Nonhyperbolic Moveout Research Articles

The Characteristic of Quartic Moveout Coefficient of P‐Wave in the TI Homogeneous Media with an Arbitrary Orientation Symmetric‐Axis

AbstractDeviations of any pure mode (non‐converted) wave reflection traveltimes from a hyperbola, called the nonhyperbolic or quartic moveout, need to be corrected properly in processing long‐spread anisotropic seismic data. Based on our exact analytic expression for the quartic moveout coefficient (A4) of pure mode (non‐converted) wave reflection on a horizontal interface in the TI media with an arbitrarily orientated symmetry axis (ATI) in space, we numerically study the characteristics of the P‐wave A4 with the CMP survey line azimuth variation. The forward modeling results show that the characteristics of the A4 are related not only to the anisotropy parameters of the TI, but also the spatial orientation of the TI. The variation characteristic of the A4 is determined by the anisotropy parameters and tilt angle of the TI symmetric‐axis, and the azimuth of the TI symmetric‐axis results in the azimuth symmetry of the A4 with the survey line variation. Our numerical results will benefit the explanation of the anisotropic seismic data and the parameter inversion.

Crack density tensor inversion for analysis of changes in rock frame architecture

SUMMARY This paper presents a development of the use of multi-axial ultrasonic data for the quantification of extrinsic, crack or grain-contact induced elastic anisotropy in core samples with application to a UKCS oil field. An approach for inversion of multi-axial velocity measurements is presented, which extends the previous work by Sayers (2002) for the determination of secondand fourth-order crack density tensors from inversion of multi-axial ultrasonic velocity data. The extensions to the inversion approach provide improved consideration of data uncertainties, by using all available P- and S-wave data and also permit the inclusion of an orthorhombic background anisotropy in the inversion [e.g. due to intrinsic lattice preferred orientation(LPO) effects]. The latter aspect leads to estimates of the extrinsic anisotropy, that is, the quantified crack density tensors, that are ‘unpolluted’ by the effects of the intrinsic anisotropy, thus permitting extrinsic and intrinsic anisotropies to be distinguished. For the samples considered, the extrinsic crack-induced anisotropy is strong relative to that of the intrinsic LPO effects, and the pre-dominant crack-set is commonly aligned parallel to the depositional fabric (which is generally horizontal). However, the LPO and extrinsic anisotropies are in general aligned, which indicates a linked origin. Furthermore, a strong correlation is observed between the degree of VTI anisotropy and the modal content of micas in the samples, which cannot be explained solely by the LPO effect. In fact, it is shown that increased horizontal (grain parallel) crack densities are associated with higher mica content. These horizontal cracks in the mica-rich samples often show moderate-to-strong variations in crack density with loading that might be detected in situ with non-hyperbolic moveout. Additionally, three samples show significant pressure sensitivity of the vertical crack sets indicating that loading-induced azimuthal anisotropy might also exist in some layers in situ and be detectable using azimuthal AVO type analyses. Analysis of the fourth-order crack density tensor allows insight into the relative sizes of the normal and tangential crack compliances, B N and B T . For one case it is found that B N ≥ B T (i.e. the crack-sets are more compliant in compression than in shear). For another sample B N < B T ; this sample had been cleaned prior to the analysis, which appears to facilitate shear in the cracks. This conclusion may have significant implications for the use of analogue samples acquired at the surface where organic products, which lead to the reduced shear compliance, are not present.

