Reflection and refraction of elastic waves at a corrugated interface

abstract The effect of a corrugated interface on wave propagation is considered by using the method that was first applied to acoustical gratings by Rayleigh. The problem is what happens when a plane P wave is incident on a corrugated interface that separates two semi-infinite media. As is well known, there are irregular (scattered) waves as well as regular waves. By assuming both the amplitude and the slope of a corrugated interface to be small, quantities of the order of the square of corrugation amplitude are taken into account. In the case of normal incidence for three models considered, the effect of corrugation on reflection is larger than the effect of corrugation on refraction; the amplitude of the regularly reflected waves decreases, and that of the regularly refracted waves and of the irregular waves increases, as the corrugation amplitude becomes larger. Generally, the larger the velocity contrast, the larger the variation of wave amplitude with the wavelength and the amplitude of corrugation. The S wave component generally becomes larger as the wavelength of corrugation becomes smaller. Boundary waves exist, depending upon the ratio of wavelength of corrugation to that of the incident wave. For a specified interface, it is possible that there is a significant difference in wave amplitude as a function of the elastic constants. In the case of oblique incidence, computation was carried out for angles of incidence smaller than 15° for one model. For these small angles of incidence, almost all results for the case of normal incidence still hold. Furthermore, it can be concluded that the effect of the angle of incidence on reflected S waves is larger than for the other waves and that large differences in the amplitudes of waves at different angles of incidence may be expected for the irregular waves.

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Scalar reverse‐time depth migration of prestack elastic seismic data

Geophysics ◽

10.1190/1.1487098 ◽

2001 ◽

Vol 66 (5) ◽

pp. 1519-1527 ◽

Cited By ~ 106

Author(s):

Robert Sun ◽

George A. McMechan

Keyword(s):

Seismic Waves ◽

Velocity Model ◽

P Wave ◽

Common Source ◽

Reverse Time ◽

S Wave ◽

Wave Source ◽

Reflected Waves ◽

S Waves ◽

Elastic Data

Reflected P‐to‐P and P‐to‐S converted seismic waves in a two‐component elastic common‐source gather generated with a P‐wave source in a two‐dimensional model can be imaged by two independent scalar reverse‐time depth migrations. The inputs to migration are pure P‐ and S‐waves that are extracted by divergence and curl calculations during (shallow) extrapolation of the elastic data recorded at the earth’s surface. For both P‐to‐P and P‐to‐S converted reflected waves, the imaging time at each point is the P‐wave traveltime from the source to that point. The extracted P‐wave is reverse‐time extrapolated and imaged with a P‐velocity model, using a finite difference solution of the scalar wave equation. The extracted S‐wave is reverse‐time extrapolated and imaged similarly, but with an S‐velocity model. Converted S‐wave data requires a polarity correction prior to migration to ensure constructive interference between data from adjacent sources. Synthetic examples show that the algorithm gives satisfactory results for laterally inhomogeneous models.

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Amplitude balancing in separating P- and S-waves in 2D and 3D elastic seismic data

Geophysics ◽

10.1190/1.3555529 ◽

2011 ◽

Vol 76 (3) ◽

pp. S103-S113 ◽

Cited By ~ 45

Author(s):

Robert Sun ◽

George A. McMechan ◽

Han-Hsiang Chuang

Keyword(s):

Wave Amplitude ◽

Amplitude Ratio ◽

Velocity Ratio ◽

Displacement Component ◽

P Wave ◽

Elastic Displacement ◽

Reverse Time ◽

S Wave ◽

Near Surface ◽

S Waves

The reflected P- and S-waves in elastic displacement component data recorded at the earth’s surface are separated by reverse-time (downward) extrapolation of the data in an elastic computational model, followed by calculations to give divergence (dilatation) and curl (rotation) at a selected reference depth. The surface data are then reconstructed by separate forward-time (upward) scalar extrapolations, from the reference depth, of the magnitude of the divergence and curl wavefields, and extraction of the separated P- and S-waves, respectively, at the top of the models. A P-wave amplitude will change by a factor that is inversely proportional to the P-velocity when it is transformed from displacement to divergence, and an S-wave amplitude will change by a factor that is inversely proportional to the S-velocity when it is transformed from displacement to curl. Consequently, the ratio of the P- to the S-wave amplitude (the P-S amplitude ratio) in the form of divergence and curl (postseparation) is different from that in the (preseparation) displacement form. This distortion can be eliminated by multiplying the separated S-wave (curl) by a relative balancing factor (which is the S- to P-velocity ratio); thus, the postseparation P-S amplitude ratio can be returned to that in the preseparation data. The absolute P- and S-wave amplitudes are also recoverable by multiplying them by a factor that depends on frequency, on the P-velocity α, and on the unit of α and is location-dependent if the near-surface P-velocity is not constant.

