Pp-wave Research Articles

DETERMINATION OF AZIMUTHAL ANISOTROPIC MEDIA ELASTIC PARAMETERS FROM MULTIWAVE AVOA DATA BY NONLINEAR OPTIMIZATION METHOD

In this paper, we investigate optimization algorithm of joint nonlinear AVOA inversion of PP+PS reflections in anisotropic media. Algorithm is based on the exact solution for PP and PS waves reflection coefficients in anisotropic HTI medium. The PP and PS wave’s reflections from the top of the anisotropic layer are examined. We use synthetic seismograms generated by ray method for the algorithm testing. We show that joint compressional and converted wave’s inversion allows increasing the robustness of the method and the accuracy of medium-parameter estimates. Coefficients of anisotropy are determined with better accuracy if signal-to-noise ratio is bigger than 5 for PP wave and bigger than 2 for PS wave.

Elastic model low- to intermediate-wavenumber inversion using reflection traveltime and waveform of multicomponent seismic data

Low-, intermediate-, and high-wavenumber components of P- and S-wave velocities jointly influence the elastic wave propagation and scattering in an isotropic medium. By taking advantage of all information in the data, elastic full-waveform inversion (E-FWI) has the potential to recover these model components. However, if the transmitted wave data are insufficient to illuminate the deeper part of the subsurface, we should rely on the solutions using reflection data. To reduce the nonlinearity of waveform inversion, we choose to decouple the effects of the model background and perturbation on the reflected waves within a linearized inversion framework. This resorts to three stages aiming to gradually fit the traveltimes and waveforms of the reflected PP and PS waves based on data or gradient preconditioning through P/S mode decomposition. For the first two stages, once the multicomponent seismograms have been separated into PP and PS reflection recordings, reflection traveltime inversion using an acoustic wave propagator (A-RTI) can successively recover the low-wavenumber components of P- and S-wave velocities. In the last stage, starting from the models having reliable low-wavenumber components, elastic reflection waveform inversion (E-RWI) can easily get out of the local minima and continue to retrieve the increasing wavenumber features sensitive to the waveform and amplitude variations. This is supported by gradient preconditioning through P/S mode decomposition of the extrapolated normal and adjoint wavefields, and alternately updating model background and high-wavenumber components in terms of linearized least-squares inversion. Numerical examples have demonstrated the performance of our E-RWI approach and the validity of the three-stage inversion workflow.

Causes for polarity reversals of PP precursor waves reflecting off the 410 km discontinuity beneath the Atlantic

We investigate the velocity and density structure of the mantle beneath the Northern Atlantic at a depth of 410 km where the olivine to wadsleyite phase transformation occurs. Measuring polarities of precursor arrivals to PP seismic waves that reflect at this transition we analyze over 1700 teleseismic seismograms from 13 different source-receiver combinations using the advantages of high-resolution array seismology methods and crossing ray paths. The final dataset consists of 45 events with Mw ≥ 5.8 with a good coverage of reflection points beneath the investigation area. We find several events where the polarity of the precursor signal is opposite to that of the PP wave, however, events with same polarity of precursor and PP wave are also observed. We test several causes for changing polarity of reflections at the 410 km discontinuity and find out that there is a dependence of the polarity on epicentral distance. We computed the reflection coefficients of the precursor wave using the Zoeppritz equations with the initial values of density, VP and VS of a pyrolite model for the olivine-wadsleyite phase transition and varied the values by ±10% above and below 410 km depth. More than 729 million combinations of density, compressional and shear wave velocities for olivine and wadsleyite layers were tested to find the combinations that can explain our polarity-distance observations. While the best-fitting models have a smaller positive contrast than pyrolite in density, a small negative contrast for P and S wave velocities is needed. Possibilities to explain these values include combination of a hydrous wadsleyite layer beneath, and anhydrous or Fe-enriched olivine above the boundary.

