Pp-wave Research Articles

Closed strings in a class of pp-wave spacetimes and the memory effect

Closed strings in a class of pp-wave spacetimes and the memory effect

Research on multi-wave joint elastic modulus inversion based on improved quantum particle swarm optimization

Young's modulus and Poisson's ratio are crucial parameters for reservoir characterization and rock brittleness evaluation. Conventional methods often rely on indirect computation or approximations of the Zoeppritz equations to estimate Young's modulus, which can introduce cumulative errors and reduce the accuracy of inversion results. To address these issues, this paper introduces the analytical solution of the Zoeppritz equation into the inversion process. The equation is re-derived and expressed in terms of Young's modulus, Poisson's ratio, and density. Within the Bayesian framework, we construct an objective function for the joint inversion of PP and PS waves. Traditional gradient-based algorithms often suffer from low precision and the computational complexity. In this study, we address limitations of conventional approaches related to low precision and complicated code by using Circle chaotic mapping, Lévy flights, and Gaussian mutation to optimize the quantum particle swarm optimization (QPSO), named improved quantum particle swarm optimization (IQPSO). The IQPSO demonstrates superior global optimization capabilities. We test the proposed inversion method with both synthetic and field data. The test results demonstrate the proposed method's feasibility and effectiveness, indicating an improvement in inversion accuracy over traditional methods.

Joint PP and PS AVA inversion using an acceleration algorithm and a multi-trace strategy

Abstract Exploration utilizing converted wave (PS) technology has garnered considerable interest in recent years owing to its ability to provide distinct insights into subsurface properties compared to compressional waves (PP). PS wave is particularly advantageous for accurately determining S-wave velocity (${V}_S$) and bulk density ($\rho $), whereas PP wave is better suited for assessing P-wave velocity (${V}_P$). Thus, theoretically, joint PP and PS AVA inversion can yield precise results for ${V}_P$, ${V}_S$, and $\rho $. However, a critical step before joint inversion is PP and PS registration, which is prone to causing lateral discontinuity in the inversion outcomes. To mitigate this issue, we present a simultaneous multi-trace joint inversion method. The method can improve the lateral consistency of inversion results by imposing lateral continuity constraints on the elastic parameters being inverted. Nevertheless, this simultaneous multi-trace inversion method inherently increases computational burden. To enhance algorithmic efficiency without sacrificing inversion precision, we implement an acceleration strategy. Synthetic and real data examples demonstrate that the proposed method offers remarkable performance in enhancing computational efficiency and lateral continuity compared to the conventional methods.

Prediction of tight sandstone reservoirs by multiwave joint prestack inversion technology: A case study of the Qiulin area in the Sichuan Basin

The Shaximiao (SXM) Formation of the Middle Triassic in the central Sichuan Basin is a river-lake sedimentary system with an average burial depth of 2800 m. The channel sand bodies of the SXM Formation in the Qiulin area are developed with large lateral variation and high- and medium-porosity sand. This leads to great differences in the seismic response and difficulty in predicting the sand bodies and reservoirs using P-wave data. We carry out well-log analysis for different types of sand bodies. The no. 8 sand body of high porosity can be effectively identified using P-wave impedance (PI). Whereas for the no. 7 sand body of relatively lower porosity (mainly due to the higher shale content in the no. 7 sand compared with the no. 8 sand), sandstone and mudstone can be distinguished using S-wave impedance (SI), and reservoirs can be recognized by the P- to S-wave velocity ratio ([Formula: see text]/[Formula: see text]). Based on the PP wave and the P-to-S converted wave (PS wave) (after matching) angle-stacked data, a prestack inversion of multicomponent data is performed. Multiwave joint inversion results (SI and [Formula: see text]/[Formula: see text]) have better accuracy and stronger stability than the PP-wave inversion in practical application. In addition, the PI of the joint inversion more clearly describes the distribution and boundaries of the channel sand and better matches the prediction and drilling data in sand 8. The SI of the joint inversion can identify subtle sand bodies (weak PI contrast) that are difficult to be detected by PP-wave inversion. The distribution of the reservoir predicted by the [Formula: see text]/[Formula: see text] of the joint inversion is clearer (the no. 7 sand body) and better than that by the PP-wave inversion. This study demonstrates the advantage of the multiwave joint inversion technology in tight sand identification.

