the result of filtering an extended volume with the calculated annihilation filter from the first step is then minimized by computing the missing data in the extended volume. This technique suffers from a limited extrapolation range and from edge effects. By making predictions with purely lateral prediction filters, the distance the data is extrapolated in the spatial directions is extended. Also, although not well understood, the use of multiple directional prediction filters appears to improve the extrapolation quality, especially near the edges of the volume where very poor predictions are made. These three-dimensional directional filters are lateral filters with the output on one corner of the three-dimensional filter. While these results are preliminary, the techniques presented here look promising.

Velocity & DMO

Statics estimation by optimization of local slant stacks

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Cole S.
Estimating statics by maximization of stack power presumes that NMO correction has flattened events. For estimating statics prior to NMO correction, or where a sufficiently accurate NMO correction is not available, an alternative is to optimize the power of local slant stacks. Long wavelength statics present a problem, as with other algorithms, but varying the number, size, and overlap of the local slant stacks can help.

Reflection tomography: Kjartansson revisited

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Claerbout J. F.
I begin experiments with Kjartansson's data set with the goal of constructing a near-but-below-surface velocity model. I test lateral extrapolation of data before tomography and propose an alternate definition of pilot trace. Traditionally, a pilot trace is defined as a stack trace. Here I define a pilot trace as the output of a three-dimensional prediction filter. This pilot trace is designed to be insensitive to velocity, dip, and shot statics. The crosscorrelation of this pilot trace with the observed trace is highly sensitive to geophone statics as well as deeper lateral velocity variations beneath the geophones. This crosscorrelation function of midpoint and offset is slant-stacked transforming offset to depth. With suitable normalization the result is a volume of coherency with axes of depth, midpoint, and slowness. The strange ``footprint'' of the 3-D prediction filter confuses the construction of a specific slowness model. A less confusing alternate is a Snell wave approach.

Toward estimating near surface lateral velocity variations

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Bevc D.
Near surface velocity irregularities distort the seismic wavefield and cause focusing anomalies in the data. These anomalies can be detected by examining plots of trace power in midpoint-offset coordinates. I generate synthetic data by upward continuing a synthetic wavefield through a v(x,z) model. The location of the velocity anomaly is determined by analyzing a plot of trace power in midpoint-offset coordinates. By performing iterative slant stacks of the power plots, I can image the velocity anomaly. Finally, I downward continue the wavefield through the velocity anomaly in order to remove distortions from the data.

A parallel implementation of Kirchhoff DMO

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Blondel P.
In this paper I compare two parallel algorithms for three-dimensional (3-D) Kirchhoff dip moveout (DMO) in a constant velocity medium. In the case of 3-D land data, the highly irregular offset geometry pleads for an algorithm where data are processed in time slices. This mapping of the data into processor memory minimizes the cost of communication, which is reduced to nearest neighbor communication of time slices, and achieves load balance, keeping eighty percent of the processors busy throughout the process. The time aliasing of the operator is solved by a spatial convolution with dip-dependent triangles in a two-dimensional parallel implementation.

How variable velocity dip moveout improves post-stack migration

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Blondel P.
Nowadays, a standard industrial seismic processing almost always involves the constant-velocity dip moveout correction. Two main features make the process attractive. First, the computational cost is low compared to prestack migration in both two-dimensional and three-dimensional processings. Secondly, because constant-velocity dip moveout is ``independent of velocity'', it may come before velocity analysis, removing the effects of dip Forel and Gardner (1988). ...


3-D Hale-McClellan migration: A salt intrusion example

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Palacharla G. and Biondi B.
McClellan transformation offers a computationally efficient scheme for 3-D post-stack depth migration. The accuracy of the response of the McClellan filter is improved by rotating and averaging the filter. We have implemented the McClellan migration with the improved circular response and tested it on real data, a salt intrusion example. The migration collapses the diffractions, thereby delineating the salt and fault boundaries. The migration results obtained using different McClellan filters show only slight differences because of the limited dip and frequency range in the dataset we used for testing the algorithm.

