S-wave singularities in tilted orthorhombic media

Geophysics ◽  
2017 ◽  
Vol 82 (4) ◽  
pp. WA11-WA21 ◽  
Author(s):  
Yuriy Ivanov ◽  
Alexey Stovas
Keyword(s):  
Numerical Models ◽  
Anomalous Behavior ◽  
Shear Mode ◽  
S Wave ◽  
Slowness Surface ◽  
S Waves ◽  
Hyperbolic Region ◽  
Conical Points ◽  
Zero Offset ◽  
Anisotropy Strength

Quasi S-wave propagation in low-symmetry anisotropic media is complicated due to the existence of point singularities (conical points) — points in the phase space at which slowness sheets of the split S-waves touch each other. At these points, two eigenvalues of the Christoffel tensor (associated with the quasi S-waves) degenerate into one and polarization directions of the S-waves, which lay in the plane orthogonal to the polarization of the quasi longitudinal wave, are not uniquely defined. In the vicinity of these points, slowness sheets of the S-waves have complicated shapes, leading to rapid variations in polarization directions, multipathing, and cusps and discontinuities of the shear wavefronts. In a tilted orthorhombic medium, the point singularities can occur close to the vertical, distorting the traveltime parameters that are defined at the zero offset. We have analyzed the influence of the singularities on these parameters by examining the derivatives of the slowness surface up to the fourth order. Using two orthorhombic numerical models of different shear anisotropy strength and with different number of singularity points, we evaluate the complexity of the slowness sheets in the vicinity of the conical points and analyze how the traveltime parameters are affected by the singularities. In particular, we observe that the hyperbolic region associated with the singularity points in a model with moderate to strong shear anisotropy spans over a big portion of the slowness surfaces and the traveltime parameters are strongly affected outside the hyperbolic region. In general, the fast shear mode is less affected by the singularities; however, the effect is still very pronounced. Moreover, the hyperbolic region associated with the singularity points on the slow S-wave affects the slowness surface of the fast mode extensively. In addition, we evaluate a relation between the slowness surface Gaussian curvature and the relative geometric spreading, which has anomalous behavior due to the singularities.

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.

Thin-bed prestack spectral inversion

Geophysics ◽  
2009 ◽  
Vol 74 (4) ◽  
pp. R49-R57 ◽  
Author(s):  
J. Germán Rubino ◽  
Danilo Velis
Keyword(s):  
Time Window ◽  
A Priori ◽  
Amplitude Spectrum ◽  
Wavelet Spectrum ◽  
Model Parameters ◽  
Data Sets ◽  
S Wave ◽  
P Waves ◽  
S Waves ◽  
Zero Offset

Prestack seismic data has been used in a new method to fully determine thin-bed properties, including the estimation of its thickness, P- and S-wave velocities, and density. The approach requires neither phase information nor normal-moveout (NMO) corrections, and assumes that the prestack seismic response of the thin layer can be isolated using an offset-dependent time window. We obtained the amplitude-versus-angle (AVA) response of the thin bed considering converted P-waves, S-waves, and all the associated multiples. We carried out the estimation of the thin-bed parameters in the frequency (amplitude spectrum) domain using simulated annealing. In contrast to using zero-offset data, the use of AVA data contributes to increase the robustness of this inverse problem under noisy conditions, as well as to significantly reduce its inherent nonuniqueness. To further reduce the nonuniqueness, and as a means to incorporate a priori geologic or geophysical information (e.g., well-log data), we imposed appropriate bounding constraints to the parameters of the media lying above and below the thin bed, which need not be known accurately. We tested the method by inverting noisy synthetic gathers corresponding to simple wedge models. In addition, we stochastically estimated the uncertainty of the solutions by inverting different data sets that share the same model parameters but are contaminated with different noise realizations. The results suggest that thin beds can be characterized fully with a moderate to high degree of confidence below tuning, even when using an approximate wavelet spectrum.

