Anisotropy 2000:

Fractures, Converted Waves, and Case Studies

Edited by L. Ikelle and A. Gangi


“This volume contains 25 papers that represent most of the best work in seismic anisotropy in 1998 and 1999. Fracture characterizations and processing of converted waves are the two main topics covered in this volume. They are addressed from both theoretical and practical viewpoints. Also included are papers describing the historical roots of seismic anisotropy.”

    1. Page 1

      The beginning of research in seismic anisotropy can be fixed precisely. When Maurycy Pius Rudzki assumed his duties as the first “Professor of Geophysics” at the Jagiellonian University in Cracow in early 1896, he stated that his research would be directed at seismology, and primarily at the propagation of seismic waves in anisotropic media. During the next 20 years he published regularly on the subject. Five major papers on different aspects of seismic anisotropy deserve to be studied even today. A year before his untimely death in 1916 he spelled out plans for future work. Several of his suggestions are now under discussion.

    2. Page 13

      Following Fermat's principle, the time during which the light covers its trajectory from a given point A to another given point B satisfies the condition

      where N denotes the index of refraction. By transformation, which is unnecessary to restate here3, one writes the above equation in this form:

      from which results the well-known differential equations4

      All of the above relate to isotropic media. In anisotropic media, N no longer denotes the index of refraction. In such media, one must distinguish between the speed of the propagation of light in the direction of the ray s and the speed in the direction normal to the wave surface q5. The index of refraction is inversely proportional to q, whereas N is inversely proportional to s. There is more; N depends also on direction, and it is a function of not only x, y, z but also of direction cosines

      If, for convenience, we denote

      the total variation of N becomes

      The three direction cosines λ, μ, v are subject to the condition

      however, it is unnecessary to introduce this additional condition, since the above relationship results simply from the equality

      which we use in the transformation of the integral (I).

      Let us calculate the variation of this integral under the assumption that N depends not only on coordinates, but also on direction cosines. With fixed limits, we obtain

      We shall transform the second (sic)6 integral in the same fashion as in the case of isotropic media, namely, by virtue of equalities

      The second integral (sic) becomes

  1. Page 21
    1. Page 21

      This paper argues that the pervasive distributions of closely-spaced stress-aligned fluid-saturated microcracks in almost all rocks are a critical system close to fracture criticality and loss of shear strength. New evidence includes three examples in which observations and modelling directly imply non-linear interactive critical systems with some form of self-organised criticality (SOC). These are a direct calibration of anisotropic poro-elasticity (APE) by monitoring and modelling the response of a reservoir to a high-pressure injection. Monitoring and modelling velocity and attenuation dispersion in a rock physics laboratory. Monitoring the effect of the build-up of stress before earthquakes and volcanic eruptions, including the successful stress forecast of the time and magnitude of an ML=5 (MS ≈ 6) earthquake in southwest Iceland.

      These new results from three very different fields strongly suggest that the earth's crust is a critical interactive non-linear system with self-organised criticality (SOC). Some effects are subtle and easily ignored. Others are so common and familiar that we have developed one-off explanations in terms of conventional deterministic physics to describe their behaviour and occurrence. We suggest that the identification of the sub-critical physical processes is one reason for the success of APE-modelling.

      Recognition of (crack) criticality leads to a new understanding of low-level (pre-fracturing) deformation that has massive implications for almost all dynamic processes in the crust. These include reservoir characterisation, hydrocarbon recovery, monitoring the progress of fluid-fluid fronts, and the build-up of stress before fracturing, faulting, and earthquakes, and the movement of magma before volcanic activity. The implications will be discussed and the arguments presented.