Conformally related vacuum gravitational waves and their symmetries

A special conformal transformation which carries a vacuum gravitational wave into another vacuum one is built by using Möbius-redefined time. It can either transform a globally defined vacuum wave into a vacuum sandwich wave, or carry the gravitational wave into itself. The first type, illustrated by linearly and circularly polarised vacuum plane gravitational waves, permutes the symmetries and the geodesics. Our second type is a pp wave with conformal O(1, 2) symmetry. An example inspired by molecular physics which seems to have escaped attention so far is an anisotropic generalisation of the familiar inverse-square profile and is reminiscent of Aichelburg-Sexl ultraboosts. The particle can escape, or perform circular periodic motion, or fall into the singularity.

Memory effect of gravitational wave pulses in PP-wave spacetimes

In this paper, we study the gravitational memory effect in pp-wave spacetimes due to the passage of a pulse having the form of a ramp profile through this spacetime. We have analyzed the effect of this pulse on the evolution of nearby geodesics, and have determined analytical solutions of the geodesic equations in the Brinkmann coordinates. We have also examined the changes in the separation between a pair of geodesics and their velocity profiles. The separation (along x or y-direction) increases monotonically from an initial constant value. In contrast, the relative velocity grows from zero and settles to a final non-zero constant value. These resulting changes are retained as memory after the pulse dies out. The nature of this memory is similar to that determined by earlier workers using Gaussian, square, and other pulse profiles, thereby validating the universality of gravitational wave memory.

Converted wave tomography based on inverse level set and adjoint formulation

SUMMARY Shear wave velocity (Vs) is a fundamental property of elastic media whose estimation from PS converted waves is challenging and requires modelling the boundary where P to S conversion occurs. This paper presents a PS tomography where seismic wave conversion/reflection points correspond to reflectors modelled with the level-set function set to zero [ϕ(x, z) = 0]. The proposed method aims for stable Vs inversion in a seismic acquisition setting using multicomponent receivers. Synthetic models simulating true Vs, Vp and the location of the geological reflector are used in the study. The inversion starts by locating a flat reflector, ϕ(x, z) = 0, which defines the zone Ω1 between the surface and the reflector, where the initial Vs and Vp fields are also set. To calculate the traveltimes of incident PT (P wave that propagates in Ω1 from source to the reflector), converted PS and reflected PP waves, for both observed and modelled data (forward problem), the methodology proposed by Rawlinson and Sambridge is adopted. This method uses the arrival times of the P waves, Tpt, from the seismic source at each reflector point as secondary sources generating the times Tps and Tpp. These times are calculated as a solution to the eikonal equation by using the Fast Marching method. The PS and PP residual times are minimized by updating Vs, Vp and ϕ(x, z) = 0 through adjoint variables designed from a formulation using Lagrange Multipliers in a variational context. The performance of the algorithm is evaluated for models with synclinal, sinusoidal and monoclinal reflector geometries using numerical tests considering the inversion of: (1) ϕ, given the true values of Vs and Vp; (2) ϕ and Vs, given the true value of Vp; (3) ϕ and Vp, given the true value of Vs and (4) the three parameters ϕ, Vs and Vp, simultaneously. Good results are obtained by inverting Vs and ϕ, given the true value of Vp. The simultaneous inversion of the three parameters exhibits promising results, despite the illumination problems caused by the different distribution of the PS, PP and PT time gradients due to the geometry of the reflectors and the acquisition setting (sources–receivers in the same plane). The proposed tomography estimates Vs and reflector positions which could help in statics corrections and improve the lithological characterization of near surface.

