Plane-wave Coefficients Research Articles

Reflectivity blurring in subsurface extended image gathers

We explain how a reverse time migration (RTM) subsurface extended-image gather (EIG) relates to the reflection function for a finite-contrast interface via a blurring function. For a plane interface between locally homogeneous media, the reflection function contains the plane-wave reflection coefficients, and so we determine how the EIG relates to amplitude versus angle. The EIG and reflection function are multidimensional; hence, the blurring function in their linear relationship is higher dimensional. We explain how it may be computed and show that it describes spatial blurring and blurring over angle of the plane-wave reflection coefficients. We determine explicitly how a slant stack of the EIG at one slowness depends on the plane-wave coefficients at nearby slownesses. This angle blurring stems from the spatial nonstationarity of the blurring function, so it should be the most significant where the illumination changes most rapidly in space. To evaluate the theory, we use finite-difference modeling in the Sigsbee 2a model to generate synthetic survey data, RTM EIGs, blurring functions, and modeled gathers for a deep reflector. Two example image points are chosen. One has good illumination, with blurring over the angle less than 5°. The other point is just under the salt body, with poorer illumination and angle blurring of almost 10°. The description and examples are for two dimensions, but the extension to three dimensions is conceptually straightforward, as is an interface that dips relative to the EIG datum level. The computational cost of blurring functions implies their targeted use for the foreseeable future, for example, in reservoir characterization. The extension to elasticity and more-complicated scatterers is also foreseeable, and we emphasize the separation of the overburden and survey-geometry blurring effects from target properties.

C2x: A tool for visualisation and input preparation for Castep and other electronic structure codes

The c2x code fills two distinct roles. Its first role is in acting as a converter between the binary format .check files from the widely-used Castep [1] electronic structure code and various visualisation programs. Its second role is to manipulate and analyse the input and output files from a variety of electronic structure codes, including Castep, Onetep and Vasp, as well as the widely-used ‘Gaussian cube’ file format. Analysis includes symmetry analysis, and manipulation arbitrary cell transformations. It continues to be under development, with growing functionality, and is written in a form which would make it easy to extend it to working directly with files from other electronic structure codes.Data which c2x is capable of extracting from Castep’s binary checkpoint files include charge densities, spin densities, wavefunctions, relaxed atomic positions, forces, the Fermi level, the total energy, and symmetry operations. It can recreate .cell input files from checkpoint files. Volumetric data can be output in formats useable by many common visualisation programs, and c2x will itself calculate integrals, expand data into supercells, and interpolate data via combinations of Fourier and trilinear interpolation. It can extract data along arbitrary lines (such as lines between atoms) as 1D output.C2x is able to convert between several common formats for describing molecules and crystals, including the .cell format of Castep. It can construct supercells, reduce cells to their primitive form, and add specified k-point meshes. It uses the spglib library [2] to report symmetry information, which it can add to .cell files.C2x is a command-line utility, so is readily included in scripts. It is available under the GPL and can be obtained from http://www.c2x.org.uk. It is believed to be the only open-source code which can read Castep’s .check files, so it will have utility in other projects. Program summaryProgram Title: c2xProgram Files doi:http://dx.doi.org/10.17632/wj5hcj7x39.1Licensing provisions: GPLv3Programming language: CNature of problem: C2x is able to extract a large variety of data from Castep’s [1] large, binary-format, output files to facilitate further processing or visualisation. These binary files are optimised for rapid input and output by Castep, but are impossible to read without detailed knowledge of their internal structure, which varies between different versions of Castep. In them wavefunctions are stored as plane wave coefficients, so need to be Fourier transformed before they can be visualised in real space. C2x can manipulate the input files of Castep, and other common crystallographic formats, performing functions useful for setting up calculations, such as supercell construction, format conversions, and symmetry analysis.Different electronic structure codes use different file formats for outputting densities. C2x is able to convert between some of the major formats, allowing visualisation and post-processing tools targeted at one electronic structure code to be used with others.Solution method: C2x is a command-line utility written in standard C. It is able to process large (multi-GB) binary files efficiently, extracting much smaller datasets. C2x has considerable knowledge of the structure of the checkpoint files as written by various versions of Castep. Wavefunctions are optionally converted to densities, and then transformed to real space using its own FFT routine. Weighted sums of densities from multiple bands can be accumulated. C2x interpolates volumetric data by using user-specified combinations of trilinear and Fourier interpolation. For symmetry analysis it relies on the existing spglib [2] library, which is also used by Castep. It supports cell transformations with the axes of the new cell expressed in absolute terms, or in terms of the original axes. It has its own internal representation of the unit cell and its contents, and has several routines for converting this to and from common crystallographic file formats.C2x is able to read densities from Castep’s binary formats, and both read and write Castep’s formatted density files, Vasp’s formatted density files, and the .cube files used by Gaussian and Onetep.Additional comments: Primary site for distribution and documentation: www.c2x.org.uk[1] S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. J. Probert, K. Refson, M. Payne, First principles methods using CASTEP, Z. Kristall. 220 (2005) 567–570.[2] A. Togo, Spglib, a library for finding and handling crystal symmetries [cited 2017-02-23].https://atztogo.github.io/spglib/

