P-wave reflection-moveout approximation for horizontally layered media of arbitrary moderate anisotropy

Geophysics ◽  
2020 ◽  
Vol 85 (2) ◽  
pp. C61-C70 ◽  
Author(s):  
Véronique Farra ◽  
Ivan Pšenčík
Keyword(s):  
Coordinate System ◽  
Transverse Isotropy ◽  
Local Coordinate System ◽  
Surface Profile ◽  
Taylor Series Expansion ◽  
Layered Media ◽  
P Wave ◽  
Orthorhombic Symmetry ◽  
Weak Anisotropy ◽  
Cartesian Coordinate

We present an approximate nonhyperbolic P-wave moveout formula applicable to horizontally layered media of moderate or weak anisotropy of arbitrary symmetry and orientation. Anisotropy symmetry and its orientation may differ from layer to layer. Instead of commonly used Taylor-series expansion of the square of the reflection traveltime in terms of the square of the offset, we use the weak-anisotropy approximation, in which the square of the reflection traveltime is expanded in terms of weak-anisotropy (WA) parameters. The resulting formula is simple, and it provides a transparent relation between the traveltimes and WA parameters. Along an arbitrarily chosen single surface profile, it depends, in each layer, on the thickness of the layer, on the reference P-wave velocity used for the construction of reference rays, and on three WA parameters specified in the Cartesian coordinate system related to the profile. In each layer, these three “profile” WA parameters depend on “local” WA parameters specifying anisotropy of a given layer in a local coordinate system and on directional cosines specifying the orientation of the local coordinate system with respect to the profile one. The number of local P-wave WA parameters may vary from three for transverse isotropy or six for orthorhombic symmetry to nine for triclinic symmetry. Our tests of the accuracy indicate that the maximum relative traveltime errors do not exceed 0.5% or 2.5% for weak or moderate P-wave anisotropy, respectively.

Reflection moveout approximations for P-waves in a moderately anisotropic homogeneous tilted transverse isotropy layer

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. C175-C185 ◽  
Author(s):  
Ivan Pšenčík ◽  
Véronique Farra
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Weak Anisotropy ◽  
P Waves ◽  
3D Space ◽  
Axis Of Symmetry ◽  
Relative Errors ◽  
Symmetry Tests ◽  
Ti Media

We have developed approximate nonhyperbolic P-wave moveout formulas applicable to weakly or moderately anisotropic media of arbitrary anisotropy symmetry and orientation. Instead of the commonly used Taylor expansion of the square of the reflection traveltime in terms of the square of the offset, we expand the square of the reflection traveltime in terms of weak-anisotropy (WA) parameters. No acoustic approximation is used. We specify the formulas designed for anisotropy of arbitrary symmetry for the transversely isotropic (TI) media with the axis of symmetry oriented arbitrarily in the 3D space. Resulting formulas depend on three P-wave WA parameters specifying the TI symmetry and two angles specifying the orientation of the axis of symmetry. Tests of the accuracy of the more accurate of the approximate formulas indicate that maximum relative errors do not exceed 0.3% or 2.5% for weak or moderate P-wave anisotropy, respectively.

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

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1305-1315 ◽  
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.

Quartic moveout coefficient for a dipping azimuthally anisotropic layer

Geophysics ◽  
2004 ◽  
Vol 69 (3) ◽  
pp. 699-707 ◽  
Author(s):  
Andrés Pech ◽  
Ilya Tsvankin
Keyword(s):  
Reservoir Characterization ◽  
Fractured Reservoirs ◽  
Symmetry Plane ◽  
Exact Expression ◽  
Naturally Fractured Reservoirs ◽  
P Wave ◽  
Orthorhombic Symmetry ◽  
Fracture Density ◽  
Weak Anisotropy ◽  
Velocity Anisotropy