AVO-sensitive semblance analysis for wide-azimuth data

Conventional semblance-based moveout analysis models prestack reflection data with events that have hyperbolic move-out and no amplitude variation with offset (AVO). Substantial amplitude variation and even phase change with offset do not significantly compromise the semblance operator. However, polarity reversal associated with a change in the sign of the reflection coefficient may cause conventional semblance to fail. An existing modification of the semblance operator that takes amplitude variations into account (so-called AK semblance) is limited to narrow-azimuth data and cannot handle nonhyperbolic moveout at large offsets. We employ a 3D nonhyperbolic moveout inversion algorithm to extend the AK semblance method to wide-azimuth data recorded on long spreads. To preserve velocity resolu-tion, the ratio [Formula: see text] of the AVO gradient and intercept is kept constant within each semblance window. In the presence of azimuthal anisotropy, however, the parameter [Formula: see text] has to be azimuthally dependent. Synthetic tests confirm that distortions in moveout analysis caused by polarity reversals become more common for long-offset data. Conventional semblance produces substantial errors in the NMO ellipse and azimuthally varying anellipticity parameter [Formula: see text] not just for class 2 AVO response but also for some models with class 1 AVO. In contrast, the AK semblance algorithm gives accurate estimates of the moveout parameters even when the position of the polarity reversal varies with azimuth. The AK method not only helps to flatten wide-azimuth reflection events prior to stacking and azimuthal AVO analysis but also provides input parameters for the anisotropic geometrical-spreading correction.

Four-parameter velocity analysis and its applications to the Ken-71 area converted-wave data

3-D converted-wave data were acquired using digital MEMS (micro-electromechanical system) three component (3C) sensors in the alternating sand and shale sequence in the overburden of the Shengli Ken-71 area. This gives rise to serious non-hyperbolic moveout effects in the converted-wave data due to both the asymmetrical ray path and anisotropic effects. Conventional velocity analysis and moveout correction based on isotropic methods do not flatten reflections events. Here, we use a four-parameter theory to evaluate these effects and process the data. These four parameters include the PS converted wave stacking velocity (VC2), the vertical velocity ratio (y0), the effective velocity ratio (yeff), and the anisotropy parameter (xeff). The method utilizes the moveout information at different offsets to estimate the different parameters and ensures that the events are properly aligned for stacking. As a result, this four-parameter theory leads to an improvement in imaging quality and correlation between the P-waves and converted-waves.

Multifocusing as a method of improving subsurface imaging

Imaging the subsurface, the ultimate goal of the seismic method, can be done in different ways and in different domains: time or depth. Although depth migration has become almost mandatory in areas of complex geology— because it accounts for traveltime nonhyperbolic moveout, it has, in fact, quite a limited purpose—to convert seismic data from one form to another for a given velocity model. Time imaging provides sufficient information for a subsurface of moderate complexity. Moreover, even for complex areas that require depth migration for correct subsurface imaging, time imaging usually constitutes a key first step that facilitates the estimation of a velocity model for depth imaging. For these reasons, improving the quality of time imaging is a focus of intensive research. A recent advance is multifocusing (MF), a method with the potential to greatly improve the quality of time imaging.

Resolving fault shadow problem by Fault Constrained Tomography

In many areas, so-called, Fault Shadows is a serious hindrance to successful seismic imaging. The major part of this problem is caused by large velocity variations in fault zones. Fault zones contain different types of geological and imaging velocity anomalies that cause seismic image distortions and non-hyperbolic moveout. Pre-stack depth migration with the proper velocity model is the only method that can solve this problem and improve the seismic image below fault zones. We have developed a special and novel technique, Fault Constrained Tomography, to build high-resolution interval velocity models that are required for fault areas. Fault Constrained Tomography takes into account the nature of velocity variation within fault zones. Combined with Pre-Stack Depth Migration it can successfully remove fault shadow distortions from seismic images.

Improving lateral and vertical resolution of seismic images by correcting for wavelet stretch in common-angle migration