Study of Converted-Wave Modeling: AVO Application for Shallow Gas Models

The application of converted-wave seismic method in hydrocarbon exploration has increased significantly. Since the conventional seismic ceases to provide an adequate result in complex geology area and it provides an ambiguous brightspot response. The main principle is that an incident P-wave produces reflected and converted P and SV wave when the downgoing P-wave impinges on an interface. Converted-wave seismic uses the multicomponent receiver that records both of vertical component and horizontal component. The vertical component is assumed to correspond to the compressional PP wave and the horizontal correspond to the PS converted-wave. In this research, a synthetic model with the shallow gas and the salt dome below are constructed. The purpose of this study is to analyze the brightspot due to the presence of shallow gas and its effect to the quality of PP and PS wave reflection below the gas zone. To achieve the goal, both vertical and horizontal seismic data processing are performed. In horizontal data processing, the best gamma function (Vp/Vs) value is estimated to produce the better and reliable image. The result shows that the brightspot response in conventional data doesn’t exist in converted-wave data and the imaging below the gas zone in converted-wave data is better than the conventional due to the attenuation and diffraction effect that caused by gas column. Processing is followed by AVO analysis to compare the AVO response of PP and PS data in characterizing gas reservoir. Both PP and PS AVO curve shows the consistency with synthetic AVO from well data. Gas reservoir is a class 1 AVO anomaly with positive intercept and negative gradient on PP data. However, PS AVO curve does not refer any anomaly. It is because S-wave is not sensitive to the existence of rock saturant.

Prestack AVA joint inversion of PP and PS waves using the vectorized reflectivity method

Most current prestack AVA joint inversion methods are based on the exact Zoeppritz equation and its various approximations. However, these equations only reflect the relation between reflection coefficients, incidence angles, and elastic parameters on either side of the interface, which means that wave-propagation effects, such as spherical spreading, attenuation, transmission loss, multiples, and event mismatching of P- and S-waves, are not considered and cannot accurately describe the true propagation characteristics of seismic waves. Conventional AVA inversion methods require that these wave-propagation effects have been fully corrected or attenuated before inversion but these requirements can hardly be satisfied in practice. Using a one-dimensional (1D) earth model, the reflectivity method can simulate the full wavefield response of seismic waves. Therefore, we propose a nonlinear multicomponent prestack AVA joint inversion method based on the vectorized reflectivity method, which uses a fast nondominated sorting genetic algorithm (NSGA II) to optimize the nonlinear multiobjective function to estimate multiple parameters, such as P- wave velocity, S-wave velocity, and density. This approach is robust because it can simultaneously cope with more than one objective function without introducing weight coefficients. Model tests prove the effectiveness of the proposed inversion method. Based on the inversion results, we find that the nonlinear prestack AVA joint inversion using the reflectivity method yields more accurate inversion results than the inversion by using the exact Zoeppritz equation when the wave-propagation effects of transmission loss and internal multiples are not completely corrected.

A class of integrable metrics. II.

Starting with a subclass of the four-dimensional spaces possessing two commuting Killing vectors and a non-trivial Killing tensor, we fully integrate Einstein's vacuum equation with a cosmological constant. Although most of the solutions happen to be already known, we have found a solution that, as far as we could search for, has not been attained before. We also characterize the geometric properties of this new solution, it is a Kundt spacetime of Petrov type II possessing a null Killing vector field and an isometry algebra that is three-dimensional and abelian. In particular, such solution becomes a pp-wave spacetime when the cosmological constant is set to zero.

Variations of the Energy of Free Particles in the pp-Wave Spacetimes

We consider the action of exact plane gravitational waves, or pp-waves, on free particles. The analysis is carried out by investigating the variations of the geodesic trajectories of the particles, before and after the passage of the wave. The initial velocities of the particles are non-vanishing. We evaluate numerically the kinetic energy per unit mass of the free particles and obtain interesting, quasi-periodic behavior of the variations of the kinetic energy with respect to the width λ of the Gaussian that represents the wave. The variation of the energy of the free particle is expected to be exactly minus the variation of the energy of the gravitational field, and therefore provides an estimation of the local variation of the gravitational energy. The investigation is carried out in the context of short bursts of gravitational waves, and of waves described by normalized Gaussians, that yield impulsive waves in a certain limit.

Axisymmetric Pure Radiation Space–Time with Causality-Violating Geodesics

We present a stationary axisymmetric space–time admitting circular closed timelike geodesics everywhere within a finite region of space. The space–time is free from curvature divergence and is locally isometric to a nonvacuum pp-wave space–time. The matter–energy content is a pure radiation field and satisfies the null energy condition (NEC), and the metric is of type N in the Petrov classification scheme. Finally, we demonstrate the existence of timelike and null circular geodesic paths for this metric.