From plane waves to local Gaussians for the simulation of correlated periodic systems.

We present a simple, robust, and black-box approach to the implementation and use of local, periodic, atom-centered Gaussian basis functions within a plane wave code, in a computationally efficient manner. The procedure outlined is based on the representation of the Gaussians within a finite bandwidth by their underlying plane wave coefficients. The core region is handled within the projected augment wave framework, by pseudizing the Gaussian functions within a cutoff radius around each nucleus, smoothing the functions so that they are faithfully represented by a plane wave basis with only moderate kinetic energy cutoff. To mitigate the effects of the basis set superposition error and incompleteness at the mean-field level introduced by the Gaussian basis, we also propose a hybrid approach, whereby the complete occupied space is first converged within a large plane wave basis, and the Gaussian basis used to construct a complementary virtual space for the application of correlated methods. We demonstrate that these pseudized Gaussians yield compact and systematically improvable spaces with an accuracy comparable to their non-pseudized Gaussian counterparts. A key advantage of the described method is its ability to efficiently capture and describe electronic correlation effects of weakly bound and low-dimensional systems, where plane waves are not sufficiently compact or able to be truncated without unphysical artifacts. We investigate the accuracy of the pseudized Gaussians for the water dimer interaction, neon solid, and water adsorption on a LiH surface, at the level of second-order Møller-Plesset perturbation theory.

Introducing k-point parallelism into VASP

For many years ab initio electronic structure calculations based upon density functional theory have been one of the main application areas in high performance computing (HPC). Typically, the Kohn–Sham equations are solved by minimisation of the total energy functional, using a plane wave basis set for valence electrons and pseudopotentials to obviate the representation of core states. One of the best known and widely used software for performing this type of calculation is the Vienna Ab initio Simulation Package, VASP, which currently offers a parallelisation strategy based on the distribution of bands and plane wave coefficients over the machine processors. We report here an improved parallelisation strategy that also distributes the k-point sampling workload over different processors, allowing much better scalability for massively parallel computers. As a result, some difficult problems requiring large k-point sampling become tractable in current computing facilities. We showcase three important applications: dielectric function of epitaxially strained indium oxide, solution energies of tetravalent dopants in metallic VO2, and hydrogen on graphene.

Long-offset AVO inversion of PP reflections from plane interfaces using effective reflection coefficients

