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Anti-aliased Kirchhoff 3-D migration: a salt intrusion example (ps 1656K) (src 8569K)
Lumley D. E.
Spatial aliasing of Kirchhoff migration imaging operators can seriously degrade the quality of subsurface reflectivity images. Sparse and/or irregular 3-D acquisition geometries can adversely magnify the problem in the presence of steep reflector dip. An anti-aliasing method is implemented to ameliorate Kirchhoff migration operator aliasing, based on local time and dip variant triangle filtering. The local anti-aliasing filters are efficiently applied as 3-point time filters after a step each of causal and acausal temporal trace integration. The anti-aliased migration is compared to a standard aperture-weighted Kirchhoff 3-D poststack time migration of a steep-dip salt intrusion from the Gulf of Mexico. Preliminary results indicate that the anti-aliased migration appears to enhance the resolution of steep salt-sediment interfaces and faults, compared to the standard migration. Both algorithms are implemented on the 32-node Connection Machine (CM5) at the Stanford Exploration Project.
3-D local monoplane annihilator (ps 183K) (src 7700K)
Claerbout J. F.
The LOMOPLAN (LOcal MOnoPLane ANnihilator) filter in three dimensions is a deconvolution filter that takes a volume in and produces two volumes out. The x-output volume results from a first order prediction-error filter on the x-axis, and the y-output volume is likewise on the y-axis. Synthetic data studies are promising.
3-D wavefield depth extrapolation by rotated McClellan filters (ps 227K) (src 3661K)
Biondi B. and Palacharla G.
The application of McClellan transformations considerably reduces the computational cost of 3-D wavefield depth extrapolation by explicit convolutional methods. The accuracy of migration methods based on McClellan transformation depends on how well the transformation filter ($k$) is approximated; errors in this approximation cause anisotropy in the extrapolator operator. This anisotropy can be greatly reduced by rotating the approximate filter by 45 degrees, and averaging the rotated filter with the original filter. The application of the rotated filter yields a migration method that images correctly very steep dips, without additional computational cost.
Interpolation, noise suppression, and velocity filtering in 3-D (ps 387K) (src 663K)
Cole S.
In SEP-73, Cole and Claerbout 1992 presented a 3-D data interpolation scheme. For each output location, a number of neighboring traces were combined along a best trajectory picked from an estimate of event coherency as a function of 3-D dip. A somewhat similar scheme is used by Reiter et al. 1993 in 2-D - ...
Equations for three-dimensional dip moveout in depth-variable velocity media (ps 56K) (src 8K)
Artley C., Blondel P., Popovici A. M., and Schwab M.
Artley 1992 introduced an original method to perform v(z) dip moveout in a two-dimensional earth model. The process uses ray tracing tables and solves a system of equations accounting for the location and the dip of the reflection point. This paper describes an extension of Artley's method and derives a new set of equations for the 3-D case.

Constant-velocity anti-aliasing three-dimensional integral dip moveout (ps 1033K) (src 3748K)
Blondel P.
A method of three-dimensional integral dip moveout processing for constant-velocity media must cope with problems related to amplitude and aliasing. The convolution of the dip moveout operator with triangle functions avoids the aliasing effect. A study of different amplitude weightings leads to the choice of the weighting scheme derived from a Fourier domain expression of dip moveout Black et al. (1993). Testing the method on a 3-D synthetic data set shows the conservation of the amplitude-versus-offset (AVO) effect throughout the dip moveout (DMO) process.
Coalbed methane: a 3-D reservoir seismic data set (ps 307K) (src 1380K)
Lumley D. E.
Coalbed methane reservoirs are becoming an increasingly interesting prospect to some major oil and gas companies. The rapid spatial heterogeneity of coal physical properties, methane saturation levels, and coal fracture systems and flowpaths makes these targets a challenging prospect for reservoir characterization research in general, and 3-D prestack seismic impedance inversion in particular. I briefly present some examples of a comprehensive 3-D land seismic data set made available to SEP by Amoco Research, and point out some interesting features, problems and prospects in the data with respect to my inversion goals. The data are characterized by a strong AVO response on the coalbed target at 0.5 seconds, and a weaker basement reflection at 1.3 seconds. The swaths are contaminated by spatially aliased air-wave noise, ground roll, and trapped surface modes, and contain apparent source-receiver amplitude and residual statics variations.
Three-dimensional map de-migration/migration (ps 57K) (src 14K)
Berlioux A.
3-D seismic wavefield can only be handled with 3-D processing, and more effectively with 3-D depth migration methods, because it contains signals from all possible directions. I assume in this study that the only data that is at my disposal are the time migrated maps. In geological areas where lateral velocity variations are present, time migrated images or maps do not represent the proper position of seismic reflections Hatton et al. (1981). Therefore, it is necessary, before ...