Long-offset or high-incident-angle seismic reflections provide us with improved velocity resolution, better leverage against multiples, less contamination by ground roll, and information that is often critical when estimating lithology and fluid product. Unfortunately, high-incident-angle seismic reflections suffer not only from nonhyperbolic moveout but also from wavelet stretch during imaging, resulting in lower-resolution images that mix the response from adjacent lithologies. For an arbitrary acoustic medium, wavelet stretch from prestack migration depends only on the cosine of the reflection angle, such that the amount of wavelet stretch will be the same for all samples of a common-reflection-angle migrated trace. Thus, we are able to implement a wavelet stretch correction by applying a simple stationary spectral shaping operation to common-angle migrated traces. We obtain such traces directly by a prestack Kirchhoff migration algorithm. Correcting for stretch effectively increases the fold of imaged data, far beyond that achieved in conventional migration, resulting in improved signal-to-noise ratio of the final stacked section. Increasing the fidelity of large incident angles results in images with improved vertical and lateral resolution and with increased angular illumination, valuable for amplitude variation with angle (AVA) and amplitude variation with offset (AVO) analysis. Finally, such large-angle images are more sensitive to and therefore provide increased leverage over errors in velocity and velocity anisotropy. These ideas were applied to prestack time migration on seismic data from the Fort Worth basin, in Texas.

Seismic critical-angle reflectometry: A method to characterize azimuthal anisotropy?

Existing anisotropic parameter-estimation algorithms that operate with long-offset data are based on the inversion of either nonhyperbolic moveout or wide-angle amplitude-variation-with-offset (AVO) response. We show that valuable information about anisotropic reservoirs can also be obtained from the critical angle of reflected waves. To explain the behavior of the critical angle, we develop weak-anisotropy approximations for vertical transverse isotropy and then use Tsvankin’s notation to extend them to azimuthally anisotropic models of orthorhombic symmetry. The P-wave critical-angle reflection in orthorhombic media is particularly sensitive to the parameters [Formula: see text] and [Formula: see text] responsible for the symmetry-plane horizontal velocity in the high-velocity layer. The azimuthal variation of the critical angle for typical orthorhombic models can reach [Formula: see text], which translates into substantial changes in the critical offset of the reflected P-wave. The main diagnostic features of the critical-angle reflection employed in our method include the rapid amplitude increase at the critical angle and the subsequent separation of the head wave. Analysis of exact synthetic seismograms, generated with the reflectivity method, confirms that the azimuthal variation of the critical offset is detectable on wide-azimuth, long-spread data and can be qualitatively described by our linearized equations. Estimation of the critical offset from the amplitude curve of the reflected wave, however, is not straightforward. Additional complications may be caused by the overburden noise train and by the influence of errors in the overburden velocity model on the computation of the critical angle. Still, critical-angle reflectometry should help to constrain the dominant fracture directions and can be combined with other methods to reduce the uncertainty in the estimated anisotropy parameters.

Seismic anisotropy as an indicator of reservoir quality in siliciclastic rocks

Abstract Improving the accuracy of subsurface imaging is commonly the main incentive for including the effects of anisotropy in seismic processing. However, the anisotropy itself holds valuable information about rock properties and, as such, can be viewed as a seismic attribute. Here we summarize results from an integrated project that explored the potential to use observations of seismic anisotropy to interpret lithological and fluid properties (the SAIL project). Our approach links detailed petrofabric analyses of reservoir rocks, laboratory based measurements of ultrasonic velocities in core samples, and reservoir-scale seismic observations. We present results for the Clair field, a Carboniferous–Devonian reservoir offshore Scotland, west of the Shetland Islands. The reservoir rocks are sandstones that are variable in composition and exhibit anisotropy on three length-scales: the crystal, grain and fracture scale. We have developed a methodology for assessing crystal-preferred-orientation (CPO) using a combination of electron back-scattered diffraction (EBSD), X-ray texture goniometry (XRTG) and image analysis. Modal proportions of individual minerals are measured using quantitative X-ray diffraction (QXRD). These measurements are used to calculate the intrinsic anisotropy due to CPO via Voigt-Reuss-Hill averaging of individual crystal elasticities and their orientations. The intrinsic anisotropy of the rock is controlled by the phyllosilicate content and to a lesser degree the orientation of quartz and feldspar; the latter can serve as a palaeoflow indicator. Our results show remarkable consistency in CPO throughout the reservoir and allow us to construct a mathematical model of reservoir anisotropy. A comparison of CPO-predicted velocities and those derived from laboratory measurements of ultrasonic signals allows the estimation of additional elastic compliance terms due to grain-boundary interactions. The results show that the CPO estimates are good proxies for the intrinsic anisotropy of the clean sandstones. The more micaceous rocks exhibit enhanced anisotropy due to interactions between the phyllosilicate grains. We then compare the lab-scale predictions with reservoir-scale measurements of seismic anisotropy, based on amplitude variation with offset and azimuth (AVOA) analysis and non-hyperbolic moveout. Our mathematical model provides a foundation for interpreting the reservoir-scale seismic data and improving the geological modelling of complex reservoirs. The observed increases in AVOA signal with depth can only be explained with an increase in fracturing beneath the major unit boundaries, rather than a change in intrinsic CPO properties. In general, the style and magnitude of anisotropy in the Clair field appears to be indicative of reservoir quality.

