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Abstract A medium (or a region of a continuum) is called anisotropic with respect to a certain parameter if this parameter changes with the direction of a measurement. If an elastic medium is anisotropic, seismic waves of a given type propagate in different directions with different velocities. This velocity anisotropy implies the existence of a certain structure (order) on the scale of seismic wavelength imposed by various physical phenomena. In typical subsurface formations, velocity changes with both spatial position and propagation direction, which makes the medium heterogeneous and anisotropic. The notions of heterogeneity and anisotropy are scaledependent, and the same medium may behave as heterogeneous for small wavelengths and as anisotropic for large wavelengths (e.g., Helbig, 1994 ). For example, such smallscale heterogeneity as fine layering detectable by well logs may create an effectively anisotropic model in the longwavelength limit. Anisotropy in sedimentary sequences is caused by the following main factors (e.g., Thomsen, 1986 ): intrinsic anisotropy due to preferred orientation of anisotropic mineral grains or the shapes of isotropic minerals thin bedding of isotropic layers on a scale small compared to the wavelength (the layers may be horizontal or tilted) vertical or dipping fractures or microcracks nonhydrostatic stress It is common to see anisotropy produced by a certain combination of these factors. For instance, systems of vertical fractures may develop in finely layered sediments, or the thin layers themselves may be intrinsically anisotropic. As a result, subsurface formations may possess several anisotropic symmetries, each with a different character of wave propagation (subsection 1.1.4 ).
Influence of anisotropy on pointsource radiation and AVO analysis
Abstract The anisotropic structure of the stiffness tensor influences not only traveltimes of seismic waves but also their polarizations and amplitudes. As follows from the Christoffel equation (1.10) , the polarization vectors of plane waves in the presence of anisotropy are not strictly parallel or perpendicular to the propagation (slowness) or ray direction. This suggests that polarization directions of body waves with realistic curved wavefronts should also be distorted by anisotropic velocity fields. Analysis of both polarization and amplitude anomalies, however, requires going beyond planewave theory and investigating 3D wavefronts generated by such sources of seismic energy as a concentrated force, explosion, and dislocation. The first section of this chapter gives a description of pointforce radiation patterns and bodywave polarizations in anisotropic media. Numerical analysis based on the evaluation of FourierBessel (Weyltype) integrals is supported by a closedform solution for the farfield displacement derived using the stationaryphase approximation. Both analytic and numerical results show that the amplitudes of Pwaves and, in particular, Swaves in anisotropic media may be substantially distorted by focusing and defocusing of energy, usually associated with maxima and minima (respectively) in the angledependent velocity. The second section is devoted to the basics of amplitudevariationwithoffset (AVO) analysis in transversely isotropic media with a vertical symmetry axis (VTI). The dependence of reflection coefficients on the incidence angle is quite sensitive to the presence of anisotropy on either side of the interface. Furthermore, if the overburden contains anisotropic (e.g., shale) layers, AVO signatures are also influenced by the amplitude focusing phenomena above
Abstract Reflection traveltimes (moveout) provide the most reliable information for building velocity models using surface seismic data, in both isotropic and anisotropic media. If the medium is anisotropic, an attempt to fit the traveltimeoffset relationship using a purely isotropic velocity field may lead to misstacking of reflection events and distortions in seismic images (see examples in chapters 6–8). Hence, understanding of the influence of anisotropy on the kinematics of reflected waves is of primary importance in seismic velocity analysis and processing. Moveout of pure (nonconverted) modes on commonmidpoint (CMP) gathers is conventionally approximated by the Taylor series expansion near the vertical (e.g., Taner and Koehler, 1969 ): where x is the sourcereceiver offset, and the coefficients are given by t 0 is the twoway zerooffset traveltime. Equation (3.1) does not include odd powers of x because CMP moveout of pure modes is symmetric with respect to zero offset (i.e., it remains the same when one interchanges the source and receiver). Later on we will replace the Taylor series (3.1) with a more accurate approximation that still includes the moveout coefficients A 0 , A 2 , and A 4 . The moveout parameter of most practical significance in exploration is the normalmoveout (NMO) velocity V nmo , responsible for the hyperbolic moveout on conventionallength spreads that do not exceed the distance between the CMP and the reflector: If the traveltime is plotted in the t 2 – x 2 coordinates, the factor 1/ V nmo 2 determines the initial slope of the moveout curve. With increasing offset, the t 2 ( x 2 ) curve deviates from a straight line due to
