Reflector Dip Research Articles

Volume-based curvature computations illuminate fracture orientations — Early to mid-Paleozoic, Central Basin Platform, west Texas

Volumetric curvature analysis is a simple but computationally intensive procedure that provides insight into fracture orientation and regional stresses. Until recently, curvature analysis has been limited to computation along horizon surfaces that may be affected by unintentional bias and picking errors introduced during the interpretation process. Volumetric curvature is best estimated in a two-step process. In the first step, we use a moving-analysis subvolume to estimate volumetric reflector dip and azimuth for the best-fit tangent plane for each sample in the full volume. In the second step, we calculate curvature from adjacent measures of dip and azimuth. We use larger curvature analysis windows to estimate longer wavelength curvatures. Such a technique allows us to output full 3D volumes of curvature values for one or more scales of analysis. We apply these techniques to a data set from the Central Basin Platform of west Texas and find lineaments not observable with other seismic attributes. These lineaments indicate that, in the lower Paleozoic interval, a left-lateral shear couple oriented due east-west controls the local stress regime. Such a model predicts that extension faulting and fractures will be oriented northeast-southwest. The example demonstrates the potential of this new technology to determine stress regimes and predict azimuths of open fractures.

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 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.

Analysis of multi‐offset GPR data: a case study in a coarse‐grained gravel aquifer

ABSTRACTMulti‐offset ground‐penetrating radar (GPR) data were collected in a coarse‐grained gravel aquifer located in a glacial delta environment within dipping foresets (at the Tettnang aquifer test site, SW Germany). We apply prestack processing techniques and normal‐moveout (NMO) velocity analysis in preparation for stacking. The estimation of propagation velocities is of considerable importance for converting time‐domain radargrams into depth‐sections and for an interpretation in terms of petrophysical properties. In our case, interval velocity determination is difficult because reflector dip angles are variable, and stacking and NMO velocities differ significantly since the antenna offset is comparable to the reflector depth. Using a synthetic two‐layer model, we systematically study possible errors in interval velocity determination. We compare the cases of decreasing velocity with depth (typical for ground‐penetrating (GPR) surveys of aquifers) and increasing velocity with depth (typical for seismic surveys). In the case of decreasing velocity with depth, the discrepancy between stacking and NMO velocity is considerable and, consequently, interval velocities calculated with the Dix equation or the more accurate 2D NMO approximation for dipping layers show unacceptably high errors. The errors are much smaller in comparable models with increasing velocity with depth. Thus, the small‐offset approximation inherent in the NMO concept is not appropriate for the first case. Consequently, it is necessary to model the true common‐midpoint (CMP) raypaths to determine realistic interval velocities from our GPR field data.Stacked GPR sections of multi‐offset data show significant improvements in imaging of reflectors and depth of investigation compared with a standard constant‐offset section. This allows a more reliable interpretation of geological structures. Simultaneous acquisition of four channels, which is possible with some commercial GPR systems, yields significantly better results than constant‐offset standard acquisition without increasing the efforts for data acquisition. Major reflectors in the GPR section can be correlated with distinct contrasts of porosity, represented by the base of open‐framework gravels and sand beds within poorly sorted gravels.

Parameter estimation in orthorhombic media using multicomponent wide-azimuth reflection data