Interpretation and inversion of azimuthally varying nonhyperbolic reflection moveout requires accounting for both velocity anisotropy and subsurface structure. Here, our previously derived exact expression for the quartic moveout coefficient A4 is applied to P‐wave reflections from a dipping interface overlaid by a medium of orthorhombic symmetry. The weak‐anisotropy approximaton for the coefficient A4 in a homogeneous orthorhombic layer is controlled by the anellipticity parameters η(1), η(2), and η(3), which are responsible for time processing of P‐wave data. If the dip plane of the reflector coincides with the vertical symmetry plane [x1, x3], A4 on the dip line is proportional to the in‐plane anellipticity parameter η(2) and always changes sign for a dip of 30○. The quartic coefficient on the strike line is a function of all three η–parameters, but for mild dips it is mostly governed by η(1)—the parameter defined in the incidence plane [x2, x3]. Whereas the magnitude of the dip line A4 typically becomes small for dips exceeding 45○, the nonhyperbolic moveout on the strike line may remain significant even for subvertical reflectors. The character of the azimuthal variation of A4 depends on reflector dip and is quite sensitive to the signs and relative magnitudes of η(1), η(2), and η(3). The analytic results and numerical modeling show that the azimuthal pattern of the quartic coefficient can contain multiple lobes, with one or two azimuths of vanishing A4 between the dip and strike directions. The strong influence of the anellipticity parameters on the azimuthally varying coefficient A4 suggests that nonhyperbolic moveout recorded in wide‐azimuth surveys can help to constrain the anisotropic velocity field. Since for typical orthorhombic models that describe naturally fractured reservoirs the parameters η(1,2,3) are closely related to the fracture density and infill, the results of azimuthal nonhyperbolic moveout analysis can also be used in reservoir characterization.

Automatic Point Clouds Registration Based on the Method of Least Squares

2009 ◽  
Vol 419-420 ◽  
pp. 305-308
Author(s):  
Fan Wen Meng ◽  
Lu Shen Wu ◽  
Qing Jin Peng
Keyword(s):  
Coordinate System ◽  
Least Squares ◽  
Least Squares Method ◽  
Local Coordinate System ◽  
Surface Profile ◽  
Point Clouds ◽  
Fringe Projection ◽  
Method Of Least Squares ◽  
Surface Patches ◽  
Euclidean Distances

An object has to be measured to recover its 3D shape in reverse engineering applications. The object surface is sampled point by point using a fringe projection. The method of least squares is used to match overlapping surfaces to estimate transformation parameters between a local coordinate system and the template coordinate system. The Gauss–Markoff model can minimize the sum of squares of Euclidean distances between surfaces for matching arbitrarily oriented 3D surface patches. This research uses the least squares method for the registration of point clouds. A relief example shows the feasibility of the proposed method. It takes about 4 seconds for the registration of 1531209 points with the error less than 0.03mm, and the iteration number is only 20. The surface profile is complete and smooth after the registration, which can meet the requirement of surface reconstruction.

Inversion of reflection traveltimes for transverse isotropy

Geophysics ◽  
1995 ◽  
Vol 60 (4) ◽  
pp. 1095-1107 ◽  
Author(s):  
Ilya Tsvankin ◽  
Leon Thomsen
Keyword(s):  
Wave Velocity ◽  
Vertical Velocity ◽  
Joint Inversion ◽  
Transverse Isotropy ◽  
Taylor Series Expansion ◽  
High Sensitivity ◽  
Transversely Isotropic ◽  
P Wave ◽  
S Wave ◽  
Quartic Term

In anisotropic media, the short‐spread stacking velocity is generally different from the root‐mean‐square vertical velocity. The influence of anisotropy makes it impossible to recover the vertical velocity (or the reflector depth) using hyperbolic moveout analysis on short‐spread, common‐midpoint (CMP) gathers, even if both P‐ and S‐waves are recorded. Hence, we examine the feasibility of inverting long‐spread (nonhyperbolic) reflection moveouts for parameters of transversely isotropic media with a vertical symmetry axis. One possible solution is to recover the quartic term of the Taylor series expansion for [Formula: see text] curves for P‐ and SV‐waves, and to use it to determine the anisotropy. However, this procedure turns out to be unstable because of the ambiguity in the joint inversion of intermediate‐spread (i.e., spreads of about 1.5 times the reflector depth) P and SV moveouts. The nonuniqueness cannot be overcome by using long spreads (twice as large as the reflector depth) if only P‐wave data are included. A general analysis of the P‐wave inverse problem proves the existence of a broad set of models with different vertical velocities, all of which provide a satisfactory fit to the exact traveltimes. This strong ambiguity is explained by a trade‐off between vertical velocity and the parameters of anisotropy on gathers with a limited angle coverage. The accuracy of the inversion procedure may be significantly increased by combining both long‐spread P and SV moveouts. The high sensitivity of the long‐spread SV moveout to the reflector depth permits a less ambiguous inversion. In some cases, the SV moveout alone may be used to recover the vertical S‐wave velocity, and hence the depth. Success of this inversion depends on the spreadlength and degree of SV‐wave velocity anisotropy, as well as on the constraints on the P‐wave vertical velocity.