Abstract Chapter 3 was devoted to the influence of anisotropy on normalmoveout velocity, which controls reflection traveltimes on spreadlengths typically limited by the distance between the CMP and the reflector. If the medium is anisotropic or heterogeneous, hyperbolic moveout equation parameterized by NMO velocity loses accuracy with increasing offset. Angledependent velocity makes reflection moveout nonhyperbolic even in a single homogeneous layer, unless the anisotropy is elliptical. Anisotropyinduced deviations from the shortspread hyperbola for a typical VTI model (Taylor Sandstone, Figure 4.1 ) are illustrated by Figures 4.2 – 4.4 . If the layer were isotropic (∈ = δ = 0), the moveout would be purely hyperbolic for any sourcereceiver offset. The residual moveout in Figure 4.2 is calculated as the difference between the exact traveltimes and the bestfit hyperbola found by the leastsquares method. Because the spreadlength x max is limited by the reflector depth z , the residuals for both P and SVwaves at all offsets are small (less than 2 ms). The error of the bestfit hyperbola, however, rapidly increases with spreadlength and for the SVwave reaches 60 ms when x max = 2 z (Figure 4.3 ). It is important to notice that even if the time residuals are small, the finitespread (effective) moveout velocity of the bestfit hyperbola may differ from the NMO velocity (Figure 4.4 ). Whereas V nmo is determined analytically in the zerospread limit, the finitespread moveout velocity is obtained by fitting generally nonhyperbolic data with a hyperbolic function. For instance, the
Abstract With recent advances in the acquisition of multicomponent data, including the technology of ocean bottom surveys, converted waves find an increasing number of applications in seismic exploration. For example, PSwaves help in imaging hydrocarbon reservoirs beneath gas clouds, where conventional Pwave methods suffer due to the high attenuation of compressional energy ( Granli et al., 1999 ; Thomsen, 1999 ). Also, converted waves provide information about shearwave velocities (including the ratio of the P and Swave vertical velocities; see Gaiser, 1996 ) and other medium parameters that cannot be constrained using Pwave data alone. This advantage of mode conversions becomes especially important in anisotropic media due to the large number of unknown parameters and the ambiguity in estimating reflector depth from surface Pwave data. It should be emphasized that the influence of anisotropy on PSwave moveout and amplitude is usually more significant than that on Pwave signatures, and isotropic processing methods often fail to produce accurate convertedwave images. A key difference between converted and pure reflections in commonmidpoint (CMP) geometry is that mode conversion can make the moveout curve asymmetric with respect to zero offset (i.e., traveltime is no longer an even function of offset). Only in the special case of horizontal reflectors and a medium with a horizontal symmetry plane, does the convertedwave (e.g., PSwave) reflection traveltime remain the same if we interchange the source and receiver. The asymmetry of the convertedwave moveout can be further enhanced by angular velocity variations in anisotropic media. Therefore, in general the
Abstract The most critical step in extending seismic inversion and processing to anisotropic media is to identify the medium parameters responsible for measured reflection signatures. In chapter 1 it was demonstrated that Thomsen notation makes it possible to reduce the number of parameters that govern Pwave kinematics in VTI media from four to three ( V P 0 , ∈, and δ). Therefore, if amplitude preservation is not an issue, for purposes of Pwave depth imaging (e.g., prestack depth migration) one needs to reconstruct the spatial distributions of those three parameters. The subject of this chapter is timedomain Pwave processing, which includes normalmoveout (NMO) and dipmoveout (DMO) corrections, prestack and poststack time migration. As shown here, for laterally homogeneous VTI models above a horizontal or a dipping target reflector, Pwave timedomain signatures depend on the interval values of just two combinations of ( V P 0 , ∈, and δ). Furthermore, both “timeprocessing” parameters can be recovered from Pwave reflection moveout alone, without using borehole information or multicomponent (S or PS) data. According to the results of chapter 4 , longspread Pwave moveout in a horizontal VTI layer is controlled by the vertical traveltime ( t P 0 ) and two medium parameters, one of which is the NMO velocity commonly measured in seismic processing: The other parameter is the anisotropy coefficient η, which quantifies the anellipticity of the medium [equation (4.14) ]: Whereas V nmo 2 (0) is responsible for the initial slope of the t 2 ( x 2 )curve, η determines the deviation of the traveltime function from a hyperbola. If the medium is elliptical (∈ = δ, η = 0),
