Do dipole sonic logs measure group or phase velocity (revisited)?

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. C311-C322
Author(s):  
Stephen Horne ◽  
Richard T. Coates ◽  
Alexei Bolshakov
Keyword(s):  
Symmetry Axis ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
Flexural Waves ◽  
S Wave ◽  
Slowness Surface ◽  
Phase Velocities ◽  
Group And Phase Velocities ◽  
Sonic Logs

We have revisited the debate about whether flexural waves from dipole sonic tools and standard processing algorithms measure group or phase velocities in anisotropic formations. We observe that much of the confusion arises from a failure to understand the different meanings of group and phase velocities. Using a transversely isotropic medium with a vertical axis of symmetry that exhibits a triplication in its S-wave group slowness surface, we generate synthetic flexural sonic waveforms corresponding to boreholes at angles of 0°–90° with respect to the anisotropy symmetry axis in 1° increments. We processed these synthetic data using standard time- and frequency-domain semblance methods. The results conclusively demonstrate that dipole sonic logs measure the group slowness for the group angle corresponding to the angle between the borehole and the anisotropic symmetry axis. In addition, data that we have evaluated suggest that current tool geometries and semblance processing may not always be sensitive enough to resolve all branches of the group slowness triplication surface.

Extraction of P‐ and S‐waves from the vertical component of the particle velocity at the sea floor

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 231-240 ◽  
Author(s):  
Lasse Amundsen ◽  
Arne Reitan
Keyword(s):  
Particle Velocity ◽  
Synthetic Data ◽  
Vertical Component ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
Elastic Stiffness ◽  
P Wave ◽  
S Wave ◽  
Sea Floor ◽  
S Waves

At the boundary between two solid media in welded contact, all three components of particle velocity and vertical traction are continuous through the boundary. Across the boundary between a fluid and a solid, however, only the vertical component of particle velocity is continuous while the horizontal components can be discontinuous. Furthermore, the pressure in the fluid is the negative of the vertical component of traction in the solid, while the horizontal components of traction vanish at the interface. Taking advantage of this latter fact, we show that total P‐ and S‐waves can be computed from the vertical component of the particle velocity recorded by single component geophones planted on the sea floor. In the case when the sea floor is transversely isotropic with a vertical axis of symmetry, the computation requires the five independent elastic stiffness components and the density. However, when the sea floor material is fully isotropic, the only material parameter needed is the local shear wave velocity. The analysis of the extraction problem is done in the slowness domain. We show, however, that the S‐wave section can be obtained by a filtering operation in the space‐frequency domain. The P‐wave section is then the difference between the vertical component of the particle velocity and the S‐wave component. A synthetic data example demonstrates the performance of the algorithm.

Geometrical spreading in a layered transversely isotropic medium with vertical symmetry axis

Geophysics ◽  
2003 ◽  
Vol 68 (6) ◽  
pp. 2082-2091 ◽  
Author(s):  
Bjørn Ursin ◽  
Ketil Hokstad
Keyword(s):  
Ray Tracing ◽  
Seismic Data ◽  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Analytical Approximation ◽  
Geometrical Spreading ◽  
S Wave ◽  
Weak Anisotropy ◽  
P Waves ◽  
Amplitude Versus

Compensation for geometrical spreading is important in prestack Kirchhoff migration and in amplitude versus offset/amplitude versus angle (AVO/AVA) analysis of seismic data. We present equations for the relative geometrical spreading of reflected and transmitted P‐ and S‐wave in horizontally layered transversely isotropic media with vertical symmetry axis (VTI). We show that relatively simple expressions are obtained when the geometrical spreading is expressed in terms of group velocities. In weakly anisotropic media, we obtain simple expressions also in terms of phase velocities. Also, we derive analytical equations for geometrical spreading based on the nonhyperbolic traveltime formula of Tsvankin and Thomsen, such that the geometrical spreading can be expressed in terms of the parameters used in time processing of seismic data. Comparison with numerical ray tracing demonstrates that the weak anisotropy approximation to geometrical spreading is accurate for P‐waves. It is less accurate for SV‐waves, but has qualitatively the correct form. For P waves, the nonhyperbolic equation for geometrical spreading compares favorably with ray‐tracing results for offset‐depth ratios less than five. For SV‐waves, the analytical approximation is accurate only at small offsets, and breaks down at offset‐depth ratios less than unity. The numerical results are in agreement with the range of validity for the nonhyperbolic traveltime equations.

