Reflection moveout and parameter estimation for horizontal transverse isotropy

Geophysics ◽  
1997 ◽  
Vol 62 (2) ◽  
pp. 614-629 ◽  
Author(s):  
Ilya Tsvankin
Keyword(s):  
Shear Wave ◽  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Shear Waves ◽  
Shear Wave Splitting ◽  
P Wave ◽  
Symmetry Direction ◽  
Wave Splitting ◽  
Hti Media ◽  
Splitting Parameter

Transverse isotropy with a horizontal axis of symmetry (HTI) is the simplest azimuthally anisotropic model used to describe fractured reservoirs that contain parallel vertical cracks. Here, I present an exact equation for normal‐moveout (NMO) velocities from horizontal reflectors valid for pure modes in HTI media with any strength of anisotropy. The azimuthally dependent P‐wave NMO velocity, which can be obtained from 3-D surveys, is controlled by the principal direction of the anisotropy (crack orientation), the P‐wave vertical velocity, and an effective anisotropic parameter equivalent to Thomsen's coefficient δ. An important parameter of fracture systems that can be constrained by seismic data is the crack density, which is usually estimated through the shear‐wave splitting coefficient γ. The formalism developed here makes it possible to obtain the shear‐wave splitting parameter using the NMO velocities of P and shear waves from horizontal reflectors. Furthermore, γ can be estimated just from the P‐wave NMO velocity in the special case of the vanishing parameter ε, corresponding to thin cracks and negligible equant porosity. Also, P‐wave moveout alone is sufficient to constrain γ if either dipping events are available or the velocity in the symmetry direction is known. Determination of the splitting parameter from P‐wave data requires, however, an estimate of the ratio of the P‐to‐S vertical velocities (either of the split shear waves can be used). Velocities and polarizations in the vertical symmetry plane of HTI media, that contains the symmetry axis, are described by the known equations for vertical transverse isotropy (VTI). Time‐related 2-D P‐wave processing (NMO, DMO, time migration) in this plane is governed by the same two parameters (the NMO velocity from a horizontal reflector and coefficient ε) as in media with a vertical symmetry axis. The analogy between vertical and horizontal transverse isotropy makes it possible to introduce Thomsen parameters of the “equivalent” VTI model, which not only control the azimuthally dependent NMO velocity, but also can be used to reconstruct phase velocity and carry out seismic processing in off‐symmetry planes.

Synthesis of multicomponent quasi-P and converted quasi-P-S seismograms for intersecting fracture systems

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1261-1271 ◽  
Author(s):  
Andrey A. Ortega ◽  
George A. McMechan
Keyword(s):  
Shear Wave ◽  
Transverse Isotropy ◽  
Normal Incidence ◽  
Shear Wave Splitting ◽  
P Wave ◽  
Orthorhombic Symmetry ◽  
S Wave ◽  
Stiffness Constant ◽  
Wave Splitting ◽  
Vertical Fractures

Dynamic ray shooting with interpolation is an economical way of computing approximate Green’s functions in 3-D heterogeneous anisotropic media. The amplitudes, traveltimes, and polarizations of the reflected rays arriving at the surface are interpolated to synthesize three‐component seismograms at the desired recording points. The algorithm is applied to investigate kinematic quasi-P-wave propagation and converted quasi-P-S-wave splitting variations produced in reflections from the bottom of a layer containing two sets of intersecting dry vertical fractures as a function of the angle between the fracture sets and of the intensity of fracturing. An analytical expression is derived for the stiffness constant C16 that extends Hudson’s second‐order scattering theory to include tetragonal-2 symmetry systems. At any offset, the amount of splitting in nonorthogonal (orthorhombic symmetry) intersecting fracture sets is larger than in orthogonal (tetragonal-1 symmetry) systems, and it increases nonlinearly as a function of the intensity of fracturing as offset increases. Such effects should be visible in field data, provided that the dominant frequency is sufficiently high and the offset is sufficiently large. The amount of shear‐wave splitting at vertical incidence increases nonlinearly as a function of the intensity of fracturing and increases nonlinearly from zero in the transition from tetragonal-1 anisotropy through orthorhombic to horizontal transverse isotropy; the latter corresponds to the two crack systems degenerating to one. The zero shear‐wave splitting corresponds to a singularity, at which the vertical velocities of the two quasi‐shear waves converge to a single value that is both predicted theoretically and illustrated numerically. For the particular case of vertical fractures, there is no P-to-S conversion of vertically propagating (zero‐offset) waves. If the fractures are not vertical, the normal incidence P-to-S reflection coefficient is not zero and thus is a potential diagnostic of fracture orientation.

