Reflection Traveltimes Research Articles

Analytic study of the geometrical spreading of P-waves in a layered transversely isotropic medium with a vertical symmetry axis

An analytical formula for geometrical spreading is derived for a horizontally layered transversely isotropic medium with a vertical symmetry axis (VTI). With this expression, geometrical spreading can be determined using only the anisotropy parameters in the first layer, the traveltime derivatives, and the source‐receiver offset. Explicit, numerically feasible expressions for geometrical spreading are obtained for special cases of transverse isotropy (weak anisotropy and elliptic anisotropy). Geometrical spreading can be calculated for transversly isotropic (TI) media by using picked traveltimes of primary nonhyperbolic P-wave reflections without having to know the actual parameters in the deeper subsurface; no ray tracing is needed. Synthetic examples verify the algorithm and show that it is numerically feasible for calculation of geometrical spreading. For media with a few (4–5) layers, relative errors in the computed geometrical spreading remain less than 0.5% for offset/depth ratios less than 1.0. Errors that change with offset are attributed to inaccuracy in the expression used for nonhyberbolic moveout. Geometrical spreading is most sensitive to errors in NMO velocity, followed by errors in zero‐offset reflection time, followed by errors in anisotropy of the surface layer. New relations between group and phase velocities and between group and phase angles are shown in appendices.

Sequential integrated inversion of refraction and wide-angle reflection traveltimes and gravity data for two-dimensional velocity structures

SUMMARY A new algorithm is presented for the integrated 2-D inversion of seismic traveltime and gravity data. The algorithm adopts the ‘maximum likelihood’ regularization scheme. We construct a ‘probability density function’ which includes three kinds of information: information derived from gravity measurements; information derived from the seismic traveltime inversion procedure applied to the model; and information on the physical correlation among the density and the velocity parameters. We assume a linear relation between density and velocity, which can be node-dependent; that is, we can choose diVerent relationships for diVerent parts of the velocity‐density grid. In addition, our procedure allows us to consider a covariance matrix related to the error propagation in linking density to velocity. We use seismic data to estimate starting velocity values and the position of boundary nodes. Subsequently, the sequential integrated inversion (SII) optimizes the layer velocities and densities for our models. The procedure is applicable, as an additional step, to any type of seismic tomographic inversion. We illustrate the method by comparing the velocity models recovered from a standard seismic traveltime inversion with those retrieved using our algorithm. The inversion of synthetic data calculated for a 2-D isotropic, laterally inhomogeneous model shows the stability and accuracy of this procedure, demonstrates the improvements to the recovery of true velocity anomalies, and proves that this technique can eYciently overcome some of the limitations of both gravity and seismic traveltime inversions, when they are used independently. An interpretation of field data from the 1994 Vesuvius test experiment is also presented. At depths down to 4.5 km, the model retrieved after a SII shows a more detailed structure than the model obtained from an interpretation of seismic traveltime only, and yields additional information for a further study of the area.

Inversion of normal moveout for monoclinic media1

Multiple vertical fracture sets, possibly combined with horizontal fine layering, produce an equivalent medium of monoclinic symmetry with a horizontal symmetry plane. Although monoclinic models may be rather common for fractured formations, they have hardly been used in seismic methods of fracture detection due to the large number of independent elements in the stiffness tensor. Here, we show that multicomponent wide‐azimuth reflection data (combined with known vertical velocity or reflector depth) or multi‐azimuth walkaway VSP surveys provide enough information to invert for all but one anisotropic parameters of monoclinic media.In order to facilitate the inversion procedure, we introduce a Thomsen‐style parametrization for monoclinic media that includes the vertical velocities of the P‐wave and one of the split S‐waves and a set of dimensionless anisotropic coefficients. Our notation, defined for the coordinate frame associated with the polarization directions of the vertically propagating shear waves, captures the combinations of the stiffnesses responsible for the normal‐moveout (NMO) ellipses of all three pure modes. The first group of the anisotropic parameters contains seven coefficients (ε(1,2), δ(1,2,3) and γ(1,2)) analogous to those defined by Tsvankin for the higher‐symmetry orthorhombic model. The parameters ε(1,2), δ(1,2) and γ(1,2) are primarily responsible for the pure‐mode NMO velocities along the coordinate axes x1 and x2 (i.e. in the shear‐wave polarization directions). The remaining coefficient δ(3) is not constrained by conventional‐spread reflection traveltimes in a horizontal monoclinic layer. The second parameter group consists of the newly introduced coefficients ζ(1,2,3) which control the rotation of the P‐, S1‐ and S2‐wave NMO ellipses with respect to the horizontal coordinate axes. Misalignment of the P‐wave NMO ellipse and shear‐wave polarization directions was recently observed on field data by Pérez et al.Our parameter‐estimation algorithm, based on NMO equations valid for any strength of the anisotropy, is designed to obtain anisotropic parameters of monoclinic media by inverting the vertical velocities and NMO ellipses of the P‐, S1‐ and S2‐waves. A Dix‐type representation of the NMO velocity of mode‐converted waves makes it possible to replace the pure shear modes in reflection surveys with the PS1‐ and PS2‐waves. Numerical tests show that our method yields stable estimates of all relevant parameters for both a single layer and a horizontally stratified monoclinic medium.