Orthorhombic models with a horizontal symmetry plane adequately describe seismic signatures recorded over many naturally fractured reservoirs. The inversion of wide-azimuth traveltimes of PP and SS (the fast [Formula: see text] and slow [Formula: see text]) reflections are discussed for Tsvankin's anisotropic parameters and the azimuths of the vertical symmetry planes of orthorhombic media. If shear waves are not excited, SS traveltimes can be found from PP and PS (converted-wave) data, which makes the method applicable to offshore surveys. The feasibility of parameter estimation is strongly dependent on reflector dip and orientation. For a horizontal reflector beneath a single orthorhombic layer, the vertical velocities and reflector depth cannot be found from conventional-spread reflection traveltimes alone. If the reflector is dipping, the inversion is ambiguous when the dip plane is close to one of the vertical symmetry planes of the orthorhombic layer above it. The parameter estimation becomes possible if the dip direction deviates by more than 10° from the nearest symmetry plane. We apply multicomponent stacking velocity tomography to perform velocity analysis for stratified orthorhombic models composed of homogeneous layers separated by plane or curved interfaces. The tomographic algorithm, which operates with the normal-moveout (NMO) ellipses, zero-offset traveltimes, and reflection time slopes of PP- and SS-waves, is designed to build the orthorhombic velocity model in the depth domain by estimating the anisotropic parameters and the shapes of the reflecting interfaces. Numerical tests show that for layered orthorhombic media, it is necessary to put constraints on the vertical velocities to avoid instability in the inversion of noise-contaminated reflection data.

Angle‐domain common‐image gathers for migration velocity analysis by wavefield‐continuation imaging

We analyze the kinematic properties of offset‐domain common image gathers (CIGs) and angle‐domain CIGs (ADCIGs) computed by wavefield‐continuation migration. Our results are valid regardless of whether the CIGs were obtained by using the correct migration velocity. They thus can be used as a theoretical basis for developing migration velocity analysis (MVA) methods that exploit the velocity information contained in ADCIGs.We demonstrate that in an ADCIG cube, the image point lies on the normal to the apparent reflector dip that passes through the point where the source ray intersects the receiver ray. The image‐point position on the normal depends on the velocity error; when the velocity is correct, the image point coincides with the point where the source ray intersects the receiver ray. Starting from this geometric result, we derive an analytical expression for the expected movements of the image points in ADCIGs as functions of the traveltime perturbation caused by velocity errors. By applying this analytical result and assuming stationary raypaths (i.e., small velocity errors), we then derive two expressions for the residual moveout (RMO) function in ADCIGs. We verify our theoretical results and test the accuracy of the proposed RMO functions by analyzing the migration results of a synthetic data set with a wide range of reflector dips.Our kinematic analysis leads also to the development of a new method for computing ADCIGs when significant geological dips cause strong artifacts in the ADCIGs computed by conventional methods. The proposed method is based on the computation of offset‐domain CIGs along the vertical‐offset axis and on the “optimal” combination of these new CIGs with conventional CIGs. We demonstrate the need for and the advantages of the proposed method on a real data set acquired in the North Sea.

Quartic moveout coefficient for a dipping azimuthally anisotropic layer

Interpretation and inversion of azimuthally varying nonhyperbolic reflection moveout requires accounting for both velocity anisotropy and subsurface structure. Here, our previously derived exact expression for the quartic moveout coefficient A4is applied to P‐wave reflections from a dipping interface overlaid by a medium of orthorhombic symmetry.The weak‐anisotropy approximaton for the coefficient A4in a homogeneous orthorhombic layer is controlled by the anellipticity parameters η(1), η(2), and η(3), which are responsible for time processing of P‐wave data. If the dip plane of the reflector coincides with the vertical symmetry plane [x1, x3], A4on the dip line is proportional to the in‐plane anellipticity parameter η(2)and always changes sign for a dip of 30○. The quartic coefficient on the strike line is a function of all three η–parameters, but for mild dips it is mostly governed by η(1)—the parameter defined in the incidence plane [x2, x3]. Whereas the magnitude of the dip line A4typically becomes small for dips exceeding 45○, the nonhyperbolic moveout on the strike line may remain significant even for subvertical reflectors. The character of the azimuthal variation of A4depends on reflector dip and is quite sensitive to the signs and relative magnitudes of η(1), η(2), and η(3). The analytic results and numerical modeling show that the azimuthal pattern of the quartic coefficient can contain multiple lobes, with one or two azimuths of vanishing A4between the dip and strike directions.The strong influence of the anellipticity parameters on the azimuthally varying coefficient A4suggests that nonhyperbolic moveout recorded in wide‐azimuth surveys can help to constrain the anisotropic velocity field. Since for typical orthorhombic models that describe naturally fractured reservoirs the parameters η(1,2,3)are closely related to the fracture density and infill, the results of azimuthal nonhyperbolic moveout analysis can also be used in reservoir characterization.