3-D seismic modeling for cracked media: Shear‐wave splitting at zero‐offset

Geophysics ◽  
2000 ◽  
Vol 65 (1) ◽  
pp. 211-221 ◽  
Author(s):  
Jaime Ramos‐Martínez ◽  
Andrey A. Ortega ◽  
George A. McMechan
Keyword(s):  
Shear Wave ◽  
Effective Properties ◽  
Transverse Isotropy ◽  
Shear Waves ◽  
Crack Density ◽  
Shear Wave Splitting ◽  
Carbonate Reservoir ◽  
Vertical Crack ◽  
Wave Splitting ◽  
Zero Offset

Splitting of zero‐offset reflected shear‐waves is measured directly from three‐component finite‐difference synthetic seismograms for media with intersecting vertical crack systems. Splitting is simulated numerically (by finite differencing) as a function of crack density, aspect ratio, fluid content, bulk density, and the angle between the crack systems. The type of anisotropy symmetry in media containing two intersecting vertical crack systems depends on the angular relation between the cracks and their relative crack densities, and it may be horizontal transverse isotropy (HTI), tetragonal, orthorhombic, or monoclinic. The transition from one symmetry to another is visible in the splitting behavior. The polarities of the reflected quasi‐shear waves polarized perpendicular and parallel to the source particle motion distinguish between HTI and orthorhombic media. The dependence of the measured amount of splitting on crack density for HTI symmetry is consistent with that predicted theoretically by the shear‐wave splitting factor. In orthorhombic media (with two orthogonal crack systems), a linear increase is observed in splitting when the difference between crack densities of the two orthogonal crack systems increases. Splitting decreases nonlinearly with the intersection angle between the two crack systems from 0° to 90°. Surface and VSP seismograms are simulated for a model with several flat homogeneous layers, each containing vertical cracks with the same and with different orientations. When the crack orientation varies with depth, previously split shear waves are split again at each interface, leading to complicated records, even for simple models. Isotropic and anisotropic three‐component S-wave zero‐offset sections are synthesized for a zero‐offset survey line over a 2.5-D model of a carbonate reservoir with a complicated geometry and two intersecting, dipping crack sets. The polarization direction of the fast shear wave, propagating obliquely through the cracked reservoir, is predicted by theoretical approximations for effective properties of anisotropic media with two nonorthogonal intersecting crack sets.

Variation of P-wave reflectivity with offset and azimuth in anisotropic media

Geophysics ◽  
1998 ◽  
Vol 63 (3) ◽  
pp. 935-947 ◽  
Author(s):  
Andreas Rüger
Keyword(s):  
Reflection Coefficient ◽  
Shear Wave ◽  
Wave Reflection ◽  
Symmetry Plane ◽  
Anisotropic Media ◽  
Shear Wave Splitting ◽  
P Wave ◽  
Reflection Coefficients ◽  
Wave Splitting ◽  
Splitting Parameter

P-wave amplitudes may be sensitive even to relatively weak anisotropy of rock mass. Recent results on symmetry‐plane P-wave reflection coefficients in azimuthally anisotropic media are extended to observations at arbitrary azimuth, large incidence angles, and lower symmetry systems. The approximate P-wave reflection coefficient in transversely isotropic media with a horizontal axis of symmetry (HTI) (typical for a system of parallel vertical cracks embedded in an isotropic matrix) shows that the amplitude versus offset (AVO) gradient varies as a function of the squared cosine of the azimuthal angle. This change can be inverted for the symmetry‐plane directions and a combination of the shear‐wave splitting parameter γ and the anisotropy coefficient [Formula: see text]. The reflection coefficient study is also extended to media of orthorhombic symmetry that are believed to be more realistic models of fractured reservoirs. The study shows the orthorhombic and HTI reflection coefficients are very similar and the azimuthal variation in the orthorhombic P-wave reflection response is a function of the shear‐wave splitting parameter γ and two anisotropy parameters describing P-wave anisotropy for near‐vertical propagation in the symmetry planes. The simple relationships between the reflection amplitudes and anisotropic coefficients given here can be regarded as helpful rules of thumb in quickly evaluating the importance of anisotropy in a particular play, integrating results of NMO and shear‐wave‐splitting analyses, planning data acquisition, and guiding more advanced numerical amplitude‐inversion procedures.