    1. Page 49

      A 3D-4C ocean bottom seismic data set, originally shot over the Valhall field (North Sea) to image through a gas cloud, was used to investigate the detection of azi-muthal anisotropy through the shear-wave splitting phenomenon. The data were processed in converted wave mode (C-waves), assuming that the conversion occurs at the reflection point and involves downgoing P-waves and upgoing S-waves. In this preliminary study two key horizons were used, the early-Miocene at about 2.8 seconds and the top chalk at about 5.5 seconds. The detection technique involves azimuth dependent analysis of radial-to-transverse energy ratios. Preliminary results show that the technique is capable of measuring azimuthal anisotropy effects despite a cross-spread acquisition geometry, where azimuth and offset are poorly sampled. The results show that the principal directions of anisotropy for both formations are close and coincide with the direction of the main fault system. In addition, due to the significant magnitude of the shear-wave splitting measured at the top chalk level, the converted-wave data show vastly improved resolution when oriented in fast and slow modes.

    2. Page 69

      Prestack depth migration is widely used as the best tool for imaging and velocity model QC and refinement, but is mostly applied isotropically, even when there is evidence of anisotropy. We outline a methodology of model-building and imaging for use in anisotropic contexts and apply it to a line of data from offshore West Africa dominated by massive shales. The results show that the anisotropy in the shale increases with depth, along with the velocity. The prestack migrated data are sufficiently sensitive to the anisotropy that a thin layer of sand showing little or no anisotropy can be detected, and they must be included in the model to get flat common image gathers everywhere. This suggests that the use of anisotropy as a lithology-discriminating attribute may be feasible on a depth scale of only a few wavelengths. In any case anisotropy determination is important for precise application of other lithoseismic methods such as AVO.

    1. Page 77

      Abstract. We analyze amplitude variations with offsets and azimuths (AVO-A) of an anisotropic half-space for P-P, P-SV and P-SH scattering. When the overburden is assumed to be isotropic, the AVO-A of each of these three scattering modes can be cast in terms of a Fourier series of azimuths,φ, as follows:

      View larger version:

      where, and are the functions that describe the seismic amplitude variations with offsets (AVO) for a given azimuth. The forms of AVO functions are similar to those of classical AVO formulae; for instance, the AVO functions corresponding to the P-P scattering mode can be interpreted in terms of the intercept and gradient, although the resulting numerical values can differ significantly from those of isotropic cases. One of the benefits of describing the AVO-A as a Fourier series is that the contribution of amplitude variations with azimuths (AVAZ) is distinguishable from that of AVO. The AVAZ is characterized by the functions {1,cosφ,sinφ,cos2φ,sin2φ,cos3φ, sin3φ, cos4φ, sin4φ}, which are mutually orthogonal. Thus, the AVO-A inversion can be formulated as a series of AVO inversions in which the AVO behaviors are represented by the functions F0,Fn,Gn.

      When the coordinate system of seismic acquisition geometry coincides with the symmetry planes of the rock formations, the series corresponding to P-P and P-SV simplify even further; they reduce to for F0 azimuthally isotropic symmetry and to F0, F2 and F4 for orthorhombic symmetry. The series corresponding to P-SH scattering is null for azimuthally isotropic symmetry and reduces to G2 and G4 for orthorhombic symmetry. Unfortunately, the coordinate system of seismic acqusition geometry rarely coincides with the symmetry planes of the rock formations; therefore, the other terms are rarely null. In particular, the functions F1 and G1 become important for P-SV and P-SH scattering because they are affected by the asymmetry of the P-S reflection. These functions are null for P-P scattering, irrespective of the symmetry planes.

      The potential ambiguity between heterogeneity and anisotropy (which can be due to a dipping interface between the isotropic and anisotropic half-spaces) in the recontruction of elastic parameters from the AVO functions F0, Fn, and Gn is also discussed, as well as the sensitivity of these AVO functions to properties of fractured rock formations, including their fluid saturation.

