Moveout inversion of P-wave data for horizontal transverse isotropy

The transversely isotropic model with a horizontal symmetry axis (HTI media) has been extensively used in seismological studies of fractured reservoirs. In this paper, a parameter‐estimation technique originally developed by Grechka and Tsvankin for the more general orthorhombic media is applied to horizontal transverse isotropy. Our methodology is based on the inversion of azimuthally‐dependent P-wave normal‐moveout (NMO) velocities from horizontal and dipping reflectors. If the NMO velocity of a given reflection event is plotted in each azimuthal direction, it forms an ellipse determined by three combinations of medium parameters. The NMO ellipse from a horizontal reflector in HTI media can be inverted for the azimuth β of the symmetry axis, the vertical velocity [Formula: see text], and the Thomsen‐type anisotropic parameter δ(V). We describe a technique for obtaining the remaining (for P-waves) anisotropic parameter η(V) (or ε(V)) from the NMO ellipse corresponding to a dipping reflector of arbitrary azimuth. The interval parameters of vertically inhomogeneous HTI media are recovered using the generalized Dix equation that operates with NMO ellipses for horizontal and dipping events. High accuracy of our method is confirmed by inverting a synthetic multiazimuth P-wave data set generated by ray tracing for a layered HTI medium with depth‐varying orientation of the symmetry axis. Although estimation of η(V) can be carried out by the algorithm developed for orthorhombic media, for more stable results the HTI model has to be used from the outset of the inversion procedure. It should be emphasized that P-wave conventional‐spread moveout data provide enough information to distinguish between HTI and lower‐symmetry models. We show that if the medium has the orthorhombic symmetry and is sufficiently different from HTI, the best‐fit HTI model cannot match the NMO ellipses for both a horizontal and a dipping event. The anisotropic coefficients responsible for P-wave moveout can be combined to estimate the crack density and predict whether the cracks are fluid‐filled or dry. A unique feature of the HTI model that distinguishes it from both vertical transverse isotropy and orthorhombic media is that moveout inversion provides not just zero‐dip NMO velocities and anisotropic coefficients, but also the true vertical velocity. As a result, reflection P-wave data acquired over HTI formations can be used to build velocity models in depth and perform anisotropic depth processing.

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Velocity analysis for tilted transversely isotropic media: A physical modeling example

Geophysics ◽

10.1190/1.1444980 ◽

2001 ◽

Vol 66 (3) ◽

pp. 904-910 ◽

Cited By ~ 20

Author(s):

Vladimir Grechka ◽

Andres Pech ◽

Ilya Tsvankin ◽

Baoniu Han

Keyword(s):

Physical Modeling ◽

Symmetry Axis ◽

Transverse Isotropy ◽

Transversely Isotropic ◽

P Wave ◽

Velocity Analysis ◽

Isotropic Layer ◽

Anisotropic Parameter ◽

Data Set ◽

Symmetry Direction

Transverse isotropy with a tilted symmetry axis (TTI media) has been recognized as a common feature of shale formations in overthrust areas, such as the Canadian Foothills. Since TTI layers cause serious problems in conventional imaging, it is important to be able to reconstruct the velocity model suitable for anisotropic depth migration. Here, we discuss the results of anisotropic parameter estimation on a physical‐modeling data set. The model represents a simplified version of a typical overthrust section from the Alberta Foothills, with a horizontal reflector overlaid by a bending transversely isotropic layer. Assuming that the TTI layer is homogeneous and the symmetry axis stays perpendicular to its boundaries, we invert P-wave normal‐moveout (NMO) velocities and zero‐offset traveltimes for the symmetry‐direction velocity V0 and the anisotropic parameters ε and δ. The coefficient ε is obtained using the traveltimes of a wave that crosses a dipping TTI block and reflects from the bottom of the model. The inversion for ε is based on analytic expressions for NMO velocity in media with intermediate dipping interfaces. Our estimates of both anisotropic coefficients are close to their actual values. The errors in the inversion, which are associated primarily with the uncertainties in picking the NMO velocities and traveltimes, can be reduced by a straighforward modification of the acquisition geometry. It should be emphasized that the moveout inversion also gives an accurate estimate of the thickness of the TTI layer, thus reconstructing the correct depth scale of the section. Although the physical model used here was relatively simple, our results demonstrate the principal feasibility of anisotropic velocity analysis and imaging in overthrust areas. The main problems in anisotropic processing for TTI models are likely to be caused by the lateral variation of the velocity field and overall structural complexity.