Splitting parameter yield (SPY): A program for semiautomatic analysis of shear-wave splitting

2012 ◽  
Vol 40 ◽  
pp. 138-145 ◽  
Author(s):  
Lucia Zaccarelli ◽  
Francesca Bianco ◽  
Riccardo Zaccarelli
Keyword(s):  
Shear Wave ◽  
Shear Wave Splitting ◽  
Wave Splitting ◽  
Splitting Parameter

Shear‐wave splitting in cross‐hole surveys: Modeling

Geophysics ◽  
1989 ◽  
Vol 54 (1) ◽  
pp. 57-65 ◽  
Author(s):  
Enru Liu ◽  
Stuart Crampin ◽  
David C. Booth
Keyword(s):  
Shear Wave ◽  
Seismic Anisotropy ◽  
Pore Space ◽  
Shear Waves ◽  
Shear Wave Splitting ◽  
Surface Layers ◽  
Near Surface ◽  
Crustal Rocks ◽  
Wave Splitting ◽  
Almost All

Shear‐wave splitting, diagnostic of some form of effective seismic anisotropy, is observed along almost all near‐vertical raypaths through the crust. The splitting is caused by propagation through distributions of stress‐aligned vertical parallel fluid‐filled cracks, microcracks, and preferentially oriented pore space that exist in most crustal rocks. Shear waves have severe interactions with the free surface and may be seriously disturbed by the surface and by near‐surface layers. In principle, cross‐hole surveys (CHSs) should be free of much of the near‐surface interference and could be used for investigating shear waves at higher frequencies and greater resolution along shorter raypaths than is possible with reflection surveys and VSPs. Synthetic seismograms are examined to estimate the effects of vertical cracks on the behavior of shear waves in CHS experiments. The azimuth of the CHS section relative to the strike of the cracks is crucial to the amount of information about seismic anisotropy that can be extracted from such surveys. Interpretation of data from only a few boreholes located at azimuths chosen from other considerations is likely to be difficult and inconclusive. Application to interpreting acoustic events generated by hydraulic pumping is likely to be more successful.

Linear‐transform techniques for processing shear‐wave anisotropy in four‐component seismic data

Geophysics ◽  
1993 ◽  
Vol 58 (2) ◽  
pp. 240-256 ◽  
Author(s):  
Xiang‐Yang Li ◽  
Stuart Crampin
Keyword(s):  
Time Series ◽  
Shear Wave ◽  
Seismic Data ◽  
Shear Waves ◽  
Shear Wave Splitting ◽  
Data Set ◽  
Reflection Data ◽  
Wave Splitting ◽  
Linear Transform ◽  
Zero Offset

Most published techniques for analyzing shear‐wave splitting tend to be computing intensive, and make assumptions, such as the orthogonality of the two split shear waves, which are not necessarily correct. We present a fast linear‐transform technique for analyzing shear‐wave splitting in four‐component (two sources/ two receivers) seismic data, which is flexible and widely applicable. We transform the four‐component data by simple linear transforms so that the complicated shear‐wave motion is linearized in a wide variety of circumstances. This allows various attributes to be measured, including the polarizations of faster split shear waves and the time delays between faster and slower split shear waves, as well as allowing the time series of the faster and slower split shear waves to be separated deterministically. In addition, with minimal assumptions, the geophone orientations can be estimated for zero‐offset verticle seismic profiles (VSPs), and the polarizations of the slower split shear waves can be measured for offset VSPs. The time series of the split shear‐waves can be separated before stack for reflection surveys. The technique has been successfully applied to a number of field VSPs and reflection data sets. Applications to a zero‐offset VSP, an offset VSP, and a reflection data set will be presented to illustrate the technique.

