PHASE SHIFTS AND RESONANCES IN THE DIRAC EQUATION

We review the analytic results for the phase shifts δl(k) in nonrelativistic scattering from a spherical well. The conditions for the existence of resonances are established in terms of time-delays. Resonances are shown to exist for p-waves (and higher angular momenta) but not for s-waves. These resonances occur when the potential is not quite strong enough to support a bound p-wave of zero energy. We then examine relativistic scattering by spherical wells and barriers in the Dirac equation. In contrast to the nonrelativistic situation, s-waves are now seen to possess resonances in scattering from both wells and barriers. When s-wave resonances occur for scattering from a well, the potential is not quite strong enough to support a zero momentum s-wave solution at E=m. Resonances resulting from scattering from a barrier can be explained in terms of the "crossing" theorem linking s-wave scattering from barriers to p-wave scattering from wells. A numerical procedure to extract phase shifts for general short range potentials is introduced and illustrated by considering relativistic scattering from a Gaussian potential well and barrier.

Download Full-text

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.

Download Full-text

Comparison of recent S-wave indicating methods

E3S Web of Conferences ◽

10.1051/e3sconf/20182900019 ◽

2018 ◽

Vol 29 ◽

pp. 00019

Author(s):

Katarzyna Hubicka ◽

Jakub Sokolowski

Keyword(s):

Real Data ◽

The Body ◽

Body Waves ◽

Identification Accuracy ◽

P Wave ◽

S Wave ◽

P Waves ◽

S Waves ◽

Mode Decomposition ◽

Wave Arrival

Seismic event consists of surface waves and body waves. Due to the fact that the body waves are faster (P-waves) and more energetic (S-waves) in literature the problem of their analysis is taken more often. The most universal information that is received from the recorded wave is its moment of arrival. When this information is obtained from at least four seismometers in different locations, the epicentre of the particular event can be estimated [1]. Since the recorded body waves may overlap in signal, the problem of wave onset moment is considered more often for faster P-wave than S-wave. This however does not mean that the issue of S-wave arrival time is not taken at all. As the process of manual picking is time-consuming, methods of automatic detection are recommended (these however may be less accurate). In this paper four recently developed methods estimating S-wave arrival are compared: the method operating on empirical mode decomposition and Teager-Kaiser operator [2], the modification of STA/LTA algorithm [3], the method using a nearest neighbour-based approach [4] and the algorithm operating on characteristic of signals’ second moments. The methods will be also compared to wellknown algorithm based on the autoregressive model [5]. The algorithms will be tested in terms of their S-wave arrival identification accuracy on real data originating from International Research Institutions for Seismology (IRIS) database.

Download Full-text

Relation between P- and S-wave corner frequencies in the seismic spectrum

Bulletin of the Seismological Society of America ◽

10.1785/bssa0640061621 ◽

1974 ◽

Vol 64 (6) ◽

pp. 1621-1627 ◽

Cited By ~ 2

Author(s):

J. C. Savage

Keyword(s):

High Frequency ◽

Body Wave ◽

Fault Model ◽

P Wave ◽

Corner Frequency ◽

Rupture Propagation ◽

S Wave ◽

P Waves ◽

Fault Surface ◽

S Waves

abstract A comprehensive set of body-wave spectra has been calculated for the Haskell fault model generalized to a circular fault surface. These spectra are used to show that in practice the P-wave corner frequency (ƒp) may exceed the S-wave corner frequency (ƒs) when near-sonic or transonic rupture propagation obtains. The explanation appears to be that in such cases ƒs is so large that it is not identified within the recorded band, but rather a secondary corner is mistaken for ƒs. As a consequence of failing to detect the true asymptotic trend, the high-frequency falloff of the spectrum with frequency is substantially less for S waves than for P waves. This explanation appears to be consistent with the demonstration by Molnar, Tucker, and Brune (1973) that ƒp may exceed ƒs.

Download Full-text

A comparison of some S-wave studies of earthquake mechanisms

Bulletin of the Seismological Society of America ◽

10.1785/bssa0510020277 ◽

1961 ◽

Vol 51 (2) ◽

pp. 277-292

Author(s):

William Stauder ◽

Adams W. M.

Keyword(s):

Fault Plane ◽

Analytical Techniques ◽

P Wave ◽

Graphical Methods ◽

S Wave ◽

Fault Plane Solutions ◽

P Waves ◽

S Waves ◽

Analytical Technique ◽

Direction Of Polarization

Abstract Graphical and analytical techniques for using S-waves in focal mechanism studies are compared. In previous applications the analytical technique has shown little or no agreement with the results of fault-plane solutions from P-waves, whereas for other groups of earthquakes the graphical methods have shown good agreement between the S-waves and the P-wave solutions. It is shown that the graphical and analytical techniques are identical in principle and that when the graphical methods are applied to the same three earthquakes to which the analytical technique had been applied the identical results are obtained. Closer examination of the graphical presentation of the data, however, shows that the disagreement between the S-waves and the fault plane solutions from P is largely apparent. The discrepancy follows upon the peculiar scatter in the S-wave data and the chance occurrence of observations of S at stations located along closely parallel planes of polarization of S. Once this is understood, it is seen that the direction of polarization of S-waves is in substantial agreement with the methods of analysis of focal mechanisms from P-waves, and that the data are consistent with a simple dipole as the point model of the earthquake focus.

Download Full-text

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

Geophysics ◽

10.1190/geo2018-0335.1 ◽

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.