Prestack slant stack migration and its application in AVO analysis

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Lin J.
One synthetic and one field dataset from the Gulf of Mexico were used to test a program of prestack slant stack migration with phase shift plus interpolation (PSPI). The migrated section showed a correct image with this algorithm, even with lateral velocity variations. I also took advantage of this migration to do amplitude-versus-offset (AVO) analysis on the migrated field dataset. The prestack slant stack migration collapses diffractions, and focuses bright spots, thus increasing confidence in AVO analysis.

Prestack phase-shift migration for separate offsets

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Popovici A. M.
The straightforward implementation of the double square root (DSR) equation in common midpoint and offset coordinates for prestack migration produces notable artifacts that are routinely attributed to Fourier domain wraparound in the offset wavenumber space. These artifacts usually preclude the use of DSR for separate constant-offset sections. I show that the artifacts are not due to Fourier wraparound. I present two phase-shift constant-offset migration algorithms free of offset artifacts. The first algorithm migrates each individual constant-offset at a time, but is slower overall than the second algorithm which requires all the constant-offset sections to be migrated at the same time. Depending on the problem to be solved one method could be more convenient than the other.

Controlled illumination by wavefront synthesis

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Ji J.
Controlled illumination by wavefront synthesis is a depth migration which is tuned by adopting the wavefront synthesis technique. This imaging method provides several advantages over conventional prestack depth migration. One is computational efficiency, the second is the quality of the images, and the third is that it provides more information about reflectors, such as angle dependent reflection coefficients.

Four-pass technique for 3-D omega-x-y migration

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Teng L. and Palacharla G.
This paper tests Ristow's four-pass method of 3-D migration as an improvement on the conventional two-pass finite-difference migration. The four-pass technique consists of applying the forty-five degree finite-difference operator along the inline and crossline directions and along the other two directions at forty-five degree azimuth to the inline and crossline directions. We have derived the dispersion relation and differential equations for the four-pass technique and implemented the four-pass technique. The results show that the four-pass operator is more isotropic than the two-pass one.

Stacking fundamentals

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Claerbout J. F.
Velocity increasing with depth causes moveout curves to cross. Here we begin from the modeling operator and its adjoint, the stacking operator. Weighting functions can be used both before and after NMO. We find that weighting functions that make the operator quasi-unitary also sharpen the stack.


AVO analysis of methane hydrate seismic data

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Ecker C. and Lumley D.
Marine seismic data from the Blake Outer Ridge offshore Florida show strong ``bottom simulating reflections'' (BSRs) associated with methane hydrate occurrence in deep marine sediments. We use a detailed amplitude versus offset (AVO) analysis of these data to explore the validity of two models regarding the origin of bottom simulating reflectors. After careful preprocessing steps including source wavelet deconvolution, nonlinear trace interpolation, amplitude and moveout traveltime calibration, and velocity analysis, we estimate the P and S impedance contrasts at all subsurface positions. This is done by applying a least-squares elastic parameter inversion method to the preprocessed CMP gathers. Our results suggest that the Blake Outer Ridge BSR is compatible with a model of methane hydrate in sediment, overlying a layer of free methane gas-saturated sediment.

Time-lapse seismic reservoir monitoring: The petrophysical basis

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Lumley D., Dvorkin J., and Nur A.
Time-lapse seismic monitoring of subsurface rock property changes incurred by reservoir production processes is increasingly being proposed as a new diagnostic tool in efficient reservoir characterization and management. We perform a model study to simulate waterflood production in a light oil reservoir of Ottawa sand, and generate synthetic time-lapse monitor seismic data both pre-flood, and at two subsequent waterflood phases. Pore pressure and oil/water pore saturation levels are simulated in the reservoir due to two water injection well galleries by diffusive fluid-flow modeling. The pressure and saturation data are converted to rock density and both bulk and shear moduli, using rock physics calibration curves derived from laboratory data. Synthetic seismic reflection data are generated from the resulting spatially variable rock physics properties at three separate waterflood stages. In the presence of realistic noise levels, stacked and prestack migrated reflection images clearly show the extent of the water-invaded zone after production. Furthermore, we apply a prestack seismic impedance inversion method and accurately track the relative P and S elastic impedance changes in the reservoir rock caused by the varying petrophysical conditions associated with the waterflood production process.