Seismic critical-angle reflectometry: A method to characterize azimuthal anisotropy?

Geophysics ◽  
2007 ◽  
Vol 72 (3) ◽  
pp. D41-D50 ◽  
Author(s):  
Martin Landrø ◽  
Ilya Tsvankin
Keyword(s):  
Critical Angle ◽  
Transverse Isotropy ◽  
Symmetry Plane ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Head Wave ◽  
Orthorhombic Symmetry ◽  
Amplitude Curve ◽  
Diagnostic Features ◽  
Azimuthal Variation

Existing anisotropic parameter-estimation algorithms that operate with long-offset data are based on the inversion of either nonhyperbolic moveout or wide-angle amplitude-variation-with-offset (AVO) response. We show that valuable information about anisotropic reservoirs can also be obtained from the critical angle of reflected waves. To explain the behavior of the critical angle, we develop weak-anisotropy approximations for vertical transverse isotropy and then use Tsvankin’s notation to extend them to azimuthally anisotropic models of orthorhombic symmetry. The P-wave critical-angle reflection in orthorhombic media is particularly sensitive to the parameters [Formula: see text] and [Formula: see text] responsible for the symmetry-plane horizontal velocity in the high-velocity layer. The azimuthal variation of the critical angle for typical orthorhombic models can reach [Formula: see text], which translates into substantial changes in the critical offset of the reflected P-wave. The main diagnostic features of the critical-angle reflection employed in our method include the rapid amplitude increase at the critical angle and the subsequent separation of the head wave. Analysis of exact synthetic seismograms, generated with the reflectivity method, confirms that the azimuthal variation of the critical offset is detectable on wide-azimuth, long-spread data and can be qualitatively described by our linearized equations. Estimation of the critical offset from the amplitude curve of the reflected wave, however, is not straightforward. Additional complications may be caused by the overburden noise train and by the influence of errors in the overburden velocity model on the computation of the critical angle. Still, critical-angle reflectometry should help to constrain the dominant fracture directions and can be combined with other methods to reduce the uncertainty in the estimated anisotropy parameters.

Impacts of kerogen content and fracture properties on the anisotropic seismic reflectivity of shales with orthorhombic symmetry

2015 ◽  
Vol 3 (3) ◽  
pp. ST1-ST7 ◽  
Author(s):  
Li Yang ◽  
Xiaoyang Wu ◽  
Mark Chapman
Keyword(s):  
Transverse Isotropy ◽  
Rock Physics ◽  
Vertical Axis ◽  
Orthorhombic Symmetry ◽  
Anisotropic Rock ◽  
Weak Anisotropy ◽  
Fracture Properties ◽  
Rock Physics Modeling ◽  
Axis Of Symmetry ◽  
Physics Modeling

Shale often has strong intrinsic anisotropy, which can be described by transverse isotropy with a vertical axis of symmetry. When vertical fractures are present, shale is likely to exhibit orthorhombic symmetry. We used anisotropic rock-physics models to describe the orthorhombic properties of fractured shale, and we determined that composition and fracture properties had an impact on the azimuthal amplitude variations. Interpretation of azimuthal reflectivity variations was often performed under simplified assumptions. Although the Rüger equation was derived for weak anisotropy and for transverse isotropy with a horizontal axis of symmetry, our results indicated that the orthorhombic response can be well described by the Rüger equation. However, ambiguities could be introduced into the interpretation of parameters. We suggested that careful rock-physics modeling was important for interpreting the anisotropic seismic response of fractured shale.