A conventional amplitude variation with offset (AVO) inversion is based on geometrical seismics which exploit plane-wave reflection coefficients to describe the reflection phenomenon. Widely exploited linearizations of plane-wave coefficients are mostly valid at pre-critical offsets for media with almost flat and weak-contrast interfaces. Existing linearizations do not account for the seismic frequency range by ignoring the frequency content of the wavelet, which is a strong assumption. Plane-wave reflection coefficients do not fully describe the reflection of seismic waves at near-critical and post-critical offsets, because reflected seismic waves are typically generated by point sources. We propose an improved approach to AVO inversion, which is based on effective reflection coefficients (ERCs). ERCs generalize plane-wave coefficients for seismic waves generated by point sources and therefore more accurately describe near-critical and post-critical reflections where head waves are generated. Moreover, they are frequency-dependent and incorporate the local curvatures of the wavefront and the reflecting interface. In our study, we neglect the effect of interface curvature and demonstrate the advantages of our approach on synthetic data for a simple model with a plane interface separating two isotropic half-spaces. A comparison of the inversion results obtained with our approach and the results from an AVO inversion method based on the exact plane-wave reflection coefficient suggests that our method is superior, in particular for long-offset ranges which extend to and beyond the critical angle. We thus propose that long offsets can be successfully exploited in an AVO inversion under the correct assumption about the reflection coefficient. Such long-offset AVO inversion shows the potential of outperforming a conventional moderate-offset AVO inversion in the accuracy of estimated model parameters.

3D diffraction modeling of singly scattered acoustic wavefields based on the combination of surface integral propagators and transmission operators

We present an improved method for modeling 3D acoustic wavefields scattered at smooth curved interfaces. The approach is based on a high-frequency approximation of surface integral propagators and a correct description of their boundary values in terms of transmission operators. The main improvement is a uniform local approximation of these operators in the form of effective reflection and transmission coefficients. We show that the effective coefficients represent a generalization of the plane-wave coefficients widely used in conventional seismic modeling, even for the case of curved reflectors, nonplanar wavefronts, and finite frequencies. The proposed method is capable of producing complex wave phenomenas, such as caustics, edge diffractions, and head waves. Seismograms modeled for even simple models reveal significant errors implicit in the plane-wave approximation. Comparison of modeling based on effective coefficients with the analytic solution reveals errors less than 4% in peak amplitude at seismic frequencies.

Reflection from a corrugated surface revisited

The problem of scattering of a plane sonic wave from a soft surface with periodic (sinusoidal) unevenness along one direction is examined by means of the Rayleigh plane-wave expansion and the Waterman extinction methods, numerically implemented by Fourier projection and expansion, respectively. The computations are done with real, double-precision, stochastic arithmetic instead of the usual complex, double-precision floating-point arithmetic in order to precisely evaluate the numerical accuracy of the results conditioned by round-off errors. It is shown that the low-order plane-wave coefficients obtained by the Rayleigh and Waterman methods are identical when obtained from matrix systems that are large enough to give ‘‘convergence’’ of these coefficients. For the same matrix size, the higher-order coefficients differ the higher the diffraction order. It is also shown that the Waterman (Fourier-series) computation of the near field is generally meaningful, whereas that of Rayleigh, involving summation of the plane waves is generally meaningless except near the points of the scattering surface first encountered by the incident wave (i.e., those in the valleys when the incident wave comes from below). Low-order plane-wave scattering coefficients, with at least two-to-three-digit accuracy, and which are identical (to this precision) to the plane-wave coefficients computed by the rigorous integral equation method, are obtained by both the Rayleigh and Waterman methods for scattering surfaces with slopes as large as 2.26 when the number of nonevanescent waves is 5. The number of significant digits decreases as the slope increases.

Plane-wave electronic-structure calculations on a parallel supercomputer.

The development of iterative solutions of Schrodinger`s equation in a plane-wave (pw) basis over the last several years has coincided with great advances in the computational power available for performing the calculations. These dual developments have enabled many new and interesting condensed matter phenomena to be studied from a first-principles approach. The authors present a detailed description of the implementation on a parallel supercomputer (hypercube) of the first-order equation-of-motion solution to Schrodinger`s equation, using plane-wave basis functions and ab initio separable pseudopotentials. By distributing the plane-waves across the processors of the hypercube many of the computations can be performed in parallel, resulting in decreases in the overall computation time relative to conventional vector supercomputers. This partitioning also provides ample memory for large Fast Fourier Transform (FFT) meshes and the storage of plane-wave coefficients for many hundreds of energy bands. The usefulness of the parallel techniques is demonstrated by benchmark timings for both the FFT`s and iterations of the self-consistent solution of Schrodinger`s equation for different sized Si unit cells of up to 512 atoms.

Defect energetics in MgO treated by first-principles methods.