Abstract The results of the previous chapters clearly demonstrate that anisotropy in general and transverse isotropy in particular have a substantial influence on the traveltimes of reflected waves and their normalmoveout velocities. Because reflection moveout is the main source of information for velocity analysis, conventional (i.e., those based on the assumption of isotropy) algorithms are bound to produce distorted velocity models and images in the presence of anisotropy. For example, as shown in chapter 3 , the difference between stacking (moveout) and vertical velocities in anisotropic media leads to misties in timetodepth conversion and the wrong depth scale of seismic sections. In addition, anisotropy causes serious difficulties in imaging of dipping reflectors, such as fault planes (chapter 6 ; also, see Lynn et al., 1991 , and examples below). Once the importance of anisotropy in seismic processing is accepted, two further impediments to taking its presence into account must be overcome. First, processing algorithms that include anisotropy are more complex than those that ignore it. Second, estimating the anisotropy parameters required by these algorithms is a highly challenging task. The second problem has always seemed to be especially intimidating for exploration seismologists. Even for the relatively simple VTI model treated in this chapter, Pwave propagation is governed by four stiffness coefficients or, alternatively, four Thomsen (1986) parameters (the vertical velocities V P 0 and V S 0 and the anisotropy coefficients ∊ and δ). This would seem to imply that velocity analysis for VTI media will not only be complicated in practice,
Abstract If one does not look for the existence of anisotropy in Pwave data, it can often go unnoticed. Still, where the subsurface is anisotropic (e.g., in the presence of shale formations), conventional Pwave processing based on the assumption of isotropy yields errors in seismic images and interpretations. Such anisotropyinduced distortions as mispositioning of both horizontal and dipping reflectors and misstacking of dipping events were discussed in chapters 3, 6, and 7, and will be further analyzed below. The most critical step in correcting for anisotropy in seismic processing is estimation of the parameters of anisotropic velocity fields. As shown in chapters 6 and 7 , Pwave time processing for VTI media with a laterally homogeneous overburden requires knowledge of two parameters [ V nmo (0) and η], which can be obtained from surface Pwave data. To estimate the vertical velocity V P 0 and build VTI models suitable for depth imaging, Pwave reflection moveout has to be combined with borehole data (e.g., check shots) or traveltimes of modeconverted or pure shear waves (see chapter 7 ). Note that if the overburden contains dipping interfaces or other kinds of lateral heterogeneity, Pwave reflection traveltimes become dependent on the individual values of V P 0 , ∈, and δ and, for a certain class of models, can be used to reconstruct the velocity field in depth ( Le Stunff et al., 2001 ). Once the needed medium parameters have been estimated, conventional isotropic Pwave processing algorithms can be extended to vertical transverse isotropy by using phasevelocity equations introduced in chapter 1 and equations for reflection moveout from chapters 3 and 4 . Here, we describe several efficient dipmoveout (DMO) and migration techniques for VTI media based on wellestablished isotropic imaging algorithms. This discussion of VTI processing is by no means exhaustive, as it does not include, for example, Kirchhoff migration. The main goal of this chapter is to show how analytic results developed in the previous chapters can be applied in seismic processing and to provide practical recipes for devising anisotropic imaging algorithms. Although DMO correction is seldom used in modern seismic processing, it retains a certain relevance because moveout of dipping events provides essential information for anisotropic parameter estimation.
Seismic Signatures and Analysis of Reflection Data in Anisotropic Media, Third Edition
Abstract This is a new edition of Ilya Tsvankin’s reference volume on seismic anisotropy and application of anisotropic models in reflection seismology. Seismic Signatures and Analysis of Reflection Data in Anisotropic Media, Geophysical References Series No. 19, provides essential background information about anisotropic wave propagation, introduces efficient notation for transversely isotropic (TI) and orthorhombic media, and identifies the key anisotropy parameters for imaging and amplitude analysis. To gain insight into the influence of anisotropy on a wide range of seismic signatures, exact solutions are simplified in the weakanisotropy approximation.