Normal moveout velocity ellipse in tilted orthorhombic media

Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. C319-C336 ◽  
Author(s):  
Yuriy Ivanov ◽  
Alexey Stovas
Keyword(s):  
Wave Mode ◽  
Initial Position ◽  
Orthorhombic Symmetry ◽  
Arbitrary Orientation ◽  
S Wave ◽  
Converted Wave ◽  
Slowness Surface ◽  
S Waves ◽  
Trade Offs ◽  
The Individual

Normal moveout (NMO) velocity is a commonly used tool in the seismic industry nowadays. In 3D surveys, the variation of the NMO velocity in a horizontal plane is elliptic in shape for the anisotropy or heterogeneity of any strength (apart from a few exotic cases). The NMO ellipse is used for Dix-type inversion and can provide important information on the strength of anisotropy and the orientation of the vertical symmetry planes, which can correspond, for example, to fractures’ orientation and compliances. To describe a vertically fractured finely layered medium (the fracture is orthogonal to the layering), an anisotropy of orthorhombic symmetry is commonly used. In areas with complicated geology and stress distribution, the orientation of the orthorhombic symmetry planes can be considerably altered from the initial position. We have derived the exact equations for the NMO ellipse in an elastic tilted orthorhombic layer with an arbitrary orientation of the symmetry planes. We have evaluated pure and converted wave modes and determined that the influence of the orientation upon the NMO ellipse for all the waves is strong. We have considered acoustic and ellipsoidal orthorhombic approximations of the NMO ellipse equations, which we used to develop a numerical inversion scheme. We determined that in the most general case of arbitrary orientation of the orthorhombic symmetry planes, the inversion results are unreliable due to significant trade-offs between the parameters. We have evaluated S-wave features such as point singularities (slowness surfaces of the split S-waves cross) and triplications (due to concaveness of the individual S-wave mode slowness surface) and their influence on the NMO ellipse.

Elastic-wavefield interferometry using P-wave source VSPs at the Wamsutter field, Wyoming

Geophysics ◽  
2012 ◽  
Vol 77 (2) ◽  
pp. Q27-Q36 ◽  
Author(s):  
James Gaiser ◽  
Ivan Vasconcelos ◽  
Rosemarie Geetan ◽  
John Faragher
Keyword(s):  
Stationary Phase ◽  
Wave Velocity ◽  
P Wave ◽  
Phase Point ◽  
S Wave ◽  
Wave Source ◽  
S Waves ◽  
Wave Splitting ◽  
Zero Offset ◽  
Tube Wave

In this study, elastic-wavefield interferometry was used to recover P- and S-waves from the 3D P-wave vibrator VSP data at Wamsutter field in Wyoming. S-wave velocity and birefringence is of particular interest for the geophysical objectives of lithology discrimination and fracture characterization in naturally fractured tight gas sand reservoirs. Because we rely on deconvolution interferometry for retrieving interreceiver P- and S-waves in the subsurface, the output fields are suitable for high-resolution, local reservoir characterization. In 1D media where the borehole is nearly vertical, data at the stationary-phase point is not conducive to conventional interferometry. Strong tube-wave noise generated by physical sources near the borehole interfere with S-wave splitting analyses. Also, converted P- to S-wave (PS-wave) polarity reversals occur at zero offset and cancel their recovery. We developed methods to eliminate tube-wave noise by removing physical sources at the stationary-phase point and perturbing the integration path in the integrand based on P-wave NMO velocity of the direct-arrival. This results in using nonphysical energy outside a Fresnel radius that could not have propagated between receivers. To limit the response near the stationary-phase point, we also applied a weighting condition to suppress energy from large offsets. For PS-waves, a derivative-like operator was applied to the physical sources at zero offset in the form of a polarity reversal. These methods resulted in effectively recovering P-wave dipole and PS-wave quadrupole pseudosource VSPs. The retrieved wavefields kinematically correspond to a vertical incidence representation of reflectivity/transmissivity and can be used for conventional P- and S-wave velocity analyses. Four-component PS-wave VSPs retrieve S-wave splitting in transmitted converted waves that provide calibration for PS-wave and P-wave azimuthal anisotropy measurements from surface-seismic data.

