SEP-92 (1996)
3-D Seismic Processing
Application of azimuth moveout to the coherent partial stacking of a 3-D marine data set
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Biondi B., Fomel S., and Chemingui N.
The application of azimuth moveout (AMO) to a marine 3-D data set shows that by including AMO in the processing flow the high-frequency steeply-dipping energy can be better preserved during partial stacking over a range of offsets and azimuths. Since the test data set requires 3-D prestack depth migration to handle strong lateral velocity variations, the results of our tests support the applicability of AMO to prestack depth imaging problems.
Handling irregular geometry in wide-azimuth surveys
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Chemingui N. and Biondi B.
The processing of seismic data for amplitude inversion often neglects an important issue which could be detrimental to amplitude preservation: the effect of sparse and irregular geometry. The problem is severe in wide-azimuth 3-D surveys acquired with the aim of studying the azimuthal variations in the AVO gradient. We present a new technique for processing wide-azimuth surveys that accurately images 3-D seismic data while preserving relative amplitudes describing the offset and azimuth-dependent reflectivity function. The method focuses on both algorithm accuracy and proper handling of irregular geometry and therefore allows for reliable AVO analysis on migrated data. The technique is based on organization of the data into common-azimuth (CA) and common-offset (CO) cubes and allows interpolation into a regular grid before imaging. The regular CA/CO cubes are then migrated using an efficient true-amplitude migration algorithm. We apply this technique in order to invert for a synthetic reflectivity model simulating a real 3-D wide-azimuth land survey over a fractured reservoir. Results show that, as expected, Kirchhoff migration is very sensitive to uneven sampling, which creates strong amplitude distortions. The azimuth moveout (AMO) transformation succeeded in organizing the data as regular gridded cubes while preserving its amplitude information for imaging. We conclude that interpolating seismic data prior to migration is effective in preserving amplitude information and that AMO can effectively correct for the irregular sampling in wide-azimuth surveys.
Revealing geological discontinuities by local plane wave suppression
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Schwab M., Holden C., and Claerbout J.
Interpreters of migrated seismic image volumes need to delineate reflector discontinuities, such as faults or buried channels. To enhance these discontinuities we applied local, data-adaptive Prediction Error (PE) Filters to the migrated image. We restricted the PE Filters to local combinations of 2-D filters, which remove local planar events and reveal rapidly varying local features. Unfortunately, while the scheme successfully enhanced geological discontinuities in one field data case, it failed in two others. In these two cases, the PE Filters erroneously predicted (and consequently removed) the geological discontinuities of interest. We were unable to tune the PE Filters to forgo a specific target bandwidth. Such a tuning may be possible if we use a sequence of two PE filtering steps of which the first filter prewhitens the image. In another case, the PE filter enhanced the image's acquisition footprint and thereby obliterated any geological details. Moreover, we do not yet know how to combine optimally the various PE Filter residuals to produce a meaningful subsurface map.
Reservoir monitoring: A multi-disciplinary feasibility study
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Biondi B., Deutsch C., GundesøR., Lumley D., Mavko G., Mukerji T., Rickett J., and Thiele M.
Monitoring reservoir changes with time-lapse seismic holds the promise to significantly improve reservoir characterization and reservoir management. To relate time-dependent changes in seismic to the underlying geological description and flow processes requires input from a number of disciplines: (1) geological modeling/geostatistics, (2) flow simulation, (3) rock physics, and (4) seismic imaging. This paper documents the start of a project that brings these disciplines together. In this initial model study we have considered the forward problem, i.e., create a detailed geological truth model, perform flow simulation, relate the dynamic rock properties to seismic properties, and, finally, image the reservoir at multiple times. We document this forward modeling exercise on a single geological relization and discuss opportunities for future work.
Modeling heterogeneous reservoirs using the first order Born Approximation
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Rickett J., Biondi B., and Lumley D.
This short-note is intended as a companion paper to Biondo et al. 1996. It aims to examine the validity of some of the assumptions necessary for the Born elastic scattering method which we used to create the synthetic seismograms, especially the assumption of weak scatterers which allows the problem to be treated as linear. Born generated synthetic seismograms for plane layer models are compared with Zoeppritz reflection coefficients, and can be seen to give good ...
Passive seismic imaging applied to synthetic data
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Rickett J. and Claerbout J.