Prestack Imaging and Processing

Partial differential equation for migration to zero-offset (ps 330K) (src 287K)
Popovici A. M.
Starting from prestack migration in offset-midpoint coordinates I obtain a partial differential equation (PDE) for migration to zero-offset in constant velocity. The PDE allows migration to zero-offset (MZO) to be performed via an algorithm similar to phase-shift migration. Since the velocity is isolated on depth intervals, this equation could theoretically be used for variable velocity migration to zero-offset. I show synthetic examples of depth variable velocity MZO which match perfectly the zero-offset section.
Amplitude and phase in DMO and MZO algorithms (ps 545K) (src 490K)
Popovici A. M.
In constant velocity, the sequence Dip Moveout (DMO), Normal Moveout (NMO), Stacking and Zero-Offset Migration (ZOM) is equivalent to Prestack Migration. In this context the Dip-Moveout correction can be expressed in two ways. First, DMO can be defined kinematically in time-space domain and implemented in Fourier domain (Hale 1983). Second, DMO can be defined as a process separated out of Prestack Migration, after eliminating the NMO, Stacking and ZOM steps. I simplify the comparison between the two methods by incorporating the sequence DMO, NMO and Stacking into a single process named Migration to Zero-Offset (MZO). MZO is a kinematic process than can be implemented in a manner similar to Hale's and Zhang's DMO, by adding the NMO step as a time shift in the Fourier domain. Second, I define MZO as the process separated out of Prestack Migration, after eliminating the Zero-Offset Migration step. I compare the amplitude and phase output of the two MZO algorithms together with Hale's and Zhang's DMO algorithms for three simple models. The results suggest that the two methods are not equivalent.
The migration of prestack slant stack with PSPI (ps 105K) (src 274K)
Lin J.
Prestack slant stack migration is implemented with both one-pass and two-pass phase shift plus interpolation (PSPI) to migrate seismic sections with lateral velocity variations. I compared impulse responses of operators with different sets of velocities to test the accuracy of the migration operator. This algorithm can handle lateral velocity variations very well.
Shot gather continuation (ps 505K) (src 2685K)
Schwab M.
Standard offset continuation transforms a constant offset gather to a gather at a different offset. The new gather is designed to yield the same migrated subsurface image as the original one. Similarly, the shot continuation operator introduced in this article extrapolates shot gathers to new shot locations, while preserving the associated migrated subsurface image. The derivation of the shot continuation method is analogous to the derivation of the offset continuation process. Unfortunately, the shot continuation operator is space and time variant and therefore more expensive than the space invariant offset continuation operator. Synthetic examples demonstrate that the shot continuation method yields shot gathers whose kinematics are correctly predicted. A novel approach for v(x,t) velocity analysis motivates the development of the shot continuation operator.
Data parallel wave equation datuming (ps 302K) (src 12424K)
Bevc D.
Seismic data gathered on land is distorted by irregular acquisition topography. Seismic imaging algorithms are generally applied to data which is redatumed to a planar surface. In regions of mild topography where the near surface velocity is much slower than the subsurface velocity, a static shift is adequate for the transformation. However, when the necessary shift increases in magnitude and when the near surface velocity is comparable to the subsurface velocity, the static approximation becomes inadequate. Under these circumstances, a static shift distorts the wave field and degrades the velocity analysis and imaging. In this case wave equation datuming is more appropriate than static shift. Wave equation datuming is much more computationally intensive than static shift and is therefore seldom applied. Furthermore, redatuming may involve multiple applications of the datuming routine to estimate the near surface velocity structure. I implement a data parallel wave equation datuming algorithm which can be applied to land data as an alternative to static shift in order to improve velocity analysis and imaging and to estimate near surface velocity structure. The Kirchhoff algorithm is efficiently implemented on a Connection Machine CM5 and makes wave equation datuming practical.
Datuming by wavefield depth extrapolation (ps 869K) (src 8259K)
Ji J.
I present a poststack and a prestack datuming scheme that uses wavefield depth extrapolation. The method I have developed allows the use of any depth extrapolation technique, such as phase-shift, split-step, and finite-difference extrapolation. I derive the datuming algorithms by transposing and taking the complex conjugate of the corresponding forward modeling operator, which does upward extrapolation from a flat surface to an irregular surface. The exact conjugate relation between the forward modeling operator and the datuming operator is demonstrated algebraically. Testing both the poststack and prestack datuming algorithms with synthetic data, using several depth extrapolation algorithms, has shown that the method works well.