    2. Page 107

      This paper investigates the use of seismic anisotropy and amplitude variation with offset and azimuth (AVOA) for fracture characterisation. Specifically the aim of this work is to provide links between rock and fracture properties, elastic modelling and the interpretation of seismic signatures to reduce the potential ambiguity when interpreting AVOA data. Analytical expressions and numerical modelling are used to highlight the sensitivity of AVOA to fracture properties. Furthermore, little prior attention has been paid to wave propagation in media with multiple fracture alignments or fractured media with a permeable matrix therefore an investigation of AVOA for these cases is included. P-wave AVOA is of obvious interest since there are more of these data. However converted wave and shear-wave AVOA are also investigated as these may provide additional insight into fracture characteristics. It is shown that P-P and P-S AVOA hold significant information about fracturing but potential ambiguity in the interpretation of these data is observed that could lead to incorrect determination of fracture orientation. This highlights the need for forward modelling with rock properties data to constrain the interpretation. Additionally, shear-wave (S-S) AVOA is shown to exhibit significant azimuthal variations which provide strong indications of fracture orientation but only at near offsets and little insight can be gained into other fracture properties.

    3. Page 145

      Solutions of many kinematic and dynamic problems regarding elastic wave propagation in transversely isotropic (TI) media have been published in the literature. In the case of more general types of anisotropy (e.g., triclinic, monoclinic or even orthorhombic), the solutions of such problems are, in general, much more complicated or simply not known. However, in some special cases the TI solutions can be generalized to the case of more complicated types of anisotropy simply by replacing the anisotropy parameters ε and δ in the TI equations by their azimuthally dependent counterparts, ε(λ) and δ(λ). This transformation is called the Azimuthaly Dependent Anisotropy Parameter Transformation, or, more simply, the ADAPT recipe. The aim of this paper is to discuss the applicability of this transformation to special cases of interest for seismic exploration.

      The study is restricted to qP-waves, constituting the great majority of data acquisition in the field in media exhibiting weak anisotropy, which is a reasonable assumption in most of the sedimentary basins. In this context, the ADAPT recipe can be applied to any 2D kinematic problem in monoclinic media (with a horizontal symmetry plane), even in the presence of the most general type of velocity/density gradient, as long as the gradient vector is confined to the investigated plane. This is successfully checked in a numerical model. When dealing with triclinic media, the recipe is applicable in 1D geometry and if the “unperturbed isotropic” seismic ray from the source to the receiver is symmetric with respect to the vertical direction.

      For dynamic problems the ADAPT recipe is not applicable in theory, except for the transmission of plane qP-waves at the plane interface between two weakly contrasted monoclinic media of moderate anisotropy strength. However, the recipe gives surprisingly good results (e.g., typical amplitude errors smaller than 6%) for the computation of the amplitudes of qP-waves radiated by a point source in a 1D multilayered model.

      The generality of some of these results is striking and is particular convenient for the straightforward adaptation of existing modeling/processing TI codes to more complicated types of anisotropy.

    4. Page 163

      Using the seismic-reflection method, we carry out a 3D study of a stack of horizontal layers characterized by contrasted anisotropic properties (from TI with a vertical axis of symmetry to monoclinic). The inversion proposed here requires two mathematical descriptions of the common mid-point reflection time-distance curve of the anisotropic medium. The first one, called the exact equation, was obtained using the P-wave phase velocity expression for weakly anisotropic media of arbitrary symmetry proposed by Mensch and Rasolofosaon (1997). This equation contains the following unknown parameters: vertical velocity, thickness and anisotropy coefficients. The second equation, also describing the P-wave travel time in the medium, is called the approximate equation. It is a four-parameter mathematical approximation whose deviation from the exact equation is smaller than the seismic time-sampling interval. It contains the measurable parameters needed for the inversion. This equation corresponds to the combination of two hyperbolae tangent at a point (x,t). The physical meaning is that the wavefronts can be described by two tangent circles in a given azimuthal plane. The common mid-point time-distance curve is thus defined by four independent parameters for each azimuth. Thirteen independent values (instead of 16) can be obtained from measurements in four azimuthal orientations (every 45 degrees), since one of the parameters, the zero-offset traveltime, is azimuthally invariant. The inverse problem consists of evaluating the 10 unknown parameters (monoclinic case) of the exact equation using the 13 independent values defined by the four approximate equations. Modeling of synthetic seismic traces in anisotropic media is achieved using the program ANRAY (Gajewski and Psencik 1987) to produce a synthetic data set. On this, the parameter measurements are made using the PSCAN theory (de Bazelaire 1988). The results from the inversion depict a good agreement with true model parameters. Finally, this technique can be applied to a stack of anisotropic media by processing the inversion individually on each of them. To achieve this, we need to find geophysical relationships which can transmit the measurements of the current layer through the upper multilayered stack using Dix-type.

