Kurt J. Marfurt

Accuracy of finite‐difference and finite‐element modeling of the scalar and elastic wave equations

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...

3-D seismic attributes using a semblance‐based coherency algorithm

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, semblance-based coherency algorithm that allows us to analyze data of lesser quality than our original three-trace cross-correlation-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.

Eigenstructure‐based coherence computations as an aid to 3-D structural and stratigraphic mapping

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.

Seismic attributes — A historical perspective

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...

Marmousi2 An elastic upgrade for Marmousi

The original Marmousi model was created by a consortium led by the Institut Francais du Petrole (IFP) in 1988. Since its creation, the model and its acoustic finite-difference 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.

3D volumetric multispectral estimates of reflector curvature and rotation

One of the most accepted geologic models is the relation between reflector curvature and the presence of open and closed fractures. Such fractures, as well as other small discontinuities, are relatively small and below the imaging rangeofconventionalseismicdata.Dependingonthetectonic regime, structural geologists link open fractures to either Gaussian curvature or to curvature in the dip or strike directions. Reflector curvature is fractal in nature, with different tectonic and lithologic effects being illuminated at the 50-m and1000-m scales.Untilnow,suchcurvatureestimateshave been limited to the analysis of picked horizons. We have developed what we feel to be the first volumetric spectral estimates of reflector curvature. We find that the most positive and negative curvatures are the most valuable in the conventional mapping of lineations — including faults, folds, and flexures.Curvatureismathematicallyindependentof,andinterpretatively complementary to, the well-established coherence geometric attribute. We find the long spectral wavelengthcurvatureestimatestobeofparticularvalueinextracting subtle, broad features in the seismic data such as folds, flexures, collapse features, fault drags, and under- and overmigrated fault terminations. We illustrate the value of these spectral curvature estimates and compare them to other attributes through application to two land data sets — a salt domefromtheonshoreLouisianaGulfCoastandafractured/ karsteddatavolumefromFortWorthbasinofNorthTexas.

Robust estimates of 3D reflector dip and azimuth

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 vol...

Efficient calculation of a partial‐derivative wavefield using reciprocity for seismic imaging and inversion

Linearized inversion of surface seismic data for a model of the earth’s subsurface requires estimating the sensitivity of the seismic response to perturbations in the earth’s subsurface. This sensitivity, or Jacobian, matrix is usually quite expensive to estimate for all but the simplest model parameterizations. We exploit the numerical structure of the finite-element method, modern sparse matrix technology, and source–receiver reciprocity to develop an algorithm that explicitly calculates the Jacobian matrix at only the cost of a forward model solution. Furthermore, we show that we can achieve improved subsurface images using only one inversion iteration through proper scaling of the image by a diagonal approximation of the Hessian matrix, as predicted by the classical Gauss-Newton method. Our method is applicable to the full suite of wave scattering problems amenable to finiteelement forward modeling. We demonstrate our method through some simple 2-D synthetic examples.

Narrow‐band spectral analysis and thin‐bed tuning

Running window seismic spectral decomposition has proven to be a very powerful tool in analyzing difficult‐to‐delineate thin‐bed tuning effects associated with variable‐thickness sand channels, fans, and bars along an interpreted seismic horizon or time slice. Unfortunately, direct application of spectral decomposition to a large 3‐D data set can result in a rather unwieldy 4‐D cube of data. We develop a suite of new seismic attributes that reduces the input 20–60 running window spectral components down to a workable subset that allows us to quickly map thin‐bed tuning effects in three dimensions. We demonstrate the effectiveness of these new attributes by applying them to a large spec survey from the Gulf of Mexico. These two thin‐bed seismic attributes provide a fast, economic tool that, when coupled with other attributes such as seismic coherence and when interpreted within the framework of geomorphology and sequence stratigraphy, can help us quickly evaluate large 3‐D seismic surveys. Ironically, in a...

Instantaneous spectral attributes to detect channels

Channels filled with porous rock and encased in a nonporousmatrixconstituteoneofthemoreimportantstratigraphic exploration plays.Although attributes such as coherence can be used to map channel width, they are relatively insensitive to channel thickness. In contrast, spectral decomposition can beusedtomapsubtlechangesinchannelthickness.Thepeak spectral frequency derived by using the short-window, discrete Fourier transform SWDFT is an excellent tool for mapping such changes along an interpreted horizon. We show that by use of instantaneous spectral attributes, we can generateequivalentmapsforcompleteseismicvolumes.Because we are often interested in mapping high-reflectivity channelsencasedinalower-reflectivitymatrix,wefindthata composite plot of the peak frequency and the above-average peak amplitude accentuates highly tuned channels. Finally, by generating a composite volume using peak frequency, peakamplitude,andcoherence,wecanestablishnotonlythe channelthickness,butalsoitswidth.Wedemonstratethevalueofsuch3Dvolumetricestimatesthroughapplicationto1 a marine survey acquired over Tertiary channels from the GulfofMexicoand2alanddatasurveyacquiredoverPaleozoic channels from the Central Basin Platform, west Texas, United States. The channels in both marine and land surveys canbehighlightedthroughcomposite-volumeanalysis.