Optimization of Nonhyperbolic Moveout Correction Equation of Long-Offset Seismic Data in VTI Media

The conventional Alkhalifah moveout correction equation is characterized by low precision. And even worse it may fail to describe relationship of traveltime-offset of long-offset seismic data in VTI media. Aiming at improving accuracy of the moveout correction equation of long-offset seismic data in VTI media, we analyze the Alkhalifah NMO equation at first. Then based on the least-error principle, a table of optimal correction coefficients is established by calculation to compensate errors resulted from long offset cases using the conventional moveout correction equation. Therefore, the optimal Alkhalifah moveout correction equation is proposed here to improve the conventional moveout correction equation. Subsequently, we derive the traveltime-offset function with a high precision from the Fomel's group velocity equation for VTI. The moveout correction equation with high precision is put forward. Above moveout correction equations are applied to inversion calculation of anisotropic parameters from long-offset seismic data. Numerical results show that the accuracy of inversion using the optimal Alkhalifah moveout equation is twice than that of inversion using the conventional moveout equation, and the inversion method using the high precision moveout correction equation is the most accurate.

Anisotropic geometrical-spreading correction for wide-azimuth P-wave reflections

Compensation for geometrical spreading along a raypath is one of the key steps in AVO (amplitude-variation-with-offset) analysis, in particular, for wide-azimuth surveys. Here, we propose an efficient methodology to correct long-spread, wide-azimuth reflection data for geometrical spreading in stratified azimuthally anisotropic media. The P-wave geometrical-spreading factor is expressed through the reflection traveltime described by a nonhyperbolic moveout equation that has the same form as in VTI (transversely isotropic with a vertical symmetry axis) media. The adapted VTI equation is parameterized by the normal-moveout (NMO) ellipse and the azimuthally varying anellipticity parameter [Formula: see text]. To estimate the moveout parameters, we apply a 3D nonhyperbolic semblance algorithm of Vasconcelos and Tsvankin that operates simultaneously with traces at all offsets andazimuths. The estimated moveout parameters are used as the input in our geometrical-spreading computation. Numerical tests for models composed of orthorhombic layers with strong, depth-varying velocity anisotropy confirm the high accuracy of our travetime-fitting procedure and, therefore, of the geometrical-spreading correction. Because our algorithm is based entirely on the kinematics of reflection arrivals, it can be incorporated readily into the processing flow of azimuthal AVO analysis. In combination with the nonhyperbolic moveout inversion, we apply our method to wide-azimuth P-wave data collected at the Weyburn field in Canada. The geometrical-spreading factor for the reflection from the top of the fractured reservoir is clearly influenced by azimuthal anisotropy in the overburden, which should cause distortions in the azimuthal AVO attributes. This case study confirms that the azimuthal variation of the geometrical-spreading factor often is comparable to or exceeds that of the reflection coefficient.