    5. Page 203

      Natural rocks usually contain discontinuities of various forms, which control fluid moment in the subsurface. Discontinuities have a wide range of scales and are a direct consequence of Earth's stress over geological history, such as faults, fractures, joints or cracks, or are due to sedimentary deposition processes, such as contact regions in pore spaces (examples include clay platelet alignment in shales and grain particle alignment in sandstones). In this study, we show that seismic waves are sensitive to fluids contained in rock discontinuities based on P and converted PS AVO analysis. We restrict ourselves to two types of rock discontinuities: shales where clay particle alignment will result in transverse isotropy with a vertical symmetry axis (TIV) and vertical aligned fractures (transverse isotropy with a horizontal symmetry axis or TIH). We establish a quantitative link between fluid saturation and interfacial or fracture compliance and derive simple analytical expressions which link seismic anisotropic measurements to pore and fracture filling fluids. The link between macroscopic parameters such as fracture compliance and physical quantities makes it possible to extract information, such as fluid saturation, from field seismic data. In particular, different behaviors of these parameters may be used to determine whether fractures are dry or saturated. The combined P-and PS-wave AVO analysis can potentially be used to differentiate fluid saturations in fractures and in shales.

    6. Page 223

      A technique is developed for distinguishing between gas and water-filled fractures using P-wave data. The method relies upon combining the azimuthal AVO signature for a fractured reservoir with its interval moveout time. A cross-plot of these attributes provides a robust and simple indicator, irrespective of the offset and azimuth distribution at the bin location. The technique is applicable to reservoirs whose production is assisted by the presence of natural fractures, where saturation maps can help to better quantify and assess pathways for flow.

    7. Page 239

      Over the years, amplitude-variation-with-offset (AVO) analysis has proved its usefulness in exploration of oil and gas reservoirs. However, the model conventionally used to interpret AVO anomalies - a single isolated interface between two isotropic half-spaces is often too simplistic. Here, I examine what can be obtained from AVO responses for a significantly more complicated reservoir model - a stack of plane azimuthally anisotropic layers. This model can be used to simulate AVO signatures over finely layered fractured reservoirs.

      I describe a processing technique which takes seismic data in the frequency - slowness (u) — p) domain, properly corrects it for slant wave propagation in a finely layered medium, and produces an instantaneous AVO intercept and an azimuthally varying AVO gradient as functions of vertical traveltime. Ideally, an obtained AVO intercept and gradient are those which would be recorded in the case of isolated interfaces and in the absence of interference between closely spaced reflections. I use an azimuthally dependent AVO gradient to obtain an instantaneous AVO azimuth, which corresponds to the direction of the greatest AVO gradient. In fractured reservoirs, the AVO azimuth is related to the orientation of vertical cracks. Therefore, fracture characterization is one of the potential applications of the described technique.

      I perform a numerical study to examine the stability of the azimuthal AVO with respect to errors in the velocity model of the reservoir, inaccuracies in wavelet estimation, and random noise in the data. The results of numerical examples indicate that azimuthal AVO is reasonably stable and suggest the possibility of detecting principal directions of azimuthal anisotropy in layers which are thinner than half of the dominant seismic wavelength.