Analysis of image gathers in factorized VTI media

Because events in image gathers generated after prestack depth migration are sensitive to the velocity field, they are often used in migration velocity analysis for isotropic media. Here, we present an analytic and numerical study of P‐wave image gathers in transversely isotropic media with a vertical symmetry axis (VTI) and establish the conditions for flattening such events and positioning them at the true reflector depth. Application of the weak‐anisotropy approximation leads to concise expressions for reflections in image gathers from homogeneous and factorized v(z) media in terms of the VTI parameters and the vertical velocity gradient kz. Flattening events in image gathers for any reflector dip requires accurate values of the zero‐dip NMO velocity at the surface [Vnmo (z = 0)], the gradient kz, and the anellipticity coefficient η. For a fixed error in Vnmo and kz, the magnitude of residual moveout of events in image gathers decreases with dip, while the moveout caused by an error in η initially increases for moderate dips but then decreases as dips approach 90°. Flat events in image gathers in VTI media, however, do not guarantee the correct depth scale of the model because reflector depth depends on the vertical migration velocity. For factorized v(x, z) media with a linear velocity variation in both the x‐ and z‐directions, the moveout on image gathers is controlled by Vnmo (x = z = 0), kz, η, and a combination of the horizontal velocity gradient kx and the Thomsen parameter δ (specifically, kx[Formula: see text]). If too large a value of any of these four quantities is used in migration, reflections in the image gathers curve downward (i.e., they are undercorrected; the inferred depth increases with offset), while a negative error results in overcorrection. Lateral heterogeneity tends to increase the sensitivity of moveout of events in image gathers to the parameter η, and errors in η may lead to measurable residual moveout of horizontal events in v(x, z) media even for offset‐to‐depth ratios close to unity. These results provide a basis for extending to VTI media conventional velocity analysis methods operating with image gathers. Although P‐wave traveltimes alone cannot be used to separate anisotropy from lateral heterogeneity (i.e., kx is coupled to δ), moveout of events in image gathers does constrain the vertical gradient kz. Hence, it may be possible to build VTI velocity models in depth by supplementing reflection data with minimal a priori information, such as the vertical velocity at the top of the factorized VTI layer.

Quartic moveout coefficient: 3D description and application to tilted TI media

Nonhyperbolic (long‐spread) moveout provides essential information for a number of seismic inversion/processing applications, particularly for parameter estimation in anisotropic media. Here, we present an analytic expression for the quartic moveout coefficient A4that controls the magnitude of nonhyperbolic moveout of pure (nonconverted) modes. Our result takes into account reflection‐point dispersal on irregular interfaces and is valid for arbitrarily anisotropic, heterogeneous media. All quantities needed to compute A4can be evaluated during the tracing of the zero‐offset ray, so long‐spread moveout can be modeled without time‐consuming multioffset, multiazimuth ray tracing.The general equation for the quartic coefficient is then used to study azimuthally varying nonhyperbolic moveout of P‐waves in a dipping transversely isotropic (TI) layer with an arbitrary tilt ν of the symmetry axis. Assuming that the symmetry axis is confined to the dip plane, we employed the weak‐anisotropy approximation to analyze the dependence of A4on the anisotropic parameters. The linearized expression for A4is proportional to the anellipticity coefficient η ≈ ε − δ and does not depend on the individual values of the Thomsen parameters. Typically, the magnitude of nonhyperbolic moveout in tilted TI media above a dipping reflector is highest near the reflector strike, whereas deviations from hyperbolic moveout on the dip line are substantial only for mild dips.The azimuthal variation of the quartic coefficient is governed by the tilt ν and reflector dip φ and has a much more complicated character than the NMO–velocity ellipse. For example, if the symmetry axis is vertical (VTI media, ν = 0) and the dip φ < 30°, A4goes to zero on two lines with different azimuths where it changes sign. If the symmetry axis is orthogonal to the reflector (this model is typical for thrust‐and‐fold belts), the strike‐line quartic coefficient is defined by the well‐known expression for a horizontal VTI layer (i.e., it is independent of dip), while the dip‐line A4is proportional to cos4φ and rapidly decreases with dip. The high sensitivity of the quartic moveout coefficient to the parameter η and the tilt of the symmetry axis can be exploited in the inversion of wide‐azimuth, long‐spread P‐wave data for the parameters of TI media.