It can be shown that for a 1-D Earth model illuminated by random plane waves from below, the cross-correlation of noise traces recorded at two points on the surface is the same as what would be recorded if one location contained a shot and the other a receiver. If this is true for real data, it could provide a way of building `pseudo-reflection seismograms' from background noise, which could then be processed and used for imaging. This conjecture is tested on synthetic data from simple 1-D and point diffractor models, and in all cases, the kinematics of observed events appear to be correct. The signal to noise ratio was found to increase as , where n is the length of the time series. The number of incident plane waves does not directly affect the signal to noise ratio; however, each plane wave contributes only its own slowness to the common shot domain, so that if complete hyperbolas are to be imaged then upcoming waves must be incident from all angles.
Fluid-flow simulation concept of methane hydrate growth and decomposition
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Ecker C.
Bottom simulating reflectors (BSR) are associated with the base of the stability zone of methane hydrates. The occurrence, stability and breakdown of these BSRs is considerably dependent on the temperature and pressure conditions in the sediment. Any P-T changes can affect the hydrate stability significantly and thus influence the behavior of the BSRs. I present the theory and development of a three-dimensional finite-difference fluid-flow simulator for hydrate decomposition/growth in porous media. I derive the equation for a 3-phase/2-component isothermal gas-water-hydrate system and discuss the code development of a two-dimensional system. The code is still in a test phase and is not working properly for all required cases. I show the results of some preliminary code testing based on simple gas-water flow. Furthermore, I discuss how to include more realistic thermodynamics into the system.
Velocities
Velocity space interpolation of aliased seismic data by preconditioning
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Crawley S.
Sparsity and irregularity of spatial sampling are common problems in seismic data. Irregularity severely limits the types of processes which may be applied to a data set, and any process will likely fail on data which are overly sparse, as data and operator aliasing become a crippling problem. Interpolation schemes seek to dealias data, but are themselves challenged by aliasing, because it is difficult for an algorithm to pick exclusively the correct dip or dips to interpolate. Transformation to velocity (or slowness) space is an attractive basis for an interpolation algorithm, because the operator is limited to 'reasonable' directions, in that it operates only along centered hyperbolas. Since data can be organized so that it is symmetric, and largely hyperbolic, in CMP gathers, this type of interpolation should greatly reduce the risk of interpolating incorrect dips. However, the velocity transform is not an exact forward/inverse-transform pair, and the smoothness and/or noisiness of the estimated velocity spectrum presents a new and serious pitfall. While the original data space may be remodeled exactly, large artifacts are likely to appear in alternate similar data spaces, such as are appropriate for interpolation or regularization. By preconditioning the inversion, the model may be made more parsimonious, resulting in improved remodeling into a new data space.
Validation and update of 3-D velocity models by inversion of seismic and well-log data
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Berlioux A.
In this paper, I propose a new procedure for estimating the validity of 3-D velocity models combining seismic data and well-log information. The method I describe gives the best least-square 3-D model for the velocity. It has even more potential for determining how accurate a velocity model is by estimating a range of possible models and giving a measure of local uncertainty about the velocity. I propose to use a 3-D prestack seismic survey as well as the well-log curves available on the site of the survey to evaluate and update a 3-D velocity model. Performing 3-D velocity analysis on the 3-D prestack seismic data, I will get an initial velocity model. From the well-log information, I plan to extract velocities at the well locations. I will then employ two different approaches to derive an accurate 3-D velocity model. The first is based on the least-square inversion of the well-log-derived velocities and uses the seismic velocities as a constraint, by applying for a conjugate gradient method. The second approach is a simulation of the velocity at each node of the 3-D model, using a sequential Gaussian simulation algorithm based on a generalized least-square inverse technique (kriging). I also intend to estimate local and global uncertainties about the velocity model I have derived.
On nonhyperbolic reflection moveout in anisotropic media
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Fomel S. and Grechka V.
The famous hyperbolic approximation of P-wave reflection moveout is strictly accurate only if the reflector is a plane, and the medium is homogeneous and isotropic. Heterogeneity, reflector curvature, and anisotropy are the three possible causes of moveout nonhyperbolicity at large offsets. In this paper, we analyze the situations where anisotropy is coupled with one of the other two effects. Using the weak anisotropy assumption for transversely isotropic media, we perform analytical derivations and comparisons. Both the case of vertical heterogeneity and the case of a curved reflector can be interpreted in terms of an effective anisotropy, though their anisotropic effects are inherently different from the effect of a homogeneous transversely isotropic model.