3D common-reflection-point-based seismic migration velocity analysis

Three-dimensional prestack depth migration and depth residual picking in common-image gathers (CIGs) are the most time-consuming parts of 3D migration velocity analysis. Most migration-based velocity analysis algorithms need spatial coordinates of reflection points and CIG depth residuals at different offsets (or angles) to provide updated velocity information. We propose a new algorithm that can analyze 3D velocity quickly and accurately. Spatial coordinates and orientations of reflection points are provided by a 3D prestack parsimonious depth migration; the migration involves only the time samples picked from the salient reflection events on one 3D common-offset volume. Ray tracing from the reflection points to the surface provides a common-reflection-point (CRP) gather for each reflection point. Predicted (nonhyperbolic) moveouts for local velocity perturbations, based on maximizing the stacked amplitude, give the estimated velocity updates for each CRP gather. Then the velocity update for each voxel in the velocity model is obtained by averaging over all predicted velocity updates for that voxel. Prior model constraints may be used to stabilize velocity updating. Compared with other migration velocity analyses, the traveltime picking is limited to only one common-offset volume (and needs to be done only once); there is no need for intensive 3D prestack depth migration. Hence, the computation time is orders of magnitude less than other migration-based velocity analyses. A 3D synthetic data test shows the algorithm works effectively and efficiently.

Velocity-independent layer stripping of PP and PS reflection traveltimes

Building accurate interval velocity models is critically important for seismic imaging and AVO (amplitude variation with offset) analysis. Here, we adapt the [Formula: see text] method to develop an exact technique for constructing the interval traveltime-offset function in a target zone beneath a horizontally layered overburden. All layers in the model can be anisotropic, with an essential assumption that the overburden has a horizontal symmetry plane (i.e., up-down symmetry). Our layer-stripping algorithm is entirely data-driven and, in contrast to the generalized Dix equations, does not require knowledge of the velocity field anywhere in the medium. Important advantages of our approach compared to the Dix-style formalism also include the ability to handle mode-converted waves, long-offset data, and laterally heterogeneous target layers with multiple, curved reflectors. Numerical tests confirm the high accuracy of the algorithm in computing the interval traveltimes of both PP- and PS-waves in a dipping, transversely isotropic layer with a tilted symmetry axis (TTI medium) beneath an anisotropic overburden. In combination with the inversion techniques developed for homogeneous TTI models, the proposed layer stripping of PP and PS data can be used to estimate the interval parameters of TTI formations in such important exploration areas as the Canadian Foothills. Potential applications of this methodology also include the dip-moveout inversion for the P-wave time-processing parameter [Formula: see text] and stable computation of the interval long-spread (nonhyperbolic) moveout for purposes of anisotropic velocity analysis.

Nonhyperbolic moveout analysis in VTI media using rational interpolation

Anisotropic velocity analysis using qP-waves in transversely isotropic media with a vertical symmetry axis (VTI) usually is done by inferring the anellipticity parameter [Formula: see text] and the normal moveout velocity [Formula: see text] from the nonhyperbolic character of the moveout. Several approximations explicit in these parameters exist with varying degrees of accuracy. Here, we present a rational interpolation approach to nonhyperbolic moveout analysis in the [Formula: see text] domain. This method has no additional computational overhead compared to using expressions explicit in [Formula: see text] and [Formula: see text]. The lack of such overhead stems from the observation that, for fixed [Formula: see text] and zero-offset two-way traveltime [Formula: see text], the moveout curve for different values of [Formula: see text] can be calculated by simple stretching of the offset axis. This observation is based on the assumptions that the traveltimes of qP-waves in transversely isotropic media mainly depend on [Formula: see text] and [Formula: see text], and that the shear-wave velocity along the symmetry axis has a negligibleinfluence on these traveltimes. The accuracy of the rational interpolation method is as good as that of these approximations. The method can be tuned accurately to any offset range of interest by increasing the order of the interpolation. We test the method using both synthetic and field data and compare it with the nonhyperbolic moveout equation of Alkhalifah and Tsvankin (1995) and the shifted hyperbola equation of Fomel (2004). Both data types confirm that for [Formula: see text], our method significantly outperforms the nonhyperbolic moveout equation in terms of combined unbiased parameter estimation with accurate moveout correction. Comparison with the shifted hyperbola equation of Fomel for Greenhorn-shale anisotropy establishes almost identical accuracy of the rational interpolation method and his equation. Even though the proposed method currently deals with homogeneous media only, results from application to synthetic and field data confirm the applicability of the proposed method to horizontally layered VTI media.