Illumination‐based normalization for wave‐equation depth migration

Illumination problems caused by finite‐recording aperture and lateral velocity lensing can bias amplitudes in migration results. In this paper, I develop a normalization scheme appropriate for wave‐equation migration algorithms that compensates for irregular illumination. I generate synthetic seismic data over a reference reflectivity model, using the adjoint of wave‐equation shot‐profile migration as the forward modeling operator. I then migrate the synthetic data with the same shot‐profile algorithm. The ratio between the synthetic migration result and the initial reference model is a measure of seismic illumination. Dividing the true data migration result by this illumination function mitigates the illumination problems. The methodology can take into account reflector dip as well as both shot and receiver geometries, and, because it is based on wave‐equation migration, it naturally models the finite‐frequency effects of wave propagation. The reference model should be as close to the true model as possible; good choices include the migrated image, or a synthetic image with a single known dip that corresponds to the expected dip of a reflector of interest. Computational shortcuts allow the illumination functions to be computed at about the cost of a single migration. Results indicate that normalization can significantly reduce amplitude distortions due to irregular subsurface illumination.

Crooked‐line 2D seismic reflection imaging in crystalline terrains: Part 2, migration

Seismic reflection surveys are frequently conducted over very complicated geological structure, but surveying often must be confined to existing crooked roads or tracks. Typically, data from such 2D crooked‐line surveys are processed using standard common midpoint (CMP) stacking techniques to obtain a 2D time section which is then 2D migrated. In Part I, we show that a reflector dip component across the processing line can cause serious problems for standard CMP stacking. We also propose a supplementary processing step in which cross‐dip is determined locally and cross‐dip moveout (CDMO) is removed from data to form an optimum cross‐dip stack. However, a crooked‐line survey is really a swath 3D survey, and ideally we would like to obtain a 3D image of reflectivity surrounding the profile. Here we investigate the potential of 3D prestack Kirchhoff migration to directly image all observed reflections; i.e., we attempt to construct a 3D image volume of all reflectors viewed by the survey. Because reflectors that face away from the acquisition line cannot return much wave energy from available sources to available receivers, they cannot be imaged even if they lie directly beneath the survey profile. Tests show that the cross‐profile spread of trace midpoints usually is sufficient to provide a useful degree of cross‐line positioning of reflection points. A very helpful image volume is thus obtained. Kirchhoff 3D prestack migration is computationally laborious. A much quicker but less complete method is to create the 3D migrated image volume from the 2D optimum cross‐dip stack and the associated set of cross‐dips. Robustness of migration methods to time errors in the prestack data traces such as poorly corrected statics is also an issue. Tests show that in difficult cases, particularly where only 2D processing is warranted, migration of trace absolute amplitude rather than standard phase data may lead to a superior result.

Palaeozoic amalgamation of Central Europe: new results from recent geological and geophysical investigations