An asymptotic inverse to the Kirchhoff-Helmholtz integral

Modelling a reflected wave by the Kirchhoff-Helmholtz (KH) integral consists of an integration along the reflector. By this, one sums the Huygens secondary-source contributions to the wavefield attached to the reflector at the observation point. The proposed asymptotic inverse KH integral, by which this modelling process is inverted, works in a completely analogous way. It consists of an integral along the reflection traveltime surface of the reflector. For a point on the reflector, one sums the reflected-wave contributions attached to the respective reflection-traveltime surface associated with the related source-receiver pair. In this way, the new integral is a more natural inverse to KH forward modelling integral than the conventional Kirchhoff migration integral that is well known in seismic reflection imaging. The new inverse integral reconstructs the Huygens sources along the reflector, thus providing their positions and amplitudes.

The null space of a generally anisotropic medium in linearized surface reflection tomography

The null space of a generally anisotropic medium in linearized surface reflection tomography

Modelling strategies and model assessment for wide-angle seismic traveltime data

Strategies for modelling seismic refraction/wide-angle reflection traveltimes to obtain 2-D velocity and interface structure are presented along with methods for assessing the reliability of the results. Emphasis is placed on using inverse methods, but a discussion of arrival picking and classification, data uncertainties, traveltime reciprocity, crooked line geometry and the selection of a starting model is also applicable to trial-and-error forward modelling. The most important advantages of an inverse method are the ability to derive simpler models for the appropriate level of fit to the data, and the ability to assess the final model in terms of resolution, parameter bounds and non-uniqueness. Given the unique characteristics of each data set and the local earth structure, there is no single approach to modelling wide-angle data that is best. This paper describes the best modelling strategies according to (1) the model parametrization, (2) the inclusion of prior information, (3) the complexity of the earth structure, (4) the characteristics of the data, and (5) the utilization of coincident seismic reflection data. There are two natural end-member inversion styles: (1) a regular, fine-grid parametrization when seeking a minimum-structure model, and (2) an irregular grid, minimum-parameter model when considering certain forms of prior information. The former style represents the ‘pure’ tomography approach. The latter style is closer to automated forward modelling, and can be applied best with a parameter-selective algorithm, that is, one that allows any subset of model parameters to be selected for inversion. If there is strong lateral heterogeneity in the near-surface only, layer stripping works well. If there is complexity at all depths, all model parameters should be determined simultaneously after careful construction of a starting model that allows the appropriate rays to be traced to all pick locations. The lateral spacing of model nodes to use will depend on the type of inversion and whether detailed prior information is included, but a general guideline based on model resolution when seeking a minimum-parameter model is a model node spacing equal to the shot spacing (receiver spacing for typical marine data), except perhaps in the upper layers where about half this may be necessary; node spacing is not an issue when using smoothing constraints, provided it is small enough to resolve the earth structure of interest. Traveltimes picked from pre-stack, unmigrated or migrated coincident reflection data can be (1) used to develop the starting model, (2) inverted simultaneously with the wide-angle data, or (3) inverted after modelling the wide-angle data to constrain interfaces that ‘float’ within the velocity model. Model assessment establishes the reliability of the final model. Presenting model statistics, traveltime fits, ray diagrams and resolution kernels is useful, but can only indirectly address this issue. Direct model assessment techniques that derive alternative models that satisfactorily fit the real data are the best means of establishing the absolute bounds on model parameters and whether a particular model feature is required by the data.