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Robust estimation of vertical symmetry axis models via joint migration inversion: Including multiples in anisotropic parameter estimation

Geophysics ◽

10.1190/geo2017-0856.1 ◽

2019 ◽

Vol 84 (1) ◽

pp. C57-C74 ◽

Cited By ~ 4

Author(s):

Abdulrahman A. Alshuhail ◽

Dirk J. Verschuur

Keyword(s):

Symmetry Axis ◽

Transverse Isotropy ◽

Half Cycle ◽

Anisotropic Parameter ◽

Data Set ◽

Trade Off ◽

Trade Offs ◽

Highly Nonlinear ◽

Wide Range ◽

Internal Multiples

Because the earth is predominately anisotropic, the anisotropy of the medium needs to be included in seismic imaging to avoid mispositioning of reflectors and unfocused images. Deriving accurate anisotropic velocities from the seismic reflection measurements is a highly nonlinear and ambiguous process. To mitigate the nonlinearity and trade-offs between parameters, we have included anisotropy in the so-called joint migration inversion (JMI) method, in which we limit ourselves to the case of transverse isotropy with a vertical symmetry axis. The JMI method is based on strictly separating the scattering effects in the data from the propagation effects. The scattering information is encoded in the reflectivity operators, whereas the phase information is encoded in the propagation operators. This strict separation enables the method to be more robust, in that it can appropriately handle a wide range of starting models, even when the differences in traveltimes are more than a half cycle away. The method also uses internal multiples in estimating reflectivities and anisotropic velocities. Including internal multiples in inversion not only reduces the crosstalk in the final image, but it can also reduce the trade-off between the anisotropic parameters because internal multiples usually have more of an imprint of the subsurface parameters compared with primaries. The inverse problem is parameterized in terms of a reflectivity, vertical velocity, horizontal velocity, and a fixed [Formula: see text] value. The method is demonstrated on several synthetic models and a marine data set from the North Sea. Our results indicate that using JMI for anisotropic inversion makes the inversion robust in terms of using highly erroneous initial models. Moreover, internal multiples can contain valuable information on the subsurface parameters, which can help to reduce the trade-off between anisotropic parameters in inversion.

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Feasibility of nonhyperbolic moveout inversion in transversely isotropic media

Geophysics ◽

10.1190/1.1444407 ◽

1998 ◽

Vol 63 (3) ◽

pp. 957-969 ◽

Cited By ~ 78

Author(s):

Vladimir Grechka ◽

Ilya Tsvankin

Keyword(s):

Vertical Velocity ◽

Transverse Isotropy ◽

Horizontal Velocity ◽

P Wave ◽

Vacuum Field ◽

Percentage Error ◽

Correlated Errors ◽

Data Set ◽

Time Migration ◽

Quartic Term

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 Vhor and normal‐moveout (NMO) velocity Vnmo and introduce a correction factor in the denominator that increases the accuracy at intermediate offsets. Then we apply this equation to obtain Vhor and 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 Vhor and Vnmo caused by the interplay between the quadratic and quartic term of the moveout series. Since the errors in Vhor and Vnmo have 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 Vhor and Vnmo that 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.

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Reflection moveout inversion for horizontal transverse isotropy: Accuracy, limitation, and acquisition

Geophysics ◽

10.1190/1.1444713 ◽

2000 ◽

Vol 65 (1) ◽

pp. 222-231 ◽

Cited By ~ 16

Author(s):

Abdulfattah Al‐Dajani ◽

Tariq Alkhalifah

Keyword(s):