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Characterization of naturally fractured Arbuckle Group in the Wellington Field, Kansas, using S-wave amplitude variation with offset

Interpretation ◽

10.1190/int-2016-0061.1 ◽

2017 ◽

Vol 5 (1) ◽

pp. T49-T63 ◽

Cited By ~ 5

Author(s):

Menal Gupta ◽

Kyle Spikes ◽

Bob Hardage

Keyword(s):

Wave Amplitude ◽

Anisotropy Parameter ◽

P Wave ◽

Fracture Density ◽

S Wave ◽

Amplitude Variation ◽

Fracture Characterization ◽

Amplitude Variation With Offset ◽

S Waves ◽

Arbuckle Group

S-wave amplitude variation with offset (AVO) analysis is sensitive to the presence of fractures and can provide a high-resolution seismic-based fracture characterization as compared with traditionally used traveltime-based methods. To determine viable attributes for estimation of properties such as spatial density and fluid fill of fractures, S-wave AVO modeling and analysis is carried out in the Wellington Field, Kansas, where 9C-2D seismic data have been acquired. Analysis is performed on the Ordovician fractured-carbonate interval called the Arbuckle Group, which is being considered for [Formula: see text] sequestration. AVO modeling of the Arbuckle interval indicates that differences in AVO intercepts of different S-wave polarizations can estimate S-wave anisotropy parameter [Formula: see text], which gives an estimate of fracture density. In addition, modeling suggests that AVO gradients of [Formula: see text] and [Formula: see text] waves can be used to derive a seismic attribute to discriminate fluid fill in fractures, provided good-quality S-wave gathers are available. The intercept anisotropy (IA) attribute obtained from AVO intercepts of S-waves provides fracture density estimates within the Arbuckle Group. These estimates are consistent with the field-wide, low-frequency observations from seismic velocities and spatially limited, high-frequency estimates obtained from drill cores and sonic and borehole-image logs. The IA attribute highlights possible high-permeability zones in the Upper and Lower Arbuckle suitable for [Formula: see text] injection. The Middle Arbuckle indicates low fracture density, potentially acting as a baffle to vertical flow and providing a seal for the Lower Arbuckle. The gradient anisotropy attribute obtained from the AVO gradient of S-waves suggests that most fractures in the Arbuckle are brine saturated. This attribute has a potential application in monitoring the movement of a [Formula: see text] plume in the Arbuckle Group when time-lapse data become available. These results demonstrate that S-wave AVO attributes can supplement the P-wave derived subsurface properties and significantly reduce uncertainties in subsurface fracture characterization.

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

Geophysics ◽

10.1190/geo2015-0265.1 ◽

2016 ◽

Vol 81 (3) ◽

pp. D283-D291 ◽

Cited By ~ 4

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.

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Green's function of the Lord–Shulman thermo-poroelasticity theory

Geophysical Journal International ◽

10.1093/gji/ggaa100 ◽

2020 ◽

Vol 221 (3) ◽

pp. 1765-1776 ◽

Cited By ~ 1

Author(s):

Jia Wei ◽

Li-Yun Fu ◽

Zhi-Wei Wang ◽

Jing Ba ◽

José M Carcione

Keyword(s):

Plane Wave ◽

Green’S Function ◽

Green's Function ◽

Heterogeneous Media ◽

Numerical Algorithms ◽

Heat Diffusion ◽

P Wave ◽

Wave Analysis ◽

S Wave ◽

S Waves

SUMMARY The Lord–Shulman thermoelasticity theory combined with Biot equations of poroelasticity, describes wave dissipation due to fluid and heat flow. This theory avoids an unphysical behaviour of the thermoelastic waves present in the classical theory based on a parabolic heat equation, that is infinite velocity. A plane-wave analysis predicts four propagation modes: the classical P and S waves and two slow waves, namely, the Biot and thermal modes. We obtain the frequency-domain Green's function in homogeneous media as the displacements-temperature solution of the thermo-poroelasticity equations. The numerical examples validate the presence of the wave modes predicted by the plane-wave analysis. The S wave is not affected by heat diffusion, whereas the P wave shows an anelastic behaviour, and the slow modes present a diffusive behaviour depending on the viscosity, frequency and thermoelasticity properties. In heterogeneous media, the P wave undergoes mesoscopic attenuation through energy conversion to the slow modes. The Green's function is useful to study the physics in thermoelastic media and test numerical algorithms.