Inversion of azimuthally dependent NMO velocity in transversely isotropic media with a tilted axis of symmetry

Geophysics ◽  
2000 ◽  
Vol 65 (1) ◽  
pp. 232-246 ◽  
Author(s):  
Vladimir Grechka ◽  
Ilya Tsvankin
Keyword(s):  
Wave Velocity ◽  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Model Parameters ◽  
Symmetry Direction ◽  
Time Migration ◽  
Inversion Procedure ◽  
Tti Media

Just as the transversely isotropic model with a vertical symmetry axis (VTI media) is typical for describing horizontally layered sediments, transverse isotropy with a tilted symmetry axis (TTI) describes dipping TI layers (such as tilted shale beds near salt domes) or crack systems. P-wave kinematic signatures in TTI media are controlled by the velocity [Formula: see text] in the symmetry direction, Thomsen’s anisotropic coefficients ε and δ, and the orientation (tilt ν and azimuth β) of the symmetry axis. Here, we show that all five parameters can be obtained from azimuthally varying P-wave NMO velocities measured for two reflectors with different dips and/or azimuths (one of the reflectors can be horizontal). The shear‐wave velocity [Formula: see text] in the symmetry direction, which has negligible influence on P-wave kinematic signatures, can be found only from the moveout of shear waves. Using the exact NMO equation, we examine the propagation of errors in observed moveout velocities into estimated values of the anisotropic parameters and establish the necessary conditions for a stable inversion procedure. Since the azimuthal variation of the NMO velocity is elliptical, each reflection event provides us with up to three constraints on the model parameters. Generally, the five parameters responsible for P-wave velocity can be obtained from two P-wave NMO ellipses, but the feasibility of the moveout inversion strongly depends on the tilt ν. If the symmetry axis is close to vertical (small ν), the P-wave NMO ellipse is largely governed by the NMO velocity from a horizontal reflector Vnmo(0) and the anellipticity coefficient η. Although for mild tilts the medium parameters cannot be determined separately, the NMO-velocity inversion provides enough information for building TTI models suitable for time processing (NMO, DMO, time migration). If the tilt of the symmetry axis exceeds 30°–40° (e.g., the symmetry axis can be horizontal), it is possible to find all P-wave kinematic parameters and construct the anisotropic model in depth. Another condition required for a stable parameter estimate is that the medium be sufficiently different from elliptical (i.e., ε cannot be close to δ). This limitation, however, can be overcome by including the SV-wave NMO ellipse from a horizontal reflector in the inversion procedure. While most of the analysis is carried out for a single layer, we also extend the inversion algorithm to vertically heterogeneous TTI media above a dipping reflector using the generalized Dix equation. A synthetic example for a strongly anisotropic, stratified TTI medium demonstrates a high accuracy of the inversion (subject to the above limitations).

Shear-wave splitting of SK(K)S-phases beneath the Upper Rhine Graben area: Indications for laterally and vertically varying seismic anisotropy

2021 ◽  
Author(s):  
Yvonne Fröhlich ◽  
Michael Grund ◽  
Joachim R. R. Ritter
Keyword(s):  
Shear Wave ◽  
Seismic Anisotropy ◽  
Symmetry Axis ◽  
Shear Wave Splitting ◽  
Small Scale ◽  
Upper Rhine Graben ◽  
Event Analysis ◽  
Horizontal Symmetry ◽  
Wave Splitting ◽  
Upper Rhine