On pp wave limit for \u03b7 deformed superstrings

In this paper, based on the notion of plane wave string/gauge theory duality, we explore the pp wave limit associated with the bosonic sector of η deformed superstrings propagating in (AdS5 × S5)η . Our analysis reveals that in the presence of NS-NS and RR fluxes, the pp wave limit associated to full ABF background satisfies type IIB equations in its standard form. However, the beta functions as well as the string Hamiltonian start receiving non trivial curvature corrections as one starts probing beyond pp wave limit which thereby takes solutions away from the standard type IIB form. Furthermore, using uniform gauge, we also explore the BMN dynamics associated with short strings and compute the corresponding Hamiltonian density. Finally, we explore the Penrose limit associated with the HT background and compute the corresponding stringy spectrum for the bosonic sector.

Kinetic energy and angular momentum of free particles in the gyratonic pp-waves space-times

Gyratonic pp-waves are exact solutions of Einstein’s equations that represent non-linear gravitational waves endowed with angular momentum. We consider gyratonic pp-waves that travel in the z direction and whose time dependence on the variable is given by Gaussians, so that the waves represent short bursts of gravitational radiation propagating in the z direction. We evaluate numerically the geodesics and velocities of free particles in the space-time of these waves, and find that after the passage of the waves both the kinetic energy and the angular momentum per unit mass of the particles are changed. Therefore there is a transfer of energy and angular momentum between the gravitational field and the free particles, so that the final values of the energy and angular momentum of the free particles may be smaller or larger in magnitude than the initial values.

Quantum cosmology of a Bianchi III LRS geometry coupled to a source free electromagnetic field

We consider a Bianchi type III axisymmetric geometry in the presence of an electromagnetic field. A first result at the classical level is that the symmetry of the geometry need not be applied on the electromagnetic tensor Fμν; the algebraic restrictions, implied by the Einstein field equations to the stress energy tensor Tμν, suffice to reduce the general Fμν to the appropriate form. The classical solution thus found contains a time dependent electric and a constant magnetic charge. The solution is also reachable from the corresponding mini-superspace action, which is strikingly similar to the Reissner-Nordstr{öm one. This points to a connection between the black hole geometry and the cosmological solution here found, which is the analog of the known correlation between the Schwarzschild and the Kantowski-Sachs metrics. The configuration space is drastically modified by the presence of the magnetic charge from a 3D flat to a 3D pp wave geometry. We map the emerging linear and quadratic classical integrals of motion, to quantum observables. Along with the Wheeler-DeWitt equation these observables provide unique, up to constants, wave functions. The employment of a Bohmian interpretation of these quantum states results in deterministic (semi-classical) geometries most of which are singularity free.

Time-lapse joint AVO inversion using generalized linear method based on exact Zoeppritz equations

The conventional method of time-lapse AVO (Amplitude Versus Offset) inversion is mainly based on the approximate expression of Zoeppritz equations. Though the approximate expression is concise and convenient to use, it has certain limitations. For example, its application condition is that the difference of elastic parameters between the upper medium and lower medium is little and the incident angle is small. In addition, the inversion of density is not stable. Therefore, we develop the method of time-lapse joint AVO inversion based on exact Zoeppritz equations. In this method, we apply exact Zoeppritz equations to calculate the reflection coefficient of PP wave. And in the construction of objective function for inversion, we use Taylor series expansion to linearize the inversion problem. Through the joint AVO inversion of seismic data in baseline survey and monitor survey, we can obtain the P-wave velocity, S-wave velocity, density in baseline survey and their time-lapse changes simultaneously. We can also estimate the oil saturation change according to inversion results. Compared with the time-lapse difference inversion, the joint inversion doesn't need certain assumptions and can estimate more parameters simultaneously. It has a better applicability. Meanwhile, by using the generalized linear method, the inversion is easily implemented and its calculation cost is small. We use the theoretical model to generate synthetic seismic records to test and analyze the influence of random noise. The results can prove the availability and anti-noise-interference ability of our method. We also apply the inversion to actual field data and prove the feasibility of our method in actual situation.