Twistor initial data characterisation of pp-waves

This note gives a concise derivation of a twistor-initial-data characterisation of pp-wave spacetimes in vacuum. The construction is based on a similar calculation for the Minkowski spacetime in Bäckdahl and Kroon [2011 Class. Quantum Grav. 28 075010]. The key difference is that for the Minkowski spacetime a necessary condition is that ∇AA′κ¯A′≠0 . In this note it is shown that if ∇AA′κ¯A′=0 then the development is a pp-wave spacetime. Furthermore, it is shown that such condition propagates off the initial hypersurface, which, in turn, gives a twistor initial data characterisation of pp-waves.

QS‐wave decoupling equation for wavefield separation in transversely isotropic media

AbstractConsidering the anisotropy of the earth media is helpful in reducing the depth error between seismic and drilling and providing reliable imaging data for seismic interpretation and inversion. Transversely isotropic media with a vertical axis of symmetry are the most common type of anisotropic media and have been under constant study. The separation of the P‐ and S‐wavefields, which affects the accuracy of elastic wave imaging and the inversion in transversely isotropic media with a vertical axis of symmetry, is a hot research topic. Among the commonly used wavefield decoupling methods, the S‐wave is typically obtained by subtracting the P‐wave from the total wavefield. However, this kind of wavefield decoupling method often leads to severe P‐wave crosstalk in the separated S‐wavefield; thus, it needs further development. In this paper, the principle that the divergence of the S‐wave is zero is employed to solve the unresolved S‐wave elastic parameters of the S‐wave stiffness matrix by utilizing the modified zero‐order pseudo‐Helmholtz decomposition operator. The obtained S‐wave elastic parameters are employed to construct the qS‐wave stress and facilitate the derivation of the relationship between the qS‐wave particle velocity and the qS‐wave stress. Furthermore, a qS‐wave decoupling first‐order velocity–stress equation, which matches the qP‐wave decoupling first‐order velocity–stress equation, is derived. By jointly using the decoupling equations of qS‐ and qP‐wave, the separation of the P‐ and S‐wavefields in transversely isotropic media with a vertical axis of symmetry is realized. The efficiency of the proposed method is demonstrated via numerical tests conducted to assess the wavefield separation. Moreover, a complex model is employed to perform elastic reverse‐time migration, resulting in the acquisition of accurate imaging results for PP, PS, SP, and SS waves, without the presence of significant artefacts. The correctness of the qS‐wave decoupling equation in transversely isotropic media with a vertical axis of symmetry is confirmed by the comparative tests using decoupling methods in isotropic media.

Seismic fluid identification method based on the joint PP‐ and SH–SH‐wave stochastic inversion

AbstractPre‐stack seismic inversion is an effective method for elastic parameter inversion using seismic data, which facilitates seismic fluid identification. However, pure PP‐wave inversion has issues of strong multi‐solution and limited prediction accuracy. Therefore, we propose a seismic fluid identification approach based on the joint PP‐ and SH–SH‐wave stochastic inversion. First, the linearized SH–SH‐wave amplitude variation with offset approximation parameterized by shear modulus and density is derived. Numerical simulations demonstrate that the SH–SH‐wave amplitude variation with offset approximation has a good accuracy. Reflection coefficient contribution analysis indicates that the new formulation has better parameter sensitivity to shear modulus and density than the PP wave amplitude variation with offset approximation derived by Russell, which helps one to improve the inversion of shear modulus and density. On this basis, we construct a joint inversion equation of PP and SH–SH waves for a Russell fluid indicator, a shear modulus and density and present a novel joint stochastic inversion method based on the ensemble smoother with multiple data assimilation. Stanford VI‐E model tests reveal that the Russell fluid indicator factor, shear modulus and density obtained from the joint PP‐ and SH–SH‐wave inversion have higher identification accuracy and smaller relative errors than those from pure PP‐wave inversion. Furthermore, field data tests indicate that this method has practical applicability in seismic fluid identification.

Approximate Mei symmetries of pp-wave spacetimes

In this paper, first order approximate Mei symmetries of the geodetic Lagrangian for some classes of the pp-wave spacetimes are obtained. These classes of pp-wave spacetimes include plane wave spacetimes in which (i). H = α 2 x 2(ii). H = α x 2 u −2 (iii). H = α 2 x 2 u −4 and for pp-wave spacetimes (iv). H = α x n (where H is called scale factor and α is a constant). After that, approximate Mei invariants are calculated corresponding to each case.

