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Velocity estimation using prestack depth migration: field data trial (ps 4366K) (src 8321K)
Etgen J. T.
Velocity information extracted from prestack migrated data can be used to update the interval velocity model by solving a filtered tomography problem. Depth migration with the new velocity model should yield an improved structural image that stacks better over offset. Initial field data results indicate the promise of the method and also point out some of the difficulties encountered when using it.
Prestack Partial Migration Analysis (ps 59K) (src 11K)
Popovici A. M.
Prestack partial migration (PSPM) is a well-known process which transforms the prestack data to zero offset. I discuss several properties of the PSPM spreading operator and of the equivalent PSPM summation operator reflected by a transformation of coordinates from (x, t) domain to (p, $$) domain, where p=2dt/dx and $$ is the NMO correction. This transformation allows for a more general representation for the PSPM operator and can explain the apparition of triplications in the DMO curve in a variable velocity medium. Then I attempt to find a partial differential equation formulation for the PSPM operator in the family of first order partial differential equations using as characteristics the curves defined by the new transformation of coordinates.
Refining the image of profile migration: residual moveout and residual migration (ps 106K) (src 362K)
Zhang L.
In complex geologic environments, imaging of seismic reflection data requires model-driven depth migration. It is well known that such migration is sensitive to velocity errors. If the velocity model used in migration does not correctly model the travel-times of reflection events, the images of reflectors on migrated shot profiles will be distorted from the true images of the earth. By solving the Eiconal equation, I find a residual-moveout equation that can be used in estimating residual velocities, and a residual-migration operator that can revise the distortion of the images on migrated shot profiles. For a constant-velocity medium, these results are precise in the kinematic sense.

Finite-difference Travel Times

Upwind finite-difference calculation of traveltimes (ps 245K) (src 379K)
Trier J. v. and Symes W. W.
Fluid flow is often described by conservation laws that define the conservation of mass, momentum, or energy in a fluid. In fluid mechanics, a standard technique for solving such laws is upwind finite differencing, a numerical method that uses different finite-difference operators depending on the direction of the fluid flow. Upwind finite-difference methods are more stable than centered finite-difference techniques because they mimic the behavior of fluid flow by only using information taken from upstream in the fluid. Seismic traveltimes can be computed with upwind finite differences by solving a transformed eikonal equation. The transformed equation is a conservation law that describes the changes in the gradient components of the traveltime field among points on the computational grid. A first-order upwind finite-difference scheme proves accurate enough for seismic applications. The method calculates single-valued traveltime functions (i.e. first arrival times) efficiently on a regular grid, and is useful both in Kirchhoff migration and modeling and in seismic tomography.
Anisotropic finite-difference traveltimes (ps 74K) (src 90K)
Dellinger J. and Trier J. V.
Finite-difference traveltime methods are an efficient way to calculate first arrival times at every point in a complex heterogeneous isotropic model. We show that Van Trier and Symes' algorithm can be extended to include anisotropy by replacing the isotropic dispersion relation inherent in their method with an anisotropic one. Unfortunately, the upwind method they use is difficult to implement in the anisotropic case, because it requires high-accuracy solutions of derivatives of complex anisotropic dispersion relations. For our preliminary results we use instead an easy-to-implement fixed-grid finite-difference method. This method works well in homogeneous anisotropic models, but seems to have trouble in heterogeneous ones.

Multiple Reflection Elimination

Generalized adaptive deconvolution (ps 567K) (src 838K)
Darche G.
In signal processing, the lattice algorithms have been widely applied to spectral analysis. Their best known example in geophysics is Burg's filtering. In fact, lattice structures can be used to construct algorithms solving any kind of prediction problems. In this paper, I derive two of these algorithms. They generalize two classical adaptive algorithms used in spectral analysis: the least-squares lattice (LSL) algorithm, and Burg's adaptive algorithm. I will apply these algorithms to the problem of multiples removal with non-stationary data; these applications will show the superiority of Burg's algorithm.
Imaging to eliminate water-bottom multiples (ps 354K) (src 886K)
Zhang L.
On a common midpoint (CMP) gather, the water-bottom multiples are characterized by their moveout of small stacking velocities. In this paper I describe a new velocity discrimination method to eliminate these low velocity events. The new method is similar in principle to the velocity filtering method. First the CMP gather is transformed into a distorted image by using a velocity-dependent operator. The velocity of the operator is chosen in such a way that the distorted images of the water-bottom multiples are separable from the distorted images of other events. Eliminating the images of the water-bottom multiples is followed by the backward transformation, to give a CMP gather free of the interference of the water-bottom multiples. The synthetic example and field-data example show that the algorithm works to a certain extent; however, several problems remain to be solved.
Fast parabolic transforms (ps 573K) (src 947K)
Darche G.
I improve here an already existing method of multiples elimination by parabolic transforms. It is based on the approximate parabolic shape of the multiples after NMO correction, and uses a parabolic transform similar in concept to the hyperbolic and slant-stack transforms. This parabolic transform is easy to express in the frequency domain. More important, I will show that its least-squares inverse can also be computed easily, because it requires only the inversion of a Toeplitz matrix. This property makes the transformations especially fast to compute, and is still valid for special cases, like irregular space sampling, or offset-dependent weighting. I will recall the interest of this method for multiple elimination, and extend it to interpolation processes to illustrate the practical advantages of the Toeplitz structure.