2-D ray‐based tomography for velocity, layer shape, and attenuation from GPR data

Tomographic inversion for near‐surface structure from multioffset ground‐penetrating radar (GPR) reflection data is implemented. The model is parameterized as layers with constant velocity and constant attenuation. The solution is divided into two parts. First, the velocity and shape of each layer are estimated from the observed reflection traveltimes; then using this geometry and the corresponding ray paths, the attenuation of each layer is estimated from the observed relative amplitudes. Solution of both tomographic problems is performed by singular value decomposition of the sensitivity matrix (to compute information density, resolution, and covariance matrices as well as the solution). Tests on synthetic data, for which the solution is known, show that the algorithm converges efficiently to the neighborhood of the correct solution. Inversion of multichannel field data in a GPR line from a fluvial/eolian environment produces a model that is consistent with structural images and velocity estimates obtained independently by traditional common‐midpoint processing.

3-D moveout inversion in azimuthally anisotropic media with lateral velocity variation: Theory and a case study

Reflection moveout recorded over an azimuthally anisotropic medium (e.g., caused by vertical or dipping fractures) varies with the azimuth of the source‐receiver line. Normal‐moveout (NMO) velocity, responsible for the reflection traveltimes on conventional‐length spreads, forms an elliptical curve in the horizontal plane. While this result remains valid in the presence of arbitrary anisotropy and heterogeneity, the inversion of the NMO ellipse for the medium parameters has been discussed so far only for horizontally homogeneous models above a horizontal or dipping reflector. Here, we develop an analytic moveout correction for weak lateral velocity variation in horizontally layered azimuthally anisotropic media. The correction term is proportional to the curvature of the zero‐offset traveltime surface at the common midpoint and, therefore, can be estimated from surface seismic data. After the influence of lateral velocity variation on the effective NMO ellipses has been stripped, the generalized Dix equation can be used to compute the interval ellipses and evaluate the magnitude of azimuthal anisotropy (measured by P-wave NMO velocity) within the layer of interest. This methodology was applied to a 3-D “wide‐azimuth” data set acquired over a fractured reservoir in the Powder River Basin, Wyoming. The processing sequence included 3-D semblance analysis (based on the elliptical NMO equation) for a grid of common‐midpoint “supergathers,” spatial smoothing of the effective NMO ellipses and zero‐offset traveltimes, correction for lateral velocity variation, and generalized Dix differentiation. Our estimates of depth‐varying fracture trends in the survey area, based on the interval P-wave NMO ellipses, are in good agreement with the results of outcrop and borehole measurements and the rotational analysis of four‐ component S-wave data.

3-D moveout velocity analysis and parameter estimation for orthorhombic media

Orthorhombic symmetry describes several azimuthally anisotropic models typical for fractured formations, such as those containing two orthogonal crack systems or parallel vertical cracks in a VTI (transversely isotropic with a vertical symmetry axis) background. Here, we present a methodology for inverting multiazimuth P-wave reflection traveltimes for the parameters of vertically inhomogeneous orthorhombic media. Our approach is based on the general analytic representation of normal‐moveout (NMO) velocity as an ellipse in the horizontal plane. A minimum of three differently oriented common‐midpoint (CMP) lines (or a “wideazimuth” 3-D survey) is needed to reconstruct the ellipse and thus obtain NMO velocity in any azimuthal direction. Then, the orientation and the semiaxes of the NMO ellipse, which are dependent on both anisotropy and heterogeneity, can be inverted for the medium parameters. Our analytic and numerical study shows that for the model of a homogeneous orthorhombic layer above a dipping reflector, the exact P-wave NMO velocity is determined by the symmetry‐plane orientation and five parameters: the NMO velocities from a horizontal reflector measured in the symmetry planes [[Formula: see text]] and three anisotropic coefficients η(1,2,3)introduced by analogy with the Alkhalifah‐Tsvankin parameter η for VTI media. The importance of the medium parameterization in terms of the η coefficients goes well beyond the NMO-velocity function. By generating migration impulse responses, we demonstrate that the parameters [Formula: see text] and η(1,2,3)are sufficient to perform all time processing steps (normal‐moveout and dip‐moveout corrections, prestack and poststack time migration) in orthorhombic models. The velocities [Formula: see text] and the orientation of the vertical symmetry planes can be found using the azimuthally dependent NMO velocity from a horizontal reflector. Then the NMO ellipse of at least one dipping event is additionally needed to obtain the coefficients η(1,2,3)that control the dip dependence of normal moveout. We discuss the stability of the inversion procedure and specify the constraints on the dip and azimuth of the reflector; for instance, for all three η coefficients to be resolved individually, the dip plane of the reflector should not coincide with either of the symmetry planes. To carry out parameter estimation in vertically inhomogeneous orthorhombic media, we apply the generalized Dix equation of Grechka, Tsvankin and Cohen, which operates with the matrices responsible for interval NMO ellipses rather than with the NMO velocities themselves. Our algorithm is designed to find the interval values of [Formula: see text] and η(1,2,3)using moveout from horizontal and dipping reflectors measured at different vertical times (i.e., only surface P-wave data are needed). Application to a synthetic multiazimuth P-wave data set over a layered orthorhombic medium with depth‐varying orientation of the symmetry planes verifies the accuracy of the inversion method.