Download Full-text

A simple method for resolving large converted‐wave (P-SV) statics

Geophysics ◽

10.1190/1.1443426 ◽

1993 ◽

Vol 58 (3) ◽

pp. 429-433 ◽

Cited By ~ 27

Author(s):

Peter W. Cary ◽

David W. S. Eaton

Keyword(s):

P Wave ◽

S Wave ◽

Converted Wave ◽

P Waves ◽

Simple Method ◽

Near Surface ◽

S Waves ◽

Static Solutions ◽

Surface Fluctuations

The processing of converted‐wave (P-SV) seismic data requires certain special considerations, such as commonconversion‐point (CCP) binning techniques (Tessmer and Behle, 1988) and a modified normal moveout formula (Slotboom, 1990), that makes it different for processing conventional P-P data. However, from the processor’s perspective, the most problematic step is often the determination of residual S‐wave statics, which are commonly two to ten times greater than the P‐wave statics for the same location (Tatham and McCormack, 1991). Conventional residualstatics algorithms often produce numerous cycle skips when attempting to resolve very large statics. Unlike P‐waves, the velocity of S‐waves is virtually unaffected by near‐surface fluctuations in the water table (Figure 1). Hence, the P‐wave and S‐wave static solutions are largely unrelated to each other, so it is generally not feasible to approximate the S‐wave statics by simply scaling the known P‐wave static values (Anno, 1986).

Download Full-text

Three-component amplitude versus offset analysis

Exploration Geophysics ◽

10.1071/eg989257 ◽

1989 ◽

Vol 20 (2) ◽

pp. 257

Author(s):

D.R. Miles ◽

G. Gassaway ◽

L. Bennett ◽

R. Brown

Keyword(s):

Seismic Data ◽

P Wave ◽

S Wave ◽

Converted Wave ◽

P Waves ◽

S Waves ◽

Avo Analysis ◽

Avo Inversion ◽

Amplitude Versus Offset ◽

Converted Waves

Three-component (3-C) amplitude versus offset (AVO) inversion is the AVO analysis of the three major energies in the seismic data, P-waves, S-waves and converted waves. For each type of energy the reflection coefficients at the boundary are a function of the contrast across the boundary in velocity, density and Poisson's ratio, and of the angle of incidence of the incoming wave. 3-C AVO analysis exploits these relationships to analyse the AVO changes in the P, S, and converted waves. 3-C AVO analysis is generally done on P, S, and converted wave data collected from a single source on 3-C geophones. Since most seismic sources generate both P and S-waves, it follows that most 3-C seismic data may be used in 3-C AVO inversion. Processing of the P-wave, S-wave and converted wave gathers is nearly the same as for single-component P-wave gathers. In split-spread shooting, the P-wave and S-wave energy on the radial component is one polarity on the forward shot and the opposite polarity on the back shot. Therefore to use both sides of the shot, the back shot must be rotated 180 degrees before it can be stacked with the forward shot. The amplitude of the returning energy is a function of all three components, not just the vertical or radial, so all three components must be stacked for P-waves, then for S-waves, and finally for converted waves. After the gathers are processed, reflectors are picked and the amplitudes are corrected for free-surface effects, spherical divergence and the shot and geophone array geometries. Next the P and S-wave interval velocities are calculated from the P and S-wave moveouts. Then the amplitude response of the P and S-wave reflections are analysed to give Poisson's ratio. The two solutions are then compared and adjusted until they match each other and the data. Three-component AVO inversion not only yields information about the lithologies and pore-fluids at a specific location; it also provides the interpreter with good correlations between the P-waves and the S-waves, and between the P and converted waves, thus greatly expanding the value of 3-C seismic data.

Download Full-text

Processing, correlating, and interpreting converted shear waves from borehole data in southern Alberta

Geophysics ◽

10.1190/1.1442878 ◽

1990 ◽

Vol 55 (6) ◽

pp. 660-669 ◽

Cited By ~ 8

Author(s):

W. T. Geis ◽

R. R. Stewart ◽

M. J. Jones ◽

P. E. Katapodis

Keyword(s):

P Wave ◽

Equation Modeling ◽

S Wave ◽

Borehole Data ◽

P Waves ◽

S Waves ◽

Sonic Log ◽

Lake Formation ◽

Southern Alberta ◽

The Time Domain

Borehole measurements coupled with phase information from Zoeppritz equation modeling has assisted in accurate correlation between a VSP converted S-wave section and both the surface and VSP P-wave sections from southern Alberta. For the most part, both the character and polarities of the sections agree; however, there are some differences. Some reflections are stronger and more distinct on the S-wave section than on the P-wave section. Spectral analysis of the time‐domain upgoing P-wave and S-wave energy shows that the frequency content of the S-waves is comparable to the P-waves. Thus, the slower velocity S-waves have a shorter wavelenght and provide better vertical resolution of some interfaces. Other upgoing S-wave modes can interfere with the P‐SV mode and contribute to the differences between the P- and S-wave sections. The match between P-wave and S-wave velocities ([Formula: see text] and [Formula: see text]), determined from VSP traveltime inversion and the full‐waveform sonic log, is best in the Paleozoic carbonate section; there is some discrepancy in Cretaceous sandstone intervals. A basal salt unit in the Paleozoic Beaverhill Lake formation has a VSP‐determined [Formula: see text] ratio of 1.97, suggesting that salt can be distinguished from carbonates using both P-wave and S-wave velocity information in this region.

Download Full-text