Azimuthal AVO for tilted TI medium

Geophysics ◽  
2017 ◽  
Vol 82 (1) ◽  
pp. C21-C33 ◽  
Author(s):  
Hongwei Wang ◽  
Suping Peng ◽  
Wenfeng Du
Keyword(s):  
Isotropic Medium ◽  
Symmetry Axis ◽  
Transversely Isotropic ◽  
P Wave ◽  
S Wave ◽  
Transverse Waves ◽  
Propagation Angle ◽  
S Waves ◽  
The Difference ◽  
Dip Angle

With the incident P-wave, we derive approximate formulas for amplitudes and polarizations of waves reflected from and transmitted through a planar, horizontal boundary between an overlying isotropic medium and an underlying tilted transversely isotropic (TTI) medium assuming that the directions of the phase and group velocities are consistent. Provided that the velocities in the isotropic medium are equal to the velocities along the symmetry axis direction, we derive the relational expression between the propagation angle in the TTI medium and the propagation angle in the hypothetical isotropic medium, under the condition that the horizontal slowness is the same, and then we update the approximate formula of the polarization in the TTI medium. Provided that the slow and fast transverse waves (qS and SH) are generated simultaneously in the anisotropic interface, we linearize for a six-order Zoeppritz equation, derive the azimuthal formula of longitudinal and S-waves, and determine their detailed expressions within the symmetry axis plane. According to the derived azimuthal AVO formula, we establish medium models, compare the derived AVO with the precision, and obtain the following conclusions: (1) The dip angle for the symmetry axis with respect to the vertical may have a sufficiently large impact on AVO, and the vertical longitudinal wave can generate an S-wave. (2) For the derived AVO formula, within the symmetry axis plane, the fitting effect of the approximate and exact formulas is good; however, within the other incident planes, taking the azimuth angle 45° as an example, the approximation is suitable for the large impedance contrast if the anisotropic parameters are set properly. (3) The error between the approximation and precision is mainly caused by the difference between the reflected and transmitted angles, the velocities’ derivation with respect to azimuth, and the division of approximation into isotropic and anisotropic parts.

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.

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.

Finite-difference frequency-domain modeling of viscoacoustic wave propagation in 2D tilted transversely isotropic (TTI) media

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. T75-T95 ◽  
Author(s):  
Stéphane Operto ◽  
Jean Virieux ◽  
A. Ribodetti ◽  
J. E. Anderson
Keyword(s):  
Finite Difference ◽  
Frequency Domain ◽  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Wave Mode ◽  
Absorbing Boundary Conditions ◽  
P Wave ◽  
Domain Modeling ◽  
S Wave ◽  
Absorbing Boundary

A 2D finite-difference, frequency-domain method was developed for modeling viscoacoustic seismic waves in transversely isotropic media with a tilted symmetry axis. The medium is parameterized by the P-wave velocity on the symmetry axis, the density, the attenuation factor, Thomsen’s anisotropic parameters [Formula: see text] and [Formula: see text], and the tilt angle. The finite-difference discretization relies on a parsimonious mixed-grid approach that designs accurate yet spatially compact stencils. The system of linear equations resulting from discretizing the time-harmonic wave equation is solved with a parallel direct solver that computes monochromatic wavefields efficiently for many sources. Dispersion analysis shows that four grid points per P-wavelength provide sufficiently accurate solutions in homogeneous media. The absorbing boundary conditions are perfectly matched layers (PMLs). The kinematic and dynamic accuracy of the method wasassessed with several synthetic examples which illustrate the propagation of S-waves excited at the source or at seismic discontinuities when [Formula: see text]. In frequency-domain modeling with absorbing boundary conditions, the unstable S-wave mode is not excited when [Formula: see text], allowing stable simulations of the P-wave mode for such anisotropic media. Some S-wave instabilities are seen in the PMLs when the symmetry axis is tilted and [Formula: see text]. These instabilities are consistent with previous theoretical analyses of PMLs in anisotropic media; they are removed if the grid interval is matched to the P-wavelength that leads to dispersive S-waves. Comparisons between seismograms computed with the frequency-domain acoustic TTI method and a finite-difference, time-domain method for the vertical transversely isotropic elastic equation show good agreement for weak to moderate anisotropy. This suggests the method can be used as a forward problem for viscoacoustic anisotropic full-waveform inversion.