Mapping prestack depth‐migrated coherent signal and noise events back to the original time gathers using Fermat’s principle

Because of its computational efficiency, prestack Kirchhoff depth migration is currently the method of choice in both 2-D and 3-D imaging of seismic data. The most algorithmically complex component of the Kirchhoff family of algorithms is the calculation and manipulation of accurate traveltime tables for each source and receiver point. Once calculated, we sum the seismic energy over all possible ray paths, allowing us to accurately image both specular and nonspecular scattered energy. Any seismic events that fall within the velocity passband, including reflected and diffracted signal, mode conversions, multiples, head waves, and aliases of surface waves, are imaged in depth. The transformation of time gathers to depth gathers can be quite complicated and nonintuitive to all but the seasoned imaging expert. In particular, easily recognized head‐wave events on common‐shot gathers are often difficult to differentiate from undermigrated coherent reflections on common‐reflection‐point depth gathers. In contrast, subsalt multiples that have propagated along complex ray paths are often easily recognized on common‐offset depth gathers but are indistinguishable from the distorted primaries on the input common‐shot or common‐midpoint time gathers. In a related area, seismic reflection traveltime tomography is currently the workhorse for 2-D and an active area of research and development for 3-D migration‐driven velocity analysis. The objective function for this “velocity inversion” problem is to either minimize the temporal difference between picked and modeled time picks, or to maximize the similarity between, or flatness of, common‐reflection‐point depth picks. Once picked and associated with the correct reflector, time picks never need to be modified during the velocity‐model updating steps that ultimately lead to a feasible solution. In practice, such time picks are nearly impossible to make in those structurally complex areas that justify the use of prestack depth migration. Instead, we almost always use the second objective function and pick reflector events in depth, where we can use our geologic insight to differentiate between signal and noise and where the difficulty of associating a picked event with the velocity/depth model horizon completely disappears. The major drawback of picking in depth is that these events need to be repicked each time any part of the overlying velocity/depth model has been updated. We show that by applying Fermat’s principle, and by reusing the same traveltime tables used in seismic prestack Kirchhoff depth imaging, we can map interpreted events on the depth gathers to corresponding interpreted events on the original time gathers. This technique, first introduced by J. van Trier in 1990, is considerably more stable and, because we reuse the already computed migration traveltime tables, more economic than two‐point ray‐trace methods. In our first application of coherent noise suppression, we show how we can relate imaging artifacts seen on the depth image to their causative coherent noise on the original time gathers. Once identified, these noise events can be safely suppressed using conventional filtering techniques. In our second application of reflection tomography, we show how we can pick partially focused reflectors in depth, and map them back to time, undoing the effect of the incorrect velocity/depth model used in prestack Kirchhoff depth migration such that the events never need to be repicked during subsequent velocity model updates.

Generalized Dix equation and analytic treatment of normal‐moveout velocity for anisotropic media*