On-axis triplications in elastic orthorhombic media

2020 ◽  
Vol 224 (1) ◽  
pp. 449-467
Author(s):  
Shibo Xu ◽  
Alexey Stovas ◽  
Hitoshi Mikada ◽  
Junichi Takekawa
Keyword(s):  
Group Velocity ◽  
Transversely Isotropic ◽  
Model Complexity ◽  
Model Parameters ◽  
S Wave ◽  
Slowness Surface ◽  
S Waves ◽  
Redundant Signal ◽  
Velocity Surface ◽  
Offset Curve

SUMMARY Triplicated traveltime curve has three arrivals at a given distance with the bowtie shape in the traveltime-offset curve. The existence of the triplication can cause a lot of problems such as several arrivals for the same wave type, anomalous amplitudes near caustics, anomalous behaviour of rays near caustics, which leads to the structure imaging deviation and redundant signal in the inversion of the model parameters. Hence, triplication prediction becomes necessary when the medium is known. The research of the triplication in transversely isotropic medium with a vertical symmetry axis (VTI) has been well investigated and it has become clear that, apart from the point singularity case, the triplicated traveltime only occurs for S wave. On contrary to the VTI case, the triplication behaviour in the orthorhombic (ORT) medium has not been well focused due to the model complexity. In this paper, we derive the second-order coefficients of the slowness surface for two S waves in the vicinity of three symmetry axes and define the elliptic form function to examine the existence of the on-axis triplication in ORT model. The existence of the on-axis triplication is found by the sign of the defined curvature coefficients. Three ORT models are defined in the numerical examples to analyse the behaviour of the on-axis triplication. The plots of the group velocity surface in the vicinity of three symmetry axes are shown for different ORT models where different shapes: convex or the saddle-shaped (concave along one direction and convex along with another) indicates the existence of the on-axis triplication. We also show the traveltime plots (associated with the group velocity surface) to illustrate the effect of the on-axis triplication.

Building tilted transversely isotropic depth models using localized anisotropic tomography with well information

Geophysics ◽  
2010 ◽  
Vol 75 (4) ◽  
pp. D27-D36 ◽  
Author(s):  
Andrey Bakulin ◽  
Marta Woodward ◽  
Dave Nichols ◽  
Konstantin Osypov ◽  
Olga Zdraveva
Keyword(s):  
Seismic Data ◽  
Model Building ◽  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Synthetic Data ◽  
Rock Physics ◽  
Transversely Isotropic ◽  
Data Set ◽  
Anisotropic Models ◽  
Well Data

Tilted transverse isotropy (TTI) is increasingly recognized as a more geologically plausible description of anisotropy in sedimentary formations than vertical transverse isotropy (VTI). Although model-building approaches for VTI media are well understood, similar approaches for TTI media are in their infancy, even when the symmetry-axis direction is assumed known. We describe a tomographic approach that builds localized anisotropic models by jointly inverting surface-seismic and well data. We present a synthetic data example of anisotropic tomography applied to a layered TTI model with a symmetry-axis tilt of 45 degrees. We demonstrate three scenarios for constraining the solution. In the first scenario, velocity along the symmetry axis is known and tomography inverts for Thomsen’s [Formula: see text] and [Formula: see text] parame-ters. In the second scenario, tomography inverts for [Formula: see text], [Formula: see text], and velocity, using surface-seismic data and vertical check-shot traveltimes. In contrast to the VTI case, both these inversions are nonunique. To combat nonuniqueness, in the third scenario, we supplement check-shot and seismic data with the [Formula: see text] profile from an offset well. This allows recovery of the correct profiles for velocity along the symmetry axis and [Formula: see text]. We conclude that TTI is more ambiguous than VTI for model building. Additional well data or rock-physics assumptions may be required to constrain the tomography and arrive at geologically plausible TTI models. Furthermore, we demonstrate that VTI models with atypical Thomsen parameters can also fit the same joint seismic and check-shot data set. In this case, although imaging with VTI models can focus the TTI data and match vertical event depths, it leads to substantial lateral mispositioning of the reflections.

Modeling and inversion of PS-wave moveout asymmetry for tilted TI media: Part 2 — Dipping TTI layer

Geophysics ◽  
2006 ◽  
Vol 71 (4) ◽  
pp. D123-D134 ◽  
Author(s):  
Pawan Dewangan ◽  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Vertical Plane ◽  
Transversely Isotropic ◽  
Estimation Algorithm ◽  
Velocity Model ◽  
Inversion Algorithm ◽  
S Wave ◽  
Weak Anisotropy ◽  
Symmetry Direction ◽  
Tti Media