Seismic vertical transversely isotropic parameter inversion from P- and S-wave cross-borehole measurements in an aquifer environment

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. D101-D116
Author(s):  
Julius K. von Ketelhodt ◽  
Musa S. D. Manzi ◽  
Raymond J. Durrheim ◽  
Thomas Fechner
Keyword(s):  
Transversely Isotropic ◽  
P Wave ◽  
Perturbation Equation ◽  
S Wave ◽  
P Waves ◽  
Data Set ◽  
Near Surface ◽  
S Waves ◽  
Wave Measurements ◽  
Wave Splitting

Joint P- and S-wave measurements for tomographic cross-borehole analysis can offer more reliable interpretational insight concerning lithologic and geotechnical parameter variations compared with P-wave measurements on their own. However, anisotropy can have a large influence on S-wave measurements, with the S-wave splitting into two modes. We have developed an inversion for parameters of transversely isotropic with a vertical symmetry axis (VTI) media. Our inversion is based on the traveltime perturbation equation, using cross-gradient constraints to ensure structural similarity for the resulting VTI parameters. We first determine the inversion on a synthetic data set consisting of P-waves and vertically and horizontally polarized S-waves. Subsequently, we evaluate inversion results for a data set comprising jointly measured P-waves and vertically and horizontally polarized S-waves that were acquired in a near-surface ([Formula: see text]) aquifer environment (the Safira research site, Germany). The inverted models indicate that the anisotropy parameters [Formula: see text] and [Formula: see text] are close to zero, with no P-wave anisotropy present. A high [Formula: see text] ratio of up to nine causes considerable SV-wave anisotropy despite the low magnitudes for [Formula: see text] and [Formula: see text]. The SH-wave anisotropy parameter [Formula: see text] is estimated to be between 0.05 and 0.15 in the clay and lignite seams. The S-wave splitting is confirmed by polarization analysis prior to the inversion. The results suggest that S-wave anisotropy may be more severe than P-wave anisotropy in near-surface environments and should be taken into account when interpreting cross-borehole S-wave data.

Reflection from a dipping plane—Transversely isotropic solid

Geophysics ◽  
1990 ◽  
Vol 55 (7) ◽  
pp. 851-855 ◽  
Author(s):  
Franklyn K. Levin
Keyword(s):  
Numerical Investigation ◽  
Elastic Constants ◽  
Wave Reflection ◽  
Symmetry Axis ◽  
Transversely Isotropic ◽  
P Wave ◽  
Isotropic Solid ◽  
Dip Angle

CMP stacking velocities for a P‐wave reflection from a dipping plane underlying a transversely isotropic solid are, after correction by the cosine of the dip angle, nearly independent of the dip angle if the symmetry axis of the solid is perpendicular to the reflector. If the symmetry axis is perpendicular to the surface, stacking velocities vary, after correction with the cosine of the dip angle, and predicting the amount and dependence on dip angle requires numerical investigation for each solid, since the stacking velocities may increase, decrease, or go through an extremum as the dip increases. The exact behavior depends on the elastic constants of the solid.

PERTURBATION OF PHASE VELOCITIES IN ELASTIC ORTHORHOMBIC MEDIA

Geophysics ◽  
2021 ◽  
pp. 1-109
Author(s):  
Alexey Stovas ◽  
Yuriy Roganov ◽  
Vyacheslav Roganov
Keyword(s):  
Seismic Waves ◽  
Transversely Isotropic ◽  
Second Order ◽  
P Wave ◽  
Order Perturbation ◽  
Perturbation Approach ◽  
S Wave ◽  
Anisotropic Models ◽  
S Waves ◽  
Phase Velocities

The parameterization of anisotropic models is very important when focusing on specific signatures of seismic waves and reducing the parameters crosstalk involved in inverting seismic data. The parameterization is strongly dependent on the problem at hand. We propose a new parameterization for an elastic orthorhombic model with on-axes P- and S-wave velocities and new symmetric anelliptic parameters. The perturbation approach is well defined for P waves in acoustic orthorhombic media. In the elastic orthorhombic media, the P-wave perturbation coefficients are very similar to their acoustic counterparts. However, the S-waves perturbation coefficients are still unknown. The perturbation coefficients can be interpreted as sensitivity coefficients, and they are important in many applications. We apply the second-order perturbation in anelliptic parameters for P, S1 and S2 wave phase velocities in elastic orthorhombic model. We show that using the conventional method some perturbation coefficients for S waves are not defined in the vicinity of the singularity point in an elliptical background model. Thus, we propose an alternative perturbation approach that overcomes this problem. We compute the first- and second-order perturbation coefficients for P and S waves. The perturbation-based approximations are very accurate for P and S waves compared with exact solutions, based on a numerical example. The reductions to transversely isotropic and acoustic orthorhombic models are also considered for analysis. We also show how perturbations in anelliptic parameters affect S-wave triplications in an elastic orthorhombic model.

scholarly journals New acoustic approximation for transversely isotropic media with a vertical symmetry axis