Inversion and Tomography

Resolution for Lanczos and Paige-Saunders inverses in tomography (ps 69K) (src 35K)
Berryman J. G.
Methods have been developed for computing both the model and data resolution matrices for iterative inverses such as those produced by the Lanczos and Paige-Saunders (LSQR) schemes for constructing linear inverses.
Tomographic estimation of elastic constants in heterogeneous transversely isotropic media (ps 393K) (src 352K)
Michelena R. J.
The procedure to estimate elastic constants of a transversely isotropic medium from limited aperture traveltimes consists of two steps. First, P- and SV-wave traveltimes are fitted with elliptical velocity functions around one of the axes of symmetry. Second, the parameters that describe the elliptical velocity functions are transformed analytically into elastic constants. When the medium is heterogeneous, the process of fitting the traveltimes with elliptical velocity functions is performed tomographically and the transformation to elastic constants is performed locally at each position in space. Cross-well synthetic and field data examples show that the procedure is accurate as long as the data aperture is properly constrained.
Angle-dependent reflectivity estimation (ps 546K) (src 1591K)
Lumley D. E.
I derive an analytical least-squares inverse solution to the problem of estimating angle-dependent reflectivity from prestack seismic reflection data. The reflection coefficients and the reflection angles are estimated simultaneously directly from the prestack seismic reflection data, without a priori knowledge of geologic structure or dip. I define a forward theory which relates angle-dependent reflection data to a generalized reflectivity model combining elements of Zoeppritz plane wave reflection and Rayleigh-Sommerfeld diffraction. The l2 inverse solutions for reflectivity and reflection angle are then derived by a standard application of stationary phase and Gauss-Newton gradient optimization, and naturally incorporate l2 compensation for limited data acquisition aperture, source and receiver directivity, geometric spreading, transmission loss, and high-frequency intrinsic Q attenuation. Uncompensated reflectivity amplitude errors include coherent noise (e.g., multiples, surface waves, etc.), steep-dip operator spatial aliasing, migration velocity error, and data mute zones. The method is validated on a synthetic data example in the presence of dips up to 30 degrees, and should prove to be a useful component of seismic prestack amplitude analysis (AVO) and impedance inversion.
Kirchhoff prestack impedance inversion: a gas reservoir pilot study (ps 721K) (src 5502K)
Lumley D. E.
Estimation of subsurface material properties and their spatial variation is important in hydrocarbon exploration and detection, and crucial to subsequent reservoir characterization and management. I conduct a pilot study to test my prestack impedance inversion theory and algorithms on a marine data set from the Gulf of Mexico, acquired over a known producing gas reservoir. First, I perform a high-resolution migration velocity analysis at every adjacent CMP location, and estimate a migration/inversion velocity model which is optimal in a Monte Carlo automatic velocity fitting technique. Next, I perform a least-squares Kirchhoff migration to estimate the angle-dependent reflectivity at each subsurface point, followed by a linearized Zoeppritz elastic parameter inversion for relative contrasts in compressional and shear wave impedance. Finally, I combine the P and S impedance contrast maps into a single image indicative of anomalous gas saturation levels. This hydrocarbon indicator correlates very well with the known gas reservoir interval, and suggests lateral heterogeneity of gas saturation along the reservoir unit.
Inverting ultrasonic data on solid/fluid mixtures for Biot-Gassmann parameters (ps 95K) (src 151K)
Berryman J. G. and Lumley D. E.
Although the Biot-Gassmann equations are known to predict acoustic data for water-saturated glass bead packings very well, these equations often seem to give anomalous results for naturally occurring rocks. The theory shows that, as the compressibility of the pore fluid increases, the inverse of the coefficients C and M in Biot's equations should increase linearly, and the theory also gives definite predictions about the values of the slope and intercept of the resulting lines. Although the anticipated linear behavior is observed in real data, the values of the slopes are smaller than expected for some rocks. Therefore, the process of inverting this data for the desired coefficients must incorporate other physical constraints and then produces, not a single value but rather, a range of possible values for each coefficient.