On the Biot slow S-wave

Geophysics ◽  
2008 ◽  
Vol 73 (4) ◽  
pp. N19-N33 ◽  
Author(s):  
Pratap N. Sahay
Keyword(s):  
Strain Rate ◽  
Constitutive Relation ◽  
Seismic Waves ◽  
Mode Conversion ◽  
Shear Mode ◽  
Biot Theory ◽  
Fluid Viscosity ◽  
S Wave ◽  
S Waves ◽  
Coupled Equations

It is accepted widely that the Biot theory predicts only one shear wave representing the in-phase/unison shear motions of the solid and fluid constituent phases (fast S-wave). The Biot theory also contains a shear mode wherein the two constituent phases essentially undergo out-of-phase shear motions (slow S-wave). From the outset of the development of the Biot framework, the existence of this mode has remained unnoticed because of an oversight in decoupling its system of two coupled equations governing shear processes. Moreover, in the absence of the fluid strain-rate term in the Biot constitutive relation, the velocity of this mode is zero. Once the Biot constitutive relation is corrected for the missing fluid strain-rate term (i.e., fluid viscosity), this mode turns out to be, in the inertial regime, a diffusive process akin to a viscous wave in a Newtonian fluid. In the viscous regime, it degenerates to a process governed by a diffusion equation with a damping term. Although this mode is damped so heavily that it dies off rapidly near its source, overlooking its existence ignores a mechanism to draw energy from seismic waves (fast P- and S-waves) via mode conversion at interfaces and at other material discontinuities and inhomogeneities. To illustrate the consequence of generating this mode at an interface, I examine the case of a horizontally polarized fast S-wave normal incident upon a planar air-water interface in a porous medium. Contrary to the classical Biot framework, which suggests that the incident wave should be transmitted practically unchanged through such an interface, the viscosity-corrected Biot framework predicts a strong, fast S-wave reflection because of the slow S-wave generated at the interface.

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.

Estimates of Q in central Asia as a function of frequency and depth using the coda of locally recorded earthquakes

1982 ◽  
Vol 72 (1) ◽  
pp. 129-149
Author(s):  
S. W. Roecker ◽  
B. Tucker ◽  
J. King ◽  
D. Hatzfeld
Keyword(s):  
Frequency Dependence ◽  
Central Asia ◽  
Wide Frequency Range ◽  
S Wave ◽  
Uppermost Mantle ◽  
Regional Heterogeneity ◽  
S Waves ◽  
Tectonic Processes ◽  
Hindu Kush ◽  
Attenuation Mechanism

abstract Digital recordings of microearthquake codas from shallow and intermediate depth earthquakes in the Hindu Kush region of Afghanistan were used to determine the attenuation factors of the S-wave coda (Qc) and primary S waves (Qβ). An anomalously rapid decay of the coda shortly after the S-wave arrival, observed also in a study of coda in central Asia by Rautian and Khalturin (1978), seems to be due primarily to depth-dependent variations in Qc. In particular, we deduce the average Qc in the crust and uppermost mantle (<100-km depth) is approximately four times lower than the deeper mantle (<400-km depth) over a wide frequency range (0.4 to 24 Hz). Further, while Qc generally increases with frequency at any depth, the degree of frequency dependence of Qc depends on depth. Except at the highest frequency studied here (∼48 Hz), the magnitude of Qc at a particular frequency increases with depth while its frequency dependence decreases. For similar depths, determinations of Qβ and Qc agree, suggesting a common wave composition and attenuation mechanism for S waves and codas. Comparison of these determinations of Qc in Afghanistan with those in other parts of the world shows that the degree of frequency dependence of Qc correlates with the expected regional heterogeneity. Such a correlation supports the prejudice that Qc is primarily influenced by scattering and suggests that tectonic processes such as folding and faulting are instrumental in creating scattering environments.

scholarly journals Array data processing techniques applied to long-period shear waves at Fennoscandian seismograph stations