Elastic Waves

Snell-driven beam-stack (ps 56K) (src 9K)
Filho C. A. C.
Combining the concepts of Snell-rays and beam-stack, I define a transform that maps prestack data from the offset-traveltime domain into the vertical-traveltime-Snell-parameter domain. The transform is defined for a common-mid-point (CMP) gather, and it requires the basic assumption that the earth can be well approximated by a stack of horizontal layers. Each output trace represents the reflections associated with the propagation of a Snell beam through the layers of the medium. Energy is summed up, inside the beam, along the reflections, and divergence effects for either a point or line source are properly compensated for by the transform.
Coupled wave propagation (ps 1104K) (src 2790K)
Karrenbach M.
Coupled wave propagation phenomena have been observed in laboratory measurements for a long time; however in field experiments observations have been less convincing. In this paper coupled wave equations allow us to model such coupling by combining elastic stress and strain, electric field strength and displacement, entropy and temperature. For piezoelectric media slowness surfaces show changes in propagation velocity and particle motion when coupling is included in the computation. Average properties of such media can be computed using the Schoenberg and Muir group theory. Possible applications of coupled wave propagation, besides earthquake prediction and mineral exploration, might be the 3D monitoring of oil fields during enhanced oil recovery. Accurate rock parameter estimation seems to be sensitive to coupling effects.
Putting Schoenberg-Muir to the test (ps 1618K) (src 3004K)
Dellinger J., Muir F., and Etgen J.
The Schoenberg-Muir averaging technique is a powerful method for calculating the bulk properties of layered and fractured elastic media. The theory is exact only for infinite layers and infinitely low frequency waves. It is already known that the approximations are accurate for infinite flat layers if the wavelength of the slowest elastic wave is long enough to contain a representative sample of the layers at any position in the medium. In this paper we run models testing the infinite flat layers assumption. We find that if the layers are much longer than they are thick this approximation is a good one and Schoenberg-Muir averaging is applicable.
A perturbation method for elastic-isotropic inversion--a proposal (ps 55K) (src 7K)
Filho C. A. C.
Several methods for elastic inversion have been applied in recent years, based on the optimization of either the fitting between the predicted and recorded data, or the fitting between the model-predicted and data-retrieved reflectivity series. In both cases, the model is usually described by a stack of layers that controls the kinematics and the dynamics of the physical process. I propose a method in which geometrical and physical effects are decoupled, so that the part of the model responsible for the dynamics can be modified without introducing any change on the kinematically-related part of the model. The model is characterized by a background medium, with smoothly-varying elastic properties, and an arbitrary number of perturbation layers. Each perturbation layer is responsible for the introduction of two reflections, whose amplitudes depend only on the contrast between the properties of the layer and the local properties of the background model. In addition to the kinematics-dynamics uncoupling, the number of unknown parameters necessary to describe the two reflections of a layer (top and bottom) is reduced to less than a half of the number required by usual model-description methods.

Novel Sources

Design considerations for experiments using novel seismic sources (ps 47K) (src 11K)
Cole S.
Two experiments using unconventional seismic sources have been studied recently at SEP, a passive seismic experiment and an experiment using a drill-bit source. The differences between these novel sources and conventional seismic sources place some added constraints on the experiment design. For instance, the ability to resolve the arrival direction of a wave in 3-D is essential if we are to discriminate the relatively weak (compared to conventional sources) source energy in the presence of nearby surface noise sources. From considerations such as this, I evaluate the design of the two experiments conducted to date and see how they could be improved upon.
Comparison of different approaches to processing ambient noise data (ps 351K) (src 1257K)
Vanyan L. and Cole S.
Various methods of processing ambient noise data and teleseismic coda waves, recorded by seismic arrays, have been developed by different groups. The main goal of these investigations is to locate endogenous sources of microseisms beneath the array (in the ambient noise case) or to detect scattering (in both cases). In the simplest processing scheme, semblance is used to measure the coherent energy exhibiting hyperbolic moveout arriving from sources (primary sources or scatterers) in the medium. The processing reveals some possible sources at depth, but the results are strongly influenced by surface sources. To eliminate the influence of these sources, a least-squares method in the frequency domain has been applied, but the method is hampered by the problem of estimating a source signal, which is required by the algorithm.