Interpolation, Aliasing and Deconvolution

Steep-dip deconvolution (ps 2416K) (src 5662K)
Claerbout J. F.
I design two-dimensional filters for deconvolving field profiles. In the spatial direction, filter coefficients are constrained to lie within a triangle defined by water velocity. (The filters were computed and applied within windows and outputs are then patched together.) I selected six Yilmaz-Cumro field profiles with low velocity noises for demonstration testing. All results were good to excellent.
Lateral prediction techniques: FX-decon versus two-D deconvolution (ps 2172K) (src 4188K)
Abma R.
The results of two-dimensional lateral deconvolution are similar to those of FX-decon, however, two-dimensional lateral deconvolution produces less artificial lineup in noise than FX-decon. Modifying the filter shape in two-dimensional deconvolution may allow more control over the process than is allowed in FX-decon.
Applications of two-D deconvolution (ps 265K) (src 503K)
Abma R. and Claerbout J.
We've found that Burg's two-dimensional filter offers too much freedom in producing lateral predictions, since the predictions within a trace appear to overwhelm the lateral predictions. Instead, we fix the filter coefficients along the output column so the form of the two-dimensional filter becomes equivalent to that of the FX-decon operator. The ability to control the length of this filter in time allows two-dimensional deconvolution to produce a result superior to the FX-decon result. A symmetry for operators calculated forward and backward in space has been found, but it only applies for filters with no free elements in the output column of that filter. The filter shapes of other two-dimensional deconvolution filters are compared and filter shapes that might allow new applications are outlined.
Integration along a line in a sampled space (ps 253K) (src 967K)
Nichols D.
Integration along a line in a sampled space can be thought of as a two stage process: first interpolate to a continuous space and then perform the integral. To compute a line integral in seismic data I first apply a weighting funtion to each trace and then stack the traces. The form of the weighting function depends on the interpolation method chosen.
Choice of integration method for anti-aliased Kirchhoff migration (ps 171K) (src 206K)
Bevc D. and Claerbout J.
In the last SEP report Bevc and Claerbout (1992), we described a fast anti-aliased Kirchhoff migration and modeling algorithm. The algorithm uses causal and anticausal integration to form triangular weighting functions which depend on operator offset, velocity, and data sampling. In this paper we examine how the particular implementation of causal and anticausal integration affects the migration and modeling results. The best result is obtained by defining the discrete integration operation ...