    8. Page 257

      We investigate the effects of fracture- and inclusion-induced seismic anisotropy in a carbonate reservoir rock and the resulting influence this anisotropy may have on surface seismic data. Whole-core velocity measurements made on a carbonate sample from the Gulf of Mexico show evidence of elastic anisotropy. Constraints on the style of this anisotropy are obtained from comparisons with effective medium modeling. The core exhibits monoclinic symmetry, which is interpreted as being caused by the combined effects of vertically-aligned drilling-induced fractures and oriented ellipsoidal vugs inclined at an of angle roughly to the vertical. The in situ anisotropy is believed to be orthorhombic, as there is evidence of natural fractures oriented orthogonally to the vugs. Surface seismic modeling is used to investigate amplitude variations with offset and azimuth (AVOA) effects due to such anisotropy. Our model is somewhat hypothetical, but consistent with velocities from the reservoir logs and the inferred in situ anisotropy. Our results suggest that for this model, P-wave AVOA will show significant azimuthal variation only at far offsets (near critical reflections). In fact, the onset of critical reflections will be dependent on the orientation of the seismic line with respect to the fracture direction. Shear modes will be more sensitive to fracture orientation at near offsets. In addition, we find that the P-P and P-S AVOA are sensitive to the presence of aligned vuggy porosity and so could provide a tool for identifying highly productive zones where fractures connect vugs.

    1. Page 271

      Numerical implementation of multicomponent true-amplitude Kirchhoff imaging in two and one-half dimensional v(x,z) media, requires the computation of several ray quantities: traveltimes, slowness, polarization vectors, and geometrical spreading. The purpose of this paper is to show that all of these weighting (Green's) functions can be computed by means of the nonhyperbolic traveltime formula of (Tsvankin and Thomsen, 1994) and (Hake et al., 1984) without any need of ray-tracing, in the case of vertically stratified VTI media. Such models are common in hydrocarbon exploration contexts, with a classical example being that of stratigraphic trap located within a horizontally layered sedimentary sequence. By means of nonhyperbolic traveltime function, we derive explicit approximations for the elastic Green's functions, applicable for multi-component true-amplitude Kirchhoff time and depth migration in vertically inhomogeneous VTI media. Numerical tests with synthetic data from 2D isotropic model show that the images computed by multicomponent true-amplitude Kirchhoff depth migration using explicit approximations of the Green's function compare well with results obtained based on dynamic ray-tracing. We have also succesfully applied the new elastic true-amplitude migration algorithm on 2D-4C OBC seismic data. The main advantage of the method is numerical efficiency rather than high accuracy. It may be of interest for prestack imaging of large data volumes and fast computation of prestack images of multicomponent data. Moreover, the proposed approach is well suited for migration velocity analysis for converted and non-converted waves in anisotropic media, where fast and repeated prestack migration is desired.

    2. Page 291

      Conventional velocity analysis, developed for pure reflection modes recorded on common-midpoint (CMP) gathers, usually cannot be directly applied to converted (PS ) waves. The problems are caused by such inherent features of PS-data as the asymmetry of PS-wave moveout in CMP geometry, polarity reversal at small offsets associated with the vanishing PS-wave reflection coefficient, and reflection-point dispersal. Whereas the moveout asymmetry precludes application of the conventional hyperbolic moveout equation, the polarity reversal reduces the accuracy of velocity-analysis methods based on coherency measures. Here, we propose a velocity-analysis technique for converted waves that overcomes some of those problems. The key idea of our method is to re-sort PS-wave data in such a way that the reflection traveltime becomes symmetric al in the vicinity of a chosen source-receiver offset. Since the traveltime at this offset typically has a minimum, we call our procedure "re-sorting to the traveltime minimum" (RTM) and the corresponding gather - the RTM gather. An important advantage of this approach is that the re-sorting algorithm operates only with the slopes of a selected reflection event and is fully independent of the velocity model. Moreover, RTM gathers can be built just for source-receiver pairs sufficiently removed from the area of polarity reversal and low reflection amplitude.