1969 ◽  
Vol 59 (5) ◽  
pp. 1863-1887
Author(s):  
James H. Whitcomb
Keyword(s):  
Data Processing ◽  
Velocity Ratio ◽  
Underground Explosion ◽  
High Signal ◽  
S Wave ◽  
Array Data ◽  
S Waves ◽  
Long Period ◽  
Normal Record ◽  
The Individual

abstract Array data processing is applied to long-period records of S waves at a network of five Fennoscandian seismograph stations (Uppsala, Umeå, Nurmijärvi, Kongsberg, Copenhagen) with a maximum separation of 1300 km. Records of five earthquakes and one underground explosion are included in the study. The S motion is resolved into SH and SV, and after appropriate time shifts the individual traces are summed, both directly and after weighting. In general, high signal correlation exists among the different stations involved resulting in more accurate time readings, especially for records which have amplitudes that are too small to be read normally. S-wave station residuals correlate with the general crustal type under each station. In addition, the Fennoscandian shield may have a higher SH/SV velocity ratio than the adjacent tectonic area to the northwest.SV-to-P conversion at the base of the crust can seriously interfere with picking the onset of Sin normal record reading. The study demonstrates that, for epicentral distances beyond about 30°, existing networks of seismograph stations can be successfully used for array processing of long-period arrivals, especially the S arrivals.

Numerical simulation of azimuthal acoustic logging in a borehole penetrating a rock formation boundary

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. D283-D291 ◽  
Author(s):  
Peng Liu ◽  
Wenxiao Qiao ◽  
Xiaohua Che ◽  
Xiaodong Ju ◽  
Junqiang Lu ◽  
...  
Keyword(s):  
Domain Model ◽  
P Wave ◽  
S Wave ◽  
Angle Variation ◽  
P Waves ◽  
Acoustic Logging ◽  
S Waves ◽  
Rock Formation ◽  
Logging Tool ◽  
Difference Time

We have developed a new 3D acoustic logging tool (3DAC). To examine the azimuthal resolution of 3DAC, we have evaluated a 3D finite-difference time-domain model to simulate a case in which the borehole penetrated a rock formation boundary when the tool worked at the azimuthal-transmitting-azimuthal-receiving mode. The results indicated that there were two types of P-waves with different slowness in waveforms: the P-wave of the harder rock (P1) and the P-wave of the softer rock (P2). The P1-wave can be observed in each azimuthal receiver, but the P2-wave appears only in the azimuthal receivers toward the softer rock. When these two types of rock are both fast formations, two types of S-waves also exist, and they have better azimuthal sensitivity compared with P-waves. The S-wave of the harder rock (S1) appears only in receivers toward the harder rock, and the S-wave of the softer rock (S2) appears only in receivers toward the softer rock. A model was simulated in which the boundary between shale and sand penetrated the borehole but not the borehole axis. The P-wave of shale and the S-wave of sand are azimuthally sensitive to the azimuth angle variation of two formations. In addition, waveforms obtained from 3DAC working at the monopole-transmitting-azimuthal-receiving mode indicate that the corresponding P-waves and S-waves are azimuthally sensitive, too. Finally, we have developed a field example of 3DAC to support our simulation results: The azimuthal variation of the P-wave slowness was observed and can thus be used to reflect the azimuthal heterogeneity of formations.

A computer program for focal mechanism determination combining P and S wave data

1969 ◽  
Vol 59 (2) ◽  
pp. 503-519
Author(s):  
Agustin Udias ◽  
Dieter Baumann
Keyword(s):  
Computer Program ◽  
Focal Mechanism ◽  
Total Error ◽  
Source Model ◽  
Polarization Angle ◽  
S Wave ◽  
Period Range ◽  
S Waves ◽  
Double Couple ◽  
The Relationship

abstract A computer program has been developed to find the orientation of a double couple source model for the mechanism of an earthquake which best satisfies the data from P and S waves. The relationship between the two axes of the solution given by the equations for the polarization angle of S is used in order to rapidly find the orientation of the source model for which a total error value involving the error of S and P data is a minimum. The program gives best results for data from homogeneous instruments of similar period range. Solutions for three earthquakes, selected because of the orientation of the source, are presented and the reliability of their solutions under ideal conditions is discussed.

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