Multidisciplinary studies of geotransects across the North European Plain and Southern North Sea, and geological reexamination of the Variscides of the North Bohemian Massif, permit a new 3-D reassessment of the relationships between the principal crustal blocks abutting Baltica along the Trans-European Suture Zone (TESZ). Accretion was in three stages: Cambrian accretion of the Bruno–Silesian, Lysogory and Malopolska terranes; end-Ordovician/early Silurian accretion of Avalonia; and early Carboniferous accretion of the Armorican Terrane Assemblage (ATA). Palaeozoic plume-influenced metabasite geochemistry in the Bohemian Massif explains the progressive rifting away of peri-Gondwanan crustal blocks before their accretion to Baltica. Geophysical data, faunal and provenance information from boreholes, and dated small inliers and cores confirm that Avalonian crust extends beyond the Anglo-Brabant Deformation Belt eastwards to northwest Poland. The location and dip of reflectors along the TESZ and beneath the North European Plain suggest that Avalonian crust overrode the Baltica passive margin, marked by a high-velocity lower crustal layer, on shallowly southwest-dipping thrust planes forming the Heligoland–Pomerania Deformation Belt. The “Variscan orocline” of southwest Poland masks two junctions between the Armorican Terrane Assemblage (ATA) and previously accreted crustal blocks. To the east is a dextrally transpressive contact with the Bruno–Silesian and Malopolska blocks, accreted in the Cambrian, while to the north is a thrust contact with easternmost Avalonia, deeply buried beneath younger sedimentary cover. In the northeast Bohemian and Rhenohercynian Massifs Devonian “early Variscide” deformation dominated by WNW and NW-directed thrusting, records closure of Ordovician–Devonian seaways between detached “islands” of the ATA and Avalonia.

A case study on using AVO walkaway VSP data - Barrow Sub-basin, Western Australia

In December 1999 the TL/1 JV partners acquired a 2D walkaway VSP survey in the Monty-2 well in the Barrow Sub-basin, Western Australia. The survey was designed to analyse AVO effects and possible later calibration of surface seismic data. The survey was recorded using CGG?s 12 level array tool, the SST-500, and comprised a total of 4 walkaway lines, with 2 azimuths at each of two array positions. Each of the 4 walkaway lines had a length of 6000 meters. Walkaway VSP data was acquired by positioning the array tool just above the two target reflectors, the North Rankin and the Mungaroo ``B' sandstone formations. The VSP data was processed using an `isotropic' sequence, which allowed for a perfect preservation of the amplitude and phase relationships between the three receiver components. AVO estimates were computed from the reflected P-wavefield after shaping deconvolution, derived from the down going P-wavefield. P-wave polarisation information was used to compute reflection angles at the target horizon and the local reflector dip. Finally, synthetic P-wave reflectivity curves were computed for the North Rankin and the Mungaroo ``B' sandstone formation and compared with the borehole seismic measurements. The AVO/AVA response at the top North Rankin reflection was found to be strongly affected by a thin overlaying carbonate layer. The best match between measured AVO curves and isotropic modeling therefore was found for a shale/carbonate model rather than a carbonate/sandstone model. In contrast, the AVO response of the Mungaroo ``B' reflection was found to be in very good agreement with isotropic modeling results.

What is DMO coverage?

‘Coverage’ or ‘fold’ is defined as the multiplicity of common-midpoint (CMP) data. For CMP stacking the coverage is consistent with the number of traces sharing a common reflection point on flat subsurface reflectors. This relationship is not true for dipping reflectors. The deficiencies of CMP stacking with respect to imaging dipping events have long been overcome by the introduction of the dip-moveout (DMO) correction. However, the concept of coverage has not yet satisfactorily been updated to a ‘DMO coverage’ consistent with DMO stacking. A definition of constant-velocity DMO coverage will be proposed here. A subsurface reflector will be illuminated from a given source and receiver location if the time difference between the reflector zero-offset traveltime and the NMO- and DMO-corrected traveltime of the reflection event is less than half a dominant wavelength. Due to the fact that a subsurface reflector location is determined by its zero-offset traveltime, its strike and its dip, the DMO coverage also depends on these three parameters. For every surface location, the proposed DMO coverage consists of a 3D fold distribution over reflector strike, dip and zero-offset traveltime.