Analytical expressions for the borehole seismic signatures excited by an explosive source buried in an elastic half‐space

ABSTRACTBorehole guided waves that are excited by explosive sources outside of the borehole are important for interpreting borehole seismic surveys and for rock property inversion workflows. Borehole seismograms are typically modelled using numerical methods of wave propagation. In order to benchmark such numerical algorithms and partially to interpret the results of modelling, an analytical methodology is presented here to compute synthetic seismograms. The specific setup is a wavefield emanating from a monopole point source embedded within a homogeneous elastic medium that interacts with a fluid‐filled borehole and a free surface. The methodology assumes that the wavelength of the seismic signal is much larger than the borehole radius. In this paper, it is supposed that there is no poroelastic coupling between the formation and the borehole.The total wavefield solution consists of P, PP, and PS body waves; the surface Rayleigh wave; and the low‐frequency guided Stoneley wave (often referred as the tube wave) within the borehole. In its turn, the tube wave consists of the partial responses generated by the incident P‐wave and the reflected PP and PS body waves at the borehole mouth and by the Rayleigh wave, as well as the Stoneley wave eigenmode. The Mach tube wave, which is a conic tube wave, additionally appears in the Mach cone in a slow formation with the tube‐wave velocity greater than the shear one. The conditions of appearance of the Mach wave in a slow formation are formulated. It is shown that the amplitude of the Mach tube wave strongly depends on Poisson's ratio of the slow surrounding formation. The amplitude of the Mach tube wave exponentially decreases when the source depth grows for weakly compressible elastic media with Poisson's ratio close to 0.5 (i.e., saturated clays and saturated clay soils). Asymptotic expressions are also provided to compute the wavefield amplitudes for different combinations of source depth and source‐well offset. These expressions allow an approximate solution of the wavefield to be computed much faster (within several seconds) than directly computing the implicit integrals arising from the analytical formulation.

Vector-based excitation amplitude imaging condition for elastic RTM

In recent years, many studies have focused on elastic reverse time migration (RTM). In response to the problems associated with elastic RTM, we propose a new procedure for 2D elastic multicomponent RTM. In this new method, decomposed P- and S-wave components are obtained from the decoupled propagation of the source and receiver wavefields, which allows the expedient calculation of the Poynting vectors and the incident and reflection angles of the P- and S-waves. In addition, we deduce the vector-based excitation amplitude imaging condition. This process automatically accounts for the particle vibration directions when determining the angle-dependent signed reflection coefficients, and does not require the sign to be determined apart from the value of the reflection coefficients. This concept was further extended to the source-normalized crosscorrelation imaging condition. The reflection coefficient of the layered model test was in agreement with the Zoeppritz theory, the PP and PS wave images of the Marmousi II model were clear, and the PS wave images had higher resolution and richer details. In addition, since the calculated reflection coefficients are angle-dependent, they can be easily used for the extraction of angle-domain common-image gathers. Moreover, the imaging condition avoids the polarization reversal in PS wave images and does not require all of the source wavefield data. Consequently, the computation and storage requirements are significantly reduced, which will facilitate the use of the elastic RTM in practice.

A hot midmantle anomaly in the area of the Indian Ocean Geoid Low

AbstractWe investigate the upper mantle seismic discontinuities at 410 and 660 km depth beneath the Indian Ocean Geoid Low (IOGL). To map the discontinuities' topography, we use differential travel times of PP and SS waves and their precursors. Our final data set consists of 37 events with Mw ≥ 5.8, which densely cover our investigation area, also with crossing ray paths. We use array methods to detect the low‐amplitude precursor signals. The best quality data show a deepened 410 km discontinuity in the center of the IOGL as well as a mostly elevated 660 km discontinuity beneath the northern Indian Ocean, which we interpret as a hot anomaly currently residing in the mantle transition zone. We conclude that the largest negative geoid anomaly might be caused by a combined effect of hot material in the midmantle below the innermost IOGL and cold material below 660 km farther south.

Classification of the Lie and Noether point symmetries for the Wave and the Klein–Gordon equations in pp-wave spacetimes

A complete classification of the Lie and Noether point symmetries for the Klein–Gordon and the wave equation in pp-wave spacetimes is obtained. The classification analysis is carried out by reducing the problem of the determination of the point symmetries to the problem of existence of conformal killing vectors on the pp-wave spacetimes. Employing the existing results for the isometry classes of the pp-wave spacetimes, the functional form of the potential is determined for which the Klein–Gordon equation admits point symmetries and Noetherian conservation law. Finally the Lie and Noether point symmetries of the wave equation are derived.