Abstract Traveltimes of reflected waves (reflection moveout) in heterogeneous anisotropic media are usually modeled by multioffset and multiazimuth ray tracing (e.g., Gajewski and Pšenčĺk, 1987 ). Whereas anisotropic raytracing codes are sufficiently fast for forward modeling, their application in moveout inversion requires repeated generation of azimuthallydependent traveltimes around many commonmidpoint (CMP) locations, which makes the inversion procedure extremely timeconsuming. Also, purely numerical solutions do not give insight into the influence of anisotropy on reflection traveltimes. This chapter is devoted to analytic treatment of conventionalspread reflection moveout in anisotropic media. For models with moderate structural complexity and spreadlengthtodepth ratios close to unity, traveltimes in CMP geometry are welldescribed by normalmoveout (NMO) velocity defined in the zerospread limit (Tsvankin and Thomsen, 1994; Tsvankin, 2005). Even in the presence of nonhyperbolic moveout, NMO velocity (Vnmo) is still responsible for the most stable, conventionaloffset portion of the moveout curve. The description of Vnmo given here provides an analytic basis for moveout inversion, helps evaluate the contribution of the anisotropy parameters to reflection traveltimes, and leads to a significant increase in the efficiency of traveltime modeling/inversion methods. A theoretical framework for 3D anisotropic moveout analysis was proposed by Grechka and Tsvankin (1998a), who showed that NMO velocity of pure (i.e., nonconverted) modes varies with azimuth as an ellipse, even if the medium is arbitrarily anisotropic and heterogeneous. This conclusion breaks down only for subsurface models in which CMP traveltime does not increase with offset (i.e., the case of reverse moveout) or cannot be described by
In this chapter, the formalism introduced in Chapter 1 is used to develop inversion and processing techniques for conventionalspread Pwave data from azimuthally anisotropic media. Stackingvelocity analysis for wideazimuth 3D surveys often ignores the azimuthal dependence of normal moveout from horizontal reflectors, which may lead to distortions in the processing results ( Lynn et al., 1996 ). A single value of stacking (NMO) velocity applied to the whole CMP gather causes underestimation of V nmo for sourcereceiver azimuths near the “fast” direction of the NMO ellipse and overestimation near the “slow” direction. Hence, mixing of different azimuths impairs the performance of the moveout correction and, therefore, the quality of the stacked section. These distortions can be avoided by reconstructing the bestfit NMO ellipse and applying the correct stacking velocity for all azimuthal directions.
Abstract In conventional seismic data processing, reflection traveltime is often assumed to be described by a hyperbolic equation with the quadratic term determined by NMO velocity. However, the influence of heterogeneity (either lateral or vertical) and nonelliptical anisotropy causes deviations from hyperbolic moveout, which typically become substantial for offsettodepth ratios exceeding unity. Practical difficulties in working with longspread data and insufficient understanding of nonhyperbolic moveout often force seismic processors to mute out the largeoffset portion of the moveout curve. Nonetheless, advances in acquisition technology have significantly increased the range of sourcereceiver offsets in 3D surveys, and nonhyperbolic moveout has proved useful in such applications as anisotropic parameter estimation, suppression of multiples, and largeangle AVO (amplitudevariationwithoffset) analysis. Among the first to recognize the benefits of employing nonhyperbolic moveout in interval anisotropic velocity parameter estimation was Sena (1991) , who developed traveltime approximations for multilayered VTI and HTI media based upon the socalled “skewed” hyperbolic moveout formulation of Byun et al. (1989) . A detailed overview of nonhyperbolic moveout analysis for pure (PP and SS) modes and PSwaves in layered VTI models can be found in Tsvankin (2005) . This chapter discusses the influence of azimuthal anisotropy on nonhyperbolic moveout and introduces an efficient moveoutinversion algorithm for wideazimuth Pwave data from orthorhombic and HTI media. We start with a brief description of nonhyperbolic moveout equations for layered anisotropic models. Deviation from hyperbolic moveout for pure reflected waves is largely governed by the quartic coefficient A4 of the traveltime series t2(x2).