      Since the PS-wave moveout of RTM gathers is locally symmetric, it can be flattened by the conventional hyperbolic moveout equation. The normal-moveout (NMO) velocity on RTM gathers depends on the velocity structure of the subsurface and therefore can be used to estimate the medium parameters. Then the subsurface model can be reconstructed by migrating -data, which solves the problem of reflection point dispersal. As an example of inverting PS -wave NMO velocities in RTM geometry, we perform parameter estimation for a homogeneous VTI (transversely isotropic with a vertical symmetry axis) layer above a dipping reflector. The results show that the traveltimes of the P-, PSV-, and PSH -waves reflected from the dipping interface can be inverted for all five medium parameters (including the P and S wave vertical velocities) and the dip and depth of the reflector.

    1. Page 311

      Transverse isotropy with a vertical symmetry axis (VTI media) is the most common anisotropic model for sedimentary basins. Here, we apply P-wave processing algorithms developed for VTI media to a 2-D synthetic data set generated by a finite difference code. The model, typical for the Gulf of Mexico, has a moderate structural complexity and includes a salt body and a dipping fault plane. Using the Alkhalifah-Tsvankin dip-moveout (DMO) inversion method, we estimate the anisotropic coefficient η responsible for the dip dependence of P-wave NMO velocity in VTI media. In combination with the normal-moveout (NMO) velocity from a horizontal reflector [Vnmo(0), the argument “0” refers to reflector dip], η is sufficient for performing all P-wave time-processing steps, including NMO and DMO corrections, prestack and poststack time migration. The NMO (stacking) velocities needed to determine Vnmo(0) and η are picked from conventional semblance velocity panels for reflections from subhorizontal interfaces, the dipping fault plane and the flank of the salt body. To mitigate the instability in the interval parameter estimation, the dependence of Vnmo(0) and η on the vertical reflection time is approximated by Chebyshev polynomials with the coefficients found by “global” fitting of all velocity picks.

      We perform prestack depth migration for the reconstructed anisotropic model and two isotropic models with different choices of the velocity field. The anisotropic migration result has a good overall quality, but reflectors are mispositioned in depth because the vertical velocity for this model cannot be obtained from surface -wave data alone. The isotropic migrated section with the NMO velocity Vnmo(0) substituted for the isotropic velocity also has the wrong depth scale and is somewhat inferior to the anisotropic result in the focusing of dipping events. Still, the image distortions are not significant because the parameter η, which controls NMO velocity for dipping reflectors, is rather small (the average value of η is about 0.05). In contrast, the isotropic section migrated with the vertical velocity has a poor quality (although the depth of the subhorizontal reflectors is correct) due to the fact that in VTI media Vo can be used to stack neither dipping nor horizontal events. The difference between vo and the zero-dip stacking velocity Vnmo(0) is determined by the anisotropic coefficient δ, which is greater than η in our model (on average δ ≈ 0.1).

    2. Page 327

      Processing of deep-offshore seismic data acquired with long streamers involves appropriate algorithms that take into account non-hyperbolic shape of the reflection curves and the inadequacy of the standard dip moveout (DMO). The seismic signatures of the effects corresponding to bedding or VTI anisotropy (Vertical Transverse Isotropy) are indistinguishable. Since the shifted hyperbola normal moveout (NMO) equation seems to describe perfectly both effects, we propose to parameterize them with the VTI anisotropy parameter η. In the case of the layered VTI model, our shifted hyperbola moveout correction using the actual effective values of NMO velocity Vnmo and anellipticity η produces a better result than Alkhalifah's VTI NMO.

      The DMO operator derived from the anelliptic shifted hyperbola equation fits quite accurately to the monotonic part of Alkhalifah VTI DMO operators. This anelliptic dip moveout depends on the effective η only. The effective η values seem to be connected to the non-hyperbolic velocity analysis occurring at the same levels. We consider that the anelliptic time processing improves the standard one for deep offshore data acquired over an anisotropic subsurface.