Fresnel zones and spatial resolution for P- and SH-waves in transversely isotropic media

In an elastically anisotropic medium where the seismic wave velocity is a function of direction, the wavefront shape is nonspherical and, in most cases, assumes a nonelliptical shape. Numerical modelling techniques have been used to calculate the Fresnel‐zone diameter for compressional (P) and shear (SH) waves in transversely isotropic and isotropic media, respectively. The size of the Fresnel zone is found to be predominantly dependent on the curvatures and wavelength of the wavefront as well as the dip angle of the reflector. In addition, the anisotropic elastic parameters δ* (critical near‐vertical anisotropy), ε (the P-wave anisotropy), and γ (the SH-wave anisotropy) are found to significantly affect the size of the Fresnel zone. Numerical modeling results show considerable differences between the Fresnel zones for anisotropic and isotropic velocity functions at various reflector dips. In addition, the Fresnel‐zone dimensions for anisotropic media exhibit asymmetry and considerable change with dip. By way of contrast, those of the corresponding isotropic velocity field exhibit symmetry and negligible variation with dip. The spatial resolution of unmigrated seismic data in an anisotropic medium would consequently be significantly different from that determined for the same medium if it is assumed to be isotropic. Physical modeling results demonstrate that anisotropy can significantly affect the spatial resolving power of seismic waves. The degree of these effects depends on the wavefront curvature, which changes with dip and orientation of the symmetry axis. This observation indicates that the spatial imaging of unmigrated reflection events from the base of thick shale sediments will be affected by anisotropy.

Common offset Continuation in Lateral Linearly Varied Velocity Media

The amplitude preserving common offset continuation methods in lateral linearly inhomogeneous media are studied in this paper. An amplitude preserving kinematically equivalent offset continuation (OC) equation is proposed for lateral linearly varied velocity media. A partial differential equation is introduced to be the amplitude-preserving OC operator, which can be regarded as an extension of the amplitude preserving OC operator from constant velocity to lateral velocity media. Consequently, with the traveltime function in linearly varied velocity media, we have proved that its kinematically equivalent equation is valid for all offsets and reflector dips. In addition, from the result of calculation on synthetic gathers and comparison with traditional DMO methods, we can conclude that the equation also preserves the amplitude of caustics in the OC process.

Coherency calculations in the presence of structural dip

We have used crosscorrelation, semblance, and eigenstructure algorithms to estimate coherency. The first two algorithms calculate coherency over a multiplicity of trial time lags or dips, with the dip having the highest coherency corresponding to the local dip of the reflector. Partially because of its greater computational cost, our original eigenstructure algorithm calculated coherency along an implicitly flat horizon. Although generalizing the eigenstructure algorithm to search over a range of test dips allowed us to image coherency in the presence of steeply dipping structures, we were somewhat surprised that this generalization concomitantly degenerated the quality of the fault images in flatter dip areas. Because it is a local estimation of reflector dip (including as few as five traces), the multidip coherency estimate provides an algorithmically correct, but interpretationally undesirable, estimate of the best apparent dip that explained the offset reflectors across a fault. We ameliorate this problem using two methods, both of which require the smoothing of a locally inaccurate estimate of regional dip. We then calculate our eigenstructure estimate of coherency only along the dip of the reflector, thereby providing maximum lateral resolution of reflector discontinuities. We are thus both better able to explain the superior results obtained by our earliest eigenstructure analysis along interpreted horizon slices, yet able to extend this resolution to steeply dipping reflectors on uninterpreted cubes of seismic data.

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.

Seismic constant‐velocity remigration

When a seismic common midpoint (CMP) stack or zero‐offset (ZO) section is depth or time migrated with different (constant) migration velocities, different reflector images of the subsurface are obtained. If the migration velocity is changed continuously, the (kinematically) migrated image of a single point on the reflector, constructed for one particular seismic ZO reflection signal, moves along a circle at depth, which we call the Thales circle. It degenerates to a vertical line for a nondipping event. For all other dips, the dislocation as a function of migration velocity depends on the reflector dip. In particular for reflectors with dips larger than 45°, the reflection point moves upward for increasing velocity. The corresponding curves in a Time‐migrated section are parabolas. These formulas will provide the seismic interpreter with a better understanding of where a reflector image might move when the velocity model is changed. Moreover, in that case, the reflector image as a whole behaves to some extent like an ensemble of body waves, which we therefore call remigration image waves. In the same way as physical waves propagate as a function of time, these image waves propagate as a function of migration velocity. Different migrated images can thus be considered as snapshots of image waves at different instants of migration velocity. By some simple plane‐wave considerations, image‐wave equations can be derived that describe the propagation of image waves as a function of the migration velocity. The Thales circles and parabolas then turn out to be the characteristics or ray trajectories for these image‐wave equations.