Ab initio total-energy calculations have been performed on a parallel computer to study the formation and migration energies of cation and anion vacancies in MgO. The calculations are made in the framework of density functional and pseudopotential theory, using the supercell method, with the valence orbitals expanded in plane waves. The relaxed ground state is determined by conjugate-gradients minimization of the total-energy functional with respect to the plane-wave coefficients of occupied orbitals. The calculated defect energies are shown to be in remarkably close agreement with experiment and with values obtained from empirical modeling. We present results for the electron distribution surrounding the vacancies that show that the distortion induced in the oxygen ions is more complex than has previously been thought.

The effects of sediment porosity on acoustic reflection and transmission at the seafloor

Abstract : The Biot theory of propagation in a porous medium provides a mathematical framework for studying acoustic interaction with the seafloor. The theory considers the two-phase porous nature of marine sediments in contrast to the classical models of wave propagation in the seafloor that consider marine sediments as an extended single-phase fluid or solid. A boundary value problem is set up and solved for a line source in a fluid medium above a poro- viscoelastic halfspace. Expressions for the reflected and transmitted field are given in integral form and asymptotic expansions in the high-frequency, far- field limit. A set of simultaneous equations is solved to give plane wave reflection and transmission (Type I, Type II and shear wave) coefficients. These equations also yield the Scholte Pseudo-Scholte, and pseudo-Rayleigh wave phase velocities and attenuations. The plane wave coefficients and the surface wave velocities and attenuations are compared with commensurate quantities from the single-phase theories. Recent experiments of high frequency transmission through a water-sand interface have indicated anomalously high transmitted energy at angles near the critical angle. The Type II wave (predicted by Biot theory but not the single-phase theories) was suspected as a possible reason for the anomalies. Calculations indicate that the contributions from the Type II wave are negligible and thus not the cause of the anomalies. Much of the anomalous behavior appeared to arise simply as a result of an incorrect compressional wave attenuation in the geoacoustic model.

Calculating the acoustical properties of fabric constructions

Architects are using fabric structures more frequently, not only because of the aesthetic appeal of such structures, but also because of their low initial and operational costs. A common application of fabric structures is in community buildings such as shopping malls and recreation centres. Fabric structures are also now used in industrial noise control applications. The constructions used in fabric structures incorporate one or more porous or impervious fabric sheets which may carry high in-plane tensions. Sometimes layers of porous materials may also be included. The acoustical performance of a fabric construction is usually adequately described by the diffuse field sound absorption cofficient, α (Δf), and the diffuse field sound reduction index, R(Δf), in frequency bands. A procedure for calculating these quantities is described. For this procedure, a fabric sheet is characterized by its mass per unit area, flexural stiffness, flexural loss factor, specific flow resistance, and the magnitudes and directions of the principal biaxial in-plane tensions. A porous layer is characterized by its thickness, flow resistivity, and position relative to the fabric sheets. The first part of the calculation procedure is concerned with finding the sound reflection and transmission coefficients for a plane wave of given frequency and direction. These calculations are based on the successive application of two types of formulae. The first type of formulae relate the surface or wave impedances across the fabric sheets and the porous layers or air spaces between them, and the second type relate the acoustic pressures across these elements. The second part of the calculation procedure involves integrating the frequency and direction dependent plane wave coefficients to give the diffuse field coefficients. The results of the application of the procedure to some representative situations are given and are compared with experimental results.

End-point contributions to synthetic seismograms

Summary. Synthetic seismograms represented by integrals generally display signals associated with the limits of integration. Sometimes these ‘end-point’ contributions are spurious (e.g. in the WKBJ seismogram) and sometimes they are the main physical interest (e.g. the Kirchhoff integral for an edge). The end-point contributions may be asymptotically approximated using integration by parts or Laplace's method and it may then be possible to reduce them if desired. We describe examples in the WKBJ seism ogram for reflected or transmitted waves in homogeneous layers and for turning waves. We also study signals due to discontinuities in reflection coefficients, by partitioning the real slowness integral so that the discontinuities lie at end points. Examples are the head wave, which is a physically correct signal, and spurious diffractions caused by using plane-wave coefficients for grazing rays in the WKBJ seismogram.