    3. Page 333

      Travel-time computation is efficiently achieved through direct numerical resolution of the eikonal equation on a 3D computational grid. Efficiency may even be improved when solving the eikonal equation using an appropriate transformation called transformation in the celerity domain. Neglecting the effects of anisotropy in the calculation of traveltime maps leads inevitably either to incorrect focusing or positioning of seismic events during the seismic imaging process. Thus, in the present paper, we propose to extend effi-cient P-wave travel-time computation to the simplified case of Vertical Transverse Isotropic (VTI) media, which is nevertheless of the most interest in surface reflection seismic. Based on an “acoustic” explicit eikonal equation for VTI media, the calculation of quasi-P travel times with our numerical resolution method in the celerity domain proves to be particularly efficient, stable and accurate. In line with seismic contractor needs, we propose a parameterization, which is based on anisotropy effects, for the quasi-P VTI model used by the eikonal solver. The three independent anisotropic parameters are the NMO velocity associated with short-spread curvatures, an ellipticity parameter equivalent to the ratio between vertical and NMO velocities, and finally the well-known anellipticity parameter η, governing the long offset behavior of NMO curves.

    4. Page 339

      Observations of polarisations and slownesses at downhole receivers in walkaway or multioffset VSP experiments can in principle be inverted to give the anisotropic seismic velocity of the earth in the vicinity of the receiver array. In practice, such inversions have been only rarely reported, possibly due to a general lack of confidence in polarisation measurements. We apply such an inversion to the direct P-wave arrivals from a 3D VSP dataset acquired at the Oseberg field in the Norwegian North Sea. The large size of this dataset ensures that the result is more stable than those from comparable inversions applied to standard 2D datasets. Comparison with the result from the more established, but more restricted, method of inversion of surface and receiver slownesses showed a large discrepancy. Further investigation revealed that this is due to acquisition-related coherent noise on the estimated surface slownesses; there is no corresponding effect on the polarisation inversion because each shot is treated independently. We therefore prefer the result from the polarisation inversion in this case.

    5. Page 349

      In TI media, exact phase-velocity equations ν(θ) are complex and difficult to exploit, in particular when derivatives with respect to some parameters have to be calculated. Moreover, traveltimes are not equally sensitive to the different parameters in the phase equations. Thus, to find simpler but still accurate equations for ν(θ), with relevant (sensitive) parameters, we examine different formulations found in the literature, and we also derive new original approximations of the phase-velocity equations. Thomsen's weak-anisotropy approximation yields simple formulas but breaks down for moderate but realistic anisotropy. We derive an approximation we call weak-anisotropy-squared approximation, which is more accurate than the former one (it respects horizontal velocities) but is still valid only for weak-anisotropy. Muir's double elliptical approximation is quite accurate, but very difficult to exploit. We derive a new empirical approximation based on Alkhalifah's ideas. This approximation is as accurate as Muir's approximation but requires fewer parameters, and it has a form allowing “easy” computation, in particular, the calculation of derivatives. Moreover, this approximation is valid for a much wider range of anisotropy parameters than the weak-anisotropy approximation. Thus we suggest using this approximation for ray-tracing and reflection tomography purposes.

  2. Page 363
    1. Page 363

      Modeling and inversion of seismic signatures in azimuthally anisotropic media is of primary importance in seismic characterization of fracture networks. The formalism developed here provides a convenient way for studying the behavior of group (ray) velocity and polarization in models with orthorhombic symmetry.

      The expressions for the group-velocity and polarization vectors become particularly simple in the coordinate system associated with the vertical plane that contains the phase-velocity vector. For example, the two “in-plane” components of the exact group-velocity vector can be obtained from phase velocity using the well-known equations for transversely isotropic media with a vertical symmetry axis (VTI). Due to the presence of azimuthal velocity variation, the group-velocity vector acquires an “out-of-plane” component that also has a concise analytic representation.