Book Three Continued

Book additions preface (ps 27K) (src 1K)
Claerbout J. F.
Additions to my third book in preparation include (1) a picking algorithm based on the operator $_x-p_t$,(2) examples and code for a missing data program, (3) prediction-error filter estimation along with missing data estimation, (4) discussion of trace interpolation beyond aliasing as an optimization problem, and (5) a pixel-precise method of velocity computation.
Univariate problems (ps 82K) (src 91K)
Claerbout J. F.

Model fitting by least squares (ps 61K) (src 24K)
Claerbout J. F.

Nonlinear problems (ps 70K) (src 54K)
Claerbout J. F.

Hyperbola tricks (ps 76K) (src 74K)
Claerbout J. F.

Velocity Analysis

Data examples of pixel-precise velocity analysis (ps 36K) (src 493K)
Ottolini R.
Velocity resolution is different using the pixel-precise analysis method on field data than using conventional analysis. The spectra do not smear out laterally. However, the signal-to-noise level can be lower.
Velocity Analysis on the Connection Machine (ps 52K) (src 7K)
Kneib G.
Implemented in slightly different ways on one of the new massively parallel computers, the Connection Machine, the performance of Jon Claerbout's pixel-precise velocity analysis algorithm can differ by much more than one order. The performance depends strongly on the way the Fortran programmer maps the data on the machine.
Anisotropic Velocity Analysis (ps 41K) (src 4K)
Ji J.
This paper examines the meaning of the anelliptic terms in relation to the processes of approximating the ray velocity surface of transversely isotropic media recently introduced by Muir (1985) and Byun et al. (1989). Compared with stacking velocity obtained by simple hyperbolic velocity analysis, the additional parameters estimated by the non-hyperbolic method contain more physically meaningful geologic information regarding the anisotropy of the subsurface. Synthetic ¶-wave model experiments demonstrate that the non-hyperbolic moveout formulas yield an excellent fit to time-distance curves over a wide range of ray angles. Thus the measurement parameters adequately reflect the characteristics of velocity dependence on ray angle, in other words, velocity anisotropy.

Missing Data

Estimation of missing data by least squares (ps 839K) (src 1628K)
Nichols D.
I formulate the problem of estimating missing data using least squares. The operators that I use are error operators that have as output that part of the interpolated data that does not fit some parametric model of the data. If all the parameters of the model are known the problem is a linear least squares problem. If the parameters of the model must be estimated at the same time as the missing data the problem is non-linear. I use a model based on local linear events to interpolate aliased data. This procedure depends on good initial estimates of the dips which can be obtained from a smoothed version of the data.

Wavelet Transforms

Wavelet decomposition of seismic data: examples and interactive XView program (ps 6K) (src 187K)
Ottolini R.
Seismic data may be decomposed into localized basis functions called wavelets. Wavelets resemble Fourier decomposition in the ability to distinguish frequency ranges and are invertible. Wavelets beat Fourier analysis in computation cost and locality. Wavelets may be inferior to the Fourier domain for propagating waves. The wavelet studied here decomposes signals by octaves. It is too coarse for studying seismic data. It might be useful for signal detection. I wrote an interactive, multi-dimensional wavelet-based bandpass application using the XView toolkit. XView combines the strengths of SunView and XWindows: powerful set of command objects, simple programming interface, portable, and network transparent.

Interactive Graphics and Processing

An interactive processing environment (ps 47K) (src 8K)
Cole S.
I have written an interactive software package that allows all SEP batch-oriented software to be used in an interactive manner, with no programming required of the end user. One can ``build'' a processing sequence, interactively modify the parameters for the various process, and see the result of each modification displayed on the workstation screen. The processing sequence and all associated parameters can be saved to a file so that the user can exit, then return at any time and pick up where he left off. With this environment it is much easier to experiment with different parameter values than in traditional batch processing. My software is written using XView, a freely-distributed X Windows toolkit developed by Sun Microsystems. Thus, like any X application, it can be ported to many machines.
Seismic movies on the XView graphics system (ps 39K) (src 41K)
Ottolini R.
XView is a graphical interaction toolkit derived from the SunView toolkit and running on top of X Windows. It allows portable seismic display software with an easy-to-use control interface.

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