Parameter Estimation ◽

Vertical Velocity ◽

Transverse Isotropy ◽

Angular Separation ◽

P Wave ◽

Velocity Anisotropy ◽

Hti Media ◽

Inversion Procedure ◽

Anisotropy Strength

Horizontal transverse isotropy (HTI) is the simplest azimuthally anisotropic model used to describe vertical fracturing in an isotropic matrix. Assuming that the subsurface is laterally homogeneous, and using the elliptical variation of P-wave NMO velocity with azimuth measured in at least three different source‐to‐receiver orientations, we can estimate three key parameters of HTI media: the vertical velocity, anisotropy, and the azimuth of the symmetry axis. Such parameter estimation is sensitive to the angular separation between the survey lines in 2-D acquisition or, equivalently, to source‐to‐receiver azimuths in 3-D acquisition and the set of azimuths used in the inversion procedure. The accuracy in estimating the azimuth, in particular, is also sensitive to the strength of anisotropy, while the accuracy in resolving vertical velocity and anisotropy is about the same for any strength of anisotropy. To maximize the accuracy and stability in parameter estimation, it is best to have the azimuths for the source‐to‐receiver directions 60° apart when only three directions are used. This requirement is feasible in land seismic data acquisition where wide azimuthal coverage can be designed. In marine streamer acquisition, however, the azimuthal data coverage is limited. Multiple survey directions are necessary to achieve such wide azimuthal coverage in streamer surveys. To perform the inversion using three azimuth directions, 60° apart, an HTI layer overlain by an azimuthally isotropic overburden should have a time thickness, relative to the total time, of at least the ratio of the error in the NMO (stacking) velocity to the interval anisotropy strength of the HTI layer. Having more than three source‐to‐receiver azimuths (e.g., full azimuthal coverage), however, provides a useful data redundancy that enhances the quality of the estimates, thus allowing acceptable parameter estimation at smaller relative thicknesses.

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VTI anisotropic corrections and effective parameter estimation after isotropic prestack depth migration

Geophysics ◽

10.1190/1.2194888 ◽

2006 ◽

Vol 71 (3) ◽

pp. D35-D43 ◽

Cited By ~ 6

Author(s):

Moshe Reshef ◽

Murray Roth

Keyword(s):

Parameter Estimation ◽

Vertical Velocity ◽

Transverse Isotropy ◽

Real Data ◽

P Wave ◽

Prestack Depth Migration ◽

Anisotropic Parameter ◽

Depth Migration ◽

Residual Correction ◽

Dip Angle

In the method for applying anisotropic corrections after isotropic prestack depth migration (PSDM), the correction, which is calculated and implemented in the depth domain, is defined as a time difference between isotropic and anisotropic traveltimes, under the assumption that the vertical velocity is known. The definition of this correction uses a special postmigration common-image-gather (CIG) ordering, which collects the migrated data according to the input-trace’s source and receiver distance from the surface CIG location. In this postmigration domain, the dip of the events can be directly related to their horizontal position in the CIG, called the imaging offset, and the separation of flat and dipping reflectors becomes easy to perform. The dependency of the seismic anisotropic effect on the subsurface dip angle is well pronounced in these CIGs. After application of an isotropic PSDM, effective anisotropic-parameter estimation is performed at selected CIG locations by using a simple two-parameter scan procedure. The optimal anisotropic parameters can be used to perform a final anisotropic PSDM or to apply a residual correction to the isotropically migrated data. We demonstrate the method for P-wave data in 2D media with vertical transverse isotropy (VTI) symmetry by using both synthetic and real data. We also present a strategy for handling the ambiguity between the vertical velocity and the anisotropic parameters.

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Reflection moveout and parameter estimation for horizontal transverse isotropy

Geophysics ◽

10.1190/1.1444170 ◽

1997 ◽

Vol 62 (2) ◽

pp. 614-629 ◽

Cited By ~ 134

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.

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

Geophysics ◽

10.1190/1.1444593 ◽

1999 ◽

Vol 64 (3) ◽

pp. 820-837 ◽

Cited By ~ 78

Author(s):

Vladimir Grechka ◽

Ilya Tsvankin

Keyword(s):

Parameter Estimation ◽

Numerical Study ◽

Symmetry Plane ◽

P Wave ◽

Inversion Method ◽

Orthorhombic Symmetry ◽

Data Set ◽

Anisotropic Models ◽

Wave Data ◽

Time Migration

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.