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Estimation of radiated energy using the KiK-net downhole records—old method for modern data

Geophysical Journal International ◽

10.1093/gji/ggaa040 ◽

2020 ◽

Vol 221 (2) ◽

pp. 1029-1042 ◽

Cited By ~ 1

Author(s):

Hiroo Kanamori ◽

Zachary E Ross ◽

Luis Rivera

Keyword(s):

Site Response ◽

Source Model ◽

Synthetic Seismograms ◽

S Wave ◽

P Waves ◽

S Waves ◽

Radiated Energy ◽

Time Integral ◽

Magnitude Range ◽

Significant Difference

SUMMARY We use KiK-net (NIED) downhole records to estimate the radiated energy, ER, of 29 Japanese inland earthquakes with a magnitude range from Mw = 5.6 to 7.0. The method is based on the work of Gutenberg and Richter in which the time integral of S-wave ground-motion velocity-squared is measured as a basic metric of the radiated energy. Only stations within a distance of 100 km are used to minimize complex path and attenuation effects. Unlike the teleseismic method that uses mainly P waves, the use of S waves which carry more than 95 per cent of the radiated energy allows us to obtain robust results. We calibrate the method using synthetic seismograms to modernize and improve the Gutenberg–Richter method. We compute synthetic seismograms for a source model of each event with a given source function (i.e. known ER), the actual mechanism and the source-station geometry. Then, we compare the given ER with the computed energy metric to correct for the unknown effect of wave propagation and the mechanism. The use of downhole records minimizes the uncertainty resulting from the site response. Our results suggest that the currently available estimates of ER from teleseismic data are probably within a factor of 3, on average, of the absolute value. The scaled energy eR ( = ER/M0) is nearly constant at about 3 × 10−5 over a magnitude range from Mw = 5.6 to 7.0 with a slight increasing trend with Mw. We found no significant difference in eR between dip-slip and strike-slip events.

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A new acoustic assumption for orthorhombic media

Geophysical Journal International ◽

10.1093/gji/ggaa367 ◽

2020 ◽

Vol 223 (2) ◽

pp. 1118-1129

Author(s):

Mohammad Mahdi Abedi ◽

Alexey Stovas

Keyword(s):

Wave Propagation ◽

Conventional Approach ◽

P Wave ◽

Model Parameters ◽

Governing Equations ◽

S Wave ◽

S Waves ◽

Exploration Seismology ◽

And Inversion ◽

Approximate Equations

SUMMARY In exploration seismology, the acquisition, processing and inversion of P-wave data is a routine. However, in orthorhombic anisotropic media, the governing equations that describe the P-wave propagation are coupled with two S waves that are considered as redundant noise. The main approach to free the P-wave signal from the S-wave noise is the acoustic assumption on the wave propagation. The conventional acoustic assumption for orthorhombic media zeros out the S-wave velocities along three orthogonal axes, but leaves significant S-wave artefacts in all other directions. The new acoustic assumption that we propose mitigates the S-wave artefacts by zeroing out their velocities along the three orthogonal symmetry planes of orthorhombic media. Similar to the conventional approach, our method reduces the number of required model parameters from nine to six. As numerical experiments on multiple orthorhombic models show, the accuracy of the new acoustic assumption also compares well to the conventional approach. On the other hand, while the conventional acoustic assumption simplifies the governing equations, the new acoustic assumption further complicates them—an issue that emphasizes the necessity of simple approximate equations. Accordingly, we also propose simpler rational approximate phase-velocity and eikonal equations for the new acoustic orthorhombic media. We show a simple ray tracing example and find out that the proposed approximate equations are still highly accurate.

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Comparison of P‐ and S‐wave velocities and Q’S from VSP and sonic log data

Geophysics ◽

10.1190/1.1443541 ◽

1994 ◽

Vol 59 (10) ◽

pp. 1512-1529 ◽

Cited By ~ 22

Author(s):

Gopa S. De ◽

Donald F. Winterstein ◽

Mark A. Meadows

Keyword(s):