Estimation of fracture parameters from reflection seismic data—Part II: Fractured models with orthorhombic symmetry

Geophysics ◽  
2000 ◽  
Vol 65 (6) ◽  
pp. 1803-1817 ◽  
Author(s):  
Andrey Bakulin ◽  
Vladimir Grechka ◽  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Fractured Reservoirs ◽  
Symmetry Plane ◽  
P Wave ◽  
Single Fracture ◽  
Orthorhombic Symmetry ◽  
Weak Anisotropy ◽  
Horizontal Symmetry ◽  
Vertical Symmetry Plane ◽  
Vertical Fractures

Existing geophysical and geological data indicate that orthorhombic media with a horizontal symmetry plane should be rather common for naturally fractured reservoirs. Here, we consider two orthorhombic models: one that contains parallel vertical fractures embedded in a transversely isotropic background with a vertical symmetry axis (VTI medium) and the other formed by two orthogonal sets of rotationally invariant vertical fractures in a purely isotropic host rock. Using the linear‐slip theory, we obtain simple analytic expressions for the anisotropic coefficients of effective orthorhombic media. Under the assumptions of weak anisotropy of the background medium (for the first model) and small compliances of the fractures, all effective anisotropic parameters reduce to the sum of the background values and the parameters associated with each fracture set. For the model with a single fracture system, this result allows us to eliminate the influence of the VTI background by evaluating the differences between the anisotropic parameters defined in the vertical symmetry planes. Subsequently, the fracture weaknesses, which carry information about the density and content of the fracture network, can be estimated in the same way as for fracture‐induced transverse isotropy with a horizontal symmetry axis (HTI media) examined in our previous paper (part I). The parameter estimation procedure can be based on the azimuthally dependent reflection traveltimes and prestack amplitudes of P-waves alone if an estimate of the ratio of the P- and S-wave vertical velocities is available. It is beneficial, however, to combine P-wave data with the vertical traveltimes, NMO velocities, or AVO gradients of mode‐converted (PS) waves. In each vertical symmetry plane of the model with two orthogonal fracture sets, the anisotropic parameters are largely governed by the weaknesses of the fractures orthogonal to this plane. For weak anisotropy, the fracture sets are essentially decoupled, and their parameters can be estimated by means of two independently performed HTI inversions. The input data for this model must include the vertical velocities (or reflector depth) to resolve the anisotropic coefficients in each vertical symmetry plane rather than just their differences. We also discuss several criteria that can be used to distinguish between the models with one and two fracture sets. For example, the semimajor axis of the P-wave NMO ellipse and the polarization direction of the vertically traveling fast shear wave are always parallel to each other for a single system of fractures, but they may become orthogonal in the medium with two fracture sets.

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.

An effective anisotropy parameter in transversely isotropic media

Geophysics ◽  
1987 ◽  
Vol 52 (12) ◽  
pp. 1654-1664 ◽  
Author(s):  
N. C. Banik
Keyword(s):  
Transverse Isotropy ◽  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
P Wave ◽  
Weak Anisotropy ◽  
Reflection Seismic ◽  
The North ◽  
Reflection Amplitude ◽  
Transversely Isotropic Media ◽  
Isotropic Media

An interesting physical meaning is presented for the anisotropy parameter δ, previously introduced by Thomsen to describe weak anisotropy in transversely isotropic media. Roughly, δ is the difference between the P-wave and SV-wave anisotropies of the medium. The observed systematic depth errors in the North Sea are reexamined in view of the new interpretation of the moveout velocity through δ. The changes in δ at an interface adequately describe the effects of transverse isotropy on the P-wave reflection amplitude, The reflection coefficient expression is linearized in terms of changes in elastic parameters. The linearized expression clearly shows that it is the variation of δ at the interface that gives the anisotropic effects at small incidence angles. Thus, δ effectively describes both the moveout velocity and the reflection amplitude variation, two very important pieces of information in reflection seismic prospecting, in the presence of transverse isotropy.

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