Geophysics ◽  
2019 ◽  
Vol 85 (1) ◽  
pp. C1-C12 ◽  
Author(s):  
Shibo Xu ◽  
Alexey Stovas ◽  
Tariq Alkhalifah ◽  
Hitoshi Mikada
Keyword(s):  
Wave Propagation ◽  
Seismic Data ◽  
Eikonal Equation ◽  
Symmetry Axis ◽  
Transversely Isotropic ◽  
P Wave ◽  
Seismic Data Processing ◽  
S Wave ◽  
P Waves ◽  
Acoustic Approximation

Seismic data processing in the elastic anisotropic model is complicated due to multiparameter dependency. Approximations to the P-wave kinematics are necessary for practical purposes. The acoustic approximation for P-waves in a transversely isotropic medium with a vertical symmetry axis (VTI) simplifies the description of wave propagation in elastic media, and as a result, it is widely adopted in seismic data processing and analysis. However, finite-difference implementations of that approximation are plagued with S-wave artifacts. Specifically, the resulting wavefield also includes artificial diamond-shaped S-waves resulting in a redundant signal for many applications that require pure P-wave data. To derive a totally S-wave-free acoustic approximation, we have developed a new acoustic approximation for pure P-waves that is totally free of S-wave artifacts in the homogeneous VTI model. To keep the S-wave velocity equal to zero, we formulate the vertical S-wave velocity to be a function of the model parameters, rather than setting it to zero. Then, the corresponding P-wave phase and group velocities for the new acoustic approximation are derived. For this new acoustic approximation, the kinematics is described by a new eikonal equation for pure P-wave propagation, which defines the new vertical slowness for the P-waves. The corresponding perturbation-based approximation for our new eikonal equation is used to compare the new equation with the original acoustic eikonal. The accuracy of our new P-wave acoustic approximation is tested on numerical examples for homogeneous and multilayered VTI models. We find that the accuracy of our new acoustic approximation is as good as the original one for the phase velocity, group velocity, and the kinematic parameters such as vertical slowness, traveltime, and relative geometric spreading. Therefore, the S-wave-free acoustic approximation could be further applied in seismic processing that requires pure P-wave data.

Empirical studies of some of the seismic phenomena of Hawaii*

1938 ◽  
Vol 28 (4) ◽  
pp. 313-337
Author(s):  
Austin E. Jones
Keyword(s):  
P Wave ◽  
Secondary Peak ◽  
S Wave ◽  
S Waves ◽  
Mauna Loa ◽  
Earthquake Records ◽  
Type K ◽  
Sea Bottom ◽  
The Difference ◽  
The Waves

Summary and Conclusions A comparison was made of all the periods of local earthquakes entered in the record books, and this showed that the P wave of 0.3-sec. period occurred a maximum of 156 times, and a secondary peak for the period of 0.5 sec. occurred 89 times. The S wave of 0.5-sec. period had a maximum of 129 occurrences, and a secondary peak for 0.8-sec. period had 100. This suggested that in any earthquake the ratio of the period of the S to the P wave was inversely as their velocities, or as the square root of three. The maxima just given appear to hold for such waves from all depths of origin. It had been noted previously that large amplitudes and periods occur together. The upper limits of the amplitudes of the P and S waves of local shocks were found to vary with the cube of the periods. Different results were found for the variation of epicentral shocks in California and Japan. The difference may be caused by the difference in physical characteristics of the underlying crustal rock. While these studies in Hawaii were made on shocks of intensities I to IV, Rossi-Forel, they show promise of giving information about the waves to be expected in destructive earthquakes. The sectorial lines may be raised by new data, but in each region should approach some unknown lines as a limit. Formulas were used to correct the observed waves to those of standard displacement and consequent period. These periods were plotted with respect to distance and depth, with no reliable result. A tendency was shown for the period of P waves to increase with distance more rapidly than the period of S waves, whereas observations of more distant earthquakes would suggest the opposite. Study of the ratios of the amplitudes of the P to the S wave (AP/SS) showed no distance effect. The formulas from the previous amplitude-period study suggest that this ratio should not vary with the local distance. For Hawaii the ratio averages about 15 per cent. About 60 per cent of the foci are less than 5 km. deep, 70 per cent less than 10 km. deep. Very few appear to have originated at 60 or more km. depth. The decline in numbers of earthquakes with depth is a rapidly decreasing exponential function. Most of the deep earthquakes are under Mauna Loa and the Kilauea southeast rift zone. A large number of the located shallow foci are in and near the Kilauea crater. Possibly this is an increase that should be expected near any active volcanic crater, but it may be due to the close network of stations about Kilauea crater. The magnitude of the shock is not a function of the location either areally or in depth; that is, large earthquakes may be expected in any part of the island and near-by sea bottom and at all depths to at least 60 km. A method of classifying the earthquake records is based on the number of P or S waves shown on the seismogram, which indicate the key number from one to seven. A map of Hawaii was constructed showing the areas in which the different types of shock had originated. The first type, K-1, occurs either central to Mauna Loa or within 50 to 60 km. radius of the seismograph. Type K-2 is not recorded from northwest Hawaii. Type K-3 does not occur close to the instruments. Types K-4 to K-7 are noted to occur at somewhat greater distances, and to date have been observed only from small outlying areas. Earthquake records of simple character are generally near the area of deep-focus shocks and near the seismographs, so that the waves come in at a steep angle. Earthquakes under Kilauea crater are generally simple. As the foci become more distant and shallow they also become more complicated in type. These criteria should help in designating phases and consequent locations, but they are not final, and may be of no help beyond 100 km. The number of phases in some of the records of outlying earthquakes suggest a complexity of structure in the island mass and the near-by sea bottom. The locations near and on the extension of rifts and in pronounced lines and zones suggest a larger and more numerous system of rifts than has previously been mapped. The resulting pattern of rifts about Mauna Loa is roughly an asterisk. The main accent is on the visible active rifts to the southwest and the east-northeast of Mokuaweoweo. These rifts have apparently controlled most of the island's seismicity in the immediate past.