Angle-dependent reflectivity estimation by Kirchhoff migration/inversion: Theory

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Lumley D. and Beydoun W. B.
We derive a new Kirchhoff forward and inverse theory for modeling and estimating P-P angle-dependent reflectivity. We replace conventional reflecting surface excitations by equivalent body force volumetric excitations, and then linearize the divergence of the elastic stress tensor wavefield with respect to smooth background material properties. To obtain the reflected wavefield, we perform a volume integral over the equivalent body force distribution, and interpret this result as a hybrid of Zoeppritz plane-wave reflection and Rayleigh-Sommerfeld elastic diffraction. We pose the inverse problem as a least-squares optimization to estimate angle-dependent reflectivity, including the reflection angles, by minimizing the squared error between the forward theory predictions and the observed seismograms. The resulting coupled normal equations are decoupled by the method of stationary phase, and then the uncoupled equations are solved by a classical Gauss-Newton gradient method with an approximate diagonal Hessian operator. The estimation for angle-dependent reflectivity requires a simultaneous calculation of four differently weighted Kirchhoff prestack depth migrations, which combine to compensate for limited acquisition aperture and enhanced reflection amplitude recovery.


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Claerbout J. and Nichols D.

Explicit schemes for estimating elastic properties of multiphase composites

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Berryman J. G.
Explicit schemes of Mori-Tanaka and Kuster-Toksöz for estimating elastic properties of multiphase composites are compared and contrasted. Both methods are known to have a limited range of validity. Reexamination of the theories and comparisons to experiment lead to the conclusion that these explicit schemes should not be used to estimate properties of systems in which the host material occupies less than 50% of the total volume. A unified approach to derivation of these explicit methods, as well as some implicit methods, is also presented.

Just in time

Elements of an exact kinematic theory for wavefronts traveling through layered anisotropic media

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Muir F.
Most conventional seismic processing steps assume either that the velocity function, v(z), is known, or that it can be determined from the data. A further, related, assumption is that the earth is isotropic. This is a major convenience and allows the use of simple, ray-based arguments for developing processes such as DMO. This Isotropic Paradigm is clearly useful where it is backed up by well control, but is unnecessary and may be quite misleading in frontier areas where surface seismic is the sole information source. In case of surface seismic data all processing steps up to and including ...

Extending Stolt migration to general dispersion relations

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Ecker C. and Muir F.
Nowadays, most data processing methods use depth dependent parameters that can not be determined in a direct way but have to be estimated indirectly. On the other hand, data processing in time requires only the knowledge of parameters that can be measured on the surface such as time, offset and horizontal velocity Muir (1993). This is of advantage especially in the case of anisotropy as the vertical velocity does not need to be estimated any longer. ...

Comparison of different interpolation methods for Stolt migration

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Lin J., Teng L., and Muir F.
In Stolt migration, it's very important to select a good algorithm for interpolation in the frequency domain. Incorrectly migrated events caused by a bad interpolator can badly distort the correctly migrated events. Harlan 1982 showed some of the artifacts produced by different interpolators. In this paper we compare the following five different interpolation methods: nearest-neighbor, linear, linear plus cork-screw, Harlan's 10-point scheme, and ...