Migration and velocity analysis by velocity continuation
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Fomel S.
Residual and cascaded migration can be described as a continuous process of velocity continuation in the post-migration domain. This process moves reflection events on the migrated seismic sections according to changes in the migration velocity. Understanding the laws of velocity continuation is crucially important for a successful application of migration velocity analysis. In this paper, I derive the kinematic laws for the case of prestack residual migration from simple trigonometric principles. The kinematic laws lead to dynamic theory via the method of characteristics. The main theoretical result is a decomposition of prestack velocity continuation into three different components corresponding to residual normal moveout, residual dip moveout, and residual zero-offset migration. The contribution and properties of each of the three components are analyzed separately.
AVO
Azimuthal behavior of P-waves in horizontal transverse isotropy
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Urdaneta H.
The behavior of P-wave reflection varies with azimuth for rocks whose material properties vary with azimuth. I study the effects of horizontal transverse isotropy in P-wave reflection patterns on a simple homogeneous model, a finely fractured shale overlying an isotropic chalk with varied amounts of fracturing. I have produced 2-D reflectivity maps that clearly illustrate the azimuthal behavior of P-waves as the aspect ratio, volume density and filling material of the cracks are varied.
Amplitude preservation for offset continuation: Confirmation for Kirchhoff data
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Fomel S. and Bleistein N.
Offset continuation (OC) is the operator that transforms common-offset seismic reflection data from one offset to another. Earlier papers by the first author presented a partial differential equation in midpoint and offset to achieve this transformation. The equation was derived from the kinematics of the continuation process. This derivation is equivalent to proposing the wave equation from knowledge of the eikonal equation. While such a method will produce a PDE with the correct traveltimes, it does not guarantee that the amplitude will be correctly propagated by the resulting second-order partial differential equation. The second author (with J. K. Cohen) proposed a dip moveout (DMO) operator for which a verification of amplitude preservation was proven for Kirchhoff data. It was observed that the solution of the OC partial differential equation produced the same DMO solution when specialized to continue data to zero offset. Synthesizing these two approaches, we present here a proof that the solution of the OC partial differential equation does propagate amplitude properly at all offsets, at least to the same order of accuracy as the Kirchhoff approximation. That is, the OC equation provides a solution with the correct traveltime and correct leading-order amplitude. ``Correct amplitude'' in this case means that the transformed amplitude exhibits the right geometrical spreading and reflection-surface-curvature effects for the new offset. The reflection coefficient of the original offset is preserved in this transformation. This result is more general than the earlier results in that it does not rely on the two-and-one-half dimensional assumption.
Optimization
Conjugate-direction Huber regression
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Claerbout J.
A straight-forward way to make conjugate-direction regressions robust (insensitive to bursty data noise) is based on the objective function of Huber (least squares for small residuals and least absolute values for large ones). The gradient is based on the clipped residual instead of the residual itself. Likewise the clipped residual is used to define the plane of the gradient and previous step (plane search). This method does not apply to the deconvolution problem because there noisy field data enters into the operator for the determination of the prediction-error filter.
Medians in multivariate regression
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Claerbout J.
Here I design filters and models of hyperbola superposition based on medians. Let ri be a residual and be a residual perturbation caused by a change in the model .With ,choosing gives us residuals where as many components as possible ...
Nonlinear least squares and regularization
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Berryman J. G.
I present and discuss some general ideas about iterative nonlinear output least-squares methods. The main result is that, if it is possible to do forward modeling on a physical problem in a way that permits the output (i.e., the predicted values of some physical parameter that could be measured) and the first derivative of the same output with respect to the model parameters (whatever they may be) to be calculated numerically, then it is possible (at least in principle) to solve the inverse problem using the method described. The main trick learned in this analysis comes from the realization that the steps in the model updates may have to be quite small in some cases for the implied guarantees of convergence to be realized.
Least-square inversion with inexact adjoints. Method of conjugate directions: A tutorial
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Fomel S.
