Numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media, and they promise to be invaluable in solving the full inverse problem. This paper quantitatively compares finite‐difference and finite‐element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time‐domain and frequency‐domain techniques. In addition to versatility and ease of implementation, it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy. To be of value, a solution technique must be able to minimize (1) numerical attenuation or amplification, (2) polarization errors, (3) numerical anisotropy, (4) errors in phase and group velocities, (5) extraneous numerical (parasitic) modes, (6) numerical diffraction and scattering, and (7) errors and transmission coefficients. This paper shows that in homogeneous media the explicit finite‐element and finite‐difference schemes are comparable when solving the scalar wave equation and when solving the elastic wave equations with Poisson’s ratio less than 0.3. Finite‐elements are superior to finite‐differences when modeling elastic media with Poisson’s ratio between 0.3 and 0.45. For both the scalar and elastic equations, the more costly implicit time integration schemes such as the Newmark scheme are inferior to the explicit central‐differences scheme, since time steps surpassing the Courant condition yield stable but highly inaccurate results. Frequency‐domain finite‐element solutions employing a weighted average of consistent and lumped masses yield the most accurate results, and they promise to be the most cost‐effective method for CDP, well log, and interactive modeling.

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Seismic coherency is a measure of lateral changes in the seismic response caused by variation in structure, stratigraphy, lithology, porosity, and the presence of hydrocarbons. Unlike shaded relief maps that allow 3-D visualization of faults and channels from horizon picks, seismic coherency operates on the seismic data itself and is therefore unencumbered by interpreter or automatic picker biases. We present a more robust, multitrace, semblancebased coherency algorithm that allows us to analyze data of lesser quality than our original three-trace crosscorrelation-based algorithm. This second-generation, semblance-based coherency algorithm provides improved vertical resolution over our original zero mean crosscorrelation algorithm, resulting in reduced mixing of overlying or underlying stratigraphic features. In general, we analyze stratigraphic features using as narrow a temporal analysis window as possible, typically determined by the highest usable frequency in the input seismic data. In the limit, one may confidently apply our new semblance-based algorithm to a one-sample-thick seismic volume extracted along a conventionally picked stratigraphic horizon corresponding to a peak or trough whose amplitudes lie sufficiently above the ambient seismic noise. In contrast, near-vertical structural features, such as faults, are better enhanced when using a longer temporal analysis window corresponding to the lowest usable frequency in the input data. The calculation of reflector dip/azimuth throughout the data volume allows us to generalize the calculation of conventional complex trace attributes (including envelope, phase, frequency, and bandwidth) to the calculation of complex reflector attributes generated by slant stacking the input data along the reflector dip within the coherency analysis window. These more robust complex reflector attribute cubes can be combined with coherency and dip/azimuth cubes using conventional geostatistical, clustering, and segmentation algorithms to provide an integrated, multiattribute analysis.

The original Marmousi model was created by a consortium led by the Institut Français du Pétrole (IFP) in 1988. Since its creation, the model and its acoustic finitedifference synthetic data have been used by hundreds of researchers throughout the world for a multitude of geophysical purposes, and to this day remains one of the most published geophysical data sets. The advancement in computer hardware capabilities since the late 1980s has made it possible to perform a major upgrade to the model and data set, thereby extending the usefulness of the model for, hopefully, some time to come. This paper outlines the creation of an updated and upgraded Marmousi model and data set which we have named Marmousi2. We based the new model on the original Marmousi structure, but extended it in width and depth, and made it fully elastic. We generated high-frequency, high-fidelity, elastic, finite-difference synthetics using a state-of-the-art modeling code made available by Lawrence Livermore National Laboratory as part of a U.S. Department of Energy research project. We simulated streamer, OBC, and VSP multicomponent shot records with offsets up to 15 km. We have found these data suitable for a wide variety of geophysical research including calibration of velocity analysis, seismic migration, AVO analysis, impedance inversion, multiple attenuation, and multicomponent imaging. As part of this project, the Marmousi2 model and data set are available to other researchers throughout the world.

A seismic attribute is a quantitative measure of a seismic characteristic of interest. Analysis of attributes has been integral to reflection seismic interpretation since the 1930s when geophysicists started to pick traveltimes to coherent reflections on seismic field records. There are now more than 50 distinct seismic attributes calculated from seismic data and applied to the interpretation of geologic structure, stratigraphy, and rock/pore fluid properties. The evolution of seismic attributes is closely linked to advances in computer technology. As examples, the advent of digital recording in the 1960s produced improved measurements of seismic amplitude and pointed out the correlation between hydrocarbon pore fluids and strong amplitudes (“bright spots”). The introduction of color printers in the early 1970s allowed color displays of reflection strength, frequency, phase, and interval velocity to be overlain routinely on black-and-white seismic records. Interpretation workstations in the 1980s provided interpreters with the ability to interact quickly with data to change scales and colors and to easily integrate seismic traces with other information such as well logs. Today, very powerful computer workstations capable of integrating large volumes of diverse data and calculating numerous seismic attributes are a routine tool used by seismic interpreters seeking geologic and reservoir engineering information from seismic data. In this review paper celebrating the 75th anniversary of the Society of Exploration Geophysicists, we reconstruct the key historical events that have lead to modern seismic attribute analysis.

Much of seismic stratigraphy is based on the morphology of seismic textures. The identification of reflector terminations and subtle changes in dip and azimuth allows us to infer coherent progradational and transgressive packages as well as more chaotic slumps, fans, and braided-stream complexes; infill of karsted terrains; gas seeps; and, of course, faults and angular unconformities. A major difficulty in estimating reflector dip and azimuth arises at discrete lateral and vertical discontinuities across which reflector dip and azimuth change. The smearing across these boundaries produced by traditional dip and azimuth estimations is avoided by using temporally and spatially shifted multiple windows that contain each analysis point. This more robust estimation of dip and azimuth leads to increased resolution of well-established algorithms such as coherence, coherent amplitude gradients, and structurally oriented filtering. More promising still is the analysis of high-resolution dip and azimuth through volumetric estimates of reflector curvature and angular unconformities. This new technique is demonstrated using two land data volumes, one from the Louisiana salt province and the other from the fractured Fort Worth basin.

Coherence measures applied to 3-D seismic data volumes have proven to be an effective method for imaging geological discontinuities such as faults and stratigraphic features. By removing the seismic wavelet from the data, seismic coherence offers interpreters a different perspective, often exposing subtle features not readily apparent in the seismic data. Several formulations exist for obtaining coherence estimates. The first three generations of coherence algorithms at Amoco are based, respectively, on cross correlation, semblance, and an eigendecomposition of the data covariance matrix. Application of these three generations to data from the Gulf of Mexico indicates that the implementation of the eigenstructure approach described in this paper produces the most robust results. This paper first introduces the basic eigenstructure approach for computing coherence followed by a comparison on data from the Gulf of Mexico. Next, Appendix A develops a theoretical connection between the well‐known semblance and the less well‐known eigenstructure measures of coherence in terms of the eigenvalues of the data covariance matrix. Appendix B further extends the analysis by comparing the semblance- and eigenstructure‐based coherence measures in the presence of additive uncorrelated noise.

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