      To understand the influence of the anisotropy parameters on the orientation of the group-velocity and polarization vectors, we derive linearized weak-anisotropy approximations based on Tsvankin's notation for orthorhombic media. Despite the influence of azimuthal anisotropy, the relationship between the -wave group-velocity and polarization directions in orthorhombic media is similar to that in TI media. The group-velocity and polarization vectors deviate from the slowness vector in the same direction (both in the vertical plane and azimuthally) and usually are close to each other; in this sense, P-wave polarization in orthorhombic media of geologic significance is almost “isotropic.” This conclusion is in agreement with existing numerical results and is further verified here by modeling for orthorhombic media with substantial anisotropy.

    2. Page 383

      Many experimental studies on the elastic properties of rocks have unambiguously established a certain number of robust results. These are the most important evidences:

    3. Page 399

      A common approach to generating fourth-order elastic-moduli tensors for linear, anisotropic elastic media is to start with the most complex case — the least symmetric case — of triclinic symmetry, in which all 81 elastic constants are non zero and 21 of them are independent. Then the relationships among the 21 independent elastic constants are determined for the higher-symmetry (less anisotropic) cases by using those coordinate transformations which, for each symmetry class, leave the material invariant. The most symmetric (least anisotropic) case is the isotropic case for which there are only two independent moduli and only 15 non zero components for the elastic-modulus tensor. Here, we take an alternate approach and start with the isotropic case which requires only two independent fourth-order tensors. We add fourth-order tensors which satisfy the lower-symmetry classes by inspection, continually increasing the number of independent con-stants as the symmetry decreases (or anisotropy increases). The selection of the fourth-order tensors which satisfy the more-anisotropic cases are shown to be easily determined and they are not restricted; that is, there are many additional fourth-order tensors that can be used to satisfy the higher-order anisotropy, but they are all related to each other and the fourth-order tensors previously used for the more-symmetric cases. For example, for the cubic-symmetry case, only one new independent elastic constant is needed and only one new independent fourth-order tensor is required compared to the isotropic case. All other possible fourth-order tensors that satisfy cubic symmetry can be composed of linear combinations of the new one and the two original ones for the isotropic case. In addition, we show that we can generate elastic-moduli tensors which are orthogonal under the double-dot product; these sets of tensors simplify the inversion of stiffness tensors into compliance tensors, and vice versa.

    4. Page 409

      The morphology of rays and wavefronts generated by a point source in a vertically heterogeneous TI elastic medium is studied. The qSV rays form two conoidal caustics with vertices at the source. A generic ray touches 0, 1 or both caustics successively. In the latter case the KMAH indices cancel. The sum of the initial phase and the phase shifts at caustics along each ray outside the caustics is 0. In the region between the caustics, the phase shifts are 0, 0 and π/2.

    5. Page 419

      Abstract It is well known from asymptotic ray theory that the complex wave amplitude along a ray undergoes a phase shift every time the ray touches a caustic surface. If the slowness surface associated with the propagation of this ray is not convex, which is possible for quasi-shear waves in anisotropic elastic media, the phase shift may be opposite in sign from the isotropic case. We present a simple explanation for this phenomenon based on the relation between slowness surfaces and the delay time function. We demonstrate that both the approaching and the receding wavefronts are convex toward the caustic in the anomalous case. This is exactly the opposite of the more familiar isotropic case. Since the magnitude of the phase shift is π/2 in both cases, neglecting the possibility of an anomalous phase shift may introduce a phase error of ±π in the calculated wavefield. Nonconvex quasi-S slowness surfaces frequently arise in naturally occurring anisotropy, and curved reflecting boundaries and other heterogeneities lead to a profusion of caustics. Therefore, encountering this problem is quite likely in practical exploration geophysics using shear-wave imaging.

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