Abstract In the presence of anisotropy, supplementing Pwave reflection data with shear waves is often needed for estimating even the parameter set responsible for Pwave kinematics. For example, reflection moveout of Pwaves in laterally homogeneous VTI media generally constrains only two parameter combinations  the zerodip NMO velocity V nmo, P and the anellipticity parameter η ( Alkhalifah and Tsvankin, 1995 ). As discussed in Chapter δ , addition of reflection traveltimes of SVwaves to Pwave moveout helps resolve the vertical P and Swave velocities and the anisotropy parameters ε and δ , provided the reflector has a mild dip and the data are acquired for a wide range of azimuths. Moreover, joint inversion of P and Swave data can be effectively used in lowersymmetry orthorhombic and monoclinic media (see Chapter 6 ). Therefore, finding practical ways of combining P and Swaves is critically important for anisotropic velocity modelbuilding. In theory, such joint processing algorithms might seem to be easy to implement because the techniques discussed in previous chapters are equally valid for both PP and SS puremode reflections. (Here we use the double indices “PP” and “SS” to emphasize that the downgoing and upgoing segments of ray trajectories correspond to the same wave type.) The applicability of algorithms originally designed for PPwaves to SS data is ensured by the reciprocity of puremode traveltimes with respect to the source and receiver positions. As a result, moveout of any puremode reflection in commonmidpoint (CMP) gathers can be described by the traveltime series
Abstract Continued progress in acquiring and processing highquality multicomponent data has provided clear evidence of the influence of anisotropy on reflection traveltimes and moveout inversion. In particular, conventional isotropic imaging methods routinely produce depth misties between PP and PS (convertedwave) sections (e.g., Nolte et al., 2000 ), which can be removed by joint anisotropic velocity analysis of PP and PS data volumes. In this chapter, PPwave reflection moveout is combined with traveltimes of modeconverted (PS) and shear (SS) waves in parameter estimation for transversely isotropic media. We begin by examining joint inversion of PP and PS (PSV) data for the simple model of a horizontal VTI layer. Although the addition of PS traveltimes makes it possible to obtain the ratio of the vertical velocities of P and Swaves and the shearwave NMO velocity, inversion for the Thomsen parameters and reflector depth remains nonunique, even for uncommonly large spreadlengthtodepth ratios. Reconstruction of the depth scale of horizontally layered VTI models from surface data requires generation of shear waves and recording of wideangle SS reflections, as demonstrated by Tsvankin and Thomsen (1995). In the second section we extend Pwave stackingvelocity inversion (tomography) described in Chapter 2 to the combination of NMO ellipses, zerooffset traveltimes, and reflection time slopes of PP and SSwaves (SS traveltimes can be computed from PP and PS data). Application of the inversion algorithm to a homogeneous TI layer above a dipping reflector shows that for a range of dips and tilt angles of the symmetry axis conventionalspread
Abstract The results of Chapter 5 demonstrate the benefits of combining Pwave data with reflection moveout of shear or modeconverted waves in parameter estimation for TI media. The advantages of multicomponent velocity analysis are even more crucial for lowersymmetry models described by a larger number of independent anisotropy parameters. This chapter is focused on moveout inversion of wideazimuth, multicomponent data from layered orthorhombic (sections 6.16.3) and monoclinic (section 6.4) media. Velocity model building using PP and PS traveltimes is especially attractive because it does not require shearwave excitation. We begin the first section by reviewing the properties of reflection moveout of converted waves for models composed of horizontal layers with a horizontal symmetry plane. As demonstrated in Chapter 1, PS traveltimes for such media are reciprocal with respect to the source and receiver positions and can be described by the conventional t2(x2) series. Furthermore, there exists a simple relationship between the NMO ellipses of pure and converted waves which, combined with the generalized Dix differentiation, makes it possible to compute the interval NMO ellipses of the split shear waves S1S1 and S2S2 from PP and PS (PS1 and PS2) data. Then the interval PP and SSwave NMO ellipses are inverted for the parameters of orthorhombic or monoclinic layers. This algorithm is tested on multicomponent (PP and PS) physicalmodeling data acquired in several azimuthal directions over a block of orthorhombic composite material. The computed symmetryplane NMO velocities of PP and SSwaves are combined with the known reflector depth to estimate eight.
Abstract The previous chapters were devoted to estimation of anisotropy from the kinematics of reflected waves. Extending Thomsen notation to symmetries lower than transverse isotropy made it possible to build realistic models of the azimuthally anisotropic subsurface. The success of the traveltime inversion was ensured by our ability to identify the combinations of the stiffness coefficients that control such key signatures measured from wideazimuth data as the NMO ellipse and nonhyperbolic moveout. Still, because reflection traveltimes are insensitive to velocity and anisotropy variations on a fine scale, moveout inversion produces models that have relatively low spatial resolution. In principle, there are two options for improving the resolution of subsurface velocity fields. One is to include information about seismic amplitudes either directly (see Chapter 8) or via fullwaveform inversion of reflection data (e.g., Tarantola, 1987; Virieux and Operto, 2009). Realizing the potential of fullwaveform inversion, however, largely belongs to the future because only preliminary (albeit encouraging) results for anisotropic media are currently available (Chang and McMechan, 2009). The second option is to make use of acquisition geometries that differ from conventional surface recording. Here, we discuss application of vertical seismic profiling (VSP) geometries in anisotropic parameter estimation and demonstrate their ability to resolve in situ anisotropy with the spatial resolution close to the dominant seismic wavelength. The term “VSP” refers to observations of elastic waves excited by sources located at or near the earth’s surface and recorded by geophones placed in a borehole. A string of such geophones tracks the evolution