Uniform circular motion in general relativity: existence and extendibility of the trajectories

The concept of uniform circular motion in a general spacetime is introduced as a particular case of a planar motion. The initial value problem of the corresponding differential equation is analysed in detail. Geometrically, an observer that obeys a uniform circular motion is characterized as a Lorentzian helix. The completeness of inextensible trajectories is studied in generalized Robertson–Walker spacetimes and in a relevant family of pp-wave spacetimes. Under reasonable assumptions, the physical interpretation of such results is that a uniform circular observer lives forever, providing the absence of the singularities defined by these timelike curves.

Generalizedppwaves in Poincaré gauge theory

Starting from the generalized pp waves that are exact vacuum solutions of general relativity with a cosmological constant, we construct a new family of exact vacuum solutions of Poincar\'e gauge theory, the generalized pp waves with torsion. The ansatz for torsion is chosen in accordance with the wave nature of the solutions. For a subfamily of these solutions, the metric is dynamically determined by the torsion.

Pre-stack separation of PP and split PS waves in HTI media

Separation of PP and split PS waves in transversely isotropic media with a horizontal axis of symmetry is crucial for imaging subsurface targets and for fracture prediction in a multicomponent seismic survey using P-wave sources. In conventional multicomponent processing, when a low velocity zone is present near the surface, it is often assumed that the vertical Z-component mainly records P modes and that the horizontal X- and Y-components record S modes, including split PS waves. However, this assumption does not hold when the ubiquitous presence of azimuthal anisotropy makes near surface velocity structures more complicated. Seismic wavefields recorded in each component therefore generally represent a complex waveform formed by PP and split PS waves, seriously distorting velocity analysis and seismic imaging. Most previous studies on wave separation have tended to separate P and S modes using pre-stack data and to separate split S modes using post-stack sections, under the assumption of orthogonal polarization. However, split S modes can hardly maintain their original orthogonal polarizations during propagation to the surface due to stratigraphic heterogeneity. Here, without assuming orthogonal polarization, we present a method for pre-stack separation of PP, PS1 and PS2 waves using all three components. The core of our method is the rotation of wave vectors from the Cartesian coordinate system established by Z-, R- and T-axes to a coordinate system established by the true PP-, PS1- and PS2-wave vector directions. Further, we propose a three-component superposition approach to obtain base wave vectors for the coordinate system transformation. Synthetic data testing results confirm that the performance of our wave separation method is stable under different noise levels. Application to field data from Southwest China reveals the potential of our proposed method.

Full Waveform Modelling for Subsurface Characterization with Converted-Wave Seismic Reflection

While a large number of reservoirs have been explored using P-waves seismic data, P-wave seismic survey ceases to provide adequate result in seismically and geologically challenging areas, like gas cloud, shallow drilling hazards, strong multiples, highly fractured, anisotropy. Most of these reservoir problems can be addressed using P and PS seismic data combination. Multicomponent seismic survey records both P-wave and S-wave unlike conventional survey that only records compressional P-wave. Under certain conditions, conventional energy source can be used to record P and PS data using the fact that compressional wave energy partly converts into shear waves at the reflector. Shear component can be recorded using down going P-wave and upcoming S-wave by placing a horizontal component geophone on the ocean floor. A synthetic model is created based on real data to analyze the effect of gas cloud existence to PP and PS wave reflections which has a similar characteristic to Sub-Volcanic imaging. The challenge within the multicomponent seismic is the different travel time between P-wave and S-wave, therefore the converted-wave seismic data should be processed with different approach. This research will provide a method to determine an optimum converted point known as Common Conversion Point (CCP) that can solve the Asymmetrical Conversion Point of PS data. The value of γ (Vp/Vs) is essential to estimate the right CCP that will be used in converted-wave seismic processing. This research will also continue to the advanced processing method of converted-wave seismic by applying Joint Inversion to PP&PS seismic. Joint Inversion is a simultaneous model-based inversion that estimates the P&S-wave impedance which are consistent with the PP&PS amplitude data. The result reveals a more complex structure mirrored in PS data below the gas cloud area. Through estimated γ section resulted from Joint Inversion, we receive a better imaging improvement below gas cloud area tribute to the converted-wave seismic as additional constrain.