Despite the complexity of wave propagation in anisotropic media, reflection moveout on conventional common‐midpoint (CMP) spreads is usually well described by the normal‐moveout (NMO) velocity defined in the zero‐offset limit. In their recent work, Grechka and Tsvankin showed that the azimuthal variation of NMO velocity around a fixed CMP location generally has an elliptical form (i.e. plotting the NMO velocity in each azimuthal direction produces an ellipse) and is determined by the spatial derivatives of the slowness vector evaluated at the CMP location. This formalism is used here to develop exact solutions for the NMO velocity in anisotropic media of arbitrary symmetry.For the model of a single homogeneous layer above a dipping reflector, we obtain an explicit NMO expression valid for all pure modes and any orientation of the CMP line with respect to the reflector strike. The contribution of anisotropy to NMO velocity is contained in the slowness components of the zero‐offset ray (along with the derivatives of the vertical slowness with respect to the horizontal slownesses) — quantities that can be found in a straightforward way from the Christoffel equation. If the medium above a dipping reflector is horizontally stratified, the effective NMO velocity is determined through a Dix‐type average of the matrices responsible for the ‘interval’ NMO ellipses in the individual layers. This generalized Dix equation provides an analytic basis for moveout inversion in vertically inhomogeneous, arbitrarily anisotropic media. For models with a throughgoing vertical symmetry plane (i.e. if the dip plane of the reflector coincides with a symmetry plane of the overburden), the semi‐axes of the NMO ellipse are found by the more conventional rms averaging of the interval NMO velocities in the dip and strike directions.Modelling of normal moveout in general heterogeneous anisotropic media requires dynamic ray tracing of only one (zero‐offset) ray. Remarkably, the expressions for geometrical spreading along the zero‐offset ray contain all the components necessary to build the NMO ellipse. This method is orders of magnitude faster than multi‐azimuth, multi‐offset ray tracing and, therefore, can be used efficiently in traveltime inversion and in devising fast dip‐moveout (DMO) processing algorithms for anisotropic media. This technique becomes especially efficient if the model consists of homogeneous layers or blocks separated by smooth interfaces.The high accuracy of our NMO expressions is illustrated by comparison with ray‐traced reflection traveltimes in piecewise‐homogeneous, azimuthally anisotropic models. We also apply the generalized Dix equation to field data collected over a fractured reservoir and show that P‐wave moveout can be used to find the depth‐dependent fracture orientation and to evaluate the magnitude of azimuthal anisotropy.

Joint inversion ofP- andPS-waves in orthorhombic media: Theory and a physical modeling study

Reflection traveltimes recorded over azimuthally anisotropic fractured media can provide valuable information for reservoir characterization. As recently shown by Grechka and Tsvankin, normal moveout (NMO) velocity of any pure (unconverted) mode depends on only three medium parameters and usually has an elliptical shape in the horizontal plane. Because of the limited information contained in the NMO ellipse of P-waves, it is advantageous to use moveout velocities of shear or converted modes in attempts to resolve the coefficients of realistic orthorhombic or lower‐symmetry fractured models. Joint inversion of P and PS traveltimes is especially attractive because it does not require shear‐wave excitation. Here, we show that for models composed of horizontal layers with a horizontal symmetry plane, the traveltime of converted waves is reciprocal with respect to the source and receiver positions (i.e., it remains the same if we interchange the source and receiver) and can be adequately described by NMO velocity on conventional‐length spreads. The azimuthal dependence of converted‐wave NMO velocity has the same form as for pure modes but requires the spatial derivatives of two-way traveltime for its determination. Using the generalized Dix equation of Grechka, Tsvankin, and Cohen, we derive a simple relationship between the NMO ellipses of pure and converted waves that provides a basis for obtaining shear‐wave information from P and PS data. For orthorhombic models, the combination of the reflection traveltimes of the P-wave and two split PS-waves makes it possible to reconstruct the azimuthally dependent NMO velocities of the pure shear modes and to find the anisotropic parameters that cannot be determined from P-wave data alone. The method is applied to a physical modeling data set acquired over a block of orthorhombic material—Phenolite XX-324. The inversion of conventional‐spread P and PS moveout data allowed us to obtain the orientation of the vertical symmetry planes and eight (out of nine) elastic parameters of the medium (the reflector depth was known). The remaining coefficient (c12or δ(3)in Tsvankin’s notation) is found from the direct P-wave arrival in the horizontal plane. The inversion results accurately predict moveout curves of the pure S-waves and are in excellent agreement with direct measurements of the horizontal velocities.

Nonlinear refraction and reflection travel time tomography

We develop a rapid nonlinear travel time tomography method that simultaneously inverts refraction and reflection travel times on a regular velocity grid. For travel time and ray path calculations, we apply a wave front method employing graph theory. The first‐arrival refraction travel times are calculated on the basis of cell velocities, and the later refraction and reflection travel times are computed using both cell velocities and given interfaces. We solve a regularized nonlinear inverse problem. A Laplacian operator is applied to regularize the model parameters (cell slownesses and reflector geometry) so that the inverse problem is valid for a continuum. The travel times are also regularized such that we invert travel time curves rather than travel time points. A conjugate gradient method is applied to minimize the nonlinear objective function. After obtaining a solution, we perform nonlinear Monte Carlo inversions for uncertainty analysis and compute the posterior model covariance. In numerical experiments, we demonstrate that combining the first arrival refraction travel times with later reflection travel times can better reconstruct the velocity field as well as the reflector geometry. This combination is particularly important for modeling crustal structures where large velocity variations occur in the upper crust. We apply this approach to model the crustal structure of the California Borderland using ocean bottom seismometer and land data collected during the Los Angeles Region Seismic Experiment along two marine survey lines. Details of our image include a high‐velocity zone under the Catalina Ridge, but a smooth gradient zone between Catalina Ridge and San Clemente Ridge. The Moho depth is about 22 km with lateral variations.