The impact of migration on AVO

Amplitude variation with offset (AVO) analysis is often limited to areas where multidimensional propagation effects such as reflector dip and diffractions from faults can be ignored. Migration‐inversion provides a framework for extending the use of seismic amplitudes to areas where structural or stratigraphic effects are important. In this procedure, sources and receivers are downward continued into the earth using uncollapsed prestack migration. Instead of stacking the data as in normal migration, the prestack migrated data are used in AVO analysis or other inversion techniques to infer local earth properties. The prestack migration can take many forms. In particular, prestack time migration of common‐angle sections provides a convenient tool for improving the lateral resolution and spatial positioning of AVO anomalies. In this approach, a plane‐wave decomposition is first applied in the offset direction, separating the wavefield into different propagating angles. The data are then gathered into common‐angle sections and migrated one angle at a time. The common‐angle migrations have a simple form and are shown to adequately preserve amplitude as a function of angle. Normal AVO analysis is then applied to the prestack migrated data. Examples using seismic lines from the Gulf of Mexico show how migration improves AVO analysis. In the first set of examples, migration is shown to improve imaging of subtle spatial variations in bright spots. Subsequent AVO analysis reveals dim spots associated with dry‐hole locations that were not resolvable using traditional processing techniques, including both conventional AVO and poststack migration. A second set of examples shows improvements in AVO response after migration is used to reduce interference from coherent noise and diffractions. A final example shows the impact of migration on the spatial location of dipping AVO anomalies. In all cases, migration improves both the signal‐to‐noise ratio and spatial resolution of AVO anomalies.

An accurate formulation of log‐stretch dip moveout in the frequency‐wavenumber domain

Dip moveout (DMO) processing is a partial prestack migration procedure that has been widely used in seismic data processing. The DMO process has been described in Deregowski (1986), Hale (1991) and Liner (1990). Many different DMO algorithms have been developed over the past decade. These algorithms have been designed to improve either the accuracy or the computational speed of the DMO process. Hale (1984) developed a method for performing DMO via Fourier transforms that is accurate for all reflector dips (assuming constant velocity). Hale’s method is computationally expensive because his DMO operator is temporally nonstationary, but its accuracy and simplicity have made it an industry standard. It has become a benchmark by which results from other DMO algorithms are judged. Of all the methods used to make the frequency‐domain DMO computationally efficient, the technique of logarithmic time stretching, first suggested in Bolondi et al. (1982), is widely used. After logarithmic stretching of the time axis, the DMO operator becomes temporally stationary which enables replacement of the slow temporal Fourier integration with a fast Fourier transform combined with a simple phase shift. Bale and Jakubowicz (1987) presented a log‐stretch DMO operator (hereafter referred to as Bale’s DMO) in the frequency‐wavenumber (F-K) domain without approximations, while Notfors and Godfrey (1987) suggested an approximate version of log‐stretch DMO operator (hereafter referred to as Notfors’s DMO). Surprisingly, Bale’s full log‐stretch DMO operator produces a less satisfactory impulse response than Notfors’s approximate log‐stretch DMO scheme (see Liner, 1990). Liner (1990) attributed this characteristic to the fact that Bale’s DMO derivation implicitly assumes that the Fourier transform frequency in the log‐stretch domain is not time‐dependant. He presented an exact log‐stretch DMO operator (hereafter referred to as Liner’s DMO) which was derived by transforming the time log‐stretched Hale’s (t, x) DMO impulse response into the Fourier domain. Its derivation is relatively complicated, but Liner has shown that his DMO does generate good DMO impulse responses.