Exponential asymptotically bounded velocity model: Part I — Effective models and velocity transformations

A new compacted-sediment, vertical-velocity model that represents the effect of gradually increasing velocities with depth is proposed. An asymptotically bounded, exponential velocity model is defined by three intuitive parameters. For a given layer, the parameters are 1) the velocity, 2) its vertical gradient at the top interface, and 3) an upper-limit velocity value. Time-depth relations, velocity transformations, hyperbolic, and nonhyperbolic moveouts are derived. The entire vertical velocity profile is composed of a set of layers, where the velocity function for each layer is defined by the proposed model. This method provides a basis for describing the subsurface using a smaller number of thick layers rather than using the classical, linear velocity model.

CRP-based seismic migration velocity analysis

A new migration velocity analysis is developed by combining the speed of parsimonious prestack depth migration with velocity adjustments estimated within and across common-reflection-point (CRP) gathers. The proposed approach is much more efficient than conventional tomographic velocity analysis because only the traces that contribute to a series of CRP gathers are depth migrated at each iteration. The local interval-velocity adjustments for each CRP are obtained by maximizing the stack amplitude over the predicted (nonhyperbolic) moveout in each CRP gather; this does not involve retracing rays. At every iteration, the velocity in each pixel is updated by averaging over all the predicted velocity updates. Finally, CRP positions and orientations are updated by parsimonious migration, and rays are retraced to define new CRP gathers for the next iteration; this ensures internal consistency between the updated velocity model and the CRP gather. Because the algorithm has a gridded-model parameterization, no explicit representation or fitting of reflectors is involved. Strong lateral-velocity variations, such as those found at salt flanks, can be handled. Application to synthetic and field data sets show that the proposed algorithm works effectively and efficiently.

Quartic reflection moveout in a weakly anisotropic dipping layer

Deviations of P-wave reflection traveltimes from a hyperbola, called the nonhyperbolic or quartic moveout, need to be handled properly while processing long-spread seismic data. As observed nonhyperbolic moveout is usually attributed to the presence of anisotropy, we devote our paper to deriving and analyzing a general formula that describes an azimuthally varying quartic moveout coefficient in a homogeneous, weakly anisotropic medium above a dipping, mildly curved reflector. To obtain the desired expression, we consistently linearize all quantities in small stiffness perturbations from a given isotropic solid. Our result incorporates all known weak-anisotropy approximations of the quartic moveout coefficient and extends them further to triclinic media. By comparing our approximation with nonhyperbolic moveout obtained from the ray-traced reflection traveltimes, we find that the former predicts azimuthal variations of the quartic moveout when its magnitude is less than 20% of the corresponding hyperbolic moveout term. We also study the influence of reflector curvature on nonhyperbolic moveout. It turns out that the curvature produces no quartic moveout in the reflector strike direction, where the anisotropy-induced moveout nonhyperbolicity is usually nonnegligible. Thus, the presence of nonhyperbolic moveout along the reflector strike might indicate effective anisotropy.