This tutorial describes the classic method of conjugate directions: the generalization of the conjugate-gradient method in iterative least-square inversion. I derive the algebraic equations of the conjugate-direction method from general optimization principles. The derivation explains the ``magic'' properties of conjugate gradients. It also justifies the use of conjugate directions in cases when these properties are distorted either by computational errors or by inexact adjoint operators. The extra cost comes from storing a larger number of previous search directions in the computer memory. A simple ratfor program and three examples illustrate the method.
Stacking operators: Adjoint versus asymptotic inverse
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Fomel S.
The paper addresses the theory of stacking operators used in seismic data processing. I compare the notion of asymptotically inverse operators with the notion of adjoint operators. These two classes of operators share the same kinematic properties, but their amplitudes (weighting functions) are defined differently. I introduce the notion of the asymptotic pseudo-unitary operator, which possesses both the property of being adjoint and the property of being asymptotically inverse. The weighting function of the asymptotic pseudo-unitary stacking operator is completely defined by its kinematics. I exemplify the general theory by considering such stacking operators as Kirchhoff datuming, migration, offset continuation, DMO, and velocity transform.
Slant Stack
Tieman's conversion of common-midpoint slant stacks to common-source
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Harlan W. S. and Claerbout J.
We derive equations for transforming slant stacks at common midpoint to plane wave stacks, i.e. slant stacks at common shotpoint or common geophone point.
Results of Tieman's conversion of common-midpoint to common-source point slant stacks on synthetic data
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Holden T. C.
I became aware of Hans Tieman`s curious method of transforming slant stacks of midpoint gathers into slant stacks of common shot gathers in January of 1996 during a presentation he gave to SEP. Inspiration to write this paper also comes from William Harlan and Jon Claerbout who expressed interest in Tieman`s work. Harlan and Claerbout presented a derivation of the Tieman transform equations during an SEP seminar soon after Tieman`s visit Harlan and Claerbout (1996). Tieman's transformation holds promise in providing an accurate method of ...
Test of wavelet-based seismic data compression software
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Sun Y. and Biondi B.
In this paper, we test a new wavelet-transform based seismic data compression technique developed by Chevron. We apply this technique to two synthetic datasets and one field dataset. Our results show that this new compression approach can virtually retain all of the important seismic information at high compression ratios. We summarize several empirical rules which will help the high performance of this software.
Reproducible Research
Making scientific computations reproducible
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Schwab M., Karrenbach M., and Claerbout J.
To organize computational scientific research and hence to conveniently transfer our technology, we impose a simple filing discipline on the authors in our laboratory. A document's makefile includes laboratory-wide standard rules that offer readers these four standard commands: make burn removes the document's result figures, make build recomputes them, make tube displays the figures, and make clean removes any intermediate files. Although we developed these standards to aid readers we discovered that authors are often the principal beneficiaries.
SEPlib90/Fortran90
SEPlib90: SEPlib for 3-D prestack data
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Biondi B., Clapp R., and Crawley S.
We developed a generalization of the SEPlib software package capable to handle data with irregular spatial sampling, such as 3-D prestack seismic data. The new SEPlib, dubbed SEPlib90, is layered on the top of the old SEPlib, and shares with it the flexibility of use and the efficiency in accessing large amount of data. These attractive characteristics derive from two principles at the basis of SEPlib90 design: separation of the geometry information from the data traces, and exploitation of existing regularity, if any, in the data geometry.
Fortran90: Introduction and use in 3-D geophysical problems
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Clapp R. G. and Crawley S.
There is a general consensus in the scientific community that object oriented programming is the correct way to perform research. The object oriented approach allows much more complex ideas to be explored by allowing the scientist to concentrate on ideas rather than algorithms. This feature is especially attractive in problems such as we encounter in 3-D processing, where dealing with the huge data size and irregularity is extremely cumbersome and prone to errors, in traditional languages such as Fortran77. We use Fortran90, built upon the existing SEP90 accessors routines, as the building blocks to an object oriented processing environment. Our initial efforts indicate that a Fortran90 base can provide the flexibility of C++, while maintaining the simplicity of Fortran, to effectively solve complex geophysical problems.
References
- Claerbout, J., 1984, Imaging the Earth's Interior: SEP-40, 398.
- Harlan, W. S., and Claerbout, J., 1996, Tieman's conversion of common-midpoint slant stacks to common-source: SEP-92, 293-298.