Anisotropy and thin layers

Relating seismic measurements on different scales (ps 732K) (src 1878K)
Karrenbach M.
Measurements carried out at different scales may not agree in their observations. An equivalent medium approach can link measurements carried out at different scales. I apply high frequency (Dix) and low frequency (Schoenberg&Muir) averages to a well log and compare the results to a conventional velocity analysis of surface seismic data.
A result in equivalent medium theory (ps 49K) (src 6K)
Muir F.
A point Green's function for a heterogeneous, one-dimensional elastic material is expanded in a Taylor series and examined term by term. As expected, the first-order term gives the kinematic result predicted by Schoenberg/Muir and like theories. However, the second-order term of this expansion (subject to a minor stationarity constraint on the heterogeneity) is unexpected in that it depends on first-order averages alone. A consequence is that ensemble average responses for materials with the same statistical properties show no second-order transmission loss. Energy transferred from the direct to the scattered field must involve higher order terms in the Taylor series expansion. It follows that there is no simple `central limit' asymptotic solution, and no equivalent homogeneous visco-elastic medium.
Dix revisited: a formalism for rays in layered media (ps 64K) (src 19K)
Dellinger J. and Muir F.
Dix shows us how to calculate the moveout velocity of a stack of isotropic layers, but what about anisotropic layers requiring higher-order paraxial approximations? The usual derivation requires a great deal of algebra even for the standard hyperbolic-moveout case. The key is to realize that the Dix equations are an equivalent-medium theory: they provide a formula for replacing a heterogeneous layer stack with an equivalent homogeneous block. Another equivalent medium theory, the Schoenberg-Muir calculus, suggests a cleaner way of deriving Dix's result. Identify layer variables that are constant through the entire stack; these are the ``knowns''. Identify layer variables that add through the stack, and express these additive parameters in terms of the known stack constants and elastic parameters in each layer. The coefficients multiplying the stack constants in this formula are the layer-group elements. Map from layer parameters to layer-group elements, sum over all layers, and map back to find the equivalent medium. For the standard case the first layer-group parameter is ``vertical traveltime'' and the second is ``moveout velocity squared''. The equivalent-medium algorithm similarly provides a direct method for calculating the analogous Dix layer-group parameters for arbitrary anisotropic systems.
Transmissivity simulation in fine layered media (ps 59K) (src 20K)
Oh H.
In a fine layered media of cyclic sedimentation, the first arrival is attenuated very quickly due to the transmission loss. As a result, the seismic information is carried by the short path multiples(O'Doherty and Anstey, 1971). In this note, I studied the average effect of many thin isotropic layers by simulating the impulse response in one dimensional random sequences of two states.


General Geophysics

Interpretation of the acoustic properties of suspensions with the diffraction theory (ps 359K) (src 510K)
Blondel P.
Though the viscoelastic theory suitably predicts the acoustic properties of suspensions, it does not help us understand the amplitude variability of the waves transmitted through media of different porosity. This paper attempts to explain two phenomena occurring in suspensions using the diffraction theory. First, lab experiments show that the amplitude of the transmitted waves decreases when the suspension becomes denser. The modeling of this experiment predicts the scattering effect on the amplitude of the waves in dense suspensions. The second phenomenon concerns the sigmoid increase of the measured P-velocities with the frequency of the measurement. This behavior can be qualitatively explained by the diffraction theory. Furthermore, a simple relationship exists between the critical frequency of the measurement and the size of the grains.
Source parameters from near regional earthquake data recorded at Garm, Tadjikistan (ps 109K) (src 72K)
Ecker C.
In recent years, numerous efforts have been made to improve the accuracy of regional event location from single station data by using travel time and azimuth information Magotra et al. (1987); Ruud et al. (1988); Thurber et al. (1989). Ruud et al. 1988, and Thurber et al. 1989 ...
Earthquake stacks at constant offset (ps 2067K) (src 4331K)
Claerbout J. F.
I show Shearer's earthquake stacks over all source-receiver locations at constant offset and compare them to exploration seismic data. This electronic document simply reads the stacks and plots them.