From pp-Waves to Galilean Spacetimes

We exhibit all spatially isotropic homogeneous Galilean spacetimes of dimension (n+1)>=4, including the novel torsional ones, as null reductions of homogeneous pp-wave spacetimes. We also show that the pp-waves are sourced by pure radiation fields and analyse their global properties.

PS reflection moveout in a homogeneous anisotropic layer of arbitrary symmetry and tilt

Use of unconverted reflected PP waves leads to moveout formulae, which relate the moveout to only a limited number of parameters specifying anisotropy. Use of converted reflected PS waves extends the number of parameters to, generally, a complete set of twenty-one parameters. It leads, on the other hand, to more complicated formulae and phenomena unknown from studies of unconverted PP waves. In anisotropic media one has to deal with the existence of two S waves and their singularities, acoustic approximation cannot be used. We present and test approximate formulae for the reflection moveout of PS waves converted at a horizontal reflector underlying a homogeneous layer of arbitrary anisotropy symmetry and orientation. For their derivation, we use the combination of weak-anisotropy approximation and A-parameters specifying anisotropy. The complete set of twenty-one A-parameters represents an alternative to twenty-one independent elements of the stiffness tensor. In addition to the moveout formulae for separate PS1 and PS2 waves, we also present moveout formula for a P to common S wave whose traveltimes represent an average of traveltimes of PS1 and PS2 waves. Use of the P to common S wave reflection leads to several simplifications. Presented tests indicate that for most offsets and models with P-wave anisotropy up-to to 26% and S-wave anisotropy over 25%, the relative traveltime errors do not exceed 2.5%.

Converted PS-wave imaging based on offset vector tiles

Stacking using offset vector tiles (OVTs) is a technique used to simultaneously retain the offset and azimuth information in a wide-azimuth seismic survey by building minimal data sets of underground homogeneous illumination. Currently, derived from the assumption that reflection points and common midpoints are in the same lateral location because the raypath is symmetric, PP-wave OVT processing has been widely applied to seismic imaging and offset and/or azimuth analysis. However, it is difficult to directly apply the PP-wave OVT gathering method to the PS wave because the PS-wave raypath is asymmetric and the conversion point varies with depth. To solve this problem, we have developed an improved PS-wave OVT imaging technique based on the precise conversion point calculation and the equivalent converted-wave hypothesis, in which the PS-wave OVT gather with homogeneous illumination can be obtained in a manner similar to the PP wave. Moreover, to further improve the imaging quality, an azimuthal anisotropic normal moveout is applied to correct the traveltime difference. With tests of synthetic and field data, it can be concluded that our method represents a new and feasible solution for PS-wave OVT gathering.

A global distinct pp-wave spacetime

This paper aims to investigate a spacetime that ensues from the generalized pp-waves metric by a specific choice of parameters. We obtain an additional off-diagonal term in the standard pp-waves line element. The spacetime is classified according to Petrov’s classification. The gravitational field equations are solved and the solutions are obtained. In order to distinguish this new spacetime from the standard pp-waves, we evaluate a Burgers vector of this spacetime in a local Burgers circuit and we show the new spacetime to be globally distinct from the standard pp-waves spacetime. Hence, we find topological defects present in a particular class of Brinkmann waves.