Velocity Dispersion ◽

P Wave ◽

S Wave ◽

S Waves ◽

Wave Velocities ◽

Sonic Log ◽

Transit Times ◽

Wave Slope ◽

Northern Alberta ◽

Q Values

We compared P‐ and S‐wave velocities and quality factors (Q’S) from vertical seismic profiling (VSP) and sonic log measurements in five wells, three from the southwest San Joaquin Basin of California, one from near Laredo, Texas, and one from northern Alberta. Our purpose was to investigate the bias between sonic log and VSP velocities and to examine to what degree this bias might be a consequence of dispersion. VSPs and sonic logs were recorded in the same well in every case. Subsurface formations were predominantly clastic. The bias found was that VSP transit times were greater than sonic log times, consistent with normal dispersion. For the San Joaquin wells, differences in S‐wave transit times averaged 1–2 percent, while differences in P‐wave transit times averaged 6–7 percent. For the Alberta well, the situation was reversed, with differences in S‐wave transit times being about 6 percent, while those for P‐waves were 2.5 percent. For the Texas well, the differences averaged about 4 percent for both P‐ and S‐waves. Drift‐curve slopes for S‐waves tended to be low where the P‐wave slopes were high and vice versa. S‐wave drift‐curve slopes in the shallow California wells were 5–10 μs/ft (16–33 μs/m) and the P‐wave slopes were 15–30 μs/ft (49–98 μs/m). The S‐wave slope in sandstones in the northern Alberta well was up to 50 μs/ft (164 μs/m), while the P‐wave slope was about 5 μs/ft (16 μs/m). In the northern Alberta well the slopes for both P‐ and S‐waves flattened in the carbonate. In the Texas well, both P‐ and S‐wave drifts were comparable. We calculated (Q’s) from a velocity dispersion formula and from spectral ratios. When the two Q’s agreed, we concluded that velocity dispersion resulted solely from absorption. These Q estimation methods were reliable only for Q values smaller than 20. We found that, even with data of generally outstanding quality, Q values determined by standard methods can have large uncertainties, and negative Q’s may be common.

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Velocity anisotropy in shale determined from crosshole seismic and vertical seismic profile data

Geophysics ◽

10.1190/1.1442856 ◽

1990 ◽

Vol 55 (4) ◽

pp. 470-479 ◽

Cited By ~ 39

Author(s):

D. F. Winterstein ◽

B. N. P. Paulsson

Keyword(s):

Seismic Profile ◽

P Wave ◽

Isotropic Model ◽

S Wave ◽

Vertical Seismic Profile ◽

Near Surface ◽

S Waves ◽

Velocity Anisotropy ◽

Wave Velocities ◽

Late Tertiary

Crosshole and vertical seismic profile (VST) data made possible accurate characterization of the elastic properties, including noticeable velocity anisotropy, of a near‐surface late Tertiary shale formation. Shear‐wave splitting was obvious in both crosshole and VSP data. In crosshole data, two orthologonally polarrized shear (S) waves arrived 19 ms in the uppermost 246 ft (75 m). Vertically traveling S waves of the VSP separated about 10 ms in the uppermost 300 ft (90 m) but remained at nearly constant separation below that level. A transversely isotropic model, which incorporates a rapid increase in S-wave velocities with depth but slow increase in P-wave velocities, closely fits the data over most of the measured interval. Elastic constants of the transvesely isotropic model show spherical P- and [Formula: see text]wave velocity surfaces but an ellipsoidal [Formula: see text]wave surface with a ratio of major to minor axes of 1.15. The magnitude of this S-wave anisotropy is consistent with and lends credence to S-wave anisotropy magnitudes deduced less directly from data of many sedimentary basins.

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Travel-Time Inversion Method of Converted Shear Waves Using RayInvr Algorithm

Applied Sciences ◽

10.3390/app11083571 ◽

2021 ◽

Vol 11 (8) ◽

pp. 3571

Author(s):

Genggeng Wen ◽

Kuiyuan Wan ◽

Shaohong Xia ◽

Huilong Xu ◽

Chaoyan Fan ◽

...

Keyword(s):

Travel Time ◽

Wave Velocity ◽

P Wave ◽

Inversion Method ◽

Time Inversion ◽

Ocean Bottom Seismometer ◽

S Wave ◽

S Waves ◽

Inversion Model ◽

Travel Time Inversion

The detailed studies of converted S-waves recorded on the Ocean Bottom Seismometer (OBS) can provide evidence for constraining lithology and geophysical properties. However, the research of converted S-waves remains a weakness, especially the S-waves’ inversion. In this study, we applied a travel-time inversion method of converted S-waves to obtain the crustal S-wave velocity along the profile NS5. The velocities of the crust are determined by the following four aspects: (1) modelling the P-wave velocity, (2) constrained sediments Vp/Vs ratios and S-wave velocity using PPS phases, (3) the correction of PSS phases’ travel-time, and (4) appropriate parameters and initial model are selected for inversion. Our results show that the vs. and Vp/Vs of the crust are 3.0–4.4 km/s and 1.71–1.80, respectively. The inversion model has a similar trend in velocity and Vp/Vs ratios with the forward model, due to a small difference with ∆Vs of 0.1 km/s and ∆Vp/Vs of 0.03 between two models. In addition, the high-resolution inversion model has revealed many details of the crustal structures, including magma conduits, which further supports our method as feasible.

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