Dipping transversely isotropic layers with a tilted symmetry axis (TTI media) cause serious imaging problems in fold-and-thrust belts and near salt domes. Here, we apply the modified [Formula: see text] method introduced in Part 1 to the inversion of long-offset PP and PS reflection data for the parameters of a TTI layer with the symmetry axis orthogonal to the bedding. The inversion algorithm combines the time- and offset-asymmetry attributes of the PSV-wave with the hyperbolic PP- and SS-wave moveout in the symmetry-axis plane (i.e., the vertical plane that contains the symmetry axis). The weak-anisotropy approximations for the moveout-asymmetry attributes, verified by numerical analysis, indicate that small-offset (leading) terms do not contain independent information for the inversion. Therefore, the parameter-estimation algorithm relies on PS data recorded at large offsets (the offset-to-depth ratio has to reach at least two), which makes the results generally less stable than those for a horizontal TTI layer in Part1. The least-resolved parameter is Thomsen’s coefficient [Formula: see text]that does not directly influence the moveout of either pure or converted modes. Still, the contribution of the PS-wave asymmetry attributes helps to constrain the TTI model for large tilts [Formula: see text] of the symmetry axis [Formula: see text]. The accuracy of the inversion for large tilts can be improved further by using wide-azimuth PP and PS reflections. With high-quality PS data, the inversion remains feasible for moderate tilts [Formula: see text], but it breaks down for models with smaller values of [Formula: see text] in which the moveout asymmetry is too weak. The tilt itself and several combinations of the medium parameters (e.g., the ratio of the P- and S-wave velocities in the symmetry direction), however, are well constrained for all symmetry-axis orientations. The results of Parts 1 and 2 show that 2D measurements of the PS-wave asymmetry attributes can be used effectively in anisotropic velocity analysis for TTI media. In addition to providing an improved velocity model for imaging beneath TTI beds, our algorithms yield information for lithology discrimination and structural interpretation.

scholarly journals Traveltime approximations for transversely isotropic media with an inhomogeneous background

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. WA31-WA42 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
Transversely Isotropic Media ◽  
Higher Order Terms ◽  
Isotropic Media ◽  
Axis Of Symmetry ◽  
Series Type ◽  
Ti Media ◽  
Taylor’S Series

A transversely isotropic (TI) model with a tilted symmetry axis is regarded as one of the most effective approximations to the Earth subsurface, especially for imaging purposes. However, we commonly utilize this model by setting the axis of symmetry normal to the reflector. This assumption may be accurate in many places, but deviations from this assumption will cause errors in the wavefield description. Using perturbation theory and Taylor’s series, I expand the solutions of the eikonal equation for 2D TI media with respect to the independent parameter [Formula: see text], the angle the tilt of the axis of symmetry makes with the vertical, in a generally inhomogeneous TI background with a vertical axis of symmetry. I do an additional expansion in terms of the independent (anellipticity) parameter [Formula: see text] in a generally inhomogeneous elliptically anisotropic background medium. These new TI traveltime solutions are given by expansions in [Formula: see text] and [Formula: see text] with coefficients extracted from solving linear first-order partial differential equations. Pade approximations are used to enhance the accuracy of the representation by predicting the behavior of the higher-order terms of the expansion. A simplification of the expansion for homogenous media provides nonhyperbolic moveout descriptions of the traveltime for TI models that are more accurate than other recently derived approximations. In addition, for 3D media, I develop traveltime approximations using Taylor’s series type of expansions in the azimuth of the axis of symmetry. The coefficients of all these expansions can also provide us with the medium sensitivity gradients (Jacobian) for nonlinear tomographic-based inversion for the tilt in the symmetry axis.

Shear‐wave polarizations and subsurface stress directions at Lost Hills field

Geophysics ◽  
1991 ◽  
Vol 56 (9) ◽  
pp. 1331-1348 ◽  
Author(s):  
D. F. Winterstein ◽  
M. A. Meadows
Keyword(s):  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Data Matrix ◽  
Wave Polarization ◽  
Polarization Direction ◽  
S Wave ◽  
Near Surface ◽  
Polarization Technique ◽  
The Matrix ◽  
Vertical Travel

2 × 2 S-wave data matrix, accomplished by computationally rotating sources and receivers. Although polarization directions obtained by assuming a homogeneous subsurface were moderately consistent with depth, considerable improvement in consistency resulted from analytically stripping off a thin near‐surface layer whose fast S-wave polarization direction was about N 6°E. S-wave birefringence for vertical travel averaged 3 percent in two zones, 200–700 ft and 1200–2100 ft (60–210 m and 370–640 m), which had closely similar S-wave polarizations. Between those zones, the polarization direction changed and the birefringence magnitude was not well defined. S-wave polarizations from two concentric rings of offset VSPs were consistent in azimuth with one another and with polarizations of the near offset VSP. This consistency argues strongly for the robustness of the S-wave polarization technique as applied in this area. The S-wave polarization pattern in offset data fits a model of vertical cracks striking N 55°E in a weakly transversely isotropic matrix, where the infinite‐fold symmetry axis of the matrix is tilted 10 degrees from the vertical towards N 70°E. Such a model is of monoclinic symmetry.

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