<p>The observed backazimuthal variations in the shear-wave splitting of core-refracted shear waves (SK(K)S-phases) at the Black Forest Observatory (BFO, SW Germany) indicate small-scale lateral and (partly) vertical variations of the elastic anisotropy in the upper mantle. However, most of the existing seismic anisotropy studies and models in the Upper Rhine Graben (URG) area are based on short-term recordings and thus suffer from a limited backazimuthal coverage and averaging over a wide or the whole backazimuth range. Hence, to find and delimit basic anisotropy regimes, also with respect to the connection to geological and tectonic processes, we carried out further SK(K)S splitting measurements at permanent (BFO, WLS, STU, ECH) and semi-permanent (TMO44, TMO07) broadband seismological recording stations.</p><p>To achieve a sufficient backazimuthal coverage and to be able to resolve and account appropriately for complex anisotropy, we analysed long-term recordings (partly > 20 yrs.). This was done manually using the MATLAB-program SplitLab (single-event analysis) together with the plugin StackSplit (multi-event analysis). The two splitting parameters, the fast polarization direction <em>Φ</em> given relative to north and the delay time <em>δt</em> accumulated between the two quasi shear waves, were determined by applying both the rotation-correlation method and the minimum-energy method for comparison. Structural anisotropy models with one layer with horizontal or tilted symmetry axis and with two layers with horizontal symmetry axes (assuming transvers isotropy with the fast axis being parallel to the symmetry axis) were tested to explain the shear-wave splitting observations, including lateral variations around a recording site.</p><p>The determined anisotropy is placed in the upper mantle due to the duration of the delay times (> 0.3 s) and missing discrepancies between SKS- and SKKS-phases (so not hints for significant lowermost mantle contributions). The spatial distribution and the lateral and backazimuthal variations of the measured (apparent) splitting parameters confirm that the anisotropy in the mantle beneath the URG area varies on small-scale laterally and partly vertically: On the east side of the URG, from the Moldanubian Zone (BFO, STU, ECH) to the Saxothuringian Zone (TMO44, TMO07) a tendency from two layers with horizontal symmetry axes to one layer is suggested. In the Moldanubian Zone, between the east side (STU, BFO) and the west side (ECH) of the URG, a change of the fast polarisation directions of the anisotropy models with two layers with horizontal symmetry axes is observed. Inconsistent measured apparent splitting parameters and the observation of numerous null measurements, especially below the URG may be at least partly related to scattering of the seismic wavefield or a modification of the mantle material.</p>

scholarly journals Deriving micro- to macro-scale seismic velocities from ice-core <i>c</i> axis orientations

2018 ◽  
Vol 12 (5) ◽  
pp. 1715-1734 ◽  
Author(s):  
Johanna Kerch ◽  
Anja Diez ◽  
Ilka Weikusat ◽  
Olaf Eisen
Keyword(s):  
Shear Wave ◽  
Seismic Anisotropy ◽  
Ice Core ◽  
Shear Wave Splitting ◽  
P Wave ◽  
Seismic Velocities ◽  
Polar Ice ◽  
S Wave ◽  
Crystal Anisotropy ◽  
Wave Splitting

Abstract. One of the great challenges in glaciology is the ability to estimate the bulk ice anisotropy in ice sheets and glaciers, which is needed to improve our understanding of ice-sheet dynamics. We investigate the effect of crystal anisotropy on seismic velocities in glacier ice and revisit the framework which is based on fabric eigenvalues to derive approximate seismic velocities by exploiting the assumed symmetry. In contrast to previous studies, we calculate the seismic velocities using the exact c axis angles describing the orientations of the crystal ensemble in an ice-core sample. We apply this approach to fabric data sets from an alpine and a polar ice core. Our results provide a quantitative evaluation of the earlier approximative eigenvalue framework. For near-vertical incidence our results differ by up to 135 m s−1 for P-wave and 200 m s−1 for S-wave velocity compared to the earlier framework (estimated 1 % difference in average P-wave velocity at the bedrock for the short alpine ice core). We quantify the influence of shear-wave splitting at the bedrock as 45 m s−1 for the alpine ice core and 59 m s−1 for the polar ice core. At non-vertical incidence we obtain differences of up to 185 m s−1 for P-wave and 280 m s−1 for S-wave velocities. Additionally, our findings highlight the variation in seismic velocity at non-vertical incidence as a function of the horizontal azimuth of the seismic plane, which can be significant for non-symmetric orientation distributions and results in a strong azimuth-dependent shear-wave splitting of max. 281 m s−1 at some depths. For a given incidence angle and depth we estimated changes in phase velocity of almost 200 m s−1 for P wave and more than 200 m s−1 for S wave and shear-wave splitting under a rotating seismic plane. We assess for the first time the change in seismic anisotropy that can be expected on a short spatial (vertical) scale in a glacier due to strong variability in crystal-orientation fabric (±50 m s−1 per 10 cm). Our investigation of seismic anisotropy based on ice-core data contributes to advancing the interpretation of seismic data, with respect to extracting bulk information about crystal anisotropy, without having to drill an ice core and with special regard to future applications employing ultrasonic sounding.

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