Seismic reflection traveltimes in two-dimensional statistically anisotropic random media

Velocity estimation remains one of the main problems when imaging the subsurface with seismic reflection data. Traveltime inversion enables us to obtain large-scale structures of the velocity field and the position of seismic reflectors. However, as the media currently under study are becoming more and more complex, we need to know the finer-scale structures. The problem is that below a certain range of velocity heterogeneities, deterministic methods become difficult to use, so we turn to a probabilistic approach. With this in view, we characterize the velocity field as a random field defined by its first and second statistical moments. Usually, a seismic random medium is defined as a homogeneous velocity background perturbed by a small random field that is assumed to be stationary. Thus, we make a link between such a random velocity medium (together with a simple reflector) and seismic reflection traveltimes. Assuming that the traveltimes are ergodic, we use 2-D seismic reflection geometry to study the decrease in the statistical traveltime fluctuations as a function of the offset (the source-receiver distance). Our formulae are based on the Rytov approximation and the parabolic approximation for acoustic waves. The validity and the limits are established for both of these approximations in statistically anisotropic random media. Finally, theoretical inversion procedures are developed for the horizontal correlation structure of the velocity heterogeneities for the simplest case of a horizontal reflector. Synthetic seismograms are then computed (on particular realizations of random media) by simulating scalar wave propagation via finite difference algorithms. There is good agreement between the theoretical and experimental results.

Seismic-reflection evidence that the Hayward fault extends into the lower crust of the San Francisco Bay area, California

Abstract This article presents deep seismic-reflection data from an experiment across San Francisco Peninsula in 1995 using large (125 to 500 kg) explosive sources. Shot gathers show a mostly nonreflective upper crust in both the Franciscan and Salinian terranes (juxtaposed across the San Andreas fault), an onset of weak lower-crustal reflectivity beginning at about 6-sec two-way travel time (TWTT) and bright southwest-dipping reflections between 11 and 13 sec TWTT. Previous studies have shown that the Moho in this area is no deeper than 25 km (∼8 to 9 sec TWTT). Three-dimensional reflection travel-time modeling of the 11 to 13 sec events from the shot gathers indicates that the bright events may be explained by reflectors 15 to 20 km into the upper mantle, northeast of the San Andreas fault. However, upper mantle reflections from these depths were not observed on marine-reflection profiles collected in San Francisco Bay, nor were they reported from a refraction prifile on San Francisco Peninsula. The most consistent interpretation of these events from 2D raytracing and 3D travel-time modeling is that they are out-of-plane reflections from a high-angle (dipping ∼70° to the southwest) impedance contrast in the lower crust that corresponds with the surface trace of the Hayward fault. These results suggest that the Hayward fault truncates the horizontal detachment fault suggested to be active beneath San Francisco Bay.

Nonhyperbolic reflection moveout for horizontal transverse isotropy

The transversely isotropic model with a horizontal axis of symmetry (HTI) has been used extensively in studies of shear‐wave splitting to describe fractured formations with a single system of parallel vertical penny‐shaped cracks. Here, we present an analytic description of longspread reflection moveout in horizontally layered HTI media with arbitrary strength of anisotropy. The hyperbolic moveout equation parameterized by the exact normal‐moveout (NMO) velocity is sufficiently accurate for P-waves on conventional‐length spreads (close to the reflector depth), although the NMO velocity is not, in general, usable for converting time to depth. However, the influence of anisotropy leads to the deviation of the moveout curve from a hyperbola with increasing spread length, even in a single‐layer model. To account for nonhyperbolic moveout, we have derived an exact expression for the azimuthally dependent quartic term of the Taylor series traveltime expansion [t2(x2)] valid for any pure mode in an HTI layer. The quartic moveout coefficient and the NMO velocity are then substituted into the nonhyperbolic moveout equation of Tsvankin and Thomsen, originally designed for vertical transverse isotropy (VTI). Numerical examples for media with both moderate and uncommonly strong nonhyperbolic moveout show that this equation accurately describes azimuthally dependent P-wave reflection traveltimes in an HTI layer, even for spread lengths twice as large as the reflector depth. In multilayered HTI media, the NMO velocity and the quartic moveout coefficient reflect the influence of layering as well as azimuthal anisotropy. We show that the conventional Dix equation for NMO velocity remains entirely valid for any azimuth in HTI media if the group‐velocity vectors (rays) for data in a common‐midpoint (CMP) gather do not deviate from the vertical incidence plane. Although this condition is not exactly satisfied in the presence of azimuthal velocity variations, rms averaging of the interval NMO velocities represents a good approximation for models with moderate azimuthal anisotropy. Furthermore, the quartic moveout coefficient for multilayered HTI media can also be calculated with acceptable accuracy using the known averaging equations for vertical transverse isotropy. This allows us to extend the nonhyperbolic moveout equation to horizontally stratified media composed of any combination of isotropic, VTI, and HTI layers. In addition to providing analytic insight into the behavior of nonhyperbolic moveout, these results can be used in modeling and inversion of reflection traveltimes in azimuthally anisotropic media.