scholarly journals TO STUDY THE POSSIBILITIES OF USING THE SEISMOLOGICAL NETWORK OF THE ALTAI-SAYAN REGION FOR REGIME VIBRO-SEISMIC OBSERVATIONS

2021 ◽  
Vol 2 (2) ◽  
pp. 289-297
Author(s):  
Victor M. Solovyev ◽  
Alexander S. Salnikov ◽  
Viktor S. Seleznev ◽  
Tatyana V. Kashubina ◽  
Natalya А. Galyova
Keyword(s):  
P Wave ◽  
S Wave ◽  
Transverse Waves ◽  
S Waves ◽  
Mountain Ranges ◽  
Wave Data ◽  
Sayan Region ◽  
Deep Seismic Studies ◽  
Seismic Sections ◽  
Profile Alignment

The results of deep seismic studies based on P - and S-wave data on the East-Stanov fragment of the reference 700-kilometer geophysical profile 8-DV are presented. Deep seismic sections of the upper crust (up to a depth of 20 km) with the distribution of the velocities of longitudinal and transverse waves are constructed. The P - wave velocities in the upper part of the section vary from 4-5 km / s within the Upper Zeya and Amur-Zeya depressions to 5.5-6.0 km/s within mountain ranges and plateaus; at depths of 10-20 km, lenses of high-velocity rocks up to 6.7-7.0 km/s are distinguished in the profile alignment. According to the S - waves in the upper part of the section, the velocity values are generally 3.0-3.2 km/s; reduced velocity values of 2.5-2.6 km / s are observed in the Upper Zey depression. At depths of 5-20 km within the section, according to the transverse wave data, a number of sections with reduced and increased velocity values are distinguished, respectively, up to 3.4-3.5 km/s and 3.75-3.8 km/s. The correlation of the selected anomalies according to the data of P-and S-waves is carried out.

scholarly journals Responses of dipole-source reflected shear waves in acoustically slow formations

2019 ◽  
Author(s):  
Hao Wang ◽  
Ning Li ◽  
Caizhi Wang ◽  
Hongliang Wu ◽  
Peng Liu ◽  
...  
Keyword(s):  
Finite Difference Time Domain ◽  
Dipole Source ◽  
Sh Wave ◽  
S Wave ◽  
High Fracture ◽  
S Waves ◽  
Angle Fracture ◽  
Dip Angle ◽  
Difference Time

Abstract In the process of dipole-source acoustic far-detection logging, the azimuth of the fracture outside the borehole can be determined with the assumption that the SH–SH wave is stronger than the SV–SV wave. However, in slow formations, the considerable borehole modulation highly complicates the dipole-source radiation of SH and SV waves. A 3D finite-difference time-domain method is used to investigate the responses of the dipole-source reflected shear wave (S–S) in slow formations and explain the relationships between the azimuth characteristics of the S–S wave and the source–receiver offset and the dip angle of the fracture outside the borehole. Results indicate that the SH–SH and SV–SV waves cannot be effectively distinguished by amplitude at some offset ranges under low- and high-fracture dip angle conditions, and the offset ranges are related to formation properties and fracture dip angle. In these cases, the fracture azimuth determined by the amplitude of the S–S wave not only has a $180^\circ $ uncertainty but may also have a $90^\circ $ difference from the actual value. Under these situations, the P–P, S–P and S–S waves can be combined to solve the problem of the $90^\circ $ difference in the azimuth determination of fractures outside the borehole, especially for a low-dip-angle fracture.

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