Modeling in polar coordinates using one way wave equations (ps 352K) (src 811K)
Nichols D.
Mono-frequency Green's functions can be calculated using several frequency domain algorithms. I model the response to a point impulse in 2-D using phase shift, PSPI and $-x$ implicit finite difference algorithms. All of these algorithms can be implemented in both rectangular and polar coordinates. In polar coordinates an outgoing wavefield can be accurately modeled using a low dip algorithm. This means that high dip accuracy can be obtained at low dip cost.
The split wave equation operator for anisotropic media (ps 111K) (src 194K)
Karrenbach M.
When applying the elastic anisotropic wave equation operator for spatially varying media, several first order derivatives have to be calculated in the process. The wave equation can formally be split into two components such that derivatives are taken with respect to medium parameters and with respect to the propagating wave field. Splitting the wave equation operator allows us to adapt derivative operators to the physical quantities to be differentiated. In particular the adaption can be guided by special properties of the observable quantity. Practical problems can arise since in general anisotropic media the derivated quantities have to be interpolated back to collocation points.
Circularly symmetric filters using McClellan transformation (ps 206K) (src 174K)
Palacharla G.
Two-dimensional filters with circular symmetry are realized by exploiting their circular symmetry. They can be realized in an efficient manner from the corresponding 1-D filter, without the 2-D coefficients being computed. Only the 1-D filter coefficients need to be stored to realize the two-dimensional filter. Transformation technique using Chebyshev polynomials (called the McClellan transformation) is used in the design. The realization of the 2-D filter using transformation is efficient compared to two-dimensional convolution.
Huygen's waveforms and half-order derivatives (ps 69K) (src 46K)
Claerbout J. F.
This short tutorial paper shows why in two dimensions, Huygen's secondary source, otherwise known as the Hankel tail, is a fractional order derivative. In three-dimensions the tail reduces to a simple derivative.


Xtpanel update: interactivity from within existing batch programs (ps 91K) (src 517K)
Cole S. and Nichols D.
Xtpanel Cole and Nichols (1992) was developed as a tool for adding a level of interactivity to existing batch seismic processing programs. From an xtpanel-built interface containing buttons, sliders, and other objects, a user can set the values of various processing parameters, then press a button that will launch a conventional background job. The job can be re-run from the panel, to try out different sets of parameters. While this approach is simpler for the user than having to re-edit ...
How to organize an interactive document (ps 158K) (src 891K)
Karrenbach M.
An interactive document is a self contained unit; it can either exist by itself or as part of a report or book. Everything necessary to recreate your paper must be within the paper directory. This document outlines the organization of the paper directory. This document follows the same organizational rules; you can view it, by going to /sepcd5/sep77/martin2 and typing "cake" . To rebuild all figures type cake figures.
How to use cake with interactive documents (ps 70K) (src 22K)
Claerbout J. F. and Karrenbach M.
At SEP we file our research in electronic documents so that any of us can rebuild a research result of any other (on a wide variety of computers) by opening an electronic report, going to an illustration and pressing a button found in the figure caption. The caption button exposes a menu of choices, often including interaction and movies. Researchers achieve this reproducibility and interactivity by filing their work using systematic naming conventions, explained here and precisely defined by a four page document SEP.idoc.rules written in ``cake'' language. A typical research directory builds 1-10 illustrations. Each directory has a cakefile containing a list (FIGLIST) of buildable figure names. All figures can be built with the command cake figures and removed with cake burn. Precious figure files (ones not considered replaceable) are named NAME.save. The LATEX macro, $$activeplot, launches the UNIX command cake NAME.menu or cake NAME.idoc. The default cake rule is to come up with an XWINDOW menu xtpanel or to show a plot with tube or show a movie with X11movie. For unusual actions, you can override the default by making a rule for the target NAME.action.

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