Improving seismic data quality in the Gippsland Basin (Australia)

Deep seismic exploration in the Gippsland Basin is hindered by strong noise below the Latrobe Group coal sequence. The reflectivity method provides a means for constructing detailed and accurate synthetic seismograms, often from little more than a partial sonic log. The noise contributions to the synthetics can then be interpreted using additional synthetics computed from variations upon the depth model and by exercising control over the wave types modeled. This approach revealed three types of persistent noise in progressively deeper parts of the subcoal image: (1) mode‐converted interbed multiples (generated within the coal sequence), (2) S-wave reflections and long‐period multiples (generated between the coal sequence and the Miocene carbonates), and (3) surface‐related multiples. The noise interpretation can also be performed upon semblance analyses of the elastic synthetics to guide a velocity analysis away from a well. This procedure helped to avoid picking the interformation long‐period multiples, whose stacking velocities were only 5 to 10% below those of the weak target zone primaries. An improve subcoal image was obtained by making full use of the versatile noise suppression offered by a τ-p domain processing stream. By separating the strong linear events at the far offsets, it is possible to stack a larger portion of the target zone reflections, provided hyperbolic velocity filtering (HVF) is applied to suppress the transform artifacts. Hyperbolic velocity filtering can be incorporated into a point‐source τ-p transform to suppress S-wave reflections and guided waves while preserving plane‐wave amplitudes to assist the subsequent deconvolution of the mode‐converted interbed multiples. Stacking in the τ-p domain is achieved using an elliptical moveout correction that reduces wavelet stretch and approximates the exact reflection traveltime better than NMO. Two regional seismic lines were reprocessed in this manner and cointerpreted with the modeling studies performed at nearby wells to avoid the noise events that still remained. Several new events appeared in the immediate target zone, passing the low‐frequency character expected following transmission through a coal sequence.

Feasibility of nonhyperbolic moveout inversion in transversely isotropic media

Inversion of reflection traveltimes in anisotropic media can provide estimates of anisotropic coefficients required for seismic processing and lithology discrimination. Nonhyperbolic P-wave moveout for transverse isotropy with a vertical symmetry axis (VTI media) is controlled by the parameter η (or, alternatively, by the horizontal velocity Vhor), which is also responsible for the influence of anisotropy on all time‐processing steps, including dip‐moveout (DMO) correction and time migration. Here, we recast the nonhyperbolic moveout equation, originally developed by Tsvankin and Thomsen, as a function of Vhorand normal‐moveout (NMO) velocity Vnmoand introduce a correction factor in the denominator that increases the accuracy at intermediate offsets. Then we apply this equation to obtain Vhorand η from nonhyperbolic semblance analysis on long common midpoint (CMP) spreads and study the accuracy and stability of the inversion procedure. Our error analysis shows that the horizontal velocity becomes relatively well‐constrained by reflection traveltimes if the spreadlength exceeds twice the reflector depth. There is, however, a certain degree of tradeoff between Vhorand Vnmocaused by the interplay between the quadratic and quartic term of the moveout series. Since the errors in Vhorand Vnmohave opposite signs, the absolute error in the parameter η (which depends on the ratio Vhor/Vnmo) turns out to be at least two times bigger than the percentage error in Vhor. Therefore, the inverted value of η is highly sensitive to small correlated errors in reflection traveltimes, with moveout distortions on the order of 3–4 ms leading to errors in η up to ±0.1—even in the simplest model of a single VTI layer. Similar conclusions apply to vertically inhomogeneous media, in which the interval horizontal velocity can be obtained with an accuracy often comparable to that of the NMO velocity, while the interval values of η are distorted by the tradeoff between Vhorand Vnmothat gets amplified by the Dix‐type differentiation procedure. We applied nonhyperbolic semblance analysis to a walkaway VSP data set acquired at Vacuum field, New Mexico, and obtained a significant value of η = 0.19 indicative of nonnegligible anisotropy in this area. Then we combined moveout inversion results with the known vertical velocity to resolve the anisotropic coefficients ε and δ. However, in agreement with our modeling results, η estimation was significantly compounded by the scatter in the measured traveltimes. Certain instability in η inversion has no influence on the results of anisotropic poststack time migration because all kinematically equivalent models obtained from nonhyperbolic moveout give an adequate description of long‐spread reflection traveltimes. Also, inversion of nonhyperbolic moveout provides a relatively accurate horizontal‐velocity function that can be combined with the vertical velocity (if it is available) to estimate the anisotropic coefficient ε. However, η represents a valuable lithology indicator that can be obtained from surface P-wave data. Therefore, for purposes of lithology discrimination, it is preferable to find η by means of the more stable DMO method of Alkhalifah and Tsvankin.