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The offset-midpoint traveltime pyramid in 3D transversely isotropic media with a horizontal symmetry axis

Geophysics ◽

10.1190/geo2013-0322.1 ◽

2015 ◽

Vol 80 (1) ◽

pp. T51-T62 ◽

Cited By ~ 3

Author(s):

Qi Hao ◽

Alexey Stovas ◽

Tariq Alkhalifah

Keyword(s):

Symmetry Axis ◽

Transversely Isotropic ◽

P Wave ◽

Analytic Representation ◽

Velocity Inversion ◽

Transversely Isotropic Media ◽

Horizontal Symmetry ◽

Time Domains ◽

Isotropic Media ◽

Hti Media

Analytic representation of the offset-midpoint traveltime equation for anisotropy is very important for prestack Kirchhoff migration and velocity inversion in anisotropic media. For transversely isotropic media with a vertical symmetry axis, the offset-midpoint traveltime resembles the shape of a Cheops’ pyramid. This is also valid for homogeneous 3D transversely isotropic media with a horizontal symmetry axis (HTI). We extended the offset-midpoint traveltime pyramid to the case of homogeneous 3D HTI. Under the assumption of weak anellipticity of HTI media, we derived an analytic representation of the P-wave traveltime equation and used Shanks transformation to improve the accuracy of horizontal and vertical slownesses. The traveltime pyramid was derived in the depth and time domains. Numerical examples confirmed the accuracy of the proposed approximation for the traveltime function in 3D HTI media.

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

Geophysics ◽

10.1190/1.1444822 ◽

2000 ◽

Vol 65 (4) ◽

pp. 1305-1315 ◽

Cited By ~ 6

Author(s):

Hongbo Zhou ◽

George A. McMechan

Keyword(s):

Isotropic Medium ◽

Symmetry Axis ◽

Transverse Isotropy ◽

Analytical Formula ◽

Transversely Isotropic ◽

P Wave ◽

Geometrical Spreading ◽

Transversely Isotropic Medium ◽

Weak Anisotropy ◽

Special Cases

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.

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Physical and numerical modeling of tuning and diffraction in azimuthally anisotropic media

Geophysics ◽

10.1190/1.1444849 ◽

2000 ◽

Vol 65 (5) ◽

pp. 1613-1621 ◽

Cited By ~ 8

Author(s):

Richard L. Gibson ◽

Stephen Theophanis ◽

M. Nafi Toksöz

Keyword(s):

Numerical Modeling ◽

Fractured Reservoirs ◽

Homogeneous Model ◽

Azimuthal Anisotropy ◽

P Wave ◽

Orthorhombic Symmetry ◽

Fractured Reservoir ◽

Sh Wave ◽

Wave Data ◽

Ray Methods

Fractured reservoirs are an important target for exploration and production geophysics, and the azimuthal anisotropy often associated with these reservoirs can strongly influence seismic wave propagation. We created a physical model of a fractured reservoir to simulate some of these propagation effects. The reservoir is represented by a phenolite disk that is thin with respect to the elastic wavelengths in the experiment, creating model dimensions that are representative of realistic reservoirs. Phenolite is strongly anisotropic with orthorhombic symmetry, which suggests that azimuthal amplitude versus offset (AVO) effects should be obvious in data. We acquired both SH- and P-wave data in common‐offset gathers with a near offset and a far offset and found that although the SH-wave data show clear azimuthal variations in AVO, the P-wave signals show no apparent changes with azimuth. We then applied numerical modeling to analyze the data. Because ray methods cannot model diffractions from the disk edge, we first used a ray‐Born technique to simulate variations in waveforms associated with such scattering. The synthetic seismograms reproduced variations in the SH-wave waveforms accurately, though the amplitude contrast between acquisition azimuths was overestimated. Assuming a laterally homogeneous model, we then applied ray methods to simulate tuning effects in SH- and P-wave data and confirmed that in spite of the large contrasts in elastic properties, the tuning of the P-wave reflections from the thin disk changed so there was negligible contrast in AVO with azimuth. Models of field scale reservoirs showed that the same effects could be expected for field applications.

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