Interpolation in Stolt migration

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Popovici A. M., Blondel P., and Muir F.
Downward continuation and imaging in the isotropic case (Gazdag, 1978) can be written as  


Parallel computing exhumes slow Fourier transform in Stolt migration (ps 233K) (src 300K)
Blondel P. and Muir F.
Stolt migration is a remapping of the data in the frequency-wave-number domain. The mapping function may be an expression of the wave equation either in isotropic media or in anisotropic media Dellinger et al. (1993); Ecker and Muir (1993). Both cases require a frequency-domain interpolation that can be a source of many numerical artifacts Harlan (1982); ...


Modeling, adjoint modeling, and migration in V(z) media, using mono-frequency Green's functions

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Nichols D.
Modeling and migration are very simple to implement if pre-tabulated Green's functions are available. I use Green's functions, calculated at a few frequencies to estimate, the Green's functions for all frequencies. In a V(z) medium the Green's function is independent of surface location therefore only one set needs to be calculated. I calculate this set by one way extrapolation in polar coordinates. They are then used to perform prestack modeling and migration of data. Because they are calculated in polar coordinates they can be used to model and image overturned waves.

Modeling nonlinear source-surface interactions

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Karrenbach M.
The interaction of a seismic source device with the free surface of the earth changes the radiation characteristic to be different from that of the equivalent body point source. Usual seismic sources are built to give enough energy to penetrate a few kilometers in the earth. It is not guaranteed that when the source is activated the surface material behavior stays within a region that can be appropriately described using infinitesimal stress strain relationships. Finite differences can easily model radiation characteristics and wave propagation when the equations of motion are no longer linear. I compare two nonlinear elastic wave equation to model such source behavior.

3-D Eikonal equation for time migration

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Berlioux A.
In the short-note I wrote in the previous SEP report Berlioux (1993b), I presented a new method to perform depth migration in 3-D, starting from time-migrated maps. Since time migration does not always produce properly migrated images or maps in areas where velocity variations are present Hatton et al. (1981), the method I proposed consists of two parts: first undo the time migration by a principle of de-migration, and then perform the depth migration. ...

3-D grid with GOCAD

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Berlioux A.
The aim of this article is two-fold. First, a small program has been developed to build a 3-D grid and fill it from geological or geophysical models using the GOCAD library of functions. This 3-D grid program has been tested in many cases and has produced good results even with very complex structural models. Second, a proposal is made to elaborate a new implementation of the GRID3_t object of GOCAD in a manner more convenient for geophysical and/or geological purposes.


  • Berlioux, A., 1993a, 3-D grid with GOCAD: SEP-79, 301-318.
  • Berlioux, A., 1993b, Three-dimensional map de-migration/migration: SEP-77, 71-76.
  • Claerbout, J. F., 1976, Fundamentals of geophysical data processing: Blackwell.
  • Hatton, L., Larner, K. L., and Gibson, B. S., 1981, Migration of seismic data from inhomogeneous media: Geophysics, 46, no. 5, 751-767.
  • Khare, V., 1991, A new ray-theoretic interpretation of time migration: 61st Annual Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1492-1496.
  • Lowenthal, D., Lu, L., Roberson, R., and Sherwood, J. W. C., 1985, The wave equation applied to migration in Gardner, G. H. F., Ed., Migration of seismic data:: Soc. Expl. Geophys., 208-227.
  • Reshef, M., and Kosloff, D., 1986, Migration of common shot gathers: Geophysics, 51, no. 2, 324-331.
  • van Trier, J., and Symes, W. W., 1990, Upwind finite difference calculation of traveltimes: SEP-65, 41-58.
  • Vidale, J., 1988, Finite-difference calculation of travel times: Bull. Seism. Soc. Am., 78, 2062-2076. 
R. Abma
A. Berlioux
J. Berryman
D. Bevc
W. Beydoun
P. Blondel
B. Biondi
J. Claerbout
S. Cole
C. Ecker
J. Ji
M. Karrenbach
L. Laane
J. Lin
D. Lumley
F. Muir
D. Nichols
H. Oh
G. Palacharla
A. Popovici
M. Schwab
L. Teng
H. Urdaneta
Publication Date