3-D description of normal moveout in anisotropic inhomogeneous media

We present a new equation for normal‐moveout (NMO) velocity that describes azimuthally dependent reflection traveltimes of pure modes from both horizontal and dipping reflectors in arbitrary anisotropic inhomogeneous media. With the exception of anomalous areas such as those where common‐midpoint (CMP) reflection time decreases with offset, the azimuthal variation of NMO velocity represents an ellipse in the horizontal plane, with the orientation of the axes determined by the properties of the medium and the direction of the reflector normal. In general, a minimum of three azimuthal measurements is necessary to reconstruct the best‐fit ellipse and obtain NMO velocity in all azimuthal directions. This result provides a simple way to correct for the azimuthal variation in stacking velocity often observed in 3-D surveys. Even more importantly, analytic expressions for the parameters of the NMO ellipse can be used in the inversion of moveout data for the anisotropic coefficients of the medium. For homogeneous transversely isotropic media with a vertical axis of symmetry (VTI media), our equation for azimuthally dependent NMO velocity from dipping reflectors becomes a relatively simple function of phase velocity and its derivatives. We show that the zero‐dip NMO velocity Vnmo(0) and the anisotropic coefficient η are sufficient to describe the P-wave NMO velocity for any orientation of the CMP line with respect to the dip plane of the reflector. Using our formalism, Vnmo(0) and η (the only parameters needed for time processing) can be found from the dip‐dependent NMO velocity at any azimuth or, alternatively, from the azimuthally dependent NMO for a single dipping reflector. We also apply this theory to more complicated azimuthally anisotropic models with the orthorhombic symmetry used to describe fractured reservoirs. For reflections from horizontal interfaces in orthorhombic media, the axes of the normal moveout ellipse are aligned with the vertical symmetry planes. Therefore, azimuthal P-wave moveout measurements can be inverted for the orientation of the symmetry planes (typically determined by the fracture direction) and the NMO velocities within them. If the vertical velocity is known, symmetry‐plane NMO velocities make it possible to estimate two anisotropic parameters equivalent to Thomsen’s coefficient δ for transversely isotropic media.

Study of out-of-plane effects in the inversion of refraction/wide-angle reflection traveltimes

In the presence of three-dimensional (3D) inhomogeneous structure, the results of 2D traveltime inversion will be in error since the effects of ‘out-of-plane’ or 3D ray sampling are ignored. We have inverted synthetic data using 2D and 3D algorithms to examine the errors caused by 3D inhomogeneities which produce significant out-of-plane ray bending. The results of inverting data from 2D experiments are compared with vertical slices through 3D models obtained by inverting data using a number of recently employed 3D recording geometries. Our results show that, even for strong 3D inhomogeneities, out-of-plane effects are relatively small, with crustal velocity errors of less than 0.15 km/s, and intra-crustal boundary depth errors generally less than 2 km. These errors are approximately equal to the uncertainties commonly assigned to crustal models derived from traveltime inversion. The artifacts are also similar in magnitude to the lateral smearing that occurs in 3D models when using relatively coarse 3D geometries. Only for a dense network of profiles will a 3D inversion using off- and in-line data provide greater lateral resolution than a 2D independent or simultaneous inversion of in-line data along each profile. 2D and 2.5D inversion of crooked-line data in the presence of strong velocity variations produces erroneous small-scale velocity structure. We conclude that most 3D crustal experiments cannot be justified on the basis that the results from a 2D experiment or a network of 2D profiles will be significantly in error due to out-of-plane effects. 3D experiments can be justified when a dense grid of shots and receivers is used or if a volume image, as opposed to a cross-sectional image, is required.