Evaluation ofStoPamplitude rations for determining focal mechanisms from regional network observations

abstractThe ratios of the amplitudes of SV to P, as recorded on vertical component seismographs near the earthquake, provide a means of determining the focal mechanism. The observed ratios are compared with values calculated on the basis of dislocation theory, with the effects of transmission across boundaries and incidence on the free surface approximately accounted for by plane-wave coefficients. A search is made for the strike and dip of the fault that provides a fit to as many of the stations as possible, for the direction of fault slip assumped. The direction of slip may also be treated as a free parameter to be determined, but the theoretical results are especially simple for pure strike-slip or pure dip-slip faulting. The technique has been applied to the analysis of events in three settings, in which pure thrusting, pure strike-slip, and pure normal faulting could be assumed. The utility and the limitations of the method are illustrated by these examples. The important limitations are that the method cannot distinguish between conjugate mechanisms or show the sense of slip on the fault. It also has a weak resolution of strike for some commonly occurring mechanisms, so that the solution may depend on the readings at only a few stations.The method has been used to pick out a group of foreshocks to an Aleutian Islands earthquake and to provide a focal mechanism for an event in the Rhine graben, at a place where such information is lacking and needed. A study of 15 central California earthquakes revealed no changes in fault plane orientation or in relative attenuation of P and S waves during 1 yr prior to a magnitude 5 earthquake.

On the adequacy of plane-wave reflection/transmission coefficients in the analysis of seismic body waves

In order to estimate the effect (on body waves) of discontinuities within the Earth, it is common practice to use the theory for plane waves incident upon the plane boundary between two homogeneous half-spaces. The resulting reflection/P~ V conversion/transmission coefficients are shown here to he inaccurate for many problems of current interest. Corrected coefficients are needed, in particular, for cases where the discontinuity (upon which boundary conditions are to be applied) is near a turning point of the P- or S-wave rays, or if one of these rays intersects the discontinuity at a near-grazing angle. Adequate corrections, based upon the Langer approximation to a full wave theory, are shown to be easily derived in practice. The method is first to write out the plane-wave coefficients as a rational polynomial, in sines and cosines of the angles of incidence upon the boundary, and second to introduce a multiplicative factor for each cosine. The new factors depart from unity only when the associated cosine tends to zero; i.e., when a turning point is approached. They incorporate all the corrections required for curvature of the boundary, frequency dependence, and Earth structure (velocity gradients) near the boundary.

A NOTE ON THE REFLECTION OF SEISMIC PULSES WITH APPLICATION TO SECOND EVENT REFRACTION SHOOTING*

AbstractRichards (1961) and several others have shown that wide angle reflections may attain large amplitudes. This note extends the plane wave calculations of Richards to include the effect of the phase changes which occur at angles greater than critical. On seismic pulses this introduces a time lead of up to one half period and alters peak‐to‐peak amplitudes by up to 15‐20 %.It is pointed out that the plane wave reflection coefficient is not applicable at angles very close to critical where the true reflection coefficient is reduced by a factor depending on (R/λ) ¼ In ultrasonic experiments this factor reduced the reflection coefficient by between ½ and ⅓.Neither does the plane wave coefficient apply at grazing incidence, when it has to be reduced by a factor depending on (R/λ)1‐0.Graphs are given of the amplitudes of wide angle reflections and head waves for two cases and it is concluded that, except for shallow refractors and angles close to critical, the reflection is always significantly greater than the head wave.

Focusing of Sound Waves in Inhomogeneous Media

The sound field from a point source in a stratified inhomogeneous medium is represented in an integral form in terms of plane wave coefficients of reflection. At sufficiently high frequencies coefficients of reflection can be calculated by means of the WKB method. Then the application of the method of steepest descent gives the ray approximation for the sound pressure. In developing this theory equations of caustics were obtained and sound pressure near caustics and in the neighborhood of a cusps of caustics was determined. The development of the field in a sum of modes [N. A. Haskell, J. Appl. Phys. 22, 157 (1951)] did not appear necessary. Examples are given. The theory of focusing action of stratified media appears to be quite analogous to the theory of focusing by optical systems when aberration is present.