Converted-wave seismology in anisotropic media Revisited, Part II: Application to parameter estimation

In transversely isotropic media with a vertical symmetry axis (VTI), the converted-wave (C-wave) moveout over intermediate-to-far offsets is determined by four parameters. These are the C-wave stacking velocity V C2, the vertical and effective velocity ratios γ 0and γ eff, and the anisotropic parameter X eff. We refer to the four parameters as the C-wave stacking velocity model. The purpose of C-wave velocity analysis is to determine this stacking velocity model. The C-wave stacking velocity model V C2, γ 0, γ geff, and X eff can be determined from P- and C-wave reflection moveout data. However, error propagation is a severe problem in C-wave reflection-moveout inversion. The current short-spread stacking velocity as deduced from hyperbolic moveout does not provide sufficient accuracy to yield meaningful inverted values for the anisotropic parameters. The non-hyperbolic moveout over intermediate-offsets (x/z from 1.0 to 1.5) is no longer negligible and can be quantified using a background γ. Non-hyperbolic analysis with a γ correction over the intermediate offsets can yield V C2 with errors less than 1% for noise free data. The procedure is very robust, allowing initial guesses of γ with up to 20% errors. It is also applicable for vertically inhomogeneous anisotropic media. This improved accuracy makes it possible to estimate anisotropic parameters using 4C seismic data. Two practical work flows are presented for this purpose: the double-scanning flow and the single-scanning flow. Applications to synthetic and real data show that the two flows yield results with similar accuracy but the single-scanning flow is more efficient than the double-scanning flow.

Geometrical spreading of P-waves in horizontally layered, azimuthally anisotropic media

For processing and inverting reflection data, it is convenient to represent geometrical spreading through the reflection traveltime measured at the earth's surface. Such expressions are particularly important for azimuthally anisotropic models in which variations of geometrical spreading with both offset and azimuth can significantly distort the results of wide-azimuth amplitude-variation-with-offset (AVO) analysis. Here, we present an equation for relative geometrical spreading in laterally homogeneous, arbitrarily anisotropic media as a simple function of the spatial derivatives of reflection traveltimes. By employing the Tsvankin-Thomsen nonhyperbolic moveout equation, the spreading is represented through the moveout coefficients, which can be estimated from surface seismic data. This formulation is then applied to P-wave reflections in an orthorhombic layer to evaluate the distortions of the geometrical spreading caused by both polar and azimuthal anisotropy. The relative geometrical spreading of P-waves in homogeneous orthorhombic media is controlled by five parameters that are also responsible for time processing. The weak-anisotropy approximation, verified by numerical tests, shows that azimuthal velocity variations contribute significantly to geometrical spreading, and the existing equations for transversely isotropic media with a vertical symmetry axis (VTI) cannot be applied even in the vertical symmetry planes. The shape of the azimuthally varying spreading factor is close to an ellipse for offsets smaller than the reflector depth but becomes more complicated for larger offset-to-depth ratios. The overall magnitude of the azimuthal variation of the geometrical spreading for the moderately anisotropic model used in the tests exceeds 25% for a wide range of offsets. While the methodology developed here is helpful in modeling and analyzing anisotropic geometrical spreading, its main practical application is in correcting the wide-azimuth AVO signature for the influence of the anisotropic overburden.

Fast model-based migration velocity analysis and reflector shape estimation

Migration velocity analysis can be made more efficient by preselecting the traces that contribute to a series of common-reflection-point (CRP) gathers and migrating only those traces. The data traces that contribute to a CRP for one reflection point on one layer are defined in a two-step procedure. First, poststack parsimonious Kirchhoff depth migration of zero-offset (or stacked) traces defines approximate reflector positions and orientations. Then, ray tracing from the reflection points for nonzero reflection angles defines the source and receiver locations of the data traces in the CRP gather. These traces are then prestack depth migrated, and the interval velocity model adjustment is obtained by fitting the velocity that maximizes the stack amplitude over the predicted (nonhyperbolic) moveout. A small number (2–3) of iterations converge to a 2D model of layer shape and interval velocity. Further efficiency is obtained by implementing layer stripping. The computation time is greatly reduced by combining parsimonious migration with processing only the salient portions of the whole seismic data set. The algorithm can handle lateral velocity variation within each layer as well as constant velocity. The computation time of the proposed algorithm is of the same order as that of the standard rms velocity scan method, but it does not have the inherent assumptions of the velocity scan method and is faster than current iterative prestack depth migration velocity analysis methods for typical field data.