On the higher order mean curvatures of spacelike hypersurfaces in pp-wave spacetimes

We study some geometric aspects of the higher order mean curvatures (or, more simply, the so-called $ r $-th mean curvatures) of a spacelike hypersurface immersed in a pp-wave spacetime, namely, in a connected Lorentzian manifold admitting a parallel and lightlike vector field. Initially, we develop general Minkowski-type integral formulas for compact (without boundary) spacelike hypersurfaces and we apply them to the study of the uniqueness and nonexistence of compact spacelike hypersurfaces in terms of their $ r $-mean curvatures. Next, we obtain a characterization of $ r $-stability for $ r $-maximal compact spacelike hypersurfaces through of the analysis of the first nonzero eigenvalue of an differential operator naturally attached to the $ r $-th mean curvature. For the noncompact case, by applying new forms of maximum principles on complete noncompact Riemannian manifolds due to Caminha [17] and Alías, Caminha and Nascimento [3], we obtain sufficient geometric conditions involving some $ r $-th mean curvature and the volume growth that allow us to establish some nonexistence results or to guarantee that a complete noncompact spacelike hypersurface is either totally geodesic, or totally umbilical, or maximal, or $ r $-maximal. We also obtain estimates for the index of minimum relative nullity of spacelike hypersurfaces.

Anisotropic Nonlinear Inversion Based on a Novel PP Wave Reflection Coefficient for VTI Media

Anisotropic Nonlinear Inversion Based on a Novel PP Wave Reflection Coefficient for VTI Media

Parallel spinors, pp-waves, and gravitational perturbations

We prove that any real, vacuum gravitational perturbation of a four-dimensional vacuum pp-wave space-time can be locally expressed, modulo gauge transformations, as the real part of a Hertz/Debye potential, where the scalar potential satisfies the wave equation. We discuss relations with complex perturbations, complex space-times, non-linear structures, and real spaces with split (ultra-hyperbolic/Kleinian) signature. Motivated by generalized notions of parallel spinors, we also discuss generalizations of the result to other space-times.

On the teleparallel frame problem

The role played by the congruence in the prediction of the gravitational energy in the teleparallel equivalent of general relativity is discussed. It is shown that some congruences yield unphysical predictions. It is also shown that the energy–momentum tensor density predicted by the proper reference frame of an arbitrary accelerated observer vanishes along the observer’s worldline, regardless of its acceleration. The latter result is discussed and arguments both against and in favor of the use of this type of frame to describe the gravitational energy are presented; arguments against the belief that the principle of equivalence is incompatible with the localization of the gravitational energy are also presented. A gauge condition is applied to the teleparallel frame and its consistence is discussed: using three different tetrads for the pp-wave spacetimes, it is shown that the most consistent prediction comes from the tetrad that satisfies this gauge. The possibility of having a well-defined concept of absolute vacuum is also discussed.

Deep carbonate reservoir and gas prediction based on multicomponent seismic amplitude attributes — A case study

Multicomponent seismic exploration provides more valuable information for the prediction of underground structures, lithology, fluids, and fractures. Most studies on lithology estimation and fluid description using PP- and PS-wave seismic data mainly focus on relatively shallow targets with high porosity. Focusing on deep carbonate (dolomite) reservoirs with low porosity (generally less than 5%), we first analyze the characteristics of the seismic response difference between PP and PS waves, and then we combine logging data and seismic forward modeling to implement the identification of fluids using the amplitude difference between PP and PS waves. Data sets are acquired over grain beach dolomite reservoirs in the Cambrian Longwangmiao Formation, the central Sichuan Basin, Western China. The reservoirs are filled with different types of fluids (e.g., gas, water, bitumen, etc.). Based on comparisons between seismic responses of PP and PS waves, we observe that, at the location of the gas-bearing layer, PP-wave amplitude exhibits strong peaks and PS-wave amplitude finds relatively weak peaks, which is different from that obtained for the water-saturated layer. However, at the location of bitumen-filled reservoirs, PP- and PS-wave amplitudes exhibit strong peaks, and the reflection amplitude of PP waves is stronger than that of PS waves. Therefore, the amplitude difference of PP and PS waves provides valuable information and feasibility for fluid identification. We verify that the maximum peak amplitude attribute of PS waves may better characterize the distribution of porous dolomite reservoirs than that of PP waves, and using the attribute of maximum peak amplitude difference between PP and PS waves, we may distinguish gas-bearing and water-saturated layers. Comparing with actual drilling results, we conclude that our PP/PS interpretation technique is feasible for fluid identification in deep carbonate reservoirs.